[{"data":1,"prerenderedAt":120521},["ShallowReactive",2],{"series-posts-Machine Learning":3,"series-data-Machine Learning":120251,"series-authors":120329,"post-count":120520},[4,196,314,455,1100,1571,2181,2472,2710,2859,3212,3332,3591,3650,3872,4142,4414,4916,5260,5602,5885,6431,6685,6850,6943,9374,13177,13311,15644,17979,21127,21257,24040,24191,24313,24476,24594,24888,32000,42206,47826,73352,73412,73507,73634,73708,73839,74025,74088,74919,74979,75325,76626,77838,78060,78194,78341,78512,79328,80780,81118,81329,81631,81996,82117,82290,82440,83258,84145,85276,86505,86659,91793,92029,92192,92403,92603,92758,98615,98847,99114,99778,100798,101062,101320,101531,101678,101929,102190,106431,106542,106727,107199,107468,107640,107748,107926,108083,108227,108602,108796,108906,109150,109406,109590,109747,109878,110040,110197,110311,110391,110557,110709,110801,110895,110998,111159,111260,111378,111452,111565,111618,111765,111845,111944,112153,112188,113011,113235,113483,113538,117133,117208,117253,118714,118856,119101,119736,119852,119961,120026,120106],{"id":5,"title":6,"author":7,"body":8,"date":183,"description":184,"draft":185,"edited_at":183,"extension":186,"featured_image":187,"meta":188,"navigation":189,"path":190,"pinned":185,"seo":191,"sitemap":192,"stem":193,"tags":194,"__hash__":195},"blog\u002Fblog\u002FDown-Syndrome.md","Down Syndrome","hibiki12141132",{"type":9,"value":10,"toc":179},"minimark",[11,19,22,35,38,41,47,59,68,81,112,121,124,127,147,156,159],[12,13,14,15,18],"p",{},"那唐氏症發生率是每1000個新生兒裡有1.3個",[16,17],"br",{},"\n母親是高齡產婦的話，風險會增加",[12,20,21],{},"這個不是遺傳疾病來的，起因有三",[23,24,25,29,32],"ul",{},[26,27,28],"li",{},"第21對染色體多了一條小的染色體：91%的唐氏症都是這種類型",[26,30,31],{},"第21對跟第22對染色體互換：4%的唐氏症是這類型",[26,33,34],{},"嵌合體：智力正常，只有外觀有特徵，臨床工作者一般會建議孩子整形",[36,37],"hr",{},[12,39,40],{},"從病理學來看",[12,42,43,44,46],{},"唐氏症的大腦比一般正常孩子的大腦重量小，大概只有正常的76%",[16,45],{},"\n形狀呈圓，短，前後較窄，外側較寬——對應了最常見的唐氏症頭形：短頭畸形",[12,48,49,50,52,53,55,56,58],{},"40%的唐氏症有先天性心臟病缺陷，常見的類型為",[16,51],{},"\n房室通道缺損(Atrioventricular canal defects)",[16,54],{},"\n心室中膈缺損(Ventriculoseptal defects)",[16,57],{},"\n情況良好的唐氏症小孩都會建議開刀，才能進行後續訓練，所以一般我們PT接到的都已經是開過刀的小孩了",[12,60,61,62,64,65,67],{},"感覺功能也會有缺失",[16,63],{},"\n在107個受試者中，有64%的唐氏症有聽力喪失",[16,66],{},"\n在75個受試者中，有41.3%有視覺缺陷",[12,69,70,71,73,74],{},"肌肉骨骼系統上也會有差異",[16,72],{},"\n低張力是主要原因，韌帶鬆馳，",[75,76,80],"span",{"className":77},[78,79],"underline","red-3","第一二節頸椎半脫位",[12,82,83,84,86,87,89,90,92,93,95,96,98,99,101,102,104,105],{},"那寰樞椎脫位是唐氏症最大問題的點，我們PT也要十分注意，症狀包括",[16,85],{},"\n步態改變（踝關節不穩定）",[16,88],{},"\n尿滯留",[16,91],{},"\n斜頸",[16,94],{},"\n不願意轉頭",[16,97],{},"\n深層肌鍵反射增加",[16,100],{},"\n2歲可以用X光進行檢查，有12~20%發現脫位，原因是齒突韌帶過鬆，所以C1C2會有十分大的活動度",[16,103],{},"\n所以，",[106,107,108],"strong",{},[75,109,111],{"className":110},[78,79],"前庭刺激手法對於唐氏症來說是禁忌",[12,113,114,115,117,118,120],{},"從肌肉上，全身呈現低張力，尤其是肢體，頸部，軀幹較明顯",[16,116],{},"\n也會有肌肉缺失問題，包括掌長肌（斷掌）和前臂屈肌",[16,119],{},"\n顴大小肌和提上唇肌這三條臉部肌肉缺乏明顯的肌腹分化",[12,122,123],{},"也會存在智力問題，通常IQ會小或等於70",[12,125,126],{},"在學習上的特徵有",[23,128,129,132,135,138,141,144],{},[26,130,131],{},"能學的不多",[26,133,134],{},"要重複性學習",[26,136,137],{},"類化不好，無法順利轉化為技巧",[26,139,140],{},"容易忘記自己學過的技巧",[26,142,143],{},"反應慢",[26,145,146],{},"受刺激時，反應少",[12,148,149,150,152,153,155],{},"所以早期介入對於唐氏症小孩是最好的，整體表現會呈現輕度跟中度，很少會演變成重度",[16,151],{},"\n一般有早期介入的唐氏症小孩在行走上是正常的",[16,154],{},"\n且因應他的智力再安排他進入不同的班級，例如資源班或者特教班",[12,157,158],{},"由於存在低張力問題，可以安排",[23,160,161,164,167,176],{},[26,162,163],{},"在幼兒期，利用抱姿或擺位，增加抗地心引力適應力和承重能力",[26,165,166],{},"也要加強抗地心引力的伸直與屈曲肌，促進關節周圍肌肉的共同收縮",[26,168,169,170,175],{},"且著重在早期的",[75,171,174],{"className":172},[78,173],"green-3","動態探索","，而不是靜態探索",[26,177,178],{},"長大後，要去加強他的體適能及靈巧性（靈敏度）",{"title":180,"searchDepth":181,"depth":181,"links":182},"",2,[],"2026-06-12","唐氏症",false,"md","\u002Fimages\u002Fuploads\u002F1781265657338-135232550_p0_master1200.webp",{},true,"\u002Fblog\u002FDown-Syndrome",{"title":6,"description":184},{"loc":190},"blog\u002FDown-Syndrome",null,"FOhG_wFbJIVxV3O-nKWt4g8KuD4rdC8E6Nyq7UBM67s",{"id":197,"title":198,"author":7,"body":199,"date":183,"description":306,"draft":185,"edited_at":183,"extension":186,"featured_image":307,"meta":308,"navigation":189,"path":309,"pinned":185,"seo":310,"sitemap":311,"stem":312,"tags":194,"__hash__":313},"blog\u002Fblog\u002FMR智能不足.md","MR智能不足",{"type":9,"value":200,"toc":304},[201,204,207,222,231,234,292],[12,202,203],{},"先鬼轉來小兒好了 東西好像有點多",[12,205,206],{},"智能不足的定義是指IQ小於等於70以下的寶寶",[12,208,209,210,212,213,215,216,218,219,221],{},"那評估的表有",[16,211],{},"\nStandford-Binet Intellectual Scale——適用於2.5歲",[16,214],{},"\n中華智力發展量表——適用於6歲",[16,217],{},"\nBayley Scale——適用於0~3歲",[16,220],{},"\nWechsler Preschool and Primary scale of intelligence——適用於學齡前",[12,223,224,225,227,228,230],{},"跟據美國智能不足協會(AAMR)定義，將適應能力分成十個項目，包括了",[16,226],{},"\n溝通，Self-care，居家生活，社交技巧，社區資源運用(community use)",[16,229],{},"\n自我導向(self-direction)，健康安全(healthy and safely)，功能性學業(functional academic)，休閒娛樂(leisure)，work",[12,232,233],{},"從IQ範圍區分出",[23,235,236,250,264,278],{},[26,237,238,239],{},"輕度智能不足：IQ 50~70",[23,240,241,244,247],{},[26,242,243],{},"學齡期前：外表看起來正常",[26,245,246],{},"學齡期：在學業上需特殊教育（可達國小六年級程度）",[26,248,249],{},"成年人：可獨立生活",[26,251,252,253],{},"中度：IQ 35~50",[23,254,255,258,261],{},[26,256,257],{},"學齡期前：社交技巧差",[26,259,260],{},"學齡期：需特殊訓練（可達國小四年級程度）",[26,262,263],{},"成年人：可從事非技術性職業",[26,265,266,267],{},"重度：IQ 20~35",[23,268,269,272,275],{},[26,270,271],{},"學齡期前：缺乏溝通能力",[26,273,274],{},"學齡期：無法學習",[26,276,277],{},"成年人：日常活動需他人協助",[26,279,280,281],{},"極嚴重：IQ\u003C20",[23,282,283,286,289],{},[26,284,285],{},"學齡期前：依賴他人照護",[26,287,288],{},"學齡期：依賴他人照護",[26,290,291],{},"成年人：依賴他人照護",[12,293,294,295,297,298,300,301,303],{},"從接受教育程度上，分成",[16,296],{},"\nEducable：可被教育，可以上一般班級或資源班",[16,299],{},"\nTrainable：可被訓練，學習一些謀生技巧",[16,302],{},"\nDependent：依賴",{"title":180,"searchDepth":181,"depth":181,"links":305},[],"Intellectual disability","\u002Fimages\u002Fuploads\u002F1781247645586-145852080_p0_master1200.webp",{},"\u002Fblog\u002FMR智能不足",{"title":198,"description":306},{"loc":309},"blog\u002FMR智能不足","NZFChlk4JSzba5twGABUXtvtVQmu1tEvseZHqvvw_z4",{"id":315,"title":316,"author":7,"body":317,"date":183,"description":446,"draft":185,"edited_at":447,"extension":186,"featured_image":448,"meta":449,"navigation":189,"path":450,"pinned":185,"seo":451,"sitemap":452,"stem":453,"tags":194,"__hash__":454},"blog\u002Fblog\u002FRett-Syndrome.md","Rett Syndrome",{"type":9,"value":318,"toc":444},[319,322,328,336,339,342,423,426,429,435,441],[12,320,321],{},"接下來講雷特氏症，先來整理一下老師給的兩頁雷特氏症的特徵",[12,323,324,325,327],{},"雷特氏症是一種發展性障礙疾病",[16,326],{},"\n好發於女性，因為是X染色體出問題導致的，有75%的典型跟非典型RS會有這個叫MECP2的異常，最新的研究這個也跟FXYD1有關係",[12,329,330,331,335],{},"主要的奇怪行為是常",[75,332,334],{"className":333},[78,79],"搓自己的手，拍自己的手，玩自己的手","，以及其他身體的怪異動作",[12,337,338],{},"雷特氏症的小孩在臨床上表現出四個時期的變化",[12,340,341],{},"RS的小孩雖然會合併自閉徵候，但RS合併MR更加明顯",[23,343,344,358,383,403],{},[26,345,346,347],{},"第一期（六個月到一歲半，早期）",[23,348,349,355],{},[26,350,351,352,354],{},"發展通常正常，這個病在這個時期會被父母忽略；父母可能會覺得小孩目光不太注視外人",[16,353],{},"\n對玩具逐漸不感興趣，較安靜乖巧，會喜歡搓手，而且頭圍成長趨緩",[26,356,357],{},"大段文字：會有一些發展遲緩問題，例如雙手協調動作未出現，臀部位移等",[26,359,360,361],{},"第二期（一歲到四歲，發展明顯遲緩及退化期，那大段文字叫快速退化期）",[23,362,363,380],{},[26,364,365,366,368,369,373,374,376,377,379],{},"小孩會顯現快速退化跟發展遲緩現象，包括語言表達逐漸喪失",[16,367],{},"\n有反覆",[75,370,372],{"className":371},[78,79],"洗手，捻手，搓手","等刻板動作，偶爾手會無理由放在背後觸摸，握緊手，張手",[16,375],{},"\n有的小孩還會吐舌，陣發喘氣，睡眠不安，或走路不穩",[16,378],{},"\n頭圍成長趨緩",[26,381,382],{},"大段文字：會喪失已學習過的技巧；會癲癇",[26,384,385,386],{},"第三期（兩歲到十歲，幼兒園到國小穩定期；那大段文字叫假性穩定期）",[23,387,388,397],{},[26,389,390,391,393,394,396],{},"小孩呈失用症，不會正常使用肢體",[16,392],{},"\n躁動，愛哭，自閉程度比之前好，會對外界事物感興趣，語言溝通較進步",[16,395],{},"\n這個時期可以維持相當一段時間",[26,398,399,400,402],{},"大段文字：手部運動功能喪失逐漸明顯；逐漸喪失自主能力",[16,401],{},"\n長達數年到數十年",[26,404,405,406],{},"第四期（五到二十五歲，運動退化晚期）",[23,407,408,417],{},[26,409,410,411,413,414,416],{},"通常會在十歲左右出現運動失能現象，逐漸不能走路，有的下肢還會強直，所以要坐輪椅",[16,412],{},"\n認知，溝通，手部動作較穩定，注視他人能力能維持",[16,415],{},"\n有些會有脊椎側彎",[26,418,419,420,422],{},"大段文字：全身性肌肉萎縮，骨骼變形，對外界無反應；最終因細菌感染，敗血病或肺炎去世",[16,421],{},"\n長達數十年",[12,424,425],{},"情緒上也會有波動，例如早上很安靜，中午很容易生氣，下午會突然很穩重，晚上會變活潑與好奇",[12,427,428],{},"他們之前學會過的任務，卻會忘記自己學過",[12,430,431,432,434],{},"在做一件任務時，興奮的動機才是誘發出正常動作的方法，他們能夠自己做出來，但假如由別人要求他再多一次出來，他就會失敗",[16,433],{},"\n所以情緒上的照顧尤為要注意，假如他不開心，就該立即停止要求他去做",[12,436,437,438,440],{},"行走能力是最關鍵的，假如喪失行走能力，會導致脊椎側彎，骨質疏鬆，甚至會影響呼吸與消化功能",[16,439],{},"\n即使時間很少，但也要著重訓練行走能力",[12,442,443],{},"學者認為，他們不是真的退化，他們還是能學到新的技巧的",{"title":180,"searchDepth":181,"depth":181,"links":445},[],"雷特氏症","2026-06-13","\u002Fimages\u002Fuploads\u002F1781344227040-145949253_p0_master1200.webp",{},"\u002Fblog\u002FRett-Syndrome",{"title":316,"description":446},{"loc":450},"blog\u002FRett-Syndrome","wp6UeSpNuWVzCMj8UKCOqphsSpW8MVmlDua_xz0r-Y0",{"id":456,"title":457,"author":7,"body":458,"date":1091,"description":1092,"draft":185,"edited_at":183,"extension":186,"featured_image":1093,"meta":1094,"navigation":189,"path":1095,"pinned":185,"seo":1096,"sitemap":1097,"stem":1098,"tags":194,"__hash__":1099},"blog\u002Fblog\u002FSCI（十五）輪椅.md","SCI（十五）輪椅",{"type":9,"value":459,"toc":1089},[460,463,466,480,482,485,500,503,506,515,524,527,529,532,594,805,807,810,842,844,847],[12,461,462],{},"剩100頁了...加油",[12,464,465],{},"這一節講到輪椅，包括了",[23,467,468,471,474,477],{},[26,469,470],{},"輪椅的選擇與配件",[26,472,473],{},"輪椅技巧",[26,475,476],{},"治療策略",[26,478,479],{},"功能性預期",[36,481],{},[12,483,484],{},"先來講輪椅，可以分成了電動輪椅跟手動輪椅",[12,486,487,488,490,491,493,494,496,497,499],{},"手動輪椅就是輕，便攜，而且可以作為運動的項目去改善體能",[16,489],{},"\n電動輪椅就更適合長距離，而且一沒辦法自己手推的輪椅的會更加適用",[16,492],{},"\n缺點也很明顯",[16,495],{},"\n手動輪椅可能會造成患者疲勞，上肢疼痛等(overuse)",[16,498],{},"\n電動輪椅也一般較大，很重，比較需要更加大的空間來用，沒辦法用一般的車輛來運送，且無法讓患者把輪椅當成運動項目",[12,501,502],{},"而電動輪椅也分成了：固定型，後傾型(tilt,整體後傾，髖關節角度不變)，後仰型(recline,只有背後仰，髖關節角度增大)，以及後傾後仰復合型",[12,504,505],{},"接下來再細說一下後傾型跟後仰型",[12,507,508,509,511,512,514],{},"首先是後傾型，像剛說的，髖關節角度不變",[16,510],{},"\n優點包括不容易產生痙攣，產生剪力較少",[16,513],{},"\n缺點包括不利於ROM，假如是腿部尿袋，會讓尿液逆流，且假如有膝上托盤，上面的東西很難固定",[12,516,517,518,520,521,523],{},"接下來講後仰型，髖關節和膝關節角度會增加",[16,519],{},"\n優點包括有利於ROM，對於間歇性導尿也更加容易",[16,522],{},"\n缺點包括可能會產生痙攣，背部可能會產生剪力",[12,525,526],{},"電動輪椅的操控有比例式（搖桿，頭部控制系統）以及按鈕式（呼吸控制，舌觸鍵盤）",[36,528],{},[12,530,531],{},"接著講輪椅的技巧，包括",[23,533,534,556,567],{},[26,535,536,537],{},"忍受直立姿勢（雖然轉位也有提到但還是再寫一下）",[23,538,539,542],{},[26,540,541],{},"主要對付的是姿勢性姿血壓（T6以上患者較為常見），所以要在學輪椅技巧之前，要先讓患者忍受得住姿勢性低血壓",[26,543,544,545],{},"應對的策略包括",[23,546,547,550,553],{},[26,548,549],{},"用後仰型輪椅慢慢回正到直立（高背輪椅：有頭配支撐，可預防姿勢性低血壓）",[26,551,552],{},"將床頭抬高雙腳垂在床邊",[26,554,555],{},"使用像彈力襪，束腹等配件",[26,557,558,559],{},"輪椅上的姿勢定位",[23,560,561,564],{},[26,562,563],{},"假如患者有肱二跟三頭肌支配，可以利用代償跟慣性控制軀幹",[26,565,566],{},"但假如兩者都沒有，會很難，所以會利用到頭肩來移動軀幹（會使用到脊椎旁肌肉，SCM，斜方肌且至少達到2\u002F2+的肌力）",[26,568,569,570],{},"減壓",[23,571,572,575],{},[26,573,574],{},"讓承重的部位恢復供血，假如是良好的坐姿壓力會集中在坐骨結節",[26,576,577,578],{},"減壓的方法有",[23,579,580,583,591],{},[26,581,582],{},"坐姿下撐起：利用肱三將臀部整個抬起",[26,584,585,586],{},"重心轉移：前傾或左右傾進行局部減壓",[23,587,588],{},[26,589,590],{},"坐輪椅時可以用手臂勾住一邊輪椅的扶手，使另一邊臀部抬高",[26,592,593],{},"後傾\u002F後仰型輪椅：就後傾或後仰，減壓完再回正",[23,595,596,648],{},[26,597,598,599],{},"平坦地面的輪椅移動能力",[23,600,601,604,612],{},[26,602,603],{},"假如是C4以上的患者，因缺乏肱二和三角肌，會推薦用電動輪椅",[26,605,606,607],{},"那有一些輪椅的手推輪圈會有突起的推把，假如手部活動有問題，肌力不足等的患者也能自己推",[23,608,609],{},[26,610,611],{},"主要利用肱二和三角肌來推（肩和肘屈曲）",[26,613,614,615],{},"還有就是一般的平滑輪圈",[23,616,617,633],{},[26,618,619,620],{},"前向推進",[23,621,622],{},[26,623,624,625],{},"假如，手指屈肌沒有支配（沒有C8）",[23,626,627,630],{},[26,628,629],{},"手掌抵住輪圈，保持肘彎曲，肩內旋",[26,631,632],{},"利用肘伸直，搭配肩內收，外旋，屈曲，用力推出去（這個動作需要強大的肌力）",[26,634,635,636],{},"後向推進",[23,637,638],{},[26,639,624,640],{},[23,641,642,645],{},[26,643,644],{},"抵住輪圈",[26,646,647],{},"利用肘伸肌將輪圈向後帶，並利用肩胛下壓繼續往後",[26,649,650,651],{},"克服障礙物",[23,652,653,661,704,758,775,788],{},[26,654,655,656],{},"斜坡路：上去這種路，會有一定的向後翻倒風險，且比起平地需要更大的力氣；而且練習不應只限應對1:12坡度的斜坡",[23,657,658],{},[26,659,660],{},"（短直角邊1 inch，長直角邊12 inch）",[26,662,663,664],{},"路緣（高低差）",[23,665,666,690],{},[26,667,668,669],{},"向前上路緣：下肢癱瘓的患者可以應付6inch(15cm)高的路緣，而C6損傷的患者可以應付2~4inch(5~10cm)",[23,670,671],{},[26,672,673,674],{},"這個動作是不需要用到肱三和手指抓握的，著重從靜止狀態（較依賴肌力，空間需求少）或利用慣性（需要助跑空間）登上去",[23,675,676],{},[26,677,678,679],{},"靜立位置登上路緣",[23,680,681,684,687],{},[26,682,683],{},"起始：距離路緣數inch",[26,685,686],{},"用力將前輪登上路緣，並將前後退後到路緣邊緣",[26,688,689],{},"把手放到輪圈後方，開始用力推，達成登上去（假如身體前傾也可以讓後輪更容易登上去）",[26,691,692,693],{},"向後下路緣：其實向前還是向後都可以，但一般向後會更簡單（一樣不需要肱三跟指握的）",[23,694,695,698,701],{},[26,696,697],{},"在下路緣時，輪椅兩個輪子平行對齊路緣邊緣，並且身體向前傾",[26,699,700],{},"要下的時候直接向後推輪子",[26,702,703],{},"之後原地小幅度的自轉，使前輪也下來路緣",[26,705,706,707],{},"樓梯：分成了上跟下",[23,708,709,723,734,742],{},[26,710,711,712],{},"將臀部放在樓梯上上樓梯：對於上肢肌力支配正常的患者來說是可行的，雖然速度較慢，但所需的的肌力也較少",[23,713,714,717,720],{},[26,715,716],{},"首先是轉位到樓梯上，將輪椅向後傾倒在樓梯上",[26,718,719],{},"自己上一階，調整自己雙腿，將輪椅拉高一階",[26,721,722],{},"輪椅穩定後，繼續下一步往復上樓梯",[26,724,725,726],{},"坐在輪椅上向後上樓梯：對於上肢肌力支配正常的患者來說是可行的，需要用到安全帶，或者將大腿近端跟輪椅坐面固定住",[23,727,728,731],{},[26,729,730],{},"先繫好安全帶（固定好），將輪椅後傾倒在樓梯上",[26,732,733],{},"雙手擺在後方上一階樓梯，用力撐，慢慢的撐上樓梯",[26,735,736,737],{},"坐在輪椅上向後下樓梯：對於上肢肌力支配正常的患者來說是可行的，雙手握住扶手，靠近扶手的那隻手握得較另一隻手低",[23,738,739],{},[26,740,741],{},"輪椅退後到樓梯邊緣，雙手握緊扶手，慢慢的下樓梯",[26,743,744,745,749,750],{},"將前輪翹起向前下樓梯：對於上肢肌力支配正常且",[75,746,748],{"className":747},[78],"精通輪椅技巧","的患者來說是可行的；跌倒風險也較高，樓梯面較寬，較短的樓梯上比較安全",[23,751,752,755],{},[26,753,754],{},"將輪椅保持翹前輪姿勢，後輪平行抵在樓梯邊緣",[26,756,757],{},"控制好，慢慢下落，途中可以推拉輪圈，以穩定輪椅",[26,759,760,761],{},"狹窄的門口",[23,762,763,766,769,772],{},[26,764,765],{},"起始貼近門口，一腳從腳踏移開，收起腳腳踏",[26,767,768],{},"用力撐起身體坐到輪椅扶手上",[26,770,771],{},"透過上拉坐墊，將輪椅變窄",[26,773,774],{},"用手拉門邊，讓輪椅被拉動通過門口",[26,776,777,778],{},"安全跌倒技巧",[23,779,780],{},[26,781,782,783],{},"向後跌倒時，身體盡量前傾低頭（下巴貼近胸口），並抓住輪圈，所以著地時，應該是輪椅推把承受重擊力，而不是背或後腦",[23,784,785],{},[26,786,787],{},"也可以一手握緊輪圈，一手握緊同一邊的扶手，避免膝蓋撞擊到臉部",[26,789,790,791],{},"跌倒後回復直立姿勢",[23,792,793,796,799,802],{},[26,794,795],{},"跌倒後，腳勾住座墊前緣（應該是指膝窩跨過座墊），先握住扶手將軀幹拉高一點",[26,797,798],{},"然後一手放在後方撐住地面，另一手握住對側的輪子",[26,800,801],{},"手用力撐起的同時，在輪子的手將輪子往後拉，慢慢的搖擺拉起來",[26,803,804],{},"慢慢拉起後，撐在地面的手也緩緩的向前移動，最後恢復直立",[36,806],{},[12,808,809],{},"然後來講策略，包括了",[23,811,812,823,834],{},[26,813,814,815],{},"翹前輪：就是前輪翹起，僅以後輪支撐",[23,816,817,820],{},[26,818,819],{},"在翹起的時候，軀幹可以稍微前傾，讓重心保持在輪椅的BOS上；在學習時也可以在有地毯的環境中，藉由地毯的阻力掌握控制翹前輪方法",[26,821,822],{},"翹起所需的角度取決於輪椅後輪軸的位置",[26,824,825,826],{},"以翹前輪姿勢向前滑行",[23,827,828,831],{},[26,829,830],{},"起始在翹前輪姿勢，接著稍微將前輪放下一點",[26,832,833],{},"之後向前推進並抬輪前輪，這樣又會保持平衡的翹前輪姿勢，並繼續因慣性前進（我猜是這樣，我腦補的）",[26,835,836,837],{},"練習坐在扶手上保持平衡",[23,838,839],{},[26,840,841],{},"用單臂支撐，坐在扶手上，保持動態平衡；治療師應在患者練習時提供保護",[36,843],{},[12,845,846],{},"最後是功能性預期，會從患者保留的level分成",[23,848,849,885,933,991,1027,1056],{},[26,850,851,852],{},"C1~C4保留",[23,853,854,877],{},[26,855,856,857],{},"典型預期表現",[23,858,859,866,869],{},[26,860,861,862],{},"因缺乏自主上肢來使用手動輪椅或搖桿式的電動輪椅，所以會使用",[75,863,865],{"className":864},[78,173],"頭部控制，呼吸控制，以及舌觸控制的電動輪椅",[26,867,868],{},"減壓部份，可以用後傾或後仰功能的輪椅以達成獨立減壓",[26,870,871,872,876],{},"擺位則",[75,873,875],{"className":874},[78],"完全依賴","他人協助",[26,878,879,880],{},"出乎預期表現",[23,881,882],{},[26,883,884],{},"C4保留且部份保留C5運動功能，就有可能可以使用手控式電動輪椅；在姿勢調整上，也可以達成直立>側傾，側傾>直立",[26,886,887,888],{},"C5保留",[23,889,890,923],{},[26,891,856,892],{},[23,893,894,901,909,920],{},[26,895,896,897],{},"在未鋪地毯的地面用手動輪椅能做到",[75,898,900],{"className":899},[78],"部份獨立",[26,902,903,904,908],{},"用",[75,905,907],{"className":906},[173],"搖桿式電動輪椅","能做到獨立",[26,910,911,912],{},"減壓部份",[23,913,914,917],{},[26,915,916],{},"後傾\u002F後仰輪椅電動輪椅能做到獨立減壓",[26,918,919],{},"手動輪椅透過側傾方法能做到部份獨立或獨立",[26,921,922],{},"調整軀幹和臀部的擺位需完全依賴或部份協助",[26,924,879,925],{},[23,926,927,930],{},[26,928,929],{},"在減壓上，不使用特殊座椅情況下，可達到獨立",[26,931,932],{},"調整軀幹和臀部的擺位可達到獨立",[26,934,935,936],{},"C6保留",[23,937,938,971],{},[26,939,856,940],{},[23,941,942,945,948,955,968],{},[26,943,944],{},"在未鋪地毯的室內環境下使用手動輪椅，可達到獨立",[26,946,947],{},"使用搖桿式電動輪椅可達到獨立",[26,949,950,951],{},"面對如1:12斜坡等輕微障礙時，需要",[75,952,954],{"className":953},[78,173],"部份協助到完全協助",[26,956,911,957],{},[23,958,959,961],{},[26,960,916],{},[26,962,963,964],{},"使用手動輪椅時，可透過撐起，前傾，側傾，達到",[75,965,967],{"className":966},[78,173],"部份協助到獨立",[26,969,970],{},"調整軀幹和臀部的擺位可達部份協助到獨立",[26,972,879,973],{},[23,974,975,978,981,988],{},[26,976,977],{},"可在社區環境中達到獨立",[26,979,980],{},"面對輕微障礙時，可達到獨立（1:12斜坡，2~4inch路緣，稍微不平坦地面等）",[26,982,983,984],{},"使用手動輪椅時，可透過撐起，前傾，側傾進行減壓，達到",[75,985,987],{"className":986},[78,173],"獨立",[26,989,990],{},"調整軀幹和臀部的擺位可達獨立",[26,992,993,994],{},"C7保留",[23,995,996,1020],{},[26,997,856,998],{},[23,999,1000,1011,1014,1017],{},[26,1001,1002,1003],{},"使用手動輪椅在室內外平坦地面可實行長距離移動，在社區中移動，皆可獨立",[23,1004,1005],{},[26,1006,1007],{},[75,1008,1010],{"className":1009},[78,79],"C7以下是個分界線，是可以獨立使用手動輪椅的界線",[26,1012,1013],{},"面對輕微障礙物可達部份協助到獨立；不平整路面或路緣需部份協助到完全協助",[26,1015,1016],{},"手撐起來的方式達到獨立減壓",[26,1018,1019],{},"調整軀幹和臀部的擺位可獨立",[26,1021,879,1022],{},[23,1023,1024],{},[26,1025,1026],{},"面對輕微障礙物能獨立，包括1:12斜坡，4 inch路緣，稍微不平坦路面",[26,1028,1029,1030],{},"C8保留",[23,1031,1032,1049],{},[26,1033,856,1034],{},[23,1035,1036,1038,1045,1047],{},[26,1037,1002],{},[26,1039,1040,1041],{},"面對輕微障礙物可達部份協助到獨立；不平整路面或路緣需",[75,1042,1044],{"className":1043},[78,173],"部份協助",[26,1046,1016],{},[26,1048,1019],{},[26,1050,879,1051],{},[23,1052,1053],{},[26,1054,1055],{},"面對輕微障礙物能獨立，包括1:12斜坡，4 inch路緣，稍微不平坦路面，狹窄門口；以及能安全跌倒",[26,1057,1058,1059],{},"T1及以下",[23,1060,1061,1078],{},[26,1062,856,1063],{},[23,1064,1065,1067,1074,1076],{},[26,1066,1002],{},[26,1068,1069,1070],{},"面對障礙物能獨立，包括比1:12更斜的斜坡，4~5inch路緣，狹窄門口；能安全跌倒，用",[75,1071,1073],{"className":1072},[78,173],"臀部坐在樓梯上上樓梯，以及握住扶手下樓梯",[26,1075,1016],{},[26,1077,1019],{},[26,1079,879,1080],{},[23,1081,1082],{},[26,1083,1084,1085],{},"面對障礙物能獨立，包括高於6 inch路緣；",[75,1086,1088],{"className":1087},[78,79],"能坐在輪椅上直接上樓梯，以及翹前輪方式下樓梯",{"title":180,"searchDepth":181,"depth":181,"links":1090},[],"2026-06-07","SCI-10","\u002Fimages\u002Fuploads\u002F1781236915013-47141636_p0_master1200.webp",{},"\u002Fblog\u002FSCI（十五）輪椅",{"title":457,"description":1092},{"loc":1095},"blog\u002FSCI（十五）輪椅","G6NdR7ijTcIge2p3lIqpTzt072yskLyJ8pnz8i1GagI",{"id":1101,"title":1102,"author":7,"body":1103,"date":1562,"description":1563,"draft":185,"edited_at":1091,"extension":186,"featured_image":1564,"meta":1565,"navigation":189,"path":1566,"pinned":185,"seo":1567,"sitemap":1568,"stem":1569,"tags":194,"__hash__":1570},"blog\u002Fblog\u002FSCI（十四）轉位.md","SCI（十四）轉位",{"type":9,"value":1104,"toc":1560},[1105,1108,1110,1113,1116,1119,1122,1125,1128,1130,1133,1211,1213,1216,1295,1297,1300,1303,1306,1328,1330,1333,1339,1355,1358,1369,1371,1374,1377,1442,1448,1451,1468,1470,1473,1476,1479,1482,1501,1504,1520,1523,1536,1539,1548,1551],[12,1106,1107],{},"剩下三份共149頁PPT了 快了快了",[36,1109],{},[12,1111,1112],{},"接下來講轉位他分成了",[12,1114,1115],{},"基礎技巧",[12,1117,1118],{},"與輪椅相關的技巧",[12,1120,1121],{},"不使用輔具的高位轉位技巧",[12,1123,1124],{},"使用輔具的高位轉位技巧",[12,1126,1127],{},"不等高轉位技巧",[36,1129],{},[12,1131,1132],{},"先來說基礎技巧，包括了",[23,1134,1135,1152,1165,1173,1184,1195,1203],{},[26,1136,1137,1138],{},"忍受直立坐姿",[23,1139,1140,1143],{},[26,1141,1142],{},"主要是因為姿勢性低血壓的問題，所以要適應，忍受直立坐姿，並慢慢將軀幹漸進到更加直立姿勢，才能繼續後續的步驟",[26,1144,1145,1146,1151],{},"首先可以利用傾斜床，假如",[75,1147,1150],{"className":1148},[78,1149],"bg-yellow-3","傾斜角度達到60度並維持15~20分鐘","，即代表可以坐輪椅",[26,1153,1154,1155],{},"利用肌肉代償伸直手肘",[23,1156,1157],{},[26,1158,1159,1160,1164],{},"這個是之前說過的，就是肱三頭肌肌力不足或沒有，這時要伸直手肘，可以利用",[75,1161,1163],{"className":1162},[78,1149],"三角肌前束和胸大肌鎖骨部","肌肉來伸直肘關節",[26,1166,1167,1168],{},"鎖定肘關節",[23,1169,1170],{},[26,1171,1172],{},"這個也是之前說過的，鎖定姿勢在肩外旋，肘和腕伸直，前臂旋後，重量壓在手掌上，並利用剛剛的肌肉代償將肘卡在伸直",[26,1174,1175,1176],{},"維持坐姿平衡",[23,1177,1178,1181],{},[26,1179,1180],{},"假如患者可以忍受直立坐姿，就可以開始平衡訓練",[26,1182,1183],{},"進階訓練包括：坐姿下雙手伸直向前撐地＞坐姿下單手伸直向前撐地＞無手撐地坐姿平衡",[26,1185,1186,1187],{},"從坐姿前傾姿勢，將軀幹拉回直立坐姿",[23,1188,1189,1192],{},[26,1190,1191],{},"這裡的重點是患者要對抗阻力（自重+重力）下伸直手肘，不管用的是肌肉代償還是肱三頭肌肌力",[26,1193,1194],{},"姿勢採短坐姿，初期的話可以在撐在前方的墊子上，進階一點可以撐在大腿遠端上方或兩側",[26,1196,1197,1198],{},"利用頭部及肩膀控制骨盆",[23,1199,1200],{},[26,1201,1202],{},"就利用頭髖關係",[26,1204,1205,1206],{},"將雙腳抬上墊子",[23,1207,1208],{},[26,1209,1210],{},"假如要讓雙腳上墊子，要一隻手繼續撐住維持平衡，另一隻手將雙腳搬上墊子",[36,1212],{},[12,1214,1215],{},"接下來是輪椅相關的技巧，包括了",[23,1217,1218,1240,1243,1265,1281,1292],{},[26,1219,1220,1221],{},"輪椅上穩定軀幹",[23,1222,1223,1226],{},[26,1224,1225],{},"因為高位損傷的患者除了向後靠背和向前靠腿以外，其他姿勢都需要用雙臂來維持穩定",[26,1227,1228,1229],{},"手臂擺放的位置取決於軀幹前傾的角度",[23,1230,1231,1234,1237],{},[26,1232,1233],{},"直立：用手臂勾住輪椅後方推把",[26,1235,1236],{},"前傾：可以靠在自己手臂上，或者一樣勾住後方推把",[26,1238,1239],{},"側傾：勾住後方推把",[26,1241,1242],{},"輪椅上移動軀幹",[26,1244,1245,1246],{},"輪椅上移動臀部",[23,1247,1248,1251],{},[26,1249,1250],{},"對於完全性損傷的患者，可以利用頭臀關係來移動臀部",[26,1252,1253,1254],{},"動作包括（例子）",[23,1255,1256,1259,1262],{},[26,1257,1258],{},"挪出去——往外\u002F往前挪，將頭向右後方扭轉，帶動臀部向左前方動",[26,1260,1261],{},"挪回來且靠左——身體前傾，將頭與軀幹往右扭，臀部後會後左動",[26,1263,1264],{},"臀部往後挪——身體前傾，雙手向前推，同時透過左右晃動頭部與肩膀完成",[26,1266,1267,1268],{},"輪椅的位置",[23,1269,1270,1273],{},[26,1271,1272],{},"從輪椅轉移到其他平面：保持約30度夾角，且盡量貼緊想要轉移的平面上",[26,1274,1275,1276],{},"跌倒時將輪椅回正",[23,1277,1278],{},[26,1279,1280],{},"首先將輪子鎖緊，用手壓下輪椅支架，將輪椅回正",[26,1282,1283,1284],{},"輪椅配件：扶手和腳踏",[23,1285,1286,1289],{},[26,1287,1288],{},"扶手最好是可以移開的那種，因為要獨立自行轉位的話，最好是扶手可以拆開，讓側邊騰出位置；不管是拆下，還是旋轉也是這樣道理",[26,1290,1291],{},"腳踏也是如此，最好是可以將腳踏摺疊，旋轉的輪椅，這樣坐輪椅的人的腳也能方便的進出",[26,1293,1294],{},"雙腿擺放",[36,1296],{},[12,1298,1299],{},"接下來不使用輔具的等高轉位",[12,1301,1302],{},"假如患者在有肱三頭肌肌力的話，會更加容易轉位，因為不需要用肌肉代償將肘卡住",[12,1304,1305],{},"但假如真沒有肱三頭肌，代表還是要將肘卡住",[23,1307,1308],{},[26,1309,1310,1311],{},"例子：轉移到等高的墊上",[23,1312,1313,1316,1319,1322,1325],{},[26,1314,1315],{},"前置：要先讓臀部向前挪動",[26,1317,1318],{},"放在輪椅的手放在大腿旁邊",[26,1320,1321],{},"放在墊上的手放越遠越好",[26,1323,1324],{},"然後手用肌肉代償方式卡成肘伸直",[26,1326,1327],{},"轉位時將重量放在雙臂上，收下巴且頭部向下向外繞，利用手臂作支點將臀部甩到墊上（看來這裡也用到頭臀關係）",[36,1329],{},[12,1331,1332],{},"使用輔具的等高轉位",[12,1334,1335,1336,1338],{},"使用轉位板，患者要抓住轉位板並將其插入大腿與臀部下方（假如身體向遠離板子方向傾斜，更容易讓板子插入）",[16,1337],{},"\n且要保持一點傾斜角度，目標高度高於現在輪椅的高度",[12,1340,1341,1342,1346,1347,1349,1350,1354],{},"接下來講直立式轉位，雖然他是利用到轉位板，但他跟",[75,1343,1345],{"className":1344},[78,173],"不使用輔具的轉位","方法基本一致，包括假如沒有肱三頭肌的話一樣使用肘卡住",[16,1348],{},"\n差別在於因為有轉位板，",[75,1351,1353],{"className":1352},[78,173],"他的臀部不需要完全抬離","，但還是需要將部份重量從臀部轉移到其他地方，才能轉移走（例如用手撐住）",[12,1356,1357],{},"那假如個案沒辦法執行直立式轉移，那我們會教導替代方法（原因可能包括肱二頭肌痙攣，平衡能力差，關節攣縮等）",[23,1359,1360,1363,1366],{},[26,1361,1362],{},"起始姿勢：利用左臂勾住後方推把，右臂勾住輪椅扶手",[26,1364,1365],{},"通過扭轉身體，將臀部向外滑動轉移到床面",[26,1367,1368],{},"最後利用肘承重完成整個動作",[36,1370],{},[12,1372,1373],{},"最後來講非等高轉位",[12,1375,1376],{},"先來講從地面轉移回輪椅上，包括了",[23,1378,1379,1402,1424],{},[26,1380,1381,1382],{},"側向（有保留部份肱三的病人可用這個）",[23,1383,1384,1387,1390],{},[26,1385,1386],{},"起始姿勢：身體側靠在輪椅旁，一手撐在地上，一手撐住輪椅",[26,1388,1389],{},"利用頭臀關係，頭向下，身體前傾，用力撐使臀部甩向輪椅坐墊",[26,1391,1392,1393],{},"接著雙手抓住兩腿，慢慢的將身體撐直",[23,1394,1395],{},[26,1396,1397,1398],{},"而這套動作需要：完整的肩伸展，外展角度，以及",[75,1399,1401],{"className":1400},[78,79],"較鬆的hamstring",[26,1403,1404,1405,1409,1410],{},"前向（這個要求",[75,1406,1408],{"className":1407},[78,79],"全部的上肢支配","，但不需要要求剛說的較鬆的hamstring）",[23,1411,1412,1415,1418,1421],{},[26,1413,1414],{},"起始姿勢：面向輪椅，一手撐住地板，一手扶住輪椅",[26,1416,1417],{},"接著手用力撐，將臀部抬離地板",[26,1419,1420],{},"切換成高跪姿",[26,1422,1423],{},"最後下壓輪椅扶手，並且將臀部帶到輪椅上",[26,1425,1426,1427,1430,1431],{},"後向（這個要求",[75,1428,1408],{"className":1429},[78,79],"，以及完整的肩膀伸直角度）",[23,1432,1433,1436,1439],{},[26,1434,1435],{},"起始姿勢：背向輪椅，雙膝微彎，兩手向後反撐在輪椅坐墊上",[26,1437,1438],{},"雙手用力撐，使臀部抬離地面",[26,1440,1441],{},"而後將臀部移到坐墊深處",[12,1443,1444,1445,1447],{},"接著是講浴室中轉位，假如沒有任何輔具的話，一樣需要上肢完全支配才能達成",[16,1446],{},"\n但假如不具備完全上肢支配的患者，可以使用洗澡轉位椅，技巧類似等高轉位",[12,1449,1450],{},"然後是車輛的轉位，包括了",[23,1452,1453,1456,1459,1462,1465],{},[26,1454,1455],{},"調整角度，盡量貼緊車輪",[26,1457,1458],{},"移開輪椅扶手和腳踏，避開障礙",[26,1460,1461],{},"一手放在車子坐墊上，一手放在輪椅",[26,1463,1464],{},"用雙臂當支點旋轉完成轉位",[26,1466,1467],{},"最後將雙腿放進車內",[36,1469],{},[12,1471,1472],{},"最後補充一下，保留到的level在轉位上的差異",[12,1474,1475],{},"首先是C1~C4，屬完全依賴類型，使用的輔具包括天花板軌道式懸吊系統，液壓式移位機等",[12,1477,1478],{},"C5level的話，屬最大協助到完全依賴類型，使用的輔具包括天花板軌道式懸吊系統，液壓式移位機，轉位板",[12,1480,1481],{},"C6level的話",[23,1483,1484,1487,1490,1493],{},[26,1485,1486],{},"等高轉位：不使用輔具可達部份協助到完全獨立",[26,1488,1489],{},"不等高轉位：部份協助或完全協助",[26,1491,1492],{},"車輛轉位：中度協助到最大協助",[26,1494,1495,1496],{},"從地面轉位：完全協助",[23,1497,1498],{},[26,1499,1500],{},"需要的輔具：轉位板",[12,1502,1503],{},"C7level的話",[23,1505,1506,1509,1512],{},[26,1507,1508],{},"等高轉位：不使用輔具可達完全獨立",[26,1510,1511],{},"不等高轉位：在輪椅與其他平面差距幾inch下可達部份協助或完全獨立",[26,1513,1514,1515],{},"車輛轉位：改裝過的車輛，可達獨立",[23,1516,1517],{},[26,1518,1519],{},"需要的輔具：可選用不用轉位板",[12,1521,1522],{},"C8level的話",[23,1524,1525,1527,1530],{},[26,1526,1508],{},[26,1528,1529],{},"不等高轉位：在輪椅與其他平面差距幾inch下可達完全獨立",[26,1531,1514,1532],{},[23,1533,1534],{},[26,1535,1519],{},[12,1537,1538],{},"T1~T9level的話",[23,1540,1541,1543,1546],{},[26,1542,1508],{},[26,1544,1545],{},"不等高轉位：從地面轉移到輪椅可能可以獨立，在輪椅與其他平面差距幾inch下可達完全獨立",[26,1547,1514],{},[12,1549,1550],{},"T10level的話",[23,1552,1553,1555,1558],{},[26,1554,1508],{},[26,1556,1557],{},"不等高轉位：從地面轉移到輪椅，以及在輪椅與其他平面差距幾inch下可達完全獨立",[26,1559,1514],{},{"title":180,"searchDepth":181,"depth":181,"links":1561},[],"2026-06-06","SCI-09","\u002Fimages\u002Fuploads\u002F1780815899663-145658642_p0_master1200.webp",{},"\u002Fblog\u002FSCI（十四）轉位",{"title":1102,"description":1563},{"loc":1566},"blog\u002FSCI（十四）轉位","V2shEkYoDoJ-k88CjsrxBqb46d3ly8iabC0qW5GBjmk",{"id":1572,"title":1573,"author":7,"body":1574,"date":2171,"description":2172,"draft":185,"edited_at":2173,"extension":186,"featured_image":2174,"meta":2175,"navigation":189,"path":2176,"pinned":185,"seo":2177,"sitemap":2178,"stem":2179,"tags":194,"__hash__":2180},"blog\u002Fblog\u002FSCI-（十三）墊上運動-ii.md","SCI （十三）墊上運動-ii",{"type":9,"value":1575,"toc":2169},[1576,1579,1675,1677,1927,1934,1936,1941,1943,2041,2043,2070,2072,2097,2099,2102,2118,2128,2138,2148,2158,2160],[12,1577,1578],{},"那墊上功能性技巧有",[23,1580,1581],{},[26,1582,1583,1584],{},"翻身",[23,1585,1586,1624,1660],{},[26,1587,1588,1589],{},"不使用器具下能躺到趴",[23,1590,1591,1594],{},[26,1592,1593],{},"假如沒有三頭肌，在甩手臂時患者要將手舉到與墊面45度；有三頭肌的話甩手臂時就要呈垂直",[26,1595,1596,1597],{},"身體條件",[23,1598,1599,1610,1621],{},[26,1600,1601,1602],{},"肌力",[23,1603,1604,1607],{},[26,1605,1606],{},"強壯的三角肌前束與後束肌力",[26,1608,1609],{},"部份的三角肌中束，棘下肌，小圓肌肌力",[26,1611,1612,1613],{},"ROM",[23,1614,1615,1618],{},[26,1616,1617],{},"正常的活動度：肩膀ER，手肘伸直",[26,1619,1620],{},"而這些關節活動障礙會影響動作：肩水平外展\u002F內收，前臂旋後",[26,1622,1623],{},"那技巧是頭手軀幹配合節奏重複的左右擺動，讓慣性或動能足夠翻身",[26,1625,1626,1627],{},"使用器具下能躺到趴",[23,1628,1629,1632,1640],{},[26,1630,1631],{},"通常，肌力不足，關節攣縮，肥胖，痙攣，認知混亂，動機低下，練習時間不足，都可能導致患者不用器具就翻不了身",[26,1633,1634,1635],{},"患者可以透過拉懸吊帶，床邊護欄，或者停在墊子\u002F床邊的輪椅來翻身（因為輪椅對於高位SCI基本上算是必備，是最理想的器具）",[23,1636,1637],{},[26,1638,1639],{},"一手固定，且利用肱二頭和三角肌前束用力，軀幹跟另一隻手同時甩向固定手，增加慣性和動量",[26,1641,1596,1642],{},[23,1643,1644,1654,1657],{},[26,1645,1601,1646],{},[23,1647,1648,1651],{},[26,1649,1650],{},"強壯的三角肌前束，肱二頭肌，肱肌，肱橈肌肌力",[26,1652,1653],{},"部份的三角肌中束\u002F後束，棘下肌，小圓肌肌力",[26,1655,1656],{},"ROM：假如肩膀彎曲，水平內收，手肘彎曲有障礙會影響動作",[26,1658,1659],{},"可用設備包括懸吊帶，輪椅，床邊護欄",[26,1661,1662,1663],{},"趴到躺：這個動作是比躺到趴簡單，患者只需要推地面，達到側躺，然後再甩一下手臂跟頭就可以躺了",[23,1664,1665],{},[26,1666,1596,1667],{},[23,1668,1669,1672],{},[26,1670,1671],{},"肌力：部份的三角肌前\u002F中\u002F後束，棘下肌，小圓肌肌力",[26,1673,1674],{},"ROM：假如肩膀彎曲，手肘伸直，前臂旋後有障礙會影響動作",[36,1676],{},[23,1678,1679,1752],{},[26,1680,1681,1682],{},"轉換至prone on elbow",[23,1683,1684,1687],{},[26,1685,1686],{},"這個姿勢是可以在床上移動的，也是切換成趴到坐的姿勢之一",[26,1688,1689,1690],{},"那分別有這三個姿勢來切換",[23,1691,1692,1716,1732],{},[26,1693,1694,1695],{},"起始趴姿，肩膀abd",[23,1696,1697,1708],{},[26,1698,1699,1700],{},"肌力需求",[23,1701,1702,1705],{},[26,1703,1704],{},"強壯的三角肌前束肌力",[26,1706,1707],{},"部份的棘下肌，小圓肌肌力",[26,1709,1710,1711,1715],{},"假如肩膀屈曲，外展，",[75,1712,1714],{"className":1713},[173,78],"外旋","，內收，以及肘彎曲角度受限，會使動作嚴重限制",[26,1717,1718,1719],{},"起始趴姿，肩膀add",[23,1720,1721,1729],{},[26,1722,1699,1723],{},[23,1724,1725,1727],{},[26,1726,1704],{},[26,1728,1707],{},[26,1730,1731],{},"假如肩膀屈曲，外展，內收，以及肘彎曲角度受限，會使動作嚴重限制",[26,1733,1734,1735],{},"起始側躺",[23,1736,1737,1747],{},[26,1738,1699,1739],{},[23,1740,1741,1744],{},[26,1742,1743],{},"強壯的三角肌後束肌力",[26,1745,1746],{},"部份的三角肌前\u002F中束，肱二頭肌，棘下肌，小圓肌肌力",[26,1748,1710,1749,1715],{},[75,1750,1714],{"className":1751},[173,78],[26,1753,1754,1755],{},"不使用器具下能坐起",[23,1756,1757,1760],{},[26,1758,1759],{},"假如肱三頭肌肌力完整的患者，是可以直接肌撐墊子從躺到坐",[26,1761,1762,1763],{},"但假如是三肌肌肌力沒有的患者，就可以用這三種方法",[23,1764,1765,1814,1862],{},[26,1766,1767,1768],{},"Roll and throw",[23,1769,1770,1790],{},[26,1771,1772,1773],{},"動作包括",[23,1774,1775,1778,1781,1784,1787],{},[26,1776,1777],{},"首先是prone on elbow姿勢",[26,1779,1780],{},"將重心轉移到一隻手上(side-lying)，另一隻未承重手將軀幹往上帶",[26,1782,1783],{},"然後未承重那隻手往後甩(應該算變成長坐姿)",[26,1785,1786],{},"變成剛剛往後甩的那隻手承重，抬起另一隻手",[26,1788,1789],{},"抬起的這隻手再往回放好（長坐姿）",[26,1791,1596,1792],{},[23,1793,1794,1803],{},[26,1795,1699,1796],{},[23,1797,1798,1801],{},[26,1799,1800],{},"強壯的三頭肌前\u002F後束肌力",[26,1802,1609],{},[26,1804,1805,1806],{},"ROM需求",[23,1807,1808,1811],{},[26,1809,1810],{},"正常的肩伸展，外旋，水平外展，手肘伸直角度",[26,1812,1813],{},"假如肩屈曲，外展，手肘屈曲，前臂旋後，手腕伸展角度受限，會使動作嚴重限制",[26,1815,1816,1817],{},"Straight from supine, hands stabilized",[23,1818,1819,1838],{},[26,1820,1772,1821],{},[23,1822,1823,1826,1829,1832,1835],{},[26,1824,1825],{},"先supine，elbow放在臀部位置",[26,1827,1828],{},"透過肘彎曲將軀幹抬高",[26,1830,1831],{},"反覆的左右轉移重心，讓手肘可以穩定的維持身體(supine on elbow)",[26,1833,1834],{},"先將重心移動到一邊手肘，另一隻手抬起",[26,1836,1837],{},"抬起的那隻手伸展並往後甩，另一隻手也是如此操作，最後一步步用雙手向前跨行，變成長坐姿",[26,1839,1596,1840],{},[23,1841,1842,1852],{},[26,1843,1699,1844],{},[23,1845,1846,1849],{},[26,1847,1848],{},"強壯的三角肌前束以及肱二頭肌肌力",[26,1850,1851],{},"部份的三角肌中\u002F後束，棘下肌，小圓肌肌力",[26,1853,1612,1854],{},[23,1855,1856,1859],{},[26,1857,1858],{},"正常的肩膀伸展，外旋，水平外展，手肘伸直角度",[26,1860,1861],{},"假如肩外展，手肘彎曲，前臂旋後，手腕伸展角度受限，會使動作嚴重限制",[26,1863,1864,1865],{},"Walking on elbows",[23,1866,1867,1901],{},[26,1868,1772,1869],{},[23,1870,1871,1874,1877,1880,1883,1886,1889,1892,1895,1898],{},[26,1872,1873],{},"起始在prone on elbow",[26,1875,1876],{},"利用雙肘向身體一側flex，側到沒有辦法再flex為止（原文是寫lateral flex）",[26,1878,1879],{},"將骨盤從趴姿朝側躺姿勢翻",[26,1881,1882],{},"雙肘向雙腿方向跨行",[26,1884,1885],{},"然後其中一隻手勾住大腿，另一隻手保持承重",[26,1887,1888],{},"承重的那隻手也算雙腿方向移動",[26,1890,1891],{},"掌心貼好，固定在墊上",[26,1893,1894],{},"然後搖晃身體，將重心轉移致雙腿正上方",[26,1896,1897],{},"最後將軀幹往上推，變成長坐姿",[26,1899,1900],{},"（我還是放圖好了，文字上我也覺得有點抽象要思考）",[26,1902,1596,1903],{},[23,1904,1905,1917],{},[26,1906,1699,1907],{},[23,1908,1909],{},[26,1910,1911,1912,1916],{},"部份的三角肌前\u002F中\u002F後束，棘下肌，小圓肌，",[75,1913,1915],{"className":1914},[78,173],"內旋肌群","，肱二頭肌肌力",[26,1918,1612,1919],{},[23,1920,1921,1924],{},[26,1922,1923],{},"正常的肩內旋角度，以及hamstring柔軟度（長坐姿嘛）",[26,1925,1926],{},"假如肩屈曲，外展，水平外展，肘屈曲，伸直角度受限，會使動作嚴重限制",[12,1928,1929],{},[1930,1931],"img",{"alt":1932,"src":1933},"1.00","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1780635044876-Screenshot_2026-06-04_215418.png",[16,1935],{},[12,1937,1938],{},[1930,1939],{"alt":1932,"src":1940},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1780635074003-Screenshot_2026-06-04_215424.png",[36,1942],{},[23,1944,1945],{},[26,1946,1947,1948],{},"使用器具下能坐起",[23,1949,1950,1992],{},[26,1951,1952,1953],{},"Loop ladder（繩圈梯，用繩綁了很多個圈）：可固定在床尾，或在平舖在墊上",[23,1954,1955,1968],{},[26,1956,1772,1957],{},[23,1958,1959,1962,1965],{},[26,1960,1961],{},"起始姿勢是躺姿",[26,1963,1964],{},"一手拉著繩圈梯，另一隻手壓住拉繩的手手肘，將軀幹撐起",[26,1966,1967],{},"原本拉繩的手手肘慢慢移動到床尾，另一隻手繼續拉著繩圈梯協助動作",[26,1969,1596,1970],{},[23,1971,1972,1982],{},[26,1973,1699,1974],{},[23,1975,1976,1979],{},[26,1977,1978],{},"強壯的三頭肌前束，肱二頭肌",[26,1980,1981],{},"部份的三頭肌中\u002F後束，棘下肌，小圓肌",[26,1983,1612,1984],{},[23,1985,1986,1989],{},[26,1987,1988],{},"需要hip flex和knee extend柔軟度",[26,1990,1991],{},"假如肩屈曲，外展，肘屈曲角度受限，會使動作嚴重限制",[26,1993,1994,1995],{},"Suspended loops（懸吊帶）：會有一個大結構在床或墊的上方，幾個繩掛在上面，很大顆，不好看，也很難搬運",[23,1996,1997,2018],{},[26,1998,1772,1999],{},[23,2000,2001,2003,2006,2009,2012,2015],{},[26,2002,1961],{},[26,2004,2005],{},"一手用前臂穿過繩圈繞緊",[26,2007,2008],{},"用力將軀幹抬離墊面",[26,2010,2011],{},"另一手用肘放在床面固定好",[26,2013,2014],{},"接著重複這個步驟繞緊第二個繩圈，將軀幹繼續抬高，負著支撐在墊面上的那隻手也是向前移動",[26,2016,2017],{},"第三個圈也是這樣，然後變成了長坐姿，一手用繩圈固定，一手繼續撐住墊面，維持平衡",[26,2019,1596,2020],{},[23,2021,2022,2031],{},[26,2023,1699,2024],{},[23,2025,2026,2029],{},[26,2027,2028],{},"強壯的三角肌前束，肱二頭肌肌力",[26,2030,1851],{},[26,2032,1612,2033],{},[23,2034,2035,2038],{},[26,2036,2037],{},"正常的手肘伸直角度",[26,2039,2040],{},"假如肩屈曲，伸展，外展，外旋，水平外展，肘屈曲，手腕伸展角度受限，會使動作嚴重限制",[36,2042],{},[23,2044,2045],{},[26,2046,2047,2048],{},"坐姿下的粗大動作活動能力",[23,2049,2050,2053],{},[26,2051,2052],{},"在長坐姿姿勢下，抬高臀部，並左右移動，實現側向移動（我沒理解錯他這兩句應該是想這樣表達）",[26,2054,1596,2055],{},[23,2056,2057,2063],{},[26,2058,1699,2059],{},[23,2060,2061],{},[26,2062,1704],{},[26,2064,1612,2065],{},[23,2066,2067],{},[26,2068,2069],{},"正常的肩外旋，肘伸直，前臂旋後，手腕伸展角度，以及髖屈曲，膝伸直柔軟度",[36,2071],{},[23,2073,2074],{},[26,2075,2076,2077],{},"下肢擺位跟管理",[23,2078,2079],{},[26,2080,1596,2081],{},[23,2082,2083,2090],{},[26,2084,1699,2085],{},[23,2086,2087],{},[26,2088,2089],{},"強壯的肱二頭肌，肱肌，肱橈肌肌力",[26,2091,1612,2092],{},[23,2093,2094],{},[26,2095,2096],{},"正常的髖屈曲和膝伸直柔軟度",[36,2098],{},[12,2100,2101],{},"對完全完性損傷的病人，我們預期恢復到的床上移動功能性目標，主要是看他傷到的level",[12,2103,2104,2105,2107,2108,2112,2114,2115,2117],{},"對於僅保留C1~C4，他們沒有任何上肢肌肉",[16,2106],{},"\nC1~C3只有頭頸部肌肉，C4多了個斜方肌跟橫膈，所以在",[75,2109,2111],{"className":2110},[78,79],"全部床上位移技巧無法達成，是完全依賴狀態",[16,2113],{},"\n但其實還能利用口腔舌頭（口含式控制），以及語音來控制治療床，也是可以達到獨立",[16,2116],{},"\n需要的是四向調整式治療床來協助照顧者執行看護任務（4-way adjustable hospital bed）",[12,2119,2120,2121,2125,2127],{},"對於保留到C5的患者，功能預期範圍從",[75,2122,2124],{"className":2123},[78,79],"最大協助到完全依賴",[16,2126],{},"\n一樣需要四向調整式治療床來協助照顧者執行看護任務",[12,2129,2130,2131,2135,2137],{},"對於保留到C6的患者，功能預期範圍從",[75,2132,2134],{"className":2133},[78,79],"輕度協助到輔具獨立",[16,2136],{},"\n可以利用四向調整式治療床；或者裝有繩圈梯的普通病床，這樣也能讓病人獨立",[12,2139,2140,2141,2145,2147],{},"對於保留到C7C8的患者，功能性預期為",[75,2142,2144],{"className":2143},[78,79],"使用輔助獨立",[16,2146],{},"\n需要的輔具為抬腿器（leg lifter）",[12,2149,2150,2151,2153,2154],{},"對於保留到T1的患者，因為上肢所有肌力保留",[16,2152],{},"\n功能預期是",[75,2155,2157],{"className":2156},[78,79],"不借助任何協助完成床上所有移動技巧",[36,2159],{},[12,2161,2162,2163,2165,2166,2168],{},"嗯，其實在墊上訓練的所有運動都是可以適用在床上的，目的是為了讓病人可以回到床上也能靠自己來實現功能性移動",[16,2164],{},"\n因為墊子比床要堅實且寬廣，所以對於動作學習來看是一個更加簡單的環境",[16,2167],{},"\n就可以利用在這個環境學習ADL技巧",{"title":180,"searchDepth":181,"depth":181,"links":2170},[],"2026-06-03","21~結束","2026-06-05","\u002Fimages\u002Fuploads\u002F1780643744732-90942932_p0.webp",{},"\u002Fblog\u002FSCI-（十三）墊上運動-ii",{"title":1573,"description":2172},{"loc":2176},"blog\u002FSCI-（十三）墊上運動-ii","ZgYJyL5XOwJ7_A3LcLz4OSADPINW0b9evpu64vZYy2k",{"id":2182,"title":2183,"author":7,"body":2184,"date":2463,"description":2464,"draft":185,"edited_at":2171,"extension":186,"featured_image":2465,"meta":2466,"navigation":189,"path":2467,"pinned":185,"seo":2468,"sitemap":2469,"stem":2470,"tags":194,"__hash__":2471},"blog\u002Fblog\u002F先天性肌肉斜頸症CMT.md","先天性肌肉斜頸症CMT",{"type":9,"value":2185,"toc":2461},[2186,2204,2216,2222,2224,2227,2344,2346,2349],[12,2187,2188,2189,2191,2192,2194,2195,2197,2198,2200,2201,2203],{},"那顧名思義，就是歪脖子，他是SCM過緊導致的（這條肌肉從耳後連到了鎖骨）",[16,2190],{},"\n每1000個新生兒中就有3到20例",[16,2193],{},"\n病因包括：",[16,2196],{},"\n胎位不正，難產，任何外傷引發斜頸，可能有腫瘤的神經學徵象",[16,2199],{},"\n而斜頸的特徵為",[16,2202],{},"\n頸向患側傾，頭向健側看；頭部跟臉部不對稱；SCM有腫塊，通常出生四周會被發現，八個月後消退（但假如沒摸到腫塊也不代表不是先天性斜頸）",[12,2205,2206,2207,2209,2210,2212,2213,2215],{},"斜頸是表徵，不是診斷結果",[16,2208],{},"\n其中又有80%是天生的",[16,2211],{},"\n斜頸可以是慢性存在，也可以是急性發作",[16,2214],{},"\n可能伴隨不對稱的眼睛位置",[12,2217,2218,2219,2221],{},"通常PT可以做牽拉患側，增加健側肌肉力量，以及擺位",[16,2220],{},"\n假如有很嚴重的畸型，可以在1歲時進行手術",[36,2223],{},[12,2225,2226],{},"現在說一下可能會引發斜頸的問題",[23,2228,2229,2237,2245,2253,2261,2269,2292,2303,2311,2319,2336],{},[26,2230,2231,2232],{},"Arnold-Chiari Malformation（小腦扁桃體下疝畸形）——特徵有頸痛（咳嗽或打噴嚏時會加劇），可能會出現視力模糊，吞嚥困難，嘔吐反射，眼球震顫",[23,2233,2234],{},[26,2235,2236],{},"原因因為小腦與第四腦室向下移位至頸椎管；可以用MRI診斷；可能需要手術擴大後顱窩",[26,2238,2239,2240],{},"Brucellosis——脖子跟喉部會有痛，壓痛，腫脹；可能是頸椎感染了布魯氏菌而引起的",[23,2241,2242],{},[26,2243,2244],{},"跟結核一樣是細菌感染；要用培養確認菌種；可利用抗生素，偶爾需要手術",[26,2246,2247,2248],{},"Clevical spine fractures and dislocations（頸椎骨折和脫臼）——有嚴重外傷史",[23,2249,2250],{},[26,2251,2252],{},"一般是車禍或高空墜落造成",[26,2254,2255,2256],{},"Chordomas（脊索瘤）——呈現下運動神經元問題，但小腦功能正常",[23,2257,2258],{},[26,2259,2260],{},"要看那顆瘤長在哪，才知道具體症狀",[26,2262,2263,2264],{},"Clavical fracture with or without BPI（鎖骨骨折伴隨或不伴隨BPI）——在Moro反射中，患側手不會動，而且移動患側手嬰兒會哭鬧",[23,2265,2266],{},[26,2267,2268],{},"因體重超重，或胎位不正導致難產",[26,2270,2271,2272],{},"Congenital muscular torticollis（先天性肌肉斜頸症CMT）——就上面提到的，頸向患側傾，頭向健側看，SCM有腫塊且緊繃，頸部活動度減少",[23,2273,2274,2277,2280,2283,2286,2289],{},[26,2275,2276],{},"SCM會處於緊繃狀態；屬慢性問題，假如不去處理會導置臉部和顱骨不對稱",[26,2278,2279],{},"SCM摸到腫塊，在二到三個月內會消退（但講義上面寫八個月內）",[26,2281,2282],{},"那跟C1C2半脫位引發的斜頸不同，這個是頭頸同向，跟CMT的頭頸反向不同",[26,2284,2285],{},"反射檢查如X光不會照到異常",[26,2287,2288],{},"治療通常是運動為主",[26,2290,2291],{},"假如是出生之後幾年才被確診的小孩，需要做手術鬆開SCM",[26,2293,2294,2295],{},"Drug intoxication（藥物中毒）——服藥過量，如抗癲癇藥物",[23,2296,2297,2300],{},[26,2298,2299],{},"那兒童的肌張力不全，一般會在腿部足部開始發作，再慢慢影響到全身",[26,2301,2302],{},"而後天性的肌張力不全，頭部可能會前傾或後仰，不會偏向特定一側",[26,2304,2305,2306],{},"Extraocular muscle Paresis（眼外肌麻痺）——無法讓眼睛全範圍運動",[23,2307,2308],{},[26,2309,2310],{},"所以小孩需要透過傾斜頭跟脖子來看清；接受了眼科矯正手術之後斜頸也會消失",[26,2312,2313,2314],{},"Juvenile Rheumatoid Arthritis（幼年型類風濕關節炎JRA）——在頭前傾姿勢十分常見",[23,2315,2316],{},[26,2317,2318],{},"單關節和少關節類型的JRA裡不太會出現斜頸；多關節類型侵犯過多之後才會累及，造成斜頸",[26,2320,2321,2322],{},"Multiple congenital cervical spine anomalies（先天性頸椎病變Klipel-Feil Disorder）——脖子短且寬，髮際線靠低，頸部活動度受損",[23,2323,2324,2327],{},[26,2325,2326],{},"患者在40~50歲會有神經系統方面的問題",[26,2328,2329],{},[2330,2331,2335],"a",{"href":2332,"rel":2333},"https:\u002F\u002Fwww.tfrd.org.tw\u002Ftfrd\u002Frare_b\u002Fview\u002Fid\u002F266",[2334],"nofollow","這個中文名字翻譯我從這裡找的",[26,2337,2338,2339],{},"Muscle strain（肌肉拉傷）——有外傷史",[23,2340,2341],{},[26,2342,2343],{},"會造成暫時性的斜頸，且沒有固定型態，要看是傷了哪一條肌肉",[36,2345],{},[12,2347,2348],{},"接下來講可能會導致斜頸的神經性運動障礙",[23,2350,2351,2359,2367,2375,2383,2391,2394,2402,2410,2418,2426,2434,2437,2445,2453],{},[26,2352,2353,2354],{},"Benign paroxysmal torticollis of infancy（良性陣發性嬰幼兒斜頸症）——類似於癲癇的症狀",[23,2355,2356],{},[26,2357,2358],{},"是多重原因導致",[26,2360,2361,2362],{},"Dystonia musculorum deformans（畸型性肌張力不全）——許多腦部疾病，如腦炎後遺症，亨丁頓氏症",[23,2363,2364],{},[26,2365,2366],{},"可以是顯性或隱性，原因不明",[26,2368,2369,2370],{},"Sandifer’s syndrome（桑迪弗氏症候群）——有胃食道逆流，突然斜頸的問題",[23,2371,2372],{},[26,2373,2374],{},"通常跟腦麻相關，有癲癇病史",[26,2376,2377,2378],{},"Pharyngeal abscess（咽後膿腫）——發炎時會有嚴重斜頸狀況",[23,2379,2380],{},[26,2381,2382],{},"口腔跟喉嚨會痠痛",[26,2384,2385,2386],{},"Posterior fossa tumor（後顱窩腫瘤）——出現小腦功能障礙問題，可以觀察到斜視，動暈症，嘔吐",[23,2387,2388],{},[26,2389,2390],{},"機制不明，初期症狀與CMT容易被混淆",[26,2392,2393],{},"Psychogenic（心因性）——非典型的頸椎問題（例如發展遲緩）",[26,2395,2396,2397],{},"Rotary subluxation of C1-C2（C1C2半脫位）——是原本正常的小孩，突然間出現斜頸狀況",[23,2398,2399],{},[26,2400,2401],{},"會維持在同一方向傾斜和旋轉（跟CMT不同），可由X光確認",[26,2403,2404,2405],{},"Spastic Tordicollis（痙攣型斜頸）——頭會轉向一邊",[23,2406,2407],{},[26,2408,2409],{},"通常伴有呼吸道感染；好發在5~11歲；治療可靠熱敷及頸圈",[26,2411,2412,2413],{},"Spinal cord tumors（脊髓瘤）——出現下運動神經無力問題，但小腦功能正常",[23,2414,2415],{},[26,2416,2417],{},"具體要看腫瘤長在哪才知道狀況（感覺跟上面脊索瘤挺像）",[26,2419,2420,2421],{},"Sprengel’s deformity（先天性肩胛骨高位症）——肩胛骨上移，頭部動作通常不受限",[23,2422,2423],{},[26,2424,2425],{},"從胚胎裡，肩胛骨會在C4長出，並在胚胎慢慢長大而下降，高位應該就是下降出問題",[26,2427,2428,2429],{},"Syringomyelia（脊髓空洞症）——感覺喪失，肌肉萎縮，肌健反射消失",[23,2430,2431],{},[26,2432,2433],{},"病況取決於病灶level",[26,2435,2436],{},"Systematic fungal infection（全身性真菌感染）——頸部和喉嚨會有疼痛，壓痛，腫脹（感覺跟上面布魯氏菌像）",[26,2438,2439,2440],{},"Thyroglossal duct cyst（甲狀舌骨囊腫）——頸部正中央會長出一塊柔軟的腫塊，而不是長在SCM上，且比起SCM的腫塊要更表淺",[23,2441,2442],{},[26,2443,2444],{},"需手術切除",[26,2446,2447,2448],{},"Tonsillitis\u002FAdenitis（扁桃腺炎）——嚴重發炎並引起斜頸，造成口腔與喉嚨發炎，可能會導致吞嚥與呼吸困難",[23,2449,2450],{},[26,2451,2452],{},"可使用抗生素治療",[26,2454,2455,2456],{},"Tuberculosis of the Cervical Spine（頸椎結核）——頸部和喉嚨會有疼痛，壓痛，腫脹",[23,2457,2458],{},[26,2459,2460],{},"通常是不對稱的傷害椎體，導致旋轉伴隨後凸",{"title":180,"searchDepth":181,"depth":181,"links":2462},[],"2026-06-01","Congenital Muscular Torticollis","\u002Fimages\u002Fuploads\u002F1780462622920-121146410_p0_master1200.webp",{},"\u002Fblog\u002F先天性肌肉斜頸症CMT",{"title":2183,"description":2464},{"loc":2467},"blog\u002F先天性肌肉斜頸症CMT","1IZxGkst_bAVHix9u2lgPuJocdxwcKhvepyQjaOjwuM",{"id":2473,"title":2474,"author":7,"body":2475,"date":2701,"description":2702,"draft":185,"edited_at":2463,"extension":186,"featured_image":2703,"meta":2704,"navigation":189,"path":2705,"pinned":185,"seo":2706,"sitemap":2707,"stem":2708,"tags":194,"__hash__":2709},"blog\u002Fblog\u002FBPI.md","BPI",{"type":9,"value":2476,"toc":2699},[2477,2483,2492,2501,2507,2513,2524,2527,2529,2532,2549,2552,2596,2598,2607,2613,2649,2652,2657,2664,2667,2681],[12,2478,2479,2480,2482],{},"BPI，臂叢傷害，在1768年在英國第一次被提出來",[16,2481],{},"\n臂叢包括了從C5到T1的五條神經",[12,2484,2485,2486,2488,2489,2491],{},"功能包括了接收來自手臂和手的訊號、傳送大腦發出的訊號",[16,2487],{},"\n那假如這條路徑受傷，訊號就傳不到大腦，反過來也傳不到手臂跟手",[16,2490],{},"\n感覺會退化，肌肉也會逐漸萎縮，進一步的變成肌肉和關節攣縮",[12,2493,2494,2495,2497,2498,2500],{},"假如神經裡面的nerve covering沒有受損，這樣的話神經是可以再生的",[16,2496],{},"\n在剛出生第一年恢復會很快，但肌肉無力還是會存在",[16,2499],{},"\n（大概每個月長1 inch\u002F每天1 mm）",[12,2502,2503,2504,2506],{},"會造成臂叢傷害，可能是因為嬰兒太大",[16,2505],{},"\n在自然分娩時頭出來但嬰兒肩膀還卡在骨盤裡",[12,2508,2509,2510,2512],{},"那為了避免頭部傷害，都會將頭部往下拉，可能會使臂叢被拉傷",[16,2511],{},"\n手臂無力在剛出生就可以被觀察到，相關併發症有",[23,2514,2515,2518,2521],{},[26,2516,2517],{},"鎖骨骨折",[26,2519,2520],{},"肱骨骨折",[26,2522,2523],{},"霍納氏症候群：眼瞼下垂，較小的瞳孔",[12,2525,2526],{},"發生率在每1000個裡有0.6~3個",[36,2528],{},[12,2530,2531],{},"按照神經受傷類型分成",[23,2533,2534,2537,2540,2543,2546],{},[26,2535,2536],{},"Avulsion撕脫——神經跟脊髓連接斷開，無法靠再生或開刀修復",[26,2538,2539],{},"Rupture斷裂——斷裂處不在脊髓，可靠開刀修復",[26,2541,2542],{},"Neuroma神經瘤——神經再生時會長出疤痕組織，會阻斷傳導，需開刀移除疤痕組織",[26,2544,2545],{},"Axonotmesis軸突斷裂——傷到了fiber，但nerve covering還在，能再生，但速度慢慢的長回支配的肌肉",[26,2547,2548],{},"Neuropraxis神經失用症——輕度傷害，沒有撕裂，在出生後三個月後才能看出明顯不同",[12,2550,2551],{},"按照傷到的哪一條神經又分成",[23,2553,2554,2572,2590,2593],{},[26,2555,2556,2557,2561,2562],{},"Erb’s Palsy or upper-plexus type（",[75,2558,2560],{"className":2559},[79,78],"最常見","）——傷到C5和C6",[23,2563,2564],{},[26,2565,2566,2567],{},"功能喪失包括：肩外展，肩屈曲，肘屈曲，腕伸展；手臂無法從身側抬高",[23,2568,2569],{},[26,2570,2571],{},"呈現下垂姿勢，在肩內收，肩內旋，前臂旋前，手指屈曲的姿勢——類似小費手",[26,2573,2574,2575,2579,2580],{},"Klumpke’s Palsy or lower-plexus type（",[75,2576,2578],{"className":2577},[78,79],"少見","）——C7到T1",[23,2581,2582],{},[26,2583,2584,2585,2589],{},"手部內在肌無力（特別是",[75,2586,2588],{"className":2587},[173],"尺側","）",[26,2591,2592],{},"Duchennes Palsy——C7",[26,2594,2595],{},"Whole arm type——C5到T1",[36,2597],{},[12,2599,2600,2601,2603,2604,2606],{},"手術處置的話，假如在出生後九個月沒有任何進步就會開刀，抑或者有些嚴重的會在出生約3到4個月就會開刀",[16,2602],{},"\n第一次開刀主要是移除疤痕組織，或者神經移植（會用不那麼重要的感覺神經做移植）",[16,2605],{},"\n（通常在一歲前肩部問題要處理好）",[12,2608,2609,2610,2612],{},"那這邊老師給了一張叫Hospital for Sick Children Active Movement Scale的量表（AMS）",[16,2611],{},"\n評分一共七個，是按照MMT改過來的（我感覺是）",[23,2614,2615,2635],{},[26,2616,2617,2618],{},"非抗重力",[23,2619,2620,2623,2626,2629,2632],{},[26,2621,2622],{},"0——沒有任何反應",[26,2624,2625],{},"1——有肌肉收縮，無動作",[26,2627,2628],{},"2——能活動範圍\u003C1\u002F2",[26,2630,2631],{},"3——能活動範圍>1\u002F2",[26,2633,2634],{},"4——full range",[26,2636,2637,2638],{},"抗重力",[23,2639,2640,2643,2646],{},[26,2641,2642],{},"5——能活動範圍\u003C1\u002F2",[26,2644,2645],{},"6——能活動範圍>1\u002F2",[26,2647,2648],{},"7——full range",[12,2650,2651],{},"這一個是Mallet classification(Mallet Scale) 錄肩部和手臂功能變化的評分系統",[12,2653,2654],{},[1930,2655],{"alt":1932,"src":2656},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1780284740177-malletslace_full.gif",[12,2658,2659],{},[2330,2660,2663],{"href":2661,"rel":2662},"https:\u002F\u002Fwww.stlouischildrens.org\u002Fconditions-treatments\u002Fneurosurgery-services\u002Fbrachial-plexus-center\u002Ftreatment-goals\u002Fmedical-treatment",[2334],"此圖我從這裡找的",[12,2665,2666],{},"評估反射的內容包括",[23,2668,2669,2672,2675,2678],{},[26,2670,2671],{},"Moro",[26,2673,2674],{},"Galant",[26,2676,2677],{},"Neck righting, hand placing",[26,2679,2680],{},"Visual tracking",[12,2682,2683,2684,2686,2687,2689,2690,2692,2693,2695],{},"那我們介入的時間，在出血跟水腫都消退後就可以開始了，當然是越早越好",[16,2685],{},"\n提供對恢復有利的環境",[16,2688],{},"\n將攣縮問題降到最低",[16,2691],{},"\n以及著手運動訓練",[16,2694],{},[75,2696,2698],{"className":2697},[78,79],"（那PT介入是無法使神經再生加快，目的是處理減少無力，攣縮等問題，還有動作訓練）",{"title":180,"searchDepth":181,"depth":181,"links":2700},[],"2026-05-29","BPI，斜頸等下再寫","\u002Fimages\u002Fuploads\u002F1780287451942-138768698_p0_master1200.webp",{},"\u002Fblog\u002FBPI",{"title":2474,"description":2702},{"loc":2705},"blog\u002FBPI","BKiUvsIFaniao4lSrn5PHOXgLG4FS10Qmaqzt74VOCs",{"id":2711,"title":2712,"author":7,"body":2713,"date":2850,"description":2851,"draft":185,"edited_at":2850,"extension":186,"featured_image":2852,"meta":2853,"navigation":189,"path":2854,"pinned":185,"seo":2855,"sitemap":2856,"stem":2857,"tags":194,"__hash__":2858},"blog\u002Fblog\u002FSCI（九）預後和早期照護.md","SCI（九）預後和早期照護",{"type":9,"value":2714,"toc":2848},[2715,2721,2749,2752,2755,2757,2766,2772,2781,2784,2822,2824,2827],[12,2716,2717,2718,2720],{},"預後，最主要跟傷在哪，還有完全性不完全性有關係",[16,2719],{},"\n較佳的預後因素有",[23,2722,2723,2734,2737,2740,2743,2746],{},[26,2724,2725,2726],{},"程度",[23,2727,2728,2731],{},[26,2729,2730],{},"不完全性損傷：恢復運動功能較高",[26,2732,2733],{},"完全性損傷：在頸椎受傷的病人，仍有70%的人可以恢復一個level",[26,2735,2736],{},"保留運動功能",[26,2738,2739],{},"保留針刺覺",[26,2741,2742],{},"損傷形態：例如中髓和Brown-Sequard的預後會比前髓症候群要來得好",[26,2744,2745],{},"早期神經功能恢復：例如在傷後1個月就有恢復會比傷後6個月才開始恢復要好",[26,2747,2748],{},"年紀",[12,2750,2751],{},"創傷當下，對患者進行固定，搬運，處置的技術，也是影響預後的關鍵",[12,2753,2754],{},"運動功能恢復，大概在受傷後12~18個月進入高原期（進步減緩）",[36,2756],{},[12,2758,2759,2760,2762,2763,2765],{},"神經功能恢復：是指喪失的運動功能或感覺會恢復",[16,2761],{},"\n絕大部份會在受傷後1年內恢復：快的6個月內，慢的有可能到2年",[16,2764],{},"\n假如只有反射恢復，沒有運動功能或感覺恢復，不會被視為是神經功能恢復",[12,2767,2768,2769,2771],{},"神經功能恢復通常分成神經根恢復和脊髓恢復，而前者恢復佔比較高",[16,2770],{},"\n具體機制未明；可能原因有出血，發炎，水腫，離子紊亂消退那些",[12,2773,2774,2775,2777,2778,2780],{},"神經功能惡化：可以透過感覺變差，肌力變弱等方式來證實",[16,2776],{},"\n大概會有1.8~10%的病人會惡化",[16,2779],{},"\n原因包括敗血症，氣管插入，早期手術介入，僵直性脊椎炎等",[12,2782,2783],{},"緊急處置：這一步十分關鍵，可以判斷後續惡化或減輕",[23,2785,2786,2806],{},[26,2787,2788,2789],{},"院前：判定由車禍，高處摔落，跳水意外，頭部創傷，發現運動或感覺功能受損，意識不清或昏迷等問題",[23,2790,2791,2794,2797,2800,2803],{},[26,2792,2793],{},"並且由專業的緊急醫療救護人員處理",[26,2795,2796],{},"主要是防止二次傷害：基本上是固定為主，並且默認脊椎已經受傷來處理",[26,2798,2799],{},"包括使用硬式頸圈，頭部固定器，加墊長背板，固定帶等，直到SCI被排除或已被治療",[26,2801,2802],{},"在不移動脊椎的前提下，保持呼吸暢通，患者可能要插管，尤其是C5以上的患者",[26,2804,2805],{},"假如生命徵象穩定，可立即轉介去脊髓損傷的創傷中心（1小時內最佳）",[26,2807,2808,2809],{},"院內：主要是處理危急生命的狀況，並且挽救神經功能",[23,2810,2811],{},[26,2812,2813,2814],{},"可施打methylprednisolone（甲基普立朗）",[23,2815,2816,2819],{},[26,2817,2818],{},"在受傷後8小時內可高劑量施打，可能會促進運動功能和感覺恢復",[26,2820,2821],{},"並且他有抑制發炎作用，減少二次損傷",[36,2823],{},[12,2825,2826],{},"骨折處置：防止進一步的脊髓傷害",[23,2828,2829,2840],{},[26,2830,2831,2832],{},"非手術處置：固定，牽拉，擺位",[23,2833,2834,2837],{},[26,2835,2836],{},"適用於治療多節段類型，穩定型SCI",[26,2838,2839],{},"初期就是透過牽拉或擺位進行固定",[26,2841,2842,2843],{},"手術處置：大概有60%SCI病人有做手術",[23,2844,2845],{},[26,2846,2847],{},"可恢復脊椎最佳的排列，也可以為神經組織減壓；通常會做脊椎融合術",{"title":180,"searchDepth":181,"depth":181,"links":2849},[],"2026-05-26","第六份議義","\u002Fimages\u002Fuploads\u002F1779768369765-97385601_p0_master1200.webp",{},"\u002Fblog\u002FSCI（九）預後和早期照護",{"title":2712,"description":2851},{"loc":2854},"blog\u002FSCI（九）預後和早期照護","eHr93Pb_aybukDadcn-jdLG6YxsHa6DnXC9bWqLkzfk",{"id":2860,"title":2861,"author":7,"body":2862,"date":2850,"description":3204,"draft":185,"edited_at":2850,"extension":186,"featured_image":3205,"meta":3206,"navigation":189,"path":3207,"pinned":185,"seo":3208,"sitemap":3209,"stem":3210,"tags":194,"__hash__":3211},"blog\u002Fblog\u002FSCI（十）物理治療介入-呼吸功能.md","SCI（十）物理治療介入-呼吸功能",{"type":9,"value":2863,"toc":3202},[2864,2867,2915,2917,2920,2923,2926,2940,2942,2945,2948,2982,2985,2988,2990,2993,3002,3017,3020,3031,3034,3048,3050,3053,3056,3072,3074,3077,3089,3102,3114,3117,3144,3156,3162,3170,3173,3184,3190,3196],[12,2865,2866],{},"首先來講呼吸功能參與的肌肉",[23,2868,2869,2891],{},[26,2870,2871,2872],{},"參與吸氣的肌肉",[23,2873,2874,2888],{},[26,2875,2876,2877],{},"平靜呼吸",[23,2878,2879,2882,2885],{},[26,2880,2881],{},"橫膈膜（膈神經）：由C345控制，負責75%吸氣量",[26,2883,2884],{},"外肋間肌（胸脊神經）：由T1~T11控制，負責25%吸氣量",[26,2886,2887],{},"輔助肌：SCM（副神經）和斜角肌（頸脊神經）",[26,2889,2890],{},"深層用力的肌肉也是如上",[26,2892,2893,2894],{},"參與呼氣的肌肉",[23,2895,2896,2899],{},[26,2897,2898],{},"平靜呼氣：一般不需要肌肉參與，單純靠回彈就夠",[26,2900,2901,2902],{},"深層用力肌肉有",[23,2903,2904,2912],{},[26,2905,2906,2907],{},"腹部肌肉：由T7~T12控制",[23,2908,2909],{},[26,2910,2911],{},"腹外斜肌，腹直肌，腹內斜肌，腹橫肌：胸脊神經",[26,2913,2914],{},"內肋間肌（胸脊神經）：由T1~T11控制",[36,2916],{},[12,2918,2919],{},"正常人",[12,2921,2922],{},"呼吸模式：般會用胸廓擴張和上腹部起伏，又稱2D2C（D橫膈，C胸腔）",[12,2924,2925],{},"咳嗽模式：會攝及聲門，吸氣，呼氣肌之間的協調，咳嗽階段包括",[23,2927,2928,2931,2934,2937],{},[26,2929,2930],{},"吸入足夠空氣",[26,2932,2933],{},"聲門關閉",[26,2935,2936],{},"在聲門關閉同時，呼氣肌用力收縮",[26,2938,2939],{},"聲門打開",[36,2941],{},[12,2943,2944],{},"SCI患者",[12,2946,2947],{},"呼吸模式，除了僅傷及腰椎和薦椎的不受影響之外，頸椎胸椎受傷都會出現某種程度的呼吸障礙",[23,2949,2950,2958,2966,2974],{},[26,2951,2952,2953],{},"高位頸椎損傷(C1~C4)會出現以下模式：",[23,2954,2955],{},[26,2956,2957],{},"1 neck 3D; 2neck 2D; 3neck 1D",[26,2959,2960,2961],{},"低位頸椎損傷(C5~C8)",[23,2962,2963],{},[26,2964,2965],{},"4D",[26,2967,2968,2969],{},"中下胸椎損傷",[23,2970,2971],{},[26,2972,2973],{},"3D 1C",[26,2975,2976,2977],{},"低位損傷",[23,2978,2979],{},[26,2980,2981],{},"接近2D 2C",[12,2983,2984],{},"在直立姿勢下：假如腹部肌肉無力，腹部內臟會往下凸，導致橫膈會被往外拉，不利於吸氣",[12,2986,2987],{},"平躺姿勢下：因為重力原因，內臟會被往頭側擠，橫膈會維持在正常休息位置",[36,2989],{},[12,2991,2992],{},"接下來講異常的呼吸模式",[12,2994,2995,2996,2998,2999,3001],{},"Paradoxical breathing pattern：意思是呼吸時肋骨動作異常",[16,2997],{},"\n在吸氣時橫膈往下，但胸廓沒有往外擴，而是一起往下往內，造成吸氣量減少",[16,3000],{},"\n隨時間推移，肋間肌會痙攣或硬化，導致矛盾不明顯",[12,3003,3004,3005,3007,3008,3010,3011,3013,3014,3016],{},"Litten氏膈現象：是指有微弱橫膈功能，但肋間肌沒有功能",[16,3006],{},"\n是胸內壓下降所產生",[16,3009],{},"\n假如存在這現象，代表橫膈有主動活動",[16,3012],{},"\n假如沒有，也不能完全說橫膈沒有動，可能會有一絲尚存，只是可能是因為其他因素遮住了",[16,3015],{},"\n在瘦子身上較易發現",[12,3018,3019],{},"高位頸椎損傷",[23,3021,3022,3025,3028],{},[26,3023,3024],{},"C1C2：部份保留SCM，斜方肌，竪直肌；必須戴呼吸機",[26,3026,3027],{},"C3：保留完整SCM，部份保留橫膈；必須戴呼吸機",[26,3029,3030],{},"C4：保留完整橫膈，但缺乏肋間肌和腹部肌肉；可能不需要戴呼吸肌最高llevel",[12,3032,3033],{},"C5以下",[23,3035,3036,3039,3042,3045],{},[26,3037,3038],{},"C5~C8：有完整橫膈，但缺乏肋間肌和腹部肌肉，吸氣和用力呼吸功能仍受損",[26,3040,3041],{},"T1~T5：有完整橫膈，部份保留外肋間肌，吸氣和用力呼吸功能輕微提升",[26,3043,3044],{},"T6~T12：有完整橫膈，部份保留外肋間肌和腹部肌肉，咳嗽能力提升",[26,3046,3047],{},"T12以下：絕大多數呼吸肌保留完好，呼吸功能基本像正常人",[36,3049],{},[12,3051,3052],{},"併發症",[12,3054,3055],{},"不論是急性期還是傷後數年，呼吸功能的併發症都是SCI的最常見死因",[23,3057,3058,3061,3069],{},[26,3059,3060],{},"肺炎：最常見，主因是高位頸椎損傷，咳嗽功能不全或喪失，導致分泌物堆積",[26,3062,3063,3064],{},"肺塌陷：因為肌肉力量不足，導致通氣減少或換氣不足",[23,3065,3066],{},[26,3067,3068],{},"因為肺泡沒有正常充氣膨漲導致，意味著血液，組織，器官可能沒辦法獲得足夠氧氣",[26,3070,3071],{},"呼吸衰竭：呼吸肌無力且更容易疲勞，進而導致衰竭",[36,3073],{},[12,3075,3076],{},"干預",[12,3078,3079,3080,3082,3083,3085,3086,3088],{},"腹式呼吸：這個方法比使用頸部肌肉更有效率，是十分重要的復健一環",[16,3081],{},"\n姿勢也可以做進階訓練",[16,3084],{},"\n從躺 半躺 坐姿 站姿 到行走",[16,3087],{},"\n並且可以增加重量做阻力訓練",[23,3090,3091],{},[26,3092,3093,3094],{},"放置重量與呼吸時機的關係",[23,3095,3096,3099],{},[26,3097,3098],{},"呼氣時：可以協助橫膈回到較高休息位置",[26,3100,3101],{},"吸氣時：上腹部需要達到完全起伏，假如頂得起來，代表橫膈發力正確",[12,3103,3104,3105,3107,3108,3110,3111,3113],{},"舌咽呼吸：利用舌頭跟嘴巴肌肉用類似吞嚥的方法將空氣吸入到肺中",[16,3106],{},"\n對於高位頸椎損傷患者十分有用，可以讓他們短時間脫離呼吸機",[16,3109],{},"\n肺活量最高可以到1000cc",[16,3112],{},"\n這個呼吸法需要患者高度配合，以及需要受訓的治療師耐心持續教導",[12,3115,3116],{},"氣體轉換手法：主要是維持並增加胸廓擴張能力，提供患者獨立自主的胸廓的方法",[23,3118,3119,3136],{},[26,3120,3121,3122],{},"步驟包括",[23,3123,3124,3127,3130,3133],{},[26,3125,3126],{},"最大吸氣（像是用腹式呼吸，腹部突出）",[26,3128,3129],{},"憋氣",[26,3131,3132],{},"橫膈放鬆（腹部會往內凹，像是把氣體往胸推）",[26,3134,3135],{},"讓氣體從下胸部轉移致上胸部",[26,3137,3138,3139],{},"一手放在病人胸部，一手放肚臍",[23,3140,3141],{},[26,3142,3143],{},"讓病人用力吸氣後，憋氣，並縮小腹",[12,3145,3146,3147,3149,3150,3152,3153,3155],{},"吸氣肌肌力增加訓練：主要訓練橫膈和吸氣輔助肌",[16,3148],{},"\n姿勢可以從躺到坐",[16,3151],{},"\n隨著訓練進程，可以搭配阻力型吸氣訓練器",[16,3154],{},"\n也可以在腹部上加沙包",[12,3157,3158,3159,3161],{},"咳嗽訓練：因為病人沒辦法功能性的去咳嗽，可以教導病人",[16,3160],{},"\n自主方式",[23,3163,3164,3167],{},[26,3165,3166],{},"第一步，吸氣，並且身體往後仰，手高舉，這樣能擴張胸廓",[26,3168,3169],{},"第二步，身體前彎，雙手抱胸，這樣可以增加腹壓協助咳嗽",[12,3171,3172],{},"假如無法自主，那就徒手幫病人",[23,3174,3175,3178,3181],{},[26,3176,3177],{},"協助者雙手放在病人腹上",[26,3179,3180],{},"讓患者用力吸氣",[26,3182,3183],{},"在患者要咳嗽時，協助者施加向內，向病人頭側施加溫和的力度",[12,3185,3186,3187,3189],{},"束腹帶：高位頸椎和胸椎患者，可以用束腹帶來改善呼吸和咳嗽功能",[16,3188],{},"\n並且還可以提供胸腔壓，以及減少姿勢性低血壓發生",[12,3191,3192,3193,3195],{},"腹部支持：利用像束腰帶或馬甲式護腰，可以將橫膈放在更有效率呼吸的位置",[16,3194],{},"\n能改善肺活量，潮氣量，血氧飽和度",[12,3197,3198,3199,3201],{},"徒手拉伸：因為肌肉功能受損和肺順應性降低，導致胸廓活動度受限",[16,3200],{},"\n可以誘過徒手的方式來增加活動度",{"title":180,"searchDepth":181,"depth":181,"links":3203},[],"第七份議義寫完呼吸","\u002Fimages\u002Fuploads\u002F1779768254538-142279337_p0_master1200.webp",{},"\u002Fblog\u002FSCI（十）物理治療介入-呼吸功能",{"title":2861,"description":3204},{"loc":3207},"blog\u002FSCI（十）物理治療介入-呼吸功能","SHP4l0TKhRBfWftrrSzrwZtambBPciGH-1Zzfu4D-84",{"id":3213,"title":3214,"author":7,"body":3215,"date":2850,"description":3324,"draft":185,"edited_at":2850,"extension":186,"featured_image":3325,"meta":3326,"navigation":189,"path":3327,"pinned":185,"seo":3328,"sitemap":3329,"stem":3330,"tags":194,"__hash__":3331},"blog\u002Fblog\u002FSCI（十一）物理治療介入-動作策略.md","SCI（十一）物理治療介入-動作策略",{"type":9,"value":3216,"toc":3322},[3217,3220,3223,3226,3263,3265,3268,3271,3273,3279,3281,3290,3313],[12,3218,3219],{},"我們會利用患者殘存的肌肉群，來教導因失去功能肌肉原本要做的事的任務（感覺我自己想的也十分拗口，等我想到再改）",[12,3221,3222],{},"有三種方法能利用：肌肉替代療法，角動力，頭臀關係",[12,3224,3225],{},"首先是代償",[23,3227,3228,3236,3244,3255],{},[26,3229,3230,3231],{},"第一種：利用協同肌",[23,3232,3233],{},[26,3234,3235],{},"可以用類以動作的肌肉來代償",[26,3237,3238,3239],{},"第二種：利用重力",[23,3240,3241],{},[26,3242,3243],{},"利用功能正常的肌肉重新調整姿勢，讓重力幫助做出想要的動作",[26,3245,3246,3247],{},"第三種：利用結構的張力",[23,3248,3249,3252],{},[26,3250,3251],{},"利用如肌腱韌帶等結構，帶出原本無法做的動作，例如wrist extend時手指會flex",[26,3253,3254],{},"這樣有利於ADL",[26,3256,3257,3258],{},"第四種：利用遠端肢體固定",[23,3259,3260],{},[26,3261,3262],{},"坐姿下手肘支撐，假如三頭肌無力，也可以靠三角肌前束將身體往內拉，將肘卡在伸直位置",[36,3264],{},[12,3266,3267],{},"角動力：意思是利用甩動或者擺動來讓肢體完成某個動作",[12,3269,3270],{},"物理特徵有：動量總和，速度，質量，力臂",[36,3272],{},[12,3274,3275,3276,3278],{},"頭臀關係：簡單來說是頭部跟臀部會是相反方向移動",[16,3277],{},"\n假如頭往前帶，臀會往後",[36,3280],{},[12,3282,3283,3284,3286,3287,3289],{},"骨科禁忌症：要注意脊椎的穩定性，不要做出會讓脊椎受損的動作",[16,3285],{},"\n頸椎不穩定的話，避免肩屈曲或外展超90度，也要避免肩部肌肉強力收縮",[16,3288],{},"\n腰椎不穩定的話，避免髖部屈曲超90度，也要避免髖部肌肉強力收縮",[12,3291,3292,3293,3295,3296,3298,3299,3301,3302,3304,3306,3307,3309,3310,3312],{},"ROM：盡可能接近完整ROM",[16,3294],{},"\n有些部位是可以接受輕微緊繃的，例如手指肌腱反應",[16,3297],{},"\n這個功能對於喪失功能是抓握是好的，所以不要過度拉伸它",[16,3300],{},"\n日常擺位在手指微彎就好",[16,3303],{},[16,3305],{},"\n下背也是一樣，輕微的緊繃可以在坐姿下保持正確的姿勢",[16,3308],{},"\n假如腿後肌太緊，就避免長坐姿",[16,3311],{},"\n大腿拉伸應該在躺姿下，而不是長坐姿下進行",[12,3314,3315,3316,3318,3319,3321],{},"最後是肌力，這應該最好理解，畢竟失去了這麼多肌肉",[16,3317],{},"\n你正常的肌肉也要去多訓練，才能應付得了代償問題",[16,3320],{},"\n像是下半身癱瘓，你要用手來支撐身體，代表你的手也要練到十分強壯才行",{"title":180,"searchDepth":181,"depth":181,"links":3323},[],"講完第七份議義","\u002Fimages\u002Fuploads\u002F1779768294564-94505483_p0_master1200.webp",{},"\u002Fblog\u002FSCI（十一）物理治療介入-動作策略",{"title":3214,"description":3324},{"loc":3327},"blog\u002FSCI（十一）物理治療介入-動作策略","TUgha_UC_4Zt_jgF1p8JbRSunSSnwVafqJveqZTVzA8",{"id":3333,"title":3334,"author":7,"body":3335,"date":2850,"description":3583,"draft":185,"edited_at":2171,"extension":186,"featured_image":3584,"meta":3585,"navigation":189,"path":3586,"pinned":185,"seo":3587,"sitemap":3588,"stem":3589,"tags":194,"__hash__":3590},"blog\u002Fblog\u002FSCI（十二）墊上運動.md","SCI（十二）墊上運動-i",{"type":9,"value":3336,"toc":3581},[3337,3346,3349,3363,3366,3369,3388,3391,3413,3436,3503,3505],[12,3338,3339,3340,3342,3343,3345],{},"墊上運動：假如患者允許活動，就可以立刻安排墊上運動",[16,3341],{},"\n會在一個堅實的墊上開始教導動作",[16,3344],{},"\n最終的目的是要讓病人能在床上活動",[12,3347,3348],{},"設計包括",[23,3350,3351,3354,3357,3360],{},[26,3352,3353],{},"確立功能性目標",[26,3355,3356],{},"決定患者用哪些方法",[26,3358,3359],{},"考量執行該動作所需要的身體與技巧先決條件",[26,3361,3362],{},"設計一套治療計劃，以培養或發展所需的身體與技巧先決條件",[12,3364,3365],{},"身體先決條件包括：肌力，ROM",[12,3367,3368],{},"技巧先決條件包括：例如從躺到趴，手肘要伸直，雙臂要甩到身體對側",[23,3370,3371],{},[26,3372,3373,3374],{},"例子：C6完全性損傷",[23,3375,3376,3379,3382,3385],{},[26,3377,3378],{},"目的：能躺到趴（從極小幫助進步到獨立）",[26,3380,3381],{},"決定患者用哪種方式：無輔具設備情況下，能躺到趴",[26,3383,3384],{},"身體先決條件：有正常的三角肌前束，後束肌力，完整的肩膀ER，跟手肘伸直ROM",[26,3386,3387],{},"技巧先決條件：手肘能擺在伸直，肩膀能上前舉維持伸直，在伸直下能將雙臂甩向對側，結合擺臂跟甩頭動作，以及慣性來完成動作",[12,3389,3390],{},"必備技巧包括了",[23,3392,3393],{},[26,3394,3395,3396],{},"手肘可以伸直",[23,3397,3398],{},[26,3399,3400,3401],{},"假如是C5C6level完全性損傷的患者來說是個問題，因為C7是控制肱三頭肌的",[23,3402,3403,3410],{},[26,3404,3405,3406],{},"常見的起始姿勢呈現手肘屈曲，內旋，並且前臂放在軀幹上，要達到肘伸直的方法是肩膀",[75,3407,3409],{"className":3408},[78,79],"後伸且外旋",[26,3411,3412],{},"另一種起始姿勢呈現手肘屈曲，外旋，要達到手肘伸直方式靠將手肘撞擊墊子同時內旋",[23,3414,3415],{},[26,3416,3417,3418],{},"肩膀前舉同時可以手肘伸直",[23,3419,3420],{},[26,3421,3422,3423],{},"這個對C5C6level完全性損傷的患者一樣是個問題",[23,3424,3425,3428],{},[26,3426,3427],{},"可以在肩膀前舉時靠重力將手肘擺成伸展姿勢下",[26,3429,3430,3431],{},"或者在前舉過程中利用外旋將手肘擺成伸展姿勢",[23,3432,3433],{},[26,3434,3435],{},"那患者要學會在手肘前舉同時放鬆肱二頭",[23,3437,3438],{},[26,3439,3440,3441],{},"利用上肢撐起身體時，具備靜態與動態控制能力",[23,3442,3443,3482],{},[26,3444,3445,3446],{},"靜態控制能力的部份，需要",[23,3447,3448,3451,3454,3457,3460,3463],{},[26,3449,3450],{},"prone on elbow",[26,3452,3453],{},"supine on elbow",[26,3455,3456],{},"supine單肘支撐",[26,3458,3459],{},"supine單肘+單手支撐",[26,3461,3462],{},"前臂支撐側躺",[26,3464,3465,3466],{},"長坐姿下單手或雙手向後支撐",[23,3467,3468],{},[26,3469,3470,3471],{},"可以遵循姿勢從簡單到難的原則",[23,3472,3473,3476,3479],{},[26,3474,3475],{},"最簡單手肘姿勢是從正下向開始，隨後手肘放在如外，前，內等地方",[26,3477,3478],{},"也可以從雙手支撐變成單手支撐來進階訓練穩定",[26,3480,3481],{},"初期訓練可以給予患者口頭提示和徒手協助",[26,3483,3484,3485],{},"動態控制的技巧包括",[23,3486,3487,3495],{},[26,3488,3489,3490],{},"雙上肢支撐下進行重心轉移",[23,3491,3492],{},[26,3493,3494],{},"姿勢包括：supine雙肘支撐，prone雙肘支撐，supine單肘+單手支撐，坐姿下雙手向後支撐",[26,3496,3497,3498],{},"單上肢平衡下進行動態肩關節控制",[23,3499,3500],{},[26,3501,3502],{},"包括：supine單肘支撐，前臂支撐側躺，長坐姿下用單手向後支撐",[12,3504,1578],{},[23,3506,3507,3571,3573,3575,3577,3579],{},[26,3508,1583,3509],{},[23,3510,3511,3539,3568],{},[26,3512,1588,3513],{},[23,3514,3515,3517],{},[26,3516,1593],{},[26,3518,1596,3519],{},[23,3520,3521,3529,3537],{},[26,3522,1601,3523],{},[23,3524,3525,3527],{},[26,3526,1606],{},[26,3528,1609],{},[26,3530,1612,3531],{},[23,3532,3533,3535],{},[26,3534,1617],{},[26,3536,1620],{},[26,3538,1623],{},[26,3540,1626,3541],{},[23,3542,3543,3545,3551],{},[26,3544,1631],{},[26,3546,1634,3547],{},[23,3548,3549],{},[26,3550,1639],{},[26,3552,1596,3553],{},[23,3554,3555,3564,3566],{},[26,3556,1601,3557],{},[23,3558,3559,3562],{},[26,3560,3561],{},"強壯的三角肌前束，肱二頭肌，肱肌，喙肱肌，肱橈肌肌力",[26,3563,1653],{},[26,3565,1656],{},[26,3567,1659],{},[26,3569,3570],{},"趴到躺",[26,3572,1681],{},[26,3574,1754],{},[26,3576,1947],{},[26,3578,2047],{},[26,3580,2076],{},{"title":180,"searchDepth":181,"depth":181,"links":3582},[],"1~20","\u002Fimages\u002Fuploads\u002F1780473923434-97639202_p0_master1200.webp",{},"\u002Fblog\u002FSCI（十二）墊上運動",{"title":3334,"description":3583},{"loc":3586},"blog\u002FSCI（十二）墊上運動","78k0NbKv6DrADSxjb_vHeUEz7d09141FGxczKd8r2mM",{"id":3592,"title":3593,"author":7,"body":3594,"date":2850,"description":3642,"draft":185,"edited_at":2850,"extension":186,"featured_image":3643,"meta":3644,"navigation":189,"path":3645,"pinned":185,"seo":3646,"sitemap":3647,"stem":3648,"tags":194,"__hash__":3649},"blog\u002Fblog\u002F基底核失調（補充）.md","基底核失調（補充）",{"type":9,"value":3595,"toc":3640},[3596,3599,3602,3620,3623],[12,3597,3598],{},"有高人指點，說這些也會出，那我也寫一下好了（昨天前天還想著偷懶）",[12,3600,3601],{},"Hoehn-Yahr classification of disability scale 來展開一下",[3603,3604,3605,3608,3611,3614,3617],"ol",{},[26,3606,3607],{},"第一階段：輕微或無症狀，若有則是單側被害",[26,3609,3610],{},"第二階段：輕微症狀或身體中線受影響，平衡能力沒受損",[26,3612,3613],{},"第三階段：翻正反應受損，轉身和從椅子坐起時不穩；仍有獨立能力",[26,3615,3616],{},"第四階段：所有症狀都變嚴重，要他人協助才能站立行走",[26,3618,3619],{},"第五階段：只能久臥，輪椅移動",[12,3621,3622],{},"亨丁頓氏症功能分期",[3603,3624,3625,3628,3631,3634,3637],{},[26,3626,3627],{},"第一期：可以正常工作(3，能處理財務(3，家庭責任也可以(2，ADL完全OK(3，假如需照護在家即可(2",[26,3629,3630],{},"第二期：開始不太能做工作(2，處理財務需協助(2，家庭責任完全可以(2，ADL完全OK(3，假如需照護在家即可(2",[26,3632,3633],{},"第三期：工作十分勉強(1，財務能力需大量協助(1，家庭責任開始受損(1，ADL輕微受損(2，假如需照護在家即可(2",[26,3635,3636],{},"第四期：無法工作(0，財務無法處理(0，家庭責任無法管理(0，ADL中度受損(1，照護可在家或到長期照護機構(1",[26,3638,3639],{},"第五期：無法工作(0，財務無法處理(0，家庭責任無法管理(0，ADL嚴重受損(0，照護只可能全方位照護機構(0",{"title":180,"searchDepth":181,"depth":181,"links":3641},[],"補充一下會出的兩個表格，看考試後有沒有我沒寫到的再補","\u002Fimages\u002Fuploads\u002F1779768181622-129631040_p0_master1200.webp",{},"\u002Fblog\u002F基底核失調（補充）",{"title":3593,"description":3642},{"loc":3645},"blog\u002F基底核失調（補充）","vBUVVxK6N-C0dQwGx5_R_jSLkcXo-94AfV67HocZ5zI",{"id":3651,"title":3652,"author":7,"body":3653,"date":3863,"description":3864,"draft":185,"edited_at":3863,"extension":186,"featured_image":3865,"meta":3866,"navigation":189,"path":3867,"pinned":185,"seo":3868,"sitemap":3869,"stem":3870,"tags":194,"__hash__":3871},"blog\u002Fblog\u002F基底核失調（七）其他的病.md","基底核失調（七）其他的病",{"type":9,"value":3654,"toc":3861},[3655,3658,3664,3670,3673,3687,3689,3692,3705,3708,3713,3716,3727,3729,3732,3734,3753,3762,3765,3768,3770,3776,3779,3782,3785,3787,3793,3796,3799,3801,3804,3818,3821,3829,3832,3834,3839,3842,3850,3853],[12,3656,3657],{},"(阿阿阿,最後了)",[12,3659,3660,3661,3663],{},"最後講到威爾森氏症",[16,3662],{},"\n原因是因為肝和豆狀核退化導致的",[12,3665,3666,3667,3669],{},"銅的毒性，會使蒼白球跟皮殼發生神經元退化",[16,3668],{},"\n皮質的灰質和小腦的齒核也會萎縮",[12,3671,3672],{},"症狀",[23,3674,3675,3678,3681,3684],{},[26,3676,3677],{},"凱撒——佛來雪倫：眼角膜會有褐綠色或褐紅色輪狀帶",[26,3679,3680],{},"20歲後發病，會有肌張力不全症，帕金森氏症類似症狀",[26,3682,3683],{},"20歲前發病，顏面跟上肢會有舞蹈徐動動作，步態接近亨丁頓氏症",[26,3685,3686],{},"不正常姿勢跟動作模式",[36,3688],{},[12,3690,3691],{},"病程",[23,3693,3694,3697],{},[26,3695,3696],{},"剛開始：人格會改變，情緒不穩定，判斷力損傷",[26,3698,3699,3700],{},"進程：失智症，肝硬化",[23,3701,3702],{},[26,3703,3704],{},"假如沒有治療的話，肌張力不全會變嚴重，無力，癲癇發作",[12,3706,3707],{},"藥物",[23,3709,3710],{},[26,3711,3712],{},"鋅鹽可以抑制銅吸收",[12,3714,3715],{},"檢查跟治療",[23,3717,3718,3721,3724],{},[26,3719,3720],{},"早期治療師比較少介入，因為早期通常能痊癒",[26,3722,3723],{},"檢查類似帕金森氏症或亨丁頓氏症",[26,3725,3726],{},"治療：重點是減輕症狀，例如減少無關動作，肌張力減少",[36,3728],{},[12,3730,3731],{},"遲發性運動障礙",[12,3733,3672],{},[23,3735,3736,3750],{},[26,3737,3738,3739],{},"運動障礙：無法產生意志動作",[23,3740,3741,3744,3747],{},[26,3742,3743],{},"會有一系列的有節律無關動作",[26,3745,3746],{},"典型會先在臉部開始：口 舌 顏面",[26,3748,3749],{},"姿勢：軀幹伸直和脊椎前凸，和頸彎曲",[26,3751,3752],{},"可能同時會有僵硬跟運動不能",[12,3754,3755,3756,3758,3759,3761],{},"病因：通常是藥物引起，會影響多巴胺代謝",[16,3757],{},"\n老化時會影響多巴胺濃度",[16,3760],{},"\n女性較危險",[12,3763,3764],{},"醫療處置：重點是預防，因為遲發性運動障礙通常無法復原",[12,3766,3767],{},"治療方式：一般依照亨丁頓氏症的治療方式",[36,3769],{},[12,3771,3772,3773,3775],{},"肌張力不全症",[16,3774],{},"\n是一種持續或間歇性的肌肉收縮導致的不正常動作或姿勢的動作障礙",[12,3777,3778],{},"主動肌跟拮抗肌過度共同收縮",[12,3780,3781],{},"肌張力通常正常，DTR也是正常",[12,3783,3784],{},"ROM跟肌力是正常",[36,3786],{},[12,3788,3789,3790,3792],{},"全身性肌張力不全",[16,3791],{},"\n動作發生時",[12,3794,3795],{},"各部位扭轉收縮：軀幹，肩部，踝，腳趾，腳掌",[12,3797,3798],{},"通常是基因問題，抑或者出生時的傷害",[36,3800],{},[12,3802,3803],{},"局部肌張力不全",[23,3805,3806,3812,3815],{},[26,3807,3808,3809,3811],{},"痙攣性斜頸最常見",[16,3810],{},"\n疼痛，無法控制頭部到一側",[26,3813,3814],{},"作家抽筋",[26,3816,3817],{},"局部手部張力不全：執行目標性任務時會有協同拮抗肌共同收縮，干擾時序，失去單關節活動能力",[12,3819,3820],{},"病因：未知，或多種問題造成",[23,3822,3823,3826],{},[26,3824,3825],{},"是學完的東西出問題，正在學的東西沒問題",[26,3827,3828],{},"常用來進行複雜音樂演奏任務的手比較常出問題",[12,3830,3831],{},"醫療處置：施打肉毒桿菌",[12,3833,3715],{},[23,3835,3836],{},[26,3837,3838],{},"查明發作時間，和驅動症狀的動作",[12,3840,3841],{},"緩解方法",[23,3843,3844,3847],{},[26,3845,3846],{},"斜頸肌張力不全：將手放在臉部，下巴後腦勺的一側，或者另一手觸摸這些地方，有時候可以減少張力不全的相關肌肉",[26,3848,3849],{},"作家抽筋：用另一隻手觸摸受影響的手，緩解症獎",[12,3851,3852],{},"動作治療",[23,3854,3855,3858],{},[26,3856,3857],{},"著重在感覺統合，以及專注注意力下的再學習，需要重複練習",[26,3859,3860],{},"斜頸的話，引導病患頸部重新回到中線，並維持動作平衡的能力",{"title":180,"searchDepth":181,"depth":181,"links":3862},[],"2026-05-25","威爾森氏症，遲發性動作遲緩，肌張力不全症，局部肌張力不全症","\u002Fimages\u002Fuploads\u002F1779680890676-100947949_p0_master1200.webp",{},"\u002Fblog\u002F基底核失調（七）其他的病",{"title":3652,"description":3864},{"loc":3867},"blog\u002F基底核失調（七）其他的病","aTp68o30irWoAPQVBtzefAofok2zAJ_4TwdWTl2Oktw",{"id":3873,"title":3874,"author":7,"body":3875,"date":3863,"description":4134,"draft":185,"edited_at":3863,"extension":186,"featured_image":4135,"meta":4136,"navigation":189,"path":4137,"pinned":185,"seo":4138,"sitemap":4139,"stem":4140,"tags":194,"__hash__":4141},"blog\u002Fblog\u002F基底核失調（六）亨丁頓氏症.md","基底核失調（六）亨丁頓氏症",{"type":9,"value":3876,"toc":4132},[3877,3880,3886,3889,3892,3895,3898,3900,3988,3990,3993,4028,4033,4036,4038,4044,4047,4083,4085,4088],[12,3878,3879],{},"(總感覺剛剛怕今生是朕嘮了很多重複的東西,我嚴重懷疑我是不是沒讀透,考試見真章吧)",[12,3881,3882,3883,3885],{},"接下來講亨丁頓氏症，又稱亨丁頓氏舞蹈症",[16,3884],{},"\n發病年紀在30歲之後",[12,3887,3888],{},"這個病是遺傳疾病，在第四對染色體短臂缺陷導致的；CAG序列重複",[12,3890,3891],{},"對於神經物質傳導的影響：乙醯膽鹼(Ach)跟它的接受器數目減少",[12,3893,3894],{},"對於基底核來說：失去Ach和GABA，影響動作",[12,3896,3897],{},"（看起來是因為Ach是用來抑制多巴胺，但失去了Ach之後反而沒人幫多巴胺煞車，導致過動（屬間接路徑），maybe）",[12,3899,3672],{},[23,3901,3902,3927,3938,3946,3971,3974,3977],{},[26,3903,3904,3905],{},"無關的動作：在肢體動時會加入一些不正常的動作",[23,3906,3907,3910,3913,3924],{},[26,3908,3909],{},"步態會變得失調(ataxic)，或舞蹈",[26,3911,3912],{},"精細動作會變笨拙跟緩慢",[26,3914,3915,3916],{},"分成了",[23,3917,3918,3921],{},[26,3919,3920],{},"舞蹈症：突然且無節奏的",[26,3922,3923],{},"徐動症：有節奏似蟲的(worm-like)",[26,3925,3926],{},"很難透過外在提示促進適當的動作",[26,3928,3929,3930],{},"機制：不明（嗯 可能剛剛猜錯了）",[23,3931,3932,3935],{},[26,3933,3934],{},"有說法是間接路徑被瓦解了",[26,3936,3937],{},"失去Ach和GABA導致無法抑制不正常pattern",[26,3939,3940,3941],{},"眼球運動：第一個出來的症狀",[23,3942,3943],{},[26,3944,3945],{},"很難啟動跟控制快速掃視，和平順的追視",[26,3947,3948,3949],{},"步態問題",[23,3950,3951,3954,3957,3960,3963],{},[26,3952,3953],{},"運動失調：有點像在跳舞",[26,3955,3956],{},"手不會擺動",[26,3958,3959],{},"步速步幅變小",[26,3961,3962],{},"步頻變小（跟帕金森氏症不同）",[26,3964,3965,3966],{},"BOS變大",[23,3967,3968],{},[26,3969,3970],{},"會有外側擺動現象",[26,3972,3973],{},"平衡問題：平衡反應會延遲",[26,3975,3976],{},"語言，呼吸，吞嚥肌肉的異常動作",[26,3978,3979,3980],{},"認知和行為問題",[23,3981,3982,3985],{},[26,3983,3984],{},"認知：智力缺損，記憶受損，語言失用症",[26,3986,3987],{},"行為：情緒偏負面",[36,3989],{},[12,3991,3992],{},"疾病進程",[23,3994,3995,4017],{},[26,3996,3997,3998],{},"剛開始：不協調，笨拙，急動",[23,3999,4000,4008,4011,4014],{},[26,4001,4002,4003],{},"測試方式：簡單握手測試",[23,4004,4005],{},[26,4006,4007],{},"患者握住PT手指，亨丁頓氏症會有milkmaid sign的特徵",[26,4009,4010],{},"顏面問題：鬼臉或無法表達複雜表情",[26,4012,4013],{},"認知問題：連自己的人設都會忘記",[26,4015,4016],{},"診斷：SPECT單光子斷層造影機",[26,4018,4019,4020],{},"進程",[23,4021,4022,4025],{},[26,4023,4024],{},"合併身心問題，影響ADL",[26,4026,4027],{},"當舞蹈型變成僵硬時，患者會久病臥床",[12,4029,4030],{},[1930,4031],{"alt":1932,"src":4032},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1779677825645-Screenshot_2026-05-25_105631.png",[12,4034,4035],{},"這是功能性活動跟進程的影響（我懶了直接上圖）",[36,4037],{},[12,4039,4040,4041,4043],{},"處置上，其實沒有完全有效的藥物",[16,4042],{},"\n一般都是用多巴胺拮抗劑，但負作用會增加自殺行為",[12,4045,4046],{},"檢查",[23,4048,4049,4052,4060,4074,4077,4080],{},[26,4050,4051],{},"Huntington's Disease health-related Quality of Life questionnaire可以測疾病進程",[26,4053,4054,4055],{},"測肌力跟ROM，尤其是軀幹跟近端肌群穩定（初期變化不會到很明顯）",[23,4056,4057],{},[26,4058,4059],{},"單一肌肉表現不佳，可能是因為患者認知功能出問題",[26,4061,4062,4063],{},"舞蹈症",[23,4064,4065,4068,4071],{},[26,4066,4067],{},"可能存在於四肢，軀幹，臉部",[26,4069,4070],{},"要評估頻率，程度",[26,4072,4073],{},"有沒有加劇或減緩程度的因素",[26,4075,4076],{},"平衡：特別是狹窄環境，或者雙重任務時",[26,4078,4079],{},"步態：需評估步頻步長",[26,4081,4082],{},"姿勢及平衡反應：要檢查有沒有聯合動作",[36,4084],{},[12,4086,4087],{},"治療目的：主要是維持最佳的生活品質",[23,4089,4090],{},[26,4091,4092,4093],{},"治療程序：不同於帕金森氏症強調動作幅度，亨丁頓氏症強調減少動作幅度",[23,4094,4095,4098,4101,4104,4107,4110,4113,4121],{},[26,4096,4097],{},"ROM運動：降低僵硬",[26,4099,4100],{},"肌力訓練：重點在軀幹跟近端肌群",[26,4102,4103],{},"促進共同活動跟軀幹穩定",[26,4105,4106],{},"步態訓練",[26,4108,4109],{},"放鬆：可以減少無關動作，初期可利用生物回饋跟傳統放鬆運動（瑜珈那些）",[26,4111,4112],{},"須額外增加營養補充：因為有無關的動作增加新陳代謝",[26,4114,4115,4116],{},"失智症對治療影響",[23,4117,4118],{},[26,4119,4120],{},"難以學習新東西；難以控制無關動作",[26,4122,4123,4124],{},"治療跟預後",[23,4125,4126,4129],{},[26,4127,4128],{},"發病早期：做肌力，柔軟度，平衡，協調，呼吸等訓練可以有效改善患者功能",[26,4130,4131],{},"步態問題可以用節拍器來改善，比音樂節奏要好",{"title":180,"searchDepth":181,"depth":181,"links":4133},[],"亨丁頓","\u002Fimages\u002Fuploads\u002F1779680603227-56661443_p0_master1200.webp",{},"\u002Fblog\u002F基底核失調（六）亨丁頓氏症",{"title":3874,"description":4134},{"loc":4137},"blog\u002F基底核失調（六）亨丁頓氏症","14IaVaIC9gY2S2L2LlqtN-timncYclo6_qpRNcYmNak",{"id":4143,"title":4144,"author":7,"body":4145,"date":4405,"description":4406,"draft":185,"edited_at":3863,"extension":186,"featured_image":4407,"meta":4408,"navigation":189,"path":4409,"pinned":185,"seo":4410,"sitemap":4411,"stem":4412,"tags":194,"__hash__":4413},"blog\u002Fblog\u002F基底核失調（五）帕金森氏症-iv.md","基底核失調（五）帕金森氏症-iv",{"type":9,"value":4146,"toc":4403},[4147,4150,4159,4162,4173,4176,4192,4195,4209,4215,4280,4283,4298,4301,4315,4318,4331,4334,4339,4342,4345,4348,4351,4354,4359,4362,4392,4395,4400],[12,4148,4149],{},"(怎麼感覺好像已經開始重複寫了很多相同的類似的東西了)",[12,4151,4152,4153,4155,4156,4158],{},"運動主要考量包括",[16,4154],{},"\n大幅度，強高度，重複的全身性運動",[16,4157],{},"\n且注意避免疲勞",[12,4160,4161],{},"放鬆運動：溫和的搖擺跟旋轉運動可以降低僵硬程度",[23,4163,4164,4167,4170],{},[26,4165,4166],{},"治療開始時先降低僵硬可以運其他運動治療更容易進行",[26,4168,4169],{},"姿勢採坐跟站，因為supine會增加僵直；且先從遠端開始放鬆",[26,4171,4172],{},"可以合併腹式呼吸，以及PNF的雙側D2動作，也能促進胸廓跟肩的ROM",[12,4174,4175],{},"伸展運動：最少每周兩到三次，每個動作做四次，每次維持伸展十五到六十少",[23,4177,4178,4189],{},[26,4179,4180,4181],{},"用PNF來做的話",[23,4182,4183,4186],{},[26,4184,4185],{},"D2 Flex，促進上胸椎伸展，減少駝背",[26,4187,4188],{},"D1 Extend，減少帕金森氏症典型彎曲姿勢",[26,4190,4191],{},"要注意不要過度伸展或彈震伸展，也要注意患者有沒有骨質疏鬆",[12,4193,4194],{},"阻力運動：原則是漸進，超過日常負荷",[23,4196,4197,4200,4203,4206],{},[26,4198,4199],{},"從低強度開始",[26,4201,4202],{},"假如患者有主動拮抗肌共同收縮問題，則不太建議使用等長收縮運動",[26,4204,4205],{},"所有肌力訓練都要軀幹肌肉",[26,4207,4208],{},"假如患者有吃藥，可以利用藥效時間進行訓練",[12,4210,4211,4212,4214],{},"功能性運動：主要提升功能性活動度，尤其是中軸結構（頭，軀幹，髖，肩）",[16,4213],{},"\n那PDF的節律動作也能促進功能性運動",[23,4216,4217,4228,4236,4250,4253,4269,4277],{},[26,4218,4219,4220],{},"床上活動",[23,4221,4222,4225],{},[26,4223,4224],{},"做側躺翻身動作：可以做分開的旋轉動作，上半身先轉，接著才到下半身",[26,4226,4227],{},"橋式運動",[26,4229,4230,4231],{},"躺到坐轉移位",[23,4232,4233],{},[26,4234,4235],{},"利用側坐，肘支撐，換成坐姿：可以訓練軀幹旋轉，跟側向屈曲",[26,4237,4238,4239],{},"坐姿",[23,4240,4241,4244,4247],{},[26,4242,4243],{},"可以增加骨盤活動度",[26,4245,4246],{},"也可以借由重量轉移來練習抓取",[26,4248,4249],{},"合併PNF：強調誘發軀幹伸展跟旋轉",[26,4251,4252],{},"坐到站轉換（中後期或藥效off期是困難的）",[26,4254,4255,4256],{},"站立",[23,4257,4258],{},[26,4259,4260,4261],{},"要先訓練以達到能完全站好，還有對稱承重才能繼續",[23,4262,4263,4266],{},[26,4264,4265],{},"可以在pelvis前方輕觸或施加小阻力，促進hip往前移動到直立",[26,4267,4268],{},"站姿+手撐牆可以增加軀幹extend",[26,4270,4271,4272],{},"仆街後爬起技巧（我已瘋了）",[23,4273,4274],{},[26,4275,4276],{},"四足跪肢＞高跪姿＞半跪姿＞站起並以上肢協助",[26,4278,4279],{},"活化顏面肌肉",[12,4281,4282],{},"平衡訓練",[23,4284,4285,4292,4295],{},[26,4286,4287,4288],{},"COM與LOS的訓練",[75,4289,4291],{"className":4290},[78],"（避免用手推患者，有可能會讓患者陷入僵直狀態）",[26,4293,4294],{},"動態穩定任務（避免外在施在干擾，患者會陷入僵直）",[26,4296,4297],{},"強化下肢ROM",[12,4299,4300],{},"移動能力訓練：減少步態缺陷，避免跌倒",[23,4302,4303,4306,4309,4312],{},[26,4304,4305],{},"促進直立對齊",[26,4307,4308],{},"改善跨步高度",[26,4310,4311],{},"強調手的擺動跟軀幹動作",[26,4313,4314],{},"外部提示（口頭提示，視覺提示，聲音提示）",[12,4316,4317],{},"減少凍結步態",[23,4319,4320],{},[26,4321,4322,4323],{},"代償方式",[23,4324,4325,4328],{},[26,4326,4327],{},"解凍方式：可以利用旋轉動作，或者花招（例如扔一紙巾在前面叫患者必須跨過去）",[26,4329,4330],{},"倒拿手杖讓患者跨過去，或者看別人走路動作",[12,4332,4333],{},"動作合併認知雙重任務",[23,4335,4336],{},[26,4337,4338],{},"雖然日常避免病人做雙重任務，但最新研究指出用雙重任務訓練可以減少凍結情形和改善步態",[12,4340,4341],{},"脊椎輔具：軀幹穩定，改善呼吸功能和生活品質",[12,4343,4344],{},"心肺復健：因為帕金森氏症可能會有醫療併發症、上呼吸道阻塞、侷限性通氣障礙、吸入性肺炎這四種問題，導致失能或死亡",[12,4346,4347],{},"語言治療",[12,4349,4350],{},"有氧運動：強度在正常訓練強度或次最大強度",[12,4352,4353],{},"群體和居家運動：例如太極拳，瑜珈那些",[23,4355,4356],{},[26,4357,4358],{},"居家運動最好是每天做，但要注意劑量，避免疲勞",[12,4360,4361],{},"適應性或支持性器具",[23,4363,4364,4367,4370,4373,4376],{},[26,4365,4366],{},"床：結實的床可以減少彎曲姿勢",[26,4368,4369],{},"坐椅：最好是有扶手跟結實的，軟的不利於保持良好姿勢；馬桶可以做增高坐墊，利於站起",[26,4371,4372],{},"鞋子：在toe加楔形墊，鞋跟平，可以減少前衝的小碎步步態",[26,4374,4375],{},"ADL：例如餐具的握把加大加粗等等",[26,4377,4378,4379],{},"拐杖或助行器",[23,4380,4381,4384],{},[26,4382,4383],{},"前中期患者可以用拐杖來，尤其是手杖，可以採倒拿的方式減少凍結步態",[26,4385,4386,4387],{},"較嚴重的患者問題，他們因為缺乏雙重任務能力，所以沒有辦法正確使用拐杖或助行器能力，且帶有輪子的助行器會增加前衝小碎步問題",[23,4388,4389],{},[26,4390,4391],{},"所以會建議用比較高的下壓煞車式有輪子助行器，這樣還可以增加軀幹伸展",[12,4393,4394],{},"心理社會議題",[23,4396,4397],{},[26,4398,4399],{},"主要是要讓患者跟家屬了解這個病；並且強調患者還能做甚麼，而不是講不能做甚麼，有助患者正面的了解病情",[12,4401,4402],{},"衛教",{"title":180,"searchDepth":181,"depth":181,"links":4404},[],"2026-05-24","講完怕今生是朕全部，等下接亨丁頓","\u002Fimages\u002Fuploads\u002F1779674513576-127240469_p0_master1200.webp",{},"\u002Fblog\u002F基底核失調（五）帕金森氏症-iv",{"title":4144,"description":4406},{"loc":4409},"blog\u002F基底核失調（五）帕金森氏症-iv","qzEbaxgVVxEhq9IdvE1epbNHEZ9TwBKYr11V5EvktjI",{"id":4415,"title":4416,"author":7,"body":4417,"date":4405,"description":4908,"draft":185,"edited_at":4405,"extension":186,"featured_image":4909,"meta":4910,"navigation":189,"path":4911,"pinned":185,"seo":4912,"sitemap":4913,"stem":4914,"tags":194,"__hash__":4915},"blog\u002Fblog\u002F基底核失調（四）帕金森氏症-iii.md","基底核失調（四）帕金森氏症-iii",{"type":9,"value":4418,"toc":4906},[4419,4422,4431,4440,4442,4445,4459,4465,4471,4474,4477,4491,4494,4496,4499,4505,4521,4523,4526,4538,4541,4544,4547,4550,4552,4555,4577,4579,4582,4731,4733,4736,4778,4780,4783,4813,4815,4818,4830,4833,4850,4852,4855,4861,4864,4875,4878,4903],[12,4420,4421],{},"(怎麼感覺這幾篇一直在提到程序性次序性 雖然是綠字沒錯啦)",[12,4423,4424,4425,4427,4428,4430],{},"接著講醫療診斷",[16,4426],{},"\n目前帕金森氏症沒有單一診斷或單一組診斷可以診斷出PD",[16,4429],{},"\n那早期靠非運動功能症狀來診斷較為顯著",[12,4432,4433,4434,4436,4437,4439],{},"基本是只能靠病史跟臨床檢查來判斷，因為帕金森氏症是會一直加重的病症",[16,4435],{},"\n有一個學派是往動作方向來探討，包括了動作徐緩+震顫or僵硬",[16,4438],{},"\n那在診斷時也要排除是不是plus版的帕金森氏症，因為這個plus版對治療帕金森氏症的藥物沒有任何幫助",[36,4441],{},[12,4443,4444],{},"進程分成四個期",[3603,4446,4447,4450,4453,4456],{},[26,4448,4449],{},"第一期——休息時顫抖，單側字體過小症",[26,4451,4452],{},"第二期——雙側僵硬，動作徐緩，姿勢改變，平衡反應能力下降",[26,4454,4455],{},"第三期——僵硬加劇（軀幹及近端），軀幹旋轉能力下降，走路無擺手，無自發性表情，動作啟動困難（若需要輔助才能行走即達到第四期）",[26,4457,4458],{},"末期\u002F危險期（Gemini寫第五期）——幻象枕頭症候群（平躺時，無枕頭亦會維持頭前flex），一次只能做一個任務，支氣管肺炎",[12,4460,4461,4462,4464],{},"疾病嚴重度跟失能分期可以靠Hoehn-Yahr classification of disability scale來評",[16,4463],{},"\n其他的就靠UPDRS（統一帕金森氏症量表）",[12,4466,4467,4468,4470],{},"關於治療部份呢，目前這帕金森氏症是沒有辦法逆轉痊癒的，只能靠藥物延緩進程（普遍靠L-dopa）",[16,4469],{},"\n(藥我就不寫了,我妄斷老師不出,出來我事後再標名一次)",[12,4472,4473],{},"營養管理——會建議患者吃高熱量低蛋白食物",[12,4475,4476],{},"手術手段——深腦刺激術",[23,4478,4479,4482,4485,4488],{},[26,4480,4481],{},"刺激蒼白球可減少運動障礙問題",[26,4483,4484],{},"刺激視丘下核可改善藥物無效(off期)的情形",[26,4486,4487],{},"刺激腳間橋腦核跟黑質可改善平衡步態問題",[26,4489,4490],{},"刺激視丘可減輕震顫",[12,4492,4493],{},"復健治療——跨科系合作",[36,4495],{},[12,4497,4498],{},"分期的復健考量",[12,4500,4501,4502,4504],{},"治療目的：依舊進程基本分成",[16,4503],{},"\n復健，預防，代償",[23,4506,4507,4510,4513],{},[26,4508,4509],{},"早期：患者多能獨立自主，這時期基本靠保持身體健康和預防為主",[26,4511,4512],{},"中期：會開始有功能性限制，不良習慣或錯誤姿勢；運動也能有效善輕中度帕金森氏症患者的動作表現",[26,4514,4515,4516],{},"後期：因藥效副作用，藥效消失，肌肉骨骼系統受損，認知功能降低，等影響導致ADL幾乎依賴",[23,4517,4518],{},[26,4519,4520],{},"PT主要預防次級併發症，代償訓練，環境改造，指導看護",[36,4522],{},[12,4524,4525],{},"PT的檢查跟評估",[12,4527,4528,4529,4531,4532,4534,4535,4537],{},"那像剛剛說的Hoehn-Yahr classification of disability scale可以評受限跟失能進程",[16,4530],{},"\n不過這個沒有評估非運動功能",[16,4533],{},"\n早期的話可以用這個表來評估",[16,4536],{},"\n後期的話會因為各種像藥物副作用的問題造成失真",[12,4539,4540],{},"亦可根據ICF模型考量",[12,4542,4543],{},"量表較常使用MDS-UPDRS，這個表運動跟非運動都有包含到",[12,4545,4546],{},"認知功能方面：可以用簡易智能測試來測，總分30分",[12,4548,4549],{},"心理社會功能：對比發病前興趣，ADL狀況等；也要關注情緒問題",[36,4551],{},[12,4553,4554],{},"感覺功能問題",[23,4556,4557,4560,4563,4574],{},[26,4558,4559],{},"觸覺跟本體感覺會遲鈍，導致功能喪失，且下肢嚴重度大於上肢，遠端大於近端",[26,4561,4562],{},"感覺異常",[26,4564,4565,4566],{},"疼痛",[23,4567,4568,4571],{},[26,4569,4570],{},"酸痛：難以定位",[26,4572,4573],{},"肌肉骨骼痛",[26,4575,4576],{},"視覺功能問題：視銳度下降，顏色辨色喪失",[36,4578],{},[12,4580,4581],{},"肌肉骨骼功能問題",[23,4583,4584,4604,4615,4659,4676,4713,4716,4719,4722,4725,4728],{},[26,4585,4586,4587],{},"僵硬：由PROM跟阻力來評",[23,4588,4589,4592,4595,4598,4601],{},[26,4590,4591],{},"0分—沒有僵硬",[26,4593,4594],{},"1分—誘發動作會僵硬",[26,4596,4597],{},"2分—不需要誘發也會僵硬，但能輕易做full ROM",[26,4599,4600],{},"3分—不需要誘發也會僵硬，吃力的做出full ROM",[26,4602,4603],{},"4分—不需要誘發也會僵硬，無法做完full ROM",[26,4605,4606,4607],{},"動作徐緩：初期動作緩慢，且缺乏節律，經常遲濟或停止",[23,4608,4609,4612],{},[26,4610,4611],{},"測試動作：快速交替動作；全身自發性動作（坐姿 站立 走路）",[26,4613,4614],{},"反應時間也要測量",[26,4616,4617,4618],{},"震顫：從部位，時間，嚴重程度來看",[23,4619,4620,4640],{},[26,4621,4622,4623],{},"嚴重程度",[23,4624,4625,4628,4631,4634,4637],{},[26,4626,4627],{},"0—沒抖",[26,4629,4630],{},"1—有抖，程度\u003C1cm",[26,4632,4633],{},"2—有抖，程度1~3cm",[26,4635,4636],{},"3—有抖，程度3~10cm",[26,4638,4639],{},"4—有抖，程度>10cm",[26,4641,4642,4643],{},"持續時間",[23,4644,4645,4647,4650,4653,4656],{},[26,4646,4627],{},[26,4648,4649],{},"1—有抖，佔檢查時長0~25%",[26,4651,4652],{},"2—有抖，佔檢查時長26~50%",[26,4654,4655],{},"3—有抖，佔檢查時長51~75%",[26,4657,4658],{},"4—有抖，佔檢查時長76~100%",[26,4660,4661,4662],{},"姿勢控制和平衡問題：觀察坐跟站",[23,4663,4664,4667,4670,4673],{},[26,4665,4666],{},"安靜站立：病患增加內外向或前後向的身體搖晃",[26,4668,4669],{},"可使用的姿勢策略：帕金森氏症傾向用髖策略",[26,4671,4672],{},"感覺系統的影響：視覺，本體感覺，前庭覺輸入影響平衡表現",[26,4674,4675],{},"雙重任務會使帕金森氏症患者不穩定（因為注意力被分散）",[26,4677,4678,4679],{},"步態問題：評估步幅，步長，速度，距離，啟動時間，安全性，穩定性，變異性",[23,4680,4681,4698,4701,4704,4707,4710],{},[26,4682,4683,4684],{},"帕金森氏症患者可能會有",[23,4685,4686,4689,4692,4695],{},[26,4687,4688],{},"拖曳步",[26,4690,4691],{},"Hip Knee Ankle 動作減少",[26,4693,4694],{},"擺手跟軀幹旋轉喪失",[26,4696,4697],{},"匆促步態",[26,4699,4700],{},"要測試不同方向的走路",[26,4702,4703],{},"走在不同的環境中",[26,4705,4706],{},"雙重任務（嗯 又是你）",[26,4708,4709],{},"是否需要輔具或看護協具",[26,4711,4712],{},"Time up and go test, 動態步態指數量表, 功能性步態測試等是測試項目",[26,4714,4715],{},"步態凍結：記錄發作頻率，原因，持續時間",[26,4717,4718],{},"跌倒風險",[26,4720,4721],{},"疲勞",[26,4723,4724],{},"動作障礙",[26,4726,4727],{},"非運動症狀",[26,4729,4730],{},"吞嚥困難和語言障礙",[36,4732],{},[12,4734,4735],{},"自主神經系統改變",[23,4737,4738,4741,4752,4755,4758,4769,4772],{},[26,4739,4740],{},"心肺耐力的問題",[26,4742,4743,4744],{},"姿勢性低血壓：頭昏眼花，蒼白，冒汗，昏厥",[23,4745,4746,4749],{},[26,4747,4748],{},"收縮壓下降20mmHg，舒張壓下降10mmHg，脈博上升10~20%",[26,4750,4751],{},"若站超過1分鐘仍有症狀代表有問題（晚期帕金森氏症）",[26,4753,4754],{},"皮膚問題：會很油，後期的會有壓瘡",[26,4756,4757],{},"功能性狀態：例如功能性轉移位，基本ADL，工具性ADL；住院患者可用FIM來測",[26,4759,4760,4761],{},"功能性表現：",[23,4762,4763,4766],{},[26,4764,4765],{},"要先分析直接跟間接影響",[26,4767,4768],{},"時間也是重點，因為會疲勞影響結果",[26,4770,4771],{},"環境與輔具評估",[26,4773,4774,4775,4777],{},"Profile of Function and Impairment Level Experience With Parkinson’s Disease",[16,4776],{},"\n（帕金森氏症功能與受損的全面概況評估）",[36,4779],{},[12,4781,4782],{},"整體健康測量：常用Rand 36-Item Health Survey(SF36)",[23,4784,4785],{},[26,4786,4787,4788],{},"疾病特異的測量",[23,4789,4790,4793,4799,4806],{},[26,4791,4792],{},"Hoehn-Yahr classification of disability",[26,4794,4795,4798],{},[75,4796,4774],{"className":4797},[78,79],"：可評估嚴重度，進程，藥物的影響",[26,4800,4801,4805],{},[75,4802,4804],{"className":4803},[78,173],"Parkinson’s Disease Questionnaire（PDQ-39）","：重點在ADL的評估",[26,4807,4808,4812],{},[75,4809,4811],{"className":4810},[78,173],"Parkinson's Disease Summary Index （PDSI）","：評估整體健康程度",[36,4814],{},[12,4816,4817],{},"介入",[12,4819,4820,4821,4823,4824,4826,4827,4829],{},"時間來看當然是第一期就可以來了，但大多數都是在2.5期（明顯有機能下降時）才開始介入",[16,4822],{},"\nPT也只能做緩和患者的問題，如延緩進程，但始終無法逆轉或終止進程",[16,4825],{},"\n可以搭配藥物一起，PT可以在ROM，心肺耐力，胸廓擴張度，日常姿勢，平衡反應這些地方下手",[16,4828],{},"\n中~強度運動可以引起多巴胺跟他的受體的結合，WHO會建議一周3次的中度運動+2次的劇烈運動",[12,4831,4832],{},"主要治療目的跟結果",[23,4834,4835,4838,4841,4844,4847],{},[26,4836,4837],{},"早期介入：預防肌肉骨骼受損，提升運動功能，運動耐力，功能表現，活動參與；並且衛教",[26,4839,4840],{},"指導協助人員",[26,4842,4843],{},"指示患者，家屬，照顧者",[26,4845,4846],{},"環境改造",[26,4848,4849],{},"支持性咨詢",[36,4851],{},[12,4853,4854],{},"動作學習策略",[12,4856,4857,4858,4860],{},"初期中期的患者可以靠練習和感覺訊改善動作品質",[16,4859],{},"\n但嚴重患者會有認知問題，訓練較難困難",[12,4862,4863],{},"訓練時強調",[23,4865,4866,4869,4872],{},[26,4867,4868],{},"大量重複訓練",[26,4870,4871],{},"要患者在非常專心情況下練習，所以要減少干擾",[26,4873,4874],{},"避免長時間且複雜的動作，也避免雙重任務",[12,4876,4877],{},"提示策略：外部提示可以讓輕中度患者提供有效的次序性動作，促進動作品質，降低動作凍結情形",[23,4879,4880],{},[26,4881,4882,4883],{},"原理：外在提示可以繞過基底核，不過並非所有患者都適用",[23,4884,4885,4888,4891,4894,4897,4900],{},[26,4886,4887],{},"視覺提示：例如可以在地面貼標籤讓患者調整步態",[26,4889,4890],{},"聽覺提示：例如可以用節拍來調整患者步頻",[26,4892,4893],{},"多重感覺提示：例如視+聽，也能改善步態",[26,4895,4896],{},"如過患者認知功能良好，可以用隨機訓練來練",[26,4898,4899],{},"心象練習也有效幫助動作",[26,4901,4902],{},"給外在提示也可以讓病人在動作凍結時將動作被促進出來",[12,4904,4905],{},"（哇靠好多",{"title":180,"searchDepth":181,"depth":181,"links":4907},[],"診斷到動作學習策略","\u002Fimages\u002Fuploads\u002F1779610578615-125497457_p0_master1200.webp",{},"\u002Fblog\u002F基底核失調（四）帕金森氏症-iii",{"title":4416,"description":4908},{"loc":4911},"blog\u002F基底核失調（四）帕金森氏症-iii","fXAz3zYJOOShbAY0aei2ggZo_fjeTbgMBGOLApD5590",{"id":4917,"title":4918,"author":7,"body":4919,"date":5251,"description":5252,"draft":185,"edited_at":4405,"extension":186,"featured_image":5253,"meta":5254,"navigation":189,"path":5255,"pinned":185,"seo":5256,"sitemap":5257,"stem":5258,"tags":194,"__hash__":5259},"blog\u002Fblog\u002F基底核失調（三）帕金森氏症-ii.md","基底核失調（三）帕金森氏症-ii",{"type":9,"value":4920,"toc":5249},[4921,4924,4927,5080,5082,5085],[12,4922,4923],{},"繼續來講怕金生是朕",[12,4925,4926],{},"次要運動症狀",[23,4928,4929,4940,5007],{},[26,4930,4931,4932],{},"肌肉表現",[23,4933,4934,4937],{},[26,4935,4936],{},"肌力下降：就無力",[26,4938,4939],{},"疲勞：肌力會隨著動作時長減少；且無力會慢慢增加，並有嗜睡感；服用L-dopa在初期會改善症狀",[26,4941,4942,4943],{},"運動功能",[23,4944,4945,4967,4974,4999],{},[26,4946,4947,4948],{},"動作規劃缺陷",[23,4949,4950,4964],{},[26,4951,4952,4953],{},"自發性及意志動作有缺陷",[23,4954,4955,4958,4961],{},[26,4956,4957],{},"速度-準確度取捨",[26,4959,4960],{},"病人會覺得難以執行複雜，次序性，雙重任務的動作",[26,4962,4963],{},"起始猶如",[26,4965,4966],{},"注意力轉移困難",[26,4968,4969,4973],{},[75,4970,4972],{"className":4971},[78],"初期","的話是可以用藥物來幫患者學習新的技巧，或精細調整",[26,4975,4976,4980,4981,4985,4986],{},[75,4977,4979],{"className":4978},[78,173],"程序性學習","缺乏則在帕金森氏症",[75,4982,4984],{"className":4983},[78],"後期","患者會發生",[23,4987,4988,4996],{},[26,4989,4990,4991,4995],{},"陳述性學習是",[75,4992,4994],{"className":4993},[173],"正常","的",[26,4997,4998],{},"假如還有隨機性的也會出問題",[26,5000,5001,5002],{},"多重動作程式會出現嚴重缺陷",[23,5003,5004],{},[26,5005,5006],{},"像是duel task，會有動作凍結",[26,5008,5009,5010],{},"步態障礙",[23,5011,5012,5053,5064],{},[26,5013,5014,5015],{},"持續性：主要是步速緩慢，動作變異大，不對稱性增加，姿勢控制不良",[23,5016,5017,5037,5040],{},[26,5018,5019,5020],{},"步速變慢",[23,5021,5022,5025,5028,5031,5034],{},[26,5023,5024],{},"步頻變慢：導致運動徐緩",[26,5026,5027],{},"步長變短：導致運動不足",[26,5029,5030],{},"僵直：因軸性僵直，患者會維持駝背姿勢",[26,5032,5033],{},"手臂擺動減少，軀幹旋轉減少：因動作徐緩或運動不足（整合較差）",[26,5035,5036],{},"整體旋轉：慢，且需更多步數才能完成",[26,5038,5039],{},"動作變異大：步幅之間的步長（前後向）及步寬（內外向）",[26,5041,5042,5043,5047,5048],{},"姿勢搖晃變大：",[75,5044,5046],{"className":5045},[173],"內外向","最明顯",[23,5049,5050],{},[26,5051,5052],{},"在LOS，疾病初期，站姿向後的LOS會下降",[26,5054,5055,5056],{},"偶發性",[23,5057,5058,5061],{},[26,5059,5060],{},"匆促推進步態：非意圖性的小短步急走，分成了向前跟向後",[26,5062,5063],{},"凍結步態",[26,5065,5066,5067],{},"其他的步態問題",[23,5068,5069,5077],{},[26,5070,5071,5072,5076],{},"Hip Ankle：Hip Flexion 減少，",[75,5073,5075],{"className":5074},[78,173],"站立後期","的Ankle planterflexion減少",[26,5078,5079],{},"拖曳步：著地flat foot，甚至toe-heel",[36,5081],{},[12,5083,5084],{},"非運動症狀：初期症狀有嗅覺喪失，便秘，清醒夢，情緒障礙，姿勢性低血壓",[23,5086,5087,5109,5117,5131,5142],{},[26,5088,5089,5090],{},"感覺問題",[23,5091,5092,5100,5103,5106],{},[26,5093,5094,5095],{},"感覺異常：非原發性感覺喪失",[23,5096,5097],{},[26,5098,5099],{},"50%的患者會有痲痺，刺痛，酸痛，異常溫度覺",[26,5101,5102],{},"痛覺異常",[26,5104,5105],{},"本體感覺異常：由本體感覺引導的意志動作受損",[26,5107,5108],{},"嗅覺喪失：初期會有嗅覺下降或喪失問題",[26,5110,5111,5112],{},"視覺問題",[23,5113,5114],{},[26,5115,5116],{},"空間技巧缺陷：空間組織的視覺感知任務做顯著較多錯誤",[26,5118,5119,5120],{},"吞嚥困難",[23,5121,5122,5125,5128],{},[26,5123,5124],{},"疾病初期就會發生",[26,5126,5127],{},"會嗆到或吸入性肺炎",[26,5129,5130],{},"流口水過多：特別是睡眠或開始講話時",[26,5132,5133,5134],{},"語言障礙",[23,5135,5136,5139],{},[26,5137,5138],{},"75~89％會有這問題",[26,5140,5141],{},"運動功能下降的發音困難：音量下降，音調無起伏，發音不正確，無法控制的講話速率",[26,5143,5144,5145],{},"認知與行為功能障礙",[23,5146,5147,5179,5199,5202,5205],{},[26,5148,5149,5150],{},"影響包括",[23,5151,5152,5155,5158,5161,5164,5167],{},[26,5153,5154],{},"注意力轉移不能",[26,5156,5157],{},"工作記憶快速存取缺陷",[26,5159,5160],{},"視覺空間的知覺跟區辨存在困難",[26,5162,5163],{},"程序學習特別受牽連",[26,5165,5166],{},"隨著病程會惡化",[26,5168,5169,5170],{},"治療方式為",[23,5171,5172],{},[26,5173,5174,5178],{},[75,5175,5177],{"className":5176},[78,173],"多練","，訓練單獨視覺輸入",[26,5180,5181,5182],{},"認知問題：認知執行功能的失能",[23,5183,5184,5187,5190,5193,5196],{},[26,5185,5186],{},"輕度認知障礙：介於正常人與早期失智症之間",[26,5188,5189],{},"失智症：許多帕金森氏症患者會在一年後發展成這個",[26,5191,5192],{},"思想遲鈍：思考過程與反應變慢，且缺乏集中力",[26,5194,5195],{},"學習缺陷：尤其是關於程序性學習的新技巧",[26,5197,5198],{},"知覺缺陷：與空間組織的視覺知覺能力缺陷有關",[26,5200,5201],{},"行為問題：偏負面情緒或表情；",[26,5203,5204],{},"睡眠問題：有睡眠障礙，在做夢時會有大的肢體動作，",[26,5206,5207,5208,2589,5212],{},"自主神經功能障礙（",[75,5209,5211],{"className":5210},[78],"自主神經內也存在路易體",[23,5213,5214,5217,5220,5223,5226,5229,5232,5240,5243,5246],{},[26,5215,5216],{},"溫度調節不正常",[26,5218,5219],{},"皮膚表面油脂分泌不正常",[26,5221,5222],{},"不正常瞳孔反應：對光反應減少且變慢",[26,5224,5225],{},"腸胃問題",[26,5227,5228],{},"失禁：夜尿症",[26,5230,5231],{},"性功能問題",[26,5233,5234,5235],{},"心血管系統問題：運動時心血管反應較為遲鈍",[23,5236,5237],{},[26,5238,5239],{},"在次最大運動時，與正常人比有較低的攝氧量，心跳，收縮壓等",[26,5241,5242],{},"姿勢性低血壓",[26,5244,5245],{},"呼吸功能受損：氣道阻塞，限制性肺功能障礙",[26,5247,5248],{},"全身性心肺功能退化：久坐，無運動",{"title":180,"searchDepth":181,"depth":181,"links":5250},[],"2026-05-23","次要運動症狀及非運動症狀","\u002Fimages\u002Fuploads\u002F1779587871440-123826866_p0_master1200.webp",{},"\u002Fblog\u002F基底核失調（三）帕金森氏症-ii",{"title":4918,"description":5252},{"loc":5255},"blog\u002F基底核失調（三）帕金森氏症-ii","5bhEAJoN1dOajwz_vecSNVLm9oj2PDkJ_VZZrhTpPJQ",{"id":5261,"title":5262,"author":7,"body":5263,"date":5251,"description":5594,"draft":185,"edited_at":5251,"extension":186,"featured_image":5595,"meta":5596,"navigation":189,"path":5597,"pinned":185,"seo":5598,"sitemap":5599,"stem":5600,"tags":194,"__hash__":5601},"blog\u002Fblog\u002F基底核失調（二）帕金森氏症-i.md","基底核失調（二）帕金森氏症-i",{"type":9,"value":5264,"toc":5592},[5265,5268,5271,5276,5279,5295,5298,5308,5311,5336,5338,5341,5349,5352,5372,5375],[12,5266,5267],{},"好耶終於進來看怕今生是朕了（大誤",[12,5269,5270],{},"先來說流行病學",[23,5272,5273],{},[26,5274,5275],{},"主要是老人問題：平均發病年紀在50到60歲，隨著年紀發生率會遞增",[12,5277,5278],{},"病因的話像上一課題到的，主要是黑質中的多巴胺退化或死亡導致（黑質的顏色來源於多巴胺，假如顏色越淺代表多巴胺死越多）",[23,5280,5281,5284],{},[26,5282,5283],{},"而在退化過程中也能發現路易體",[26,5285,5286,5287],{},"當多巴胺退化到30到60%會出現顥著的臨床徵狀，並會有兩種臨床表現",[23,5288,5289,5292],{},[26,5290,5291],{},"PIGD：主要是姿勢不穩，步態障礙等問題",[26,5293,5294],{},"震顫為主：主要是靜止震顫等問題，較少會出現非運動症狀",[12,5296,5297],{},"那會導致多巴胺減少的原因至今不明，目前大部份共識是環境（有毒性環境中），遺傳，老化",[23,5299,5300],{},[26,5301,5302,5303],{},"像上一課提到的，多巴胺同時有興奮跟抑制作用",[23,5304,5305],{},[26,5306,5307],{},"假如缺失了，會導致興奮視丘的輸入下降，對周邊的抑制也會下降",[12,5309,5310],{},"那帕金森氏症也分成以下類型",[23,5312,5313,5316,5330,5333],{},[26,5314,5315],{},"原發性：佔78%，病因不明或因遺傳導致",[26,5317,5318,5319],{},"次發性：",[23,5320,5321,5324,5327],{},[26,5322,5323],{},"感染後：在流行性感冒合併嗜睡性腦炎，多年後會惡化成帕金森氏症的症狀",[26,5325,5326],{},"中毒性：暴露在例如殺蟲劑，或錳，或一氧化環境中，其中錳佔最大宗",[26,5328,5329],{},"藥物性：吃下會干援多巴胺的藥；會導致錐體外失調",[26,5331,5332],{},"代謝性：臨床上最少；",[26,5334,5335],{},"Plus版：會產生類似帕金森氏症的症狀，但使用抗帕金森藥物無效（例如L-dopa，是補充多巴胺的物藥）",[36,5337],{},[12,5339,5340],{},"基底核出問題會有兩種表現",[23,5342,5343,5346],{},[26,5344,5345],{},"運動過量障礙",[26,5347,5348],{},"運動不足障礙",[12,5350,5351],{},"帕金森氏症分成了六期",[3603,5353,5354,5357,5360,5363,5366,5369],{},[26,5355,5356],{},"嗅球、嗅核、第九第十對腦神經的傷害",[26,5358,5359],{},"橋腦被蓋區的傷害",[26,5361,5362],{},"黑質紋狀體的傷害（這裡症狀開始顯著）——黑質緻密部最明顯",[26,5364,5365],{},"下視丘及部份視丘傷害",[26,5367,5368],{},"新皮質，感覺聯合區，前額葉新皮質區傷害",[26,5370,5371],{},"初級感覺聯合區及運動前區傷害",[12,5373,5374],{},"主要特徵有：僵直，動作徐緩，震顫，姿勢不穩定",[23,5376,5377,5434,5478,5498,5531,5570],{},[26,5378,5379,5380],{},"僵直：主述感到沉重跟僵硬感",[23,5381,5382,5400,5411,5414,5417,5420],{},[26,5383,5384,5389],{},[75,5385,5388],{"className":5386},[78,5387],"bg-red-3","定義",[23,5390,5391,5394,5397],{},[26,5392,5393],{},"在做被動動作時會有阻力，但這個阻力跟速度無關",[26,5395,5396],{},"不管主動肌或拮抗肌在做動作都會有阻力",[26,5398,5399],{},"脊髓牽張反射是正常的",[26,5401,5402,5403],{},"類型",[23,5404,5405,5408],{},[26,5406,5407],{},"鉛管式：被動運動時整段阻力一致",[26,5409,5410],{},"齒輪式，被動運動時感覺會有jerky（顛簸）；機轉向是鉛管式+震顫",[26,5412,5413],{},"進程：首先是近端被害，接著是臉部跟遠端，最後是全身；且僵直會越來越嚴重",[26,5415,5416],{},"其他影響因素：主動動作，在想東西（注意力集中），情緒問題",[26,5418,5419],{},"併發症：ROM下降，能量消耗上升（疲勞）",[26,5421,5422,5423],{},"可能會導致僵直的原因",[23,5424,5425,5428,5431],{},[26,5426,5427],{},"與以下無關——伽瑪運動神經元亢進；周期性抑制一下降；運動系統產生普遍性興奮",[26,5429,5430],{},"MLR或LLR亢進導致肌張力異常；而SLR屬正常（如DTR）",[26,5432,5433],{},"CNS反射增益喪失：因為缺乏改變能力，變成做所有事興奮性一致",[26,5435,5436,5437],{},"動作徐緩或動作不能",[23,5438,5439,5461,5475],{},[26,5440,5441,5442],{},"動作徐緩：動作緩慢且難以緩持動作；不過外在提示可以改善這問題",[23,5443,5444],{},[26,5445,5446,5447],{},"可能會導致動作徐緩的原因",[23,5448,5449,5452,5455,5458],{},[26,5450,5451],{},"震顫，無力，僵硬會影響但不是主因",[26,5453,5454],{},"思想遲鈍可能會導致",[26,5456,5457],{},"大腦皮質活化不足",[26,5459,5460],{},"運動單元方面——並非僵直或無法放鬆導致；EMG模式正確，但強度不夠，不能快速徵召肌肉產生動作",[26,5462,5463,5464],{},"動作不能：起始動作困難；缺乏自發性動作（僕克臉），缺乏聯合動作，動作凍結（靠外在提示可克服）",[23,5465,5466,5469,5472],{},[26,5467,5468],{},"當整合兩個動作程式時難以保持繼續活動",[26,5470,5471],{},"動作轉換時會導致凍結",[26,5473,5474],{},"到具挑戰環境時也會凍結",[26,5476,5477],{},"運動功能衰退：持續動作時，動作會越來越小，例如字體過小症",[26,5479,5480,5481],{},"動作影響範圍：全方面的（包括起始，改動，停止），品質也是",[23,5482,5483,5486,5489,5492,5495],{},[26,5484,5485],{},"複雜的動作影響較大",[26,5487,5488],{},"次序性動作執行難度增加",[26,5490,5491],{},"雙重任務執行難度增加",[26,5493,5494],{},"難以在任務間轉變",[26,5496,5497],{},"在動作學習上，有次序性的動作也是較難學習",[26,5499,5500,5501],{},"震顫，也分成了三種",[23,5502,5503,5525,5528],{},[26,5504,5505,5506],{},"休息時震顫：休息時會出現，但有意志動作時會消失；顫抖頻率大概在4到7次\u002F秒",[23,5507,5508,5511],{},[26,5509,5510],{},"手部最常見；腳，嘴，吞頭，下巴較少見",[26,5512,5513,5514],{},"影響因素包括",[23,5515,5516,5519,5522],{},[26,5517,5518],{},"情緒壓力或疲勞：震顫上升",[26,5520,5521],{},"休息或意志動作：震顫下降",[26,5523,5524],{},"睡眠：震顫消失",[26,5526,5527],{},"姿勢性震顫：維持抗重力姿勢時會出現的震顫",[26,5529,5530],{},"動作性震顫：做動作時會抖，代表會嚴重影響日常生活",[26,5532,5533,5534],{},"姿勢不穩定：因為無法正常透過改變姿勢而維持平行，患者自身會採取降低重心來維持",[23,5535,5536,5547,5550],{},[26,5537,5538,5539],{},"在無障礙環境+注意力集中時，表現跟常人無異",[23,5540,5541,5544],{},[26,5542,5543],{},"單腳站或注意力分心時，不穩定表現急增",[26,5545,5546],{},"動態不穩定活動如功能性伸展，走路，轉彎等也會讓維持姿勢難度上升",[26,5548,5549],{},"帕金森氏症的跌倒是非帕金森氏症病人的兩倍，且後遺症率上升",[26,5551,5552,5553],{},"可能會影響的原因",[23,5554,5555,5558,5561,5564,5567],{},[26,5556,5557],{},"運動覺跟本體感覺缺陷",[26,5559,5560],{},"無法選擇平衡策略：基本上採單一策略，無法混用策略",[26,5562,5563],{},"意志動作時，身體會變得僵硬",[26,5565,5566],{},"當面對不穩定時，會採取異常的全身性共同收縮",[26,5568,5569],{},"其他因素包括：僵直，無力，ROM下降，失智症，沮喪，姿勢性低血壓",[26,5571,5572,5573],{},"肌肉骨骼改變",[23,5574,5575,5583,5586,5589],{},[26,5576,5577,5578],{},"全身性失能：因為缺乏運動導致",[23,5579,5580],{},[26,5581,5582],{},"導致：肌肉無力，失去柔軟度，骨質疏鬆",[26,5584,5585],{},"畸型：如駝背（最常見），或側彎",[26,5587,5588],{},"撲克臉：很少眨眼或缺乏表情；笑容也需要自主意識參與才能達成",[26,5590,5591],{},"缺少運動：整體的動作下降，如旋轉減少，自動化動作缺失，身心疲勞",{"title":180,"searchDepth":181,"depth":181,"links":5593},[],"講完了四個病徵","\u002Fimages\u002Fuploads\u002F1779503304400-136888213_p0_master1200.webp",{},"\u002Fblog\u002F基底核失調（二）帕金森氏症-i",{"title":5262,"description":5594},{"loc":5597},"blog\u002F基底核失調（二）帕金森氏症-i","D-WK0iAAokCR17cPGmoNlPZQp-q8e08c4TOyHlvwxHc",{"id":5603,"title":5604,"author":7,"body":5605,"date":5876,"description":5877,"draft":185,"edited_at":5251,"extension":186,"featured_image":5878,"meta":5879,"navigation":189,"path":5880,"pinned":185,"seo":5881,"sitemap":5882,"stem":5883,"tags":194,"__hash__":5884},"blog\u002Fblog\u002F基底核失調（一）解剖.md","基底核失調（一）解剖",{"type":9,"value":5606,"toc":5874},[5607,5613,5616,5619,5629,5632,5637,5640,5642,5645,5670,5672,5675,5696,5698,5701,5712,5714,5717,5720,5735,5738,5753,5755,5758,5774,5776,5779],[12,5608,5609],{},[5610,5611,5612],"del",{},"(哇佬欸 又一周教了快兩百頁ppt 快去世了",[12,5614,5615],{},"先來說基底核，功能是用來整合和調整大腦皮質跟視丘發出的訊息；與自主運動產生，姿勢調整控制有關",[12,5617,5618],{},"先來說背側基底核，它包含了",[23,5620,5621],{},[26,5622,5623,5624],{},"尾狀核，皮殼，蒼白球",[23,5625,5626],{},[26,5627,5628],{},"而這三者被合稱為紋狀體；而尾狀核與皮殼又被合稱為新紋狀體",[12,5630,5631],{},"而腦幹的黑質及視丘下核也一同被歸類在基底核系統中",[23,5633,5634],{},[26,5635,5636],{},"其中腦間橋腦核與基底核做連結，並負責與背景肌肉張力（完全放鬆時也會有一絲張力來維持姿勢不會軟癱）調節",[12,5638,5639],{},"腹側基底核則與邊緣系統有關",[36,5641],{},[12,5643,5644],{},"接著來講輸入路徑，由尾狀核及皮殼負責；輸入的來源包括",[23,5646,5647],{},[26,5648,5649,5650],{},"大腦皮質，視丘，腦幹",[23,5651,5652],{},[26,5653,5654,5655],{},"先從大腦皮質發出，經過紋狀體，視丘，再回傳到大腦皮質，呈現軀體形態的排列方式",[23,5656,5657],{},[26,5658,5659,5660],{},"其中，皮殼是連結感覺運動的功能；而尾狀核是連結聯合功能，包括認知功能",[23,5661,5662],{},[26,5663,5664,5665],{},"而路徑再細分分為：皮質紋狀體徑、視丘紋狀體徑、黑質紋狀體徑、腦幹紋狀體徑",[23,5666,5667],{},[26,5668,5669],{},"其中黑質紋狀體徑是由黑質的緻密部傳遞到紋狀體；而黑質的緻密部屬多巴胺神經元，當其退化，會引發帕金森氏症",[36,5671],{},[12,5673,5674],{},"然後是輸出路徑，由蒼白球及黑質組成",[23,5676,5677],{},[26,5678,5679,5680],{},"蒼白球分成內外側區；黑質分成背側緻密部，及腹側網狀部",[23,5681,5682,5685],{},[26,5683,5684],{},"路徑是由input傳入，經過尾狀核及皮殼，蒼白球及黑質(GPi和SNr)，最後輸出到視丘",[26,5686,5687,5688],{},"尾狀核及皮殼是接收區域；輸出的區域為蒼白球及黑質",[23,5689,5690,5693],{},[26,5691,5692],{},"蒼白球，傳至視丘的腹外核及腹前核的外側，最後傳到大腦皮質",[26,5694,5695],{},"黑質，傳至視丘的腹外核及腹前核的內側，最後傳到大腦皮質",[36,5697],{},[12,5699,5700],{},"基底核並沒有直接連入脊髓，但依舊會經過三條路徑對運動系統作用；",[23,5702,5703,5706,5709],{},[26,5704,5705],{},"往上：經由視丘的腹外核及腹前核，傳到了運動前皮質區（對皮質脊髓區有影響）",[26,5707,5708],{},"中層：經由四疊體上丘核，傳了四疊體脊髓徑（負責頭動作及眼睛的快速掃視動作）",[26,5710,5711],{},"往下：經由蒼白球及黑質，網狀系統，傳到了網狀脊髓徑（負責軀幹及肢體肌肉張和牽張反射）",[36,5713],{},[12,5715,5716],{},"基底核在運動感覺系統分成直接與間接路徑",[12,5718,5719],{},"先來講直接路徑",[23,5721,5722],{},[26,5723,5724,5725],{},"路線是從運動皮質出發，經過皮殼，蒼白球內側部及黑質網狀部，視丘的腹外核，回到大腦皮質",[23,5726,5727],{},[26,5728,5729,5730],{},"直接路徑是去抑制作用，用於活化視丘皮質徑",[23,5731,5732],{},[26,5733,5734],{},"功能是協助起始的意志動作（自主動作）",[12,5736,5737],{},"接下來是間接路徑",[23,5739,5740],{},[26,5741,5742,5743],{},"路徑由運動皮質出發，經由皮殼，蒼白球外側部，視丘下核，蒼白球內側部，黑質網狀部，最後傳到視丘、四疊體上丘、中腦被蓋區",[23,5744,5745],{},[26,5746,5747,5748],{},"間接路徑是抑制作用，用於抑制視丘皮質徑",[23,5749,5750],{},[26,5751,5752],{},"主要抑制其他不適合的動作（讓動作精確的做出）",[36,5754],{},[12,5756,5757],{},"那基底核路徑的傳導物質有",[23,5759,5760,5768,5771],{},[26,5761,5762,5763],{},"多巴胺——主要是黑質紋狀體徑的傳導物質；由黑質緻密部產生；",[23,5764,5765],{},[26,5766,5767],{},"同時具備興奮及抑制性：興奮直接路徑，抑制間接路徑；若失去多巴胺則會使視丘皮質徑活化降低（帕金森氏症）",[26,5769,5770],{},"乙醯膽鹼——位於尾狀核及皮殼之間小神經元的傳導物質；用於抑制多巴胺分泌",[26,5772,5773],{},"GABA——由尾狀核及皮殼合成；用於影響蒼白球及黑質；而亨丁頓氏症主因為GABA缺乏",[36,5775],{},[12,5777,5778],{},"接下來講基底核的生理",[23,5780,5781,5784,5806],{},[26,5782,5783],{},"蒼白球的放電方式是高頻電訊號，而尾狀核及皮殼則屬於低頻電訊號",[26,5785,5786,5787],{},"而基底核可以修改輸入訊息，再傳給大腦皮質",[23,5788,5789,5792,5795,5798],{},[26,5790,5791],{},"Mink：基底核輸入到大腦皮質，能活化最需要的動作的訊號，以及抑制其他不需要的動作的訊號",[26,5793,5794],{},"假如直接路徑活化過度，間接路徑活化不足：動作徐緩，運動不能",[26,5796,5797],{},"假如直接路徑活化不足，間接路徑活化過度：無關的動作（不自主運動）",[26,5799,5800,5801],{},"基底核可能會影響肌肉活動的臨床問題有",[23,5802,5803],{},[26,5804,5805],{},"僵硬，動作徐緩或動作不能，無關的動作",[26,5807,5808,5809],{},"自動化運動和姿勢有跟基底核有關",[23,5810,5811,5825,5841,5854],{},[26,5812,5813,5814],{},"自動化運動",[23,5815,5816,5819,5822],{},[26,5817,5818],{},"移除兩側紋狀體，會產生強迫或無法克制的持續；假如移除一側則不受影響",[26,5820,5821],{},"破壞蒼白球，會讓脊椎彎曲的突起面側向受傷側",[26,5823,5824],{},"給予高電量刺激，會有全身性動作模式產生",[26,5826,5827,5828],{},"姿勢控制——是在遠端動作出來之前，指導身體姿勢的調整",[23,5829,5830,5833],{},[26,5831,5832],{},"假如基底核受損，患者會採取屈曲固定姿勢",[26,5834,5835,5836],{},"帕金森氏症患者在遮眼時平衡反應表現中，靜態的平衡能力不佳，也不能做出正常的姿勢移動",[23,5837,5838],{},[26,5839,5840],{},"且運動覺跟本體感覺也會受損",[26,5842,5843,5844],{},"也跟動作起始有關",[23,5845,5846],{},[26,5847,5848,5849],{},"研究表明：帕金森氏症患者缺乏準備電位（readiness potential），而這種問題在做複雜動作時比起簡單動作較為顯著",[23,5850,5851],{},[26,5852,5853],{},"在肌電圖下幀測肌肉還沒達到50到200ms時，大腦皮質，基底核，視丘，小腦等就已經有神經活動",[26,5855,5856,5857],{},"知覺與認知問題",[23,5858,5859,5862,5865,5868,5871],{},[26,5860,5861],{},"假如基底核受損，也會對已學習動作造成影響，比對reflex影響較大",[26,5863,5864],{},"且改變反應會受影響，所以更傾向維持原本的反應",[26,5866,5867],{},"對於時間次序的記憶或學習任務有缺陷",[26,5869,5870],{},"在矯正錯誤中表現較差",[26,5872,5873],{},"整合身體兩側的動作會有缺陷",{"title":180,"searchDepth":181,"depth":181,"links":5875},[],"2026-05-22","基底核解剖到生理","\u002Fimages\u002Fuploads\u002F1779503031306-141129344_p0_master1200.webp",{},"\u002Fblog\u002F基底核失調（一）解剖",{"title":5604,"description":5877},{"loc":5880},"blog\u002F基底核失調（一）解剖","MytbMFnGxbL4IK56mozK6sUInopGmTGSxXGBdQz0HbQ",{"id":5886,"title":5887,"author":7,"body":5888,"date":6422,"description":6423,"draft":185,"edited_at":3863,"extension":186,"featured_image":6424,"meta":6425,"navigation":189,"path":6426,"pinned":185,"seo":6427,"sitemap":6428,"stem":6429,"tags":194,"__hash__":6430},"blog\u002Fblog\u002FSCI（八）相關症狀.md","SCI（八）相關症狀",{"type":9,"value":5889,"toc":6420},[5890,5893,5940,5942,5979,5981,5986,5988,6006,6008,6055,6057,6076,6078,6179,6181,6252,6254,6366,6368],[12,5891,5892],{},"接下來講相關症狀",[23,5894,5895],{},[26,5896,5897,5898,5900,5901],{},"脊髓休克——脊髓因外傷而發生的暫時性現象；會在病灶以下失去所有功能",[16,5899],{},"\n通常需要數周才能完全恢復",[23,5902,5903,5911,5924,5932],{},[26,5904,5905,5906],{},"完全癱瘓在受傷1天內",[23,5907,5908],{},[26,5909,5910],{},"無反射，並癱瘓",[26,5912,5913,5914],{},"反射恢復在受傷後1到3天",[23,5915,5916],{},[26,5917,5918,5919],{},"反射剛開始回來，延遲出現的皮膚蹠反射和球海綿體反射",[23,5920,5921],{},[26,5922,5923],{},"球海綿體反射——當捏壓龜頭或輕拉導尿管時，若引起肛門括約肌收縮，則代表已進入這一期",[26,5925,5926,5927],{},"反射漸增在受傷後一到四周",[23,5928,5929],{},[26,5930,5931],{},"深腱反射會重新回來，開始出現痙攣跟反射增強",[26,5933,5934,5935],{},"反射過強在受傷後一到六個月",[23,5936,5937],{},[26,5938,5939],{},"痙攣和反射在這時期最強",[36,5941],{},[23,5943,5944,5968],{},[26,5945,5946,5947],{},"自主運動功能——下行運動神經路徑、前角細胞或神根經受損，都會導致自主肌肉癱瘓",[23,5948,5949,5957,5965],{},[26,5950,5951,5952],{},"LMN損傷：因前角細胞或其周邊神經受損而引起；臨床表現為無張力，肌肉鬆馳",[23,5953,5954],{},[26,5955,5956],{},"始於脊髓，由此發送訊號致肌肉與腺體",[26,5958,5959,5960],{},"UMN損傷：因大腦或下運動神經路徑受損而引起；臨床表現為高張力，肌肉痙攣",[23,5961,5962],{},[26,5963,5964],{},"始於大腦皮質，並一路往下傳到脊髓",[26,5966,5967],{},"多數的脊髓損傷：是UMN和LMN的混合型；因為在受傷的level會有白質和灰質同時被影響",[26,5969,5970,5971],{},"肌肉張力——在脊髓休克時呈無反射狀態，會隨著時間恢復而讓反射功能變強達到痙攣狀態；好發在頸椎或上胸椎的不完全性損傷病人（AIS B或C）",[23,5972,5973,5976],{},[26,5974,5975],{},"LMN損傷：遲緩性癱瘓、DTR減弱或消失",[26,5977,5978],{},"UMN損傷：痙攣，陣攣，DTR增強或過強，肌肉痙攣",[36,5980],{},[23,5982,5983],{},[26,5984,5985],{},"ROM下降——主要因為卧床過久或被固定不動導致，會影響患者的功能性活動，或有外觀上的問題",[36,5987],{},[23,5989,5990],{},[26,5991,5992,5993],{},"骨質疏鬆症及骨折——骨質疏鬆症會發生在病灶區域以下",[23,5994,5995,6003],{},[26,5996,5997,5998],{},"主要原因為：靜脈血流滯留，肌肉活動能力喪失，失去負重能力的結果",[23,5999,6000],{},[26,6001,6002],{},"靜脈血液滯留會增加血球細胞分泌化學介質，這些介質會活化破骨細胞",[26,6004,6005],{},"危險因子有：運動功能完全性損傷，白人，受傷後時間較長，女性",[36,6007],{},[23,6009,6010],{},[26,6011,6012,6013],{},"疼痛(pain)——疼痛基本上分為兩類",[23,6014,6015,6036],{},[26,6016,6017,6018],{},"傷害感受性疼痛：原因由結構性功能障礙所引起；類型包括軀體痛（肌肉骨骼痛）和內臟痛",[23,6019,6020,6028],{},[26,6021,6022,6023],{},"肌肉骨骼痛：源於肌肉骨骼裡的感受器被刺激導致；病因有外傷，機械性壓力，發炎反應以及肌肉痙攣",[23,6024,6025],{},[26,6026,6027],{},"慢性疼痛通常是overuse所引起",[26,6029,6030,6031],{},"內臟痛：源於臟器中的感受器受到刺激所引起；",[23,6032,6033],{},[26,6034,6035],{},"下半身癱瘓的患者感覺與受傷前一致；四肢癱瘓的患者則感覺到會有模糊不清的不適感",[26,6037,6038,6039],{},"神經痛：原因源於脊髓本身的損傷，病因尚未明確；可以是持續性或陣發性，也可能發生在受傷level以下、同一level、受傷level以上",[23,6040,6041],{},[26,6042,6043,6044],{},"可以是CNS或者PNS；因為沒有單一治療可以對神經痛造成適合有效的方法，所以臨床上具挑戰性",[23,6045,6046,6049,6052],{},[26,6047,6048],{},"受傷在病灶以下的：很有可能是脊髓本身受損引起；可以是持續或波動（時大時小的痛）性的；可能會有異樣感或痛覺過敏問題",[26,6050,6051],{},"受傷在病灶那裡的：可能由神經根或脊髓受損或受迫引起",[26,6053,6054],{},"受傷在病灶以上的：可能由神經夾擠或周邊神經被壓迫引起",[36,6056],{},[23,6058,6059],{},[26,6060,6061,6062],{},"異位性骨化——是指在肌肉或其他組織長出骨頭的現象；有16到53%的SCI病人會有這問題；常發生在髖，膝，肘",[23,6063,6064,6067,6070,6073],{},[26,6065,6066],{},"一到四個月之後會出現，並且在12到18個月會成熟",[26,6068,6069],{},"假如是小範圍的通常不會有大問題；但假如是大問題的會有發炎反應的所有症狀（發紅，腫脹，發熱，ROM下降）",[26,6071,6072],{},"基本上是影響ROM，最後導致受限甚至失能，產生壓瘡等",[26,6074,6075],{},"危險因子：高齡，完全性SCI，男性，痙攣狀態",[36,6077],{},[23,6079,6080],{},[26,6081,6082,6083,6085,6086,6090,6091,6093,6094],{},"心肺功能問題——交感神經傳出源於T1到L2或L3；副交感神經就從腦幹的迷走神經傳出",[16,6084],{},"\n那假如SCI出問題，變成只剩迷走神經在運作，所以心跳會變慢（在",[75,6087,6089],{"className":6088},[78,79],"T6以上","損傷顯著）",[16,6092],{},"\n數周內會緩解",[23,6095,6096,6104,6115,6118,6129,6152,6166],{},[26,6097,6098,6099],{},"神經性休克（最常見）——會危害生命，發生在交感神經突然喪失，且副交感良好，導致自律神經不穩定",[23,6100,6101],{},[26,6102,6103],{},"會有因血管擴張引起的低血壓，心跳過緩性心律不整，以及體溫調節失調等問題",[26,6105,6106,6107],{},"姿勢性低血壓——收縮壓突然下降20mmHg，舒張壓下降10mmHg",[23,6108,6109,6112],{},[26,6110,6111],{},"所有SCI患者都會出現，但T6以上損傷更常見",[26,6113,6114],{},"處置方式：讓患者躺下或將傾斜床躺平，穿戴過膝彈性襪促進下肢靜脈血回流",[26,6116,6117],{},"血管迷走神經反應：主動的周邊血管擴張，血壓突然下降，心率減慢",[26,6119,6120,6121],{},"自主神經反射異常：血壓突然上升，心跳過緩，頭痛，臉部潮紅，受傷部位以上出汗，焦慮",[23,6122,6123,6126],{},[26,6124,6125],{},"主要是在T6或以上損傷的患者，損傷後六個月會出現",[26,6127,6128],{},"處置：假如是躺著發生血壓上升，那就立刻讓患者坐起來；鬆開衣著或束縛器具",[26,6130,6131,6132],{},"深層靜脈血栓：常見於急性期",[23,6133,6134,6137,6140,6143,6146,6149],{},[26,6135,6136],{},"常見於周邊血管擴張，下肢肌力喪失或減少，久臥",[26,6138,6139],{},"風險因子有：男性，下肢癱瘓，完全性損傷",[26,6141,6142],{},"也會有肺栓塞：最初數周會有",[26,6144,6145],{},"臨床癥狀有：腿步疼痛，小腿壓痛，小腿發熱",[26,6147,6148],{},"預防：吃抗凝血藥物，穿壓力襪；嚴重時可置入靜脈過濾器；也可以做主動\u002F被動運動（除非是禁忌症）",[26,6150,6151],{},"發現了就要立即處置",[26,6153,6154,6155],{},"體溫調節問題：其中一條就是靠自主神經管理，SCI會破壞下視丘跟脊髓的傳導，導致失常",[23,6156,6157,6160,6163],{},[26,6158,6159],{},"常見有失溫和體溫過高",[26,6161,6162],{},"受傷節段以下溫度調節喪失",[26,6164,6165],{},"要看是完全性還是不完全性損傷；假如是C level的完全性損傷最嚴重",[26,6167,5225,6168],{},[23,6169,6170,6173,6176],{},[26,6171,6172],{},"有2.2%到22%的SCI在急性期患有十二指腸壓力性潰瘍",[26,6174,6175],{},"腸胃道出血在受傷後一個月會有",[26,6177,6178],{},"在脊髓休克期間可能會有胃擴張問題，會影響橫膈功能，嘔吐或吸入性肺炎",[36,6180],{},[23,6182,6183,6210],{},[26,6184,6185,6186],{},"男性性功能問題",[23,6187,6188,6191,6202],{},[26,6189,6190],{},"分為UMN(S2~S4正常，受傷部位在其之上)LMN(薦髓功能被破壞)",[26,6192,6193,6194],{},"勃起障礙：UMN受傷患者勃起功能較佳，不完全性損傷較佳",[23,6195,6196,6199],{},[26,6197,6198],{},"反射性勃起：需要有完整反射弧S2~S4",[26,6200,6201],{},"心因性勃起：由大腦皮層調節",[26,6203,6204,6205],{},"射精障礙：通常較少發生，LMN射精功能保留較佳，不完全性損傷較佳",[23,6206,6207],{},[26,6208,6209],{},"原因主要是因為精子生成功能受損，其次才是因為射精反射失能",[26,6211,6212,6213],{},"女性性功能問題",[23,6214,6215,6226,6229,6232,6235,6249],{},[26,6216,6217,6218],{},"UMN：反射弧保留完整",[23,6219,6220,6223],{},[26,6221,6222],{},"性奮生理反應如陰道潤滑是靠反射性刺激產生",[26,6224,6225],{},"心因性反應喪失",[26,6227,6228],{},"LMN：對 就剛剛的反過來",[26,6230,6231],{},"月經問題：在受傷時會停止，在五到十二個月後會恢復",[26,6233,6234],{},"生育能力：受孕潛力不影響",[26,6236,6237,6238],{},"懷孕問題：在密切關注下是可行的",[23,6239,6240,6243,6246],{},[26,6241,6242],{},"會有一些併發症如泌尿道感染，貧血，深層靜脈血栓",[26,6244,6245],{},"T10以上損傷患者會在無感覺情況下進入臨盤",[26,6247,6248],{},"T7以上患者會在分娩時產生宮縮（危）",[26,6250,6251],{},"性功能問題要跟患者密切交流，提供轉介",[36,6253],{},[23,6255,6256],{},[26,6257,6258,6259],{},"泌尿問題",[23,6260,6261,6275,6283,6291,6322,6341],{},[26,6262,6263,6264],{},"控制",[23,6265,6266,6269,6272],{},[26,6267,6268],{},"交感神經：逼尿肌被抑制，放鬆；內括約肌被興奮，收縮（忍尿）",[26,6270,6271],{},"副交感神經：逼尿肌被興奮，收縮（排尿）",[26,6273,6274],{},"軀體神經：外括約肌和尿道周圍骨盆底肌",[26,6276,6277,6278],{},"排尿反射",[23,6279,6280],{},[26,6281,6282],{},"當膀胱容量超過200到400時，就會有神經衝動傳到脊髓，並且副交感神經會回傳回膀胱，引發逼尿肌收縮",[26,6284,6285,6286],{},"那忍尿也是按照上面說的，排尿反射會被抑制，外括約肌和尿道周圍骨盆底肌會收縮",[23,6287,6288],{},[26,6289,6290],{},"排尿是外括約肌和尿道周圍骨盆底肌放鬆才能達成",[26,6292,6293,6294],{},"功能保留",[23,6295,6296],{},[26,6297,6298,6299],{},"薦椎的反射弧以及它跟大腦的連結（完全性與不完全性損傷），決定膀胱保留能力",[23,6300,6301],{},[26,6302,6303,6304],{},"分類",[23,6305,6306,6314],{},[26,6307,6308,6309],{},"無反射性\u002F癱瘓型神經性膀胱：LMN受損",[23,6310,6311],{},[26,6312,6313],{},"脊髓休克期間，受傷害3~6周後，無反射活動（尿滯留）",[26,6315,6316,6317],{},"反射性\u002F痙攣型神經性膀胱：UMN受損",[23,6318,6319],{},[26,6320,6321],{},"因為失去了大腦抑制的路徑（尿失禁）",[26,6323,6324,6325],{},"逼尿肌外括約肌共濟不全",[23,6326,6327,6330,6333],{},[26,6328,6329],{},"就是一起收縮，導致尿液被卡住",[26,6331,6332],{},"許多反射性患者會有這個問題",[26,6334,6335,6336],{},"供發症有尿道感染，敗血症，腎衰竭等",[23,6337,6338],{},[26,6339,6340],{},"尿道感染也是最主要死因",[26,6342,6343,6344],{},"膀胱管理：就是定期排空尿液",[23,6345,6346,6349],{},[26,6347,6348],{},"早期設置導尿管",[26,6350,6351,6352],{},"亞急性處置包括",[23,6353,6354,6357,6360,6363],{},[26,6355,6356],{},"留置導尿管：最多30天",[26,6358,6359],{},"間歇性導尿：每四到六小時一次",[26,6361,6362],{},"壓腹排尿與閉氣用力法：適合弛緩型患者",[26,6364,6365],{},"反射型的反射性膀胱排尿",[36,6367],{},[23,6369,6370],{},[26,6371,6372,6373],{},"皮膚照護",[23,6374,6375],{},[26,6376,6377,6378],{},"壓瘡：主要因為皮膚血液循環問題，以及運動功能問題引起",[23,6379,6380,6383,6386,6403],{},[26,6381,6382],{},"是常見的併發症，85％的SCI會有這問題",[26,6384,6385],{},"物理力量為壓力（主要）和剪力",[26,6387,6388,6389],{},"分期",[23,6390,6391,6394,6397,6400],{},[26,6392,6393],{},"第一期：只有表皮層被害，局部的不可壓白髮紅，皮膚完整",[26,6395,6396],{},"第二期：表皮層和真皮層被害，已經有表淺開放性傷口",[26,6398,6399],{},"第三層：表皮層真皮層和皮下組織被害，皮下脂肪已經能看見，肌肉骨骼還沒看見",[26,6401,6402],{},"第四期：能看見肌肉骨骼",[26,6404,6405,6406],{},"預防",[23,6407,6408,6411,6414,6417],{},[26,6409,6410],{},"墊上運動：部位是肘，手段是使用護肘",[26,6412,6413],{},"移位：部位是坐骨結節，薦椎，尾骨，手段是穿著合適的衣服，避免鈍傷或剪力",[26,6415,6416],{},"輪椅：部位是坐骨結節、薦椎、手掌，手段是調整合適的輪椅，使用坐墊，戴手套，15到20分鐘減壓一次",[26,6418,6419],{},"步行：部位足跟後側、內踝、與輔具接觸的區域，手段是穿著合適的輔木",{"title":180,"searchDepth":181,"depth":181,"links":6421},[],"2026-05-20","講義第五份","\u002Fimages\u002Fuploads\u002F1779724865862-103686804_p0.webp",{},"\u002Fblog\u002FSCI（八）相關症狀",{"title":5887,"description":6423},{"loc":6426},"blog\u002FSCI（八）相關症狀","brwjHj-eNSxaTTuci41hkYSp7lhiKLfkRId-af0aOac",{"id":6432,"title":6433,"author":7,"body":6434,"date":6676,"description":6677,"draft":185,"edited_at":6422,"extension":186,"featured_image":6678,"meta":6679,"navigation":189,"path":6680,"pinned":185,"seo":6681,"sitemap":6682,"stem":6683,"tags":194,"__hash__":6684},"blog\u002Fblog\u002FSCI（七）感覺和運動檢查.md","SCI（七）感覺和運動檢查",{"type":9,"value":6435,"toc":6674},[6436,6439,6450,6453,6458,6467,6478,6481,6495,6497,6500,6503,6514,6516,6525,6527,6530,6616,6618,6642],[12,6437,6438],{},"感覺檢查有三個個是必要的，分別是",[23,6440,6441,6444,6447],{},[26,6442,6443],{},"針刺覺（鈍感和尖銳）——評估脊髓丘腦徑",[26,6445,6446],{},"輕觸覺——評估DCML",[26,6448,6449],{},"肛門深部壓力覺感知——如果上兩者都沒有感覺，就來測這個，用來分辨是否完全性損傷",[12,6451,6452],{},"而也有一個是非必要的，叫關節位置覺，測的是本體覺",[12,6454,6455],{},[1930,6456],{"alt":1932,"src":6457},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1779067725003-Screenshot_2026-05-18_092817.png",[12,6459,6460,6461,6463,6464,6466],{},"先來說針刺覺，我們會利用迴紋針來測，用尖端跟鈍端來測",[16,6462],{},"\n跟臉部對比，因為臉部是由三叉神經所管控，一般都預設患者的三叉神經是正常的，除非患者原本就有腦神經病症就不會拿來做對比",[16,6465],{},"\n我們會先在臉部給患者一次正常的感覺做為標準，然後閉上雙眼去測其他key point，會說以下語句",[23,6468,6469,6472,6475],{},[26,6470,6471],{},"有沒有感覺",[26,6473,6474],{},"跟臉部感覺是否相同",[26,6476,6477],{},"你分不分得清這是鈍感還是尖感",[12,6479,6480],{},"那得分標準就長這樣",[23,6482,6483,6486,6489,6492],{},[26,6484,6485],{},"0分——無法在相應的key point感覺到任何東西",[26,6487,6488],{},"1分——在相應的key point有感覺到甚麼但跟臉部比起來有差異",[26,6490,6491],{},"2分——在相應key point感覺到的跟臉部感覺到的一致",[26,6493,6494],{},"NT(無法測試)——臉部本身就有受損，無法作為標準評測",[36,6496],{},[12,6498,6499],{},"接下來是輕觸覺，利用棉花棒棉花那一端來觸碰，範圍一次不超過1cm，對比方法和問的問題與針刺覺一致（除了尖端鈍端那個不用問）",[12,6501,6502],{},"得分標準為",[23,6504,6505,6507,6509,6511],{},[26,6506,6485],{},[26,6508,6488],{},[26,6510,6491],{},[26,6512,6513],{},"NT——臉部本身就有受損，無法作為標準評測",[36,6515],{},[12,6517,6518,6519,6521,6522,6524],{},"最後是DAP(肛門深部壓力覺)，若當上兩者沒有感覺，則利用DAP來檢查有沒有殘留感覺，只有DAP在，則代表不完全性損傷",[16,6520],{},"\n醫師需帶手套，然後伸進直腸，在直腸壁上施加壓力，並詢問患者有無感覺",[16,6523],{},"\n評分只有存在與缺失兩種",[36,6526],{},[12,6528,6529],{},"也標上28個Key point來好了",[23,6531,6532,6535,6538,6541,6544,6547,6550,6553,6556,6559,6562,6565,6568,6571,6574,6577,6580,6583,6586,6589,6592,6595,6598,6601,6604,6607,6610,6613],{},[26,6533,6534],{},"C2——耳後3cm",[26,6536,6537],{},"C3——鎖骨中線上窩",[26,6539,6540],{},"C4——AC Joint（肩鎖關節）",[26,6542,6543],{},"C5——肘窩橈側",[26,6545,6546],{},"C6——背側的大拇指近端指骨",[26,6548,6549],{},"C7——背側的中指近端指骨",[26,6551,6552],{},"C8——背側的小指近端指骨",[26,6554,6555],{},"T1——肘窩尺側",[26,6557,6558],{},"T2——腋窩最中間",[26,6560,6561],{},"T3——在鎖骨中線找到第三肋間",[26,6563,6564],{},"T4——在鎖骨中線找到第四肋間（與乳頭同一水平線上）",[26,6566,6567],{},"T5——T4與T6中間",[26,6569,6570],{},"T6——劍突水平線的鎖骨中線上",[26,6572,6573],{},"T7——劍突到T8中間的鎖骨中線上",[26,6575,6576],{},"T8——劍突與肚臍的中間的鎖骨中線上",[26,6578,6579],{},"T9——T8跟肚臍的鎖骨中線上",[26,6581,6582],{},"T10——肚臍水平線的鎖骨中線上",[26,6584,6585],{},"T11——T10跟T12中間",[26,6587,6588],{},"T12——在腹股溝韌帶中點",[26,6590,6591],{},"L1——在T12跟L2中間",[26,6593,6594],{},"L2——大腿前內側中間，在膝蓋內側垂直線上\u002FT12跟L3中間（？）",[26,6596,6597],{},"L3——在膝蓋內側",[26,6599,6600],{},"L4——在內踝",[26,6602,6603],{},"L5——在足背第三蹠趾關節",[26,6605,6606],{},"S1——在足跟外側",[26,6608,6609],{},"S2——在膝窩正中央",[26,6611,6612],{},"S3——在坐骨結節",[26,6614,6615],{},"S4\u002FS5——肛門周邊區域",[36,6617],{},[12,6619,6620,6621,6625,6626,6630,6632,6633,6635,6636,6638,6639,6641],{},"最後是十條Key muscle，而這個測試",[75,6622,6624],{"className":6623},[78,5387],"不允許評分有+跟-","這兩個表示，直接標出012345分，",[75,6627,6629],{"className":6628},[78,5387],"而且必須強制在supine位置",[16,6631],{},"\n是MMT的測試方法跟評分標準",[16,6634],{},"\n而還有另外兩個得分",[16,6637],{},"\n5*是代表，假如不存在某些會抑制肌力干擾因素，肌肉本身能正常展現抗阻力能力",[16,6640],{},"\nNT是代表，目前患者沒辦法可靠地用力而做出肌肉相應的動作，例如打石膏的限制狀態中，或已截肢",[23,6643,6644,6647,6650,6653,6656,6659,6662,6665,6668,6671],{},[26,6645,6646],{},"C5——Elbow flexors，肱二頭肌和肱肌",[26,6648,6649],{},"C6——Wrist extensors，橈側伸腕長肌和橈側伸腕短肌",[26,6651,6652],{},"C7——Elbow extensors，肱三頭肌",[26,6654,6655],{},"C8——Long finger flexors，屈指深肌",[26,6657,6658],{},"T1——Small finger abductor，外展小指肌",[26,6660,6661],{},"L2——Hip flexors，髂腰肌",[26,6663,6664],{},"L3——Knee Extensors，股四頭肌",[26,6666,6667],{},"L4——Ankle dorsiflexors，脛前肌",[26,6669,6670],{},"L5——Long toe extensors，伸拇趾長肌",[26,6672,6673],{},"S1——Ankle planterflexors，腓腸肌和比目魚肌",{"title":180,"searchDepth":181,"depth":181,"links":6675},[],"2026-05-18","感覺跟運動檢查","\u002Fimages\u002Fuploads\u002F1779164485654-107860162_p0_master1200.webp",{},"\u002Fblog\u002FSCI（七）感覺和運動檢查",{"title":6433,"description":6677},{"loc":6680},"blog\u002FSCI（七）感覺和運動檢查","E31jha7Hmtl2sPO59r-GF2JGFeenPgCgbWqmPLoAeCQ",{"id":6686,"title":6687,"author":7,"body":6688,"date":6841,"description":6842,"draft":185,"edited_at":6676,"extension":186,"featured_image":6843,"meta":6844,"navigation":189,"path":6845,"pinned":185,"seo":6846,"sitemap":6847,"stem":6848,"tags":194,"__hash__":6849},"blog\u002Fblog\u002FSCI（六）分類--ASIA.md","SCI（六）分類--ASIA",{"type":9,"value":6689,"toc":6839},[6690,6693,6696,6716,6733,6751,6773,6776,6779,6810,6836],[12,6691,6692],{},"呵呵，終於來到SCI的大魔王了，這裡可是要記好區分，我記得我那一年跑台也要填這個的筆試，老師在跑台時也會出題問ASIA跟後面會教的處置方式",[12,6694,6695],{},"那ASIA有六個步驟要看",[3603,6697,6698,6701,6704,6707,6710,6713],{},[26,6699,6700],{},"感覺level——左右兩邊的感覺保留層級到多高",[26,6702,6703],{},"運動level——左右兩邊的運動功能保留層級到多高",[26,6705,6706],{},"NLI——感覺跟運動level同時保留的最低節段",[26,6708,6709],{},"完全或不完全性損傷——判定S4，S5功能還有沒有存在",[26,6711,6712],{},"AIS分級——受傷分級",[26,6714,6715],{},"ZPP——部份保留",[12,6717,6718,6719,6723,6724,6726,6727,6729],{},"先來看感覺保留的level，身體左右各有28個Key point，分別要去測",[75,6720,6722],{"className":6721},[5387,78],"針刺覺跟輕觸覺","，最高兩分，左右兩邊分開，所以兩種感覺一共是28x2x2=112",[16,6725],{},"\n兩分是指感覺正常；一分是指感覺比較不太一樣（有感覺但異常），一般都是跟臉部感覺做比較；零分就是沒有",[16,6728],{},[75,6730,6732],{"className":6731},[78,1149],"要兩邊都有正常的感覺保留level才是最低脊髓level",[12,6734,6735,6736,6738,6739,6743,6744,6748,6750],{},"接下來看運動保留的level，而分數是靠MMT來測的(死去的記憶該回來了哦)，3分就是抗重力不能抗阻力那個",[16,6737],{},"\n算法是",[75,6740,6742],{"className":6741},[78,5387],"最低level有3分，3分以上的level全都是5分","，那3分的那個level才是最低運動保留的level",[75,6745,6747],{"className":6746},[78,1149],"（4分也不行哦，一定要5分）",[16,6749],{},"\n左右各有10條肌肉要測，測的是C5到T1，和L2到S1，這十條key muscle",[12,6752,6753,6754,6756,6757,6762,6764,6765,6767,6768,6772],{},"那像剛剛說的，只有十條肌肉能夠去測而這十條對應的level是C5到T1（C5 C6 C7 C8 T1）和L2到S1（L2 L3 L4 L5 S1）",[16,6755],{},"\n而",[75,6758,6761],{"className":6759},[78,6760],"bg-green-3","其他沒有提及的就是沒有關鍵肌肉能夠測量，就去看感覺level保留到哪",[16,6763],{},"\n而這就是第三點要看的NLI",[16,6766],{},"\n看",[75,6769,6771],{"className":6770},[78,5387],"運動跟感覺都有，左右兩邊都存在的","，最低level（完全性損傷只有25%到30%的患者感覺跟運動都在同一個level）",[12,6774,6775],{},"然後就是來看這位病人是完全性還是不完全性損傷，看的是S4 S5還有沒有絲毫感覺或運動能力（DAP和VAC，肛門深部感覺和肛門自主收縮）",[12,6777,6778],{},"接下來是判定ABCDE分級了",[23,6780,6781,6784,6801,6804,6807],{},[26,6782,6783],{},"A——完全性損傷，無DAPVAC也沒有S4S5感覺保留",[26,6785,6786,6787],{},"B——不完全性損傷，無VAC但有DAP\u002FS4S5感覺保留任一",[23,6788,6789],{},[26,6790,6791,6792,6796,6797],{},"假如運動平面以下",[75,6793,6795],{"className":6794},[78],"大於","三個level保留運動功能，則從B升格到C，",[75,6798,6800],{"className":6799},[78],"等於三個level不予升格",[26,6802,6803],{},"C——不完全性損傷，有DAP\u002FVAC且NLI以下的Key muscle少於一半到達3分或以上",[26,6805,6806],{},"D——不完全性損傷，有DAP\u002FVAC且NLI以下的Key muscle大於一半到達3分或以上",[26,6808,6809],{},"E——正常人",[12,6811,6812,6813,6815,6816,6818,6819,6821,6822,6824,6825,6827,6828,6830,6831,6835],{},"最後就是來看ZPP部份保留的部份，看他左右兩邊的感覺跟運動還有沒有一些保留",[16,6814],{},"\n假如",[16,6817],{},"\n感覺功能是S4S5沒有感覺加上DAP是沒有，就要去寫",[16,6820],{},"\n運動功能是沒有VAC就要寫",[16,6823],{},"\n寫的是看左右兩邊感覺跟運動，從下往上找，有一個不是0分的就去填他上去，找最低的",[16,6826],{},"\n然後再看",[16,6829],{},"\n假如是在",[75,6832,6834],{"className":6833},[78,5387],"沒有Key muscle的地方，有感覺保留(針刺覺和輕觸覺同時)最低level兩分的","，那ZPP motor就填在那個level",[12,6837,6838],{},"（這裡老師是有給例題做的，先做做看，我是覺得也有太多例外跟特點要記了，反而覺得例題不夠寫）",{"title":180,"searchDepth":181,"depth":181,"links":6840},[],"2026-05-16","ASIA全部","\u002Fimages\u002Fuploads\u002F1779066022115-94972865_p0_master1200.webp",{},"\u002Fblog\u002FSCI（六）分類--ASIA",{"title":6687,"description":6842},{"loc":6845},"blog\u002FSCI（六）分類--ASIA","X3eimk7Vs9dMJmKcXl_0QWwNARWQZCpH4Ot3iiJZKzY",{"id":6851,"title":6852,"author":7,"body":6853,"date":6934,"description":6935,"draft":185,"edited_at":6841,"extension":186,"featured_image":6936,"meta":6937,"navigation":189,"path":6938,"pinned":185,"seo":6939,"sitemap":6940,"stem":6941,"tags":194,"__hash__":6942},"blog\u002Fblog\u002FSCI（五）類型--病症.md","SCI（五）分類--病症",{"type":9,"value":6854,"toc":6932},[6855,6858,6863,6869,6889,6898,6903,6909,6917,6920,6926],[12,6856,6857],{},"先從受傷區域來看病症分類，那先從這裡看脊椎的神經傳導區域",[12,6859,6860],{},[1930,6861],{"alt":1932,"src":6862},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1778829557684-Screenshot_2026-05-15_151736.png",[12,6864,6865,6866,6868],{},"Brown-Sequard Syndrome——他是脊髓半側損傷的病症(半切症候群)",[16,6867],{},"\n真正的半側損傷很稀有，基本上都是不完全的半側損傷，意思是一邊比另一邊嚴重",[23,6870,6871],{},[26,6872,6873,6874,6878,6879,6883,6884],{},"患傷側會有",[75,6875,6877],{"className":6876},[78,79],"運動功能以及本體覺","受損（兩者進到脊髓沒交叉，都在延腦處交叉）；健側會有",[75,6880,6882],{"className":6881},[78,79],"針刺覺（痛覺）跟溫度覺","受損（進去脊椎後直接交叉）",[23,6885,6886],{},[26,6887,6888],{},"而在復健過程中，患者的神經功能是有機會恢復的，並擁有獨立自理的能力",[12,6890,6891,6892,6894,6895,6897],{},"前髓症候群——前側跟前外側受損，僅有後側是被保留的；",[16,6893],{},"\n基本上由屈曲型的淚滴狀骨折或者爆裂性骨折造成",[16,6896],{},"\n本體覺會保留（因為DCML在後側），而痛溫覺及運動功能會有不同程度的喪失",[23,6899,6900],{},[26,6901,6902],{},"前脊髓動脈症候群：畢竟前脊髓動脈管前2\u002F3區域的供血，是前髓症候群的一種，會導致受傷level以下雙側偏癱跟痛覺喪失，但本體覺保留（原因同上）",[12,6904,6905,6906,6908],{},"中髓症候群——脊髓正中間，幾乎只發生在頸椎節段；最常見是伸展型損傷",[16,6907],{},"\n上肢影響程度比下肢大，薦椎也會保留（如上圖，上肢在內側，下肢在外側）",[23,6910,6911,6914],{},[26,6912,6913],{},"老人跌倒的SCI重要原因是因為椎管狹窄或有骨刺，頸部後仰，從而刺傷脊髓",[26,6915,6916],{},"出血或者壞死導致（受傷在前脊髓動脈或者微動脈）",[12,6918,6919],{},"後髓症候群——就是後1\u002F3受損，只有本體覺喪失，其他功能正常；十分稀有",[12,6921,6922,6923,6925],{},"馬尾症候群——就是脊髓圓錐末端的腰薦椎神經根受損，基本上屬不完全性損傷",[16,6924],{},"\n是下運動神經元損傷（周邊神經受損）；臨床表徵有下肢無力，失去排泄反射，鞍區麻木（騎在馬鞍上的接觸馬鞍的區域）",[12,6927,6928,6929,6931],{},"脊髓圓錐症候群——這次就是傷到脊髓圓錐了，不過他的形態是LMN（無力）和UMN（反射變強）的混合型；",[16,6930],{},"\n與剛剛一樣下肢無力，失去排泄反射；不過有時候會保留薦椎反射",{"title":180,"searchDepth":181,"depth":181,"links":6933},[],"2026-05-15","損傷的部位跟病症差別","\u002Fimages\u002Fuploads\u002F1778897957991-121218399_p0_master1200.webp",{},"\u002Fblog\u002FSCI（五）類型--病症",{"title":6852,"description":6935},{"loc":6938},"blog\u002FSCI（五）類型--病症","p4r5BzeJFYZ3MOk4mAb5kuYAYVJnluvG6FeNSlSDc8k",{"id":6944,"title":6945,"author":6946,"body":6947,"date":9366,"description":9367,"draft":185,"edited_at":9366,"extension":186,"featured_image":194,"meta":9368,"navigation":189,"path":9369,"pinned":185,"seo":9370,"sitemap":9371,"stem":9372,"tags":194,"__hash__":9373},"blog\u002Fblog\u002FQueues.md","Queues","chinono",{"type":9,"value":6948,"toc":9335},[6949,6960,6963,6968,6971,6981,6984,6996,7003,7010,7014,7025,7029,7032,7077,7081,7084,7143,7165,7170,7181,7184,7188,7198,7201,7211,7223,7232,7244,7247,7253,7263,7266,7273,7280,7303,7306,7317,7330,7341,7344,7350,7363,7371,7374,7611,7614,7658,7662,7806,7817,7821,7824,7827,7834,7846,7860,7864,7875,7878,7885,7891,7894,7907,7914,8036,8067,8071,8074,8109,8113,8496,8502,8506,8513,8523,8770,8779,9004,9007,9013,9024,9064,9068,9071,9109,9112,9145,9149,9160,9189,9314,9317,9321,9324,9331],[12,6950,6951,6952,6955,6956,6959],{},"If you've ever stood in line at a bubble tea shop, you already understand a ",[106,6953,6954],{},"queue",". That's basically the whole idea — but in code, this simple concept turns into one of the most useful data structures you'll meet. In this post we'll walk through what a queue is, how to implement one in Java, and then level it up into a ",[106,6957,6958],{},"priority queue",".",[12,6961,6962],{},"This is written for someone seeing it for the first time, so I won't assume much beyond \"you've written a few Java classes.\"",[6964,6965,6967],"h2",{"id":6966},"_1-the-mental-model","1. The mental model",[12,6969,6970],{},"Picture a checkout line at the supermarket:",[6972,6973,6978],"pre",{"className":6974,"code":6976,"language":6977},[6975],"language-text","  front                                 rear\n   ↓                                     ↓\n [ A ] ← [ B ] ← [ C ] ← [ D ] ← (new person joins here)\n   ↑\n (cashier serves whoever is here next)\n","text",[6979,6980,6976],"code",{"__ignoreMap":180},[12,6982,6983],{},"Two rules govern this line:",[23,6985,6986,6991],{},[26,6987,6988],{},[106,6989,6990],{},"New people join at the rear.",[26,6992,6993],{},[106,6994,6995],{},"The cashier always serves whoever is at the front.",[12,6997,6998,6999,7002],{},"That's it. That's a queue. And the rule it follows has a famous three-letter name: ",[106,7000,7001],{},"FIFO — First In, First Out."," The first person in line is the first person out.",[12,7004,7005,7006,7009],{},"Compare this to a ",[106,7007,7008],{},"stack",", which is LIFO (Last In, First Out) — like a stack of plates where you only take the top one. Same family of \"linear\" structures, different access rule.",[6964,7011,7013],{"id":7012},"_2-where-queues-live-in-java","2. Where queues live in Java",[12,7015,7016,7017,7020,7021,7024],{},"Before we build one, let's see where ",[6979,7018,7019],{},"Queue"," sits in Java's official ",[106,7022,7023],{},"Collections Framework",":",[7026,7027],"custom-html",{"code":7028},"\u003C!DOCTYPE html>\n\u003Chtml lang=\"en\">\n\u003Chead>\n\u003Cmeta charset=\"UTF-8\">\n\u003Ctitle>Java Collection Framework Hierarchy\u003C\u002Ftitle>\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.googleapis.com\">\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.gstatic.com\" crossorigin>\n\u003Clink href=\"https:\u002F\u002Ffonts.googleapis.com\u002Fcss2?family=Fraunces:opsz,wght@9..144,400;9..144,500;9..144,600;9..144,700&family=JetBrains+Mono:wght@400;500;600&display=swap\" rel=\"stylesheet\">\n\u003Cstyle>\n  :root {\n    --bg: #f5efe4;\n    --paper: #fbf6ec;\n    --ink: #1a1612;\n    --ink-soft: #6b6258;\n    --rule: #d8cfc0;\n    --accent: #c44536;\n    --accent-soft: #e8b9b3;\n    --teal: #2c5f5d;\n    --gold: #b08e3c;\n  }\n  * { box-sizing: border-box; margin: 0; padding: 0; }\n  body {\n    background: var(--bg);\n    color: var(--ink);\n    font-family: 'Fraunces', Georgia, serif;\n    padding: 2rem 1rem;\n    min-height: 100vh;\n    display: flex;\n    flex-direction: column;\n    align-items: center;\n  }\n  .widget {\n    width: 100%;\n    max-width: 820px;\n    background: var(--paper);\n    border: 1px solid var(--rule);\n    border-radius: 4px;\n    padding: 2rem 1.5rem 2.5rem;\n    position: relative;\n    box-shadow: 0 1px 0 rgba(0,0,0,0.02), 0 20px 40px -20px rgba(26,22,18,0.12);\n  }\n  .widget::before {\n    content: \"FIG. 01\";\n    position: absolute;\n    top: 1rem; right: 1.25rem;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.7rem;\n    letter-spacing: 0.15em;\n    color: var(--ink-soft);\n  }\n  h1 {\n    font-family: 'Fraunces', serif;\n    font-weight: 600;\n    font-size: 1.5rem;\n    font-variation-settings: \"opsz\" 96;\n    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stroke: var(--ink); stroke-width: 1.5; }\n  .node-bg-subject { fill: var(--accent); stroke: var(--accent); stroke-width: 1.5; }\n  .node-label-dark { fill: var(--ink); font-family: 'JetBrains Mono', monospace; font-size: 12px; font-weight: 500; }\n  .node-label-light { fill: var(--paper); font-family: 'JetBrains Mono', monospace; font-size: 12px; font-weight: 500; }\n  .connector { stroke: var(--ink-soft); stroke-width: 1; fill: none; }\n  .connector-accent { stroke: var(--accent); stroke-width: 1.5; fill: none; }\n \n  .annotation {\n    font-family: 'Fraunces', serif;\n    font-style: italic;\n    font-size: 13px;\n    fill: var(--accent);\n  }\n  .caption {\n    margin-top: 1.5rem;\n    padding-top: 1rem;\n    border-top: 1px solid var(--rule);\n    font-size: 0.95rem;\n    line-height: 1.55;\n    color: var(--ink-soft);\n    font-style: italic;\n  }\n  .caption strong { color: var(--ink); font-style: normal; font-weight: 600; }\n  \n  @keyframes pulse-subject {\n    0%, 100% { transform: scale(1); }\n    50% { transform: scale(1.05); }\n  }\n  #subject-node {\n    transform-origin: center;\n    transform-box: fill-box;\n    animation: pulse-subject 2.5s ease-in-out infinite;\n  }\n\u003C\u002Fstyle>\n\u003C\u002Fhead>\n\u003Cbody>\n\u003Cdiv class=\"widget\">\n  \u003Ch1>Java Collection Framework\u003C\u002Fh1>\n  \u003Cdiv class=\"subtitle\">— Where Queue Lives —\u003C\u002Fdiv>\n \n  \u003Cdiv class=\"legend\">\n    \u003Cspan class=\"legend-item\">\u003Cspan class=\"legend-swatch swatch-interface\">\u003C\u002Fspan> Interface\u003C\u002Fspan>\n    \u003Cspan class=\"legend-item\">\u003Cspan class=\"legend-swatch swatch-class\">\u003C\u002Fspan> Class\u003C\u002Fspan>\n    \u003Cspan class=\"legend-item\">\u003Cspan class=\"legend-swatch swatch-subject\">\u003C\u002Fspan> Our subject\u003C\u002Fspan>\n  \u003C\u002Fdiv>\n \n  \u003Csvg viewBox=\"0 0 800 560\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\">\n    \u003C!-- Iterable -->\n    \u003Cg transform=\"translate(340, 10)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">Iterable\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003C!-- arrow Iterable -> Collection -->\n    \u003Cpath class=\"connector\" d=\"M 400 44 L 400 70\"\u002F>\n    \u003Cpolygon points=\"396,68 400,76 404,68\" fill=\"#6b6258\"\u002F>\n \n    \u003C!-- Collection -->\n    \u003Cg transform=\"translate(340, 76)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">Collection\u003C\u002Ftext>\n    \u003C\u002Fg>\n \n    \u003C!-- Branches from Collection -->\n    \u003Cpath class=\"connector\" d=\"M 400 110 L 400 130 L 160 130 L 160 150\"\u002F>\n    \n    \u003C!-- Accent connector for Queue -->\n    \u003Cpath class=\"connector-accent\" d=\"M 400 130 L 400 146\"\u002F>\n    \u003Cpolygon points=\"395,144 400,156 405,144\" fill=\"#c44536\"\u002F>\n    \n    \u003Cpath class=\"connector\" d=\"M 400 130 L 640 130 L 640 150\"\u002F>\n    \u003Cpolygon points=\"156,148 160,156 164,148\" fill=\"#6b6258\"\u002F>\n    \u003Cpolygon points=\"636,148 640,156 644,148\" fill=\"#6b6258\"\u002F>\n \n    \u003C!-- List -->\n    \u003Cg transform=\"translate(100, 156)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">List\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \n    \u003C!-- Queue (Highlighted & Wrapped to protect translate from scale animation) -->\n    \u003Cg transform=\"translate(340, 156)\">\n      \u003Cg id=\"subject-node\">\n        \u003Crect class=\"node-bg-subject\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n        \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">Queue\u003C\u002Ftext>\n      \u003C\u002Fg>\n    \u003C\u002Fg>\n    \u003Ctext class=\"annotation\" x=\"472\" y=\"178\">← you are here\u003C\u002Ftext>\n\n    \u003C!-- Set -->\n    \u003Cg transform=\"translate(580, 156)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">Set\u003C\u002Ftext>\n    \u003C\u002Fg>\n \n    \u003C!-- LIST CHILDREN -->\n    \u003Cpath class=\"connector\" d=\"M 160 190 L 160 210\"\u002F>\n    \u003Cpolygon points=\"156,208 160,216 164,208\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(100, 216)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">ArrayList\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \n    \u003Cpath class=\"connector\" d=\"M 160 250 L 160 270\"\u002F>\n    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transform=\"translate(100, 396)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">Stack\u003C\u002Ftext>\n    \u003C\u002Fg>\n \n    \u003C!-- QUEUE CHILDREN -->\n    \u003Cpath class=\"connector\" d=\"M 400 190 L 400 210\"\u002F>\n    \u003Cpolygon points=\"396,208 400,216 404,208\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(340, 216)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">PriorityQueue\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \n    \u003Cpath class=\"connector\" d=\"M 400 250 L 400 276\"\u002F>\n    \u003Cpolygon points=\"396,274 400,282 404,274\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(340, 282)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">Deque\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \n    \u003Cpath class=\"connector\" d=\"M 400 316 L 400 336\"\u002F>\n    \u003Cpolygon points=\"396,334 400,342 404,334\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(340, 342)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">ArrayDeque\u003C\u002Ftext>\n    \u003C\u002Fg>\n \n    \u003C!-- SET CHILDREN -->\n    \u003Cpath class=\"connector\" d=\"M 640 190 L 640 210\"\u002F>\n    \u003Cpolygon points=\"636,208 640,216 644,208\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(580, 216)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">HashSet\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \n    \u003Cpath class=\"connector\" d=\"M 640 250 L 640 270\"\u002F>\n    \u003Cpolygon points=\"636,268 640,276 644,268\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(580, 276)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">LinkedHashSet\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \n    \u003Cpath class=\"connector\" d=\"M 640 310 L 640 336\"\u002F>\n    \u003Cpolygon points=\"636,334 640,342 644,334\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(580, 342)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">SortedSet\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \n    \u003Cpath class=\"connector\" d=\"M 640 376 L 640 396\"\u002F>\n    \u003Cpolygon points=\"636,394 640,402 644,394\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(580, 402)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">TreeSet\u003C\u002Ftext>\n    \u003C\u002Fg>\n  \u003C\u002Fsvg>\n \n  \u003Cdiv class=\"caption\">\n    Notice how \u003Cstrong>Queue\u003C\u002Fstrong> lives directly under \u003Cstrong>Collection\u003C\u002Fstrong>. It represents a fundamental collection designed for holding elements prior to processing, typically enforcing strict First-In-First-Out (FIFO) rules.\n  \u003C\u002Fdiv>\n\u003C\u002Fdiv>\n\u003C\u002Fbody>\n\u003C\u002Fhtml>",[12,7030,7031],{},"A few takeaways:",[23,7033,7034,7051,7071],{},[26,7035,7036,7038,7039,7042,7043,7046,7047,7050],{},[6979,7037,7019],{}," is a sibling of ",[6979,7040,7041],{},"List"," and ",[6979,7044,7045],{},"Set"," — they're all ",[6979,7048,7049],{},"Collection","s.",[26,7052,7053,7055,7056,7059,7060,7063,7064,7067,7068,6959],{},[6979,7054,7019],{}," is an ",[106,7057,7058],{},"interface",", not a concrete class. You don't ",[6979,7061,7062],{},"new Queue()",". You use an implementation like ",[6979,7065,7066],{},"LinkedList"," or ",[6979,7069,7070],{},"PriorityQueue",[26,7072,7073,7076],{},[6979,7074,7075],{},"Deque"," (double-ended queue) is a more flexible cousin that allows add\u002Fremove at both ends. We won't focus on it here.",[6964,7078,7080],{"id":7079},"_3-the-two-operations-that-define-a-queue","3. The two operations that define a queue",[12,7082,7083],{},"You only really need two:",[7085,7086,7087,7103],"table",{},[7088,7089,7090],"thead",{},[7091,7092,7093,7097,7100],"tr",{},[7094,7095,7096],"th",{},"Operation",[7094,7098,7099],{},"What it does",[7094,7101,7102],{},"Where it happens",[7104,7105,7106,7127],"tbody",{},[7091,7107,7108,7114,7121],{},[7109,7110,7111],"td",{},[6979,7112,7113],{},"enqueue(x)",[7109,7115,7116,7117,7120],{},"Add ",[6979,7118,7119],{},"x"," to the queue",[7109,7122,7123,7124],{},"At the ",[106,7125,7126],{},"rear",[7091,7128,7129,7134,7137],{},[7109,7130,7131],{},[6979,7132,7133],{},"dequeue()",[7109,7135,7136],{},"Remove and return the next element",[7109,7138,7139,7140],{},"From the ",[106,7141,7142],{},"front",[12,7144,7145,7146,7149,7150,7153,7154,7157,7158,7042,7161,7164],{},"You'll often also see helpers like ",[6979,7147,7148],{},"peek()"," (look at the front without removing), ",[6979,7151,7152],{},"size()",", and ",[6979,7155,7156],{},"isEmpty()",". But ",[6979,7159,7160],{},"enqueue",[6979,7162,7163],{},"dequeue"," are the heart of it.",[7166,7167,7169],"h3",{"id":7168},"try-it-yourself","Try it yourself",[12,7171,7172,7173,7176,7177,7180],{},"I built a tiny visualizer — type a value, hit ",[6979,7174,7175],{},"ENQUEUE",", watch it appear at the rear. Hit ",[6979,7178,7179],{},"DEQUEUE"," to pull from the front. The borders highlight which node is the front (red) and which is the rear (blue).",[7026,7182],{"code":7183},"\u003C!DOCTYPE html>\n\u003Chtml lang=\"en\">\n\u003Chead>\n\u003Cmeta charset=\"UTF-8\" \u002F>\n\u003Cmeta name=\"viewport\" content=\"width=device-width, initial-scale=1.0\" \u002F>\n\u003Ctitle>Queue Visualizer\u003C\u002Ftitle>\n\u003Cstyle>\n  :root {\n    --paper: #f6f1e8;\n    --ink: #2b2a26;\n    --muted: #7a7468;\n    --line: #1a1916;\n    --front: #c9412c;\n    --rear:  #2563a8;\n    --accent: #d9a441;\n    --shadow: 4px 4px 0 var(--line);\n  }\n  * { box-sizing: border-box; }\n  html, body {\n    margin: 0; padding: 0;\n    font-family: \"Georgia\", \"Cambria\", serif;\n    background: var(--paper);\n    color: var(--ink);\n  }\n  body {\n    background-image:\n      radial-gradient(circle at 20% 10%, rgba(0,0,0,0.03) 0, transparent 60%),\n      radial-gradient(circle at 80% 90%, rgba(0,0,0,0.03) 0, transparent 60%);\n    padding: 32px 20px;\n  }\n  .wrap {\n    max-width: 760px; margin: 0 auto;\n    border: 2px solid var(--line);\n    background: #fffdf7;\n    padding: 28px 28px 24px;\n    box-shadow: var(--shadow);\n    position: relative;\n  }\n  .wrap::before {\n    content: \"FIFO\";\n    position: absolute; top: -14px; left: 22px;\n    background: var(--accent);\n    color: var(--line);\n    font-family: \"Courier New\", monospace;\n    font-weight: 700;\n    font-size: 12px;\n    letter-spacing: 2px;\n    padding: 3px 10px;\n    border: 2px solid var(--line);\n  }\n  h1 {\n    font-family: \"Georgia\", serif;\n    font-weight: 700;\n    font-size: 26px;\n    margin: 4px 0 4px;\n    letter-spacing: -0.5px;\n  }\n  .sub {\n    font-style: italic;\n    color: var(--muted);\n    margin: 0 0 20px;\n    font-size: 14px;\n  }\n  .controls {\n    display: flex; flex-wrap: wrap; gap: 8px;\n    align-items: center;\n    padding: 14px;\n    background: #f1ead9;\n    border: 1.5px dashed var(--line);\n    margin-bottom: 22px;\n  }\n  .controls input[type=\"text\"] {\n    flex: 1; min-width: 120px;\n    padding: 8px 10px;\n    font-family: \"Courier New\", monospace;\n    font-size: 14px;\n    border: 2px solid var(--line);\n    background: #fffdf7;\n    outline: none;\n  }\n  .controls input[type=\"text\"]:focus {\n    background: #fff;\n    box-shadow: 2px 2px 0 var(--line);\n    transform: translate(-1px, -1px);\n  }\n  button {\n    font-family: \"Courier New\", monospace;\n    font-size: 13px;\n    font-weight: 700;\n    letter-spacing: 1px;\n    padding: 8px 14px;\n    border: 2px solid var(--line);\n    background: #fffdf7;\n    color: var(--ink);\n    cursor: pointer;\n    transition: transform 0.08s ease, box-shadow 0.08s ease;\n  }\n  button:hover { transform: translate(-1px, -1px); box-shadow: 2px 2px 0 var(--line); }\n  button:active { transform: translate(1px, 1px); box-shadow: none; }\n  .btn-enq { background: #d9eedd; }\n  .btn-deq { background: #f4d3c9; }\n  .btn-clr { background: #fffdf7; }\n  .stage {\n    position: relative;\n    min-height: 180px;\n    padding: 28px 12px 12px;\n    border: 2px solid var(--line);\n    background: #fffdf7;\n    overflow: hidden;\n  }\n  .labels {\n    position: absolute; top: 6px; left: 12px; right: 12px;\n    display: flex; justify-content: space-between;\n    font-family: \"Courier New\", monospace;\n    font-size: 11px;\n    letter-spacing: 1.5px;\n    color: var(--muted);\n  }\n  .labels .front-l { color: var(--front); }\n  .labels .rear-l  { color: var(--rear);  }\n  .queue {\n    display: flex; align-items: center; justify-content: flex-start;\n    gap: 6px;\n    \u002F* FIX: Increased bottom padding so text doesn't get clipped by the scroll area *\u002F\n    padding: 18px 6px 32px; \n    min-height: 110px; \u002F* adjusted to account for new padding *\u002F\n    overflow-x: auto;\n  }\n  .queue::-webkit-scrollbar { height: 6px; }\n  .queue::-webkit-scrollbar-thumb { background: var(--muted); }\n  .node {\n    flex-shrink: 0;\n    min-width: 60px;\n    height: 56px;\n    display: flex; align-items: center; justify-content: center;\n    font-family: \"Courier New\", monospace;\n    font-weight: 700;\n    font-size: 16px;\n    background: #fffdf7;\n    border: 2px solid var(--line);\n    box-shadow: 3px 3px 0 var(--line);\n    position: relative;\n    animation: slideIn 0.35s cubic-bezier(.2,.9,.3,1.2);\n  }\n  .node.first { border-color: var(--front); }\n  .node.last  { border-color: var(--rear);  }\n  .node.first::after,\n  .node.last::after {\n    position: absolute;\n    \u002F* FIX: Pushed down slightly & added nowrap to prevent overlapping *\u002F\n    bottom: -28px; \n    left: 50%; transform: translateX(-50%);\n    font-family: \"Courier New\", monospace;\n    font-size: 10px;\n    letter-spacing: 1px;\n    white-space: nowrap; \n  }\n  .node.first::after { content: \"FRONT ↑\"; color: var(--front); }\n  .node.last::after  { content: \"REAR ↑\";  color: var(--rear);  }\n  .node.first.last::after { content: \"FRONT & REAR ↑\"; color: var(--ink); }\n  .node.leaving {\n    animation: slideOut 0.3s ease forwards;\n  }\n  @keyframes slideIn {\n    0%   { transform: translateX(30px) scale(0.7); opacity: 0; }\n    100% { transform: translateX(0) scale(1); opacity: 1; }\n  }\n  @keyframes slideOut {\n    0%   { transform: translateX(0) scale(1); opacity: 1; }\n    100% { transform: translateX(-50px) scale(0.7); opacity: 0; }\n  }\n  .empty {\n    color: var(--muted);\n    font-style: italic;\n    font-family: \"Georgia\", serif;\n    text-align: center;\n    width: 100%;\n    padding: 24px 0;\n  }\n  .log {\n    margin-top: 18px;\n    font-family: \"Courier New\", monospace;\n    font-size: 12px;\n    background: var(--line);\n    color: #f3e9c8;\n    padding: 12px 14px;\n    height: 110px;\n    overflow-y: auto;\n    border: 2px solid var(--line);\n  }\n  .log div { margin-bottom: 2px; }\n  .log .ok  { color: #9be4a3; }\n  .log .err { color: #f49d8e; }\n  .log .info { color: #f3e9c8; }\n  .stats {\n    margin-top: 12px;\n    display: flex; gap: 18px;\n    font-family: \"Courier New\", monospace;\n    font-size: 13px;\n    color: var(--muted);\n  }\n  .stats b { color: var(--ink); }\n\u003C\u002Fstyle>\n\u003C\u002Fhead>\n\u003Cbody>\n\u003Cdiv class=\"wrap\">\n  \u003Ch1>Queue Visualizer\u003C\u002Fh1>\n  \u003Cp class=\"sub\">Watch elements line up at the rear and leave from the front.\u003C\u002Fp>\n \n  \u003Cdiv class=\"controls\">\n    \u003Cinput id=\"val\" type=\"text\" placeholder=\"Enter a value (e.g. Tom)\" maxlength=\"10\" \u002F>\n    \u003Cbutton class=\"btn-enq\" onclick=\"enqueue()\">ENQUEUE\u003C\u002Fbutton>\n    \u003Cbutton class=\"btn-deq\" onclick=\"dequeue()\">DEQUEUE\u003C\u002Fbutton>\n    \u003Cbutton class=\"btn-clr\" onclick=\"clearQueue()\">CLEAR\u003C\u002Fbutton>\n  \u003C\u002Fdiv>\n \n  \u003Cdiv class=\"stage\">\n    \u003Cdiv class=\"labels\">\n      \u003Cspan class=\"front-l\">← front (remove here)\u003C\u002Fspan>\n      \u003Cspan class=\"rear-l\">rear (add here) →\u003C\u002Fspan>\n    \u003C\u002Fdiv>\n    \u003Cdiv id=\"queue\" class=\"queue\">\n      \u003Cdiv class=\"empty\">— queue is empty —\u003C\u002Fdiv>\n    \u003C\u002Fdiv>\n  \u003C\u002Fdiv>\n \n  \u003Cdiv class=\"stats\">\n    \u003Cspan>size: \u003Cb id=\"size\">0\u003C\u002Fb>\u003C\u002Fspan>\n    \u003Cspan>front: \u003Cb id=\"frontEl\">∅\u003C\u002Fb>\u003C\u002Fspan>\n    \u003Cspan>rear: \u003Cb id=\"rearEl\">∅\u003C\u002Fb>\u003C\u002Fspan>\n  \u003C\u002Fdiv>\n \n  \u003Cdiv class=\"log\" id=\"log\">\u003C\u002Fdiv>\n\u003C\u002Fdiv>\n \n\u003Cscript>\n  const queue = [];\n  const queueEl = document.getElementById(\"queue\");\n  const logEl = document.getElementById(\"log\");\n  const valEl = document.getElementById(\"val\");\n  let counter = 0;\n \n  function log(msg, type = \"info\") {\n    const line = document.createElement(\"div\");\n    line.className = type;\n    line.textContent = \"> \" + msg;\n    logEl.appendChild(line);\n    logEl.scrollTop = logEl.scrollHeight;\n  }\n \n  function render() {\n    queueEl.innerHTML = \"\";\n    if (queue.length === 0) {\n      const empty = document.createElement(\"div\");\n      empty.className = \"empty\";\n      empty.textContent = \"— queue is empty —\";\n      queueEl.appendChild(empty);\n    } else {\n      queue.forEach((item, i) => {\n        const node = document.createElement(\"div\");\n        node.className = \"node\";\n        if (i === 0) node.classList.add(\"first\");\n        if (i === queue.length - 1) node.classList.add(\"last\");\n        node.textContent = item.value;\n        node.dataset.id = item.id;\n        queueEl.appendChild(node);\n      });\n    }\n    document.getElementById(\"size\").textContent = queue.length;\n    document.getElementById(\"frontEl\").textContent = queue[0]?.value ?? \"∅\";\n    document.getElementById(\"rearEl\").textContent = queue[queue.length - 1]?.value ?? \"∅\";\n  }\n \n  function enqueue() {\n    const v = valEl.value.trim();\n    if (!v) {\n      log(\"enqueue failed — value is empty\", \"err\");\n      return;\n    }\n    queue.push({ value: v, id: counter++ });\n    log(\"enqueue(\\\"\" + v + \"\\\")  →  added at rear\", \"ok\");\n    valEl.value = \"\";\n    valEl.focus();\n    render();\n  }\n \n  function dequeue() {\n    if (queue.length === 0) {\n      log(\"dequeue failed — queue is empty\", \"err\");\n      return;\n    }\n    const removed = queue[0];\n    const firstNode = queueEl.querySelector(\".node.first\");\n    if (firstNode) {\n      firstNode.classList.add(\"leaving\");\n      setTimeout(() => {\n        queue.shift();\n        log(\"dequeue()  →  removed \\\"\" + removed.value + \"\\\" from front\", \"ok\");\n        render();\n      }, 280);\n    } else {\n      queue.shift();\n      render();\n    }\n  }\n \n  function clearQueue() {\n    if (queue.length === 0) { log(\"queue already empty\", \"info\"); return; }\n    queue.length = 0;\n    log(\"cleared the queue\", \"info\");\n    render();\n  }\n \n  valEl.addEventListener(\"keydown\", (e) => {\n    if (e.key === \"Enter\") enqueue();\n  });\n \n  \u002F\u002F Seed with a small demo\n  [\"Tom\", \"Susan\", \"Kim\"].forEach(v => queue.push({ value: v, id: counter++ }));\n  render();\n  log(\"queue initialized with 3 elements\", \"info\");\n\u003C\u002Fscript>\n\u003C\u002Fbody>\n\u003C\u002Fhtml>",[6964,7185,7187],{"id":7186},"_4-implementing-a-queue-why-a-linked-list-wins","4. Implementing a queue: why a linked list wins",[12,7189,7190,7191,7194,7195,7197],{},"Here's a subtle but important question: should I use an ",[6979,7192,7193],{},"ArrayList"," or a ",[6979,7196,7066],{}," under the hood?",[12,7199,7200],{},"Let's think about what each operation costs.",[12,7202,7203,7206,7207],{},[106,7204,7205],{},"With an"," ",[106,7208,7209,7024],{},[6979,7210,7193],{},[23,7212,7213,7218],{},[26,7214,7215,7217],{},[6979,7216,7160],{}," (add at end) → cheap, just appends. ✅",[26,7219,7220,7222],{},[6979,7221,7163],{}," (remove from index 0) → ⚠️ expensive! Every other element has to shift left by one. That's an O(n) hit on every removal.",[12,7224,7225,7206,7228],{},[106,7226,7227],{},"With a",[106,7229,7230,7024],{},[6979,7231,7066],{},[23,7233,7234,7239],{},[26,7235,7236,7238],{},[6979,7237,7160],{}," (add at tail) → O(1). ✅",[26,7240,7241,7243],{},[6979,7242,7163],{}," (remove from head) → O(1). ✅",[12,7245,7246],{},"Both queue operations are constant-time with a linked list. That's why the textbook recommends a linked list as the backing structure for a queue.",[6972,7248,7251],{"className":7249,"code":7250,"language":6977},[6975],"   Queue conceptually          Queue implemented as a LinkedList\n \n   ┌───┬───┬───┐                ┌───┐   ┌───┐   ┌───┐\n   │ A │ B │ C │                │ A │ → │ B │ → │ C │\n   └───┴───┴───┘                └───┘   └───┘   └───┘\n    front     rear                first           last\n                                  (dequeue        (enqueue\n                                   from here)      to here)\n",[6979,7252,7250],{"__ignoreMap":180},[6964,7254,7256,7257,7260,7261],{"id":7255},"_5-two-ways-to-wire-a-genericqueue-to-a-linkedlist","5. Two ways to wire a ",[6979,7258,7259],{},"GenericQueue"," to a ",[6979,7262,7066],{},[12,7264,7265],{},"Once you've decided to use a linked list, you still have a design choice. Java's textbook author Y. Daniel Liang lays out two paths.",[7166,7267,7269,7270],{"id":7268},"option-a-inheritance-genericqueue-extends-linkedlist","Option A: Inheritance — ",[6979,7271,7272],{},"GenericQueue extends LinkedList",[12,7274,7275,7276,7279],{},"Your queue ",[106,7277,7278],{},"is-a"," linked list, and inherits all of its methods.",[23,7281,7282,7285],{},[26,7283,7284],{},"✅ Easy to write.",[26,7286,7287,7288,7206,7292,7294,7295,7298,7299,7302],{},"❌ But… your queue now exposes ",[7289,7290,7291],"em",{},"every",[6979,7293,7066],{}," method (like ",[6979,7296,7297],{},"add(int index, E e)",", ",[6979,7300,7301],{},"get(i)",", etc.).",[12,7304,7305],{},"That breaks the queue's contract — users can sneakily insert in the middle, peek at index 5, all sorts of things a real queue shouldn't allow.",[7166,7307,7309,7310,7206,7312,7206,7315],{"id":7308},"option-b-composition-genericqueue-has-a-linkedlist","Option B: Composition — ",[6979,7311,7259],{},[7289,7313,7314],{},"has-a",[6979,7316,7066],{},[12,7318,7275,7319,7322,7323,7298,7325,7298,7327,6959],{},[106,7320,7321],{},"owns"," a private linked list and only exposes ",[6979,7324,7160],{},[6979,7326,7163],{},[6979,7328,7329],{},"getSize",[23,7331,7332,7335],{},[26,7333,7334],{},"✅ Encapsulates the implementation. The user can only do queue things.",[26,7336,7337,7338,7340],{},"✅ You could swap the internal ",[6979,7339,7066],{}," for something else later without breaking callers.",[12,7342,7343],{},"This is the preferred design.",[6972,7345,7348],{"className":7346,"code":7347,"language":6977},[6975],"   (a) Inheritance                    (b) Composition\n \n   LinkedList ◁──── GenericQueue      GenericQueue ◆──── LinkedList\n                                       (holds one privately)\n",[6979,7349,7347],{"__ignoreMap":180},[7351,7352,7353],"blockquote",{},[12,7354,7355,7358,7359,7362],{},[106,7356,7357],{},"Rule of thumb:"," Prefer composition over inheritance unless your subclass is genuinely a more specific ",[7289,7360,7361],{},"kind"," of the parent. A queue isn't a \"kind of linked list\" — it's a different abstraction that happens to use one internally.",[6964,7364,7366,7367,7370],{"id":7365},"_6-genericqueuee-the-code","6. ",[6979,7368,7369],{},"GenericQueue\u003CE>"," — the code",[12,7372,7373],{},"Here's the textbook implementation using composition. It's short enough to memorize:",[6972,7375,7379],{"className":7376,"code":7377,"language":7378,"meta":180,"style":180},"language-java shiki shiki-themes github-light github-dark","public class GenericQueue\u003CE> {\n    private java.util.LinkedList\u003CE> list = new java.util.LinkedList\u003C>();\n \n    public void enqueue(E e) {\n        list.addLast(e);          \u002F\u002F add at rear\n    }\n \n    public E dequeue() {\n        return list.removeFirst(); \u002F\u002F remove from front\n    }\n \n    public int getSize() {\n        return list.size();\n    }\n \n    @Override\n    public String toString() {\n        return \"Queue: \" + list.toString();\n    }\n}\n","java",[6979,7380,7381,7406,7428,7434,7456,7472,7478,7483,7496,7514,7519,7524,7537,7550,7555,7560,7569,7582,7600,7605],{"__ignoreMap":180},[75,7382,7385,7389,7392,7396,7400,7403],{"class":7383,"line":7384},"line",1,[75,7386,7388],{"class":7387},"szBVR","public",[75,7390,7391],{"class":7387}," class",[75,7393,7395],{"class":7394},"sScJk"," GenericQueue",[75,7397,7399],{"class":7398},"sVt8B","\u003C",[75,7401,7402],{"class":7387},"E",[75,7404,7405],{"class":7398},"> {\n",[75,7407,7408,7411,7414,7416,7419,7422,7425],{"class":7383,"line":181},[75,7409,7410],{"class":7387},"    private",[75,7412,7413],{"class":7398}," java.util.LinkedList\u003C",[75,7415,7402],{"class":7387},[75,7417,7418],{"class":7398},"> list ",[75,7420,7421],{"class":7387},"=",[75,7423,7424],{"class":7387}," new",[75,7426,7427],{"class":7398}," java.util.LinkedList\u003C>();\n",[75,7429,7431],{"class":7383,"line":7430},3,[75,7432,7433],{"class":7398}," \n",[75,7435,7437,7440,7443,7446,7449,7453],{"class":7383,"line":7436},4,[75,7438,7439],{"class":7387},"    public",[75,7441,7442],{"class":7387}," void",[75,7444,7445],{"class":7394}," enqueue",[75,7447,7448],{"class":7398},"(E ",[75,7450,7452],{"class":7451},"s4XuR","e",[75,7454,7455],{"class":7398},") {\n",[75,7457,7459,7462,7465,7468],{"class":7383,"line":7458},5,[75,7460,7461],{"class":7398},"        list.",[75,7463,7464],{"class":7394},"addLast",[75,7466,7467],{"class":7398},"(e);          ",[75,7469,7471],{"class":7470},"sJ8bj","\u002F\u002F add at rear\n",[75,7473,7475],{"class":7383,"line":7474},6,[75,7476,7477],{"class":7398},"    }\n",[75,7479,7481],{"class":7383,"line":7480},7,[75,7482,7433],{"class":7398},[75,7484,7486,7488,7491,7493],{"class":7383,"line":7485},8,[75,7487,7439],{"class":7387},[75,7489,7490],{"class":7398}," E ",[75,7492,7163],{"class":7394},[75,7494,7495],{"class":7398},"() {\n",[75,7497,7499,7502,7505,7508,7511],{"class":7383,"line":7498},9,[75,7500,7501],{"class":7387},"        return",[75,7503,7504],{"class":7398}," list.",[75,7506,7507],{"class":7394},"removeFirst",[75,7509,7510],{"class":7398},"(); ",[75,7512,7513],{"class":7470},"\u002F\u002F remove from front\n",[75,7515,7517],{"class":7383,"line":7516},10,[75,7518,7477],{"class":7398},[75,7520,7522],{"class":7383,"line":7521},11,[75,7523,7433],{"class":7398},[75,7525,7527,7529,7532,7535],{"class":7383,"line":7526},12,[75,7528,7439],{"class":7387},[75,7530,7531],{"class":7387}," int",[75,7533,7534],{"class":7394}," getSize",[75,7536,7495],{"class":7398},[75,7538,7540,7542,7544,7547],{"class":7383,"line":7539},13,[75,7541,7501],{"class":7387},[75,7543,7504],{"class":7398},[75,7545,7546],{"class":7394},"size",[75,7548,7549],{"class":7398},"();\n",[75,7551,7553],{"class":7383,"line":7552},14,[75,7554,7477],{"class":7398},[75,7556,7558],{"class":7383,"line":7557},15,[75,7559,7433],{"class":7398},[75,7561,7563,7566],{"class":7383,"line":7562},16,[75,7564,7565],{"class":7398},"    @",[75,7567,7568],{"class":7387},"Override\n",[75,7570,7572,7574,7577,7580],{"class":7383,"line":7571},17,[75,7573,7439],{"class":7387},[75,7575,7576],{"class":7398}," String ",[75,7578,7579],{"class":7394},"toString",[75,7581,7495],{"class":7398},[75,7583,7585,7587,7591,7594,7596,7598],{"class":7383,"line":7584},18,[75,7586,7501],{"class":7387},[75,7588,7590],{"class":7589},"sZZnC"," \"Queue: \"",[75,7592,7593],{"class":7387}," +",[75,7595,7504],{"class":7398},[75,7597,7579],{"class":7394},[75,7599,7549],{"class":7398},[75,7601,7603],{"class":7383,"line":7602},19,[75,7604,7477],{"class":7398},[75,7606,7608],{"class":7383,"line":7607},20,[75,7609,7610],{"class":7398},"}\n",[12,7612,7613],{},"A few things worth noticing:",[23,7615,7616,7637,7647],{},[26,7617,7618,7621,7622,7625,7626,7298,7629,7632,7633,7636],{},[6979,7619,7620],{},"\u003CE>"," is a ",[106,7623,7624],{},"generic type parameter",". It lets you build ",[6979,7627,7628],{},"GenericQueue\u003CString>",[6979,7630,7631],{},"GenericQueue\u003CInteger>",", or ",[6979,7634,7635],{},"GenericQueue\u003CMyOwnClass>"," from the same definition.",[26,7638,7639,7042,7641,7643,7644,7646],{},[6979,7640,7464],{},[6979,7642,7507],{}," are methods Java's ",[6979,7645,7066],{}," already provides. We're just renaming them to the queue vocabulary.",[26,7648,7649,7650,7653,7654,7657],{},"The internal ",[6979,7651,7652],{},"list"," is ",[6979,7655,7656],{},"private"," — outside code can't reach in and break invariants.",[7166,7659,7661],{"id":7660},"testing-it","Testing it",[6972,7663,7665],{"className":7376,"code":7664,"language":7378,"meta":180,"style":180},"GenericQueue\u003CString> queue = new GenericQueue\u003C>();\n \nqueue.enqueue(\"Tom\");          \u002F\u002F Queue: [Tom]\nqueue.enqueue(\"Susan\");        \u002F\u002F Queue: [Tom, Susan]\nqueue.enqueue(\"Kim\");\nqueue.enqueue(\"Michael\");      \u002F\u002F Queue: [Tom, Susan, Kim, Michael]\n \nSystem.out.println(queue.dequeue()); \u002F\u002F Tom\nSystem.out.println(queue.dequeue()); \u002F\u002F Susan\nSystem.out.println(queue);           \u002F\u002F Queue: [Kim, Michael]\n",[6979,7666,7667,7685,7689,7708,7725,7739,7756,7760,7779,7794],{"__ignoreMap":180},[75,7668,7669,7672,7675,7678,7680,7682],{"class":7383,"line":7384},[75,7670,7671],{"class":7398},"GenericQueue\u003C",[75,7673,7674],{"class":7387},"String",[75,7676,7677],{"class":7398},"> queue ",[75,7679,7421],{"class":7387},[75,7681,7424],{"class":7387},[75,7683,7684],{"class":7398}," GenericQueue\u003C>();\n",[75,7686,7687],{"class":7383,"line":181},[75,7688,7433],{"class":7398},[75,7690,7691,7694,7696,7699,7702,7705],{"class":7383,"line":7430},[75,7692,7693],{"class":7398},"queue.",[75,7695,7160],{"class":7394},[75,7697,7698],{"class":7398},"(",[75,7700,7701],{"class":7589},"\"Tom\"",[75,7703,7704],{"class":7398},");          ",[75,7706,7707],{"class":7470},"\u002F\u002F Queue: [Tom]\n",[75,7709,7710,7712,7714,7716,7719,7722],{"class":7383,"line":7436},[75,7711,7693],{"class":7398},[75,7713,7160],{"class":7394},[75,7715,7698],{"class":7398},[75,7717,7718],{"class":7589},"\"Susan\"",[75,7720,7721],{"class":7398},");        ",[75,7723,7724],{"class":7470},"\u002F\u002F Queue: [Tom, Susan]\n",[75,7726,7727,7729,7731,7733,7736],{"class":7383,"line":7458},[75,7728,7693],{"class":7398},[75,7730,7160],{"class":7394},[75,7732,7698],{"class":7398},[75,7734,7735],{"class":7589},"\"Kim\"",[75,7737,7738],{"class":7398},");\n",[75,7740,7741,7743,7745,7747,7750,7753],{"class":7383,"line":7474},[75,7742,7693],{"class":7398},[75,7744,7160],{"class":7394},[75,7746,7698],{"class":7398},[75,7748,7749],{"class":7589},"\"Michael\"",[75,7751,7752],{"class":7398},");      ",[75,7754,7755],{"class":7470},"\u002F\u002F Queue: [Tom, Susan, Kim, Michael]\n",[75,7757,7758],{"class":7383,"line":7480},[75,7759,7433],{"class":7398},[75,7761,7762,7765,7768,7771,7773,7776],{"class":7383,"line":7485},[75,7763,7764],{"class":7398},"System.out.",[75,7766,7767],{"class":7394},"println",[75,7769,7770],{"class":7398},"(queue.",[75,7772,7163],{"class":7394},[75,7774,7775],{"class":7398},"()); ",[75,7777,7778],{"class":7470},"\u002F\u002F Tom\n",[75,7780,7781,7783,7785,7787,7789,7791],{"class":7383,"line":7498},[75,7782,7764],{"class":7398},[75,7784,7767],{"class":7394},[75,7786,7770],{"class":7398},[75,7788,7163],{"class":7394},[75,7790,7775],{"class":7398},[75,7792,7793],{"class":7470},"\u002F\u002F Susan\n",[75,7795,7796,7798,7800,7803],{"class":7383,"line":7516},[75,7797,7764],{"class":7398},[75,7799,7767],{"class":7394},[75,7801,7802],{"class":7398},"(queue);           ",[75,7804,7805],{"class":7470},"\u002F\u002F Queue: [Kim, Michael]\n",[12,7807,7808,7809,7812,7813,7816],{},"Notice how Tom — the ",[106,7810,7811],{},"first one in"," — is also the ",[106,7814,7815],{},"first one out",". FIFO in action.",[6964,7818,7820],{"id":7819},"_7-priority-queues-when-fifo-isnt-enough","7. Priority Queues — when FIFO isn't enough",[12,7822,7823],{},"Now imagine an emergency room.",[12,7825,7826],{},"A patient with a stab wound arrives at 10:00 AM. A patient with a sprained ankle arrives at 10:05 AM. Should the ankle be treated first because they arrived first? Of course not.",[12,7828,7829,7830,7833],{},"That's the limitation of a plain queue: it doesn't care ",[106,7831,7832],{},"how important"," an element is. In real life, sometimes you need to jump the line.",[7351,7835,7836],{},[12,7837,7838,7841,7842,7845],{},[106,7839,7840],{},"Priority Queue:"," A queue where each element has a ",[7289,7843,7844],{},"priority",", and the highest-priority element comes out first — regardless of insertion order.",[12,7847,7848,7849,7852,7853,7067,7856,7859],{},"Some people call this ",[106,7850,7851],{},"\"largest-in, first-out\""," behavior, though as you'll see, \"largest\" can mean either ",[7289,7854,7855],{},"biggest number",[7289,7857,7858],{},"smallest number"," depending on how you set it up.",[7166,7861,7863],{"id":7862},"visualizing-it","Visualizing it",[12,7865,7866,7867,7870,7871,7874],{},"Here's the classic textbook picture: elements are dumped into a container with priorities, and removal always pulls out the highest-priority one. With ",[6979,7868,7869],{},"8, 3, 18, 15, 13"," in the bowl (higher number = higher priority), ",[6979,7872,7873],{},"18"," leaves first.",[7026,7876],{"code":7877},"\u003C!DOCTYPE html>\n\u003Chtml lang=\"en\">\n\u003Chead>\n\u003Cmeta charset=\"UTF-8\" \u002F>\n\u003Cmeta name=\"viewport\" content=\"width=device-width, initial-scale=1.0\" \u002F>\n\u003Ctitle>Priority Queue Visualizer\u003C\u002Ftitle>\n\u003Cstyle>\n  :root {\n    --paper: #f6f1e8;\n    --ink: #2b2a26;\n    --muted: #7a7468;\n    --line: #1a1916;\n    --hot: #c9412c;\n    --cold: #5a8fbf;\n    --accent: #d9a441;\n    --shadow: 4px 4px 0 var(--line);\n  }\n  * { box-sizing: border-box; }\n  html, body {\n    margin: 0; padding: 0;\n    font-family: \"Georgia\", \"Cambria\", serif;\n    background: var(--paper);\n    color: var(--ink);\n  }\n  body {\n    background-image:\n      radial-gradient(circle at 20% 10%, rgba(0,0,0,0.03) 0, transparent 60%),\n      radial-gradient(circle at 80% 90%, rgba(0,0,0,0.03) 0, transparent 60%);\n    padding: 32px 20px;\n  }\n  .wrap {\n    max-width: 760px; margin: 0 auto;\n    border: 2px solid var(--line);\n    background: #fffdf7;\n    padding: 28px 28px 24px;\n    box-shadow: var(--shadow);\n    position: relative;\n  }\n  .wrap::before {\n    content: \"PRIORITY\";\n    position: absolute; top: -14px; left: 22px;\n    background: var(--accent);\n    color: var(--line);\n    font-family: \"Courier New\", monospace;\n    font-weight: 700;\n    font-size: 12px;\n    letter-spacing: 2px;\n    padding: 3px 10px;\n    border: 2px solid var(--line);\n  }\n  h1 {\n    font-family: \"Georgia\", serif;\n    font-weight: 700;\n    font-size: 26px;\n    margin: 4px 0 4px;\n    letter-spacing: -0.5px;\n  }\n  .sub {\n    font-style: italic;\n    color: var(--muted);\n    margin: 0 0 16px;\n    font-size: 14px;\n  }\n  .mode-toggle {\n    display: inline-flex;\n    border: 2px solid var(--line);\n    margin-bottom: 18px;\n    font-family: \"Courier New\", monospace;\n    font-size: 12px;\n    font-weight: 700;\n  }\n  .mode-toggle button {\n    background: #fffdf7;\n    border: none;\n    padding: 6px 14px;\n    cursor: pointer;\n    color: var(--muted);\n    letter-spacing: 1px;\n  }\n  .mode-toggle button.active {\n    background: var(--ink);\n    color: var(--accent);\n  }\n  .controls {\n    display: flex; flex-wrap: wrap; gap: 8px;\n    align-items: center;\n    padding: 14px;\n    background: #f1ead9;\n    border: 1.5px dashed var(--line);\n    margin-bottom: 22px;\n  }\n  .controls input {\n    padding: 8px 10px;\n    font-family: \"Courier New\", monospace;\n    font-size: 14px;\n    border: 2px solid var(--line);\n    background: #fffdf7;\n    outline: none;\n  }\n  .controls input:focus {\n    background: #fff;\n    box-shadow: 2px 2px 0 var(--line);\n    transform: translate(-1px, -1px);\n  }\n  .controls input.val   { flex: 2; min-width: 100px; }\n  .controls input.prio  { width: 80px; }\n  button {\n    font-family: \"Courier New\", monospace;\n    font-size: 13px;\n    font-weight: 700;\n    letter-spacing: 1px;\n    padding: 8px 14px;\n    border: 2px solid var(--line);\n    background: #fffdf7;\n    color: var(--ink);\n    cursor: pointer;\n    transition: transform 0.08s ease, box-shadow 0.08s ease;\n  }\n  button:hover { transform: translate(-1px, -1px); box-shadow: 2px 2px 0 var(--line); }\n  button:active { transform: translate(1px, 1px); box-shadow: none; }\n  .btn-add  { background: #d9eedd; }\n  .btn-poll { background: #f4d3c9; }\n  .btn-clr  { background: #fffdf7; }\n  .stage {\n    position: relative;\n    min-height: 200px;\n    padding: 28px 12px 36px;\n    border: 2px solid var(--line);\n    background: #fffdf7;\n  }\n  .stage-label {\n    position: absolute; top: 6px; left: 12px;\n    font-family: \"Courier New\", monospace;\n    font-size: 11px;\n    letter-spacing: 1.5px;\n    color: var(--muted);\n  }\n  .stage-label b { color: var(--hot); }\n  .queue {\n    display: flex; flex-wrap: wrap;\n    gap: 10px;\n    padding: 18px 6px 6px;\n    min-height: 120px;\n    align-items: center;\n    justify-content: center;\n  }\n  .node {\n    min-width: 90px;\n    padding: 10px 8px 6px;\n    display: flex; flex-direction: column; align-items: center;\n    font-family: \"Courier New\", monospace;\n    background: #fffdf7;\n    border: 2px solid var(--line);\n    box-shadow: 3px 3px 0 var(--line);\n    position: relative;\n    transition: transform 0.4s cubic-bezier(.2,.9,.3,1.2);\n    animation: pop 0.35s cubic-bezier(.2,.9,.3,1.2);\n  }\n  .node .value {\n    font-weight: 700;\n    font-size: 15px;\n  }\n  .node .prio {\n    font-size: 11px;\n    color: var(--muted);\n    margin-top: 4px;\n    letter-spacing: 1px;\n  }\n  .node.next {\n    border-color: var(--hot);\n    background: #fff4ef;\n  }\n  .node.next::after {\n    content: \"↑ NEXT OUT\";\n    position: absolute;\n    bottom: -22px; left: 50%; transform: translateX(-50%);\n    font-size: 10px;\n    color: var(--hot);\n    letter-spacing: 1px;\n    white-space: nowrap;\n  }\n  .node.leaving {\n    animation: fly 0.4s ease forwards;\n  }\n  @keyframes pop {\n    0%   { transform: scale(0.5); opacity: 0; }\n    100% { transform: scale(1); opacity: 1; }\n  }\n  @keyframes fly {\n    0%   { transform: translateY(0) scale(1); opacity: 1; }\n    100% { transform: translateY(-40px) scale(0.6); opacity: 0; }\n  }\n  .empty {\n    color: var(--muted);\n    font-style: italic;\n    font-family: \"Georgia\", serif;\n    text-align: center;\n    width: 100%;\n    padding: 24px 0;\n  }\n  .log {\n    margin-top: 18px;\n    font-family: \"Courier New\", monospace;\n    font-size: 12px;\n    background: var(--line);\n    color: #f3e9c8;\n    padding: 12px 14px;\n    height: 110px;\n    overflow-y: auto;\n    border: 2px solid var(--line);\n  }\n  .log div { margin-bottom: 2px; }\n  .log .ok  { color: #9be4a3; }\n  .log .err { color: #f49d8e; }\n  .log .info { color: #f3e9c8; }\n\u003C\u002Fstyle>\n\u003C\u002Fhead>\n\u003Cbody>\n\u003Cdiv class=\"wrap\">\n  \u003Ch1>Priority Queue Visualizer\u003C\u002Fh1>\n  \u003Cp class=\"sub\">Each item has a priority. The one with the highest priority leaves first — order in doesn't matter.\u003C\u002Fp>\n \n  \u003Cdiv class=\"mode-toggle\">\n    \u003Cbutton id=\"hiBtn\" class=\"active\" onclick=\"setMode('high')\">HIGHER = HIGHER PRIORITY\u003C\u002Fbutton>\n    \u003Cbutton id=\"loBtn\" onclick=\"setMode('low')\">LOWER = HIGHER PRIORITY\u003C\u002Fbutton>\n  \u003C\u002Fdiv>\n \n  \u003Cdiv class=\"controls\">\n    \u003Cinput class=\"val\"  id=\"val\"  type=\"text\"   placeholder=\"value (e.g. patient name)\" maxlength=\"14\" \u002F>\n    \u003Cinput class=\"prio\" id=\"prio\" type=\"number\" placeholder=\"priority\" \u002F>\n    \u003Cbutton class=\"btn-add\"  onclick=\"addItem()\">OFFER\u003C\u002Fbutton>\n    \u003Cbutton class=\"btn-poll\" onclick=\"pollItem()\">POLL\u003C\u002Fbutton>\n    \u003Cbutton class=\"btn-clr\"  onclick=\"clearQ()\">CLEAR\u003C\u002Fbutton>\n  \u003C\u002Fdiv>\n \n  \u003Cdiv class=\"stage\">\n    \u003Cdiv class=\"stage-label\">order shown by priority — \u003Cb>highlighted = poll() returns this next\u003C\u002Fb>\u003C\u002Fdiv>\n    \u003Cdiv id=\"queue\" class=\"queue\">\n      \u003Cdiv class=\"empty\">— priority queue is empty —\u003C\u002Fdiv>\n    \u003C\u002Fdiv>\n  \u003C\u002Fdiv>\n \n  \u003Cdiv class=\"log\" id=\"log\">\u003C\u002Fdiv>\n\u003C\u002Fdiv>\n \n\u003Cscript>\n  let queue = [];\n  let mode = \"high\"; \u002F\u002F \"high\" = larger number wins, \"low\" = smaller number wins\n  let counter = 0;\n  const queueEl = document.getElementById(\"queue\");\n  const logEl = document.getElementById(\"log\");\n  const valEl = document.getElementById(\"val\");\n  const prioEl = document.getElementById(\"prio\");\n \n  function log(msg, type = \"info\") {\n    const line = document.createElement(\"div\");\n    line.className = type;\n    line.textContent = \"> \" + msg;\n    logEl.appendChild(line);\n    logEl.scrollTop = logEl.scrollHeight;\n  }\n \n  function compare(a, b) {\n    return mode === \"high\" ? b.priority - a.priority : a.priority - b.priority;\n  }\n \n  function sortedView() {\n    return [...queue].sort(compare);\n  }\n \n  function render() {\n    queueEl.innerHTML = \"\";\n    const view = sortedView();\n    if (view.length === 0) {\n      const empty = document.createElement(\"div\");\n      empty.className = \"empty\";\n      empty.textContent = \"— priority queue is empty —\";\n      queueEl.appendChild(empty);\n      return;\n    }\n    view.forEach((item, i) => {\n      const node = document.createElement(\"div\");\n      node.className = \"node\";\n      if (i === 0) node.classList.add(\"next\");\n      node.dataset.id = item.id;\n      node.innerHTML = `\u003Cdiv class=\"value\">${escapeHtml(item.value)}\u003C\u002Fdiv>\n                        \u003Cdiv class=\"prio\">p = ${item.priority}\u003C\u002Fdiv>`;\n      queueEl.appendChild(node);\n    });\n  }\n \n  function escapeHtml(s) {\n    return String(s).replace(\u002F[&\u003C>\"']\u002Fg, c => ({\n      \"&\":\"&\",\"\u003C\":\"<\",\">\":\">\",\"\\\"\":\""\",\"'\":\"'\"\n    }[c]));\n  }\n \n  function addItem() {\n    const v = valEl.value.trim();\n    const p = parseInt(prioEl.value, 10);\n    if (!v) { log(\"offer failed — value is empty\", \"err\"); return; }\n    if (Number.isNaN(p)) { log(\"offer failed — priority must be a number\", \"err\"); return; }\n    queue.push({ value: v, priority: p, id: counter++ });\n    log(`offer(\"${v}\", priority=${p})`, \"ok\");\n    valEl.value = \"\"; prioEl.value = \"\";\n    valEl.focus();\n    render();\n  }\n \n  function pollItem() {\n    if (queue.length === 0) { log(\"poll failed — queue is empty\", \"err\"); return; }\n    const sorted = sortedView();\n    const winner = sorted[0];\n    \u002F\u002F Find and animate\n    const winnerNode = queueEl.querySelector(`.node[data-id=\"${winner.id}\"]`);\n    if (winnerNode) {\n      winnerNode.classList.add(\"leaving\");\n      setTimeout(() => {\n        queue = queue.filter(x => x.id !== winner.id);\n        log(`poll() → \"${winner.value}\" (priority ${winner.priority})`, \"ok\");\n        render();\n      }, 380);\n    } else {\n      queue = queue.filter(x => x.id !== winner.id);\n      render();\n    }\n  }\n \n  function clearQ() {\n    if (queue.length === 0) { log(\"queue already empty\", \"info\"); return; }\n    queue = [];\n    log(\"cleared\", \"info\");\n    render();\n  }\n \n  function setMode(m) {\n    mode = m;\n    document.getElementById(\"hiBtn\").classList.toggle(\"active\", m === \"high\");\n    document.getElementById(\"loBtn\").classList.toggle(\"active\", m === \"low\");\n    log(`mode → ${m === \"high\" ? \"larger number wins\" : \"smaller number wins\"}`, \"info\");\n    render();\n  }\n \n  prioEl.addEventListener(\"keydown\", (e) => {\n    if (e.key === \"Enter\") addItem();\n  });\n  valEl.addEventListener(\"keydown\", (e) => {\n    if (e.key === \"Enter\") prioEl.focus();\n  });\n \n  \u002F\u002F Seed with the hospital example\n  [\n    { value: \"Donald\", priority: 3 },\n    { value: \"Chong\",  priority: 1 },\n    { value: \"Ali\",    priority: 2 },\n    { value: \"Bala\",   priority: 4 },\n  ].forEach(x => queue.push({ ...x, id: counter++ }));\n  render();\n  log(\"seeded with 4 patients — Bala has highest priority\", \"info\");\n\u003C\u002Fscript>\n\u003C\u002Fbody>\n\u003C\u002Fhtml>",[7351,7879,7880],{},[12,7881,7882],{},[7289,7883,7884],{},"(Try toggling the \"higher = higher priority\" \u002F \"lower = higher priority\" mode at the top — same data, different winner.)",[6964,7886,7888,7889],{"id":7887},"_8-javas-built-in-priorityqueue","8. Java's built-in ",[6979,7890,7070],{},[12,7892,7893],{},"Good news: Java ships one. You don't have to build it yourself.",[6972,7895,7897],{"className":7376,"code":7896,"language":7378,"meta":180,"style":180},"import java.util.PriorityQueue;\n",[6979,7898,7899],{"__ignoreMap":180},[75,7900,7901,7904],{"class":7383,"line":7384},[75,7902,7903],{"class":7387},"import",[75,7905,7906],{"class":7398}," java.util.PriorityQueue;\n",[12,7908,7909,7910,7913],{},"It implements the ",[6979,7911,7912],{},"Queue\u003CE>"," interface, so the method names look familiar:",[7085,7915,7916,7928],{},[7088,7917,7918],{},[7091,7919,7920,7923,7925],{},[7094,7921,7922],{},"Method",[7094,7924,7099],{},[7094,7926,7927],{},"Returns",[7104,7929,7930,7945,7964,7979,7993,8007,8021],{},[7091,7931,7932,7937,7940],{},[7109,7933,7934],{},[6979,7935,7936],{},"offer(E e)",[7109,7938,7939],{},"Add an element",[7109,7941,7942],{},[6979,7943,7944],{},"boolean",[7091,7946,7947,7952,7955],{},[7109,7948,7949],{},[6979,7950,7951],{},"poll()",[7109,7953,7954],{},"Remove and return the highest-priority element",[7109,7956,7957,7959,7960,7963],{},[6979,7958,7402],{}," (or ",[6979,7961,7962],{},"null",")",[7091,7965,7966,7970,7973],{},[7109,7967,7968],{},[6979,7969,7148],{},[7109,7971,7972],{},"Look at the highest-priority element without removing",[7109,7974,7975,7959,7977,7963],{},[6979,7976,7402],{},[6979,7978,7962],{},[7091,7980,7981,7986,7989],{},[7109,7982,7983],{},[6979,7984,7985],{},"remove(Object o)",[7109,7987,7988],{},"Remove a specific object",[7109,7990,7991],{},[6979,7992,7944],{},[7091,7994,7995,8000,8003],{},[7109,7996,7997],{},[6979,7998,7999],{},"contains(Object o)",[7109,8001,8002],{},"Check if it's in the queue",[7109,8004,8005],{},[6979,8006,7944],{},[7091,8008,8009,8013,8016],{},[7109,8010,8011],{},[6979,8012,7152],{},[7109,8014,8015],{},"How many elements",[7109,8017,8018],{},[6979,8019,8020],{},"int",[7091,8022,8023,8028,8031],{},[7109,8024,8025],{},[6979,8026,8027],{},"clear()",[7109,8029,8030],{},"Empty it",[7109,8032,8033],{},[6979,8034,8035],{},"void",[7351,8037,8038],{},[12,8039,8040,7206,8043,8046,8047,8046,8050,8053,8054,7067,8056,8059,8060,8046,8063,8066],{},[106,8041,8042],{},"A note on naming:",[6979,8044,8045],{},"offer"," \u002F ",[6979,8048,8049],{},"poll",[6979,8051,8052],{},"peek"," are the queue-aware versions that return ",[6979,8055,7962],{},[6979,8057,8058],{},"false"," instead of throwing on edge cases (empty queue, capacity full). The older ",[6979,8061,8062],{},"add",[6979,8064,8065],{},"remove"," exist too but can throw exceptions. Prefer the new names.",[7166,8068,8070],{"id":8069},"how-does-it-know-what-highest-priority-means","How does it know what \"highest priority\" means?",[12,8072,8073],{},"Two ways:",[3603,8075,8076,8098],{},[26,8077,8078,8081,8082,7298,8085,8087,8088,8090,8091,8094,8095,6959],{},[106,8079,8080],{},"Natural ordering."," If your elements implement ",[6979,8083,8084],{},"Comparable\u003CE>",[6979,8086,7070],{}," uses that. For ",[6979,8089,7674],{},", natural order is alphabetical, so ",[6979,8092,8093],{},"\"Georgia\""," comes out before ",[6979,8096,8097],{},"\"Texas\"",[26,8099,8100,7206,8103,8108],{},[106,8101,8102],{},"Custom",[106,8104,8105,6959],{},[6979,8106,8107],{},"Comparator"," Pass one to the constructor and you control the order.",[6964,8110,8112],{"id":8111},"_9-example-1-strings-two-orderings","9. Example 1 — strings, two orderings",[6972,8114,8116],{"className":7376,"code":8115,"language":7378,"meta":180,"style":180},"import java.util.*;\n \npublic class PriorityQueueDemo {\n  public static void main(String[] args) {\n    \u002F\u002F Default: natural ordering (alphabetical for Strings)\n    PriorityQueue\u003CString> queue1 = new PriorityQueue\u003C>();\n    queue1.offer(\"Oklahoma\");\n    queue1.offer(\"Indiana\");\n    queue1.offer(\"Georgia\");\n    queue1.offer(\"Texas\");\n \n    System.out.println(\"Priority queue using Comparable:\");\n    while (queue1.size() > 0) {\n      System.out.print(queue1.poll() + \" \");\n    }\n    \u002F\u002F → Georgia Indiana Oklahoma Texas\n \n    \u002F\u002F Reversed order via Collections.reverseOrder()\n    PriorityQueue\u003CString> queue2 =\n        new PriorityQueue\u003C>(4, Collections.reverseOrder());\n    queue2.offer(\"Oklahoma\");\n    queue2.offer(\"Indiana\");\n    queue2.offer(\"Georgia\");\n    queue2.offer(\"Texas\");\n \n    System.out.println(\"\\nPriority queue using Comparator:\");\n    while (queue2.size() > 0) {\n      System.out.print(queue2.poll() + \" \");\n    }\n    \u002F\u002F → Texas Oklahoma Indiana Georgia\n  }\n}\n",[6979,8117,8118,8132,8136,8148,8173,8178,8195,8209,8222,8234,8246,8250,8264,8285,8308,8312,8317,8321,8326,8338,8358,8372,8385,8398,8411,8416,8436,8454,8474,8479,8485,8491],{"__ignoreMap":180},[75,8119,8120,8122,8125,8129],{"class":7383,"line":7384},[75,8121,7903],{"class":7387},[75,8123,8124],{"class":7398}," java.util.",[75,8126,8128],{"class":8127},"sj4cs","*",[75,8130,8131],{"class":7398},";\n",[75,8133,8134],{"class":7383,"line":181},[75,8135,7433],{"class":7398},[75,8137,8138,8140,8142,8145],{"class":7383,"line":7430},[75,8139,7388],{"class":7387},[75,8141,7391],{"class":7387},[75,8143,8144],{"class":7394}," PriorityQueueDemo",[75,8146,8147],{"class":7398}," {\n",[75,8149,8150,8153,8156,8158,8161,8163,8165,8168,8171],{"class":7383,"line":7436},[75,8151,8152],{"class":7387},"  public",[75,8154,8155],{"class":7387}," static",[75,8157,7442],{"class":7387},[75,8159,8160],{"class":7394}," main",[75,8162,7698],{"class":7398},[75,8164,7674],{"class":7387},[75,8166,8167],{"class":7398},"[] ",[75,8169,8170],{"class":7451},"args",[75,8172,7455],{"class":7398},[75,8174,8175],{"class":7383,"line":7458},[75,8176,8177],{"class":7470},"    \u002F\u002F Default: natural ordering (alphabetical for Strings)\n",[75,8179,8180,8183,8185,8188,8190,8192],{"class":7383,"line":7474},[75,8181,8182],{"class":7398},"    PriorityQueue\u003C",[75,8184,7674],{"class":7387},[75,8186,8187],{"class":7398},"> queue1 ",[75,8189,7421],{"class":7387},[75,8191,7424],{"class":7387},[75,8193,8194],{"class":7398}," PriorityQueue\u003C>();\n",[75,8196,8197,8200,8202,8204,8207],{"class":7383,"line":7480},[75,8198,8199],{"class":7398},"    queue1.",[75,8201,8045],{"class":7394},[75,8203,7698],{"class":7398},[75,8205,8206],{"class":7589},"\"Oklahoma\"",[75,8208,7738],{"class":7398},[75,8210,8211,8213,8215,8217,8220],{"class":7383,"line":7485},[75,8212,8199],{"class":7398},[75,8214,8045],{"class":7394},[75,8216,7698],{"class":7398},[75,8218,8219],{"class":7589},"\"Indiana\"",[75,8221,7738],{"class":7398},[75,8223,8224,8226,8228,8230,8232],{"class":7383,"line":7498},[75,8225,8199],{"class":7398},[75,8227,8045],{"class":7394},[75,8229,7698],{"class":7398},[75,8231,8093],{"class":7589},[75,8233,7738],{"class":7398},[75,8235,8236,8238,8240,8242,8244],{"class":7383,"line":7516},[75,8237,8199],{"class":7398},[75,8239,8045],{"class":7394},[75,8241,7698],{"class":7398},[75,8243,8097],{"class":7589},[75,8245,7738],{"class":7398},[75,8247,8248],{"class":7383,"line":7521},[75,8249,7433],{"class":7398},[75,8251,8252,8255,8257,8259,8262],{"class":7383,"line":7526},[75,8253,8254],{"class":7398},"    System.out.",[75,8256,7767],{"class":7394},[75,8258,7698],{"class":7398},[75,8260,8261],{"class":7589},"\"Priority queue using Comparable:\"",[75,8263,7738],{"class":7398},[75,8265,8266,8269,8272,8274,8277,8280,8283],{"class":7383,"line":7539},[75,8267,8268],{"class":7387},"    while",[75,8270,8271],{"class":7398}," (queue1.",[75,8273,7546],{"class":7394},[75,8275,8276],{"class":7398},"() ",[75,8278,8279],{"class":7387},">",[75,8281,8282],{"class":8127}," 0",[75,8284,7455],{"class":7398},[75,8286,8287,8290,8293,8296,8298,8300,8303,8306],{"class":7383,"line":7552},[75,8288,8289],{"class":7398},"      System.out.",[75,8291,8292],{"class":7394},"print",[75,8294,8295],{"class":7398},"(queue1.",[75,8297,8049],{"class":7394},[75,8299,8276],{"class":7398},[75,8301,8302],{"class":7387},"+",[75,8304,8305],{"class":7589}," \" \"",[75,8307,7738],{"class":7398},[75,8309,8310],{"class":7383,"line":7557},[75,8311,7477],{"class":7398},[75,8313,8314],{"class":7383,"line":7562},[75,8315,8316],{"class":7470},"    \u002F\u002F → Georgia Indiana Oklahoma Texas\n",[75,8318,8319],{"class":7383,"line":7571},[75,8320,7433],{"class":7398},[75,8322,8323],{"class":7383,"line":7584},[75,8324,8325],{"class":7470},"    \u002F\u002F Reversed order via Collections.reverseOrder()\n",[75,8327,8328,8330,8332,8335],{"class":7383,"line":7602},[75,8329,8182],{"class":7398},[75,8331,7674],{"class":7387},[75,8333,8334],{"class":7398},"> queue2 ",[75,8336,8337],{"class":7387},"=\n",[75,8339,8340,8343,8346,8349,8352,8355],{"class":7383,"line":7607},[75,8341,8342],{"class":7387},"        new",[75,8344,8345],{"class":7398}," PriorityQueue\u003C>(",[75,8347,8348],{"class":8127},"4",[75,8350,8351],{"class":7398},", Collections.",[75,8353,8354],{"class":7394},"reverseOrder",[75,8356,8357],{"class":7398},"());\n",[75,8359,8361,8364,8366,8368,8370],{"class":7383,"line":8360},21,[75,8362,8363],{"class":7398},"    queue2.",[75,8365,8045],{"class":7394},[75,8367,7698],{"class":7398},[75,8369,8206],{"class":7589},[75,8371,7738],{"class":7398},[75,8373,8375,8377,8379,8381,8383],{"class":7383,"line":8374},22,[75,8376,8363],{"class":7398},[75,8378,8045],{"class":7394},[75,8380,7698],{"class":7398},[75,8382,8219],{"class":7589},[75,8384,7738],{"class":7398},[75,8386,8388,8390,8392,8394,8396],{"class":7383,"line":8387},23,[75,8389,8363],{"class":7398},[75,8391,8045],{"class":7394},[75,8393,7698],{"class":7398},[75,8395,8093],{"class":7589},[75,8397,7738],{"class":7398},[75,8399,8401,8403,8405,8407,8409],{"class":7383,"line":8400},24,[75,8402,8363],{"class":7398},[75,8404,8045],{"class":7394},[75,8406,7698],{"class":7398},[75,8408,8097],{"class":7589},[75,8410,7738],{"class":7398},[75,8412,8414],{"class":7383,"line":8413},25,[75,8415,7433],{"class":7398},[75,8417,8419,8421,8423,8425,8428,8431,8434],{"class":7383,"line":8418},26,[75,8420,8254],{"class":7398},[75,8422,7767],{"class":7394},[75,8424,7698],{"class":7398},[75,8426,8427],{"class":7589},"\"",[75,8429,8430],{"class":8127},"\\n",[75,8432,8433],{"class":7589},"Priority queue using Comparator:\"",[75,8435,7738],{"class":7398},[75,8437,8439,8441,8444,8446,8448,8450,8452],{"class":7383,"line":8438},27,[75,8440,8268],{"class":7387},[75,8442,8443],{"class":7398}," (queue2.",[75,8445,7546],{"class":7394},[75,8447,8276],{"class":7398},[75,8449,8279],{"class":7387},[75,8451,8282],{"class":8127},[75,8453,7455],{"class":7398},[75,8455,8457,8459,8461,8464,8466,8468,8470,8472],{"class":7383,"line":8456},28,[75,8458,8289],{"class":7398},[75,8460,8292],{"class":7394},[75,8462,8463],{"class":7398},"(queue2.",[75,8465,8049],{"class":7394},[75,8467,8276],{"class":7398},[75,8469,8302],{"class":7387},[75,8471,8305],{"class":7589},[75,8473,7738],{"class":7398},[75,8475,8477],{"class":7383,"line":8476},29,[75,8478,7477],{"class":7398},[75,8480,8482],{"class":7383,"line":8481},30,[75,8483,8484],{"class":7470},"    \u002F\u002F → Texas Oklahoma Indiana Georgia\n",[75,8486,8488],{"class":7383,"line":8487},31,[75,8489,8490],{"class":7398},"  }\n",[75,8492,8494],{"class":7383,"line":8493},32,[75,8495,7610],{"class":7398},[12,8497,8498,8499,6959],{},"Same elements, opposite outputs — just by swapping in ",[6979,8500,8501],{},"Collections.reverseOrder()",[6964,8503,8505],{"id":8504},"_10-example-2-your-own-objects","10. Example 2 — your own objects",[12,8507,8508,8509,8512],{},"What if you want to queue something Java doesn't know how to compare? Like a ",[6979,8510,8511],{},"Customer","?",[12,8514,8515,8516,8519,8520,7024],{},"You have to teach it. Implement ",[6979,8517,8518],{},"Comparable\u003CCustomer>"," and define ",[6979,8521,8522],{},"compareTo",[6972,8524,8526],{"className":7376,"code":8525,"language":7378,"meta":180,"style":180},"public class Customer implements Comparable\u003CCustomer> {\n  private Integer id;\n  private String name;\n \n  public Customer(Integer id, String name) {\n    this.id = id;\n    this.name = name;\n  }\n \n  public Integer getID() { return id; }\n  public String  getName() { return name; }\n \n  @Override\n  public int compareTo(Customer c) {\n    return this.getID().compareTo(c.getID());\n  }\n \n  @Override\n  public String toString() {\n    return \"Customer [id=\" + id + \", name=\" + name + \"]\";\n  }\n}\n",[6979,8527,8528,8549,8557,8564,8568,8588,8601,8613,8617,8621,8640,8657,8661,8668,8685,8709,8713,8717,8723,8733,8762,8766],{"__ignoreMap":180},[75,8529,8530,8532,8534,8537,8540,8543,8545,8547],{"class":7383,"line":7384},[75,8531,7388],{"class":7387},[75,8533,7391],{"class":7387},[75,8535,8536],{"class":7394}," Customer",[75,8538,8539],{"class":7387}," implements",[75,8541,8542],{"class":7394}," Comparable",[75,8544,7399],{"class":7398},[75,8546,8511],{"class":7387},[75,8548,7405],{"class":7398},[75,8550,8551,8554],{"class":7383,"line":181},[75,8552,8553],{"class":7387},"  private",[75,8555,8556],{"class":7398}," Integer id;\n",[75,8558,8559,8561],{"class":7383,"line":7430},[75,8560,8553],{"class":7387},[75,8562,8563],{"class":7398}," String name;\n",[75,8565,8566],{"class":7383,"line":7436},[75,8567,7433],{"class":7398},[75,8569,8570,8572,8574,8577,8580,8583,8586],{"class":7383,"line":7458},[75,8571,8152],{"class":7387},[75,8573,8536],{"class":7394},[75,8575,8576],{"class":7398},"(Integer ",[75,8578,8579],{"class":7451},"id",[75,8581,8582],{"class":7398},", String ",[75,8584,8585],{"class":7451},"name",[75,8587,7455],{"class":7398},[75,8589,8590,8593,8596,8598],{"class":7383,"line":7474},[75,8591,8592],{"class":8127},"    this",[75,8594,8595],{"class":7398},".id ",[75,8597,7421],{"class":7387},[75,8599,8600],{"class":7398}," id;\n",[75,8602,8603,8605,8608,8610],{"class":7383,"line":7480},[75,8604,8592],{"class":8127},[75,8606,8607],{"class":7398},".name ",[75,8609,7421],{"class":7387},[75,8611,8612],{"class":7398}," name;\n",[75,8614,8615],{"class":7383,"line":7485},[75,8616,8490],{"class":7398},[75,8618,8619],{"class":7383,"line":7498},[75,8620,7433],{"class":7398},[75,8622,8623,8625,8628,8631,8634,8637],{"class":7383,"line":7516},[75,8624,8152],{"class":7387},[75,8626,8627],{"class":7398}," Integer ",[75,8629,8630],{"class":7394},"getID",[75,8632,8633],{"class":7398},"() { ",[75,8635,8636],{"class":7387},"return",[75,8638,8639],{"class":7398}," id; }\n",[75,8641,8642,8644,8647,8650,8652,8654],{"class":7383,"line":7521},[75,8643,8152],{"class":7387},[75,8645,8646],{"class":7398}," String  ",[75,8648,8649],{"class":7394},"getName",[75,8651,8633],{"class":7398},[75,8653,8636],{"class":7387},[75,8655,8656],{"class":7398}," name; }\n",[75,8658,8659],{"class":7383,"line":7526},[75,8660,7433],{"class":7398},[75,8662,8663,8666],{"class":7383,"line":7539},[75,8664,8665],{"class":7398},"  @",[75,8667,7568],{"class":7387},[75,8669,8670,8672,8674,8677,8680,8683],{"class":7383,"line":7552},[75,8671,8152],{"class":7387},[75,8673,7531],{"class":7387},[75,8675,8676],{"class":7394}," compareTo",[75,8678,8679],{"class":7398},"(Customer ",[75,8681,8682],{"class":7451},"c",[75,8684,7455],{"class":7398},[75,8686,8687,8690,8693,8695,8697,8700,8702,8705,8707],{"class":7383,"line":7557},[75,8688,8689],{"class":7387},"    return",[75,8691,8692],{"class":8127}," this",[75,8694,6959],{"class":7398},[75,8696,8630],{"class":7394},[75,8698,8699],{"class":7398},"().",[75,8701,8522],{"class":7394},[75,8703,8704],{"class":7398},"(c.",[75,8706,8630],{"class":7394},[75,8708,8357],{"class":7398},[75,8710,8711],{"class":7383,"line":7562},[75,8712,8490],{"class":7398},[75,8714,8715],{"class":7383,"line":7571},[75,8716,7433],{"class":7398},[75,8718,8719,8721],{"class":7383,"line":7584},[75,8720,8665],{"class":7398},[75,8722,7568],{"class":7387},[75,8724,8725,8727,8729,8731],{"class":7383,"line":7602},[75,8726,8152],{"class":7387},[75,8728,7576],{"class":7398},[75,8730,7579],{"class":7394},[75,8732,7495],{"class":7398},[75,8734,8735,8737,8740,8742,8745,8747,8750,8752,8755,8757,8760],{"class":7383,"line":7607},[75,8736,8689],{"class":7387},[75,8738,8739],{"class":7589}," \"Customer [id=\"",[75,8741,7593],{"class":7387},[75,8743,8744],{"class":7398}," id ",[75,8746,8302],{"class":7387},[75,8748,8749],{"class":7589}," \", name=\"",[75,8751,7593],{"class":7387},[75,8753,8754],{"class":7398}," name ",[75,8756,8302],{"class":7387},[75,8758,8759],{"class":7589}," \"]\"",[75,8761,8131],{"class":7398},[75,8763,8764],{"class":7383,"line":8360},[75,8765,8490],{"class":7398},[75,8767,8768],{"class":7383,"line":8374},[75,8769,7610],{"class":7398},[12,8771,8772,8773,8775,8776,8778],{},"Now you can drop ",[6979,8774,8511],{}," objects into a ",[6979,8777,7070],{}," and it'll know what to do:",[6972,8780,8782],{"className":7376,"code":8781,"language":7378,"meta":180,"style":180},"PriorityQueue\u003CCustomer> customerQueue =\n    new PriorityQueue\u003C>(Collections.reverseOrder());\n \ncustomerQueue.add(new Customer(3, \"Donald\"));\ncustomerQueue.add(new Customer(1, \"Chong\"));\ncustomerQueue.add(new Customer(2, \"Ali\"));\ncustomerQueue.add(new Customer(4, \"Bala\"));\n \nCustomer c = customerQueue.peek();\nif (c != null) {\n  System.out.println(c.getName() + \" is in the queue\");\n  while ((c = customerQueue.poll()) != null) {\n    System.out.println(c);\n  }\n}\n",[6979,8783,8784,8796,8808,8812,8839,8863,8887,8910,8914,8928,8944,8964,8987,8996,9000],{"__ignoreMap":180},[75,8785,8786,8789,8791,8794],{"class":7383,"line":7384},[75,8787,8788],{"class":7398},"PriorityQueue\u003C",[75,8790,8511],{"class":7387},[75,8792,8793],{"class":7398},"> customerQueue ",[75,8795,8337],{"class":7387},[75,8797,8798,8801,8804,8806],{"class":7383,"line":181},[75,8799,8800],{"class":7387},"    new",[75,8802,8803],{"class":7398}," PriorityQueue\u003C>(Collections.",[75,8805,8354],{"class":7394},[75,8807,8357],{"class":7398},[75,8809,8810],{"class":7383,"line":7430},[75,8811,7433],{"class":7398},[75,8813,8814,8817,8819,8821,8824,8826,8828,8831,8833,8836],{"class":7383,"line":7436},[75,8815,8816],{"class":7398},"customerQueue.",[75,8818,8062],{"class":7394},[75,8820,7698],{"class":7398},[75,8822,8823],{"class":7387},"new",[75,8825,8536],{"class":7394},[75,8827,7698],{"class":7398},[75,8829,8830],{"class":8127},"3",[75,8832,7298],{"class":7398},[75,8834,8835],{"class":7589},"\"Donald\"",[75,8837,8838],{"class":7398},"));\n",[75,8840,8841,8843,8845,8847,8849,8851,8853,8856,8858,8861],{"class":7383,"line":7458},[75,8842,8816],{"class":7398},[75,8844,8062],{"class":7394},[75,8846,7698],{"class":7398},[75,8848,8823],{"class":7387},[75,8850,8536],{"class":7394},[75,8852,7698],{"class":7398},[75,8854,8855],{"class":8127},"1",[75,8857,7298],{"class":7398},[75,8859,8860],{"class":7589},"\"Chong\"",[75,8862,8838],{"class":7398},[75,8864,8865,8867,8869,8871,8873,8875,8877,8880,8882,8885],{"class":7383,"line":7474},[75,8866,8816],{"class":7398},[75,8868,8062],{"class":7394},[75,8870,7698],{"class":7398},[75,8872,8823],{"class":7387},[75,8874,8536],{"class":7394},[75,8876,7698],{"class":7398},[75,8878,8879],{"class":8127},"2",[75,8881,7298],{"class":7398},[75,8883,8884],{"class":7589},"\"Ali\"",[75,8886,8838],{"class":7398},[75,8888,8889,8891,8893,8895,8897,8899,8901,8903,8905,8908],{"class":7383,"line":7480},[75,8890,8816],{"class":7398},[75,8892,8062],{"class":7394},[75,8894,7698],{"class":7398},[75,8896,8823],{"class":7387},[75,8898,8536],{"class":7394},[75,8900,7698],{"class":7398},[75,8902,8348],{"class":8127},[75,8904,7298],{"class":7398},[75,8906,8907],{"class":7589},"\"Bala\"",[75,8909,8838],{"class":7398},[75,8911,8912],{"class":7383,"line":7485},[75,8913,7433],{"class":7398},[75,8915,8916,8919,8921,8924,8926],{"class":7383,"line":7498},[75,8917,8918],{"class":7398},"Customer c ",[75,8920,7421],{"class":7387},[75,8922,8923],{"class":7398}," customerQueue.",[75,8925,8052],{"class":7394},[75,8927,7549],{"class":7398},[75,8929,8930,8933,8936,8939,8942],{"class":7383,"line":7516},[75,8931,8932],{"class":7387},"if",[75,8934,8935],{"class":7398}," (c ",[75,8937,8938],{"class":7387},"!=",[75,8940,8941],{"class":8127}," null",[75,8943,7455],{"class":7398},[75,8945,8946,8949,8951,8953,8955,8957,8959,8962],{"class":7383,"line":7521},[75,8947,8948],{"class":7398},"  System.out.",[75,8950,7767],{"class":7394},[75,8952,8704],{"class":7398},[75,8954,8649],{"class":7394},[75,8956,8276],{"class":7398},[75,8958,8302],{"class":7387},[75,8960,8961],{"class":7589}," \" is in the queue\"",[75,8963,7738],{"class":7398},[75,8965,8966,8969,8972,8974,8976,8978,8981,8983,8985],{"class":7383,"line":7526},[75,8967,8968],{"class":7387},"  while",[75,8970,8971],{"class":7398}," ((c ",[75,8973,7421],{"class":7387},[75,8975,8923],{"class":7398},[75,8977,8049],{"class":7394},[75,8979,8980],{"class":7398},"()) ",[75,8982,8938],{"class":7387},[75,8984,8941],{"class":8127},[75,8986,7455],{"class":7398},[75,8988,8989,8991,8993],{"class":7383,"line":7539},[75,8990,8254],{"class":7398},[75,8992,7767],{"class":7394},[75,8994,8995],{"class":7398},"(c);\n",[75,8997,8998],{"class":7383,"line":7552},[75,8999,8490],{"class":7398},[75,9001,9002],{"class":7383,"line":7557},[75,9003,7610],{"class":7398},[12,9005,9006],{},"Output:",[6972,9008,9011],{"className":9009,"code":9010,"language":6977},[6975],"Bala is in the queue\nCustomer [id=4, name=Bala]\nCustomer [id=3, name=Donald]\nCustomer [id=2, name=Ali]\nCustomer [id=1, name=Chong]\n",[6979,9012,9010],{"__ignoreMap":180},[12,9014,9015,9016,9019,9020,9023],{},"Because we used ",[6979,9017,9018],{},"reverseOrder()",", the ",[106,9021,9022],{},"biggest id wins",". Bala (id=4) is the VIP.",[7351,9025,9026,9035,9058],{},[12,9027,9028,7206,9032],{},[106,9029,9030],{},[6979,9031,8522],{},[106,9033,9034],{},"cheat sheet:",[23,9036,9037,9047,9050],{},[26,9038,9039,9040,9043,9044],{},"Returns negative → ",[6979,9041,9042],{},"this"," is \"less than\" ",[6979,9045,9046],{},"other",[26,9048,9049],{},"Returns zero → equal priority",[26,9051,9052,9053,9055,9056],{},"Returns positive → ",[6979,9054,9042],{}," is \"greater than\" ",[6979,9057,9046],{},[12,9059,9060,9061,9063],{},"By default, \"greater\" = \"later out\". Reverse it with a ",[6979,9062,8107],{}," if you want the largest first.",[6964,9065,9067],{"id":9066},"_11-quick-check-can-you-answer-these","11. Quick check — can you answer these?",[12,9069,9070],{},"Before moving on, see if you can answer these without scrolling back up.",[3603,9072,9073,9076,9079,9088,9095],{},[26,9074,9075],{},"Where in a queue do you insert? Where do you delete?",[26,9077,9078],{},"What does FIFO stand for?",[26,9080,9081,9082,9084,9085,9087],{},"Why is a ",[6979,9083,7066],{}," a better choice than an ",[6979,9086,7193],{}," to implement a queue?",[26,9089,9090,9091,9094],{},"What are the two preferred methods to add and remove in ",[6979,9092,9093],{},"java.util.PriorityQueue"," (the ones that don't throw)?",[26,9096,9097,9098,9101,9102,9105,9106,9108],{},"If you put ",[6979,9099,9100],{},"5, 1, 8, 3"," into a default ",[6979,9103,9104],{},"PriorityQueue\u003CInteger>"," and call ",[6979,9107,7951],{}," four times, what comes out and in what order?",[12,9110,9111],{},"Answers",[3603,9113,9114,9117,9120,9126,9135],{},[26,9115,9116],{},"Insert at the rear (back\u002Ftail). Delete from the front (head).",[26,9118,9119],{},"First In, First Out.",[26,9121,9122,9123,9125],{},"Removing from the front of an ",[6979,9124,7193],{}," is O(n) because every other element has to shift. With a doubly-linked list, head removal is O(1).",[26,9127,9128,9131,9132,9134],{},[6979,9129,9130],{},"offer(e)"," to add, ",[6979,9133,7951],{}," to remove.",[26,9136,9137,9140,9141,9144],{},[6979,9138,9139],{},"1, 3, 5, 8"," — natural ordering for ",[6979,9142,9143],{},"Integer"," is ascending, so smallest comes out first.",[6964,9146,9148],{"id":9147},"_12-exercise-build-your-own","12. Exercise — build your own",[12,9150,9151,9152,9155,9156,9159],{},"Want a workout? Try implementing a queue using an ",[106,9153,9154],{},"array"," instead of a linked list. Call it ",[6979,9157,9158],{},"ArrayQueue\u003CE>",". You'll need to think about:",[23,9161,9162,9165,9182],{},[26,9163,9164],{},"Two index pointers — one for the front, one for the rear.",[26,9166,9167,9168,9170,9171,9174,9175,9177,9178,9181],{},"What happens when ",[6979,9169,7126],{}," reaches the end of the array? (Hint: a ",[106,9172,9173],{},"circular array"," wraps ",[6979,9176,7126],{}," back to ",[6979,9179,9180],{},"0",".)",[26,9183,9184,9185,9188],{},"How and when to ",[6979,9186,9187],{},"resize()"," when the array fills up.\nSketch of the API:",[6972,9190,9192],{"className":7376,"code":9191,"language":7378,"meta":180,"style":180},"public class ArrayQueue\u003CE> {\n    public ArrayQueue();\n    public ArrayQueue(int initialCapacity);\n    public void enqueue(E e);\n    public E    dequeue();\n    public E    getElement();   \u002F\u002F peek\n    public boolean isEmpty();\n    public int  size();\n    private void resize();      \u002F\u002F grow when full\n}\n",[6979,9193,9194,9209,9217,9232,9246,9257,9272,9284,9295,9310],{"__ignoreMap":180},[75,9195,9196,9198,9200,9203,9205,9207],{"class":7383,"line":7384},[75,9197,7388],{"class":7387},[75,9199,7391],{"class":7387},[75,9201,9202],{"class":7394}," ArrayQueue",[75,9204,7399],{"class":7398},[75,9206,7402],{"class":7387},[75,9208,7405],{"class":7398},[75,9210,9211,9213,9215],{"class":7383,"line":181},[75,9212,7439],{"class":7387},[75,9214,9202],{"class":7394},[75,9216,7549],{"class":7398},[75,9218,9219,9221,9223,9225,9227,9230],{"class":7383,"line":7430},[75,9220,7439],{"class":7387},[75,9222,9202],{"class":7394},[75,9224,7698],{"class":7398},[75,9226,8020],{"class":7387},[75,9228,9229],{"class":7451}," initialCapacity",[75,9231,7738],{"class":7398},[75,9233,9234,9236,9238,9240,9242,9244],{"class":7383,"line":7436},[75,9235,7439],{"class":7387},[75,9237,7442],{"class":7387},[75,9239,7445],{"class":7394},[75,9241,7448],{"class":7398},[75,9243,7452],{"class":7451},[75,9245,7738],{"class":7398},[75,9247,9248,9250,9253,9255],{"class":7383,"line":7458},[75,9249,7439],{"class":7387},[75,9251,9252],{"class":7398}," E    ",[75,9254,7163],{"class":7394},[75,9256,7549],{"class":7398},[75,9258,9259,9261,9263,9266,9269],{"class":7383,"line":7474},[75,9260,7439],{"class":7387},[75,9262,9252],{"class":7398},[75,9264,9265],{"class":7394},"getElement",[75,9267,9268],{"class":7398},"();   ",[75,9270,9271],{"class":7470},"\u002F\u002F peek\n",[75,9273,9274,9276,9279,9282],{"class":7383,"line":7480},[75,9275,7439],{"class":7387},[75,9277,9278],{"class":7387}," boolean",[75,9280,9281],{"class":7394}," isEmpty",[75,9283,7549],{"class":7398},[75,9285,9286,9288,9290,9293],{"class":7383,"line":7485},[75,9287,7439],{"class":7387},[75,9289,7531],{"class":7387},[75,9291,9292],{"class":7394},"  size",[75,9294,7549],{"class":7398},[75,9296,9297,9299,9301,9304,9307],{"class":7383,"line":7498},[75,9298,7410],{"class":7387},[75,9300,7442],{"class":7387},[75,9302,9303],{"class":7394}," resize",[75,9305,9306],{"class":7398},"();      ",[75,9308,9309],{"class":7470},"\u002F\u002F grow when full\n",[75,9311,9312],{"class":7383,"line":7516},[75,9313,7610],{"class":7398},[12,9315,9316],{},"This is a classic interview-style exercise. You learn a lot by getting the circular-array bookkeeping right.",[6964,9318,9320],{"id":9319},"_13-where-to-go-next","13. Where to go next",[12,9322,9323],{},"Queues show up everywhere once you start looking — task schedulers, BFS in graph algorithms, print spoolers, message brokers, request handling in web servers. Priority queues power Dijkstra's shortest-path algorithm and A* search. The simple FIFO idea is doing a lot of work behind the scenes in real systems.",[12,9325,9326,9327,9330],{},"Coming up next in this series: ",[106,9328,9329],{},"trees"," — the structure that makes priority queues fast under the hood (via something called a binary heap), and a whole lot more.",[9332,9333,9334],"style",{},"html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}",{"title":180,"searchDepth":181,"depth":181,"links":9336},[9337,9338,9339,9342,9343,9350,9354,9357,9361,9362,9363,9364,9365],{"id":6966,"depth":181,"text":6967},{"id":7012,"depth":181,"text":7013},{"id":7079,"depth":181,"text":7080,"children":9340},[9341],{"id":7168,"depth":7430,"text":7169},{"id":7186,"depth":181,"text":7187},{"id":7255,"depth":181,"text":9344,"children":9345},"5. Two ways to wire a GenericQueue to a LinkedList",[9346,9348],{"id":7268,"depth":7430,"text":9347},"Option A: Inheritance — GenericQueue extends LinkedList",{"id":7308,"depth":7430,"text":9349},"Option B: Composition — GenericQueue has-a LinkedList",{"id":7365,"depth":181,"text":9351,"children":9352},"6. GenericQueue\u003CE> — the code",[9353],{"id":7660,"depth":7430,"text":7661},{"id":7819,"depth":181,"text":7820,"children":9355},[9356],{"id":7862,"depth":7430,"text":7863},{"id":7887,"depth":181,"text":9358,"children":9359},"8. Java's built-in PriorityQueue",[9360],{"id":8069,"depth":7430,"text":8070},{"id":8111,"depth":181,"text":8112},{"id":8504,"depth":181,"text":8505},{"id":9066,"depth":181,"text":9067},{"id":9147,"depth":181,"text":9148},{"id":9319,"depth":181,"text":9320},"2026-05-14","If you've ever stood in line at a bubble tea shop, you already understand a queue. That's basically the whole idea — but in code, this simple concept turns into one of the most useful data structures you'll meet. In this post we'll walk through what a queue is, how to implement one in Java, and then level it up into a priority queue.",{},"\u002Fblog\u002FQueues",{"title":6945,"description":9367},{"loc":9369},"blog\u002FQueues","fSS6ss1_gzB_hU6o1S-NfDhbxydCl3wC9KEyOZLDRiw",{"id":9375,"title":9376,"author":6946,"body":9377,"date":13169,"description":13170,"draft":185,"edited_at":13169,"extension":186,"featured_image":194,"meta":13171,"navigation":189,"path":13172,"pinned":185,"seo":13173,"sitemap":13174,"stem":13175,"tags":194,"__hash__":13176},"blog\u002Fblog\u002FLinked-Lists-&-Doubly-Linked-Lists.md","Linked Lists & Doubly Linked Lists",{"type":9,"value":9378,"toc":13103},[9379,9395,9399,9405,9420,9426,9432,9453,9480,9484,9489,9509,9520,9539,9543,9546,9551,9574,9579,9594,9597,9601,9604,9615,9621,9627,9634,9638,9641,9655,9730,9733,9739,9742,9760,9769,9777,9781,9793,9797,9838,9842,9873,9876,9882,9890,9894,9923,9932,9938,9942,9969,9974,9983,9987,9998,10056,10059,10067,10077,10142,10145,10152,10253,10265,10272,10373,10379,10382,10579,10594,10601,10704,10711,10724,10934,10941,10948,11139,11142,11146,11163,11166,11170,11187,11193,11196,11200,11211,11217,11221,11231,11237,11241,11253,11256,11262,11269,11273,11276,11300,11304,11307,11319,11323,11327,11483,11487,11599,11603,11606,11631,11638,11647,11764,11771,11774,11886,11893,12113,12136,12139,12163,12167,12171,12259,12263,12342,12351,12355,12358,12364,12480,12486,12493,12602,12605,12611,12866,12872,12901,12903,12910,13037,13040,13044,13047,13093,13100],[12,9380,9381,9382,9384,9385,9387,9388,9390,9391,9394],{},"If you've ever used ",[6979,9383,7193],{}," in Java and wondered what the ",[7289,9386,9046],{}," list — ",[6979,9389,7066],{}," — is doing differently, this post is for you. We're going to walk through what linked lists are, why they exist alongside arrays, and how to build one from scratch. Then we'll level up to ",[106,9392,9393],{},"doubly linked lists",", which are a small twist on the idea with surprisingly nice properties.",[6964,9396,9398],{"id":9397},"_1-where-linked-lists-live-the-java-collection-framework","1. Where Linked Lists Live: The Java Collection Framework",[12,9400,9401,9402,9404],{},"Before diving in, it helps to know where ",[6979,9403,7066],{}," sits in Java's standard library.",[12,9406,9407,9408,9411,9412,9415,9416,9419],{},"A ",[106,9409,9410],{},"collection"," is just a container object that holds a group of other objects (called ",[7289,9413,9414],{},"elements","). Java's ",[6979,9417,9418],{},"java.util"," package has an entire hierarchy of these:",[6972,9421,9424],{"className":9422,"code":9423,"language":6977},[6975],"Iterable (interface)\n   └── Collection (interface)\n         ├── List (interface)\n         │     ├── ArrayList\n         │     ├── LinkedList   ← our star today\n         │     └── Vector → Stack\n         ├── Queue (interface)\n         │     ├── PriorityQueue\n         │     └── Deque → ArrayDeque\n         └── Set (interface)\n               ├── HashSet\n               ├── LinkedHashSet\n               └── SortedSet → TreeSet\n",[6979,9425,9423],{"__ignoreMap":180},[12,9427,9428],{},[1930,9429],{"alt":9430,"src":9431},"0.54","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1778666021685-Screenshot_2026-05-13_at_5.53.18_PM.png",[12,9433,9434,9435,9438,9439,9442,9443,9446,9447,9449,9450,9452],{},"The orange boxes in the picture are ",[106,9436,9437],{},"interfaces"," (contracts that say ",[7289,9440,9441],{},"what"," operations exist), and the blue ones are ",[106,9444,9445],{},"classes"," (actual implementations). ",[6979,9448,7066],{}," is one concrete way to implement the ",[6979,9451,7041],{}," interface.",[12,9454,9455,9456,9458,9459,7298,9461,7298,9463,7298,9466,7298,9468,7298,9471,9474,9475,7067,9477,9479],{},"The ",[6979,9457,7049],{}," interface defines operations every collection should support — things like ",[6979,9460,8062],{},[6979,9462,8065],{},[6979,9464,9465],{},"contains",[6979,9467,7546],{},[6979,9469,9470],{},"isEmpty",[6979,9472,9473],{},"iterator",", and so on. Whether you use ",[6979,9476,7193],{},[6979,9478,7066],{},", you get all these methods.",[6964,9481,9483],{"id":9482},"_2-what-is-a-list-conceptually","2. What Is a List, Conceptually?",[12,9485,9407,9486,9488],{},[106,9487,7652],{}," is an Abstract Data Type (ADT) that stores data in sequential order. Think of any everyday list — a class roster, a to-do list, books on a shelf. The common operations we care about are:",[23,9490,9491,9494,9497,9500,9503,9506],{},[26,9492,9493],{},"Retrieve an element",[26,9495,9496],{},"Insert a new element",[26,9498,9499],{},"Delete an element",[26,9501,9502],{},"Find how many elements are in the list",[26,9504,9505],{},"Check if an element exists",[26,9507,9508],{},"Check if the list is empty",[12,9510,9511,9512,9515,9516,9519],{},"The key word above is ",[106,9513,9514],{},"abstract",". A \"list\" doesn't say ",[7289,9517,9518],{},"how"," the data is stored — only what you can do with it. There are two main ways Java actually implements lists under the hood:",[3603,9521,9522,9531],{},[26,9523,9524,9527,9528,9530],{},[106,9525,9526],{},"Using an array"," — ",[6979,9529,7193],{},". The array grows dynamically: when it gets full, Java creates a bigger one and copies everything over.",[26,9532,9533,9527,9536,9538],{},[106,9534,9535],{},"Using a linked structure",[6979,9537,7066],{},". Each element lives in its own little container (a \"node\"), and the nodes are chained together with references.",[7166,9540,9542],{"id":9541},"array-or-linked-list-which-should-you-use","Array or Linked List? Which Should You Use?",[12,9544,9545],{},"Both work. They have different trade-offs:",[12,9547,9548],{},[106,9549,9550],{},"Use an array (ArrayList) when:",[23,9552,9553,9560,9567],{},[26,9554,9555,9556,9559],{},"You need fast access by index (",[6979,9557,9558],{},"get(5)"," is instant — O(1))",[26,9561,9562,9563,9566],{},"You mostly add to the ",[7289,9564,9565],{},"end"," of the list",[26,9568,9569,9570,9573],{},"❌ Inserting or deleting in the ",[7289,9571,9572],{},"middle"," is slow, because everything after it has to shift",[12,9575,9576],{},[106,9577,9578],{},"Use a linked list when:",[23,9580,9581,9588,9591],{},[26,9582,9583,9584,9587],{},"You frequently add or remove elements ",[7289,9585,9586],{},"anywhere"," in the list",[26,9589,9590],{},"You don't need random access by index very often",[26,9592,9593],{},"❌ Getting the 500th element requires walking through 500 nodes",[12,9595,9596],{},"A good rule of thumb: arrays are like books on a shelf (easy to grab the 5th one, painful to insert a new book at position 2). Linked lists are like a chain of paper clips (easy to splice in or pull out anywhere, but you have to start from one end to find the middle one).",[6964,9598,9600],{"id":9599},"_3-introducing-the-linked-list-the-pop-chain-analogy","3. Introducing the Linked List: The Pop-Chain Analogy",[12,9602,9603],{},"Imagine a child's toy \"pop chain\" — each piece has a connector on one side and a socket on the other. To build a chain, you push the connector of one piece into the socket of the next.",[12,9605,9606,9607,9610,9611,9614],{},"A linked list works exactly like this. Each ",[106,9608,9609],{},"node"," is one piece, and each piece is ",[7289,9612,9613],{},"linked"," to the next.",[12,9616,9617,9620],{},[106,9618,9619],{},"Inserting"," a piece anywhere in the chain: you just break one connection, pop in the new piece, and reconnect. The other pieces don't move.",[12,9622,9623,9626],{},[106,9624,9625],{},"Removing"," a piece: break the two connections around it, take it out, then bridge the two neighbors back together.",[12,9628,9629,9630,9633],{},"This is the magic of linked lists: ",[106,9631,9632],{},"insertion and deletion are local operations",". You only need to update the links right around the change — the rest of the list is undisturbed. With an array, inserting at the start means shifting every other element over by one. With a linked list, no shifting required.",[6964,9635,9637],{"id":9636},"_4-nodes-the-building-blocks","4. Nodes: The Building Blocks",[12,9639,9640],{},"Each node in a linked list has two parts:",[3603,9642,9643,9649],{},[26,9644,9455,9645,9648],{},[106,9646,9647],{},"element"," (the actual data — a String, an Integer, whatever)",[26,9650,9407,9651,9654],{},[106,9652,9653],{},"reference"," (or \"pointer\") to the next node\nIn Java, a node looks like this:",[6972,9656,9658],{"className":7376,"code":9657,"language":7378,"meta":180,"style":180},"class Node\u003CE> {\n    E element;        \u002F\u002F the data\n    Node\u003CE> next;     \u002F\u002F reference to the next node\n \n    public Node(E o) {\n        element = o;\n    }\n}\n",[6979,9659,9660,9674,9682,9695,9699,9712,9722,9726],{"__ignoreMap":180},[75,9661,9662,9665,9668,9670,9672],{"class":7383,"line":7384},[75,9663,9664],{"class":7387},"class",[75,9666,9667],{"class":7394}," Node",[75,9669,7399],{"class":7398},[75,9671,7402],{"class":7387},[75,9673,7405],{"class":7398},[75,9675,9676,9679],{"class":7383,"line":181},[75,9677,9678],{"class":7398},"    E element;        ",[75,9680,9681],{"class":7470},"\u002F\u002F the data\n",[75,9683,9684,9687,9689,9692],{"class":7383,"line":7430},[75,9685,9686],{"class":7398},"    Node\u003C",[75,9688,7402],{"class":7387},[75,9690,9691],{"class":7398},"> next;     ",[75,9693,9694],{"class":7470},"\u002F\u002F reference to the next node\n",[75,9696,9697],{"class":7383,"line":7436},[75,9698,7433],{"class":7398},[75,9700,9701,9703,9705,9707,9710],{"class":7383,"line":7458},[75,9702,7439],{"class":7387},[75,9704,9667],{"class":7394},[75,9706,7448],{"class":7398},[75,9708,9709],{"class":7451},"o",[75,9711,7455],{"class":7398},[75,9713,9714,9717,9719],{"class":7383,"line":7474},[75,9715,9716],{"class":7398},"        element ",[75,9718,7421],{"class":7387},[75,9720,9721],{"class":7398}," o;\n",[75,9723,9724],{"class":7383,"line":7480},[75,9725,7477],{"class":7398},[75,9727,9728],{"class":7383,"line":7485},[75,9729,7610],{"class":7398},[12,9731,9732],{},"A whole list, then, is a chain of these:",[6972,9734,9737],{"className":9735,"code":9736,"language":6977},[6975],"head → [Chicago | next] → [Denver | next] → [Dallas | null] ← tail\n",[6979,9738,9736],{"__ignoreMap":180},[12,9740,9741],{},"We keep two convenience references:",[23,9743,9744,9752],{},[26,9745,9746,9751],{},[106,9747,9748],{},[6979,9749,9750],{},"head"," — points to the first node",[26,9753,9754,9759],{},[106,9755,9756],{},[6979,9757,9758],{},"tail"," — points to the last node",[12,9761,9762,9763,7653,9766,9768],{},"The very last node's ",[6979,9764,9765],{},"next",[6979,9767,7962],{}," — that's how we know we've reached the end.",[7351,9770,9771],{},[12,9772,9773,9774,9776],{},"Heads up: each node can physically live anywhere in memory. The ",[6979,9775,9765],{}," pointers are what tie them together logically. This is different from an array, where elements are packed side-by-side in memory.",[6964,9778,9780],{"id":9779},"_5-building-a-list-step-by-step","5. Building a List Step by Step",[12,9782,9783,9784,7298,9787,7298,9790,6959],{},"Let's create a list of three city names: ",[6979,9785,9786],{},"\"Chicago\"",[6979,9788,9789],{},"\"Denver\"",[6979,9791,9792],{},"\"Dallas\"",[7166,9794,9796],{"id":9795},"step-1-start-empty","Step 1: Start empty",[6972,9798,9800],{"className":7376,"code":9799,"language":7378,"meta":180,"style":180},"Node\u003CString> head = null;\nNode\u003CString> tail = null;\n\u002F\u002F The list is empty.\n",[6979,9801,9802,9818,9833],{"__ignoreMap":180},[75,9803,9804,9807,9809,9812,9814,9816],{"class":7383,"line":7384},[75,9805,9806],{"class":7398},"Node\u003C",[75,9808,7674],{"class":7387},[75,9810,9811],{"class":7398},"> head ",[75,9813,7421],{"class":7387},[75,9815,8941],{"class":8127},[75,9817,8131],{"class":7398},[75,9819,9820,9822,9824,9827,9829,9831],{"class":7383,"line":181},[75,9821,9806],{"class":7398},[75,9823,7674],{"class":7387},[75,9825,9826],{"class":7398},"> tail ",[75,9828,7421],{"class":7387},[75,9830,8941],{"class":8127},[75,9832,8131],{"class":7398},[75,9834,9835],{"class":7383,"line":7430},[75,9836,9837],{"class":7470},"\u002F\u002F The list is empty.\n",[7166,9839,9841],{"id":9840},"step-2-add-the-first-node-chicago","Step 2: Add the first node (\"Chicago\")",[6972,9843,9845],{"className":7376,"code":9844,"language":7378,"meta":180,"style":180},"head = new Node\u003C>(\"Chicago\");\ntail = head;\n",[6979,9846,9847,9863],{"__ignoreMap":180},[75,9848,9849,9852,9854,9856,9859,9861],{"class":7383,"line":7384},[75,9850,9851],{"class":7398},"head ",[75,9853,7421],{"class":7387},[75,9855,7424],{"class":7387},[75,9857,9858],{"class":7398}," Node\u003C>(",[75,9860,9786],{"class":7589},[75,9862,7738],{"class":7398},[75,9864,9865,9868,9870],{"class":7383,"line":181},[75,9866,9867],{"class":7398},"tail ",[75,9869,7421],{"class":7387},[75,9871,9872],{"class":7398}," head;\n",[12,9874,9875],{},"After this:",[6972,9877,9880],{"className":9878,"code":9879,"language":6977},[6975],"head → [Chicago | null] ← tail\n",[6979,9881,9879],{"__ignoreMap":180},[12,9883,9884,9885,7042,9887,9889],{},"Both ",[6979,9886,9750],{},[6979,9888,9758],{}," point to the same single node.",[7166,9891,9893],{"id":9892},"step-3-add-the-second-node-denver","Step 3: Add the second node (\"Denver\")",[6972,9895,9897],{"className":7376,"code":9896,"language":7378,"meta":180,"style":180},"tail.next = new Node\u003C>(\"Denver\");\ntail = tail.next;\n",[6979,9898,9899,9914],{"__ignoreMap":180},[75,9900,9901,9904,9906,9908,9910,9912],{"class":7383,"line":7384},[75,9902,9903],{"class":7398},"tail.next ",[75,9905,7421],{"class":7387},[75,9907,7424],{"class":7387},[75,9909,9858],{"class":7398},[75,9911,9789],{"class":7589},[75,9913,7738],{"class":7398},[75,9915,9916,9918,9920],{"class":7383,"line":181},[75,9917,9867],{"class":7398},[75,9919,7421],{"class":7387},[75,9921,9922],{"class":7398}," tail.next;\n",[12,9924,9925,9926,9928,9929,9931],{},"We attach the new node to the current tail's ",[6979,9927,9765],{},", then move ",[6979,9930,9758],{}," forward.",[6972,9933,9936],{"className":9934,"code":9935,"language":6977},[6975],"head → [Chicago | next] → [Denver | null] ← tail\n",[6979,9937,9935],{"__ignoreMap":180},[7166,9939,9941],{"id":9940},"step-4-add-the-third-node-dallas","Step 4: Add the third node (\"Dallas\")",[6972,9943,9945],{"className":7376,"code":9944,"language":7378,"meta":180,"style":180},"tail.next = new Node\u003C>(\"Dallas\");\ntail = tail.next;\n",[6979,9946,9947,9961],{"__ignoreMap":180},[75,9948,9949,9951,9953,9955,9957,9959],{"class":7383,"line":7384},[75,9950,9903],{"class":7398},[75,9952,7421],{"class":7387},[75,9954,7424],{"class":7387},[75,9956,9858],{"class":7398},[75,9958,9792],{"class":7589},[75,9960,7738],{"class":7398},[75,9962,9963,9965,9967],{"class":7383,"line":181},[75,9964,9867],{"class":7398},[75,9966,7421],{"class":7387},[75,9968,9922],{"class":7398},[6972,9970,9972],{"className":9971,"code":9736,"language":6977},[6975],[6979,9973,9736],{"__ignoreMap":180},[12,9975,9976,9977,9979,9980,9982],{},"That's a linked list! Three nodes, chained together, with ",[6979,9978,9750],{}," at the front and ",[6979,9981,9758],{}," at the back.",[6964,9984,9986],{"id":9985},"_6-traversing-the-list","6. Traversing the List",[12,9988,9989,9990,9992,9993,9995,9996,7024],{},"To visit every node, we start at ",[6979,9991,9750],{}," and follow ",[6979,9994,9765],{}," pointers until we hit ",[6979,9997,7962],{},[6972,9999,10001],{"className":7376,"code":10000,"language":7378,"meta":180,"style":180},"Node\u003CE> current = head;\nwhile (current != null) {\n    System.out.println(current.element);\n    current = current.next;  \u002F\u002F step forward\n}\n",[6979,10002,10003,10016,10030,10039,10052],{"__ignoreMap":180},[75,10004,10005,10007,10009,10012,10014],{"class":7383,"line":7384},[75,10006,9806],{"class":7398},[75,10008,7402],{"class":7387},[75,10010,10011],{"class":7398},"> current ",[75,10013,7421],{"class":7387},[75,10015,9872],{"class":7398},[75,10017,10018,10021,10024,10026,10028],{"class":7383,"line":181},[75,10019,10020],{"class":7387},"while",[75,10022,10023],{"class":7398}," (current ",[75,10025,8938],{"class":7387},[75,10027,8941],{"class":8127},[75,10029,7455],{"class":7398},[75,10031,10032,10034,10036],{"class":7383,"line":7430},[75,10033,8254],{"class":7398},[75,10035,7767],{"class":7394},[75,10037,10038],{"class":7398},"(current.element);\n",[75,10040,10041,10044,10046,10049],{"class":7383,"line":7436},[75,10042,10043],{"class":7398},"    current ",[75,10045,7421],{"class":7387},[75,10047,10048],{"class":7398}," current.next;  ",[75,10050,10051],{"class":7470},"\u002F\u002F step forward\n",[75,10053,10054],{"class":7383,"line":7458},[75,10055,7610],{"class":7398},[12,10057,10058],{},"This is the bread-and-butter pattern of working with linked lists. Almost every operation uses some variant of this walk.",[6964,10060,10062,10063,10066],{"id":10061},"_7-building-our-own-mylinkedlist-class","7. Building Our Own ",[6979,10064,10065],{},"MyLinkedList"," Class",[12,10068,10069,10070,7298,10072,7153,10074,10076],{},"Let's wrap all this up into a proper class. It'll keep track of ",[6979,10071,9750],{},[6979,10073,9758],{},[6979,10075,7546],{},", and offer methods to add\u002Fremove elements:",[6972,10078,10080],{"className":7376,"code":10079,"language":7378,"meta":180,"style":180},"public class MyLinkedList\u003CE> {\n    private Node\u003CE> head;\n    private Node\u003CE> tail;\n    private int size;\n \n    \u002F\u002F ... methods below\n}\n",[6979,10081,10082,10097,10109,10120,10129,10133,10138],{"__ignoreMap":180},[75,10083,10084,10086,10088,10091,10093,10095],{"class":7383,"line":7384},[75,10085,7388],{"class":7387},[75,10087,7391],{"class":7387},[75,10089,10090],{"class":7394}," MyLinkedList",[75,10092,7399],{"class":7398},[75,10094,7402],{"class":7387},[75,10096,7405],{"class":7398},[75,10098,10099,10101,10104,10106],{"class":7383,"line":181},[75,10100,7410],{"class":7387},[75,10102,10103],{"class":7398}," Node\u003C",[75,10105,7402],{"class":7387},[75,10107,10108],{"class":7398},"> head;\n",[75,10110,10111,10113,10115,10117],{"class":7383,"line":7430},[75,10112,7410],{"class":7387},[75,10114,10103],{"class":7398},[75,10116,7402],{"class":7387},[75,10118,10119],{"class":7398},"> tail;\n",[75,10121,10122,10124,10126],{"class":7383,"line":7436},[75,10123,7410],{"class":7387},[75,10125,7531],{"class":7387},[75,10127,10128],{"class":7398}," size;\n",[75,10130,10131],{"class":7383,"line":7458},[75,10132,7433],{"class":7398},[75,10134,10135],{"class":7383,"line":7474},[75,10136,10137],{"class":7470},"    \u002F\u002F ... methods below\n",[75,10139,10140],{"class":7383,"line":7480},[75,10141,7610],{"class":7398},[12,10143,10144],{},"Now let's implement the core operations.",[7166,10146,10148,10151],{"id":10147},"addfirste-e-add-to-the-head",[6979,10149,10150],{},"addFirst(E e)"," — Add to the Head",[6972,10153,10155],{"className":7376,"code":10154,"language":7378,"meta":180,"style":180},"public void addFirst(E e) {\n    Node\u003CE> newNode = new Node\u003C>(e);\n    newNode.next = head;        \u002F\u002F new node points to current head\n    head = newNode;             \u002F\u002F new node becomes the head\n    size++;\n    if (tail == null)           \u002F\u002F if list was empty, new node is also tail\n        tail = head;\n}\n",[6979,10156,10157,10169,10185,10198,10211,10221,10240,10249],{"__ignoreMap":180},[75,10158,10159,10161,10163,10166],{"class":7383,"line":7384},[75,10160,7388],{"class":7387},[75,10162,7442],{"class":7387},[75,10164,10165],{"class":7394}," addFirst",[75,10167,10168],{"class":7398},"(E e) {\n",[75,10170,10171,10173,10175,10178,10180,10182],{"class":7383,"line":181},[75,10172,9686],{"class":7398},[75,10174,7402],{"class":7387},[75,10176,10177],{"class":7398},"> newNode ",[75,10179,7421],{"class":7387},[75,10181,7424],{"class":7387},[75,10183,10184],{"class":7398}," Node\u003C>(e);\n",[75,10186,10187,10190,10192,10195],{"class":7383,"line":7430},[75,10188,10189],{"class":7398},"    newNode.next ",[75,10191,7421],{"class":7387},[75,10193,10194],{"class":7398}," head;        ",[75,10196,10197],{"class":7470},"\u002F\u002F new node points to current head\n",[75,10199,10200,10203,10205,10208],{"class":7383,"line":7436},[75,10201,10202],{"class":7398},"    head ",[75,10204,7421],{"class":7387},[75,10206,10207],{"class":7398}," newNode;             ",[75,10209,10210],{"class":7470},"\u002F\u002F new node becomes the head\n",[75,10212,10213,10216,10219],{"class":7383,"line":7458},[75,10214,10215],{"class":7398},"    size",[75,10217,10218],{"class":7387},"++",[75,10220,8131],{"class":7398},[75,10222,10223,10226,10229,10232,10234,10237],{"class":7383,"line":7474},[75,10224,10225],{"class":7387},"    if",[75,10227,10228],{"class":7398}," (tail ",[75,10230,10231],{"class":7387},"==",[75,10233,8941],{"class":8127},[75,10235,10236],{"class":7398},")           ",[75,10238,10239],{"class":7470},"\u002F\u002F if list was empty, new node is also tail\n",[75,10241,10242,10245,10247],{"class":7383,"line":7480},[75,10243,10244],{"class":7398},"        tail ",[75,10246,7421],{"class":7387},[75,10248,9872],{"class":7398},[75,10250,10251],{"class":7383,"line":7485},[75,10252,7610],{"class":7398},[12,10254,10255,10256,7206,10258,10261,10262,10264],{},"The trick is to set the new node's ",[6979,10257,9765],{},[7289,10259,10260],{},"before"," reassigning ",[6979,10263,9750],{},". Otherwise we'd lose the rest of the list.",[7166,10266,10268,10271],{"id":10267},"addlaste-e-add-to-the-tail",[6979,10269,10270],{},"addLast(E e)"," — Add to the Tail",[6972,10273,10275],{"className":7376,"code":10274,"language":7378,"meta":180,"style":180},"public void addLast(E e) {\n    if (tail == null) {                  \u002F\u002F list is empty\n        head = tail = new Node\u003C>(e);\n    } else {\n        tail.next = new Node\u003C>(e);       \u002F\u002F old tail now points to new node\n        tail = tail.next;                \u002F\u002F new node becomes the tail\n    }\n    size++;\n}\n",[6979,10276,10277,10288,10304,10320,10330,10345,10357,10361,10369],{"__ignoreMap":180},[75,10278,10279,10281,10283,10286],{"class":7383,"line":7384},[75,10280,7388],{"class":7387},[75,10282,7442],{"class":7387},[75,10284,10285],{"class":7394}," addLast",[75,10287,10168],{"class":7398},[75,10289,10290,10292,10294,10296,10298,10301],{"class":7383,"line":181},[75,10291,10225],{"class":7387},[75,10293,10228],{"class":7398},[75,10295,10231],{"class":7387},[75,10297,8941],{"class":8127},[75,10299,10300],{"class":7398},") {                  ",[75,10302,10303],{"class":7470},"\u002F\u002F list is empty\n",[75,10305,10306,10309,10311,10314,10316,10318],{"class":7383,"line":7430},[75,10307,10308],{"class":7398},"        head ",[75,10310,7421],{"class":7387},[75,10312,10313],{"class":7398}," tail ",[75,10315,7421],{"class":7387},[75,10317,7424],{"class":7387},[75,10319,10184],{"class":7398},[75,10321,10322,10325,10328],{"class":7383,"line":7436},[75,10323,10324],{"class":7398},"    } ",[75,10326,10327],{"class":7387},"else",[75,10329,8147],{"class":7398},[75,10331,10332,10335,10337,10339,10342],{"class":7383,"line":7458},[75,10333,10334],{"class":7398},"        tail.next ",[75,10336,7421],{"class":7387},[75,10338,7424],{"class":7387},[75,10340,10341],{"class":7398}," Node\u003C>(e);       ",[75,10343,10344],{"class":7470},"\u002F\u002F old tail now points to new node\n",[75,10346,10347,10349,10351,10354],{"class":7383,"line":7474},[75,10348,10244],{"class":7398},[75,10350,7421],{"class":7387},[75,10352,10353],{"class":7398}," tail.next;                ",[75,10355,10356],{"class":7470},"\u002F\u002F new node becomes the tail\n",[75,10358,10359],{"class":7383,"line":7480},[75,10360,7477],{"class":7398},[75,10362,10363,10365,10367],{"class":7383,"line":7485},[75,10364,10215],{"class":7398},[75,10366,10218],{"class":7387},[75,10368,8131],{"class":7398},[75,10370,10371],{"class":7383,"line":7498},[75,10372,7610],{"class":7398},[7166,10374,10376,10378],{"id":10375},"addint-index-e-e-insert-at-a-specific-position",[6979,10377,7297],{}," — Insert at a Specific Position",[12,10380,10381],{},"This is the most general case. We handle the easy cases first, then walk to the position:",[6972,10383,10385],{"className":7376,"code":10384,"language":7378,"meta":180,"style":180},"public void add(int index, E e) {\n    if (index == 0) {\n        addFirst(e);\n    } else if (index >= size) {\n        addLast(e);\n    } else {\n        Node\u003CE> current = head;\n        for (int i = 1; i \u003C index; i++) {\n            current = current.next;       \u002F\u002F walk to the node BEFORE the index\n        }\n        Node\u003CE> temp = current.next;      \u002F\u002F remember what comes after\n        current.next = new Node\u003C>(e);     \u002F\u002F splice new node in\n        current.next.next = temp;         \u002F\u002F new node points to the rest\n        size++;\n    }\n}\n",[6979,10386,10387,10403,10416,10424,10441,10448,10456,10469,10499,10512,10517,10534,10549,10562,10571,10575],{"__ignoreMap":180},[75,10388,10389,10391,10393,10396,10398,10400],{"class":7383,"line":7384},[75,10390,7388],{"class":7387},[75,10392,7442],{"class":7387},[75,10394,10395],{"class":7394}," add",[75,10397,7698],{"class":7398},[75,10399,8020],{"class":7387},[75,10401,10402],{"class":7398}," index, E e) {\n",[75,10404,10405,10407,10410,10412,10414],{"class":7383,"line":181},[75,10406,10225],{"class":7387},[75,10408,10409],{"class":7398}," (index ",[75,10411,10231],{"class":7387},[75,10413,8282],{"class":8127},[75,10415,7455],{"class":7398},[75,10417,10418,10421],{"class":7383,"line":7430},[75,10419,10420],{"class":7394},"        addFirst",[75,10422,10423],{"class":7398},"(e);\n",[75,10425,10426,10428,10430,10433,10435,10438],{"class":7383,"line":7436},[75,10427,10324],{"class":7398},[75,10429,10327],{"class":7387},[75,10431,10432],{"class":7387}," if",[75,10434,10409],{"class":7398},[75,10436,10437],{"class":7387},">=",[75,10439,10440],{"class":7398}," size) {\n",[75,10442,10443,10446],{"class":7383,"line":7458},[75,10444,10445],{"class":7394},"        addLast",[75,10447,10423],{"class":7398},[75,10449,10450,10452,10454],{"class":7383,"line":7474},[75,10451,10324],{"class":7398},[75,10453,10327],{"class":7387},[75,10455,8147],{"class":7398},[75,10457,10458,10461,10463,10465,10467],{"class":7383,"line":7480},[75,10459,10460],{"class":7398},"        Node\u003C",[75,10462,7402],{"class":7387},[75,10464,10011],{"class":7398},[75,10466,7421],{"class":7387},[75,10468,9872],{"class":7398},[75,10470,10471,10474,10477,10479,10482,10484,10487,10490,10492,10495,10497],{"class":7383,"line":7485},[75,10472,10473],{"class":7387},"        for",[75,10475,10476],{"class":7398}," (",[75,10478,8020],{"class":7387},[75,10480,10481],{"class":7398}," i ",[75,10483,7421],{"class":7387},[75,10485,10486],{"class":8127}," 1",[75,10488,10489],{"class":7398},"; i ",[75,10491,7399],{"class":7387},[75,10493,10494],{"class":7398}," index; i",[75,10496,10218],{"class":7387},[75,10498,7455],{"class":7398},[75,10500,10501,10504,10506,10509],{"class":7383,"line":7498},[75,10502,10503],{"class":7398},"            current ",[75,10505,7421],{"class":7387},[75,10507,10508],{"class":7398}," current.next;       ",[75,10510,10511],{"class":7470},"\u002F\u002F walk to the node BEFORE the index\n",[75,10513,10514],{"class":7383,"line":7516},[75,10515,10516],{"class":7398},"        }\n",[75,10518,10519,10521,10523,10526,10528,10531],{"class":7383,"line":7521},[75,10520,10460],{"class":7398},[75,10522,7402],{"class":7387},[75,10524,10525],{"class":7398},"> temp ",[75,10527,7421],{"class":7387},[75,10529,10530],{"class":7398}," current.next;      ",[75,10532,10533],{"class":7470},"\u002F\u002F remember what comes after\n",[75,10535,10536,10539,10541,10543,10546],{"class":7383,"line":7526},[75,10537,10538],{"class":7398},"        current.next ",[75,10540,7421],{"class":7387},[75,10542,7424],{"class":7387},[75,10544,10545],{"class":7398}," Node\u003C>(e);     ",[75,10547,10548],{"class":7470},"\u002F\u002F splice new node in\n",[75,10550,10551,10554,10556,10559],{"class":7383,"line":7539},[75,10552,10553],{"class":7398},"        current.next.next ",[75,10555,7421],{"class":7387},[75,10557,10558],{"class":7398}," temp;         ",[75,10560,10561],{"class":7470},"\u002F\u002F new node points to the rest\n",[75,10563,10564,10567,10569],{"class":7383,"line":7552},[75,10565,10566],{"class":7398},"        size",[75,10568,10218],{"class":7387},[75,10570,8131],{"class":7398},[75,10572,10573],{"class":7383,"line":7557},[75,10574,7477],{"class":7398},[75,10576,10577],{"class":7383,"line":7562},[75,10578,7610],{"class":7398},[12,10580,10581,10582,10585,10586,10589,10590,10593],{},"The key idea: to insert ",[7289,10583,10584],{},"at"," index ",[6979,10587,10588],{},"i",", we stop ",[7289,10591,10592],{},"one node before",". We then \"splice\" — break the existing link and reconnect on both sides of the new node.",[7166,10595,10597,10600],{"id":10596},"removefirst-remove-the-head",[6979,10598,10599],{},"removeFirst()"," — Remove the Head",[6972,10602,10604],{"className":7376,"code":10603,"language":7378,"meta":180,"style":180},"public E removeFirst() {\n    if (size == 0) return null;\n    Node\u003CE> temp = head;\n    head = head.next;       \u002F\u002F second node becomes head\n    size--;\n    if (head == null) tail = null;   \u002F\u002F list is now empty\n    return temp.element;\n}\n",[6979,10605,10606,10616,10636,10648,10660,10669,10693,10700],{"__ignoreMap":180},[75,10607,10608,10610,10612,10614],{"class":7383,"line":7384},[75,10609,7388],{"class":7387},[75,10611,7490],{"class":7398},[75,10613,7507],{"class":7394},[75,10615,7495],{"class":7398},[75,10617,10618,10620,10623,10625,10627,10630,10632,10634],{"class":7383,"line":181},[75,10619,10225],{"class":7387},[75,10621,10622],{"class":7398}," (size ",[75,10624,10231],{"class":7387},[75,10626,8282],{"class":8127},[75,10628,10629],{"class":7398},") ",[75,10631,8636],{"class":7387},[75,10633,8941],{"class":8127},[75,10635,8131],{"class":7398},[75,10637,10638,10640,10642,10644,10646],{"class":7383,"line":7430},[75,10639,9686],{"class":7398},[75,10641,7402],{"class":7387},[75,10643,10525],{"class":7398},[75,10645,7421],{"class":7387},[75,10647,9872],{"class":7398},[75,10649,10650,10652,10654,10657],{"class":7383,"line":7436},[75,10651,10202],{"class":7398},[75,10653,7421],{"class":7387},[75,10655,10656],{"class":7398}," head.next;       ",[75,10658,10659],{"class":7470},"\u002F\u002F second node becomes head\n",[75,10661,10662,10664,10667],{"class":7383,"line":7458},[75,10663,10215],{"class":7398},[75,10665,10666],{"class":7387},"--",[75,10668,8131],{"class":7398},[75,10670,10671,10673,10676,10678,10680,10683,10685,10687,10690],{"class":7383,"line":7474},[75,10672,10225],{"class":7387},[75,10674,10675],{"class":7398}," (head ",[75,10677,10231],{"class":7387},[75,10679,8941],{"class":8127},[75,10681,10682],{"class":7398},") tail ",[75,10684,7421],{"class":7387},[75,10686,8941],{"class":8127},[75,10688,10689],{"class":7398},";   ",[75,10691,10692],{"class":7470},"\u002F\u002F list is now empty\n",[75,10694,10695,10697],{"class":7383,"line":7480},[75,10696,8689],{"class":7387},[75,10698,10699],{"class":7398}," temp.element;\n",[75,10701,10702],{"class":7383,"line":7485},[75,10703,7610],{"class":7398},[7166,10705,10707,10710],{"id":10706},"removelast-remove-the-tail",[6979,10708,10709],{},"removeLast()"," — Remove the Tail",[12,10712,10713,10714,10717,10718,10720,10721,10723],{},"Trickier in a singly linked list! Why? Because to remove the tail, we need to update the ",[7289,10715,10716],{},"second-to-last"," node's ",[6979,10719,9765],{}," to ",[6979,10722,7962],{},". But the tail doesn't have a \"previous\" pointer, so we have to walk from the head to find it.",[6972,10725,10727],{"className":7376,"code":10726,"language":7378,"meta":180,"style":180},"public E removeLast() {\n    if (size == 0) return null;\n    if (size == 1) {                     \u002F\u002F only one node\n        Node\u003CE> temp = head;\n        head = tail = null;\n        size = 0;\n        return temp.element;\n    }\n    Node\u003CE> current = head;\n    for (int i = 0; i \u003C size - 2; i++)   \u002F\u002F stop at second-to-last\n        current = current.next;\n    Node\u003CE> temp = tail;\n    tail = current;\n    tail.next = null;\n    size--;\n    return temp.element;\n}\n",[6979,10728,10729,10740,10758,10774,10786,10800,10811,10817,10821,10833,10872,10882,10895,10905,10916,10924,10930],{"__ignoreMap":180},[75,10730,10731,10733,10735,10738],{"class":7383,"line":7384},[75,10732,7388],{"class":7387},[75,10734,7490],{"class":7398},[75,10736,10737],{"class":7394},"removeLast",[75,10739,7495],{"class":7398},[75,10741,10742,10744,10746,10748,10750,10752,10754,10756],{"class":7383,"line":181},[75,10743,10225],{"class":7387},[75,10745,10622],{"class":7398},[75,10747,10231],{"class":7387},[75,10749,8282],{"class":8127},[75,10751,10629],{"class":7398},[75,10753,8636],{"class":7387},[75,10755,8941],{"class":8127},[75,10757,8131],{"class":7398},[75,10759,10760,10762,10764,10766,10768,10771],{"class":7383,"line":7430},[75,10761,10225],{"class":7387},[75,10763,10622],{"class":7398},[75,10765,10231],{"class":7387},[75,10767,10486],{"class":8127},[75,10769,10770],{"class":7398},") {                     ",[75,10772,10773],{"class":7470},"\u002F\u002F only one node\n",[75,10775,10776,10778,10780,10782,10784],{"class":7383,"line":7436},[75,10777,10460],{"class":7398},[75,10779,7402],{"class":7387},[75,10781,10525],{"class":7398},[75,10783,7421],{"class":7387},[75,10785,9872],{"class":7398},[75,10787,10788,10790,10792,10794,10796,10798],{"class":7383,"line":7458},[75,10789,10308],{"class":7398},[75,10791,7421],{"class":7387},[75,10793,10313],{"class":7398},[75,10795,7421],{"class":7387},[75,10797,8941],{"class":8127},[75,10799,8131],{"class":7398},[75,10801,10802,10805,10807,10809],{"class":7383,"line":7474},[75,10803,10804],{"class":7398},"        size ",[75,10806,7421],{"class":7387},[75,10808,8282],{"class":8127},[75,10810,8131],{"class":7398},[75,10812,10813,10815],{"class":7383,"line":7480},[75,10814,7501],{"class":7387},[75,10816,10699],{"class":7398},[75,10818,10819],{"class":7383,"line":7485},[75,10820,7477],{"class":7398},[75,10822,10823,10825,10827,10829,10831],{"class":7383,"line":7498},[75,10824,9686],{"class":7398},[75,10826,7402],{"class":7387},[75,10828,10011],{"class":7398},[75,10830,7421],{"class":7387},[75,10832,9872],{"class":7398},[75,10834,10835,10838,10840,10842,10844,10846,10848,10850,10852,10855,10858,10861,10864,10866,10869],{"class":7383,"line":7516},[75,10836,10837],{"class":7387},"    for",[75,10839,10476],{"class":7398},[75,10841,8020],{"class":7387},[75,10843,10481],{"class":7398},[75,10845,7421],{"class":7387},[75,10847,8282],{"class":8127},[75,10849,10489],{"class":7398},[75,10851,7399],{"class":7387},[75,10853,10854],{"class":7398}," size ",[75,10856,10857],{"class":7387},"-",[75,10859,10860],{"class":8127}," 2",[75,10862,10863],{"class":7398},"; i",[75,10865,10218],{"class":7387},[75,10867,10868],{"class":7398},")   ",[75,10870,10871],{"class":7470},"\u002F\u002F stop at second-to-last\n",[75,10873,10874,10877,10879],{"class":7383,"line":7521},[75,10875,10876],{"class":7398},"        current ",[75,10878,7421],{"class":7387},[75,10880,10881],{"class":7398}," current.next;\n",[75,10883,10884,10886,10888,10890,10892],{"class":7383,"line":7526},[75,10885,9686],{"class":7398},[75,10887,7402],{"class":7387},[75,10889,10525],{"class":7398},[75,10891,7421],{"class":7387},[75,10893,10894],{"class":7398}," tail;\n",[75,10896,10897,10900,10902],{"class":7383,"line":7539},[75,10898,10899],{"class":7398},"    tail ",[75,10901,7421],{"class":7387},[75,10903,10904],{"class":7398}," current;\n",[75,10906,10907,10910,10912,10914],{"class":7383,"line":7552},[75,10908,10909],{"class":7398},"    tail.next ",[75,10911,7421],{"class":7387},[75,10913,8941],{"class":8127},[75,10915,8131],{"class":7398},[75,10917,10918,10920,10922],{"class":7383,"line":7557},[75,10919,10215],{"class":7398},[75,10921,10666],{"class":7387},[75,10923,8131],{"class":7398},[75,10925,10926,10928],{"class":7383,"line":7562},[75,10927,8689],{"class":7387},[75,10929,10699],{"class":7398},[75,10931,10932],{"class":7383,"line":7571},[75,10933,7610],{"class":7398},[12,10935,10936,10937,10940],{},"This is O(n) — not great. This is exactly the kind of problem that the ",[106,10938,10939],{},"doubly linked list"," (coming later) solves.",[7166,10942,10944,10947],{"id":10943},"removeint-index-remove-at-a-specific-position",[6979,10945,10946],{},"remove(int index)"," — Remove at a Specific Position",[6972,10949,10951],{"className":7376,"code":10950,"language":7378,"meta":180,"style":180},"public E remove(int index) {\n    if (index \u003C 0 || index >= size) return null;\n    if (index == 0) return removeFirst();\n    if (index == size - 1) return removeLast();\n \n    Node\u003CE> previous = head;\n    for (int i = 1; i \u003C index; i++)\n        previous = previous.next;        \u002F\u002F stop just before the target\n    Node\u003CE> current = previous.next;     \u002F\u002F this is the node to remove\n    previous.next = current.next;        \u002F\u002F bypass it\n    size--;\n    return current.element;\n}\n",[6979,10952,10953,10968,10994,11013,11036,11040,11053,11078,11091,11107,11120,11128,11135],{"__ignoreMap":180},[75,10954,10955,10957,10959,10961,10963,10965],{"class":7383,"line":7384},[75,10956,7388],{"class":7387},[75,10958,7490],{"class":7398},[75,10960,8065],{"class":7394},[75,10962,7698],{"class":7398},[75,10964,8020],{"class":7387},[75,10966,10967],{"class":7398}," index) {\n",[75,10969,10970,10972,10974,10976,10978,10981,10983,10985,10988,10990,10992],{"class":7383,"line":181},[75,10971,10225],{"class":7387},[75,10973,10409],{"class":7398},[75,10975,7399],{"class":7387},[75,10977,8282],{"class":8127},[75,10979,10980],{"class":7387}," ||",[75,10982,10585],{"class":7398},[75,10984,10437],{"class":7387},[75,10986,10987],{"class":7398}," size) ",[75,10989,8636],{"class":7387},[75,10991,8941],{"class":8127},[75,10993,8131],{"class":7398},[75,10995,10996,10998,11000,11002,11004,11006,11008,11011],{"class":7383,"line":7430},[75,10997,10225],{"class":7387},[75,10999,10409],{"class":7398},[75,11001,10231],{"class":7387},[75,11003,8282],{"class":8127},[75,11005,10629],{"class":7398},[75,11007,8636],{"class":7387},[75,11009,11010],{"class":7394}," removeFirst",[75,11012,7549],{"class":7398},[75,11014,11015,11017,11019,11021,11023,11025,11027,11029,11031,11034],{"class":7383,"line":7436},[75,11016,10225],{"class":7387},[75,11018,10409],{"class":7398},[75,11020,10231],{"class":7387},[75,11022,10854],{"class":7398},[75,11024,10857],{"class":7387},[75,11026,10486],{"class":8127},[75,11028,10629],{"class":7398},[75,11030,8636],{"class":7387},[75,11032,11033],{"class":7394}," removeLast",[75,11035,7549],{"class":7398},[75,11037,11038],{"class":7383,"line":7458},[75,11039,7433],{"class":7398},[75,11041,11042,11044,11046,11049,11051],{"class":7383,"line":7474},[75,11043,9686],{"class":7398},[75,11045,7402],{"class":7387},[75,11047,11048],{"class":7398},"> previous ",[75,11050,7421],{"class":7387},[75,11052,9872],{"class":7398},[75,11054,11055,11057,11059,11061,11063,11065,11067,11069,11071,11073,11075],{"class":7383,"line":7480},[75,11056,10837],{"class":7387},[75,11058,10476],{"class":7398},[75,11060,8020],{"class":7387},[75,11062,10481],{"class":7398},[75,11064,7421],{"class":7387},[75,11066,10486],{"class":8127},[75,11068,10489],{"class":7398},[75,11070,7399],{"class":7387},[75,11072,10494],{"class":7398},[75,11074,10218],{"class":7387},[75,11076,11077],{"class":7398},")\n",[75,11079,11080,11083,11085,11088],{"class":7383,"line":7485},[75,11081,11082],{"class":7398},"        previous ",[75,11084,7421],{"class":7387},[75,11086,11087],{"class":7398}," previous.next;        ",[75,11089,11090],{"class":7470},"\u002F\u002F stop just before the target\n",[75,11092,11093,11095,11097,11099,11101,11104],{"class":7383,"line":7498},[75,11094,9686],{"class":7398},[75,11096,7402],{"class":7387},[75,11098,10011],{"class":7398},[75,11100,7421],{"class":7387},[75,11102,11103],{"class":7398}," previous.next;     ",[75,11105,11106],{"class":7470},"\u002F\u002F this is the node to remove\n",[75,11108,11109,11112,11114,11117],{"class":7383,"line":7516},[75,11110,11111],{"class":7398},"    previous.next ",[75,11113,7421],{"class":7387},[75,11115,11116],{"class":7398}," current.next;        ",[75,11118,11119],{"class":7470},"\u002F\u002F bypass it\n",[75,11121,11122,11124,11126],{"class":7383,"line":7521},[75,11123,10215],{"class":7398},[75,11125,10666],{"class":7387},[75,11127,8131],{"class":7398},[75,11129,11130,11132],{"class":7383,"line":7526},[75,11131,8689],{"class":7387},[75,11133,11134],{"class":7398}," current.element;\n",[75,11136,11137],{"class":7383,"line":7539},[75,11138,7610],{"class":7398},[12,11140,11141],{},"Notice we orphan the removed node by skipping over it. Java's garbage collector will clean it up.",[6964,11143,11145],{"id":11144},"_8-variations-singly-circular-doubly","8. Variations: Singly, Circular, Doubly",[12,11147,11148,11149,11152,11153,11155,11156,11158,11159,11162],{},"What we built above is called a ",[106,11150,11151],{},"singly linked list"," — each node has exactly one pointer (",[6979,11154,9765],{},"), and traversal can only go forward. To get to any node, you must start at ",[6979,11157,9750],{}," and walk one step at a time. It's ",[106,11160,11161],{},"not a direct access structure"," (unlike arrays).",[12,11164,11165],{},"There are a few common variations:",[7166,11167,11169],{"id":11168},"circular-linked-list","Circular Linked List",[12,11171,11172,11173,7206,11176,7206,11180,11183,11184,11186],{},"Just like a singly linked list, except the ",[106,11174,11175],{},"last node's",[106,11177,11178],{},[6979,11179,9765],{},[106,11181,11182],{},"points back to the first node"," (instead of ",[6979,11185,7962],{},"). The list becomes a loop.",[6972,11188,11191],{"className":11189,"code":11190,"language":6977},[6975],"head → [A | next] → [B | next] → [C | next] ┐\n        ↑___________________________________│\n",[6979,11192,11190],{"__ignoreMap":180},[12,11194,11195],{},"Useful for things like round-robin schedulers or any \"cyclic\" data.",[7166,11197,11199],{"id":11198},"doubly-linked-list","Doubly Linked List",[12,11201,11202,11203,11206,11207,11210],{},"Each node has ",[106,11204,11205],{},"two"," pointers: one to the next node, and one to the previous node. You can traverse the list in ",[7289,11208,11209],{},"both"," directions.",[6972,11212,11215],{"className":11213,"code":11214,"language":6977},[6975],"null ← [prev | A | next] ⇄ [prev | B | next] ⇄ [prev | C | next] → null\n",[6979,11216,11214],{"__ignoreMap":180},[7166,11218,11220],{"id":11219},"circular-doubly-linked-list","Circular Doubly Linked List",[12,11222,11223,11224,11226,11227,11230],{},"Doubly linked, plus the ends connect to each other (last's ",[6979,11225,9765],{}," → first, first's ",[6979,11228,11229],{},"prev"," → last). A loop you can walk in either direction.",[12,11232,11233,11234,11236],{},"The rest of this post focuses on the doubly linked list, since it's the most useful variation in practice (it's actually what Java's built-in ",[6979,11235,7066],{}," class uses internally).",[6964,11238,11240],{"id":11239},"_9-doubly-linked-lists-the-real-workhorse","9. Doubly Linked Lists: The Real Workhorse",[12,11242,9407,11243,11245,11246,11248,11249,11252],{},[106,11244,10939],{}," is a linked structure where each node has two reference fields — one pointing to the ",[106,11247,9765],{}," node, one pointing to the ",[106,11250,11251],{},"previous"," node.",[12,11254,11255],{},"Visually:",[6972,11257,11260],{"className":11258,"code":11259,"language":6977},[6975],"        ┌──────┐ next  ┌──────┐ next  ┌──────┐\nnull  ← │  12  │  ⇄    │  99  │  ⇄    │  37  │ → null\n        └──────┘ prev  └──────┘ prev  └──────┘\n",[6979,11261,11259],{"__ignoreMap":180},[12,11263,11264,11265,11268],{},"You can think of it as ",[106,11266,11267],{},"two singly linked lists"," layered on the same data — one going forward, one going backward.",[7166,11270,11272],{"id":11271},"why-bother","Why bother?",[12,11274,11275],{},"The two-pointer design gives us two big wins:",[3603,11277,11278,11288],{},[26,11279,11280,11283,11284,11287],{},[106,11281,11282],{},"Bidirectional traversal."," You can iterate forward ",[7289,11285,11286],{},"or"," backward.",[26,11289,11290,11293,11294,11296,11297,11299],{},[106,11291,11292],{},"Simpler local modifications."," When inserting or deleting a node, you don't have to walk the list to find its \"previous\" — you already have it via ",[6979,11295,11229],{},". This is what makes ",[6979,11298,10709],{}," fast in a doubly linked list.",[7166,11301,11303],{"id":11302},"the-trade-offs","The trade-offs",[12,11305,11306],{},"It's not all free:",[23,11308,11309,11312],{},[26,11310,11311],{},"Each node needs extra memory (one more pointer per node).",[26,11313,11314,11315,11318],{},"Insertion and deletion update more pointers (4 link updates instead of 2), so each operation does slightly more work — though it's still O(1) when you already have a reference to the node.\nFor most use cases, the benefits win. That's why Java's standard ",[6979,11316,11317],{},"LinkedList\u003CE>"," is doubly linked.",[6964,11320,11322],{"id":11321},"_10-building-a-doubly-linked-list","10. Building a Doubly Linked List",[7166,11324,11326],{"id":11325},"the-node-class","The Node class",[6972,11328,11330],{"className":7376,"code":11329,"language":7378,"meta":180,"style":180},"public class Node\u003CE> {\n    E element;\n    Node\u003CE> next;\n    Node\u003CE> prev;\n \n    public Node(E element, Node\u003CE> next, Node\u003CE> prev) {\n        this.element = element;\n        this.next = next;\n        this.prev = prev;\n    }\n \n    public Node(E element) {\n        this(element, null, null);\n    }\n}\n",[6979,11331,11332,11346,11351,11360,11369,11373,11403,11416,11428,11440,11444,11448,11460,11475,11479],{"__ignoreMap":180},[75,11333,11334,11336,11338,11340,11342,11344],{"class":7383,"line":7384},[75,11335,7388],{"class":7387},[75,11337,7391],{"class":7387},[75,11339,9667],{"class":7394},[75,11341,7399],{"class":7398},[75,11343,7402],{"class":7387},[75,11345,7405],{"class":7398},[75,11347,11348],{"class":7383,"line":181},[75,11349,11350],{"class":7398},"    E element;\n",[75,11352,11353,11355,11357],{"class":7383,"line":7430},[75,11354,9686],{"class":7398},[75,11356,7402],{"class":7387},[75,11358,11359],{"class":7398},"> next;\n",[75,11361,11362,11364,11366],{"class":7383,"line":7436},[75,11363,9686],{"class":7398},[75,11365,7402],{"class":7387},[75,11367,11368],{"class":7398},"> prev;\n",[75,11370,11371],{"class":7383,"line":7458},[75,11372,7433],{"class":7398},[75,11374,11375,11377,11379,11381,11383,11386,11388,11391,11393,11395,11397,11399,11401],{"class":7383,"line":7474},[75,11376,7439],{"class":7387},[75,11378,9667],{"class":7394},[75,11380,7448],{"class":7398},[75,11382,9647],{"class":7451},[75,11384,11385],{"class":7398},", Node\u003C",[75,11387,7402],{"class":7387},[75,11389,11390],{"class":7398},"> ",[75,11392,9765],{"class":7451},[75,11394,11385],{"class":7398},[75,11396,7402],{"class":7387},[75,11398,11390],{"class":7398},[75,11400,11229],{"class":7451},[75,11402,7455],{"class":7398},[75,11404,11405,11408,11411,11413],{"class":7383,"line":7480},[75,11406,11407],{"class":8127},"        this",[75,11409,11410],{"class":7398},".element ",[75,11412,7421],{"class":7387},[75,11414,11415],{"class":7398}," element;\n",[75,11417,11418,11420,11423,11425],{"class":7383,"line":7485},[75,11419,11407],{"class":8127},[75,11421,11422],{"class":7398},".next ",[75,11424,7421],{"class":7387},[75,11426,11427],{"class":7398}," next;\n",[75,11429,11430,11432,11435,11437],{"class":7383,"line":7498},[75,11431,11407],{"class":8127},[75,11433,11434],{"class":7398},".prev ",[75,11436,7421],{"class":7387},[75,11438,11439],{"class":7398}," prev;\n",[75,11441,11442],{"class":7383,"line":7516},[75,11443,7477],{"class":7398},[75,11445,11446],{"class":7383,"line":7521},[75,11447,7433],{"class":7398},[75,11449,11450,11452,11454,11456,11458],{"class":7383,"line":7526},[75,11451,7439],{"class":7387},[75,11453,9667],{"class":7394},[75,11455,7448],{"class":7398},[75,11457,9647],{"class":7451},[75,11459,7455],{"class":7398},[75,11461,11462,11464,11467,11469,11471,11473],{"class":7383,"line":7539},[75,11463,11407],{"class":8127},[75,11465,11466],{"class":7398},"(element, ",[75,11468,7962],{"class":8127},[75,11470,7298],{"class":7398},[75,11472,7962],{"class":8127},[75,11474,7738],{"class":7398},[75,11476,11477],{"class":7383,"line":7552},[75,11478,7477],{"class":7398},[75,11480,11481],{"class":7383,"line":7557},[75,11482,7610],{"class":7398},[7166,11484,11486],{"id":11485},"the-list-class","The List class",[6972,11488,11490],{"className":7376,"code":11489,"language":7378,"meta":180,"style":180},"public class DoublyLinkedList\u003CE> {\n    private Node\u003CE> head;\n    private Node\u003CE> tail;\n    private int size;\n \n    public DoublyLinkedList() {\n        size = 0;\n        this.head = null;\n        this.tail = null;\n    }\n \n    \u002F\u002F ... methods below\n}\n",[6979,11491,11492,11507,11517,11527,11535,11539,11547,11557,11570,11583,11587,11591,11595],{"__ignoreMap":180},[75,11493,11494,11496,11498,11501,11503,11505],{"class":7383,"line":7384},[75,11495,7388],{"class":7387},[75,11497,7391],{"class":7387},[75,11499,11500],{"class":7394}," DoublyLinkedList",[75,11502,7399],{"class":7398},[75,11504,7402],{"class":7387},[75,11506,7405],{"class":7398},[75,11508,11509,11511,11513,11515],{"class":7383,"line":181},[75,11510,7410],{"class":7387},[75,11512,10103],{"class":7398},[75,11514,7402],{"class":7387},[75,11516,10108],{"class":7398},[75,11518,11519,11521,11523,11525],{"class":7383,"line":7430},[75,11520,7410],{"class":7387},[75,11522,10103],{"class":7398},[75,11524,7402],{"class":7387},[75,11526,10119],{"class":7398},[75,11528,11529,11531,11533],{"class":7383,"line":7436},[75,11530,7410],{"class":7387},[75,11532,7531],{"class":7387},[75,11534,10128],{"class":7398},[75,11536,11537],{"class":7383,"line":7458},[75,11538,7433],{"class":7398},[75,11540,11541,11543,11545],{"class":7383,"line":7474},[75,11542,7439],{"class":7387},[75,11544,11500],{"class":7394},[75,11546,7495],{"class":7398},[75,11548,11549,11551,11553,11555],{"class":7383,"line":7480},[75,11550,10804],{"class":7398},[75,11552,7421],{"class":7387},[75,11554,8282],{"class":8127},[75,11556,8131],{"class":7398},[75,11558,11559,11561,11564,11566,11568],{"class":7383,"line":7485},[75,11560,11407],{"class":8127},[75,11562,11563],{"class":7398},".head ",[75,11565,7421],{"class":7387},[75,11567,8941],{"class":8127},[75,11569,8131],{"class":7398},[75,11571,11572,11574,11577,11579,11581],{"class":7383,"line":7498},[75,11573,11407],{"class":8127},[75,11575,11576],{"class":7398},".tail ",[75,11578,7421],{"class":7387},[75,11580,8941],{"class":8127},[75,11582,8131],{"class":7398},[75,11584,11585],{"class":7383,"line":7516},[75,11586,7477],{"class":7398},[75,11588,11589],{"class":7383,"line":7521},[75,11590,7433],{"class":7398},[75,11592,11593],{"class":7383,"line":7526},[75,11594,10137],{"class":7470},[75,11596,11597],{"class":7383,"line":7539},[75,11598,7610],{"class":7398},[6964,11600,11602],{"id":11601},"_11-doubly-linked-list-insertion","11. Doubly Linked List: Insertion",[12,11604,11605],{},"There are three cases (same as singly linked):",[23,11607,11608,11617,11624],{},[26,11609,11610,11611,9527,11614],{},"Insert at the ",[106,11612,11613],{},"beginning",[6979,11615,11616],{},"addFirst(E)",[26,11618,11610,11619,9527,11621],{},[106,11620,9565],{},[6979,11622,11623],{},"addLast(E)",[26,11625,11610,11626,9527,11628],{},[106,11627,9572],{},[6979,11629,11630],{},"add(int index, E)",[7166,11632,11634,11637],{"id":11633},"addfirste-element-insert-at-the-beginning",[6979,11635,11636],{},"addFirst(E element)"," — Insert at the Beginning",[12,11639,11640,11641,11643,11644,11646],{},"Two changes are needed: the new node's ",[6979,11642,9765],{}," should point to the old head, and the old head's ",[6979,11645,11229],{}," should point to the new node.",[6972,11648,11650],{"className":7376,"code":11649,"language":7378,"meta":180,"style":180},"public void addFirst(E element) {\n    Node\u003CE> tmp = new Node\u003C>(element, head, null);\n    if (head != null) head.prev = tmp;     \u002F\u002F old head's prev now points back\n    head = tmp;                            \u002F\u002F new node is the head\n    if (tail == null) tail = tmp;          \u002F\u002F if list was empty, also the tail\n    size++;\n    System.out.println(\"adding: \" + element);\n}\n",[6979,11651,11652,11663,11683,11704,11716,11736,11744,11760],{"__ignoreMap":180},[75,11653,11654,11656,11658,11660],{"class":7383,"line":7384},[75,11655,7388],{"class":7387},[75,11657,7442],{"class":7387},[75,11659,10165],{"class":7394},[75,11661,11662],{"class":7398},"(E element) {\n",[75,11664,11665,11667,11669,11672,11674,11676,11679,11681],{"class":7383,"line":181},[75,11666,9686],{"class":7398},[75,11668,7402],{"class":7387},[75,11670,11671],{"class":7398},"> tmp ",[75,11673,7421],{"class":7387},[75,11675,7424],{"class":7387},[75,11677,11678],{"class":7398}," Node\u003C>(element, head, ",[75,11680,7962],{"class":8127},[75,11682,7738],{"class":7398},[75,11684,11685,11687,11689,11691,11693,11696,11698,11701],{"class":7383,"line":7430},[75,11686,10225],{"class":7387},[75,11688,10675],{"class":7398},[75,11690,8938],{"class":7387},[75,11692,8941],{"class":8127},[75,11694,11695],{"class":7398},") head.prev ",[75,11697,7421],{"class":7387},[75,11699,11700],{"class":7398}," tmp;     ",[75,11702,11703],{"class":7470},"\u002F\u002F old head's prev now points back\n",[75,11705,11706,11708,11710,11713],{"class":7383,"line":7436},[75,11707,10202],{"class":7398},[75,11709,7421],{"class":7387},[75,11711,11712],{"class":7398}," tmp;                            ",[75,11714,11715],{"class":7470},"\u002F\u002F new node is the head\n",[75,11717,11718,11720,11722,11724,11726,11728,11730,11733],{"class":7383,"line":7458},[75,11719,10225],{"class":7387},[75,11721,10228],{"class":7398},[75,11723,10231],{"class":7387},[75,11725,8941],{"class":8127},[75,11727,10682],{"class":7398},[75,11729,7421],{"class":7387},[75,11731,11732],{"class":7398}," tmp;          ",[75,11734,11735],{"class":7470},"\u002F\u002F if list was empty, also the tail\n",[75,11737,11738,11740,11742],{"class":7383,"line":7474},[75,11739,10215],{"class":7398},[75,11741,10218],{"class":7387},[75,11743,8131],{"class":7398},[75,11745,11746,11748,11750,11752,11755,11757],{"class":7383,"line":7480},[75,11747,8254],{"class":7398},[75,11749,7767],{"class":7394},[75,11751,7698],{"class":7398},[75,11753,11754],{"class":7589},"\"adding: \"",[75,11756,7593],{"class":7387},[75,11758,11759],{"class":7398}," element);\n",[75,11761,11762],{"class":7383,"line":7485},[75,11763,7610],{"class":7398},[7166,11765,11767,11770],{"id":11766},"addlaste-element-insert-at-the-end",[6979,11768,11769],{},"addLast(E element)"," — Insert at the End",[12,11772,11773],{},"Mirror image:",[6972,11775,11777],{"className":7376,"code":11776,"language":7378,"meta":180,"style":180},"public void addLast(E element) {\n    Node\u003CE> tmp = new Node\u003C>(element, null, tail);\n    if (tail != null) tail.next = tmp;     \u002F\u002F old tail's next points forward\n    tail = tmp;                            \u002F\u002F new node is the tail\n    if (head == null) head = tmp;          \u002F\u002F if list was empty, also the head\n    size++;\n    System.out.println(\"adding: \" + element);\n}\n",[6979,11778,11779,11789,11809,11829,11840,11860,11868,11882],{"__ignoreMap":180},[75,11780,11781,11783,11785,11787],{"class":7383,"line":7384},[75,11782,7388],{"class":7387},[75,11784,7442],{"class":7387},[75,11786,10285],{"class":7394},[75,11788,11662],{"class":7398},[75,11790,11791,11793,11795,11797,11799,11801,11804,11806],{"class":7383,"line":181},[75,11792,9686],{"class":7398},[75,11794,7402],{"class":7387},[75,11796,11671],{"class":7398},[75,11798,7421],{"class":7387},[75,11800,7424],{"class":7387},[75,11802,11803],{"class":7398}," Node\u003C>(element, ",[75,11805,7962],{"class":8127},[75,11807,11808],{"class":7398},", tail);\n",[75,11810,11811,11813,11815,11817,11819,11822,11824,11826],{"class":7383,"line":7430},[75,11812,10225],{"class":7387},[75,11814,10228],{"class":7398},[75,11816,8938],{"class":7387},[75,11818,8941],{"class":8127},[75,11820,11821],{"class":7398},") tail.next ",[75,11823,7421],{"class":7387},[75,11825,11700],{"class":7398},[75,11827,11828],{"class":7470},"\u002F\u002F old tail's next points forward\n",[75,11830,11831,11833,11835,11837],{"class":7383,"line":7436},[75,11832,10899],{"class":7398},[75,11834,7421],{"class":7387},[75,11836,11712],{"class":7398},[75,11838,11839],{"class":7470},"\u002F\u002F new node is the tail\n",[75,11841,11842,11844,11846,11848,11850,11853,11855,11857],{"class":7383,"line":7458},[75,11843,10225],{"class":7387},[75,11845,10675],{"class":7398},[75,11847,10231],{"class":7387},[75,11849,8941],{"class":8127},[75,11851,11852],{"class":7398},") head ",[75,11854,7421],{"class":7387},[75,11856,11732],{"class":7398},[75,11858,11859],{"class":7470},"\u002F\u002F if list was empty, also the head\n",[75,11861,11862,11864,11866],{"class":7383,"line":7474},[75,11863,10215],{"class":7398},[75,11865,10218],{"class":7387},[75,11867,8131],{"class":7398},[75,11869,11870,11872,11874,11876,11878,11880],{"class":7383,"line":7480},[75,11871,8254],{"class":7398},[75,11873,7767],{"class":7394},[75,11875,7698],{"class":7398},[75,11877,11754],{"class":7589},[75,11879,7593],{"class":7387},[75,11881,11759],{"class":7398},[75,11883,11884],{"class":7383,"line":7485},[75,11885,7610],{"class":7398},[7166,11887,11889,11892],{"id":11888},"addint-index-e-element-insert-in-the-middle",[6979,11890,11891],{},"add(int index, E element)"," — Insert in the Middle",[6972,11894,11896],{"className":7376,"code":11895,"language":7378,"meta":180,"style":180},"public void add(int index, E element) {\n    if (index \u003C 0 || index > size)\n        throw new IndexOutOfBoundsException();\n \n    if (index == 0) {\n        addFirst(element);\n    } else if (index == size) {\n        addLast(element);\n    } else {\n        \u002F\u002F Walk to the node currently at the target index\n        Node\u003CE> temp = head;\n        for (int i = 0; i \u003C index; i++) {\n            temp = temp.next;\n        }\n        \u002F\u002F Create new node, sitting between temp.prev and temp\n        Node\u003CE> insert = new Node\u003C>(element, temp, temp.prev);\n        temp.prev.next = insert;   \u002F\u002F link 1: prev's next → new node\n        temp.prev = insert;        \u002F\u002F link 2: temp's prev → new node\n        size++;\n    }\n}\n",[6979,11897,11898,11913,11932,11944,11948,11960,11967,11981,11987,11995,12000,12012,12036,12046,12050,12055,12071,12084,12097,12105,12109],{"__ignoreMap":180},[75,11899,11900,11902,11904,11906,11908,11910],{"class":7383,"line":7384},[75,11901,7388],{"class":7387},[75,11903,7442],{"class":7387},[75,11905,10395],{"class":7394},[75,11907,7698],{"class":7398},[75,11909,8020],{"class":7387},[75,11911,11912],{"class":7398}," index, E element) {\n",[75,11914,11915,11917,11919,11921,11923,11925,11927,11929],{"class":7383,"line":181},[75,11916,10225],{"class":7387},[75,11918,10409],{"class":7398},[75,11920,7399],{"class":7387},[75,11922,8282],{"class":8127},[75,11924,10980],{"class":7387},[75,11926,10585],{"class":7398},[75,11928,8279],{"class":7387},[75,11930,11931],{"class":7398}," size)\n",[75,11933,11934,11937,11939,11942],{"class":7383,"line":7430},[75,11935,11936],{"class":7387},"        throw",[75,11938,7424],{"class":7387},[75,11940,11941],{"class":7394}," IndexOutOfBoundsException",[75,11943,7549],{"class":7398},[75,11945,11946],{"class":7383,"line":7436},[75,11947,7433],{"class":7398},[75,11949,11950,11952,11954,11956,11958],{"class":7383,"line":7458},[75,11951,10225],{"class":7387},[75,11953,10409],{"class":7398},[75,11955,10231],{"class":7387},[75,11957,8282],{"class":8127},[75,11959,7455],{"class":7398},[75,11961,11962,11964],{"class":7383,"line":7474},[75,11963,10420],{"class":7394},[75,11965,11966],{"class":7398},"(element);\n",[75,11968,11969,11971,11973,11975,11977,11979],{"class":7383,"line":7480},[75,11970,10324],{"class":7398},[75,11972,10327],{"class":7387},[75,11974,10432],{"class":7387},[75,11976,10409],{"class":7398},[75,11978,10231],{"class":7387},[75,11980,10440],{"class":7398},[75,11982,11983,11985],{"class":7383,"line":7485},[75,11984,10445],{"class":7394},[75,11986,11966],{"class":7398},[75,11988,11989,11991,11993],{"class":7383,"line":7498},[75,11990,10324],{"class":7398},[75,11992,10327],{"class":7387},[75,11994,8147],{"class":7398},[75,11996,11997],{"class":7383,"line":7516},[75,11998,11999],{"class":7470},"        \u002F\u002F Walk to the node currently at the target index\n",[75,12001,12002,12004,12006,12008,12010],{"class":7383,"line":7521},[75,12003,10460],{"class":7398},[75,12005,7402],{"class":7387},[75,12007,10525],{"class":7398},[75,12009,7421],{"class":7387},[75,12011,9872],{"class":7398},[75,12013,12014,12016,12018,12020,12022,12024,12026,12028,12030,12032,12034],{"class":7383,"line":7526},[75,12015,10473],{"class":7387},[75,12017,10476],{"class":7398},[75,12019,8020],{"class":7387},[75,12021,10481],{"class":7398},[75,12023,7421],{"class":7387},[75,12025,8282],{"class":8127},[75,12027,10489],{"class":7398},[75,12029,7399],{"class":7387},[75,12031,10494],{"class":7398},[75,12033,10218],{"class":7387},[75,12035,7455],{"class":7398},[75,12037,12038,12041,12043],{"class":7383,"line":7539},[75,12039,12040],{"class":7398},"            temp ",[75,12042,7421],{"class":7387},[75,12044,12045],{"class":7398}," temp.next;\n",[75,12047,12048],{"class":7383,"line":7552},[75,12049,10516],{"class":7398},[75,12051,12052],{"class":7383,"line":7557},[75,12053,12054],{"class":7470},"        \u002F\u002F Create new node, sitting between temp.prev and temp\n",[75,12056,12057,12059,12061,12064,12066,12068],{"class":7383,"line":7562},[75,12058,10460],{"class":7398},[75,12060,7402],{"class":7387},[75,12062,12063],{"class":7398},"> insert ",[75,12065,7421],{"class":7387},[75,12067,7424],{"class":7387},[75,12069,12070],{"class":7398}," Node\u003C>(element, temp, temp.prev);\n",[75,12072,12073,12076,12078,12081],{"class":7383,"line":7571},[75,12074,12075],{"class":7398},"        temp.prev.next ",[75,12077,7421],{"class":7387},[75,12079,12080],{"class":7398}," insert;   ",[75,12082,12083],{"class":7470},"\u002F\u002F link 1: prev's next → new node\n",[75,12085,12086,12089,12091,12094],{"class":7383,"line":7584},[75,12087,12088],{"class":7398},"        temp.prev ",[75,12090,7421],{"class":7387},[75,12092,12093],{"class":7398}," insert;        ",[75,12095,12096],{"class":7470},"\u002F\u002F link 2: temp's prev → new node\n",[75,12098,12099,12101,12103],{"class":7383,"line":7602},[75,12100,10566],{"class":7398},[75,12102,10218],{"class":7387},[75,12104,8131],{"class":7398},[75,12106,12107],{"class":7383,"line":7607},[75,12108,7477],{"class":7398},[75,12110,12111],{"class":7383,"line":8360},[75,12112,7610],{"class":7398},[12,12114,12115,12116,9019,12118,7206,12121,12125,12126,12128,12129,12132,12133,12135],{},"The trick: when inserting at index ",[6979,12117,10588],{},[106,12119,12120],{},"new node will become the node at index",[106,12122,12123],{},[6979,12124,10588],{},", and the existing node at index ",[6979,12127,10588],{}," shifts to index ",[6979,12130,12131],{},"i+1",". So we walk to the existing node at index ",[6979,12134,10588],{},", then splice the new one in just before it.",[12,12137,12138],{},"Four pointer updates happen here:",[3603,12140,12141,12147,12152,12158],{},[26,12142,12143,12144,12146],{},"New node's ",[6979,12145,9765],{}," → existing node (set in constructor)",[26,12148,12143,12149,12151],{},[6979,12150,11229],{}," → existing node's prev (set in constructor)",[26,12153,12154,12155,12157],{},"The previous node's ",[6979,12156,9765],{}," → new node",[26,12159,12160,12161,12157],{},"The existing node's ",[6979,12162,11229],{},[6964,12164,12166],{"id":12165},"_12-doubly-linked-list-traversal","12. Doubly Linked List: Traversal",[7166,12168,12170],{"id":12169},"forward-same-as-singly-linked","Forward (same as singly linked):",[6972,12172,12174],{"className":7376,"code":12173,"language":7378,"meta":180,"style":180},"public void iterateForward() {\n    System.out.println(\"iterating forward..\");\n    Node\u003CE> tmp = head;\n    while (tmp != null) {\n        System.out.print(tmp.element + \" \");\n        tmp = tmp.next;\n    }\n}\n",[6979,12175,12176,12187,12200,12212,12225,12241,12251,12255],{"__ignoreMap":180},[75,12177,12178,12180,12182,12185],{"class":7383,"line":7384},[75,12179,7388],{"class":7387},[75,12181,7442],{"class":7387},[75,12183,12184],{"class":7394}," iterateForward",[75,12186,7495],{"class":7398},[75,12188,12189,12191,12193,12195,12198],{"class":7383,"line":181},[75,12190,8254],{"class":7398},[75,12192,7767],{"class":7394},[75,12194,7698],{"class":7398},[75,12196,12197],{"class":7589},"\"iterating forward..\"",[75,12199,7738],{"class":7398},[75,12201,12202,12204,12206,12208,12210],{"class":7383,"line":7430},[75,12203,9686],{"class":7398},[75,12205,7402],{"class":7387},[75,12207,11671],{"class":7398},[75,12209,7421],{"class":7387},[75,12211,9872],{"class":7398},[75,12213,12214,12216,12219,12221,12223],{"class":7383,"line":7436},[75,12215,8268],{"class":7387},[75,12217,12218],{"class":7398}," (tmp ",[75,12220,8938],{"class":7387},[75,12222,8941],{"class":8127},[75,12224,7455],{"class":7398},[75,12226,12227,12230,12232,12235,12237,12239],{"class":7383,"line":7458},[75,12228,12229],{"class":7398},"        System.out.",[75,12231,8292],{"class":7394},[75,12233,12234],{"class":7398},"(tmp.element ",[75,12236,8302],{"class":7387},[75,12238,8305],{"class":7589},[75,12240,7738],{"class":7398},[75,12242,12243,12246,12248],{"class":7383,"line":7474},[75,12244,12245],{"class":7398},"        tmp ",[75,12247,7421],{"class":7387},[75,12249,12250],{"class":7398}," tmp.next;\n",[75,12252,12253],{"class":7383,"line":7480},[75,12254,7477],{"class":7398},[75,12256,12257],{"class":7383,"line":7485},[75,12258,7610],{"class":7398},[7166,12260,12262],{"id":12261},"backward-this-is-where-doubly-linked-shines","Backward — this is where doubly linked shines:",[6972,12264,12266],{"className":7376,"code":12265,"language":7378,"meta":180,"style":180},"public void iterateBackward() {\n    System.out.println(\"iterating backward..\");\n    Node\u003CE> tmp = tail;\n    while (tmp != null) {\n        System.out.println(tmp.element);\n        tmp = tmp.prev;\n    }\n}\n",[6979,12267,12268,12279,12292,12304,12316,12325,12334,12338],{"__ignoreMap":180},[75,12269,12270,12272,12274,12277],{"class":7383,"line":7384},[75,12271,7388],{"class":7387},[75,12273,7442],{"class":7387},[75,12275,12276],{"class":7394}," iterateBackward",[75,12278,7495],{"class":7398},[75,12280,12281,12283,12285,12287,12290],{"class":7383,"line":181},[75,12282,8254],{"class":7398},[75,12284,7767],{"class":7394},[75,12286,7698],{"class":7398},[75,12288,12289],{"class":7589},"\"iterating backward..\"",[75,12291,7738],{"class":7398},[75,12293,12294,12296,12298,12300,12302],{"class":7383,"line":7430},[75,12295,9686],{"class":7398},[75,12297,7402],{"class":7387},[75,12299,11671],{"class":7398},[75,12301,7421],{"class":7387},[75,12303,10894],{"class":7398},[75,12305,12306,12308,12310,12312,12314],{"class":7383,"line":7436},[75,12307,8268],{"class":7387},[75,12309,12218],{"class":7398},[75,12311,8938],{"class":7387},[75,12313,8941],{"class":8127},[75,12315,7455],{"class":7398},[75,12317,12318,12320,12322],{"class":7383,"line":7458},[75,12319,12229],{"class":7398},[75,12321,7767],{"class":7394},[75,12323,12324],{"class":7398},"(tmp.element);\n",[75,12326,12327,12329,12331],{"class":7383,"line":7474},[75,12328,12245],{"class":7398},[75,12330,7421],{"class":7387},[75,12332,12333],{"class":7398}," tmp.prev;\n",[75,12335,12336],{"class":7383,"line":7480},[75,12337,7477],{"class":7398},[75,12339,12340],{"class":7383,"line":7485},[75,12341,7610],{"class":7398},[12,12343,12344,12345,12347,12348,12350],{},"Start from ",[6979,12346,9758],{},", follow ",[6979,12349,11229],{}," pointers. Impossible to do efficiently in a singly linked list.",[6964,12352,12354],{"id":12353},"_13-doubly-linked-list-deletion","13. Doubly Linked List: Deletion",[12,12356,12357],{},"Three cases again.",[7166,12359,12361,12363],{"id":12360},"removefirst-delete-the-head",[6979,12362,10599],{}," — Delete the Head",[6972,12365,12367],{"className":7376,"code":12366,"language":7378,"meta":180,"style":180},"public E removeFirst() {\n    if (size == 0) throw new NoSuchElementException();\n    Node\u003CE> tmp = head;\n    head = head.next;            \u002F\u002F second node becomes head\n    if (head != null) head.prev = null;   \u002F\u002F new head has no previous\n    size--;\n    System.out.println(\"deleted: \" + tmp.element);\n    return tmp.element;\n}\n",[6979,12368,12369,12379,12401,12413,12424,12445,12453,12469,12476],{"__ignoreMap":180},[75,12370,12371,12373,12375,12377],{"class":7383,"line":7384},[75,12372,7388],{"class":7387},[75,12374,7490],{"class":7398},[75,12376,7507],{"class":7394},[75,12378,7495],{"class":7398},[75,12380,12381,12383,12385,12387,12389,12391,12394,12396,12399],{"class":7383,"line":181},[75,12382,10225],{"class":7387},[75,12384,10622],{"class":7398},[75,12386,10231],{"class":7387},[75,12388,8282],{"class":8127},[75,12390,10629],{"class":7398},[75,12392,12393],{"class":7387},"throw",[75,12395,7424],{"class":7387},[75,12397,12398],{"class":7394}," NoSuchElementException",[75,12400,7549],{"class":7398},[75,12402,12403,12405,12407,12409,12411],{"class":7383,"line":7430},[75,12404,9686],{"class":7398},[75,12406,7402],{"class":7387},[75,12408,11671],{"class":7398},[75,12410,7421],{"class":7387},[75,12412,9872],{"class":7398},[75,12414,12415,12417,12419,12422],{"class":7383,"line":7436},[75,12416,10202],{"class":7398},[75,12418,7421],{"class":7387},[75,12420,12421],{"class":7398}," head.next;            ",[75,12423,10659],{"class":7470},[75,12425,12426,12428,12430,12432,12434,12436,12438,12440,12442],{"class":7383,"line":7458},[75,12427,10225],{"class":7387},[75,12429,10675],{"class":7398},[75,12431,8938],{"class":7387},[75,12433,8941],{"class":8127},[75,12435,11695],{"class":7398},[75,12437,7421],{"class":7387},[75,12439,8941],{"class":8127},[75,12441,10689],{"class":7398},[75,12443,12444],{"class":7470},"\u002F\u002F new head has no previous\n",[75,12446,12447,12449,12451],{"class":7383,"line":7474},[75,12448,10215],{"class":7398},[75,12450,10666],{"class":7387},[75,12452,8131],{"class":7398},[75,12454,12455,12457,12459,12461,12464,12466],{"class":7383,"line":7480},[75,12456,8254],{"class":7398},[75,12458,7767],{"class":7394},[75,12460,7698],{"class":7398},[75,12462,12463],{"class":7589},"\"deleted: \"",[75,12465,7593],{"class":7387},[75,12467,12468],{"class":7398}," tmp.element);\n",[75,12470,12471,12473],{"class":7383,"line":7485},[75,12472,8689],{"class":7387},[75,12474,12475],{"class":7398}," tmp.element;\n",[75,12477,12478],{"class":7383,"line":7498},[75,12479,7610],{"class":7398},[7166,12481,12483,12485],{"id":12482},"removelast-delete-the-tail",[6979,12484,10709],{}," — Delete the Tail",[12,12487,12488,12489,12492],{},"Much easier than in a singly linked list! No need to walk the list — ",[6979,12490,12491],{},"tail.prev"," is right there:",[6972,12494,12496],{"className":7376,"code":12495,"language":7378,"meta":180,"style":180},"public E removeLast() {\n    if (size == 0) throw new NoSuchElementException();\n    Node\u003CE> tmp = tail;\n    tail = tail.prev;            \u002F\u002F second-to-last becomes tail\n    if (tail != null) tail.next = null;\n    size--;\n    System.out.println(\"deleted: \" + tmp.element);\n    return tmp.element;\n}\n",[6979,12497,12498,12508,12528,12540,12552,12570,12578,12592,12598],{"__ignoreMap":180},[75,12499,12500,12502,12504,12506],{"class":7383,"line":7384},[75,12501,7388],{"class":7387},[75,12503,7490],{"class":7398},[75,12505,10737],{"class":7394},[75,12507,7495],{"class":7398},[75,12509,12510,12512,12514,12516,12518,12520,12522,12524,12526],{"class":7383,"line":181},[75,12511,10225],{"class":7387},[75,12513,10622],{"class":7398},[75,12515,10231],{"class":7387},[75,12517,8282],{"class":8127},[75,12519,10629],{"class":7398},[75,12521,12393],{"class":7387},[75,12523,7424],{"class":7387},[75,12525,12398],{"class":7394},[75,12527,7549],{"class":7398},[75,12529,12530,12532,12534,12536,12538],{"class":7383,"line":7430},[75,12531,9686],{"class":7398},[75,12533,7402],{"class":7387},[75,12535,11671],{"class":7398},[75,12537,7421],{"class":7387},[75,12539,10894],{"class":7398},[75,12541,12542,12544,12546,12549],{"class":7383,"line":7436},[75,12543,10899],{"class":7398},[75,12545,7421],{"class":7387},[75,12547,12548],{"class":7398}," tail.prev;            ",[75,12550,12551],{"class":7470},"\u002F\u002F second-to-last becomes tail\n",[75,12553,12554,12556,12558,12560,12562,12564,12566,12568],{"class":7383,"line":7458},[75,12555,10225],{"class":7387},[75,12557,10228],{"class":7398},[75,12559,8938],{"class":7387},[75,12561,8941],{"class":8127},[75,12563,11821],{"class":7398},[75,12565,7421],{"class":7387},[75,12567,8941],{"class":8127},[75,12569,8131],{"class":7398},[75,12571,12572,12574,12576],{"class":7383,"line":7474},[75,12573,10215],{"class":7398},[75,12575,10666],{"class":7387},[75,12577,8131],{"class":7398},[75,12579,12580,12582,12584,12586,12588,12590],{"class":7383,"line":7480},[75,12581,8254],{"class":7398},[75,12583,7767],{"class":7394},[75,12585,7698],{"class":7398},[75,12587,12463],{"class":7589},[75,12589,7593],{"class":7387},[75,12591,12468],{"class":7398},[75,12593,12594,12596],{"class":7383,"line":7485},[75,12595,8689],{"class":7387},[75,12597,12475],{"class":7398},[75,12599,12600],{"class":7383,"line":7498},[75,12601,7610],{"class":7398},[12,12603,12604],{},"This is O(1) — instant. That's the doubly linked list paying its rent.",[7166,12606,12608,12610],{"id":12607},"removeint-index-delete-in-the-middle",[6979,12609,10946],{}," — Delete in the Middle",[6972,12612,12614],{"className":7376,"code":12613,"language":7378,"meta":180,"style":180},"public E remove(int index) {\n    E element = null;\n    if (index \u003C 0 || index >= size)\n        throw new IndexOutOfBoundsException();\n \n    if (index == 0) {\n        return removeFirst();\n    } else if (index == size - 1) {\n        return removeLast();\n    } else {\n        Node\u003CE> temp = head;\n        for (int i = 0; i \u003C index; i++)\n            temp = temp.next;          \u002F\u002F walk to the target node\n        element = temp.element;\n \n        \u002F\u002F Splice it out\n        temp.next.prev = temp.prev;    \u002F\u002F next node's prev skips over temp\n        temp.prev.next = temp.next;    \u002F\u002F previous node's next skips over temp\n        temp.next = null;              \u002F\u002F clean up references\n        temp.prev = null;\n        size--;\n    }\n    return element;\n}\n",[6979,12615,12616,12630,12641,12659,12669,12673,12685,12693,12713,12721,12729,12741,12765,12777,12785,12789,12794,12807,12819,12834,12844,12852,12856,12862],{"__ignoreMap":180},[75,12617,12618,12620,12622,12624,12626,12628],{"class":7383,"line":7384},[75,12619,7388],{"class":7387},[75,12621,7490],{"class":7398},[75,12623,8065],{"class":7394},[75,12625,7698],{"class":7398},[75,12627,8020],{"class":7387},[75,12629,10967],{"class":7398},[75,12631,12632,12635,12637,12639],{"class":7383,"line":181},[75,12633,12634],{"class":7398},"    E element ",[75,12636,7421],{"class":7387},[75,12638,8941],{"class":8127},[75,12640,8131],{"class":7398},[75,12642,12643,12645,12647,12649,12651,12653,12655,12657],{"class":7383,"line":7430},[75,12644,10225],{"class":7387},[75,12646,10409],{"class":7398},[75,12648,7399],{"class":7387},[75,12650,8282],{"class":8127},[75,12652,10980],{"class":7387},[75,12654,10585],{"class":7398},[75,12656,10437],{"class":7387},[75,12658,11931],{"class":7398},[75,12660,12661,12663,12665,12667],{"class":7383,"line":7436},[75,12662,11936],{"class":7387},[75,12664,7424],{"class":7387},[75,12666,11941],{"class":7394},[75,12668,7549],{"class":7398},[75,12670,12671],{"class":7383,"line":7458},[75,12672,7433],{"class":7398},[75,12674,12675,12677,12679,12681,12683],{"class":7383,"line":7474},[75,12676,10225],{"class":7387},[75,12678,10409],{"class":7398},[75,12680,10231],{"class":7387},[75,12682,8282],{"class":8127},[75,12684,7455],{"class":7398},[75,12686,12687,12689,12691],{"class":7383,"line":7480},[75,12688,7501],{"class":7387},[75,12690,11010],{"class":7394},[75,12692,7549],{"class":7398},[75,12694,12695,12697,12699,12701,12703,12705,12707,12709,12711],{"class":7383,"line":7485},[75,12696,10324],{"class":7398},[75,12698,10327],{"class":7387},[75,12700,10432],{"class":7387},[75,12702,10409],{"class":7398},[75,12704,10231],{"class":7387},[75,12706,10854],{"class":7398},[75,12708,10857],{"class":7387},[75,12710,10486],{"class":8127},[75,12712,7455],{"class":7398},[75,12714,12715,12717,12719],{"class":7383,"line":7498},[75,12716,7501],{"class":7387},[75,12718,11033],{"class":7394},[75,12720,7549],{"class":7398},[75,12722,12723,12725,12727],{"class":7383,"line":7516},[75,12724,10324],{"class":7398},[75,12726,10327],{"class":7387},[75,12728,8147],{"class":7398},[75,12730,12731,12733,12735,12737,12739],{"class":7383,"line":7521},[75,12732,10460],{"class":7398},[75,12734,7402],{"class":7387},[75,12736,10525],{"class":7398},[75,12738,7421],{"class":7387},[75,12740,9872],{"class":7398},[75,12742,12743,12745,12747,12749,12751,12753,12755,12757,12759,12761,12763],{"class":7383,"line":7526},[75,12744,10473],{"class":7387},[75,12746,10476],{"class":7398},[75,12748,8020],{"class":7387},[75,12750,10481],{"class":7398},[75,12752,7421],{"class":7387},[75,12754,8282],{"class":8127},[75,12756,10489],{"class":7398},[75,12758,7399],{"class":7387},[75,12760,10494],{"class":7398},[75,12762,10218],{"class":7387},[75,12764,11077],{"class":7398},[75,12766,12767,12769,12771,12774],{"class":7383,"line":7539},[75,12768,12040],{"class":7398},[75,12770,7421],{"class":7387},[75,12772,12773],{"class":7398}," temp.next;          ",[75,12775,12776],{"class":7470},"\u002F\u002F walk to the target node\n",[75,12778,12779,12781,12783],{"class":7383,"line":7552},[75,12780,9716],{"class":7398},[75,12782,7421],{"class":7387},[75,12784,10699],{"class":7398},[75,12786,12787],{"class":7383,"line":7557},[75,12788,7433],{"class":7398},[75,12790,12791],{"class":7383,"line":7562},[75,12792,12793],{"class":7470},"        \u002F\u002F Splice it out\n",[75,12795,12796,12799,12801,12804],{"class":7383,"line":7571},[75,12797,12798],{"class":7398},"        temp.next.prev ",[75,12800,7421],{"class":7387},[75,12802,12803],{"class":7398}," temp.prev;    ",[75,12805,12806],{"class":7470},"\u002F\u002F next node's prev skips over temp\n",[75,12808,12809,12811,12813,12816],{"class":7383,"line":7584},[75,12810,12075],{"class":7398},[75,12812,7421],{"class":7387},[75,12814,12815],{"class":7398}," temp.next;    ",[75,12817,12818],{"class":7470},"\u002F\u002F previous node's next skips over temp\n",[75,12820,12821,12824,12826,12828,12831],{"class":7383,"line":7602},[75,12822,12823],{"class":7398},"        temp.next ",[75,12825,7421],{"class":7387},[75,12827,8941],{"class":8127},[75,12829,12830],{"class":7398},";              ",[75,12832,12833],{"class":7470},"\u002F\u002F clean up references\n",[75,12835,12836,12838,12840,12842],{"class":7383,"line":7607},[75,12837,12088],{"class":7398},[75,12839,7421],{"class":7387},[75,12841,8941],{"class":8127},[75,12843,8131],{"class":7398},[75,12845,12846,12848,12850],{"class":7383,"line":8360},[75,12847,10566],{"class":7398},[75,12849,10666],{"class":7387},[75,12851,8131],{"class":7398},[75,12853,12854],{"class":7383,"line":8374},[75,12855,7477],{"class":7398},[75,12857,12858,12860],{"class":7383,"line":8387},[75,12859,8689],{"class":7387},[75,12861,11415],{"class":7398},[75,12863,12864],{"class":7383,"line":8400},[75,12865,7610],{"class":7398},[12,12867,12868,12869,7024],{},"The two key lines are the ",[106,12870,12871],{},"splice",[23,12873,12874,12886],{},[26,12875,12876,12879,12880,12883,12884],{},[6979,12877,12878],{},"temp.next.prev = temp.prev"," — the node after ",[6979,12881,12882],{},"temp"," now points back to the node before ",[6979,12885,12882],{},[26,12887,12888,12891,12892,12894,12895,12897,12898,12900],{},[6979,12889,12890],{},"temp.prev.next = temp.next"," — the node before ",[6979,12893,12882],{}," now points forward to the node after ",[6979,12896,12882],{},"\nTogether, they \"skip over\" ",[6979,12899,12882],{},", removing it from the chain.",[36,12902],{},[6964,12904,12906,12907,12909],{"id":12905},"_14-bonus-clear-wipe-the-list","14. Bonus: ",[6979,12908,8027],{}," — Wipe the List",[6972,12911,12913],{"className":7376,"code":12912,"language":7378,"meta":180,"style":180},"public void clear() {\n    Node\u003CE> temp = head;\n    while (head != null) {\n        temp = head.next;\n        head.prev = head.next = null;   \u002F\u002F disconnect the current head\n        head = temp;                    \u002F\u002F advance\n    }\n    temp = null;\n    tail.prev = tail.next = null;\n    size = 0;\n}\n",[6979,12914,12915,12926,12938,12950,12960,12979,12991,12995,13006,13022,13033],{"__ignoreMap":180},[75,12916,12917,12919,12921,12924],{"class":7383,"line":7384},[75,12918,7388],{"class":7387},[75,12920,7442],{"class":7387},[75,12922,12923],{"class":7394}," clear",[75,12925,7495],{"class":7398},[75,12927,12928,12930,12932,12934,12936],{"class":7383,"line":181},[75,12929,9686],{"class":7398},[75,12931,7402],{"class":7387},[75,12933,10525],{"class":7398},[75,12935,7421],{"class":7387},[75,12937,9872],{"class":7398},[75,12939,12940,12942,12944,12946,12948],{"class":7383,"line":7430},[75,12941,8268],{"class":7387},[75,12943,10675],{"class":7398},[75,12945,8938],{"class":7387},[75,12947,8941],{"class":8127},[75,12949,7455],{"class":7398},[75,12951,12952,12955,12957],{"class":7383,"line":7436},[75,12953,12954],{"class":7398},"        temp ",[75,12956,7421],{"class":7387},[75,12958,12959],{"class":7398}," head.next;\n",[75,12961,12962,12965,12967,12970,12972,12974,12976],{"class":7383,"line":7458},[75,12963,12964],{"class":7398},"        head.prev ",[75,12966,7421],{"class":7387},[75,12968,12969],{"class":7398}," head.next ",[75,12971,7421],{"class":7387},[75,12973,8941],{"class":8127},[75,12975,10689],{"class":7398},[75,12977,12978],{"class":7470},"\u002F\u002F disconnect the current head\n",[75,12980,12981,12983,12985,12988],{"class":7383,"line":7474},[75,12982,10308],{"class":7398},[75,12984,7421],{"class":7387},[75,12986,12987],{"class":7398}," temp;                    ",[75,12989,12990],{"class":7470},"\u002F\u002F advance\n",[75,12992,12993],{"class":7383,"line":7480},[75,12994,7477],{"class":7398},[75,12996,12997,13000,13002,13004],{"class":7383,"line":7485},[75,12998,12999],{"class":7398},"    temp ",[75,13001,7421],{"class":7387},[75,13003,8941],{"class":8127},[75,13005,8131],{"class":7398},[75,13007,13008,13011,13013,13016,13018,13020],{"class":7383,"line":7498},[75,13009,13010],{"class":7398},"    tail.prev ",[75,13012,7421],{"class":7387},[75,13014,13015],{"class":7398}," tail.next ",[75,13017,7421],{"class":7387},[75,13019,8941],{"class":8127},[75,13021,8131],{"class":7398},[75,13023,13024,13027,13029,13031],{"class":7383,"line":7516},[75,13025,13026],{"class":7398},"    size ",[75,13028,7421],{"class":7387},[75,13030,8282],{"class":8127},[75,13032,8131],{"class":7398},[75,13034,13035],{"class":7383,"line":7521},[75,13036,7610],{"class":7398},[12,13038,13039],{},"We walk through and explicitly null out every pointer. This helps the garbage collector reclaim memory promptly.",[6964,13041,13043],{"id":13042},"_15-wrapping-up","15. Wrapping Up",[12,13045,13046],{},"Here's the mental model to take with you:",[23,13048,13049,13055,13061,13070,13079,13087,13090],{},[26,13050,13051,13054],{},[106,13052,13053],{},"A linked list is a chain of nodes connected by references."," No contiguous memory required, no shifting needed when inserting in the middle.",[26,13056,13057,13060],{},[106,13058,13059],{},"Singly linked"," = forward pointers only. Cheap, but you can't go backward, and finding \"the previous node\" requires walking from the head.",[26,13062,13063,13066,13067,13069],{},[106,13064,13065],{},"Doubly linked"," = pointers in both directions. A bit more memory and slightly more work per update, but everything becomes simpler and ",[6979,13068,10709],{}," becomes O(1).",[26,13071,13072,13075,13076,13078],{},[106,13073,13074],{},"Circular"," = the ends loop back. Useful for cyclic data.\nWhen should you reach for a linked list in real code? Honestly, in most everyday Java, ",[6979,13077,7193],{}," wins — it's faster in practice because modern CPUs love contiguous memory. But linked lists shine when:",[26,13080,13081,13082,13084,13085,7963],{},"You're frequently inserting\u002Fremoving at known positions (especially at both ends — use ",[6979,13083,7075],{},"\u002F",[6979,13086,7066],{},[26,13088,13089],{},"You're building other data structures on top (queues, stacks, LRU caches, adjacency lists for graphs)",[26,13091,13092],{},"The teaching value alone: understanding linked lists makes trees, graphs, and many other structures click.",[12,13094,13095,13096,13099],{},"That last point is the real reason this is taught early. Linked lists are the gateway to thinking about ",[106,13097,13098],{},"structures made of nodes and references"," — and that mental model is everywhere in computer science.",[9332,13101,13102],{},"html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}",{"title":180,"searchDepth":181,"depth":181,"links":13104},[13105,13106,13109,13110,13111,13117,13118,13133,13138,13142,13146,13154,13158,13166,13168],{"id":9397,"depth":181,"text":9398},{"id":9482,"depth":181,"text":9483,"children":13107},[13108],{"id":9541,"depth":7430,"text":9542},{"id":9599,"depth":181,"text":9600},{"id":9636,"depth":181,"text":9637},{"id":9779,"depth":181,"text":9780,"children":13112},[13113,13114,13115,13116],{"id":9795,"depth":7430,"text":9796},{"id":9840,"depth":7430,"text":9841},{"id":9892,"depth":7430,"text":9893},{"id":9940,"depth":7430,"text":9941},{"id":9985,"depth":181,"text":9986},{"id":10061,"depth":181,"text":13119,"children":13120},"7. Building Our Own MyLinkedList Class",[13121,13123,13125,13127,13129,13131],{"id":10147,"depth":7430,"text":13122},"addFirst(E e) — Add to the Head",{"id":10267,"depth":7430,"text":13124},"addLast(E e) — Add to the Tail",{"id":10375,"depth":7430,"text":13126},"add(int index, E e) — Insert at a Specific Position",{"id":10596,"depth":7430,"text":13128},"removeFirst() — Remove the Head",{"id":10706,"depth":7430,"text":13130},"removeLast() — Remove the Tail",{"id":10943,"depth":7430,"text":13132},"remove(int index) — Remove at a Specific Position",{"id":11144,"depth":181,"text":11145,"children":13134},[13135,13136,13137],{"id":11168,"depth":7430,"text":11169},{"id":11198,"depth":7430,"text":11199},{"id":11219,"depth":7430,"text":11220},{"id":11239,"depth":181,"text":11240,"children":13139},[13140,13141],{"id":11271,"depth":7430,"text":11272},{"id":11302,"depth":7430,"text":11303},{"id":11321,"depth":181,"text":11322,"children":13143},[13144,13145],{"id":11325,"depth":7430,"text":11326},{"id":11485,"depth":7430,"text":11486},{"id":11601,"depth":181,"text":11602,"children":13147},[13148,13150,13152],{"id":11633,"depth":7430,"text":13149},"addFirst(E element) — Insert at the Beginning",{"id":11766,"depth":7430,"text":13151},"addLast(E element) — Insert at the End",{"id":11888,"depth":7430,"text":13153},"add(int index, E element) — Insert in the Middle",{"id":12165,"depth":181,"text":12166,"children":13155},[13156,13157],{"id":12169,"depth":7430,"text":12170},{"id":12261,"depth":7430,"text":12262},{"id":12353,"depth":181,"text":12354,"children":13159},[13160,13162,13164],{"id":12360,"depth":7430,"text":13161},"removeFirst() — Delete the Head",{"id":12482,"depth":7430,"text":13163},"removeLast() — Delete the Tail",{"id":12607,"depth":7430,"text":13165},"remove(int index) — Delete in the Middle",{"id":12905,"depth":181,"text":13167},"14. Bonus: clear() — Wipe the List",{"id":13042,"depth":181,"text":13043},"2026-05-13","If you've ever used ArrayList in Java and wondered what the other list — LinkedList — is doing differently, this post is for you. We're going to walk through what linked lists are, why they exist alongside arrays, and how to build one from scratch. Then we'll level up to doubly linked lists, which are a small twist on the idea with surprisingly nice properties.",{},"\u002Fblog\u002FLinked-Lists-&-Doubly-Linked-Lists",{"title":9376,"description":13170},{"loc":13172},"blog\u002FLinked-Lists-&-Doubly-Linked-Lists","X5It7DnQ1OY4lgsHR7NDVM9Apq6O2Ox5tcqCudDSxzA",{"id":13178,"title":13179,"author":7,"body":13180,"date":13169,"description":13303,"draft":185,"edited_at":6934,"extension":186,"featured_image":13304,"meta":13305,"navigation":189,"path":13306,"pinned":185,"seo":13307,"sitemap":13308,"stem":13309,"tags":194,"__hash__":13310},"blog\u002Fblog\u002FSCI（四）病理.md","SCI（四）病理",{"type":9,"value":13181,"toc":13301},[13182,13185],[12,13183,13184],{},"下一章來講病理，首先是過程",[23,13186,13187,13210],{},[26,13188,13189,13190],{},"原發性損傷：意思是受傷當下的物理性破壞",[23,13191,13192,13195],{},[26,13193,13194],{},"若受傷在細胞體內最嚴重，也有可能發生在軸突處",[26,13196,13197,13198,13202],{},"挫傷一般情況都是因為移位的骨頭、軟組織或兩者共同壓迫導致，",[75,13199,13201],{"className":13200},[78,79],"導致脊髓瘀血或出血，可以造成完全性損傷且永久",[23,13203,13204],{},[26,13205,13206],{},[75,13207,13209],{"className":13208},[5387,78],"脊髓不一定要完全斷裂，也能造成不可逆之損傷",[26,13211,13212,13213],{},"次發性損傷：脊髓內部啟動的進行性組織破壞（SCI大多數都是這一種造成的）",[23,13214,13215,13218],{},[26,13216,13217],{},"定義是在原發性創傷發生內數小時，在脊髓發生的所有損傷誘發的破壞性事件；通常持續數天或四周",[26,13219,13220,13221],{},"機制尚未能完全掌握；目前大概可能有這幾種",[23,13222,13223,13239,13263,13279],{},[26,13224,13225,13226],{},"缺血：因外傷及血管痙攣導致血流中斷；定義是受傷區域血流量減少",[23,13227,13228,13236],{},[26,13229,13230,13231],{},"主要原因是前溝動脈(anterior sulcal arteries)以及微動脈(離心系統)受損",[23,13232,13233],{},[26,13234,13235],{},"血管痙攣會讓前溝動脈收縮；例如上去甲腎上腺素導致的收縮",[26,13237,13238],{},"其他原因包括破裂、血栓、靜脈系統受損、代謝紊亂、水腫",[26,13240,13241,13242,13246,13247],{},"發炎：",[75,13243,13245],{"className":13244},[78,79],"嗜中性球及微膠細胞\u002F巨噬細胞","會執行吞噬作用清除碎片(微膠細胞相當於中樞系統的巨噬細胞)",[23,13248,13249,13252,13255],{},[26,13250,13251],{},"是在48小時內擴大病灶的原因",[26,13253,13254],{},"活性氧自由基也會使其進一步惡化，他會進一步攻擊核酸、蛋白質、磷脂",[26,13256,13257,13258],{},"發炎是好壞參半的機制",[23,13259,13260],{},[26,13261,13262],{},"壞處像剛提到的會讓次發性損傷惡化；好處它會保留一部份好的神經元，在修復過程中也扮演關鍵角色",[26,13264,13265,13266,13270,13271],{},"離子濃度失調：鉀離子流出，鈉離子流入；使鈣離子堆積過多，導致蛋白質跟磷脂分解，引發",[75,13267,13269],{"className":13268},[79,78],"脫髓鞘","現象發生",[23,13272,13273],{},[26,13274,13275],{},[75,13276,13278],{"className":13277},[5387,78],"神經元內鈉離子濃度是SCI次發性損傷的關鍵因素之一",[26,13280,13281,13282],{},"細胞凋亡（Apoptosis是自發性的，類以自殺）：由創傷直接誘發",[23,13283,13284,13292,13295,13298],{},[26,13285,13286,13287],{},"正常的話像胚胎發育也是如此，但不正常就像CNS受損",[23,13288,13289],{},[26,13290,13291],{},"而不正常的叫細胞壞死（necrosis），是異常有害的",[26,13293,13294],{},"會在受傷後4到6小時開始，在24小時內結束（受傷的level）；會蔓延到附近level，最多持續三周",[26,13296,13297],{},"受影響細胞為神經元跟神經膠細胞",[26,13299,13300],{},"而這個程序目前還沒有完全掌握",{"title":180,"searchDepth":181,"depth":181,"links":13302},[],"病理全部","\u002Fimages\u002Fuploads\u002F1778827085830-144702095_p0_master1200.webp",{},"\u002Fblog\u002FSCI（四）病理",{"title":13179,"description":13303},{"loc":13306},"blog\u002FSCI（四）病理","0CCgRiM1FuIHmbk04n9g_kY74EsMC1-OjL-SUMfLDIk",{"id":13312,"title":13313,"author":6946,"body":13314,"date":13169,"description":180,"draft":185,"edited_at":9366,"extension":186,"featured_image":194,"meta":15638,"navigation":189,"path":15639,"pinned":185,"seo":15640,"sitemap":15641,"stem":15642,"tags":194,"__hash__":15643},"blog\u002Fblog\u002FStack.md","Stack",{"type":9,"value":13315,"toc":15611},[13316,13320,13336,13339,13342,13363,13367,13372,13383,13386,13414,13421,13434,13437,13449,13453,13456,13469,13472,13476,13494,13497,13503,13506,13509,13520,13524,13527,13582,13585,13604,13607,13611,13614,13620,13623,13626,13644,13658,13662,13665,13669,13687,13690,13693,13696,13702,13706,13713,13715,13718,13724,13728,13731,14005,14008,14050,14054,14066,14069,14162,14169,14173,14400,14402,14408,14419,14423,14426,14430,14433,14439,14453,14468,14474,14477,14497,14501,14571,14578,14582,14585,14591,14594,14597,14600,14603,14607,14613,14618,14624,14630,14636,14641,14646,14652,14655,14659,14666,15453,15455,15461,15465,15468,15505,15511,15533,15537,15540,15608],[6964,13317,13319],{"id":13318},"a-quick-recap-where-stacks-fit-in","A Quick Recap: Where Stacks Fit In",[12,13321,13322,13323,13326,13327,7298,13329,7298,13331,7298,13333,13335],{},"Before we get to stacks, let's place them on the map. In Java, almost every data structure lives inside something called the ",[106,13324,13325],{},"Collection Framework",". It's a family tree of interfaces and classes that all share the same basic vocabulary — things like ",[6979,13328,8062],{},[6979,13330,8065],{},[6979,13332,7546],{},[6979,13334,9465],{},", and so on.",[12,13337,13338],{},"Here's a simplified view of the hierarchy:",[7026,13340],{"code":13341},"\u003C!DOCTYPE html>\n\u003Chtml lang=\"en\">\n\u003Chead>\n\u003Cmeta charset=\"UTF-8\">\n\u003Ctitle>Java Collection Framework Hierarchy\u003C\u002Ftitle>\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.googleapis.com\">\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.gstatic.com\" crossorigin>\n\u003Clink href=\"https:\u002F\u002Ffonts.googleapis.com\u002Fcss2?family=Fraunces:opsz,wght@9..144,400;9..144,500;9..144,600;9..144,700&family=JetBrains+Mono:wght@400;500;600&display=swap\" rel=\"stylesheet\">\n\u003Cstyle>\n  :root {\n    --bg: #f5efe4;\n    --paper: #fbf6ec;\n    --ink: #1a1612;\n    --ink-soft: #6b6258;\n    --rule: #d8cfc0;\n    --accent: #c44536;\n    --accent-soft: #e8b9b3;\n    --teal: #2c5f5d;\n    --gold: #b08e3c;\n  }\n  * { box-sizing: border-box; margin: 0; padding: 0; }\n  body {\n    background: var(--bg);\n    color: var(--ink);\n    font-family: 'Fraunces', Georgia, serif;\n    padding: 2rem 1rem;\n    min-height: 100vh;\n    display: flex;\n    flex-direction: column;\n    align-items: center;\n  }\n  .widget {\n    width: 100%;\n    max-width: 820px;\n    background: var(--paper);\n    border: 1px solid var(--rule);\n    border-radius: 4px;\n    padding: 2rem 1.5rem 2.5rem;\n    position: relative;\n    box-shadow: 0 1px 0 rgba(0,0,0,0.02), 0 20px 40px -20px rgba(26,22,18,0.12);\n  }\n  .widget::before {\n    content: \"FIG. 01\";\n    position: absolute;\n    top: 1rem; right: 1.25rem;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.7rem;\n    letter-spacing: 0.15em;\n    color: var(--ink-soft);\n  }\n  h1 {\n    font-family: 'Fraunces', serif;\n    font-weight: 600;\n    font-size: 1.5rem;\n    font-variation-settings: \"opsz\" 96;\n    margin-bottom: 0.25rem;\n    letter-spacing: -0.01em;\n  }\n  .subtitle {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.75rem;\n    color: var(--ink-soft);\n    letter-spacing: 0.05em;\n    margin-bottom: 1.5rem;\n    text-transform: uppercase;\n  }\n  .legend {\n    display: flex;\n    gap: 1.5rem;\n    margin-bottom: 1rem;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.72rem;\n    color: var(--ink-soft);\n  }\n  .legend-item { display: flex; align-items: center; gap: 0.4rem; }\n  .legend-swatch {\n    display: inline-block;\n    width: 14px; height: 14px;\n    border: 1.5px solid var(--ink);\n  }\n  .swatch-interface { background: var(--paper); }\n  .swatch-class { background: var(--ink); }\n  .swatch-stack { background: var(--accent); border-color: var(--accent); }\n \n  svg { width: 100%; height: auto; display: block; }\n  .node-bg-interface { fill: var(--paper); stroke: var(--ink); stroke-width: 1.5; }\n  .node-bg-class { fill: var(--ink); stroke: var(--ink); stroke-width: 1.5; }\n  .node-bg-stack { fill: var(--accent); stroke: var(--accent); stroke-width: 1.5; }\n  .node-label-dark { fill: var(--ink); font-family: 'JetBrains Mono', monospace; font-size: 12px; font-weight: 500; }\n  .node-label-light { fill: var(--paper); font-family: 'JetBrains Mono', monospace; font-size: 12px; font-weight: 500; }\n  .connector { stroke: var(--ink-soft); stroke-width: 1; fill: none; }\n  .connector-accent { stroke: var(--accent); stroke-width: 1.5; fill: none; }\n \n  .annotation {\n    font-family: 'Fraunces', serif;\n    font-style: italic;\n    font-size: 13px;\n    fill: var(--accent);\n  }\n  .caption {\n    margin-top: 1.5rem;\n    padding-top: 1rem;\n    border-top: 1px solid var(--rule);\n    font-size: 0.95rem;\n    line-height: 1.55;\n    color: var(--ink-soft);\n    font-style: italic;\n  }\n  .caption strong { color: var(--ink); font-style: normal; font-weight: 600; }\n  @keyframes pulse-stack {\n    0%, 100% { transform: scale(1); }\n    50% { transform: scale(1.05); }\n  }\n  #stack-node {\n    transform-origin: center;\n    transform-box: fill-box;\n    animation: pulse-stack 2.5s ease-in-out infinite;\n  }\n\u003C\u002Fstyle>\n\u003C\u002Fhead>\n\u003Cbody>\n\u003Cdiv class=\"widget\">\n  \u003Ch1>Java Collection Framework\u003C\u002Fh1>\n  \u003Cdiv class=\"subtitle\">— Where Stack Lives —\u003C\u002Fdiv>\n \n  \u003Cdiv class=\"legend\">\n    \u003Cspan class=\"legend-item\">\u003Cspan class=\"legend-swatch swatch-interface\">\u003C\u002Fspan> Interface\u003C\u002Fspan>\n    \u003Cspan class=\"legend-item\">\u003Cspan class=\"legend-swatch swatch-class\">\u003C\u002Fspan> Class\u003C\u002Fspan>\n    \u003Cspan class=\"legend-item\">\u003Cspan class=\"legend-swatch swatch-stack\">\u003C\u002Fspan> Our subject\u003C\u002Fspan>\n  \u003C\u002Fdiv>\n \n  \u003Csvg viewBox=\"0 0 800 560\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\">\n    \u003C!-- Iterable -->\n    \u003Cg transform=\"translate(340, 10)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">Iterable\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003C!-- arrow Iterable -> Collection -->\n    \u003Cpath class=\"connector\" d=\"M 400 44 L 400 70\"\u002F>\n    \u003Cpolygon points=\"396,68 400,76 404,68\" fill=\"#6b6258\"\u002F>\n \n    \u003C!-- Collection -->\n    \u003Cg transform=\"translate(340, 76)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">Collection\u003C\u002Ftext>\n    \u003C\u002Fg>\n \n    \u003C!-- Branches from Collection -->\n    \u003Cpath class=\"connector\" d=\"M 400 110 L 400 130 L 160 130 L 160 150\"\u002F>\n    \u003Cpath class=\"connector\" d=\"M 400 130 L 400 150\"\u002F>\n    \u003Cpath class=\"connector\" d=\"M 400 130 L 640 130 L 640 150\"\u002F>\n    \u003Cpolygon points=\"156,148 160,156 164,148\" fill=\"#6b6258\"\u002F>\n    \u003Cpolygon points=\"396,148 400,156 404,148\" fill=\"#6b6258\"\u002F>\n    \u003Cpolygon points=\"636,148 640,156 644,148\" fill=\"#6b6258\"\u002F>\n \n    \u003C!-- List -->\n    \u003Cg transform=\"translate(100, 156)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">List\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003C!-- Queue -->\n    \u003Cg transform=\"translate(340, 156)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">Queue\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003C!-- Set -->\n    \u003Cg transform=\"translate(580, 156)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">Set\u003C\u002Ftext>\n    \u003C\u002Fg>\n \n    \u003C!-- LIST CHILDREN -->\n    \u003Cpath class=\"connector\" d=\"M 160 190 L 160 210\"\u002F>\n    \u003Cpolygon points=\"156,208 160,216 164,208\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(100, 216)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">ArrayList\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003Cpath class=\"connector\" d=\"M 160 250 L 160 270\"\u002F>\n    \u003Cpolygon points=\"156,268 160,276 164,268\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(100, 276)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">LinkedList\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003Cpath class=\"connector\" d=\"M 160 310 L 160 330\"\u002F>\n    \u003Cpolygon points=\"156,328 160,336 164,328\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(100, 336)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">Vector\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003C!-- Vector -> Stack with accent -->\n    \u003Cpath class=\"connector-accent\" d=\"M 160 370 L 160 396\"\u002F>\n    \u003Cpolygon points=\"155,394 160,404 165,394\" fill=\"#c44536\"\u002F>\n    \n    \u003C!-- Fix: Group handles translation, inner #stack-node handles CSS scale transform -->\n    \u003Cg transform=\"translate(100, 404)\">\n      \u003Cg id=\"stack-node\">\n        \u003Crect class=\"node-bg-stack\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n        \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">Stack\u003C\u002Ftext>\n      \u003C\u002Fg>\n    \u003C\u002Fg>\n    \n    \u003Ctext class=\"annotation\" x=\"232\" y=\"426\">← you are here\u003C\u002Ftext>\n \n    \u003C!-- QUEUE CHILDREN -->\n    \u003Cpath class=\"connector\" d=\"M 400 190 L 400 210\"\u002F>\n    \u003Cpolygon points=\"396,208 400,216 404,208\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(340, 216)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">PriorityQueue\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003Cpath class=\"connector\" d=\"M 400 250 L 400 276\"\u002F>\n    \u003Cpolygon points=\"396,274 400,282 404,274\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(340, 282)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">Deque\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003Cpath class=\"connector\" d=\"M 400 316 L 400 336\"\u002F>\n    \u003Cpolygon points=\"396,334 400,342 404,334\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(340, 342)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">ArrayDeque\u003C\u002Ftext>\n    \u003C\u002Fg>\n \n    \u003C!-- SET CHILDREN -->\n    \u003Cpath class=\"connector\" d=\"M 640 190 L 640 210\"\u002F>\n    \u003Cpolygon points=\"636,208 640,216 644,208\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(580, 216)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">HashSet\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003Cpath class=\"connector\" d=\"M 640 250 L 640 270\"\u002F>\n    \u003Cpolygon points=\"636,268 640,276 644,268\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(580, 276)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">LinkedHashSet\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003Cpath class=\"connector\" d=\"M 640 310 L 640 336\"\u002F>\n    \u003Cpolygon points=\"636,334 640,342 644,334\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(580, 342)\">\n      \u003Crect class=\"node-bg-interface\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-dark\" x=\"60\" y=\"22\" text-anchor=\"middle\">SortedSet\u003C\u002Ftext>\n    \u003C\u002Fg>\n    \u003Cpath class=\"connector\" d=\"M 640 376 L 640 396\"\u002F>\n    \u003Cpolygon points=\"636,394 640,402 644,394\" fill=\"#6b6258\"\u002F>\n    \u003Cg transform=\"translate(580, 402)\">\n      \u003Crect class=\"node-bg-class\" width=\"120\" height=\"34\" rx=\"2\"\u002F>\n      \u003Ctext class=\"node-label-light\" x=\"60\" y=\"22\" text-anchor=\"middle\">TreeSet\u003C\u002Ftext>\n    \u003C\u002Fg>\n  \u003C\u002Fsvg>\n \n  \u003Cdiv class=\"caption\">\n    Notice how \u003Cstrong>Stack\u003C\u002Fstrong> lives under \u003Cstrong>Vector\u003C\u002Fstrong>, which lives under \u003Cstrong>List\u003C\u002Fstrong>. That's because a stack is, fundamentally, a kind of list — just one with strict rules about where you can poke at it.\n  \u003C\u002Fdiv>\n\u003C\u002Fdiv>\n\u003C\u002Fbody>\n\u003C\u002Fhtml>",[12,13343,13344,13345,7206,13349,7206,13352,7206,13358,13362],{},"The piece I want you to notice: ",[106,13346,13347],{},[6979,13348,13313],{},[106,13350,13351],{},"lives under",[106,13353,13354,13357],{},[6979,13355,13356],{},"Vector",", which lives under",[106,13359,13360],{},[6979,13361,7041],{},". That's a hint about what a stack actually is.",[7166,13364,13366],{"id":13365},"whats-a-list-again","What's a List, Again?",[12,13368,9407,13369,13371],{},[106,13370,7041],{}," is an Abstract Data Type (ADT) that stores items in a sequential order. Think of:",[23,13373,13374,13377,13380],{},[26,13375,13376],{},"a list of students in a class",[26,13378,13379],{},"a list of available hotel rooms",[26,13381,13382],{},"a list of books on a shelf",[12,13384,13385],{},"The common operations you can do on a list:",[23,13387,13388,13394,13400,13405,13408,13411],{},[26,13389,13390,13393],{},[106,13391,13392],{},"Retrieve"," an element",[26,13395,13396,13399],{},[106,13397,13398],{},"Insert"," a new element",[26,13401,13402,13393],{},[106,13403,13404],{},"Delete",[26,13406,13407],{},"Check how many elements it has",[26,13409,13410],{},"Check whether some element is inside",[26,13412,13413],{},"Check if it's empty",[12,13415,13416,13417,13420],{},"There are two classic ways to ",[7289,13418,13419],{},"implement"," a list under the hood:",[3603,13422,13423,13428],{},[26,13424,13425,13427],{},[106,13426,9526],{}," — fast random access by index, but inserting\u002Fremoving from the middle is slow (you have to shift everything).",[26,13429,13430,13433],{},[106,13431,13432],{},"Using a linked list"," — fast insertion\u002Fdeletion anywhere, but slower to jump to a specific index.",[12,13435,13436],{},"Keep this trade-off in your back pocket. It's going to come up again very soon.",[13438,13439,13443],"div",{"className":13440},[13441,13442],"info-box","info-box-info",[12,13444,13445,13448],{},[106,13446,13447],{},"Quick mental model:"," an array is like a row of numbered lockers — you can open locker #47 instantly, but rearranging them is a pain. A linked list is like a treasure hunt — each clue points to the next, so inserting a new clue is easy, but finding the 47th clue means following 46 others first.",[6964,13450,13452],{"id":13451},"introducing-the-stack","Introducing the Stack",[12,13454,13455],{},"Here's the one-sentence definition you should remember forever:",[13438,13457,13459],{"className":13458},[13441,13442],[12,13460,13461,7206,13464,6959],{},[106,13462,13463],{},"A stack is a list where elements are accessed, inserted, and deleted only from one end — called the",[7289,13465,13466],{},[106,13467,13468],{},"top",[12,13470,13471],{},"That's it. Everything else about stacks follows from this constraint.",[7166,13473,13475],{"id":13474},"the-lifo-principle","The LIFO Principle",[12,13477,13478,13479,13482,13483,13486,13487,13490,13491,6959],{},"Because you can only add and remove from the top, the ",[7289,13480,13481],{},"last"," item you put in is always the ",[7289,13484,13485],{},"first"," item you take out. We call this ",[106,13488,13489],{},"LIFO",": ",[106,13492,13493],{},"Last-In, First-Out",[12,13495,13496],{},"Think of a stack of plates in a cafeteria:",[6972,13498,13501],{"className":13499,"code":13500,"language":6977},[6975],"        🍽️   ← top (last one placed, first one taken)\n        🍽️\n        🍽️\n        🍽️   ← bottom\n",[6979,13502,13500],{"__ignoreMap":180},[12,13504,13505],{},"You don't pull a plate from the middle of the stack — you take the one on top. If you want the bottom plate, tough luck, you have to remove all the ones above it first.",[12,13507,13508],{},"Other real-world examples:",[23,13510,13511,13514,13517],{},[26,13512,13513],{},"A pile of pancakes (the one on top gets eaten first)",[26,13515,13516],{},"The browser's back button history (last page you visited is the first one you go back to)",[26,13518,13519],{},"The \"undo\" function in your text editor (your last action is the first to be undone)",[7166,13521,13523],{"id":13522},"the-three-core-methods","The Three Core Methods",[12,13525,13526],{},"Every stack supports three fundamental operations:",[7085,13528,13529,13537],{},[7088,13530,13531],{},[7091,13532,13533,13535],{},[7094,13534,7922],{},[7094,13536,7099],{},[7104,13538,13539,13552,13566],{},[7091,13540,13541,13546],{},[7109,13542,13543],{},[6979,13544,13545],{},"push(x)",[7109,13547,13548,13549,13551],{},"Adds element ",[6979,13550,7119],{}," to the top",[7091,13553,13554,13559],{},[7109,13555,13556],{},[6979,13557,13558],{},"pop()",[7109,13560,13561,13562,13565],{},"Removes ",[106,13563,13564],{},"and returns"," the top element",[7091,13567,13568,13575],{},[7109,13569,13570,7959,13572,7963],{},[6979,13571,7148],{},[6979,13573,13574],{},"top()",[7109,13576,13577,13578,13581],{},"Returns the top element ",[7289,13579,13580],{},"without"," removing it",[12,13583,13584],{},"Sometimes you'll also see:",[23,13586,13587,13596],{},[26,13588,13589,13591,13592,13595],{},[6979,13590,7156],{}," — returns ",[6979,13593,13594],{},"true"," if the stack has no elements",[26,13597,13598,7959,13600,13603],{},[6979,13599,7152],{},[6979,13601,13602],{},"getSize()",") — returns how many elements are in the stack",[12,13605,13606],{},"That's the entire stack interface. Beautifully small.",[7166,13608,13610],{"id":13609},"a-worked-example","A Worked Example",[12,13612,13613],{},"Let's trace through what happens with the following operations:",[6972,13615,13618],{"className":13616,"code":13617,"language":6977},[6975],"1. Push A\n2. Push B\n3. Push C\n4. Pop  (removes C)\n5. Pop  (removes B)\n6. Push D\n",[6979,13619,13617],{"__ignoreMap":180},[12,13621,13622],{},"Here's what the stack looks like after each step:",[7026,13624],{"code":13625},"\u003C!DOCTYPE html>\n\u003Chtml lang=\"en\">\n\u003Chead>\n\u003Cmeta charset=\"UTF-8\">\n\u003Ctitle>Stack Operations Walkthrough\u003C\u002Ftitle>\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.googleapis.com\">\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.gstatic.com\" crossorigin>\n\u003Clink href=\"https:\u002F\u002Ffonts.googleapis.com\u002Fcss2?family=Fraunces:opsz,wght@9..144,400;9..144,500;9..144,600;9..144,700&family=JetBrains+Mono:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\n\u003Cstyle>\n  :root {\n    --bg: #f5efe4;\n    --paper: #fbf6ec;\n    --ink: #1a1612;\n    --ink-soft: #6b6258;\n    --rule: #d8cfc0;\n    --accent: #c44536;\n    --accent-soft: #e8b9b3;\n    --teal: #2c5f5d;\n    --gold: #b08e3c;\n    --green: #4a7c4d;\n  }\n  * { box-sizing: border-box; margin: 0; padding: 0; }\n  body {\n    background: var(--bg);\n    color: var(--ink);\n    font-family: 'Fraunces', Georgia, serif;\n    padding: 2rem 1rem;\n    min-height: 100vh;\n    display: flex;\n    flex-direction: column;\n    align-items: center;\n  }\n  .widget {\n    width: 100%;\n    max-width: 820px;\n    background: var(--paper);\n    border: 1px solid var(--rule);\n    border-radius: 4px;\n    padding: 2rem 1.5rem 2.5rem;\n    position: relative;\n    box-shadow: 0 1px 0 rgba(0,0,0,0.02), 0 20px 40px -20px rgba(26,22,18,0.12);\n  }\n  .widget::before {\n    content: \"FIG. 02 · INTERACTIVE\";\n    position: absolute;\n    top: 1rem; right: 1.25rem;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.7rem;\n    letter-spacing: 0.15em;\n    color: var(--ink-soft);\n  }\n  h1 {\n    font-family: 'Fraunces', serif;\n    font-weight: 600;\n    font-size: 1.5rem;\n    font-variation-settings: \"opsz\" 96;\n    margin-bottom: 0.25rem;\n    letter-spacing: -0.01em;\n  }\n  .subtitle {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.75rem;\n    color: var(--ink-soft);\n    letter-spacing: 0.05em;\n    margin-bottom: 1.5rem;\n    text-transform: uppercase;\n  }\n \n  .timeline {\n    display: grid;\n    grid-template-columns: repeat(6, 1fr);\n    gap: 6px;\n    margin-bottom: 1.75rem;\n  }\n  .step {\n    padding: 0.5rem 0.4rem;\n    border: 1px solid var(--rule);\n    background: transparent;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.7rem;\n    text-align: center;\n    color: var(--ink-soft);\n    transition: all 0.3s ease;\n  }\n  .step.done {\n    background: var(--ink);\n    color: var(--paper);\n    border-color: var(--ink);\n  }\n  .step.active {\n    background: var(--accent);\n    color: var(--paper);\n    border-color: var(--accent);\n    transform: translateY(-2px);\n    box-shadow: 0 4px 8px -2px rgba(196,69,54,0.4);\n  }\n  .step-num {\n    display: block;\n    font-size: 0.6rem;\n    opacity: 0.6;\n    margin-bottom: 2px;\n  }\n \n  .main {\n    display: grid;\n    grid-template-columns: 1fr 1.2fr;\n    gap: 2rem;\n    align-items: center;\n  }\n  @media (max-width: 600px) {\n    .main { grid-template-columns: 1fr; }\n  }\n \n  .stack-zone {\n    display: flex;\n    flex-direction: column;\n    align-items: center;\n    min-height: 280px;\n    justify-content: flex-end;\n    position: relative;\n  }\n  .top-label {\n    font-family: 'Fraunces', serif;\n    font-style: italic;\n    font-size: 0.95rem;\n    color: var(--accent);\n    margin-bottom: 0.5rem;\n    height: 1.5em;\n    transition: opacity 0.3s;\n    display: flex;\n    align-items: center;\n    gap: 0.4rem;\n  }\n  .top-label::after {\n    content: \"↓\";\n    font-size: 1.2rem;\n  }\n  .top-label.hidden { opacity: 0; }\n  .stack-container {\n    width: 100px;\n    border-left: 2px solid var(--ink);\n    border-right: 2px solid var(--ink);\n    border-bottom: 2px solid var(--ink);\n    min-height: 200px;\n    display: flex;\n    flex-direction: column-reverse;\n    padding: 4px;\n    background: var(--bg);\n  }\n  .stack-cell {\n    height: 44px;\n    background: var(--ink);\n    color: var(--paper);\n    display: flex;\n    align-items: center;\n    justify-content: center;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 1.2rem;\n    font-weight: 600;\n    margin-top: 4px;\n    border-radius: 2px;\n    animation: slideIn 0.4s cubic-bezier(0.34, 1.56, 0.64, 1);\n  }\n  .stack-cell.popping {\n    animation: slideOut 0.4s ease-in forwards;\n  }\n  .stack-cell.top-cell {\n    background: var(--accent);\n  }\n  @keyframes slideIn {\n    from { transform: translateY(-60px); opacity: 0; }\n    to { transform: translateY(0); opacity: 1; }\n  }\n  @keyframes slideOut {\n    to { transform: translateY(-60px); opacity: 0; }\n  }\n  .empty-msg {\n    font-family: 'Fraunces', serif;\n    font-style: italic;\n    color: var(--ink-soft);\n    align-self: center;\n    margin: auto;\n  }\n \n  .narration {\n    padding: 1.5rem;\n    background: var(--bg);\n    border: 1px solid var(--rule);\n    border-radius: 4px;\n  }\n  .narration-action {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 1.1rem;\n    font-weight: 600;\n    color: var(--ink);\n    margin-bottom: 0.75rem;\n  }\n  .narration-action .verb {\n    color: var(--accent);\n  }\n  .narration-text {\n    font-size: 0.95rem;\n    line-height: 1.55;\n    color: var(--ink-soft);\n  }\n  .narration-text strong { color: var(--ink); font-weight: 600; }\n \n  .controls {\n    display: flex;\n    gap: 0.6rem;\n    margin-top: 1.75rem;\n    padding-top: 1.5rem;\n    border-top: 1px solid var(--rule);\n    flex-wrap: wrap;\n  }\n  button {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.8rem;\n    font-weight: 600;\n    letter-spacing: 0.04em;\n    text-transform: uppercase;\n    padding: 0.7rem 1.2rem;\n    border: 1px solid var(--ink);\n    background: var(--paper);\n    color: var(--ink);\n    cursor: pointer;\n    transition: all 0.15s ease;\n    border-radius: 2px;\n  }\n  button:hover:not(:disabled) {\n    background: var(--ink);\n    color: var(--paper);\n  }\n  button.primary {\n    background: var(--ink);\n    color: var(--paper);\n  }\n  button.primary:hover:not(:disabled) {\n    background: var(--accent);\n    border-color: var(--accent);\n  }\n  button:disabled {\n    opacity: 0.35;\n    cursor: not-allowed;\n  }\n  .progress {\n    margin-left: auto;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.75rem;\n    color: var(--ink-soft);\n    align-self: center;\n  }\n\u003C\u002Fstyle>\n\u003C\u002Fhead>\n\u003Cbody>\n\u003Cdiv class=\"widget\">\n  \u003Ch1>Stack Operations, Step by Step\u003C\u002Fh1>\n  \u003Cdiv class=\"subtitle\">— Push A · B · C · Pop C · Pop B · Push D —\u003C\u002Fdiv>\n \n  \u003Cdiv class=\"timeline\" id=\"timeline\">\u003C\u002Fdiv>\n \n  \u003Cdiv class=\"main\">\n    \u003Cdiv class=\"stack-zone\">\n      \u003Cdiv class=\"top-label hidden\" id=\"top-label\">top\u003C\u002Fdiv>\n      \u003Cdiv class=\"stack-container\" id=\"stack\">\u003C\u002Fdiv>\n    \u003C\u002Fdiv>\n \n    \u003Cdiv class=\"narration\">\n      \u003Cdiv class=\"narration-action\" id=\"action\">Ready to begin\u003C\u002Fdiv>\n      \u003Cdiv class=\"narration-text\" id=\"explanation\">\n        We'll trace through six operations on an initially empty stack. Click \u003Cstrong>Next step\u003C\u002Fstrong> to advance.\n      \u003C\u002Fdiv>\n    \u003C\u002Fdiv>\n  \u003C\u002Fdiv>\n \n  \u003Cdiv class=\"controls\">\n    \u003Cbutton id=\"prev-btn\" disabled>← Back\u003C\u002Fbutton>\n    \u003Cbutton id=\"next-btn\" class=\"primary\">Next step →\u003C\u002Fbutton>\n    \u003Cbutton id=\"reset-btn\">Reset\u003C\u002Fbutton>\n    \u003Cspan class=\"progress\" id=\"progress\">0 \u002F 6\u003C\u002Fspan>\n  \u003C\u002Fdiv>\n\u003C\u002Fdiv>\n \n\u003Cscript>\n  const operations = [\n    { verb: 'push', value: 'A', text: 'We push \u003Cstrong>A\u003C\u002Fstrong> onto an empty stack. It sits at the bottom — but for now, it\\'s also the top.' },\n    { verb: 'push', value: 'B', text: 'We push \u003Cstrong>B\u003C\u002Fstrong>. It goes on top of A. The \"top\" pointer now refers to B.' },\n    { verb: 'push', value: 'C', text: 'We push \u003Cstrong>C\u003C\u002Fstrong>. The stack now reads (bottom → top): A, B, C.' },\n    { verb: 'pop',  value: 'C', text: 'We pop. Out comes \u003Cstrong>C\u003C\u002Fstrong> — the most recent arrival. This is LIFO: the last one in is the first one out.' },\n    { verb: 'pop',  value: 'B', text: 'Another pop. \u003Cstrong>B\u003C\u002Fstrong> leaves. A is now alone at the bottom, and also the top.' },\n    { verb: 'push', value: 'D', text: 'Finally we push \u003Cstrong>D\u003C\u002Fstrong>. The stack now holds A at the bottom and D at the top. B and C are gone forever.' }\n  ];\n \n  let step = 0; \u002F\u002F 0 = before any operation; 1-6 = after operations[step-1]\n  let stack = [];\n \n  const stackEl = document.getElementById('stack');\n  const topLabel = document.getElementById('top-label');\n  const actionEl = document.getElementById('action');\n  const explainEl = document.getElementById('explanation');\n  const prevBtn = document.getElementById('prev-btn');\n  const nextBtn = document.getElementById('next-btn');\n  const resetBtn = document.getElementById('reset-btn');\n  const progressEl = document.getElementById('progress');\n  const timeline = document.getElementById('timeline');\n \n  \u002F\u002F Build timeline\n  operations.forEach((op, i) => {\n    const div = document.createElement('div');\n    div.className = 'step';\n    div.dataset.index = i;\n    div.innerHTML = `\u003Cspan class=\"step-num\">STEP ${i+1}\u003C\u002Fspan>${op.verb === 'push' ? 'Push' : 'Pop'} ${op.value}`;\n    timeline.appendChild(div);\n  });\n \n  function render(animate = true) {\n    \u002F\u002F Render stack\n    stackEl.innerHTML = '';\n    if (stack.length === 0) {\n      const empty = document.createElement('div');\n      empty.className = 'empty-msg';\n      empty.textContent = '(empty)';\n      stackEl.appendChild(empty);\n      topLabel.classList.add('hidden');\n    } else {\n      stack.forEach((v, i) => {\n        const cell = document.createElement('div');\n        cell.className = 'stack-cell';\n        if (i === stack.length - 1) cell.classList.add('top-cell');\n        if (!animate) cell.style.animation = 'none';\n        cell.textContent = v;\n        stackEl.appendChild(cell);\n      });\n      topLabel.classList.remove('hidden');\n    }\n \n    \u002F\u002F Narration\n    if (step === 0) {\n      actionEl.innerHTML = 'Ready to begin';\n      explainEl.innerHTML = 'We\\'ll trace through six operations on an initially empty stack. Click \u003Cstrong>Next step\u003C\u002Fstrong> to advance.';\n    } else {\n      const op = operations[step - 1];\n      actionEl.innerHTML = `\u003Cspan class=\"verb\">${op.verb === 'push' ? 'PUSH' : 'POP'}\u003C\u002Fspan> ${op.value}`;\n      explainEl.innerHTML = op.text;\n    }\n \n    \u002F\u002F Timeline\n    document.querySelectorAll('.step').forEach((el, i) => {\n      el.classList.remove('done', 'active');\n      if (i \u003C step - 1) el.classList.add('done');\n      if (i === step - 1) el.classList.add('active');\n    });\n \n    \u002F\u002F Controls\n    prevBtn.disabled = step === 0;\n    nextBtn.disabled = step === operations.length;\n    progressEl.textContent = `${step} \u002F ${operations.length}`;\n  }\n \n  function next() {\n    if (step >= operations.length) return;\n    const op = operations[step];\n    if (op.verb === 'push') stack.push(op.value);\n    else stack.pop();\n    step++;\n    render();\n  }\n \n  function prev() {\n    if (step === 0) return;\n    step--;\n    \u002F\u002F Replay from scratch\n    stack = [];\n    for (let i = 0; i \u003C step; i++) {\n      const op = operations[i];\n      if (op.verb === 'push') stack.push(op.value);\n      else stack.pop();\n    }\n    render(false);\n  }\n \n  function reset() {\n    step = 0;\n    stack = [];\n    render(false);\n  }\n \n  nextBtn.addEventListener('click', next);\n  prevBtn.addEventListener('click', prev);\n  resetBtn.addEventListener('click', reset);\n \n  render(false);\n\u003C\u002Fscript>\n\u003C\u002Fbody>\n\u003C\u002Fhtml>",[12,13627,13628,13629,13632,13633,13636,13637,7042,13640,13643],{},"After step 6, the stack contains ",[6979,13630,13631],{},"A"," at the bottom and ",[6979,13634,13635],{},"D"," at the top. Notice that ",[6979,13638,13639],{},"B",[6979,13641,13642],{},"C"," are gone — they were popped off and are no longer in the structure.",[7351,13645,13646],{},[12,13647,13648,13651,13652,13657],{},[106,13649,13650],{},"Try it yourself:"," there's a great interactive visualization at ",[2330,13653,13656],{"href":13654,"rel":13655},"https:\u002F\u002Fyongdanielliang.github.io\u002Fanimation\u002Fweb\u002FStack.html",[2334],"Daniel Liang's Stack Animation",". Highly recommended for getting an intuitive feel.",[6964,13659,13661],{"id":13660},"implementing-a-stack-in-java","Implementing a Stack in Java",[12,13663,13664],{},"Now for the fun part — building one ourselves.",[7166,13666,13668],{"id":13667},"why-an-arraylist-works-so-well","Why an ArrayList Works So Well",[12,13670,13671,13672,13675,13676,13679,13680,7042,13683,13686],{},"Recall the trade-off between arrays and linked lists. With a stack, we ",[106,13673,13674],{},"only ever add and remove from one end",". That's exactly the operation that arrays are ",[7289,13677,13678],{},"fast"," at. We don't need fancy middle-insertion — we just need a fast ",[6979,13681,13682],{},"add to end",[6979,13684,13685],{},"remove from end",", both of which are O(1) on an ArrayList.",[12,13688,13689],{},"So for a stack, an ArrayList is the more efficient backing structure.",[12,13691,13692],{},"Conceptually:",[7026,13694],{"code":13695},"\u003C!DOCTYPE html>\n\u003Chtml lang=\"en\">\n\u003Chead>\n\u003Cmeta charset=\"UTF-8\">\n\u003Ctitle>Stack as ArrayList\u003C\u002Ftitle>\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.googleapis.com\">\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.gstatic.com\" crossorigin>\n\u003Clink href=\"https:\u002F\u002Ffonts.googleapis.com\u002Fcss2?family=Fraunces:opsz,wght@9..144,400;9..144,500;9..144,600;9..144,700&family=JetBrains+Mono:wght@400;500;600&display=swap\" rel=\"stylesheet\">\n\u003Cstyle>\n  :root {\n    --bg: #f5efe4;\n    --paper: #fbf6ec;\n    --ink: #1a1612;\n    --ink-soft: #6b6258;\n    --rule: #d8cfc0;\n    --accent: #c44536;\n    --teal: #2c5f5d;\n  }\n  * { box-sizing: border-box; margin: 0; padding: 0; }\n  body {\n    background: var(--bg);\n    color: var(--ink);\n    font-family: 'Fraunces', Georgia, serif;\n    padding: 2rem 1rem;\n    min-height: 100vh;\n    display: flex;\n    flex-direction: column;\n    align-items: center;\n  }\n  .widget {\n    width: 100%;\n    max-width: 820px;\n    background: var(--paper);\n    border: 1px solid var(--rule);\n    border-radius: 4px;\n    padding: 2rem 1.5rem 2.5rem;\n    position: relative;\n    box-shadow: 0 1px 0 rgba(0,0,0,0.02), 0 20px 40px -20px rgba(26,22,18,0.12);\n  }\n  .widget::before {\n    content: \"FIG. 03\";\n    position: absolute;\n    top: 1rem; right: 1.25rem;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.7rem;\n    letter-spacing: 0.15em;\n    color: var(--ink-soft);\n  }\n  h1 {\n    font-family: 'Fraunces', serif;\n    font-weight: 600;\n    font-size: 1.5rem;\n    font-variation-settings: \"opsz\" 96;\n    margin-bottom: 0.25rem;\n    letter-spacing: -0.01em;\n  }\n  .subtitle {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.75rem;\n    color: var(--ink-soft);\n    letter-spacing: 0.05em;\n    margin-bottom: 2rem;\n    text-transform: uppercase;\n  }\n  svg { width: 100%; height: auto; display: block; }\n  .label-title {\n    font-family: 'Fraunces', serif;\n    font-style: italic;\n    font-size: 16px;\n    fill: var(--ink);\n  }\n  .label-small {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 11px;\n    fill: var(--ink-soft);\n  }\n  .label-cell {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 14px;\n    font-weight: 600;\n    fill: var(--paper);\n  }\n  .label-top {\n    font-family: 'Fraunces', serif;\n    font-style: italic;\n    font-size: 14px;\n    fill: var(--accent);\n  }\n  .cell { fill: var(--ink); }\n  .cell-top { fill: var(--accent); }\n  .border { stroke: var(--ink); stroke-width: 2; fill: none; }\n  .pointer { stroke: var(--accent); stroke-width: 1.5; fill: none; }\n  .pointer-dashed {\n    stroke: var(--accent);\n    stroke-width: 1.2;\n    fill: none;\n    stroke-dasharray: 4 3;\n    opacity: 0.7;\n  }\n  .caption {\n    margin-top: 1.5rem;\n    padding-top: 1rem;\n    border-top: 1px solid var(--rule);\n    font-size: 0.95rem;\n    line-height: 1.6;\n    color: var(--ink-soft);\n  }\n  .caption strong { color: var(--ink); font-weight: 600; }\n  .caption .accent { color: var(--accent); font-weight: 600; }\n\u003C\u002Fstyle>\n\u003C\u002Fhead>\n\u003Cbody>\n\u003Cdiv class=\"widget\">\n  \u003Ch1>How a Stack Maps onto an ArrayList\u003C\u002Fh1>\n  \u003Cdiv class=\"subtitle\">— Conceptual model · Implementation —\u003C\u002Fdiv>\n \n  \u003Csvg viewBox=\"0 0 800 320\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\">\n    \u003C!-- LEFT: Conceptual vertical stack -->\n    \u003Ctext class=\"label-title\" x=\"120\" y=\"30\" text-anchor=\"middle\">Conceptually\u003C\u002Ftext>\n    \u003Ctext class=\"label-small\" x=\"120\" y=\"48\" text-anchor=\"middle\">a vertical stack\u003C\u002Ftext>\n \n    \u003Ctext class=\"label-top\" x=\"58\" y=\"82\" text-anchor=\"end\">top →\u003C\u002Ftext>\n    \u003C!-- container with open top -->\n    \u003Cpath class=\"border\" d=\"M 70 70 L 70 220 L 170 220 L 170 70\"\u002F>\n    \u003C!-- cells: A bottom, B middle, C top -->\n    \u003Crect class=\"cell\" x=\"74\" y=\"174\" width=\"92\" height=\"42\" rx=\"2\"\u002F>\n    \u003Ctext class=\"label-cell\" x=\"120\" y=\"201\" text-anchor=\"middle\">A\u003C\u002Ftext>\n    \u003Crect class=\"cell\" x=\"74\" y=\"128\" width=\"92\" height=\"42\" rx=\"2\"\u002F>\n    \u003Ctext class=\"label-cell\" x=\"120\" y=\"155\" text-anchor=\"middle\">B\u003C\u002Ftext>\n    \u003Crect class=\"cell-top\" x=\"74\" y=\"82\" width=\"92\" height=\"42\" rx=\"2\"\u002F>\n    \u003Ctext class=\"label-cell\" x=\"120\" y=\"109\" text-anchor=\"middle\">C\u003C\u002Ftext>\n    \u003Ctext class=\"label-small\" x=\"120\" y=\"240\" text-anchor=\"middle\">↑ bottom\u003C\u002Ftext>\n \n    \u003C!-- MIDDLE: Mapping arrow + equals -->\n    \u003Cg transform=\"translate(260, 145)\">\n      \u003Ctext x=\"0\" y=\"0\" class=\"label-title\" style=\"font-size:20px;\">≡\u003C\u002Ftext>\n      \u003Ctext class=\"label-small\" x=\"0\" y=\"22\">same data,\u003C\u002Ftext>\n      \u003Ctext class=\"label-small\" x=\"0\" y=\"36\">rotated 90°\u003C\u002Ftext>\n    \u003C\u002Fg>\n \n    \u003C!-- RIGHT: Horizontal ArrayList -->\n    \u003Ctext class=\"label-title\" x=\"540\" y=\"30\" text-anchor=\"middle\">In Java\u003C\u002Ftext>\n    \u003Ctext class=\"label-small\" x=\"540\" y=\"48\" text-anchor=\"middle\">a horizontal ArrayList\u003C\u002Ftext>\n \n    \u003C!-- Array cells -->\n    \u003Cg transform=\"translate(370, 120)\">\n      \u003Crect class=\"cell\" x=\"0\" y=\"0\" width=\"60\" height=\"50\" rx=\"2\"\u002F>\n      \u003Ctext class=\"label-cell\" x=\"30\" y=\"32\" text-anchor=\"middle\">A\u003C\u002Ftext>\n      \u003Ctext class=\"label-small\" x=\"30\" y=\"68\" text-anchor=\"middle\">[0]\u003C\u002Ftext>\n \n      \u003Crect class=\"cell\" x=\"68\" y=\"0\" width=\"60\" height=\"50\" rx=\"2\"\u002F>\n      \u003Ctext class=\"label-cell\" x=\"98\" y=\"32\" text-anchor=\"middle\">B\u003C\u002Ftext>\n      \u003Ctext class=\"label-small\" x=\"98\" y=\"68\" text-anchor=\"middle\">[1]\u003C\u002Ftext>\n \n      \u003Crect class=\"cell-top\" x=\"136\" y=\"0\" width=\"60\" height=\"50\" rx=\"2\"\u002F>\n      \u003Ctext class=\"label-cell\" x=\"166\" y=\"32\" text-anchor=\"middle\">C\u003C\u002Ftext>\n      \u003Ctext class=\"label-small\" x=\"166\" y=\"68\" text-anchor=\"middle\">[2]\u003C\u002Ftext>\n \n      \u003Crect x=\"204\" y=\"0\" width=\"60\" height=\"50\" rx=\"2\" fill=\"none\" stroke=\"#d8cfc0\" stroke-width=\"1.5\" stroke-dasharray=\"3 3\"\u002F>\n      \u003Ctext class=\"label-small\" x=\"234\" y=\"32\" text-anchor=\"middle\" style=\"opacity:0.5\">?\u003C\u002Ftext>\n      \u003Ctext class=\"label-small\" x=\"234\" y=\"68\" text-anchor=\"middle\">[3]\u003C\u002Ftext>\n \n      \u003Crect x=\"272\" y=\"0\" width=\"60\" height=\"50\" rx=\"2\" fill=\"none\" stroke=\"#d8cfc0\" stroke-width=\"1.5\" stroke-dasharray=\"3 3\"\u002F>\n      \u003Ctext class=\"label-small\" x=\"302\" y=\"32\" text-anchor=\"middle\" style=\"opacity:0.5\">?\u003C\u002Ftext>\n      \u003Ctext class=\"label-small\" x=\"302\" y=\"68\" text-anchor=\"middle\">[4]\u003C\u002Ftext>\n    \u003C\u002Fg>\n \n    \u003C!-- Pointers labels for array -->\n    \u003Ctext class=\"label-small\" x=\"400\" y=\"206\" text-anchor=\"middle\" style=\"fill:var(--ink-soft)\">front\u003C\u002Ftext>\n    \u003Ctext class=\"label-top\" x=\"536\" y=\"206\" text-anchor=\"middle\">back = top\u003C\u002Ftext>\n \n    \u003C!-- Curly bracket showing \"size\" -->\n    \u003Cpath d=\"M 370 230 Q 370 245, 380 245 L 460 245 Q 470 245, 470 255 Q 470 245, 480 245 L 560 245 Q 570 245, 570 230\" class=\"pointer-dashed\" style=\"opacity:0.5\"\u002F>\n    \u003Ctext class=\"label-small\" x=\"470\" y=\"270\" text-anchor=\"middle\" style=\"font-style:italic; font-family:'Fraunces',serif;\">size = 3\u003C\u002Ftext>\n \n    \u003C!-- Dashed correspondence lines from conceptual to array -->\n    \u003Cpath class=\"pointer-dashed\" d=\"M 168 104 C 240 100, 320 130, 530 138\"\u002F>\n    \u003Cpath class=\"pointer-dashed\" d=\"M 168 150 C 240 155, 320 155, 460 138\"\u002F>\n    \u003Cpath class=\"pointer-dashed\" d=\"M 168 196 C 240 195, 320 175, 396 138\"\u002F>\n  \u003C\u002Fsvg>\n \n  \u003Cdiv class=\"caption\">\n    The picture you draw in your head — a vertical pile with the newest item on top — is exactly the same data as a horizontal ArrayList. The \u003Cspan class=\"accent\">top of the stack\u003C\u002Fspan> corresponds to the \u003Cspan class=\"accent\">back of the array\u003C\u002Fspan>. Push appends to the end. Pop removes from the end. Both are O(1) operations, which is why an ArrayList is the natural choice for backing a stack.\n  \u003C\u002Fdiv>\n\u003C\u002Fdiv>\n\u003C\u002Fbody>\n\u003C\u002Fhtml>",[12,13697,13698,13699,13701],{},"The top of the stack corresponds to the ",[7289,13700,9565],{}," of the array list. The bottom is index 0.",[7166,13703,13705],{"id":13704},"two-design-choices-inheritance-vs-composition","Two Design Choices: Inheritance vs Composition",[12,13707,13708,13709,13712],{},"When designing the ",[6979,13710,13711],{},"GenericStack"," class, we have two ways to use ArrayList under the hood:",[16,13714],{},[7026,13716],{"code":13717},"\u003C!DOCTYPE html>\n\u003Chtml lang=\"en\">\n\u003Chead>\n\u003Cmeta charset=\"UTF-8\">\n\u003Ctitle>Inheritance vs Composition\u003C\u002Ftitle>\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.googleapis.com\">\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.gstatic.com\" crossorigin>\n\u003Clink href=\"https:\u002F\u002Ffonts.googleapis.com\u002Fcss2?family=Fraunces:opsz,wght@9..144,400;9..144,500;9..144,600;9..144,700&family=JetBrains+Mono:wght@400;500;600&display=swap\" rel=\"stylesheet\">\n\u003Cstyle>\n  :root {\n    --bg: #f5efe4;\n    --paper: #fbf6ec;\n    --ink: #1a1612;\n    --ink-soft: #6b6258;\n    --rule: #d8cfc0;\n    --accent: #c44536;\n    --teal: #2c5f5d;\n    --green: #4a7c4d;\n  }\n  * { box-sizing: border-box; margin: 0; padding: 0; }\n  body {\n    background: var(--bg);\n    color: var(--ink);\n    font-family: 'Fraunces', Georgia, serif;\n    padding: 2rem 1rem;\n    min-height: 100vh;\n    display: flex;\n    flex-direction: column;\n    align-items: center;\n  }\n  .widget {\n    width: 100%;\n    max-width: 820px;\n    background: var(--paper);\n    border: 1px solid var(--rule);\n    border-radius: 4px;\n    padding: 2rem 1.5rem 2.5rem;\n    position: relative;\n    box-shadow: 0 1px 0 rgba(0,0,0,0.02), 0 20px 40px -20px rgba(26,22,18,0.12);\n  }\n  .widget::before {\n    content: \"FIG. 04\";\n    position: absolute;\n    top: 1rem; right: 1.25rem;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.7rem;\n    letter-spacing: 0.15em;\n    color: var(--ink-soft);\n  }\n  h1 {\n    font-family: 'Fraunces', serif;\n    font-weight: 600;\n    font-size: 1.5rem;\n    font-variation-settings: \"opsz\" 96;\n    margin-bottom: 0.25rem;\n    letter-spacing: -0.01em;\n  }\n  .subtitle {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.75rem;\n    color: var(--ink-soft);\n    letter-spacing: 0.05em;\n    margin-bottom: 2rem;\n    text-transform: uppercase;\n  }\n  .grid {\n    display: grid;\n    grid-template-columns: 1fr 1fr;\n    gap: 1.5rem;\n  }\n  @media (max-width: 600px) {\n    .grid { grid-template-columns: 1fr; }\n  }\n  .approach {\n    border: 1px solid var(--rule);\n    padding: 1.5rem 1.25rem;\n    background: var(--bg);\n    border-radius: 3px;\n    display: flex;\n    flex-direction: column;\n  }\n  .approach-tag {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.68rem;\n    letter-spacing: 0.12em;\n    color: var(--ink-soft);\n    margin-bottom: 0.3rem;\n    text-transform: uppercase;\n  }\n  .approach-name {\n    font-family: 'Fraunces', serif;\n    font-style: italic;\n    font-size: 1.3rem;\n    margin-bottom: 0.2rem;\n    color: var(--ink);\n  }\n  .approach-relation {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.75rem;\n    color: var(--ink-soft);\n    margin-bottom: 1.25rem;\n  }\n  .approach-relation .key {\n    color: var(--accent);\n    font-weight: 600;\n  }\n  .diagram {\n    background: var(--paper);\n    border: 1px solid var(--rule);\n    padding: 1rem 0.5rem;\n    margin-bottom: 1.25rem;\n  }\n  .verdict {\n    font-size: 0.92rem;\n    line-height: 1.5;\n    color: var(--ink-soft);\n    margin-top: auto;\n  }\n  .verdict .good { color: var(--green); font-weight: 600; font-style: normal; }\n  .verdict .bad { color: var(--accent); font-weight: 600; font-style: normal; }\n \n  svg { width: 100%; height: auto; display: block; }\n  .box { fill: var(--paper); stroke: var(--ink); stroke-width: 1.5; }\n  .box-label {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 12px;\n    fill: var(--ink);\n    font-weight: 500;\n  }\n  .arrow { stroke: var(--ink-soft); stroke-width: 1.2; fill: none; }\n \n  .conclusion {\n    margin-top: 1.5rem;\n    padding: 1.25rem;\n    background: var(--ink);\n    color: var(--paper);\n    border-radius: 3px;\n    font-size: 0.95rem;\n    line-height: 1.55;\n  }\n  .conclusion strong {\n    color: var(--accent);\n    font-family: 'Fraunces', serif;\n    font-weight: 600;\n    font-style: italic;\n  }\n\u003C\u002Fstyle>\n\u003C\u002Fhead>\n\u003Cbody>\n\u003Cdiv class=\"widget\">\n  \u003Ch1>Two Ways to Build a Stack on Top of ArrayList\u003C\u002Fh1>\n  \u003Cdiv class=\"subtitle\">— Inheritance vs Composition —\u003C\u002Fdiv>\n \n  \u003Cdiv class=\"grid\">\n \n    \u003C!-- INHERITANCE -->\n    \u003Cdiv class=\"approach\">\n      \u003Cdiv class=\"approach-tag\">Approach 01\u003C\u002Fdiv>\n      \u003Cdiv class=\"approach-name\">Inheritance\u003C\u002Fdiv>\n      \u003Cdiv class=\"approach-relation\">GenericStack \u003Cspan class=\"key\">IS-A\u003C\u002Fspan> ArrayList\u003C\u002Fdiv>\n \n      \u003Cdiv class=\"diagram\">\n        \u003Csvg viewBox=\"0 0 280 100\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\">\n          \u003Crect class=\"box\" x=\"10\" y=\"30\" width=\"100\" height=\"40\" rx=\"2\"\u002F>\n          \u003Ctext class=\"box-label\" x=\"60\" y=\"55\" text-anchor=\"middle\">ArrayList\u003C\u002Ftext>\n \n          \u003C!-- empty triangle arrow (inheritance) -->\n          \u003Cline class=\"arrow\" x1=\"110\" y1=\"50\" x2=\"160\" y2=\"50\"\u002F>\n          \u003Cpolygon points=\"160,42 160,58 175,50\" fill=\"none\" stroke=\"#6b6258\" stroke-width=\"1.2\"\u002F>\n \n          \u003Crect class=\"box\" x=\"175\" y=\"30\" width=\"100\" height=\"40\" rx=\"2\"\u002F>\n          \u003Ctext class=\"box-label\" x=\"225\" y=\"55\" text-anchor=\"middle\">GenericStack\u003C\u002Ftext>\n \n          \u003Ctext x=\"143\" y=\"22\" text-anchor=\"middle\" style=\"font-family:'Fraunces',serif; font-style:italic; font-size:11px; fill:#6b6258;\">extends\u003C\u002Ftext>\n        \u003C\u002Fsvg>\n      \u003C\u002Fdiv>\n \n      \u003Cdiv class=\"verdict\">\n        \u003Cspan class=\"bad\">×\u003C\u002Fspan> Inherits \u003Cem>everything\u003C\u002Fem> from ArrayList — including \u003Ccode style=\"font-family:'JetBrains Mono',monospace; font-size:0.85em;\">add(index, o)\u003C\u002Fcode> and \u003Ccode style=\"font-family:'JetBrains Mono',monospace; font-size:0.85em;\">remove(index)\u003C\u002Fcode>. A user could insert in the middle and break the stack discipline.\n      \u003C\u002Fdiv>\n    \u003C\u002Fdiv>\n \n    \u003C!-- COMPOSITION -->\n    \u003Cdiv class=\"approach\">\n      \u003Cdiv class=\"approach-tag\">Approach 02 · Preferred\u003C\u002Fdiv>\n      \u003Cdiv class=\"approach-name\">Composition\u003C\u002Fdiv>\n      \u003Cdiv class=\"approach-relation\">GenericStack \u003Cspan class=\"key\">HAS-A\u003C\u002Fspan> ArrayList\u003C\u002Fdiv>\n \n      \u003Cdiv class=\"diagram\">\n        \u003Csvg viewBox=\"0 0 280 100\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\">\n          \u003Crect class=\"box\" x=\"10\" y=\"30\" width=\"100\" height=\"40\" rx=\"2\"\u002F>\n          \u003Ctext class=\"box-label\" x=\"60\" y=\"55\" text-anchor=\"middle\">GenericStack\u003C\u002Ftext>\n \n          \u003C!-- diamond arrow (composition) -->\n          \u003Cline class=\"arrow\" x1=\"110\" y1=\"50\" x2=\"160\" y2=\"50\"\u002F>\n          \u003Cpolygon points=\"110,50 122,44 134,50 122,56\" fill=\"#1a1612\" stroke=\"#1a1612\" stroke-width=\"1.2\"\u002F>\n \n          \u003Crect class=\"box\" x=\"175\" y=\"30\" width=\"100\" height=\"40\" rx=\"2\"\u002F>\n          \u003Ctext class=\"box-label\" x=\"225\" y=\"55\" text-anchor=\"middle\">ArrayList\u003C\u002Ftext>\n \n          \u003Ctext x=\"143\" y=\"22\" text-anchor=\"middle\" style=\"font-family:'Fraunces',serif; font-style:italic; font-size:11px; fill:#6b6258;\">private field\u003C\u002Ftext>\n        \u003C\u002Fsvg>\n      \u003C\u002Fdiv>\n \n      \u003Cdiv class=\"verdict\">\n        \u003Cspan class=\"good\">✓\u003C\u002Fspan> ArrayList is hidden inside as a private field. Only \u003Ccode style=\"font-family:'JetBrains Mono',monospace; font-size:0.85em;\">push\u003C\u002Fcode>, \u003Ccode style=\"font-family:'JetBrains Mono',monospace; font-size:0.85em;\">pop\u003C\u002Fcode>, \u003Ccode style=\"font-family:'JetBrains Mono',monospace; font-size:0.85em;\">peek\u003C\u002Fcode> are exposed. The stack contract stays intact.\n      \u003C\u002Fdiv>\n    \u003C\u002Fdiv>\n \n  \u003C\u002Fdiv>\n \n  \u003Cdiv class=\"conclusion\">\n    The principle: \u003Cstrong>composition gives encapsulation; inheritance leaks the abstraction.\u003C\u002Fstrong> When the parent class has more capabilities than your subclass should expose, prefer holding it as a field instead of inheriting from it.\n  \u003C\u002Fdiv>\n\u003C\u002Fdiv>\n\u003C\u002Fbody>\n\u003C\u002Fhtml>",[12,13719,13720,13721],{},"Composition lets us hide the ArrayList completely and only expose the stack operations we want. ",[106,13722,13723],{},"Composition gives us encapsulation; inheritance leaks the abstraction.",[7166,13725,13727],{"id":13726},"the-genericstack-class","The GenericStack Class",[12,13729,13730],{},"Here's the full implementation using composition:",[6972,13732,13734],{"className":7376,"code":13733,"language":7378,"meta":180,"style":180},"public class GenericStack\u003CE> {\n    private java.util.ArrayList\u003CE> list = new java.util.ArrayList\u003C>();\n \n    public int getSize() {\n        return list.size();\n    }\n \n    public E peek() {\n        return list.get(getSize() - 1);\n    }\n \n    public void push(E o) {\n        list.add(o);\n    }\n \n    public E pop() {\n        E o = list.get(getSize() - 1);\n        list.remove(getSize() - 1);\n        return o;\n    }\n \n    public boolean isEmpty() {\n        return list.isEmpty();\n    }\n \n    @Override\n    public String toString() {\n        return \"stack: \" + list.toString();\n    }\n}\n",[6979,13735,13736,13751,13769,13773,13783,13793,13797,13801,13811,13832,13836,13840,13855,13864,13868,13872,13883,13906,13924,13930,13934,13938,13948,13958,13962,13966,13972,13982,13997,14001],{"__ignoreMap":180},[75,13737,13738,13740,13742,13745,13747,13749],{"class":7383,"line":7384},[75,13739,7388],{"class":7387},[75,13741,7391],{"class":7387},[75,13743,13744],{"class":7394}," GenericStack",[75,13746,7399],{"class":7398},[75,13748,7402],{"class":7387},[75,13750,7405],{"class":7398},[75,13752,13753,13755,13758,13760,13762,13764,13766],{"class":7383,"line":181},[75,13754,7410],{"class":7387},[75,13756,13757],{"class":7398}," java.util.ArrayList\u003C",[75,13759,7402],{"class":7387},[75,13761,7418],{"class":7398},[75,13763,7421],{"class":7387},[75,13765,7424],{"class":7387},[75,13767,13768],{"class":7398}," java.util.ArrayList\u003C>();\n",[75,13770,13771],{"class":7383,"line":7430},[75,13772,7433],{"class":7398},[75,13774,13775,13777,13779,13781],{"class":7383,"line":7436},[75,13776,7439],{"class":7387},[75,13778,7531],{"class":7387},[75,13780,7534],{"class":7394},[75,13782,7495],{"class":7398},[75,13784,13785,13787,13789,13791],{"class":7383,"line":7458},[75,13786,7501],{"class":7387},[75,13788,7504],{"class":7398},[75,13790,7546],{"class":7394},[75,13792,7549],{"class":7398},[75,13794,13795],{"class":7383,"line":7474},[75,13796,7477],{"class":7398},[75,13798,13799],{"class":7383,"line":7480},[75,13800,7433],{"class":7398},[75,13802,13803,13805,13807,13809],{"class":7383,"line":7485},[75,13804,7439],{"class":7387},[75,13806,7490],{"class":7398},[75,13808,8052],{"class":7394},[75,13810,7495],{"class":7398},[75,13812,13813,13815,13817,13820,13822,13824,13826,13828,13830],{"class":7383,"line":7498},[75,13814,7501],{"class":7387},[75,13816,7504],{"class":7398},[75,13818,13819],{"class":7394},"get",[75,13821,7698],{"class":7398},[75,13823,7329],{"class":7394},[75,13825,8276],{"class":7398},[75,13827,10857],{"class":7387},[75,13829,10486],{"class":8127},[75,13831,7738],{"class":7398},[75,13833,13834],{"class":7383,"line":7516},[75,13835,7477],{"class":7398},[75,13837,13838],{"class":7383,"line":7521},[75,13839,7433],{"class":7398},[75,13841,13842,13844,13846,13849,13851,13853],{"class":7383,"line":7526},[75,13843,7439],{"class":7387},[75,13845,7442],{"class":7387},[75,13847,13848],{"class":7394}," push",[75,13850,7448],{"class":7398},[75,13852,9709],{"class":7451},[75,13854,7455],{"class":7398},[75,13856,13857,13859,13861],{"class":7383,"line":7539},[75,13858,7461],{"class":7398},[75,13860,8062],{"class":7394},[75,13862,13863],{"class":7398},"(o);\n",[75,13865,13866],{"class":7383,"line":7552},[75,13867,7477],{"class":7398},[75,13869,13870],{"class":7383,"line":7557},[75,13871,7433],{"class":7398},[75,13873,13874,13876,13878,13881],{"class":7383,"line":7562},[75,13875,7439],{"class":7387},[75,13877,7490],{"class":7398},[75,13879,13880],{"class":7394},"pop",[75,13882,7495],{"class":7398},[75,13884,13885,13888,13890,13892,13894,13896,13898,13900,13902,13904],{"class":7383,"line":7571},[75,13886,13887],{"class":7398},"        E o ",[75,13889,7421],{"class":7387},[75,13891,7504],{"class":7398},[75,13893,13819],{"class":7394},[75,13895,7698],{"class":7398},[75,13897,7329],{"class":7394},[75,13899,8276],{"class":7398},[75,13901,10857],{"class":7387},[75,13903,10486],{"class":8127},[75,13905,7738],{"class":7398},[75,13907,13908,13910,13912,13914,13916,13918,13920,13922],{"class":7383,"line":7584},[75,13909,7461],{"class":7398},[75,13911,8065],{"class":7394},[75,13913,7698],{"class":7398},[75,13915,7329],{"class":7394},[75,13917,8276],{"class":7398},[75,13919,10857],{"class":7387},[75,13921,10486],{"class":8127},[75,13923,7738],{"class":7398},[75,13925,13926,13928],{"class":7383,"line":7602},[75,13927,7501],{"class":7387},[75,13929,9721],{"class":7398},[75,13931,13932],{"class":7383,"line":7607},[75,13933,7477],{"class":7398},[75,13935,13936],{"class":7383,"line":8360},[75,13937,7433],{"class":7398},[75,13939,13940,13942,13944,13946],{"class":7383,"line":8374},[75,13941,7439],{"class":7387},[75,13943,9278],{"class":7387},[75,13945,9281],{"class":7394},[75,13947,7495],{"class":7398},[75,13949,13950,13952,13954,13956],{"class":7383,"line":8387},[75,13951,7501],{"class":7387},[75,13953,7504],{"class":7398},[75,13955,9470],{"class":7394},[75,13957,7549],{"class":7398},[75,13959,13960],{"class":7383,"line":8400},[75,13961,7477],{"class":7398},[75,13963,13964],{"class":7383,"line":8413},[75,13965,7433],{"class":7398},[75,13967,13968,13970],{"class":7383,"line":8418},[75,13969,7565],{"class":7398},[75,13971,7568],{"class":7387},[75,13973,13974,13976,13978,13980],{"class":7383,"line":8438},[75,13975,7439],{"class":7387},[75,13977,7576],{"class":7398},[75,13979,7579],{"class":7394},[75,13981,7495],{"class":7398},[75,13983,13984,13986,13989,13991,13993,13995],{"class":7383,"line":8456},[75,13985,7501],{"class":7387},[75,13987,13988],{"class":7589}," \"stack: \"",[75,13990,7593],{"class":7387},[75,13992,7504],{"class":7398},[75,13994,7579],{"class":7394},[75,13996,7549],{"class":7398},[75,13998,13999],{"class":7383,"line":8476},[75,14000,7477],{"class":7398},[75,14002,14003],{"class":7383,"line":8481},[75,14004,7610],{"class":7398},[12,14006,14007],{},"Let's read through it:",[23,14009,14010,14027,14037,14045],{},[26,14011,9455,14012,14014,14015,14018,14019,14022,14023,14026],{},[6979,14013,7620],{}," makes it ",[106,14016,14017],{},"generic"," — you can have a ",[6979,14020,14021],{},"GenericStack\u003CString>",", a ",[6979,14024,14025],{},"GenericStack\u003CInteger>",", whatever you need.",[26,14028,14029,14032,14033,14036],{},[6979,14030,14031],{},"push(o)"," is literally just ",[6979,14034,14035],{},"list.add(o)"," — adding to the end.",[26,14038,14039,7653,14041,14044],{},[6979,14040,7148],{},[6979,14042,14043],{},"list.get(size - 1)"," — looking at the last element.",[26,14046,14047,14049],{},[6979,14048,13558],{}," reads the last element, removes it from the list, and returns it.",[7166,14051,14053],{"id":14052},"handling-the-empty-case","Handling the Empty Case",[12,14055,14056,14057,7067,14059,14061,14062,14065],{},"There's a subtle problem: what if someone calls ",[6979,14058,13558],{},[6979,14060,7148],{}," on an empty stack? You'd get an ",[6979,14063,14064],{},"IndexOutOfBoundsException"," — not very informative.",[12,14067,14068],{},"A cleaner approach is to check first and throw a meaningful exception:",[6972,14070,14072],{"className":7376,"code":14071,"language":7378,"meta":180,"style":180},"public E pop() {\n    if (isEmpty()) {\n        throw new EmptyStackException();\n    }\n    E o = list.get(getSize() - 1);\n    list.remove(getSize() - 1);\n    return o;\n}\n",[6979,14073,14074,14084,14095,14106,14110,14133,14152,14158],{"__ignoreMap":180},[75,14075,14076,14078,14080,14082],{"class":7383,"line":7384},[75,14077,7388],{"class":7387},[75,14079,7490],{"class":7398},[75,14081,13880],{"class":7394},[75,14083,7495],{"class":7398},[75,14085,14086,14088,14090,14092],{"class":7383,"line":181},[75,14087,10225],{"class":7387},[75,14089,10476],{"class":7398},[75,14091,9470],{"class":7394},[75,14093,14094],{"class":7398},"()) {\n",[75,14096,14097,14099,14101,14104],{"class":7383,"line":7430},[75,14098,11936],{"class":7387},[75,14100,7424],{"class":7387},[75,14102,14103],{"class":7394}," EmptyStackException",[75,14105,7549],{"class":7398},[75,14107,14108],{"class":7383,"line":7436},[75,14109,7477],{"class":7398},[75,14111,14112,14115,14117,14119,14121,14123,14125,14127,14129,14131],{"class":7383,"line":7458},[75,14113,14114],{"class":7398},"    E o ",[75,14116,7421],{"class":7387},[75,14118,7504],{"class":7398},[75,14120,13819],{"class":7394},[75,14122,7698],{"class":7398},[75,14124,7329],{"class":7394},[75,14126,8276],{"class":7398},[75,14128,10857],{"class":7387},[75,14130,10486],{"class":8127},[75,14132,7738],{"class":7398},[75,14134,14135,14138,14140,14142,14144,14146,14148,14150],{"class":7383,"line":7474},[75,14136,14137],{"class":7398},"    list.",[75,14139,8065],{"class":7394},[75,14141,7698],{"class":7398},[75,14143,7329],{"class":7394},[75,14145,8276],{"class":7398},[75,14147,10857],{"class":7387},[75,14149,10486],{"class":8127},[75,14151,7738],{"class":7398},[75,14153,14154,14156],{"class":7383,"line":7480},[75,14155,8689],{"class":7387},[75,14157,9721],{"class":7398},[75,14159,14160],{"class":7383,"line":7485},[75,14161,7610],{"class":7398},[12,14163,14164,14165,14168],{},"This is what Java's own built-in ",[6979,14166,14167],{},"java.util.Stack"," does. When you write production code, you'd want this kind of defensive check.",[7166,14170,14172],{"id":14171},"testing-it-out","Testing It Out",[6972,14174,14176],{"className":7376,"code":14175,"language":7378,"meta":180,"style":180},"public class TestGenericStack {\n    public static void main(String[] args) {\n        GenericStack\u003CString> stack = new GenericStack\u003C>();\n \n        stack.push(\"Tom\");\n        System.out.println(\"(1) \" + stack);\n \n        stack.push(\"Susan\");\n        System.out.println(\"(2) \" + stack);\n \n        stack.push(\"Kim\");\n        stack.push(\"Michael\");\n        System.out.println(\"(3) \" + stack);\n \n        System.out.println(\"(4) \" + stack.pop());\n        System.out.println(\"(5) \" + stack.pop());\n        System.out.println(\"(6) \" + stack);\n    }\n}\n",[6979,14177,14178,14189,14209,14226,14230,14244,14260,14264,14276,14291,14295,14307,14319,14334,14338,14358,14377,14392,14396],{"__ignoreMap":180},[75,14179,14180,14182,14184,14187],{"class":7383,"line":7384},[75,14181,7388],{"class":7387},[75,14183,7391],{"class":7387},[75,14185,14186],{"class":7394}," TestGenericStack",[75,14188,8147],{"class":7398},[75,14190,14191,14193,14195,14197,14199,14201,14203,14205,14207],{"class":7383,"line":181},[75,14192,7439],{"class":7387},[75,14194,8155],{"class":7387},[75,14196,7442],{"class":7387},[75,14198,8160],{"class":7394},[75,14200,7698],{"class":7398},[75,14202,7674],{"class":7387},[75,14204,8167],{"class":7398},[75,14206,8170],{"class":7451},[75,14208,7455],{"class":7398},[75,14210,14211,14214,14216,14219,14221,14223],{"class":7383,"line":7430},[75,14212,14213],{"class":7398},"        GenericStack\u003C",[75,14215,7674],{"class":7387},[75,14217,14218],{"class":7398},"> stack ",[75,14220,7421],{"class":7387},[75,14222,7424],{"class":7387},[75,14224,14225],{"class":7398}," GenericStack\u003C>();\n",[75,14227,14228],{"class":7383,"line":7436},[75,14229,7433],{"class":7398},[75,14231,14232,14235,14238,14240,14242],{"class":7383,"line":7458},[75,14233,14234],{"class":7398},"        stack.",[75,14236,14237],{"class":7394},"push",[75,14239,7698],{"class":7398},[75,14241,7701],{"class":7589},[75,14243,7738],{"class":7398},[75,14245,14246,14248,14250,14252,14255,14257],{"class":7383,"line":7474},[75,14247,12229],{"class":7398},[75,14249,7767],{"class":7394},[75,14251,7698],{"class":7398},[75,14253,14254],{"class":7589},"\"(1) \"",[75,14256,7593],{"class":7387},[75,14258,14259],{"class":7398}," stack);\n",[75,14261,14262],{"class":7383,"line":7480},[75,14263,7433],{"class":7398},[75,14265,14266,14268,14270,14272,14274],{"class":7383,"line":7485},[75,14267,14234],{"class":7398},[75,14269,14237],{"class":7394},[75,14271,7698],{"class":7398},[75,14273,7718],{"class":7589},[75,14275,7738],{"class":7398},[75,14277,14278,14280,14282,14284,14287,14289],{"class":7383,"line":7498},[75,14279,12229],{"class":7398},[75,14281,7767],{"class":7394},[75,14283,7698],{"class":7398},[75,14285,14286],{"class":7589},"\"(2) \"",[75,14288,7593],{"class":7387},[75,14290,14259],{"class":7398},[75,14292,14293],{"class":7383,"line":7516},[75,14294,7433],{"class":7398},[75,14296,14297,14299,14301,14303,14305],{"class":7383,"line":7521},[75,14298,14234],{"class":7398},[75,14300,14237],{"class":7394},[75,14302,7698],{"class":7398},[75,14304,7735],{"class":7589},[75,14306,7738],{"class":7398},[75,14308,14309,14311,14313,14315,14317],{"class":7383,"line":7526},[75,14310,14234],{"class":7398},[75,14312,14237],{"class":7394},[75,14314,7698],{"class":7398},[75,14316,7749],{"class":7589},[75,14318,7738],{"class":7398},[75,14320,14321,14323,14325,14327,14330,14332],{"class":7383,"line":7539},[75,14322,12229],{"class":7398},[75,14324,7767],{"class":7394},[75,14326,7698],{"class":7398},[75,14328,14329],{"class":7589},"\"(3) \"",[75,14331,7593],{"class":7387},[75,14333,14259],{"class":7398},[75,14335,14336],{"class":7383,"line":7552},[75,14337,7433],{"class":7398},[75,14339,14340,14342,14344,14346,14349,14351,14354,14356],{"class":7383,"line":7557},[75,14341,12229],{"class":7398},[75,14343,7767],{"class":7394},[75,14345,7698],{"class":7398},[75,14347,14348],{"class":7589},"\"(4) \"",[75,14350,7593],{"class":7387},[75,14352,14353],{"class":7398}," stack.",[75,14355,13880],{"class":7394},[75,14357,8357],{"class":7398},[75,14359,14360,14362,14364,14366,14369,14371,14373,14375],{"class":7383,"line":7562},[75,14361,12229],{"class":7398},[75,14363,7767],{"class":7394},[75,14365,7698],{"class":7398},[75,14367,14368],{"class":7589},"\"(5) \"",[75,14370,7593],{"class":7387},[75,14372,14353],{"class":7398},[75,14374,13880],{"class":7394},[75,14376,8357],{"class":7398},[75,14378,14379,14381,14383,14385,14388,14390],{"class":7383,"line":7571},[75,14380,12229],{"class":7398},[75,14382,7767],{"class":7394},[75,14384,7698],{"class":7398},[75,14386,14387],{"class":7589},"\"(6) \"",[75,14389,7593],{"class":7387},[75,14391,14259],{"class":7398},[75,14393,14394],{"class":7383,"line":7584},[75,14395,7477],{"class":7398},[75,14397,14398],{"class":7383,"line":7602},[75,14399,7610],{"class":7398},[12,14401,9006],{},[6972,14403,14406],{"className":14404,"code":14405,"language":6977},[6975],"(1) stack: [Tom]\n(2) stack: [Tom, Susan]\n(3) stack: [Tom, Susan, Kim, Michael]\n(4) Michael\n(5) Kim\n(6) stack: [Tom, Susan]\n",[6979,14407,14405],{"__ignoreMap":180},[12,14409,14410,14411,14414,14415,14418],{},"Notice how ",[6979,14412,14413],{},"Michael"," came out first (he was pushed last), then ",[6979,14416,14417],{},"Kim",". That's LIFO in action.",[6964,14420,14422],{"id":14421},"postfix-evaluation-a-real-use-case-for-stacks","Postfix Evaluation: A Real Use Case for Stacks",[12,14424,14425],{},"This is where stacks earn their keep. Let me show you a beautiful application.",[7166,14427,14429],{"id":14428},"what-is-postfix-notation","What Is Postfix Notation?",[12,14431,14432],{},"You've been writing math your whole life like this:",[6972,14434,14437],{"className":14435,"code":14436,"language":6977},[6975],"3 + 4\n",[6979,14438,14436],{"__ignoreMap":180},[12,14440,14441,14442,14445,14446,14448,14449,14452],{},"That's called ",[106,14443,14444],{},"infix"," notation — the operator (",[6979,14447,8302],{},") is ",[7289,14450,14451],{},"in between"," the operands. It's how humans read math.",[12,14454,14455,14456,14459,14460,14463,14464,14467],{},"But there's another way to write expressions: ",[106,14457,14458],{},"postfix"," notation, also called ",[106,14461,14462],{},"Reverse Polish Notation (RPN)",", where the operator comes ",[7289,14465,14466],{},"after"," its operands:",[6972,14469,14472],{"className":14470,"code":14471,"language":6977},[6975],"3 4 +\n",[6979,14473,14471],{"__ignoreMap":180},[12,14475,14476],{},"It looks weird at first, but RPN has three wonderful properties:",[3603,14478,14479,14485,14491],{},[26,14480,14481,14484],{},[106,14482,14483],{},"No precedence rules needed."," You don't have to memorize PEMDAS \u002F BODMAS.",[26,14486,14487,14490],{},[106,14488,14489],{},"No parentheses needed."," Ever.",[26,14492,14493,14496],{},[106,14494,14495],{},"Faster for computers to evaluate"," — fewer memory accesses.",[7166,14498,14500],{"id":14499},"infix-postfix-examples","Infix → Postfix Examples",[7085,14502,14503,14516],{},[7088,14504,14505],{},[7091,14506,14507,14510,14513],{},[7094,14508,14509],{},"Infix",[7094,14511,14512],{},"Postfix (RPN)",[7094,14514,14515],{},"Why",[7104,14517,14518,14538,14556],{},[7091,14519,14520,14525,14530],{},[7109,14521,14522],{},[6979,14523,14524],{},"a + b * c",[7109,14526,14527],{},[6979,14528,14529],{},"a b c * +",[7109,14531,14532,14534,14535,14537],{},[6979,14533,8128],{}," binds tighter than ",[6979,14536,8302],{},", so it's computed first",[7091,14539,14540,14545,14550],{},[7109,14541,14542],{},[6979,14543,14544],{},"(a + b) * c",[7109,14546,14547],{},[6979,14548,14549],{},"a b + c *",[7109,14551,14552,14553,14555],{},"Parentheses force ",[6979,14554,8302],{}," first",[7091,14557,14558,14563,14568],{},[7109,14559,14560],{},[6979,14561,14562],{},"(a*b + c) \u002F d + e",[7109,14564,14565],{},[6979,14566,14567],{},"a b * c + d \u002F e +",[7109,14569,14570],{},"Inner subexpression first, then division, then final addition",[12,14572,14573,14574,14577],{},"Read each postfix expression left to right and you'll notice: ",[106,14575,14576],{},"every operator immediately follows its two operands",". That's the rule.",[7166,14579,14581],{"id":14580},"the-stack-based-algorithm","The Stack-Based Algorithm",[12,14583,14584],{},"Here's the magical part. Evaluating a postfix expression is shockingly simple with a stack:",[6972,14586,14589],{"className":14587,"code":14588,"language":6977},[6975],"1. Scan the expression left to right, token by token.\n2. If the token is an OPERAND (a number):\n       → push it onto the stack.\n3. If the token is an OPERATOR (+, -, *, \u002F, ^):\n       → pop two operands off the stack\n         (the first pop is the right operand,\n          the second pop is the left operand)\n       → compute: left OPERATOR right\n       → push the result back onto the stack.\n4. When done, the single value left on the stack is the answer.\n",[6979,14590,14588],{"__ignoreMap":180},[12,14592,14593],{},"That's the whole algorithm.",[7166,14595,14596],{"id":7168},"Try It Yourself",[12,14598,14599],{},"Below is a fully interactive postfix evaluator. Type any expression (with spaces between every token) or pick one of the presets, then step through it. The stack on the right updates with each step. The two red presets are intentionally malformed — see how the algorithm catches the errors.",[7026,14601],{"code":14602},"\u003C!DOCTYPE html>\n\u003Chtml lang=\"en\">\n\u003Chead>\n\u003Cmeta charset=\"UTF-8\">\n\u003Ctitle>Postfix Evaluator\u003C\u002Ftitle>\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.googleapis.com\">\n\u003Clink rel=\"preconnect\" href=\"https:\u002F\u002Ffonts.gstatic.com\" crossorigin>\n\u003Clink href=\"https:\u002F\u002Ffonts.googleapis.com\u002Fcss2?family=Fraunces:opsz,wght@9..144,400;9..144,500;9..144,600;9..144,700&family=JetBrains+Mono:wght@400;500;600;700&display=swap\" rel=\"stylesheet\">\n\u003Cstyle>\n  :root {\n    --bg: #f5efe4;\n    --paper: #fbf6ec;\n    --ink: #1a1612;\n    --ink-soft: #6b6258;\n    --rule: #d8cfc0;\n    --accent: #c44536;\n    --accent-soft: #e8b9b3;\n    --teal: #2c5f5d;\n    --green: #4a7c4d;\n    --gold: #b08e3c;\n  }\n  * { box-sizing: border-box; margin: 0; padding: 0; }\n  body {\n    background: var(--bg);\n    color: var(--ink);\n    font-family: 'Fraunces', Georgia, serif;\n    padding: 2rem 1rem;\n    min-height: 100vh;\n    display: flex;\n    flex-direction: column;\n    align-items: center;\n  }\n  .widget {\n    width: 100%;\n    max-width: 880px;\n    background: var(--paper);\n    border: 1px solid var(--rule);\n    border-radius: 4px;\n    padding: 2rem 1.5rem 2.5rem;\n    position: relative;\n    box-shadow: 0 1px 0 rgba(0,0,0,0.02), 0 20px 40px -20px rgba(26,22,18,0.12);\n  }\n  .widget::before {\n    content: \"FIG. 05 · INTERACTIVE\";\n    position: absolute;\n    top: 1rem; right: 1.25rem;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.7rem;\n    letter-spacing: 0.15em;\n    color: var(--ink-soft);\n  }\n  h1 {\n    font-family: 'Fraunces', serif;\n    font-weight: 600;\n    font-size: 1.5rem;\n    font-variation-settings: \"opsz\" 96;\n    margin-bottom: 0.25rem;\n    letter-spacing: -0.01em;\n  }\n  .subtitle {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.75rem;\n    color: var(--ink-soft);\n    letter-spacing: 0.05em;\n    margin-bottom: 1.5rem;\n    text-transform: uppercase;\n  }\n \n  .input-row {\n    display: flex;\n    gap: 0.5rem;\n    margin-bottom: 0.75rem;\n    flex-wrap: wrap;\n  }\n  input[type=\"text\"] {\n    flex: 1;\n    min-width: 200px;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 1rem;\n    padding: 0.75rem 0.9rem;\n    border: 1px solid var(--ink);\n    background: var(--paper);\n    color: var(--ink);\n    border-radius: 2px;\n    outline: none;\n  }\n  input[type=\"text\"]:focus {\n    border-color: var(--accent);\n    box-shadow: 0 0 0 3px rgba(196,69,54,0.15);\n  }\n  button {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.78rem;\n    font-weight: 600;\n    letter-spacing: 0.04em;\n    text-transform: uppercase;\n    padding: 0.7rem 1.1rem;\n    border: 1px solid var(--ink);\n    background: var(--paper);\n    color: var(--ink);\n    cursor: pointer;\n    transition: all 0.15s ease;\n    border-radius: 2px;\n  }\n  button:hover:not(:disabled) {\n    background: var(--ink);\n    color: var(--paper);\n  }\n  button.primary {\n    background: var(--ink);\n    color: var(--paper);\n  }\n  button.primary:hover:not(:disabled) {\n    background: var(--accent);\n    border-color: var(--accent);\n  }\n  button:disabled {\n    opacity: 0.3;\n    cursor: not-allowed;\n  }\n \n  .presets {\n    display: flex;\n    flex-wrap: wrap;\n    gap: 0.4rem;\n    margin-bottom: 1.5rem;\n    align-items: center;\n  }\n  .presets-label {\n    font-family: 'Fraunces', serif;\n    font-style: italic;\n    font-size: 0.85rem;\n    color: var(--ink-soft);\n    margin-right: 0.3rem;\n  }\n  .preset-chip {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.72rem;\n    padding: 0.35rem 0.7rem;\n    background: transparent;\n    border: 1px solid var(--rule);\n    color: var(--ink-soft);\n    border-radius: 99px;\n    cursor: pointer;\n    transition: all 0.15s ease;\n  }\n  .preset-chip:hover {\n    background: var(--ink);\n    color: var(--paper);\n    border-color: var(--ink);\n  }\n  .preset-chip.error-preset {\n    border-color: var(--accent-soft);\n    color: var(--accent);\n  }\n  .preset-chip.error-preset:hover {\n    background: var(--accent);\n    color: var(--paper);\n    border-color: var(--accent);\n  }\n \n  .stage {\n    display: grid;\n    grid-template-columns: 1.4fr 1fr;\n    gap: 1.5rem;\n    margin-top: 1rem;\n  }\n  @media (max-width: 640px) {\n    .stage { grid-template-columns: 1fr; }\n  }\n \n  .left-pane {\n    display: flex;\n    flex-direction: column;\n    gap: 1rem;\n  }\n \n  .tokens {\n    padding: 1rem;\n    background: var(--bg);\n    border: 1px solid var(--rule);\n    border-radius: 3px;\n  }\n  .tokens-label {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.7rem;\n    letter-spacing: 0.12em;\n    color: var(--ink-soft);\n    margin-bottom: 0.6rem;\n    text-transform: uppercase;\n  }\n  .tokens-row {\n    display: flex;\n    flex-wrap: wrap;\n    gap: 0.4rem;\n  }\n  .token {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 1.05rem;\n    font-weight: 600;\n    padding: 0.45rem 0.75rem;\n    min-width: 42px;\n    text-align: center;\n    border: 1.5px solid var(--rule);\n    background: var(--paper);\n    color: var(--ink-soft);\n    border-radius: 3px;\n    transition: all 0.3s cubic-bezier(0.34, 1.56, 0.64, 1);\n  }\n  .token.done {\n    border-color: var(--ink);\n    color: var(--ink);\n    opacity: 0.5;\n  }\n  .token.current {\n    background: var(--accent);\n    color: var(--paper);\n    border-color: var(--accent);\n    transform: translateY(-3px) scale(1.08);\n    box-shadow: 0 6px 12px -4px rgba(196,69,54,0.4);\n  }\n  .token.error {\n    background: var(--accent);\n    color: var(--paper);\n    border-color: var(--accent);\n    animation: shake 0.4s ease-in-out;\n  }\n  @keyframes shake {\n    0%, 100% { transform: translateX(0); }\n    25% { transform: translateX(-4px); }\n    75% { transform: translateX(4px); }\n  }\n \n  .narration {\n    padding: 1rem 1.25rem;\n    background: var(--bg);\n    border: 1px solid var(--rule);\n    border-radius: 3px;\n    min-height: 90px;\n  }\n  .narration-action {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.95rem;\n    font-weight: 600;\n    color: var(--ink);\n    margin-bottom: 0.4rem;\n  }\n  .narration-text {\n    font-size: 0.9rem;\n    line-height: 1.5;\n    color: var(--ink-soft);\n  }\n  .narration-text .num {\n    font-family: 'JetBrains Mono', monospace;\n    color: var(--ink);\n    font-weight: 600;\n  }\n  .narration-text .op {\n    color: var(--accent);\n    font-weight: 600;\n  }\n  .narration.error .narration-action { color: var(--accent); }\n  .narration.success .narration-action { color: var(--green); }\n \n  .result-box {\n    padding: 1rem 1.25rem;\n    background: var(--ink);\n    color: var(--paper);\n    border-radius: 3px;\n    display: none;\n  }\n  .result-box.visible { display: block; }\n  .result-box.error { background: var(--accent); }\n  .result-label {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.7rem;\n    letter-spacing: 0.12em;\n    opacity: 0.7;\n    margin-bottom: 0.3rem;\n    text-transform: uppercase;\n  }\n  .result-value {\n    font-family: 'Fraunces', serif;\n    font-size: 1.8rem;\n    font-weight: 600;\n    font-variation-settings: \"opsz\" 96;\n  }\n \n  .right-pane {\n    display: flex;\n    flex-direction: column;\n    align-items: center;\n  }\n  .stack-label {\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.7rem;\n    letter-spacing: 0.12em;\n    color: var(--ink-soft);\n    margin-bottom: 0.4rem;\n    text-transform: uppercase;\n    align-self: flex-start;\n    margin-left: 0.5rem;\n  }\n  .top-marker {\n    font-family: 'Fraunces', serif;\n    font-style: italic;\n    font-size: 0.9rem;\n    color: var(--accent);\n    height: 1.4em;\n    display: flex;\n    align-items: center;\n    gap: 0.3rem;\n    opacity: 0;\n    transition: opacity 0.3s;\n  }\n  .top-marker.visible { opacity: 1; }\n  .top-marker::after { content: \"↓\"; }\n  .stack-bucket {\n    width: 120px;\n    border-left: 2px solid var(--ink);\n    border-right: 2px solid var(--ink);\n    border-bottom: 2px solid var(--ink);\n    min-height: 240px;\n    display: flex;\n    flex-direction: column-reverse;\n    padding: 5px;\n    background: var(--bg);\n  }\n  .stack-item {\n    height: 42px;\n    background: var(--ink);\n    color: var(--paper);\n    display: flex;\n    align-items: center;\n    justify-content: center;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 1rem;\n    font-weight: 600;\n    margin-top: 5px;\n    border-radius: 2px;\n    animation: stackIn 0.35s cubic-bezier(0.34, 1.56, 0.64, 1);\n  }\n  .stack-item.is-top { background: var(--accent); }\n  @keyframes stackIn {\n    from { transform: translateY(-50px); opacity: 0; }\n    to { transform: translateY(0); opacity: 1; }\n  }\n  .stack-empty {\n    margin: auto;\n    font-family: 'Fraunces', serif;\n    font-style: italic;\n    color: var(--ink-soft);\n    font-size: 0.85rem;\n  }\n \n  .controls {\n    display: flex;\n    gap: 0.6rem;\n    margin-top: 1.5rem;\n    padding-top: 1.25rem;\n    border-top: 1px solid var(--rule);\n    flex-wrap: wrap;\n    align-items: center;\n  }\n  .progress {\n    margin-left: auto;\n    font-family: 'JetBrains Mono', monospace;\n    font-size: 0.72rem;\n    color: var(--ink-soft);\n  }\n\u003C\u002Fstyle>\n\u003C\u002Fhead>\n\u003Cbody>\n\u003Cdiv class=\"widget\">\n  \u003Ch1>Postfix Expression Evaluator\u003C\u002Fh1>\n  \u003Cdiv class=\"subtitle\">— Type or pick an expression, then step through it —\u003C\u002Fdiv>\n \n  \u003Cdiv class=\"input-row\">\n    \u003Cinput type=\"text\" id=\"expr-input\" value=\"4 3 5 * +\" placeholder=\"e.g.  4 3 5 * +\" autocomplete=\"off\"\u002F>\n    \u003Cbutton id=\"load-btn\" class=\"primary\">Load\u003C\u002Fbutton>\n  \u003C\u002Fdiv>\n \n  \u003Cdiv class=\"presets\">\n    \u003Cspan class=\"presets-label\">Try:\u003C\u002Fspan>\n    \u003Cbutton class=\"preset-chip\" data-expr=\"4 3 5 * +\">4 3 5 * +\u003C\u002Fbutton>\n    \u003Cbutton class=\"preset-chip\" data-expr=\"2 3 + 4 * 5 -\">2 3 + 4 * 5 -\u003C\u002Fbutton>\n    \u003Cbutton class=\"preset-chip\" data-expr=\"2 3 * 4 2 - \u002F 5 6 * +\">2 3 * 4 2 - \u002F 5 6 * +\u003C\u002Fbutton>\n    \u003Cbutton class=\"preset-chip\" data-expr=\"2 4 - 3 ^ 5 +\">2 4 - 3 ^ 5 +\u003C\u002Fbutton>\n    \u003Cbutton class=\"preset-chip error-preset\" data-expr=\"3 8 + * 9\">3 8 + * 9\u003C\u002Fbutton>\n    \u003Cbutton class=\"preset-chip error-preset\" data-expr=\"9 8 + 7\">9 8 + 7\u003C\u002Fbutton>\n  \u003C\u002Fdiv>\n \n  \u003Cdiv class=\"stage\">\n    \u003Cdiv class=\"left-pane\">\n      \u003Cdiv class=\"tokens\">\n        \u003Cdiv class=\"tokens-label\">Expression tokens\u003C\u002Fdiv>\n        \u003Cdiv class=\"tokens-row\" id=\"tokens-row\">\u003C\u002Fdiv>\n      \u003C\u002Fdiv>\n \n      \u003Cdiv class=\"narration\" id=\"narration\">\n        \u003Cdiv class=\"narration-action\">Ready\u003C\u002Fdiv>\n        \u003Cdiv class=\"narration-text\">Press \u003Cstrong>Next\u003C\u002Fstrong> to begin processing token by token. Operands push onto the stack; operators pop two values, compute, and push the result.\u003C\u002Fdiv>\n      \u003C\u002Fdiv>\n \n      \u003Cdiv class=\"result-box\" id=\"result-box\">\n        \u003Cdiv class=\"result-label\" id=\"result-label\">Final result\u003C\u002Fdiv>\n        \u003Cdiv class=\"result-value\" id=\"result-value\">—\u003C\u002Fdiv>\n      \u003C\u002Fdiv>\n    \u003C\u002Fdiv>\n \n    \u003Cdiv class=\"right-pane\">\n      \u003Cdiv class=\"stack-label\">Operand stack\u003C\u002Fdiv>\n      \u003Cdiv class=\"top-marker\" id=\"top-marker\">top\u003C\u002Fdiv>\n      \u003Cdiv class=\"stack-bucket\" id=\"stack-bucket\">\n        \u003Cdiv class=\"stack-empty\">(empty)\u003C\u002Fdiv>\n      \u003C\u002Fdiv>\n    \u003C\u002Fdiv>\n  \u003C\u002Fdiv>\n \n  \u003Cdiv class=\"controls\">\n    \u003Cbutton id=\"prev-btn\" disabled>← Back\u003C\u002Fbutton>\n    \u003Cbutton id=\"next-btn\" class=\"primary\">Next step →\u003C\u002Fbutton>\n    \u003Cbutton id=\"reset-btn\">Reset\u003C\u002Fbutton>\n    \u003Cspan class=\"progress\" id=\"progress\">0 \u002F 0\u003C\u002Fspan>\n  \u003C\u002Fdiv>\n\u003C\u002Fdiv>\n \n\u003Cscript>\n  const exprInput = document.getElementById('expr-input');\n  const loadBtn = document.getElementById('load-btn');\n  const tokensRow = document.getElementById('tokens-row');\n  const narration = document.getElementById('narration');\n  const resultBox = document.getElementById('result-box');\n  const resultLabel = document.getElementById('result-label');\n  const resultValue = document.getElementById('result-value');\n  const stackBucket = document.getElementById('stack-bucket');\n  const topMarker = document.getElementById('top-marker');\n  const prevBtn = document.getElementById('prev-btn');\n  const nextBtn = document.getElementById('next-btn');\n  const resetBtn = document.getElementById('reset-btn');\n  const progressEl = document.getElementById('progress');\n \n  let tokens = [];\n  let step = 0;\n  let stack = [];\n  let error = null;\n  let finished = false;\n \n  function isNumeric(s) {\n    if (s === undefined || s === null || s === '') return false;\n    return !isNaN(parseFloat(s)) && isFinite(s);\n  }\n  function isOperator(s) {\n    return ['+', '-', '*', '\u002F', '^'].includes(s);\n  }\n  function compute(a, b, op) {\n    switch (op) {\n      case '+': return a + b;\n      case '-': return a - b;\n      case '*': return a * b;\n      case '\u002F': return a \u002F b;\n      case '^': return Math.pow(a, b);\n    }\n  }\n  function fmt(n) {\n    if (Number.isInteger(n)) return n.toString();\n    return Number(n.toFixed(4)).toString();\n  }\n \n  function load(expr) {\n    exprInput.value = expr;\n    tokens = expr.trim().split(\u002F\\s+\u002F).filter(t => t.length > 0);\n    step = 0;\n    stack = [];\n    error = null;\n    finished = false;\n    renderTokens();\n    render();\n  }\n \n  function renderTokens() {\n    tokensRow.innerHTML = '';\n    tokens.forEach((t, i) => {\n      const el = document.createElement('div');\n      el.className = 'token';\n      el.dataset.index = i;\n      el.textContent = t;\n      tokensRow.appendChild(el);\n    });\n  }\n \n  function render() {\n    \u002F\u002F Update tokens\n    document.querySelectorAll('.token').forEach((el, i) => {\n      el.classList.remove('done', 'current', 'error');\n      if (i \u003C step) el.classList.add('done');\n      if (i === step && !finished && !error) el.classList.add('current');\n      if (error && error.tokenIndex === i) el.classList.add('error');\n    });\n \n    \u002F\u002F Update stack\n    stackBucket.innerHTML = '';\n    if (stack.length === 0) {\n      const empty = document.createElement('div');\n      empty.className = 'stack-empty';\n      empty.textContent = '(empty)';\n      stackBucket.appendChild(empty);\n      topMarker.classList.remove('visible');\n    } else {\n      stack.forEach((v, i) => {\n        const cell = document.createElement('div');\n        cell.className = 'stack-item';\n        if (i === stack.length - 1) cell.classList.add('is-top');\n        cell.textContent = fmt(v);\n        stackBucket.appendChild(cell);\n      });\n      topMarker.classList.add('visible');\n    }\n \n    \u002F\u002F Progress\n    progressEl.textContent = `${Math.min(step, tokens.length)} \u002F ${tokens.length}`;\n \n    \u002F\u002F Controls\n    prevBtn.disabled = step === 0;\n    nextBtn.disabled = finished || error !== null;\n \n    \u002F\u002F Result box\n    if (error) {\n      resultBox.classList.add('visible', 'error');\n      resultLabel.textContent = 'Error';\n      resultValue.textContent = error.message;\n    } else if (finished) {\n      resultBox.classList.add('visible');\n      resultBox.classList.remove('error');\n      resultLabel.textContent = 'Final result';\n      resultValue.textContent = fmt(stack[0]);\n    } else {\n      resultBox.classList.remove('visible', 'error');\n    }\n  }\n \n  function nextStep() {\n    if (step >= tokens.length) {\n      \u002F\u002F End of expression - check final state\n      if (stack.length === 1) {\n        finished = true;\n        narration.className = 'narration success';\n        narration.innerHTML = `\n          \u003Cdiv class=\"narration-action\">✓ Done\u003C\u002Fdiv>\n          \u003Cdiv class=\"narration-text\">All tokens processed. The stack holds a single value, \u003Cspan class=\"num\">${fmt(stack[0])}\u003C\u002Fspan>, which is the answer.\u003C\u002Fdiv>\n        `;\n      } else if (stack.length > 1) {\n        error = {\n          tokenIndex: -1,\n          message: 'Too many operands'\n        };\n        narration.className = 'narration error';\n        narration.innerHTML = `\n          \u003Cdiv class=\"narration-action\">⚠ Too many operands\u003C\u002Fdiv>\n          \u003Cdiv class=\"narration-text\">The expression finished but the stack still contains \u003Cspan class=\"num\">${stack.length}\u003C\u002Fspan> values. A valid postfix expression should reduce to exactly one. Some operand was never consumed by an operator.\u003C\u002Fdiv>\n        `;\n      } else {\n        error = {\n          tokenIndex: -1,\n          message: 'Empty stack at end'\n        };\n        narration.className = 'narration error';\n        narration.innerHTML = `\u003Cdiv class=\"narration-action\">⚠ No result\u003C\u002Fdiv>\u003Cdiv class=\"narration-text\">The stack is empty after processing all tokens.\u003C\u002Fdiv>`;\n      }\n      render();\n      return;\n    }\n \n    const token = tokens[step];\n    if (isNumeric(token)) {\n      const v = parseFloat(token);\n      stack.push(v);\n      step++;\n      narration.className = 'narration';\n      narration.innerHTML = `\n        \u003Cdiv class=\"narration-action\">PUSH \u003Cspan style=\"color:var(--accent)\">${token}\u003C\u002Fspan>\u003C\u002Fdiv>\n        \u003Cdiv class=\"narration-text\">Token \u003Cspan class=\"num\">${token}\u003C\u002Fspan> is an operand. Push it onto the stack.\u003C\u002Fdiv>\n      `;\n    } else if (isOperator(token)) {\n      if (stack.length \u003C 2) {\n        \u002F\u002F Error: too few operands\n        error = {\n          tokenIndex: step,\n          message: 'Too few operands'\n        };\n        narration.className = 'narration error';\n        narration.innerHTML = `\n          \u003Cdiv class=\"narration-action\">⚠ Too few operands\u003C\u002Fdiv>\n          \u003Cdiv class=\"narration-text\">Operator \u003Cspan class=\"op\">${token}\u003C\u002Fspan> needs two operands, but the stack has only \u003Cspan class=\"num\">${stack.length}\u003C\u002Fspan>. The expression is malformed — likely too many operators.\u003C\u002Fdiv>\n        `;\n        render();\n        return;\n      }\n      const right = stack.pop();\n      const left = stack.pop();\n      const result = compute(left, right, token);\n      stack.push(result);\n      step++;\n      narration.className = 'narration';\n      narration.innerHTML = `\n        \u003Cdiv class=\"narration-action\">APPLY \u003Cspan style=\"color:var(--accent)\">${token}\u003C\u002Fspan>\u003C\u002Fdiv>\n        \u003Cdiv class=\"narration-text\">Pop \u003Cspan class=\"num\">${fmt(right)}\u003C\u002Fspan> (right operand), then pop \u003Cspan class=\"num\">${fmt(left)}\u003C\u002Fspan> (left). Compute \u003Cspan class=\"num\">${fmt(left)}\u003C\u002Fspan> \u003Cspan class=\"op\">${token}\u003C\u002Fspan> \u003Cspan class=\"num\">${fmt(right)}\u003C\u002Fspan> = \u003Cspan class=\"num\">${fmt(result)}\u003C\u002Fspan>. Push the result back.\u003C\u002Fdiv>\n      `;\n    } else {\n      error = {\n        tokenIndex: step,\n        message: 'Unknown token'\n      };\n      narration.className = 'narration error';\n      narration.innerHTML = `\u003Cdiv class=\"narration-action\">⚠ Unknown token\u003C\u002Fdiv>\u003Cdiv class=\"narration-text\">\"${token}\" is neither a number nor a supported operator (+ - * \u002F ^).\u003C\u002Fdiv>`;\n    }\n    render();\n  }\n \n  function prevStep() {\n    if (step === 0) return;\n    \u002F\u002F Replay from scratch up to step-1\n    const targetStep = step - 1;\n    step = 0;\n    stack = [];\n    error = null;\n    finished = false;\n    for (let i = 0; i \u003C targetStep; i++) {\n      const token = tokens[i];\n      if (isNumeric(token)) {\n        stack.push(parseFloat(token));\n        step++;\n      } else if (isOperator(token)) {\n        if (stack.length \u003C 2) {\n          error = { tokenIndex: i, message: 'Too few operands' };\n          break;\n        }\n        const right = stack.pop();\n        const left = stack.pop();\n        stack.push(compute(left, right, token));\n        step++;\n      }\n    }\n    narration.className = 'narration';\n    if (step === 0) {\n      narration.innerHTML = `\u003Cdiv class=\"narration-action\">Ready\u003C\u002Fdiv>\u003Cdiv class=\"narration-text\">Press \u003Cstrong>Next\u003C\u002Fstrong> to begin processing token by token.\u003C\u002Fdiv>`;\n    } else {\n      narration.innerHTML = `\u003Cdiv class=\"narration-action\">Rewound to step ${step}\u003C\u002Fdiv>\u003Cdiv class=\"narration-text\">Press \u003Cstrong>Next\u003C\u002Fstrong> to continue forward.\u003C\u002Fdiv>`;\n    }\n    render();\n  }\n \n  function reset() {\n    load(exprInput.value);\n    narration.className = 'narration';\n    narration.innerHTML = `\u003Cdiv class=\"narration-action\">Ready\u003C\u002Fdiv>\u003Cdiv class=\"narration-text\">Press \u003Cstrong>Next\u003C\u002Fstrong> to begin processing token by token. Operands push onto the stack; operators pop two values, compute, and push the result.\u003C\u002Fdiv>`;\n  }\n \n  loadBtn.addEventListener('click', () => {\n    load(exprInput.value);\n    narration.className = 'narration';\n    narration.innerHTML = `\u003Cdiv class=\"narration-action\">Loaded\u003C\u002Fdiv>\u003Cdiv class=\"narration-text\">Expression ready. Press \u003Cstrong>Next\u003C\u002Fstrong> to step through.\u003C\u002Fdiv>`;\n  });\n  exprInput.addEventListener('keydown', e => {\n    if (e.key === 'Enter') loadBtn.click();\n  });\n  nextBtn.addEventListener('click', nextStep);\n  prevBtn.addEventListener('click', prevStep);\n  resetBtn.addEventListener('click', reset);\n \n  document.querySelectorAll('.preset-chip').forEach(chip => {\n    chip.addEventListener('click', () => {\n      load(chip.dataset.expr);\n      narration.className = 'narration';\n      narration.innerHTML = `\u003Cdiv class=\"narration-action\">Loaded\u003C\u002Fdiv>\u003Cdiv class=\"narration-text\">Expression ready. Press \u003Cstrong>Next\u003C\u002Fstrong> to step through.\u003C\u002Fdiv>`;\n    });\n  });\n \n  \u002F\u002F Initial load\n  load('4 3 5 * +');\n\u003C\u002Fscript>\n\u003C\u002Fbody>\n\u003C\u002Fhtml>",[7166,14604,14606],{"id":14605},"when-things-go-wrong-error-detection","When Things Go Wrong: Error Detection",[12,14608,14609,14610],{},"A beautiful side-effect of this algorithm: ",[106,14611,14612],{},"the state of the stack tells you about errors.",[12,14614,14615],{},[106,14616,14617],{},"Error case 1: Too many operators",[12,14619,14620,14621],{},"Expression: ",[6979,14622,14623],{},"3 8 + * 9",[6972,14625,14628],{"className":14626,"code":14627,"language":6977},[6975],"Push 3, push 8   →   stack: [3, 8]\nSee \"+\", pop both, compute 3+8 = 11, push   →   stack: [11]\nSee \"*\"  →   need to pop two operands, but stack has only ONE element!\n",[6979,14629,14627],{"__ignoreMap":180},[12,14631,14632,14633,14635],{},"The moment we try to apply ",[6979,14634,8128],{}," with only one thing on the stack, we know the input is malformed.",[12,14637,14638],{},[106,14639,14640],{},"Error case 2: Too many operands",[12,14642,14620,14643],{},[6979,14644,14645],{},"9 8 + 7",[6972,14647,14650],{"className":14648,"code":14649,"language":6977},[6975],"Push 9, push 8   →   stack: [9, 8]\nSee \"+\", pop, compute 17, push   →   stack: [17]\nPush 7   →   stack: [17, 7]\nEnd of input — but the stack has TWO elements!\n",[6979,14651,14649],{"__ignoreMap":180},[12,14653,14654],{},"A valid postfix expression always ends with exactly one number on the stack. If there's more, something was left dangling.",[7166,14656,14658],{"id":14657},"java-implementation","Java Implementation",[12,14660,14661,14662,14665],{},"Here's a working ",[6979,14663,14664],{},"PostfixEvaluation"," class. I've kept it close to the lecture version but added comments:",[6972,14667,14669],{"className":7376,"code":14668,"language":7378,"meta":180,"style":180},"public class PostfixEvaluation {\n    public static void main(String[] args) {\n        System.out.println(\"Testing PostfixEvaluation:\\n\");\n        System.out.println(\"2 3 + 4 * 5 - : \"\n                + evaluatePostfix(\"2 3 + 4 * 5 -\") + \"\\n\");\n        System.out.println(\"2 3 * 4 2 - \u002F 5 6 * + : \"\n                + evaluatePostfix(\"2 3 * 4 2 - \u002F 5 6 * +\") + \"\\n\");\n        System.out.println(\"2 4 - 3 ^ 5 + : \"\n                + evaluatePostfix(\"2 4 - 3 ^ 5 +\") + \"\\n\");\n        System.out.println(\"Done.\");\n    }\n \n    \u002F** Evaluates a postfix expression and returns the result. *\u002F\n    public static double evaluatePostfix(String postfix) {\n        GenericStack\u003CDouble> valueStack = new GenericStack\u003C>();\n        String[] tokens = postfix.split(\" \");\n \n        for (String token : tokens) {\n            if (isNumeric(token)) {\n                valueStack.push(new Double(token));\n            } else if (token.equals(\"+\") || token.equals(\"-\")\n                    || token.equals(\"*\") || token.equals(\"\u002F\")\n                    || token.equals(\"^\")) {\n                Double operandTwo = valueStack.pop();   \u002F\u002F right operand\n                Double operandOne = valueStack.pop();   \u002F\u002F left operand\n                Double result = compute(operandOne, operandTwo, token);\n                valueStack.push(result);\n            }\n        }\n \n        return valueStack.peek();\n    }\n \n    public static boolean isNumeric(String str) {\n        try {\n            double d = Double.parseDouble(str);\n        } catch (NumberFormatException nfe) {\n            return false;\n        }\n        return true;\n    }\n \n    private static Double compute(Double operandOne, Double operandTwo,\n                                  String operator) {\n        double result;\n        switch (operator) {\n            case \"+\": result = operandOne + operandTwo; break;\n            case \"-\": result = operandOne - operandTwo; break;\n            case \"*\": result = operandOne * operandTwo; break;\n            case \"\u002F\": result = operandOne \u002F operandTwo; break;\n            case \"^\": result = Math.pow(operandOne, operandTwo); break;\n            default:  result = 0; break;  \u002F\u002F unexpected character\n        }\n        return result;\n    }\n}\n",[6979,14670,14671,14682,14702,14719,14730,14756,14767,14790,14801,14824,14837,14841,14845,14850,14868,14884,14907,14911,14923,14936,14953,14990,15019,15035,15052,15068,15081,15090,15095,15099,15103,15113,15117,15122,15141,15149,15169,15186,15197,15202,15212,15217,15222,15250,15261,15270,15279,15308,15332,15356,15380,15407,15431,15436,15443,15448],{"__ignoreMap":180},[75,14672,14673,14675,14677,14680],{"class":7383,"line":7384},[75,14674,7388],{"class":7387},[75,14676,7391],{"class":7387},[75,14678,14679],{"class":7394}," PostfixEvaluation",[75,14681,8147],{"class":7398},[75,14683,14684,14686,14688,14690,14692,14694,14696,14698,14700],{"class":7383,"line":181},[75,14685,7439],{"class":7387},[75,14687,8155],{"class":7387},[75,14689,7442],{"class":7387},[75,14691,8160],{"class":7394},[75,14693,7698],{"class":7398},[75,14695,7674],{"class":7387},[75,14697,8167],{"class":7398},[75,14699,8170],{"class":7451},[75,14701,7455],{"class":7398},[75,14703,14704,14706,14708,14710,14713,14715,14717],{"class":7383,"line":7430},[75,14705,12229],{"class":7398},[75,14707,7767],{"class":7394},[75,14709,7698],{"class":7398},[75,14711,14712],{"class":7589},"\"Testing PostfixEvaluation:",[75,14714,8430],{"class":8127},[75,14716,8427],{"class":7589},[75,14718,7738],{"class":7398},[75,14720,14721,14723,14725,14727],{"class":7383,"line":7436},[75,14722,12229],{"class":7398},[75,14724,7767],{"class":7394},[75,14726,7698],{"class":7398},[75,14728,14729],{"class":7589},"\"2 3 + 4 * 5 - : \"\n",[75,14731,14732,14735,14738,14740,14743,14745,14747,14750,14752,14754],{"class":7383,"line":7458},[75,14733,14734],{"class":7387},"                +",[75,14736,14737],{"class":7394}," evaluatePostfix",[75,14739,7698],{"class":7398},[75,14741,14742],{"class":7589},"\"2 3 + 4 * 5 -\"",[75,14744,10629],{"class":7398},[75,14746,8302],{"class":7387},[75,14748,14749],{"class":7589}," \"",[75,14751,8430],{"class":8127},[75,14753,8427],{"class":7589},[75,14755,7738],{"class":7398},[75,14757,14758,14760,14762,14764],{"class":7383,"line":7474},[75,14759,12229],{"class":7398},[75,14761,7767],{"class":7394},[75,14763,7698],{"class":7398},[75,14765,14766],{"class":7589},"\"2 3 * 4 2 - \u002F 5 6 * + : \"\n",[75,14768,14769,14771,14773,14775,14778,14780,14782,14784,14786,14788],{"class":7383,"line":7480},[75,14770,14734],{"class":7387},[75,14772,14737],{"class":7394},[75,14774,7698],{"class":7398},[75,14776,14777],{"class":7589},"\"2 3 * 4 2 - \u002F 5 6 * +\"",[75,14779,10629],{"class":7398},[75,14781,8302],{"class":7387},[75,14783,14749],{"class":7589},[75,14785,8430],{"class":8127},[75,14787,8427],{"class":7589},[75,14789,7738],{"class":7398},[75,14791,14792,14794,14796,14798],{"class":7383,"line":7485},[75,14793,12229],{"class":7398},[75,14795,7767],{"class":7394},[75,14797,7698],{"class":7398},[75,14799,14800],{"class":7589},"\"2 4 - 3 ^ 5 + : \"\n",[75,14802,14803,14805,14807,14809,14812,14814,14816,14818,14820,14822],{"class":7383,"line":7498},[75,14804,14734],{"class":7387},[75,14806,14737],{"class":7394},[75,14808,7698],{"class":7398},[75,14810,14811],{"class":7589},"\"2 4 - 3 ^ 5 +\"",[75,14813,10629],{"class":7398},[75,14815,8302],{"class":7387},[75,14817,14749],{"class":7589},[75,14819,8430],{"class":8127},[75,14821,8427],{"class":7589},[75,14823,7738],{"class":7398},[75,14825,14826,14828,14830,14832,14835],{"class":7383,"line":7516},[75,14827,12229],{"class":7398},[75,14829,7767],{"class":7394},[75,14831,7698],{"class":7398},[75,14833,14834],{"class":7589},"\"Done.\"",[75,14836,7738],{"class":7398},[75,14838,14839],{"class":7383,"line":7521},[75,14840,7477],{"class":7398},[75,14842,14843],{"class":7383,"line":7526},[75,14844,7433],{"class":7398},[75,14846,14847],{"class":7383,"line":7539},[75,14848,14849],{"class":7470},"    \u002F** Evaluates a postfix expression and returns the result. *\u002F\n",[75,14851,14852,14854,14856,14859,14861,14864,14866],{"class":7383,"line":7552},[75,14853,7439],{"class":7387},[75,14855,8155],{"class":7387},[75,14857,14858],{"class":7387}," double",[75,14860,14737],{"class":7394},[75,14862,14863],{"class":7398},"(String ",[75,14865,14458],{"class":7451},[75,14867,7455],{"class":7398},[75,14869,14870,14872,14875,14878,14880,14882],{"class":7383,"line":7557},[75,14871,14213],{"class":7398},[75,14873,14874],{"class":7387},"Double",[75,14876,14877],{"class":7398},"> valueStack ",[75,14879,7421],{"class":7387},[75,14881,7424],{"class":7387},[75,14883,14225],{"class":7398},[75,14885,14886,14889,14892,14894,14897,14900,14902,14905],{"class":7383,"line":7562},[75,14887,14888],{"class":7387},"        String",[75,14890,14891],{"class":7398},"[] tokens ",[75,14893,7421],{"class":7387},[75,14895,14896],{"class":7398}," postfix.",[75,14898,14899],{"class":7394},"split",[75,14901,7698],{"class":7398},[75,14903,14904],{"class":7589},"\" \"",[75,14906,7738],{"class":7398},[75,14908,14909],{"class":7383,"line":7571},[75,14910,7433],{"class":7398},[75,14912,14913,14915,14918,14920],{"class":7383,"line":7584},[75,14914,10473],{"class":7387},[75,14916,14917],{"class":7398}," (String token ",[75,14919,7024],{"class":7387},[75,14921,14922],{"class":7398}," tokens) {\n",[75,14924,14925,14928,14930,14933],{"class":7383,"line":7602},[75,14926,14927],{"class":7387},"            if",[75,14929,10476],{"class":7398},[75,14931,14932],{"class":7394},"isNumeric",[75,14934,14935],{"class":7398},"(token)) {\n",[75,14937,14938,14941,14943,14945,14947,14950],{"class":7383,"line":7607},[75,14939,14940],{"class":7398},"                valueStack.",[75,14942,14237],{"class":7394},[75,14944,7698],{"class":7398},[75,14946,8823],{"class":7387},[75,14948,14949],{"class":7394}," Double",[75,14951,14952],{"class":7398},"(token));\n",[75,14954,14955,14958,14960,14962,14965,14968,14970,14973,14975,14978,14981,14983,14985,14988],{"class":7383,"line":8360},[75,14956,14957],{"class":7398},"            } ",[75,14959,10327],{"class":7387},[75,14961,10432],{"class":7387},[75,14963,14964],{"class":7398}," (token.",[75,14966,14967],{"class":7394},"equals",[75,14969,7698],{"class":7398},[75,14971,14972],{"class":7589},"\"+\"",[75,14974,10629],{"class":7398},[75,14976,14977],{"class":7387},"||",[75,14979,14980],{"class":7398}," token.",[75,14982,14967],{"class":7394},[75,14984,7698],{"class":7398},[75,14986,14987],{"class":7589},"\"-\"",[75,14989,11077],{"class":7398},[75,14991,14992,14995,14997,14999,15001,15004,15006,15008,15010,15012,15014,15017],{"class":7383,"line":8374},[75,14993,14994],{"class":7387},"                    ||",[75,14996,14980],{"class":7398},[75,14998,14967],{"class":7394},[75,15000,7698],{"class":7398},[75,15002,15003],{"class":7589},"\"*\"",[75,15005,10629],{"class":7398},[75,15007,14977],{"class":7387},[75,15009,14980],{"class":7398},[75,15011,14967],{"class":7394},[75,15013,7698],{"class":7398},[75,15015,15016],{"class":7589},"\"\u002F\"",[75,15018,11077],{"class":7398},[75,15020,15021,15023,15025,15027,15029,15032],{"class":7383,"line":8387},[75,15022,14994],{"class":7387},[75,15024,14980],{"class":7398},[75,15026,14967],{"class":7394},[75,15028,7698],{"class":7398},[75,15030,15031],{"class":7589},"\"^\"",[75,15033,15034],{"class":7398},")) {\n",[75,15036,15037,15040,15042,15045,15047,15049],{"class":7383,"line":8400},[75,15038,15039],{"class":7398},"                Double operandTwo ",[75,15041,7421],{"class":7387},[75,15043,15044],{"class":7398}," valueStack.",[75,15046,13880],{"class":7394},[75,15048,9268],{"class":7398},[75,15050,15051],{"class":7470},"\u002F\u002F right operand\n",[75,15053,15054,15057,15059,15061,15063,15065],{"class":7383,"line":8413},[75,15055,15056],{"class":7398},"                Double operandOne ",[75,15058,7421],{"class":7387},[75,15060,15044],{"class":7398},[75,15062,13880],{"class":7394},[75,15064,9268],{"class":7398},[75,15066,15067],{"class":7470},"\u002F\u002F left operand\n",[75,15069,15070,15073,15075,15078],{"class":7383,"line":8418},[75,15071,15072],{"class":7398},"                Double result ",[75,15074,7421],{"class":7387},[75,15076,15077],{"class":7394}," compute",[75,15079,15080],{"class":7398},"(operandOne, operandTwo, token);\n",[75,15082,15083,15085,15087],{"class":7383,"line":8438},[75,15084,14940],{"class":7398},[75,15086,14237],{"class":7394},[75,15088,15089],{"class":7398},"(result);\n",[75,15091,15092],{"class":7383,"line":8456},[75,15093,15094],{"class":7398},"            }\n",[75,15096,15097],{"class":7383,"line":8476},[75,15098,10516],{"class":7398},[75,15100,15101],{"class":7383,"line":8481},[75,15102,7433],{"class":7398},[75,15104,15105,15107,15109,15111],{"class":7383,"line":8487},[75,15106,7501],{"class":7387},[75,15108,15044],{"class":7398},[75,15110,8052],{"class":7394},[75,15112,7549],{"class":7398},[75,15114,15115],{"class":7383,"line":8493},[75,15116,7477],{"class":7398},[75,15118,15120],{"class":7383,"line":15119},33,[75,15121,7433],{"class":7398},[75,15123,15125,15127,15129,15131,15134,15136,15139],{"class":7383,"line":15124},34,[75,15126,7439],{"class":7387},[75,15128,8155],{"class":7387},[75,15130,9278],{"class":7387},[75,15132,15133],{"class":7394}," isNumeric",[75,15135,14863],{"class":7398},[75,15137,15138],{"class":7451},"str",[75,15140,7455],{"class":7398},[75,15142,15144,15147],{"class":7383,"line":15143},35,[75,15145,15146],{"class":7387},"        try",[75,15148,8147],{"class":7398},[75,15150,15152,15155,15158,15160,15163,15166],{"class":7383,"line":15151},36,[75,15153,15154],{"class":7387},"            double",[75,15156,15157],{"class":7398}," d ",[75,15159,7421],{"class":7387},[75,15161,15162],{"class":7398}," Double.",[75,15164,15165],{"class":7394},"parseDouble",[75,15167,15168],{"class":7398},"(str);\n",[75,15170,15172,15175,15178,15181,15184],{"class":7383,"line":15171},37,[75,15173,15174],{"class":7398},"        } ",[75,15176,15177],{"class":7387},"catch",[75,15179,15180],{"class":7398}," (NumberFormatException ",[75,15182,15183],{"class":7451},"nfe",[75,15185,7455],{"class":7398},[75,15187,15189,15192,15195],{"class":7383,"line":15188},38,[75,15190,15191],{"class":7387},"            return",[75,15193,15194],{"class":8127}," false",[75,15196,8131],{"class":7398},[75,15198,15200],{"class":7383,"line":15199},39,[75,15201,10516],{"class":7398},[75,15203,15205,15207,15210],{"class":7383,"line":15204},40,[75,15206,7501],{"class":7387},[75,15208,15209],{"class":8127}," true",[75,15211,8131],{"class":7398},[75,15213,15215],{"class":7383,"line":15214},41,[75,15216,7477],{"class":7398},[75,15218,15220],{"class":7383,"line":15219},42,[75,15221,7433],{"class":7398},[75,15223,15225,15227,15229,15232,15235,15238,15241,15244,15247],{"class":7383,"line":15224},43,[75,15226,7410],{"class":7387},[75,15228,8155],{"class":7387},[75,15230,15231],{"class":7398}," Double ",[75,15233,15234],{"class":7394},"compute",[75,15236,15237],{"class":7398},"(Double ",[75,15239,15240],{"class":7451},"operandOne",[75,15242,15243],{"class":7398},", Double ",[75,15245,15246],{"class":7451},"operandTwo",[75,15248,15249],{"class":7398},",\n",[75,15251,15253,15256,15259],{"class":7383,"line":15252},44,[75,15254,15255],{"class":7398},"                                  String ",[75,15257,15258],{"class":7451},"operator",[75,15260,7455],{"class":7398},[75,15262,15264,15267],{"class":7383,"line":15263},45,[75,15265,15266],{"class":7387},"        double",[75,15268,15269],{"class":7398}," result;\n",[75,15271,15273,15276],{"class":7383,"line":15272},46,[75,15274,15275],{"class":7387},"        switch",[75,15277,15278],{"class":7398}," (operator) {\n",[75,15280,15282,15285,15288,15290,15293,15295,15298,15300,15303,15306],{"class":7383,"line":15281},47,[75,15283,15284],{"class":7387},"            case",[75,15286,15287],{"class":7589}," \"+\"",[75,15289,7024],{"class":7387},[75,15291,15292],{"class":7398}," result ",[75,15294,7421],{"class":7387},[75,15296,15297],{"class":7398}," operandOne ",[75,15299,8302],{"class":7387},[75,15301,15302],{"class":7398}," operandTwo; ",[75,15304,15305],{"class":7387},"break",[75,15307,8131],{"class":7398},[75,15309,15311,15313,15316,15318,15320,15322,15324,15326,15328,15330],{"class":7383,"line":15310},48,[75,15312,15284],{"class":7387},[75,15314,15315],{"class":7589}," \"-\"",[75,15317,7024],{"class":7387},[75,15319,15292],{"class":7398},[75,15321,7421],{"class":7387},[75,15323,15297],{"class":7398},[75,15325,10857],{"class":7387},[75,15327,15302],{"class":7398},[75,15329,15305],{"class":7387},[75,15331,8131],{"class":7398},[75,15333,15335,15337,15340,15342,15344,15346,15348,15350,15352,15354],{"class":7383,"line":15334},49,[75,15336,15284],{"class":7387},[75,15338,15339],{"class":7589}," \"*\"",[75,15341,7024],{"class":7387},[75,15343,15292],{"class":7398},[75,15345,7421],{"class":7387},[75,15347,15297],{"class":7398},[75,15349,8128],{"class":7387},[75,15351,15302],{"class":7398},[75,15353,15305],{"class":7387},[75,15355,8131],{"class":7398},[75,15357,15359,15361,15364,15366,15368,15370,15372,15374,15376,15378],{"class":7383,"line":15358},50,[75,15360,15284],{"class":7387},[75,15362,15363],{"class":7589}," \"\u002F\"",[75,15365,7024],{"class":7387},[75,15367,15292],{"class":7398},[75,15369,7421],{"class":7387},[75,15371,15297],{"class":7398},[75,15373,13084],{"class":7387},[75,15375,15302],{"class":7398},[75,15377,15305],{"class":7387},[75,15379,8131],{"class":7398},[75,15381,15383,15385,15388,15390,15392,15394,15397,15400,15403,15405],{"class":7383,"line":15382},51,[75,15384,15284],{"class":7387},[75,15386,15387],{"class":7589}," \"^\"",[75,15389,7024],{"class":7387},[75,15391,15292],{"class":7398},[75,15393,7421],{"class":7387},[75,15395,15396],{"class":7398}," Math.",[75,15398,15399],{"class":7394},"pow",[75,15401,15402],{"class":7398},"(operandOne, operandTwo); ",[75,15404,15305],{"class":7387},[75,15406,8131],{"class":7398},[75,15408,15410,15413,15416,15418,15420,15423,15425,15428],{"class":7383,"line":15409},52,[75,15411,15412],{"class":7387},"            default:",[75,15414,15415],{"class":7398},"  result ",[75,15417,7421],{"class":7387},[75,15419,8282],{"class":8127},[75,15421,15422],{"class":7398},"; ",[75,15424,15305],{"class":7387},[75,15426,15427],{"class":7398},";  ",[75,15429,15430],{"class":7470},"\u002F\u002F unexpected character\n",[75,15432,15434],{"class":7383,"line":15433},53,[75,15435,10516],{"class":7398},[75,15437,15439,15441],{"class":7383,"line":15438},54,[75,15440,7501],{"class":7387},[75,15442,15269],{"class":7398},[75,15444,15446],{"class":7383,"line":15445},55,[75,15447,7477],{"class":7398},[75,15449,15451],{"class":7383,"line":15450},56,[75,15452,7610],{"class":7398},[12,15454,9006],{},[6972,15456,15459],{"className":15457,"code":15458,"language":6977},[6975],"Testing PostfixEvaluation:\n \n2 3 + 4 * 5 - : 15.0\n \n2 3 * 4 2 - \u002F 5 6 * + : 33.0\n \n2 4 - 3 ^ 5 + : -3.0\n \nDone.\n",[6979,15460,15458],{"__ignoreMap":180},[7166,15462,15464],{"id":15463},"the-critical-pop-order","The Critical Pop Order",[12,15466,15467],{},"One thing I want to highlight, because it's a common bug:",[6972,15469,15471],{"className":7376,"code":15470,"language":7378,"meta":180,"style":180},"Double operandTwo = valueStack.pop();   \u002F\u002F FIRST pop = RIGHT operand\nDouble operandOne = valueStack.pop();   \u002F\u002F SECOND pop = LEFT operand\n",[6979,15472,15473,15489],{"__ignoreMap":180},[75,15474,15475,15478,15480,15482,15484,15486],{"class":7383,"line":7384},[75,15476,15477],{"class":7398},"Double operandTwo ",[75,15479,7421],{"class":7387},[75,15481,15044],{"class":7398},[75,15483,13880],{"class":7394},[75,15485,9268],{"class":7398},[75,15487,15488],{"class":7470},"\u002F\u002F FIRST pop = RIGHT operand\n",[75,15490,15491,15494,15496,15498,15500,15502],{"class":7383,"line":181},[75,15492,15493],{"class":7398},"Double operandOne ",[75,15495,7421],{"class":7387},[75,15497,15044],{"class":7398},[75,15499,13880],{"class":7394},[75,15501,9268],{"class":7398},[75,15503,15504],{"class":7470},"\u002F\u002F SECOND pop = LEFT operand\n",[12,15506,15507,15508,6959],{},"Why does order matter? Because subtraction and division aren't commutative. ",[6979,15509,15510],{},"5 - 3 ≠ 3 - 5",[12,15512,15513,15514,15517,15518,15520,15521,15524,15525,15528,15529,15532],{},"When evaluating ",[6979,15515,15516],{},"5 3 -",", you push 5, then push 3. The first thing you pop is ",[106,15519,8830],{},", which is the ",[7289,15522,15523],{},"right"," side of the subtraction (",[6979,15526,15527],{},"5 - 3","). The second pop is ",[106,15530,15531],{},"5",", the left side. Get this wrong and your calculator returns negative answers for everything.",[6964,15534,15536],{"id":15535},"wrap-up","Wrap-Up",[12,15538,15539],{},"Let's recap what we covered:",[23,15541,15542,15549,15555,15565,15575,15586,15589,15592,15595,15598,15601],{},[26,15543,9407,15544,15546,15547,6959],{},[106,15545,7008],{}," is a list restricted to insertions and deletions at one end — the ",[106,15548,13468],{},[26,15550,15551,15552,15554],{},"It follows the ",[106,15553,13489],{}," principle: last in, first out.",[26,15556,15557,15558,7298,15560,7153,15562,15564],{},"The core operations are ",[6979,15559,14237],{},[6979,15561,13880],{},[6979,15563,8052],{},", all of which are O(1).",[26,15566,15567,15568,15570,15571,15574],{},"We implement stacks efficiently using an ",[106,15569,7193],{},", preferably via ",[106,15572,15573],{},"composition"," rather than inheritance, so we don't accidentally expose list operations that violate the stack discipline.",[26,15576,15577,15578,15581,15582,15585],{},"A classic application is evaluating ",[106,15579,15580],{},"postfix expressions"," (Reverse Polish Notation). The algorithm is short and elegant, and the stack's state at any point also helps detect malformed input.\nStacks show up ",[7289,15583,15584],{},"everywhere"," once you start looking:",[26,15587,15588],{},"Function call stacks in every programming language",[26,15590,15591],{},"Undo\u002Fredo systems",[26,15593,15594],{},"Browser history",[26,15596,15597],{},"Expression parsing in compilers",[26,15599,15600],{},"Backtracking algorithms (think: solving a maze)",[26,15602,15603,15604,15607],{},"Balanced parenthesis checking\nNext time you press ",[6979,15605,15606],{},"Ctrl+Z",", give a small nod to the stack working quietly behind the scenes.",[9332,15609,15610],{},"html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}",{"title":180,"searchDepth":181,"depth":181,"links":15612},[15613,15616,15621,15628,15637],{"id":13318,"depth":181,"text":13319,"children":15614},[15615],{"id":13365,"depth":7430,"text":13366},{"id":13451,"depth":181,"text":13452,"children":15617},[15618,15619,15620],{"id":13474,"depth":7430,"text":13475},{"id":13522,"depth":7430,"text":13523},{"id":13609,"depth":7430,"text":13610},{"id":13660,"depth":181,"text":13661,"children":15622},[15623,15624,15625,15626,15627],{"id":13667,"depth":7430,"text":13668},{"id":13704,"depth":7430,"text":13705},{"id":13726,"depth":7430,"text":13727},{"id":14052,"depth":7430,"text":14053},{"id":14171,"depth":7430,"text":14172},{"id":14421,"depth":181,"text":14422,"children":15629},[15630,15631,15632,15633,15634,15635,15636],{"id":14428,"depth":7430,"text":14429},{"id":14499,"depth":7430,"text":14500},{"id":14580,"depth":7430,"text":14581},{"id":7168,"depth":7430,"text":14596},{"id":14605,"depth":7430,"text":14606},{"id":14657,"depth":7430,"text":14658},{"id":15463,"depth":7430,"text":15464},{"id":15535,"depth":181,"text":15536},{},"\u002Fblog\u002FStack",{"title":13313,"description":180},{"loc":15639},"blog\u002FStack","NbqFtFEqLNon0GwWdOlYQa7IKqYmX80wSt965Y6nMYg",{"id":15645,"title":15646,"author":6946,"body":15647,"date":17971,"description":17972,"draft":185,"edited_at":13169,"extension":186,"featured_image":194,"meta":17973,"navigation":189,"path":17974,"pinned":185,"seo":17975,"sitemap":17976,"stem":17977,"tags":194,"__hash__":17978},"blog\u002Fblog\u002FAbstract-Data-Types-&-The-Bag.md","Abstract Data Types & The Bag",{"type":9,"value":15648,"toc":17944},[15649,15666,15669,15683,15687,15691,15694,15732,15736,15739,15764,15775,15779,15786,15806,15818,15821,15844,15848,15851,15929,15932,15967,15971,15974,16013,16017,16020,16062,16066,16069,16086,16090,16096,16100,16103,16130,16137,16140,16144,16147,16199,16203,16214,16220,16229,16233,16240,16246,16258,16261,16378,16382,16391,16437,16457,16461,16468,16743,16750,16754,16760,16771,16777,16781,16787,17124,17127,17133,17140,17144,17297,17300,17550,17552,17558,17573,17577,17583,17637,17640,17647,17653,17775,17782,17786,17789,17889,17903,17907,17910,17941],[12,15650,15651,15652,15655,15656,15658,15659,15661,15662,15665],{},"If you're new to data structures, the very first hurdle isn't a fancy algorithm — it's a way of ",[7289,15653,15654],{},"thinking",". Programmers separate ",[106,15657,9441],{}," a thing does from ",[106,15660,9518],{}," it's built. That's the whole idea behind an ",[106,15663,15664],{},"Abstract Data Type (ADT)",", and once it clicks, the rest of the course starts to feel much more organized.",[12,15667,15668],{},"In this post, we'll cover two things:",[3603,15670,15671,15677],{},[26,15672,15673,15676],{},[106,15674,15675],{},"What an ADT actually is"," (and how it differs from a data structure).",[26,15678,15679,15682],{},[106,15680,15681],{},"The Bag"," — one of the simplest possible ADTs, and a great way to see all of this in action.\nLet's go.",[6964,15684,15686],{"id":15685},"part-1-abstract-data-types-adts","Part 1: Abstract Data Types (ADTs)",[7166,15688,15690],{"id":15689},"data-is-everywhere-and-we-already-organize-it","Data is everywhere — and we already organize it",[12,15692,15693],{},"Before we touch any code, look around you. Real-world data is already structured in ways you use every day:",[23,15695,15696,15702,15708,15714,15720],{},[26,15697,9407,15698,15701],{},[106,15699,15700],{},"to-do list"," — items in some order, you add and cross off.",[26,15703,9407,15704,15707],{},[106,15705,15706],{},"dictionary"," — words you can look up quickly.",[26,15709,9407,15710,15713],{},[106,15711,15712],{},"stack of books"," — you take from the top.",[26,15715,9407,15716,15719],{},[106,15717,15718],{},"queue for tickets"," — first in, first out.",[26,15721,9407,15722,15725,15726,15729,15730,6959],{},[106,15723,15724],{},"road map"," — connections between places.\nEach of these is a ",[7289,15727,15728],{},"collection of things"," with rules about how you add to it, remove from it, and look stuff up. Programmers do the exact same thing with data — and the formal name for these patterns is an ",[106,15731,15664],{},[7166,15733,15735],{"id":15734},"collection-vs-container","Collection vs. Container",[12,15737,15738],{},"These two words get used a lot, and they're easy to mix up:",[23,15740,15741,15755],{},[26,15742,9407,15743,15745,15746,15749,15750,15752,15753,6959],{},[106,15744,9410],{}," is a general idea — ",[7289,15747,15748],{},"\"a group of objects with some operations.\""," It says ",[106,15751,9441],{}," you can do (add, remove, search), not ",[106,15754,9518],{},[26,15756,9407,15757,15760,15761,15763],{},[106,15758,15759],{},"container"," is the actual ",[106,15762,9664],{}," in a programming language that implements a collection. It's the concrete thing.",[7351,15765,15766],{},[12,15767,15768,15769,15772,15773,6959],{},"Think of \"collection\" as the ",[7289,15770,15771],{},"concept"," and \"container\" as the ",[7289,15774,6979],{},[7166,15776,15778],{"id":15777},"so-what-exactly-is-an-adt","So what exactly is an ADT?",[12,15780,15781,15782,15785],{},"An ",[106,15783,15784],{},"ADT is a conceptual model"," that defines two things:",[3603,15787,15788,15794],{},[26,15789,15790,15793],{},[106,15791,15792],{},"The type of data"," it stores.",[26,15795,15796,15799,15800,15802,15803,15805],{},[106,15797,15798],{},"The operations"," that can be performed on that data.\nAnd critically, it says nothing about ",[106,15801,9518],{}," any of it is implemented. No arrays, no linked lists, no Java — just ",[7289,15804,9441],{}," the thing does.",[7351,15807,15808],{},[12,15809,15810,15813,15814,15817],{},[7289,15811,15812],{},"Abstract"," here means ",[106,15815,15816],{},"irrelevant details are ignored"," — like how the data is stored in memory, or what algorithm runs underneath.",[12,15819,15820],{},"A useful way to phrase it:",[23,15822,15823,15832],{},[26,15824,15825,15828,15829,15831],{},[106,15826,15827],{},"What"," operations can be performed → ",[106,15830,7388],{}," (the client can see this).",[26,15833,15834,15837,15838,15840,15841,15843],{},[106,15835,15836],{},"How"," those operations are implemented → ",[106,15839,7656],{}," (hidden from the client).\nThe user of an ADT only interacts with its ",[106,15842,7058],{},", never its internal logic.",[7166,15845,15847],{"id":15846},"adt-vs-data-structure-whats-the-difference","ADT vs. Data Structure — what's the difference?",[12,15849,15850],{},"This is the question that trips up almost every beginner, so let's nail it down:",[7085,15852,15853,15871],{},[7088,15854,15855],{},[7091,15856,15857,15861,15866],{},[7094,15858,15859],{},[16,15860],{},[7094,15862,15863],{},[106,15864,15865],{},"ADT",[7094,15867,15868],{},[106,15869,15870],{},"Data Structure",[7104,15872,15873,15886,15903,15916],{},[7091,15874,15875,15880,15883],{},[7109,15876,15877],{},[106,15878,15879],{},"Nature",[7109,15881,15882],{},"Conceptual \u002F abstract",[7109,15884,15885],{},"Concrete \u002F actual code",[7091,15887,15888,15893,15898],{},[7109,15889,15890],{},[106,15891,15892],{},"Focus",[7109,15894,15895,15897],{},[7289,15896,15827],{}," it does",[7109,15899,15900,15902],{},[7289,15901,15836],{}," it does it",[7091,15904,15905,15910,15913],{},[7109,15906,15907],{},[106,15908,15909],{},"Language?",[7109,15911,15912],{},"Language-independent",[7109,15914,15915],{},"Language-specific",[7091,15917,15918,15923,15926],{},[7109,15919,15920],{},[106,15921,15922],{},"Example",[7109,15924,15925],{},"\"List ADT supports add\u002Fremove\u002Faccess\"",[7109,15927,15928],{},"\"Use an array or linked list to implement it\"",[12,15930,15931],{},"A few classic ADTs you'll meet soon:",[23,15933,15934,15940,15950],{},[26,15935,15936,15939],{},[106,15937,15938],{},"List ADT"," → supports adding, removing, and accessing elements.",[26,15941,15942,15945,15946,7042,15948,6959],{},[106,15943,15944],{},"Stack ADT"," → Last-In-First-Out (LIFO). Think ",[6979,15947,14237],{},[6979,15949,13880],{},[26,15951,15952,15955,15956,7042,15958,15960,15961,7194,15963,15966],{},[106,15953,15954],{},"Queue ADT"," → First-In-First-Out (FIFO). Think ",[6979,15957,7160],{},[6979,15959,7163],{},".\nAll three of these could be implemented using either an ",[106,15962,9154],{},[106,15964,15965],{},"linked list"," — same ADT, different data structures.",[7166,15968,15970],{"id":15969},"an-adt-in-plain-english-a-cinema-reservation-system","An ADT in plain English: a cinema reservation system",[12,15972,15973],{},"Let's design an ADT for booking cinema seats — without writing a single line of code.",[23,15975,15976,15988,16007],{},[26,15977,15978,15981,15982,7067,15985,6959],{},[106,15979,15980],{},"Data?"," Seats. Each seat is either ",[7289,15983,15984],{},"reserved",[7289,15986,15987],{},"available",[26,15989,15990,15993],{},[106,15991,15992],{},"Operations?",[23,15994,15995,15998,16001,16004],{},[26,15996,15997],{},"Check if a seat is available",[26,15999,16000],{},"Reserve a seat",[26,16002,16003],{},"Cancel a reservation",[26,16005,16006],{},"Find a block of available seats together",[26,16008,16009,16012],{},[106,16010,16011],{},"Implementation?"," Don't care. Not now.\nThat's it — that's an ADT specification. Whether the seats end up stored in a 2D array, a database, or scribbled on paper is somebody else's problem.",[7166,16014,16016],{"id":16015},"a-common-thread-example-cards-contacts-and-decks","A common-thread example: cards, contacts, and decks",[12,16018,16019],{},"Consider these unrelated things:",[23,16021,16022,16025,16028,16031,16037,16040,16043,16046],{},[26,16023,16024],{},"A deck of playing cards",[26,16026,16027],{},"A box of index cards with birthdays",[26,16029,16030],{},"The contacts list on your phone. They all share the same underlying behavior:",[26,16032,16033,16034,6959],{},"Each is a ",[106,16035,16036],{},"collection of elements",[26,16038,16039],{},"There's a first element, a second, ..., a last.",[26,16041,16042],{},"Given any element, there's a \"next\" one (unless it's the last).",[26,16044,16045],{},"Given any element, there's a \"previous\" one (unless it's the first).",[26,16047,16048,16049,16051,16052,16054,16055,16058,16059,16061],{},"You can ",[106,16050,8062],{}," an element, ",[106,16053,8065],{}," one, and ",[106,16056,16057],{},"search"," for one by going through systematically.\nThis shared structure is what would later be called the ",[106,16060,15938],{},". Once you spot the pattern, the same ADT covers all three.",[7166,16063,16065],{"id":16064},"why-bother-with-adts-at-all","Why bother with ADTs at all?",[12,16067,16068],{},"Two big wins:",[3603,16070,16071,16077],{},[26,16072,16073,16076],{},[106,16074,16075],{},"Shared data and operations"," — design once, reuse everywhere. The same Bag ADT works for a shopping cart and a piggy bank.",[26,16078,16079,16082,16083,16085],{},[106,16080,16081],{},"Information hiding"," — users of the ADT don't need to know ",[7289,16084,9518],{}," it's implemented. The implementer can swap an array for a linked list, optimize for speed, fix bugs — and as long as the interface stays the same, no client code breaks.\nThis is the heart of good software design. Hold onto it.",[6964,16087,16089],{"id":16088},"part-2-the-bag","Part 2: The Bag",[12,16091,16092,16093,6959],{},"Now let's see all of this in action with a really simple ADT: the ",[106,16094,16095],{},"Bag",[7166,16097,16099],{"id":16098},"whats-a-bag","What's a bag?",[12,16101,16102],{},"Picture a real bag — a backpack, a tote, a grocery bag. Now ask yourself:",[23,16104,16105,16115,16124],{},[26,16106,16107,16110,16111,16114],{},[7289,16108,16109],{},"Should the items be stored in a specific order?"," → ",[106,16112,16113],{},"No",", you just toss stuff in.",[26,16116,16117,16110,16120,16123],{},[7289,16118,16119],{},"Can you keep repetitive items in the same bag?",[106,16121,16122],{},"Yes",", two apples are fine.",[26,16125,16126,16129],{},[7289,16127,16128],{},"Is there a standard size limit?"," → It varies. Some bags are fixed, some stretch.\nSo a \"bag\" in programming has the same vibe:",[7351,16131,16132],{},[12,16133,16134],{},[106,16135,16136],{},"The ADT Bag is a finite collection of objects in no particular order, possibly with duplicates.",[12,16138,16139],{},"That's the whole definition. No sorting, no positions, no uniqueness — just stuff in a bag.",[7166,16141,16143],{"id":16142},"what-can-you-do-with-a-bag","What can you do with a bag?",[12,16145,16146],{},"The standard behaviors are:",[23,16148,16149,16155,16161,16168,16174,16180,16186,16192],{},[26,16150,16151,16154],{},[106,16152,16153],{},"Add"," an object to the bag.",[26,16156,16157,16160],{},[106,16158,16159],{},"Remove"," an object from the bag.",[26,16162,16163,16164,16167],{},"Get the ",[106,16165,16166],{},"number of items"," inside.",[26,16169,16170,16171,6959],{},"Check if the bag is ",[106,16172,16173],{},"empty",[26,16175,16176,16177,6959],{},"Count how often a specific object ",[106,16178,16179],{},"appears",[26,16181,16182,16183,16185],{},"Check whether the bag ",[106,16184,9465],{}," a specific object.",[26,16187,16188,16191],{},[106,16189,16190],{},"Clear"," the bag.",[26,16193,16194,16195,16198],{},"Get ",[106,16196,16197],{},"all items"," as an array.",[7166,16200,16202],{"id":16201},"designing-the-bag-with-a-crc-card","Designing the bag with a CRC card",[12,16204,16205,16206,16209,16210,16213],{},"Before writing code, designers often use a ",[106,16207,16208],{},"CRC card"," — short for ",[106,16211,16212],{},"Class–Responsibility–Collaboration",". It's a sticky-note-sized summary of one class:",[6972,16215,16218],{"className":16216,"code":16217,"language":6977},[6975],"┌──────────────────────────────────────────────────────────┐\n│                          Bag                             │\n├──────────────────────────────────────────────────────────┤\n│ Responsibilities:                                        │\n│   - Get the number of items currently in the bag         │\n│   - See whether the bag is empty                         │\n│   - Add a given object to the bag                        │\n│   - Remove an unspecified object from the bag            │\n│   - Remove an occurrence of a particular object          │\n│   - Remove all objects from the bag                      │\n│   - Count how many times an object occurs in the bag     │\n│   - Test whether the bag contains a particular object    │\n│   - Look at all objects that are in the bag              │\n├──────────────────────────────────────────────────────────┤\n│ Collaborations:                                          │\n│   - The class of objects that the bag can contain        │\n└──────────────────────────────────────────────────────────┘\n",[6979,16219,16217],{"__ignoreMap":180},[12,16221,16222,16223,7042,16226,6959],{},"Notice we're still not writing any code. We're just listing ",[106,16224,16225],{},"what the class is responsible for",[106,16227,16228],{},"who it works with",[7166,16230,16232],{"id":16231},"uml-notation-the-same-idea-more-formal","UML notation — the same idea, more formal",[12,16234,16235,16236,16239],{},"Once we have the CRC card, the next step is to specify each method precisely — its name, parameters, and return type. ",[106,16237,16238],{},"UML"," (Unified Modeling Language) gives us a clean diagram for this:",[6972,16241,16244],{"className":16242,"code":16243,"language":6977},[6975],"┌────────────────────────────────────────┐\n│                  Bag                   │\n├────────────────────────────────────────┤\n│                                        │\n├────────────────────────────────────────┤\n│ +getCurrentSize(): integer             │\n│ +isEmpty(): boolean                    │\n│ +add(newEntry: T): boolean             │\n│ +remove(): T                           │\n│ +remove(anEntry: T): boolean           │\n│ +clear(): void                         │\n│ +getFrequencyOf(anEntry: T): integer   │\n│ +contains(anEntry: T): boolean         │\n│ +toArray(): T[]                        │\n└────────────────────────────────────────┘\n",[6979,16245,16243],{"__ignoreMap":180},[12,16247,9455,16248,16250,16251,16253,16254,16257],{},[6979,16249,8302],{}," means ",[7289,16252,7388],{},". The ",[6979,16255,16256],{},"T"," is a generic type — the bag can hold any kind of object.",[12,16259,16260],{},"Here's what each method does:",[7085,16262,16263,16272],{},[7088,16264,16265],{},[7091,16266,16267,16269],{},[7094,16268,7922],{},[7094,16270,16271],{},"Purpose",[7104,16273,16274,16284,16295,16308,16324,16337,16346,16356,16368],{},[7091,16275,16276,16281],{},[7109,16277,16278],{},[6979,16279,16280],{},"getCurrentSize()",[7109,16282,16283],{},"How many objects are in the bag right now.",[7091,16285,16286,16290],{},[7109,16287,16288],{},[6979,16289,7156],{},[7109,16291,16292,16294],{},[6979,16293,13594],{}," if the bag has nothing in it.",[7091,16296,16297,16302],{},[7109,16298,16299],{},[6979,16300,16301],{},"add(newEntry)",[7109,16303,16304,16305,16307],{},"Add an object. Returns ",[6979,16306,13594],{}," on success.",[7091,16309,16310,16315],{},[7109,16311,16312],{},[6979,16313,16314],{},"remove()",[7109,16316,16317,16318,16321,16322,6959],{},"Remove ",[7289,16319,16320],{},"some"," (unspecified) object. Returns it, or ",[6979,16323,7962],{},[7091,16325,16326,16331],{},[7109,16327,16328],{},[6979,16329,16330],{},"remove(anEntry)",[7109,16332,16333,16334,16336],{},"Remove one occurrence of a specific object. Returns ",[6979,16335,13594],{}," if found and removed.",[7091,16338,16339,16343],{},[7109,16340,16341],{},[6979,16342,8027],{},[7109,16344,16345],{},"Empty the bag.",[7091,16347,16348,16353],{},[7109,16349,16350],{},[6979,16351,16352],{},"getFrequencyOf(anEntry)",[7109,16354,16355],{},"How many times this object appears.",[7091,16357,16358,16363],{},[7109,16359,16360],{},[6979,16361,16362],{},"contains(anEntry)",[7109,16364,16365,16367],{},[6979,16366,13594],{}," if the object is in the bag.",[7091,16369,16370,16375],{},[7109,16371,16372],{},[6979,16373,16374],{},"toArray()",[7109,16376,16377],{},"Return all items as an array.",[7166,16379,16381],{"id":16380},"design-decisions-handling-weird-situations","Design decisions: handling weird situations",[12,16383,16384,16385,16387,16388,7024],{},"What if someone calls ",[6979,16386,16314],{}," on an empty bag? What should happen? There's no single right answer — there are ",[7289,16389,16390],{},"choices",[23,16392,16393,16399,16405,16411,16420,16431],{},[26,16394,16395,16398],{},[106,16396,16397],{},"Assume it won't happen"," (risky).",[26,16400,16401,16404],{},[106,16402,16403],{},"Ignore invalid situations"," (silently do nothing).",[26,16406,16407,16410],{},[106,16408,16409],{},"Guess the client's intention"," (also risky).",[26,16412,16413,16416,16417,16419],{},[106,16414,16415],{},"Return a special value"," that signals a problem (like ",[6979,16418,7962],{},").",[26,16421,16422,9527,16425,16427,16428,16430],{},[106,16423,16424],{},"Return a boolean",[6979,16426,13594],{}," for success, ",[6979,16429,8058],{}," for failure.",[26,16432,16433,16436],{},[106,16434,16435],{},"Throw an exception"," — the most explicit, but interrupts flow.",[12,16438,16439,16440,16443,16444,16446,16447,16449,16450,16452,16453,16456],{},"Good ADT design means making these decisions ",[7289,16441,16442],{},"deliberately"," and documenting them. The Bag we'll look at uses ",[6979,16445,7962],{}," for ",[6979,16448,16314],{}," on empty, and ",[6979,16451,8058],{}," from ",[6979,16454,16455],{},"add()"," if it fails.",[7166,16458,16460],{"id":16459},"from-specification-to-java-interface","From specification to Java interface",[12,16462,16463,16464,16467],{},"Once the methods are specified, we can write them as a ",[106,16465,16466],{},"Java interface"," — a contract that any Bag implementation must follow:",[6972,16469,16471],{"className":7376,"code":16470,"language":7378,"meta":180,"style":180},"\u002F**\n * An interface that describes the operations of a bag of objects.\n * @author Frank M. Carrano\n *\u002F\npublic interface BagInterface\u003CT> {\n    \u002F** Gets the current number of entries in this bag. *\u002F\n    public int getCurrentSize();\n \n    \u002F** Sees whether this bag is empty. *\u002F\n    public boolean isEmpty();\n \n    \u002F** Adds a new entry to this bag.\n     *  @return True if the addition is successful, false if not. *\u002F\n    public boolean add(T newEntry);\n \n    \u002F** Removes one unspecified entry from this bag, if possible.\n     *  @return Either the removed entry, if the removal was successful, or null. *\u002F\n    public T remove();\n \n    \u002F** Removes one occurrence of a given entry from this bag, if possible. *\u002F\n    public boolean remove(T anEntry);\n \n    \u002F** Removes all entries from this bag. *\u002F\n    public void clear();\n \n    \u002F** Counts the number of times a given entry appears in this bag. *\u002F\n    public int getFrequencyOf(T anEntry);\n \n    \u002F** Tests whether this bag contains a given entry. *\u002F\n    public boolean contains(T anEntry);\n \n    \u002F** Retrieves all entries that are in this bag.\n     *  @return A newly allocated array of all the entries in the bag. *\u002F\n    public T[] toArray();\n}\n",[6979,16472,16473,16478,16483,16494,16499,16515,16520,16531,16535,16540,16550,16554,16559,16570,16586,16590,16595,16604,16615,16619,16624,16640,16644,16649,16659,16663,16668,16683,16687,16692,16707,16711,16716,16725,16739],{"__ignoreMap":180},[75,16474,16475],{"class":7383,"line":7384},[75,16476,16477],{"class":7470},"\u002F**\n",[75,16479,16480],{"class":7383,"line":181},[75,16481,16482],{"class":7470}," * An interface that describes the operations of a bag of objects.\n",[75,16484,16485,16488,16491],{"class":7383,"line":7430},[75,16486,16487],{"class":7470}," * ",[75,16489,16490],{"class":7387},"@author",[75,16492,16493],{"class":7470}," Frank M. Carrano\n",[75,16495,16496],{"class":7383,"line":7436},[75,16497,16498],{"class":7470}," *\u002F\n",[75,16500,16501,16503,16506,16509,16511,16513],{"class":7383,"line":7458},[75,16502,7388],{"class":7387},[75,16504,16505],{"class":7387}," interface",[75,16507,16508],{"class":7394}," BagInterface",[75,16510,7399],{"class":7398},[75,16512,16256],{"class":7387},[75,16514,7405],{"class":7398},[75,16516,16517],{"class":7383,"line":7474},[75,16518,16519],{"class":7470},"    \u002F** Gets the current number of entries in this bag. *\u002F\n",[75,16521,16522,16524,16526,16529],{"class":7383,"line":7480},[75,16523,7439],{"class":7387},[75,16525,7531],{"class":7387},[75,16527,16528],{"class":7394}," getCurrentSize",[75,16530,7549],{"class":7398},[75,16532,16533],{"class":7383,"line":7485},[75,16534,7433],{"class":7398},[75,16536,16537],{"class":7383,"line":7498},[75,16538,16539],{"class":7470},"    \u002F** Sees whether this bag is empty. *\u002F\n",[75,16541,16542,16544,16546,16548],{"class":7383,"line":7516},[75,16543,7439],{"class":7387},[75,16545,9278],{"class":7387},[75,16547,9281],{"class":7394},[75,16549,7549],{"class":7398},[75,16551,16552],{"class":7383,"line":7521},[75,16553,7433],{"class":7398},[75,16555,16556],{"class":7383,"line":7526},[75,16557,16558],{"class":7470},"    \u002F** Adds a new entry to this bag.\n",[75,16560,16561,16564,16567],{"class":7383,"line":7539},[75,16562,16563],{"class":7470},"     *  ",[75,16565,16566],{"class":7387},"@return",[75,16568,16569],{"class":7470}," True if the addition is successful, false if not. *\u002F\n",[75,16571,16572,16574,16576,16578,16581,16584],{"class":7383,"line":7552},[75,16573,7439],{"class":7387},[75,16575,9278],{"class":7387},[75,16577,10395],{"class":7394},[75,16579,16580],{"class":7398},"(T ",[75,16582,16583],{"class":7451},"newEntry",[75,16585,7738],{"class":7398},[75,16587,16588],{"class":7383,"line":7557},[75,16589,7433],{"class":7398},[75,16591,16592],{"class":7383,"line":7562},[75,16593,16594],{"class":7470},"    \u002F** Removes one unspecified entry from this bag, if possible.\n",[75,16596,16597,16599,16601],{"class":7383,"line":7571},[75,16598,16563],{"class":7470},[75,16600,16566],{"class":7387},[75,16602,16603],{"class":7470}," Either the removed entry, if the removal was successful, or null. *\u002F\n",[75,16605,16606,16608,16611,16613],{"class":7383,"line":7584},[75,16607,7439],{"class":7387},[75,16609,16610],{"class":7398}," T ",[75,16612,8065],{"class":7394},[75,16614,7549],{"class":7398},[75,16616,16617],{"class":7383,"line":7602},[75,16618,7433],{"class":7398},[75,16620,16621],{"class":7383,"line":7607},[75,16622,16623],{"class":7470},"    \u002F** Removes one occurrence of a given entry from this bag, if possible. *\u002F\n",[75,16625,16626,16628,16630,16633,16635,16638],{"class":7383,"line":8360},[75,16627,7439],{"class":7387},[75,16629,9278],{"class":7387},[75,16631,16632],{"class":7394}," remove",[75,16634,16580],{"class":7398},[75,16636,16637],{"class":7451},"anEntry",[75,16639,7738],{"class":7398},[75,16641,16642],{"class":7383,"line":8374},[75,16643,7433],{"class":7398},[75,16645,16646],{"class":7383,"line":8387},[75,16647,16648],{"class":7470},"    \u002F** Removes all entries from this bag. *\u002F\n",[75,16650,16651,16653,16655,16657],{"class":7383,"line":8400},[75,16652,7439],{"class":7387},[75,16654,7442],{"class":7387},[75,16656,12923],{"class":7394},[75,16658,7549],{"class":7398},[75,16660,16661],{"class":7383,"line":8413},[75,16662,7433],{"class":7398},[75,16664,16665],{"class":7383,"line":8418},[75,16666,16667],{"class":7470},"    \u002F** Counts the number of times a given entry appears in this bag. *\u002F\n",[75,16669,16670,16672,16674,16677,16679,16681],{"class":7383,"line":8438},[75,16671,7439],{"class":7387},[75,16673,7531],{"class":7387},[75,16675,16676],{"class":7394}," getFrequencyOf",[75,16678,16580],{"class":7398},[75,16680,16637],{"class":7451},[75,16682,7738],{"class":7398},[75,16684,16685],{"class":7383,"line":8456},[75,16686,7433],{"class":7398},[75,16688,16689],{"class":7383,"line":8476},[75,16690,16691],{"class":7470},"    \u002F** Tests whether this bag contains a given entry. *\u002F\n",[75,16693,16694,16696,16698,16701,16703,16705],{"class":7383,"line":8481},[75,16695,7439],{"class":7387},[75,16697,9278],{"class":7387},[75,16699,16700],{"class":7394}," contains",[75,16702,16580],{"class":7398},[75,16704,16637],{"class":7451},[75,16706,7738],{"class":7398},[75,16708,16709],{"class":7383,"line":8487},[75,16710,7433],{"class":7398},[75,16712,16713],{"class":7383,"line":8493},[75,16714,16715],{"class":7470},"    \u002F** Retrieves all entries that are in this bag.\n",[75,16717,16718,16720,16722],{"class":7383,"line":15119},[75,16719,16563],{"class":7470},[75,16721,16566],{"class":7387},[75,16723,16724],{"class":7470}," A newly allocated array of all the entries in the bag. *\u002F\n",[75,16726,16727,16729,16732,16734,16737],{"class":7383,"line":15124},[75,16728,7439],{"class":7387},[75,16730,16731],{"class":7387}," T",[75,16733,8167],{"class":7398},[75,16735,16736],{"class":7394},"toArray",[75,16738,7549],{"class":7398},[75,16740,16741],{"class":7383,"line":15143},[75,16742,7610],{"class":7398},[12,16744,16745,16746,16749],{},"This is ",[7289,16747,16748],{},"still"," the ADT side. There's no actual storage logic yet — just a list of method signatures.",[7166,16751,16753],{"id":16752},"implementing-the-adt-and-why-we-dont-care-how","Implementing the ADT — and why we don't care how",[12,16755,16756,16757,16759],{},"Now imagine we hand this interface to a programmer and ask: \"Build me a ",[6979,16758,16095],{}," class in Java that implements this.\"",[12,16761,16762,16763,16766,16767,16770],{},"They might use an array. They might use a linked list. They might use a hash table or some wild custom structure. ",[106,16764,16765],{},"It doesn't matter to us."," As long as the class implements ",[6979,16768,16769],{},"BagInterface\u003CT>"," correctly, every method does what its specification says, and we can use it.",[12,16772,16773,16774],{},"This is the magic of ADTs in practice: ",[106,16775,16776],{},"you can change implementations without breaking client code.",[7166,16778,16780],{"id":16779},"example-1-an-online-shopping-cart","Example 1: An online shopping cart",[12,16782,16783,16784,16786],{},"Here's a small program that uses ",[6979,16785,16095],{}," as a shopping cart:",[6972,16788,16790],{"className":7376,"code":16789,"language":7378,"meta":180,"style":180},"public class OnlineShopper {\n    public static void main(String[] args) {\n        Item[] items = {\n            new Item(\"Bird feeder\", 2050),\n            new Item(\"Squirrel guard\", 1547),\n            new Item(\"Bird bath\", 4499),\n            new Item(\"Sunflower seeds\", 1295)\n        };\n \n        BagInterface\u003CItem> shoppingCart = new Bag\u003C>();\n        int totalCost = 0;\n \n        \u002F\u002F Add selected items to the cart\n        for (int index = 0; index \u003C items.length; index++) {\n            Item nextItem = items[index];\n            shoppingCart.add(nextItem);\n            totalCost = totalCost + nextItem.getPrice();\n        }\n \n        \u002F\u002F Simulate checkout\n        while (!shoppingCart.isEmpty())\n            System.out.println(shoppingCart.remove());\n \n        System.out.println(\"Total cost: \\t$\" + totalCost \u002F 100 + \".\" + totalCost % 100);\n    }\n}\n",[6979,16791,16792,16803,16823,16835,16856,16874,16892,16910,16915,16919,16937,16951,16955,16960,16986,16996,17006,17025,17029,17033,17038,17056,17070,17074,17116,17120],{"__ignoreMap":180},[75,16793,16794,16796,16798,16801],{"class":7383,"line":7384},[75,16795,7388],{"class":7387},[75,16797,7391],{"class":7387},[75,16799,16800],{"class":7394}," OnlineShopper",[75,16802,8147],{"class":7398},[75,16804,16805,16807,16809,16811,16813,16815,16817,16819,16821],{"class":7383,"line":181},[75,16806,7439],{"class":7387},[75,16808,8155],{"class":7387},[75,16810,7442],{"class":7387},[75,16812,8160],{"class":7394},[75,16814,7698],{"class":7398},[75,16816,7674],{"class":7387},[75,16818,8167],{"class":7398},[75,16820,8170],{"class":7451},[75,16822,7455],{"class":7398},[75,16824,16825,16828,16831,16833],{"class":7383,"line":7430},[75,16826,16827],{"class":7387},"        Item",[75,16829,16830],{"class":7398},"[] items ",[75,16832,7421],{"class":7387},[75,16834,8147],{"class":7398},[75,16836,16837,16840,16843,16845,16848,16850,16853],{"class":7383,"line":7436},[75,16838,16839],{"class":7387},"            new",[75,16841,16842],{"class":7394}," Item",[75,16844,7698],{"class":7398},[75,16846,16847],{"class":7589},"\"Bird feeder\"",[75,16849,7298],{"class":7398},[75,16851,16852],{"class":8127},"2050",[75,16854,16855],{"class":7398},"),\n",[75,16857,16858,16860,16862,16864,16867,16869,16872],{"class":7383,"line":7458},[75,16859,16839],{"class":7387},[75,16861,16842],{"class":7394},[75,16863,7698],{"class":7398},[75,16865,16866],{"class":7589},"\"Squirrel guard\"",[75,16868,7298],{"class":7398},[75,16870,16871],{"class":8127},"1547",[75,16873,16855],{"class":7398},[75,16875,16876,16878,16880,16882,16885,16887,16890],{"class":7383,"line":7474},[75,16877,16839],{"class":7387},[75,16879,16842],{"class":7394},[75,16881,7698],{"class":7398},[75,16883,16884],{"class":7589},"\"Bird bath\"",[75,16886,7298],{"class":7398},[75,16888,16889],{"class":8127},"4499",[75,16891,16855],{"class":7398},[75,16893,16894,16896,16898,16900,16903,16905,16908],{"class":7383,"line":7480},[75,16895,16839],{"class":7387},[75,16897,16842],{"class":7394},[75,16899,7698],{"class":7398},[75,16901,16902],{"class":7589},"\"Sunflower seeds\"",[75,16904,7298],{"class":7398},[75,16906,16907],{"class":8127},"1295",[75,16909,11077],{"class":7398},[75,16911,16912],{"class":7383,"line":7485},[75,16913,16914],{"class":7398},"        };\n",[75,16916,16917],{"class":7383,"line":7498},[75,16918,7433],{"class":7398},[75,16920,16921,16924,16927,16930,16932,16934],{"class":7383,"line":7516},[75,16922,16923],{"class":7398},"        BagInterface\u003C",[75,16925,16926],{"class":7387},"Item",[75,16928,16929],{"class":7398},"> shoppingCart ",[75,16931,7421],{"class":7387},[75,16933,7424],{"class":7387},[75,16935,16936],{"class":7398}," Bag\u003C>();\n",[75,16938,16939,16942,16945,16947,16949],{"class":7383,"line":7521},[75,16940,16941],{"class":7387},"        int",[75,16943,16944],{"class":7398}," totalCost ",[75,16946,7421],{"class":7387},[75,16948,8282],{"class":8127},[75,16950,8131],{"class":7398},[75,16952,16953],{"class":7383,"line":7526},[75,16954,7433],{"class":7398},[75,16956,16957],{"class":7383,"line":7539},[75,16958,16959],{"class":7470},"        \u002F\u002F Add selected items to the cart\n",[75,16961,16962,16964,16966,16968,16970,16972,16974,16977,16979,16982,16984],{"class":7383,"line":7552},[75,16963,10473],{"class":7387},[75,16965,10476],{"class":7398},[75,16967,8020],{"class":7387},[75,16969,10585],{"class":7398},[75,16971,7421],{"class":7387},[75,16973,8282],{"class":8127},[75,16975,16976],{"class":7398},"; index ",[75,16978,7399],{"class":7387},[75,16980,16981],{"class":7398}," items.length; index",[75,16983,10218],{"class":7387},[75,16985,7455],{"class":7398},[75,16987,16988,16991,16993],{"class":7383,"line":7557},[75,16989,16990],{"class":7398},"            Item nextItem ",[75,16992,7421],{"class":7387},[75,16994,16995],{"class":7398}," items[index];\n",[75,16997,16998,17001,17003],{"class":7383,"line":7562},[75,16999,17000],{"class":7398},"            shoppingCart.",[75,17002,8062],{"class":7394},[75,17004,17005],{"class":7398},"(nextItem);\n",[75,17007,17008,17011,17013,17015,17017,17020,17023],{"class":7383,"line":7571},[75,17009,17010],{"class":7398},"            totalCost ",[75,17012,7421],{"class":7387},[75,17014,16944],{"class":7398},[75,17016,8302],{"class":7387},[75,17018,17019],{"class":7398}," nextItem.",[75,17021,17022],{"class":7394},"getPrice",[75,17024,7549],{"class":7398},[75,17026,17027],{"class":7383,"line":7584},[75,17028,10516],{"class":7398},[75,17030,17031],{"class":7383,"line":7602},[75,17032,7433],{"class":7398},[75,17034,17035],{"class":7383,"line":7607},[75,17036,17037],{"class":7470},"        \u002F\u002F Simulate checkout\n",[75,17039,17040,17043,17045,17048,17051,17053],{"class":7383,"line":8360},[75,17041,17042],{"class":7387},"        while",[75,17044,10476],{"class":7398},[75,17046,17047],{"class":7387},"!",[75,17049,17050],{"class":7398},"shoppingCart.",[75,17052,9470],{"class":7394},[75,17054,17055],{"class":7398},"())\n",[75,17057,17058,17061,17063,17066,17068],{"class":7383,"line":8374},[75,17059,17060],{"class":7398},"            System.out.",[75,17062,7767],{"class":7394},[75,17064,17065],{"class":7398},"(shoppingCart.",[75,17067,8065],{"class":7394},[75,17069,8357],{"class":7398},[75,17071,17072],{"class":7383,"line":8387},[75,17073,7433],{"class":7398},[75,17075,17076,17078,17080,17082,17085,17088,17091,17093,17095,17097,17100,17102,17105,17107,17109,17112,17114],{"class":7383,"line":8400},[75,17077,12229],{"class":7398},[75,17079,7767],{"class":7394},[75,17081,7698],{"class":7398},[75,17083,17084],{"class":7589},"\"Total cost: ",[75,17086,17087],{"class":8127},"\\t",[75,17089,17090],{"class":7589},"$\"",[75,17092,7593],{"class":7387},[75,17094,16944],{"class":7398},[75,17096,13084],{"class":7387},[75,17098,17099],{"class":8127}," 100",[75,17101,7593],{"class":7387},[75,17103,17104],{"class":7589}," \".\"",[75,17106,7593],{"class":7387},[75,17108,16944],{"class":7398},[75,17110,17111],{"class":7387},"%",[75,17113,17099],{"class":8127},[75,17115,7738],{"class":7398},[75,17117,17118],{"class":7383,"line":8413},[75,17119,7477],{"class":7398},[75,17121,17122],{"class":7383,"line":8418},[75,17123,7610],{"class":7398},[12,17125,17126],{},"Sample output:",[6972,17128,17131],{"className":17129,"code":17130,"language":6977},[6975],"Sunflower seeds $12.95\nBird bath       $44.99\nSquirrel guard  $15.47\nBird feeder     $20.50\nTotal cost:     $93.91\n",[6979,17132,17130],{"__ignoreMap":180},[12,17134,17135,17136,17139],{},"Notice the items come out in a different order than they went in. ",[106,17137,17138],{},"That's fine"," — a Bag has no specified order. The shopping cart still works.",[7166,17141,17143],{"id":17142},"example-2-a-piggy-bank","Example 2: A piggy bank",[6972,17145,17147],{"className":7376,"code":17146,"language":7378,"meta":180,"style":180},"public class PiggyBank {\n    private BagInterface\u003CCoin> coins;\n \n    public PiggyBank() {\n        coins = new Bag\u003C>();\n    }\n \n    public boolean add(Coin aCoin) {\n        return coins.add(aCoin);\n    }\n \n    public Coin remove() {\n        return coins.remove();\n    }\n \n    public boolean isEmpty() {\n        return coins.isEmpty();\n    }\n}\n",[6979,17148,17149,17160,17173,17177,17185,17196,17200,17204,17220,17232,17236,17240,17251,17261,17265,17269,17279,17289,17293],{"__ignoreMap":180},[75,17150,17151,17153,17155,17158],{"class":7383,"line":7384},[75,17152,7388],{"class":7387},[75,17154,7391],{"class":7387},[75,17156,17157],{"class":7394}," PiggyBank",[75,17159,8147],{"class":7398},[75,17161,17162,17164,17167,17170],{"class":7383,"line":181},[75,17163,7410],{"class":7387},[75,17165,17166],{"class":7398}," BagInterface\u003C",[75,17168,17169],{"class":7387},"Coin",[75,17171,17172],{"class":7398},"> coins;\n",[75,17174,17175],{"class":7383,"line":7430},[75,17176,7433],{"class":7398},[75,17178,17179,17181,17183],{"class":7383,"line":7436},[75,17180,7439],{"class":7387},[75,17182,17157],{"class":7394},[75,17184,7495],{"class":7398},[75,17186,17187,17190,17192,17194],{"class":7383,"line":7458},[75,17188,17189],{"class":7398},"        coins ",[75,17191,7421],{"class":7387},[75,17193,7424],{"class":7387},[75,17195,16936],{"class":7398},[75,17197,17198],{"class":7383,"line":7474},[75,17199,7477],{"class":7398},[75,17201,17202],{"class":7383,"line":7480},[75,17203,7433],{"class":7398},[75,17205,17206,17208,17210,17212,17215,17218],{"class":7383,"line":7485},[75,17207,7439],{"class":7387},[75,17209,9278],{"class":7387},[75,17211,10395],{"class":7394},[75,17213,17214],{"class":7398},"(Coin ",[75,17216,17217],{"class":7451},"aCoin",[75,17219,7455],{"class":7398},[75,17221,17222,17224,17227,17229],{"class":7383,"line":7498},[75,17223,7501],{"class":7387},[75,17225,17226],{"class":7398}," coins.",[75,17228,8062],{"class":7394},[75,17230,17231],{"class":7398},"(aCoin);\n",[75,17233,17234],{"class":7383,"line":7516},[75,17235,7477],{"class":7398},[75,17237,17238],{"class":7383,"line":7521},[75,17239,7433],{"class":7398},[75,17241,17242,17244,17247,17249],{"class":7383,"line":7526},[75,17243,7439],{"class":7387},[75,17245,17246],{"class":7398}," Coin ",[75,17248,8065],{"class":7394},[75,17250,7495],{"class":7398},[75,17252,17253,17255,17257,17259],{"class":7383,"line":7539},[75,17254,7501],{"class":7387},[75,17256,17226],{"class":7398},[75,17258,8065],{"class":7394},[75,17260,7549],{"class":7398},[75,17262,17263],{"class":7383,"line":7552},[75,17264,7477],{"class":7398},[75,17266,17267],{"class":7383,"line":7557},[75,17268,7433],{"class":7398},[75,17270,17271,17273,17275,17277],{"class":7383,"line":7562},[75,17272,7439],{"class":7387},[75,17274,9278],{"class":7387},[75,17276,9281],{"class":7394},[75,17278,7495],{"class":7398},[75,17280,17281,17283,17285,17287],{"class":7383,"line":7571},[75,17282,7501],{"class":7387},[75,17284,17226],{"class":7398},[75,17286,9470],{"class":7394},[75,17288,7549],{"class":7398},[75,17290,17291],{"class":7383,"line":7584},[75,17292,7477],{"class":7398},[75,17294,17295],{"class":7383,"line":7602},[75,17296,7610],{"class":7398},[12,17298,17299],{},"And a quick demo:",[6972,17301,17303],{"className":7376,"code":17302,"language":7378,"meta":180,"style":180},"PiggyBank myBank = new PiggyBank();\naddCoin(new Coin(1, 2010), myBank);   \u002F\u002F penny\naddCoin(new Coin(5, 2011), myBank);   \u002F\u002F nickel\naddCoin(new Coin(10, 2000), myBank);  \u002F\u002F dime\naddCoin(new Coin(25, 2012), myBank);  \u002F\u002F quarter\n \nSystem.out.println(\"Removing all the coins:\");\nint amountRemoved = 0;\nwhile (!myBank.isEmpty()) {\n    Coin removedCoin = myBank.remove();\n    System.out.println(\"Removed a \" + removedCoin.getCoinName() + \".\");\n    amountRemoved = amountRemoved + removedCoin.getValue();\n}\nSystem.out.println(\"All done. Removed \" + amountRemoved + \" cents.\");\n",[6979,17304,17305,17318,17345,17369,17395,17420,17424,17437,17450,17465,17479,17506,17524,17528],{"__ignoreMap":180},[75,17306,17307,17310,17312,17314,17316],{"class":7383,"line":7384},[75,17308,17309],{"class":7398},"PiggyBank myBank ",[75,17311,7421],{"class":7387},[75,17313,7424],{"class":7387},[75,17315,17157],{"class":7394},[75,17317,7549],{"class":7398},[75,17319,17320,17323,17325,17327,17330,17332,17334,17336,17339,17342],{"class":7383,"line":181},[75,17321,17322],{"class":7394},"addCoin",[75,17324,7698],{"class":7398},[75,17326,8823],{"class":7387},[75,17328,17329],{"class":7394}," Coin",[75,17331,7698],{"class":7398},[75,17333,8855],{"class":8127},[75,17335,7298],{"class":7398},[75,17337,17338],{"class":8127},"2010",[75,17340,17341],{"class":7398},"), myBank);   ",[75,17343,17344],{"class":7470},"\u002F\u002F penny\n",[75,17346,17347,17349,17351,17353,17355,17357,17359,17361,17364,17366],{"class":7383,"line":7430},[75,17348,17322],{"class":7394},[75,17350,7698],{"class":7398},[75,17352,8823],{"class":7387},[75,17354,17329],{"class":7394},[75,17356,7698],{"class":7398},[75,17358,15531],{"class":8127},[75,17360,7298],{"class":7398},[75,17362,17363],{"class":8127},"2011",[75,17365,17341],{"class":7398},[75,17367,17368],{"class":7470},"\u002F\u002F nickel\n",[75,17370,17371,17373,17375,17377,17379,17381,17384,17386,17389,17392],{"class":7383,"line":7436},[75,17372,17322],{"class":7394},[75,17374,7698],{"class":7398},[75,17376,8823],{"class":7387},[75,17378,17329],{"class":7394},[75,17380,7698],{"class":7398},[75,17382,17383],{"class":8127},"10",[75,17385,7298],{"class":7398},[75,17387,17388],{"class":8127},"2000",[75,17390,17391],{"class":7398},"), myBank);  ",[75,17393,17394],{"class":7470},"\u002F\u002F dime\n",[75,17396,17397,17399,17401,17403,17405,17407,17410,17412,17415,17417],{"class":7383,"line":7458},[75,17398,17322],{"class":7394},[75,17400,7698],{"class":7398},[75,17402,8823],{"class":7387},[75,17404,17329],{"class":7394},[75,17406,7698],{"class":7398},[75,17408,17409],{"class":8127},"25",[75,17411,7298],{"class":7398},[75,17413,17414],{"class":8127},"2012",[75,17416,17391],{"class":7398},[75,17418,17419],{"class":7470},"\u002F\u002F quarter\n",[75,17421,17422],{"class":7383,"line":7474},[75,17423,7433],{"class":7398},[75,17425,17426,17428,17430,17432,17435],{"class":7383,"line":7480},[75,17427,7764],{"class":7398},[75,17429,7767],{"class":7394},[75,17431,7698],{"class":7398},[75,17433,17434],{"class":7589},"\"Removing all the coins:\"",[75,17436,7738],{"class":7398},[75,17438,17439,17441,17444,17446,17448],{"class":7383,"line":7485},[75,17440,8020],{"class":7387},[75,17442,17443],{"class":7398}," amountRemoved ",[75,17445,7421],{"class":7387},[75,17447,8282],{"class":8127},[75,17449,8131],{"class":7398},[75,17451,17452,17454,17456,17458,17461,17463],{"class":7383,"line":7498},[75,17453,10020],{"class":7387},[75,17455,10476],{"class":7398},[75,17457,17047],{"class":7387},[75,17459,17460],{"class":7398},"myBank.",[75,17462,9470],{"class":7394},[75,17464,14094],{"class":7398},[75,17466,17467,17470,17472,17475,17477],{"class":7383,"line":7516},[75,17468,17469],{"class":7398},"    Coin removedCoin ",[75,17471,7421],{"class":7387},[75,17473,17474],{"class":7398}," myBank.",[75,17476,8065],{"class":7394},[75,17478,7549],{"class":7398},[75,17480,17481,17483,17485,17487,17490,17492,17495,17498,17500,17502,17504],{"class":7383,"line":7521},[75,17482,8254],{"class":7398},[75,17484,7767],{"class":7394},[75,17486,7698],{"class":7398},[75,17488,17489],{"class":7589},"\"Removed a \"",[75,17491,7593],{"class":7387},[75,17493,17494],{"class":7398}," removedCoin.",[75,17496,17497],{"class":7394},"getCoinName",[75,17499,8276],{"class":7398},[75,17501,8302],{"class":7387},[75,17503,17104],{"class":7589},[75,17505,7738],{"class":7398},[75,17507,17508,17511,17513,17515,17517,17519,17522],{"class":7383,"line":7526},[75,17509,17510],{"class":7398},"    amountRemoved ",[75,17512,7421],{"class":7387},[75,17514,17443],{"class":7398},[75,17516,8302],{"class":7387},[75,17518,17494],{"class":7398},[75,17520,17521],{"class":7394},"getValue",[75,17523,7549],{"class":7398},[75,17525,17526],{"class":7383,"line":7539},[75,17527,7610],{"class":7398},[75,17529,17530,17532,17534,17536,17539,17541,17543,17545,17548],{"class":7383,"line":7552},[75,17531,7764],{"class":7398},[75,17533,7767],{"class":7394},[75,17535,7698],{"class":7398},[75,17537,17538],{"class":7589},"\"All done. Removed \"",[75,17540,7593],{"class":7387},[75,17542,17443],{"class":7398},[75,17544,8302],{"class":7387},[75,17546,17547],{"class":7589}," \" cents.\"",[75,17549,7738],{"class":7398},[12,17551,9006],{},[6972,17553,17556],{"className":17554,"code":17555,"language":6977},[6975],"Added a PENNY.\nAdded a NICKEL.\nAdded a DIME.\nAdded a QUARTER.\nRemoving all the coins:\nRemoved a QUARTER.\nRemoved a DIME.\nRemoved a NICKEL.\nRemoved a PENNY.\nAll done. Removed 41 cents.\n",[6979,17557,17555],{"__ignoreMap":180},[12,17559,9884,17560,7042,17563,17566,17567,17569,17570],{},[6979,17561,17562],{},"OnlineShopper",[6979,17564,17565],{},"PiggyBank"," use the same ",[6979,17568,16095],{}," class with totally different domains. ",[106,17571,17572],{},"One ADT, many uses.",[7166,17574,17576],{"id":17575},"the-vending-machine-analogy","The vending machine analogy",[12,17578,17579,17580],{},"Here's the mental model I always come back to: ",[106,17581,17582],{},"using an ADT is like using a vending machine.",[7085,17584,17585,17595],{},[7088,17586,17587],{},[7091,17588,17589,17592],{},[7094,17590,17591],{},"Vending Machine",[7094,17593,17594],{},"ADT Bag",[7104,17596,17597,17605,17613,17621,17629],{},[7091,17598,17599,17602],{},[7109,17600,17601],{},"You can only do what the buttons let you do.",[7109,17603,17604],{},"You can only call the methods the ADT exposes.",[7091,17606,17607,17610],{},[7109,17608,17609],{},"You must understand what each button does.",[7109,17611,17612],{},"You must follow the method specifications.",[7091,17614,17615,17618],{},[7109,17616,17617],{},"You can't open the machine and rearrange things.",[7109,17619,17620],{},"You can't poke at the internal data directly.",[7091,17622,17623,17626],{},[7109,17624,17625],{},"You can use it without knowing what's inside.",[7109,17627,17628],{},"You can use it without knowing how it stores data.",[7091,17630,17631,17634],{},[7109,17632,17633],{},"Still works even if they restock or rewire it.",[7109,17635,17636],{},"Still works if the implementation changes.",[12,17638,17639],{},"That's it. That's the whole philosophy.",[7166,17641,17643,17644,17646],{"id":17642},"a-peek-ahead-how-javas-set-interface-looks","A peek ahead: how Java's ",[6979,17645,7045],{}," interface looks",[12,17648,17649,17650,17652],{},"Just for context, Java's own ",[6979,17651,7045],{}," interface is structured almost identically:",[6972,17654,17656],{"className":7376,"code":17655,"language":7378,"meta":180,"style":180},"public interface SetInterface\u003CT> {\n    public int getCurrentSize();\n    public boolean isEmpty();\n    public boolean add(T newEntry);     \u002F\u002F avoids duplicates\n    public boolean remove(T anEntry);\n    public T remove();\n    public void clear();\n    public boolean contains(T anEntry);\n    public T[] toArray();\n}\n",[6979,17657,17658,17673,17683,17693,17711,17725,17735,17745,17759,17771],{"__ignoreMap":180},[75,17659,17660,17662,17664,17667,17669,17671],{"class":7383,"line":7384},[75,17661,7388],{"class":7387},[75,17663,16505],{"class":7387},[75,17665,17666],{"class":7394}," SetInterface",[75,17668,7399],{"class":7398},[75,17670,16256],{"class":7387},[75,17672,7405],{"class":7398},[75,17674,17675,17677,17679,17681],{"class":7383,"line":181},[75,17676,7439],{"class":7387},[75,17678,7531],{"class":7387},[75,17680,16528],{"class":7394},[75,17682,7549],{"class":7398},[75,17684,17685,17687,17689,17691],{"class":7383,"line":7430},[75,17686,7439],{"class":7387},[75,17688,9278],{"class":7387},[75,17690,9281],{"class":7394},[75,17692,7549],{"class":7398},[75,17694,17695,17697,17699,17701,17703,17705,17708],{"class":7383,"line":7436},[75,17696,7439],{"class":7387},[75,17698,9278],{"class":7387},[75,17700,10395],{"class":7394},[75,17702,16580],{"class":7398},[75,17704,16583],{"class":7451},[75,17706,17707],{"class":7398},");     ",[75,17709,17710],{"class":7470},"\u002F\u002F avoids duplicates\n",[75,17712,17713,17715,17717,17719,17721,17723],{"class":7383,"line":7458},[75,17714,7439],{"class":7387},[75,17716,9278],{"class":7387},[75,17718,16632],{"class":7394},[75,17720,16580],{"class":7398},[75,17722,16637],{"class":7451},[75,17724,7738],{"class":7398},[75,17726,17727,17729,17731,17733],{"class":7383,"line":7474},[75,17728,7439],{"class":7387},[75,17730,16610],{"class":7398},[75,17732,8065],{"class":7394},[75,17734,7549],{"class":7398},[75,17736,17737,17739,17741,17743],{"class":7383,"line":7480},[75,17738,7439],{"class":7387},[75,17740,7442],{"class":7387},[75,17742,12923],{"class":7394},[75,17744,7549],{"class":7398},[75,17746,17747,17749,17751,17753,17755,17757],{"class":7383,"line":7485},[75,17748,7439],{"class":7387},[75,17750,9278],{"class":7387},[75,17752,16700],{"class":7394},[75,17754,16580],{"class":7398},[75,17756,16637],{"class":7451},[75,17758,7738],{"class":7398},[75,17760,17761,17763,17765,17767,17769],{"class":7383,"line":7498},[75,17762,7439],{"class":7387},[75,17764,16731],{"class":7387},[75,17766,8167],{"class":7398},[75,17768,16736],{"class":7394},[75,17770,7549],{"class":7398},[75,17772,17773],{"class":7383,"line":7516},[75,17774,7610],{"class":7398},[12,17776,17777,17778,17781],{},"The big difference: ",[106,17779,17780],{},"a Set doesn't allow duplicates",". The Bag does. Same shape, different rules.",[7166,17783,17785],{"id":17784},"looking-ahead-arraybag-and-linkedbag","Looking ahead: ArrayBag and LinkedBag",[12,17787,17788],{},"Now that the ADT is fully specified, we can implement it two different ways — and this is what the next part of the course covers:",[23,17790,17791,17805],{},[26,17792,17793,17798,17799],{},[106,17794,17795],{},[6979,17796,17797],{},"ArrayBag"," — stores entries in a fixed-size array. Simple, fast access, but capacity is bounded.",[6972,17800,17803],{"className":17801,"code":17802,"language":6977},[6975],"┌───────────────────────────┐\n│         ArrayBag          │\n├───────────────────────────┤\n│ -bag: T[]                 │\n│ -numberOfEntries: integer │\n│ -DEFAULT_CAPACITY: integer│\n├───────────────────────────┤\n│ (all the BagInterface     │\n│  methods, plus...)        │\n│ -isArrayFull(): boolean   │\n└───────────────────────────┘\n",[6979,17804,17802],{"__ignoreMap":180},[26,17806,17807,17812,17813],{},[106,17808,17809],{},[6979,17810,17811],{},"LinkedBag"," — stores entries in a chain of linked nodes. No size limit, but slightly more overhead per item.",[6972,17814,17816],{"className":7376,"code":17815,"language":7378,"meta":180,"style":180},"public final class LinkedBag\u003CT> implements BagInterface\u003CT> {\n    private Node firstNode;\n    private int numberOfEntries;\n    \u002F\u002F ...\n    private class Node { \u002F* inner class *\u002F }\n}\n",[6979,17817,17818,17847,17854,17863,17868,17885],{"__ignoreMap":180},[75,17819,17820,17822,17825,17827,17830,17832,17834,17836,17839,17841,17843,17845],{"class":7383,"line":7384},[75,17821,7388],{"class":7387},[75,17823,17824],{"class":7387}," final",[75,17826,7391],{"class":7387},[75,17828,17829],{"class":7394}," LinkedBag",[75,17831,7399],{"class":7398},[75,17833,16256],{"class":7387},[75,17835,11390],{"class":7398},[75,17837,17838],{"class":7387},"implements",[75,17840,16508],{"class":7394},[75,17842,7399],{"class":7398},[75,17844,16256],{"class":7387},[75,17846,7405],{"class":7398},[75,17848,17849,17851],{"class":7383,"line":181},[75,17850,7410],{"class":7387},[75,17852,17853],{"class":7398}," Node firstNode;\n",[75,17855,17856,17858,17860],{"class":7383,"line":7430},[75,17857,7410],{"class":7387},[75,17859,7531],{"class":7387},[75,17861,17862],{"class":7398}," numberOfEntries;\n",[75,17864,17865],{"class":7383,"line":7436},[75,17866,17867],{"class":7470},"    \u002F\u002F ...\n",[75,17869,17870,17872,17874,17876,17879,17882],{"class":7383,"line":7458},[75,17871,7410],{"class":7387},[75,17873,7391],{"class":7387},[75,17875,9667],{"class":7394},[75,17877,17878],{"class":7398}," { ",[75,17880,17881],{"class":7470},"\u002F* inner class *\u002F",[75,17883,17884],{"class":7398}," }\n",[75,17886,17887],{"class":7383,"line":7474},[75,17888,7610],{"class":7398},[12,17890,17891,17892,17894,17895,17898,17899,17902],{},"Both implement ",[6979,17893,16769],{},". Both are valid bags. The choice between them is an ",[106,17896,17897],{},"implementation decision"," — and the rest of the client code doesn't care which one you use. ",[7289,17900,17901],{},"That's"," the power of an ADT.",[6964,17904,17906],{"id":17905},"quick-recap","Quick recap",[12,17908,17909],{},"If you remember just five things from this post:",[3603,17911,17912,17922,17928,17933,17938],{},[26,17913,15781,17914,7653,17916,17918,17919,17921],{},[106,17915,15865],{},[7289,17917,9441],{}," a data type does, not ",[7289,17920,9518],{}," it does it.",[26,17923,9407,17924,17927],{},[106,17925,17926],{},"data structure"," is the actual implementation of an ADT in code.",[26,17929,17930,17932],{},[106,17931,16081],{}," lets the implementation change without breaking users.",[26,17934,9407,17935,17937],{},[106,17936,16095],{}," is a finite, unordered collection that allows duplicates.",[26,17939,17940],{},"The same Bag ADT can power a shopping cart, a piggy bank, or anything else — that's the whole point.",[9332,17942,17943],{},"html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html pre.shiki code .sZZnC, html code.shiki .sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}",{"title":180,"searchDepth":181,"depth":181,"links":17945},[17946,17955,17970],{"id":15685,"depth":181,"text":15686,"children":17947},[17948,17949,17950,17951,17952,17953,17954],{"id":15689,"depth":7430,"text":15690},{"id":15734,"depth":7430,"text":15735},{"id":15777,"depth":7430,"text":15778},{"id":15846,"depth":7430,"text":15847},{"id":15969,"depth":7430,"text":15970},{"id":16015,"depth":7430,"text":16016},{"id":16064,"depth":7430,"text":16065},{"id":16088,"depth":181,"text":16089,"children":17956},[17957,17958,17959,17960,17961,17962,17963,17964,17965,17966,17967,17969],{"id":16098,"depth":7430,"text":16099},{"id":16142,"depth":7430,"text":16143},{"id":16201,"depth":7430,"text":16202},{"id":16231,"depth":7430,"text":16232},{"id":16380,"depth":7430,"text":16381},{"id":16459,"depth":7430,"text":16460},{"id":16752,"depth":7430,"text":16753},{"id":16779,"depth":7430,"text":16780},{"id":17142,"depth":7430,"text":17143},{"id":17575,"depth":7430,"text":17576},{"id":17642,"depth":7430,"text":17968},"A peek ahead: how Java's Set interface looks",{"id":17784,"depth":7430,"text":17785},{"id":17905,"depth":181,"text":17906},"2026-05-12","If you're new to data structures, the very first hurdle isn't a fancy algorithm — it's a way of thinking. Programmers separate what a thing does from how it's built. That's the whole idea behind an Abstract Data Type (ADT), and once it clicks, the rest of the course starts to feel much more organized.",{},"\u002Fblog\u002FAbstract-Data-Types-&-The-Bag",{"title":15646,"description":17972},{"loc":17974},"blog\u002FAbstract-Data-Types-&-The-Bag","7lon_BVLnmwmYuV6r5uFdPUQvP1uFFpTCUfTL1nd9L8",{"id":17980,"title":17981,"author":6946,"body":17982,"date":17971,"description":180,"draft":185,"edited_at":17971,"extension":186,"featured_image":194,"meta":21121,"navigation":189,"path":21122,"pinned":185,"seo":21123,"sitemap":21124,"stem":21125,"tags":194,"__hash__":21126},"blog\u002Fblog\u002FJava-Generics.md","Java Generics",{"type":9,"value":17983,"toc":21080},[17984,17992,17997,18007,18070,18082,18144,18157,18161,18164,18222,18225,18251,18320,18330,18337,18351,18354,18360,18380,18425,18429,18544,18548,18651,18655,18662,18752,18756,18814,18843,18851,18857,18886,18893,19079,19082,19164,19188,19192,19265,19268,19296,19300,19307,19312,19339,19348,19455,19458,19579,19586,19594,19604,19611,19639,19666,19852,19856,19872,19876,19882,19902,19916,19920,19923,19981,19984,20011,20018,20103,20115,20118,20122,20133,20148,20208,20218,20222,20283,20291,20295,20325,20329,20335,20345,20348,20368,20416,20419,20464,20471,20475,20547,20572,20576,20583,20589,20611,20621,20625,20648,20651,20681,20684,20688,20755,20765,20769,20799,20815,20819,21077],[6964,17985,17987,17988,17991],{"id":17986},"_1-what-even-is-a-generic","1. What even ",[7289,17989,17990],{},"is"," a generic?",[12,17993,17994],{},[106,17995,17996],{},"Generic = the ability to parameterize types.",[12,17998,17999,18000,18003,18004,6959],{},"That sounds abstract, so let me translate. You already know about ",[106,18001,18002],{},"parameters in methods",": you write a method once and pass in different ",[7289,18005,18006],{},"values",[6972,18008,18010],{"className":7376,"code":18009,"language":7378,"meta":180,"style":180},"public int square(int x) { return x * x; }\n \nsquare(3);   \u002F\u002F works\nsquare(10);  \u002F\u002F works\n",[6979,18011,18012,18038,18042,18057],{"__ignoreMap":180},[75,18013,18014,18016,18018,18021,18023,18025,18028,18030,18033,18035],{"class":7383,"line":7384},[75,18015,7388],{"class":7387},[75,18017,7531],{"class":7387},[75,18019,18020],{"class":7394}," square",[75,18022,7698],{"class":7398},[75,18024,8020],{"class":7387},[75,18026,18027],{"class":7398}," x) { ",[75,18029,8636],{"class":7387},[75,18031,18032],{"class":7398}," x ",[75,18034,8128],{"class":7387},[75,18036,18037],{"class":7398}," x; }\n",[75,18039,18040],{"class":7383,"line":181},[75,18041,7433],{"class":7398},[75,18043,18044,18047,18049,18051,18054],{"class":7383,"line":7430},[75,18045,18046],{"class":7394},"square",[75,18048,7698],{"class":7398},[75,18050,8830],{"class":8127},[75,18052,18053],{"class":7398},");   ",[75,18055,18056],{"class":7470},"\u002F\u002F works\n",[75,18058,18059,18061,18063,18065,18068],{"class":7383,"line":7436},[75,18060,18046],{"class":7394},[75,18062,7698],{"class":7398},[75,18064,17383],{"class":8127},[75,18066,18067],{"class":7398},");  ",[75,18069,18056],{"class":7470},[12,18071,9407,18072,18074,18075,18078,18079,6959],{},[106,18073,14017],{}," does the same thing — but instead of passing in a ",[7289,18076,18077],{},"value",", you pass in a ",[106,18080,18081],{},"type",[6972,18083,18085],{"className":7376,"code":18084,"language":7378,"meta":180,"style":180},"ArrayList\u003CString>  cities  = new ArrayList\u003C>();   \u002F\u002F a list of Strings\nArrayList\u003CInteger> scores  = new ArrayList\u003C>();   \u002F\u002F a list of Integers\nArrayList\u003CCircle>  shapes  = new ArrayList\u003C>();   \u002F\u002F a list of Circles\n",[6979,18086,18087,18107,18125],{"__ignoreMap":180},[75,18088,18089,18092,18094,18097,18099,18101,18104],{"class":7383,"line":7384},[75,18090,18091],{"class":7398},"ArrayList\u003C",[75,18093,7674],{"class":7387},[75,18095,18096],{"class":7398},">  cities  ",[75,18098,7421],{"class":7387},[75,18100,7424],{"class":7387},[75,18102,18103],{"class":7398}," ArrayList\u003C>();   ",[75,18105,18106],{"class":7470},"\u002F\u002F a list of Strings\n",[75,18108,18109,18111,18113,18116,18118,18120,18122],{"class":7383,"line":181},[75,18110,18091],{"class":7398},[75,18112,9143],{"class":7387},[75,18114,18115],{"class":7398},"> scores  ",[75,18117,7421],{"class":7387},[75,18119,7424],{"class":7387},[75,18121,18103],{"class":7398},[75,18123,18124],{"class":7470},"\u002F\u002F a list of Integers\n",[75,18126,18127,18129,18132,18135,18137,18139,18141],{"class":7383,"line":7430},[75,18128,18091],{"class":7398},[75,18130,18131],{"class":7387},"Circle",[75,18133,18134],{"class":7398},">  shapes  ",[75,18136,7421],{"class":7387},[75,18138,7424],{"class":7387},[75,18140,18103],{"class":7398},[75,18142,18143],{"class":7470},"\u002F\u002F a list of Circles\n",[12,18145,18146,18147,18149,18150,18153,18154,6959],{},"The class ",[6979,18148,7193],{}," is written ",[7289,18151,18152],{},"once",". You decide what kind of thing it stores by plugging in a type between the ",[6979,18155,18156],{},"\u003C>",[6964,18158,18160],{"id":18159},"_2-why-bother-aka-the-why-generics-rant","2. Why bother? (a.k.a. the \"Why Generics?\" rant)",[12,18162,18163],{},"Before Java 1.5, you had to write things like this:",[6972,18165,18167],{"className":7376,"code":18166,"language":7378,"meta":180,"style":180},"ArrayList list = new ArrayList();    \u002F\u002F raw type — no \u003C>\nlist.add(\"hello\");\nString s = (String) list.get(0);     \u002F\u002F manual cast required\n",[6979,18168,18169,18187,18201],{"__ignoreMap":180},[75,18170,18171,18174,18176,18178,18181,18184],{"class":7383,"line":7384},[75,18172,18173],{"class":7398},"ArrayList list ",[75,18175,7421],{"class":7387},[75,18177,7424],{"class":7387},[75,18179,18180],{"class":7394}," ArrayList",[75,18182,18183],{"class":7398},"();    ",[75,18185,18186],{"class":7470},"\u002F\u002F raw type — no \u003C>\n",[75,18188,18189,18192,18194,18196,18199],{"class":7383,"line":181},[75,18190,18191],{"class":7398},"list.",[75,18193,8062],{"class":7394},[75,18195,7698],{"class":7398},[75,18197,18198],{"class":7589},"\"hello\"",[75,18200,7738],{"class":7398},[75,18202,18203,18206,18208,18211,18213,18215,18217,18219],{"class":7383,"line":7430},[75,18204,18205],{"class":7398},"String s ",[75,18207,7421],{"class":7387},[75,18209,18210],{"class":7398}," (String) list.",[75,18212,13819],{"class":7394},[75,18214,7698],{"class":7398},[75,18216,9180],{"class":8127},[75,18218,17707],{"class":7398},[75,18220,18221],{"class":7470},"\u002F\u002F manual cast required\n",[12,18223,18224],{},"Two problems:",[3603,18226,18227,18233],{},[26,18228,18229,18232],{},[106,18230,18231],{},"You had to cast everything."," Ugly.",[26,18234,18235,18238,18239,18242,18243,18246,18247,18250],{},[106,18236,18237],{},"No type safety."," You could ",[6979,18240,18241],{},"list.add(\"hello\")"," and then ",[6979,18244,18245],{},"list.add(42)"," and the compiler wouldn't say a word — until your program crashed at runtime with a ",[6979,18248,18249],{},"ClassCastException",".\nWith generics, both problems disappear:",[6972,18252,18254],{"className":7376,"code":18253,"language":7378,"meta":180,"style":180},"ArrayList\u003CString> list = new ArrayList\u003C>();\nlist.add(\"hello\");\nString s = list.get(0);              \u002F\u002F ✅ no cast needed\nlist.add(42);                        \u002F\u002F ❌ compile error — Java stops you immediately\n",[6979,18255,18256,18271,18283,18303],{"__ignoreMap":180},[75,18257,18258,18260,18262,18264,18266,18268],{"class":7383,"line":7384},[75,18259,18091],{"class":7398},[75,18261,7674],{"class":7387},[75,18263,7418],{"class":7398},[75,18265,7421],{"class":7387},[75,18267,7424],{"class":7387},[75,18269,18270],{"class":7398}," ArrayList\u003C>();\n",[75,18272,18273,18275,18277,18279,18281],{"class":7383,"line":181},[75,18274,18191],{"class":7398},[75,18276,8062],{"class":7394},[75,18278,7698],{"class":7398},[75,18280,18198],{"class":7589},[75,18282,7738],{"class":7398},[75,18284,18285,18287,18289,18291,18293,18295,18297,18300],{"class":7383,"line":7430},[75,18286,18205],{"class":7398},[75,18288,7421],{"class":7387},[75,18290,7504],{"class":7398},[75,18292,13819],{"class":7394},[75,18294,7698],{"class":7398},[75,18296,9180],{"class":8127},[75,18298,18299],{"class":7398},");              ",[75,18301,18302],{"class":7470},"\u002F\u002F ✅ no cast needed\n",[75,18304,18305,18307,18309,18311,18314,18317],{"class":7383,"line":7436},[75,18306,18191],{"class":7398},[75,18308,8062],{"class":7394},[75,18310,7698],{"class":7398},[75,18312,18313],{"class":8127},"42",[75,18315,18316],{"class":7398},");                        ",[75,18318,18319],{"class":7470},"\u002F\u002F ❌ compile error — Java stops you immediately\n",[12,18321,18322,18325,18326,18329],{},[106,18323,18324],{},"The big win:"," errors get caught at ",[106,18327,18328],{},"compile time",", not after your program is running in production. That's the entire selling point.",[6964,18331,18333,18334,18336],{"id":18332},"_3-meeting-arraylist-properly","3. Meeting ",[6979,18335,7193],{}," properly",[12,18338,18339,18340,18343,18344,18346,18347,18350],{},"Arrays in Java have a fixed size — once you do ",[6979,18341,18342],{},"new int[10]",", you've got 10 slots and that's it. ",[6979,18345,7193],{}," solves this by being a ",[106,18348,18349],{},"resizable"," container.",[12,18352,18353],{},"Here's the class signature you'll see in the docs:",[6972,18355,18358],{"className":18356,"code":18357,"language":6977},[6975],"java.util.ArrayList\u003CE>\n",[6979,18359,18357],{"__ignoreMap":180},[12,18361,18362,18363,18365,18366,18369,18370,18372,18373,18376,18377,18379],{},"That ",[6979,18364,7620],{}," is the ",[106,18367,18368],{},"formal generic type",". The letter ",[6979,18371,7402],{}," stands for ",[7289,18374,18375],{},"Element",". When you actually use it, you replace ",[6979,18378,7402],{}," with a real type:",[6972,18381,18383],{"className":7376,"code":18382,"language":7378,"meta":180,"style":180},"ArrayList\u003CString> cities = new ArrayList\u003CString>();\n\u002F\u002F or, since Java 7, the shorter \"diamond\" form:\nArrayList\u003CString> cities = new ArrayList\u003C>();\n",[6979,18384,18385,18406,18411],{"__ignoreMap":180},[75,18386,18387,18389,18391,18394,18396,18398,18401,18403],{"class":7383,"line":7384},[75,18388,18091],{"class":7398},[75,18390,7674],{"class":7387},[75,18392,18393],{"class":7398},"> cities ",[75,18395,7421],{"class":7387},[75,18397,7424],{"class":7387},[75,18399,18400],{"class":7398}," ArrayList\u003C",[75,18402,7674],{"class":7387},[75,18404,18405],{"class":7398},">();\n",[75,18407,18408],{"class":7383,"line":181},[75,18409,18410],{"class":7470},"\u002F\u002F or, since Java 7, the shorter \"diamond\" form:\n",[75,18412,18413,18415,18417,18419,18421,18423],{"class":7383,"line":7430},[75,18414,18091],{"class":7398},[75,18416,7674],{"class":7387},[75,18418,18393],{"class":7398},[75,18420,7421],{"class":7387},[75,18422,7424],{"class":7387},[75,18424,18270],{"class":7398},[7166,18426,18428],{"id":18427},"useful-methods","Useful methods",[7085,18430,18431,18439],{},[7088,18432,18433],{},[7091,18434,18435,18437],{},[7094,18436,7922],{},[7094,18438,7099],{},[7104,18440,18441,18451,18461,18471,18481,18490,18499,18507,18516,18526,18535],{},[7091,18442,18443,18448],{},[7109,18444,18445],{},[6979,18446,18447],{},"add(E o)",[7109,18449,18450],{},"Append an element to the end",[7091,18452,18453,18458],{},[7109,18454,18455],{},[6979,18456,18457],{},"add(int index, E o)",[7109,18459,18460],{},"Insert at a specific position",[7091,18462,18463,18468],{},[7109,18464,18465],{},[6979,18466,18467],{},"get(int index)",[7109,18469,18470],{},"Retrieve element at index",[7091,18472,18473,18478],{},[7109,18474,18475],{},[6979,18476,18477],{},"set(int index, E o)",[7109,18479,18480],{},"Replace element at index",[7091,18482,18483,18487],{},[7109,18484,18485],{},[6979,18486,7985],{},[7109,18488,18489],{},"Remove first occurrence",[7091,18491,18492,18496],{},[7109,18493,18494],{},[6979,18495,10946],{},[7109,18497,18498],{},"Remove by index",[7091,18500,18501,18505],{},[7109,18502,18503],{},[6979,18504,7152],{},[7109,18506,8015],{},[7091,18508,18509,18513],{},[7109,18510,18511],{},[6979,18512,7999],{},[7109,18514,18515],{},"Does it contain this?",[7091,18517,18518,18523],{},[7109,18519,18520],{},[6979,18521,18522],{},"indexOf(Object o)",[7109,18524,18525],{},"Position of first match",[7091,18527,18528,18532],{},[7109,18529,18530],{},[6979,18531,7156],{},[7109,18533,18534],{},"true if size == 0",[7091,18536,18537,18541],{},[7109,18538,18539],{},[6979,18540,8027],{},[7109,18542,18543],{},"Wipe everything",[7166,18545,18547],{"id":18546},"array-vs-arraylist-side-by-side","Array vs ArrayList — side by side",[7085,18549,18550,18561],{},[7088,18551,18552],{},[7091,18553,18554,18556,18559],{},[7094,18555,7096],{},[7094,18557,18558],{},"Array",[7094,18560,7193],{},[7104,18562,18563,18578,18593,18608,18623,18637],{},[7091,18564,18565,18568,18573],{},[7109,18566,18567],{},"Create",[7109,18569,18570],{},[6979,18571,18572],{},"String[] a = new String[10];",[7109,18574,18575],{},[6979,18576,18577],{},"ArrayList\u003CString> list = new ArrayList\u003C>();",[7091,18579,18580,18583,18588],{},[7109,18581,18582],{},"Access",[7109,18584,18585],{},[6979,18586,18587],{},"a[index]",[7109,18589,18590],{},[6979,18591,18592],{},"list.get(index)",[7091,18594,18595,18598,18603],{},[7109,18596,18597],{},"Update",[7109,18599,18600],{},[6979,18601,18602],{},"a[index] = \"London\";",[7109,18604,18605],{},[6979,18606,18607],{},"list.set(index, \"London\");",[7091,18609,18610,18613,18618],{},[7109,18611,18612],{},"Size",[7109,18614,18615],{},[6979,18616,18617],{},"a.length",[7109,18619,18620],{},[6979,18621,18622],{},"list.size()",[7091,18624,18625,18627,18632],{},[7109,18626,16153],{},[7109,18628,18629],{},[7289,18630,18631],{},"(not possible — fixed size!)",[7109,18633,18634],{},[6979,18635,18636],{},"list.add(\"London\");",[7091,18638,18639,18641,18646],{},[7109,18640,16159],{},[7109,18642,18643],{},[7289,18644,18645],{},"(not possible)",[7109,18647,18648],{},[6979,18649,18650],{},"list.remove(index);",[6964,18652,18654],{"id":18653},"_4-the-naming-convention-youll-see-everywhere","4. The naming convention you'll see everywhere",[12,18656,18657,18658,18661],{},"Generic type parameters are written as ",[106,18659,18660],{},"single uppercase letters",". This is a convention, not a rule, but everyone follows it because otherwise you can't tell type parameters apart from real class names.",[7085,18663,18664,18674],{},[7088,18665,18666],{},[7091,18667,18668,18671],{},[7094,18669,18670],{},"Letter",[7094,18672,18673],{},"Meaning",[7104,18675,18676,18690,18705,18719,18731,18742],{},[7091,18677,18678,18682],{},[7109,18679,18680],{},[6979,18681,7402],{},[7109,18683,18684,18686,18687,7963],{},[106,18685,7402],{},"lement (used in collections — ",[6979,18688,18689],{},"List\u003CE>",[7091,18691,18692,18697],{},[7109,18693,18694],{},[6979,18695,18696],{},"K",[7109,18698,18699,18701,18702,7963],{},[106,18700,18696],{},"ey (used in maps — ",[6979,18703,18704],{},"Map\u003CK, V>",[7091,18706,18707,18712],{},[7109,18708,18709],{},[6979,18710,18711],{},"V",[7109,18713,18714,18716,18717,7963],{},[106,18715,18711],{},"alue (used in maps — ",[6979,18718,18704],{},[7091,18720,18721,18726],{},[7109,18722,18723],{},[6979,18724,18725],{},"N",[7109,18727,18728,18730],{},[106,18729,18725],{},"umber",[7091,18732,18733,18737],{},[7109,18734,18735],{},[6979,18736,16256],{},[7109,18738,18739,18741],{},[106,18740,16256],{},"ype (general placeholder)",[7091,18743,18744,18749],{},[7109,18745,18746],{},[6979,18747,18748],{},"S, U, V…",[7109,18750,18751],{},"Extra type params when you need more than one",[6964,18753,18755],{"id":18754},"_5-one-important-restriction-only-reference-types","5. One important restriction: only reference types",[6972,18757,18759],{"className":7376,"code":18758,"language":7378,"meta":180,"style":180},"ArrayList\u003CInteger> a = new ArrayList\u003C>();   \u002F\u002F ✅ ok\nArrayList\u003CDouble>  b = new ArrayList\u003C>();   \u002F\u002F ✅ ok\nArrayList\u003Cint>     c = new ArrayList\u003C>();   \u002F\u002F ❌ compile error\n",[6979,18760,18761,18779,18796],{"__ignoreMap":180},[75,18762,18763,18765,18767,18770,18772,18774,18776],{"class":7383,"line":7384},[75,18764,18091],{"class":7398},[75,18766,9143],{"class":7387},[75,18768,18769],{"class":7398},"> a ",[75,18771,7421],{"class":7387},[75,18773,7424],{"class":7387},[75,18775,18103],{"class":7398},[75,18777,18778],{"class":7470},"\u002F\u002F ✅ ok\n",[75,18780,18781,18783,18785,18788,18790,18792,18794],{"class":7383,"line":181},[75,18782,18091],{"class":7398},[75,18784,14874],{"class":7387},[75,18786,18787],{"class":7398},">  b ",[75,18789,7421],{"class":7387},[75,18791,7424],{"class":7387},[75,18793,18103],{"class":7398},[75,18795,18778],{"class":7470},[75,18797,18798,18800,18802,18805,18807,18809,18811],{"class":7383,"line":7430},[75,18799,18091],{"class":7398},[75,18801,8020],{"class":7387},[75,18803,18804],{"class":7398},">     c ",[75,18806,7421],{"class":7387},[75,18808,7424],{"class":7387},[75,18810,18103],{"class":7398},[75,18812,18813],{"class":7470},"\u002F\u002F ❌ compile error\n",[12,18815,18816,18817,18820,18821,7298,18823,7298,18826,7298,18829,18831,18832,7298,18834,7298,18836,7298,18839,18842],{},"The type parameter ",[106,18818,18819],{},"must be a reference type"," (a class). Primitives (",[6979,18822,8020],{},[6979,18824,18825],{},"double",[6979,18827,18828],{},"char",[6979,18830,7944],{},") are not allowed. Use their wrapper classes (",[6979,18833,9143],{},[6979,18835,14874],{},[6979,18837,18838],{},"Character",[6979,18840,18841],{},"Boolean",") instead.",[6964,18844,18846,18847,18850],{"id":18845},"_6-writing-your-own-generic-class","6. Writing your ",[7289,18848,18849],{},"own"," generic class",[12,18852,18853,18854,18856],{},"You're not limited to using ",[6979,18855,7193],{},". You can write your own generic class. The syntax:",[6972,18858,18860],{"className":7376,"code":18859,"language":7378,"meta":180,"style":180},"public class ClassName\u003CE> {\n    \u002F\u002F use E like a normal type inside\n}\n",[6979,18861,18862,18877,18882],{"__ignoreMap":180},[75,18863,18864,18866,18868,18871,18873,18875],{"class":7383,"line":7384},[75,18865,7388],{"class":7387},[75,18867,7391],{"class":7387},[75,18869,18870],{"class":7394}," ClassName",[75,18872,7399],{"class":7398},[75,18874,7402],{"class":7387},[75,18876,7405],{"class":7398},[75,18878,18879],{"class":7383,"line":181},[75,18880,18881],{"class":7470},"    \u002F\u002F use E like a normal type inside\n",[75,18883,18884],{"class":7383,"line":7430},[75,18885,7610],{"class":7398},[12,18887,18888,18889,18892],{},"Here's a tiny example — a ",[6979,18890,18891],{},"Box"," that can hold any single thing:",[6972,18894,18896],{"className":7376,"code":18895,"language":7378,"meta":180,"style":180},"public class GenericBox\u003CT> {\n    private T item;\n    private boolean full;\n \n    public GenericBox() { full = false; }\n \n    public void store(T a) {\n        this.item = a;\n        full = true;\n    }\n \n    public void remove() {\n        item = null;\n        full = false;\n    }\n \n    public String toString() {\n        return full ? item.toString() : \"nothing\";\n    }\n}\n",[6979,18897,18898,18913,18920,18929,18933,18949,18953,18968,18980,18991,18995,18999,19009,19020,19030,19034,19038,19048,19071,19075],{"__ignoreMap":180},[75,18899,18900,18902,18904,18907,18909,18911],{"class":7383,"line":7384},[75,18901,7388],{"class":7387},[75,18903,7391],{"class":7387},[75,18905,18906],{"class":7394}," GenericBox",[75,18908,7399],{"class":7398},[75,18910,16256],{"class":7387},[75,18912,7405],{"class":7398},[75,18914,18915,18917],{"class":7383,"line":181},[75,18916,7410],{"class":7387},[75,18918,18919],{"class":7398}," T item;\n",[75,18921,18922,18924,18926],{"class":7383,"line":7430},[75,18923,7410],{"class":7387},[75,18925,9278],{"class":7387},[75,18927,18928],{"class":7398}," full;\n",[75,18930,18931],{"class":7383,"line":7436},[75,18932,7433],{"class":7398},[75,18934,18935,18937,18939,18942,18944,18946],{"class":7383,"line":7458},[75,18936,7439],{"class":7387},[75,18938,18906],{"class":7394},[75,18940,18941],{"class":7398},"() { full ",[75,18943,7421],{"class":7387},[75,18945,15194],{"class":8127},[75,18947,18948],{"class":7398},"; }\n",[75,18950,18951],{"class":7383,"line":7474},[75,18952,7433],{"class":7398},[75,18954,18955,18957,18959,18962,18964,18966],{"class":7383,"line":7480},[75,18956,7439],{"class":7387},[75,18958,7442],{"class":7387},[75,18960,18961],{"class":7394}," store",[75,18963,16580],{"class":7398},[75,18965,2330],{"class":7451},[75,18967,7455],{"class":7398},[75,18969,18970,18972,18975,18977],{"class":7383,"line":7485},[75,18971,11407],{"class":8127},[75,18973,18974],{"class":7398},".item ",[75,18976,7421],{"class":7387},[75,18978,18979],{"class":7398}," a;\n",[75,18981,18982,18985,18987,18989],{"class":7383,"line":7498},[75,18983,18984],{"class":7398},"        full ",[75,18986,7421],{"class":7387},[75,18988,15209],{"class":8127},[75,18990,8131],{"class":7398},[75,18992,18993],{"class":7383,"line":7516},[75,18994,7477],{"class":7398},[75,18996,18997],{"class":7383,"line":7521},[75,18998,7433],{"class":7398},[75,19000,19001,19003,19005,19007],{"class":7383,"line":7526},[75,19002,7439],{"class":7387},[75,19004,7442],{"class":7387},[75,19006,16632],{"class":7394},[75,19008,7495],{"class":7398},[75,19010,19011,19014,19016,19018],{"class":7383,"line":7539},[75,19012,19013],{"class":7398},"        item ",[75,19015,7421],{"class":7387},[75,19017,8941],{"class":8127},[75,19019,8131],{"class":7398},[75,19021,19022,19024,19026,19028],{"class":7383,"line":7552},[75,19023,18984],{"class":7398},[75,19025,7421],{"class":7387},[75,19027,15194],{"class":8127},[75,19029,8131],{"class":7398},[75,19031,19032],{"class":7383,"line":7557},[75,19033,7477],{"class":7398},[75,19035,19036],{"class":7383,"line":7562},[75,19037,7433],{"class":7398},[75,19039,19040,19042,19044,19046],{"class":7383,"line":7571},[75,19041,7439],{"class":7387},[75,19043,7576],{"class":7398},[75,19045,7579],{"class":7394},[75,19047,7495],{"class":7398},[75,19049,19050,19052,19055,19057,19060,19062,19064,19066,19069],{"class":7383,"line":7584},[75,19051,7501],{"class":7387},[75,19053,19054],{"class":7398}," full ",[75,19056,8512],{"class":7387},[75,19058,19059],{"class":7398}," item.",[75,19061,7579],{"class":7394},[75,19063,8276],{"class":7398},[75,19065,7024],{"class":7387},[75,19067,19068],{"class":7589}," \"nothing\"",[75,19070,8131],{"class":7398},[75,19072,19073],{"class":7383,"line":7602},[75,19074,7477],{"class":7398},[75,19076,19077],{"class":7383,"line":7607},[75,19078,7610],{"class":7398},[12,19080,19081],{},"And in use:",[6972,19083,19085],{"className":7376,"code":19084,"language":7378,"meta":180,"style":180},"GenericBox\u003CString>  box1 = new GenericBox\u003C>();\nGenericBox\u003CInteger> box2 = new GenericBox\u003C>();\n \nbox1.store(\"Hello World\");\nbox2.store(100);\n \n\u002F\u002F box1.store(100);   \u002F\u002F ❌ compile error — box1 only takes Strings\n",[6979,19086,19087,19104,19119,19123,19138,19152,19156],{"__ignoreMap":180},[75,19088,19089,19092,19094,19097,19099,19101],{"class":7383,"line":7384},[75,19090,19091],{"class":7398},"GenericBox\u003C",[75,19093,7674],{"class":7387},[75,19095,19096],{"class":7398},">  box1 ",[75,19098,7421],{"class":7387},[75,19100,7424],{"class":7387},[75,19102,19103],{"class":7398}," GenericBox\u003C>();\n",[75,19105,19106,19108,19110,19113,19115,19117],{"class":7383,"line":181},[75,19107,19091],{"class":7398},[75,19109,9143],{"class":7387},[75,19111,19112],{"class":7398},"> box2 ",[75,19114,7421],{"class":7387},[75,19116,7424],{"class":7387},[75,19118,19103],{"class":7398},[75,19120,19121],{"class":7383,"line":7430},[75,19122,7433],{"class":7398},[75,19124,19125,19128,19131,19133,19136],{"class":7383,"line":7436},[75,19126,19127],{"class":7398},"box1.",[75,19129,19130],{"class":7394},"store",[75,19132,7698],{"class":7398},[75,19134,19135],{"class":7589},"\"Hello World\"",[75,19137,7738],{"class":7398},[75,19139,19140,19143,19145,19147,19150],{"class":7383,"line":7458},[75,19141,19142],{"class":7398},"box2.",[75,19144,19130],{"class":7394},[75,19146,7698],{"class":7398},[75,19148,19149],{"class":8127},"100",[75,19151,7738],{"class":7398},[75,19153,19154],{"class":7383,"line":7474},[75,19155,7433],{"class":7398},[75,19157,19158,19161],{"class":7383,"line":7480},[75,19159,19160],{"class":7470},"\u002F\u002F box1.store(100);",[75,19162,19163],{"class":7470},"   \u002F\u002F ❌ compile error — box1 only takes Strings\n",[12,19165,19166,19167,19170,19171,19174,19175,19177,19178,19180,19181,19184,19185,19187],{},"Compare this to the old non-generic version using ",[6979,19168,19169],{},"Comparable item"," — that one would ",[7289,19172,19173],{},"compile"," even if you mixed a ",[6979,19176,7674],{}," and an ",[6979,19179,9143],{},", then ",[106,19182,19183],{},"explode at runtime"," when you tried ",[6979,19186,8522],{},". Generics catch it earlier. Beautiful.",[7166,19189,19191],{"id":19190},"generic-interfaces-work-the-same-way","Generic interfaces work the same way",[6972,19193,19195],{"className":7376,"code":19194,"language":7378,"meta":180,"style":180},"public interface InterfaceName\u003CE> { \u002F* ... *\u002F }\n \n\u002F\u002F real-world examples you already know:\npublic interface Comparable\u003CE> { \u002F* ... *\u002F }\npublic interface Edible\u003CE>     { \u002F* ... *\u002F }\n",[6979,19196,19197,19218,19222,19227,19245],{"__ignoreMap":180},[75,19198,19199,19201,19203,19206,19208,19210,19213,19216],{"class":7383,"line":7384},[75,19200,7388],{"class":7387},[75,19202,16505],{"class":7387},[75,19204,19205],{"class":7394}," InterfaceName",[75,19207,7399],{"class":7398},[75,19209,7402],{"class":7387},[75,19211,19212],{"class":7398},"> { ",[75,19214,19215],{"class":7470},"\u002F* ... *\u002F",[75,19217,17884],{"class":7398},[75,19219,19220],{"class":7383,"line":181},[75,19221,7433],{"class":7398},[75,19223,19224],{"class":7383,"line":7430},[75,19225,19226],{"class":7470},"\u002F\u002F real-world examples you already know:\n",[75,19228,19229,19231,19233,19235,19237,19239,19241,19243],{"class":7383,"line":7436},[75,19230,7388],{"class":7387},[75,19232,16505],{"class":7387},[75,19234,8542],{"class":7394},[75,19236,7399],{"class":7398},[75,19238,7402],{"class":7387},[75,19240,19212],{"class":7398},[75,19242,19215],{"class":7470},[75,19244,17884],{"class":7398},[75,19246,19247,19249,19251,19254,19256,19258,19261,19263],{"class":7383,"line":7458},[75,19248,7388],{"class":7387},[75,19250,16505],{"class":7387},[75,19252,19253],{"class":7394}," Edible",[75,19255,7399],{"class":7398},[75,19257,7402],{"class":7387},[75,19259,19260],{"class":7398},">     { ",[75,19262,19215],{"class":7470},[75,19264,17884],{"class":7398},[12,19266,19267],{},"And you can declare a class as a subtype of a generic interface:",[6972,19269,19271],{"className":7376,"code":19270,"language":7378,"meta":180,"style":180},"public class String implements Comparable\u003CString> { \u002F* ... *\u002F }\n",[6979,19272,19273],{"__ignoreMap":180},[75,19274,19275,19277,19279,19282,19284,19286,19288,19290,19292,19294],{"class":7383,"line":7384},[75,19276,7388],{"class":7387},[75,19278,7391],{"class":7387},[75,19280,19281],{"class":7394}," String",[75,19283,8539],{"class":7387},[75,19285,8542],{"class":7394},[75,19287,7399],{"class":7398},[75,19289,7674],{"class":7387},[75,19291,19212],{"class":7398},[75,19293,19215],{"class":7470},[75,19295,17884],{"class":7398},[6964,19297,19299],{"id":19298},"_7-generic-methods","7. Generic methods",[12,19301,19302,19303,19306],{},"You can make a ",[7289,19304,19305],{},"single method"," generic, even if its class isn't generic.",[12,19308,19309],{},[106,19310,19311],{},"Syntax:",[6972,19313,19315],{"className":7376,"code":19314,"language":7378,"meta":180,"style":180},"public static \u003CE> returnType methodName(E parameter) { ... }\n",[6979,19316,19317],{"__ignoreMap":180},[75,19318,19319,19321,19323,19326,19328,19330,19333,19336],{"class":7383,"line":7384},[75,19320,7388],{"class":7387},[75,19322,8155],{"class":7387},[75,19324,19325],{"class":7387}," \u003C",[75,19327,7402],{"class":7398},[75,19329,8279],{"class":7387},[75,19331,19332],{"class":7398}," returnType ",[75,19334,19335],{"class":7394},"methodName",[75,19337,19338],{"class":7398},"(E parameter) { ... }\n",[12,19340,9455,19341,19343,19344,19347],{},[6979,19342,7620],{}," sits ",[106,19345,19346],{},"right before the return type",". Examples:",[6972,19349,19351],{"className":7376,"code":19350,"language":7378,"meta":180,"style":180},"public static \u003CE> void print(E[] list) {\n    for (int i = 0; i \u003C list.length; i++)\n        System.out.print(list[i] + \" \");\n    System.out.println();\n}\n \npublic \u003CE> boolean isFilled(E filled) { \u002F* ... *\u002F }\n",[6979,19352,19353,19377,19402,19417,19425,19429,19433],{"__ignoreMap":180},[75,19354,19355,19357,19359,19361,19363,19365,19367,19370,19372,19374],{"class":7383,"line":7384},[75,19356,7388],{"class":7387},[75,19358,8155],{"class":7387},[75,19360,19325],{"class":7387},[75,19362,7402],{"class":7398},[75,19364,8279],{"class":7387},[75,19366,7442],{"class":7387},[75,19368,19369],{"class":7394}," print",[75,19371,7698],{"class":7398},[75,19373,7402],{"class":7387},[75,19375,19376],{"class":7398},"[] list) {\n",[75,19378,19379,19381,19383,19385,19387,19389,19391,19393,19395,19398,19400],{"class":7383,"line":181},[75,19380,10837],{"class":7387},[75,19382,10476],{"class":7398},[75,19384,8020],{"class":7387},[75,19386,10481],{"class":7398},[75,19388,7421],{"class":7387},[75,19390,8282],{"class":8127},[75,19392,10489],{"class":7398},[75,19394,7399],{"class":7387},[75,19396,19397],{"class":7398}," list.length; i",[75,19399,10218],{"class":7387},[75,19401,11077],{"class":7398},[75,19403,19404,19406,19408,19411,19413,19415],{"class":7383,"line":7430},[75,19405,12229],{"class":7398},[75,19407,8292],{"class":7394},[75,19409,19410],{"class":7398},"(list[i] ",[75,19412,8302],{"class":7387},[75,19414,8305],{"class":7589},[75,19416,7738],{"class":7398},[75,19418,19419,19421,19423],{"class":7383,"line":7436},[75,19420,8254],{"class":7398},[75,19422,7767],{"class":7394},[75,19424,7549],{"class":7398},[75,19426,19427],{"class":7383,"line":7458},[75,19428,7610],{"class":7398},[75,19430,19431],{"class":7383,"line":7474},[75,19432,7433],{"class":7398},[75,19434,19435,19437,19439,19441,19443,19445,19448,19451,19453],{"class":7383,"line":7480},[75,19436,7388],{"class":7387},[75,19438,19325],{"class":7387},[75,19440,7402],{"class":7398},[75,19442,8279],{"class":7387},[75,19444,9278],{"class":7387},[75,19446,19447],{"class":7394}," isFilled",[75,19449,19450],{"class":7398},"(E filled) { ",[75,19452,19215],{"class":7470},[75,19454,17884],{"class":7398},[12,19456,19457],{},"Calling it:",[6972,19459,19461],{"className":7376,"code":19460,"language":7378,"meta":180,"style":180},"Integer[] integers = {1, 2, 3, 4, 5};\nString[]  strings  = {\"London\", \"Paris\", \"New York\", \"Austin\"};\n \nGenericMethodDemo.\u003CInteger>print(integers);   \u002F\u002F explicit type\nGenericMethodDemo.\u003CString>print(strings);     \u002F\u002F explicit type\nGenericMethodDemo.print(integers);            \u002F\u002F or just let Java figure it out\n",[6979,19462,19463,19496,19527,19531,19550,19567],{"__ignoreMap":180},[75,19464,19465,19467,19470,19472,19475,19477,19479,19481,19483,19485,19487,19489,19491,19493],{"class":7383,"line":7384},[75,19466,9143],{"class":7387},[75,19468,19469],{"class":7398},"[] integers ",[75,19471,7421],{"class":7387},[75,19473,19474],{"class":7398}," {",[75,19476,8855],{"class":8127},[75,19478,7298],{"class":7398},[75,19480,8879],{"class":8127},[75,19482,7298],{"class":7398},[75,19484,8830],{"class":8127},[75,19486,7298],{"class":7398},[75,19488,8348],{"class":8127},[75,19490,7298],{"class":7398},[75,19492,15531],{"class":8127},[75,19494,19495],{"class":7398},"};\n",[75,19497,19498,19500,19503,19505,19507,19510,19512,19515,19517,19520,19522,19525],{"class":7383,"line":181},[75,19499,7674],{"class":7387},[75,19501,19502],{"class":7398},"[]  strings  ",[75,19504,7421],{"class":7387},[75,19506,19474],{"class":7398},[75,19508,19509],{"class":7589},"\"London\"",[75,19511,7298],{"class":7398},[75,19513,19514],{"class":7589},"\"Paris\"",[75,19516,7298],{"class":7398},[75,19518,19519],{"class":7589},"\"New York\"",[75,19521,7298],{"class":7398},[75,19523,19524],{"class":7589},"\"Austin\"",[75,19526,19495],{"class":7398},[75,19528,19529],{"class":7383,"line":7430},[75,19530,7433],{"class":7398},[75,19532,19533,19536,19538,19540,19542,19544,19547],{"class":7383,"line":7436},[75,19534,19535],{"class":7398},"GenericMethodDemo.",[75,19537,7399],{"class":7387},[75,19539,9143],{"class":7398},[75,19541,8279],{"class":7387},[75,19543,8292],{"class":7394},[75,19545,19546],{"class":7398},"(integers);   ",[75,19548,19549],{"class":7470},"\u002F\u002F explicit type\n",[75,19551,19552,19554,19556,19558,19560,19562,19565],{"class":7383,"line":7458},[75,19553,19535],{"class":7398},[75,19555,7399],{"class":7387},[75,19557,7674],{"class":7398},[75,19559,8279],{"class":7387},[75,19561,8292],{"class":7394},[75,19563,19564],{"class":7398},"(strings);     ",[75,19566,19549],{"class":7470},[75,19568,19569,19571,19573,19576],{"class":7383,"line":7474},[75,19570,19535],{"class":7398},[75,19572,8292],{"class":7394},[75,19574,19575],{"class":7398},"(integers);            ",[75,19577,19578],{"class":7470},"\u002F\u002F or just let Java figure it out\n",[12,19580,19581,19582,19585],{},"The compiler is usually smart enough to infer the type from the argument, so the explicit ",[6979,19583,19584],{},"\u003CInteger>"," is optional in most cases.",[6964,19587,19589,19590,19593],{"id":19588},"_8-bounded-generics-i-want-any-type-but-only-certain-ones","8. Bounded generics — \"I want ",[7289,19591,19592],{},"any"," type, but only certain ones\"",[12,19595,19596,19597,19599,19600,19603],{},"Sometimes you want a generic that accepts ",[7289,19598,19592],{}," type — ",[7289,19601,19602],{},"as long as it has certain capabilities",". Example: a method that only makes sense for numbers.",[12,19605,19606,19607,19610],{},"You use the ",[6979,19608,19609],{},"extends"," keyword:",[6972,19612,19614],{"className":7376,"code":19613,"language":7378,"meta":180,"style":180},"public \u003CU extends Number> void inspect(U u) { \u002F* ... *\u002F }\n",[6979,19615,19616],{"__ignoreMap":180},[75,19617,19618,19620,19622,19625,19627,19629,19632,19635,19637],{"class":7383,"line":7384},[75,19619,7388],{"class":7387},[75,19621,19325],{"class":7387},[75,19623,19624],{"class":7398},"U extends Number",[75,19626,8279],{"class":7387},[75,19628,7442],{"class":7387},[75,19630,19631],{"class":7394}," inspect",[75,19633,19634],{"class":7398},"(U u) { ",[75,19636,19215],{"class":7470},[75,19638,17884],{"class":7398},[12,19640,19641,19642,19645,19646,7206,19649,19654,19655,7298,19657,7298,19659,19662,19663,19665],{},"This says: \"",[6979,19643,19644],{},"U"," can be any type, ",[106,19647,19648],{},"but it has to be a subtype of",[106,19650,19651],{},[6979,19652,19653],{},"Number",".\" So ",[6979,19656,9143],{},[6979,19658,14874],{},[6979,19660,19661],{},"Float"," all work — ",[6979,19664,7674],{}," does not.",[6972,19667,19669],{"className":7376,"code":19668,"language":7378,"meta":180,"style":180},"public class BoundedGeneric2\u003CT extends Number> {\n    T data;\n    public BoundedGeneric2(T t) { data = t; }\n    void display() {\n        System.out.println(\"Value is : \" + data);\n        System.out.println(\" and type is \" + data.getClass().getName());\n    }\n}\n \n\u002F\u002F Usage:\nBoundedGeneric2\u003CInteger> b1 = new BoundedGeneric2\u003C>(3);      \u002F\u002F ✅\nBoundedGeneric2\u003CDouble>  b2 = new BoundedGeneric2\u003C>(3.14);   \u002F\u002F ✅\nBoundedGeneric2\u003CString>  b3 = new BoundedGeneric2\u003C>(\"Hi\");   \u002F\u002F ❌ compile error\n",[6979,19670,19671,19692,19697,19716,19726,19742,19767,19771,19775,19779,19784,19808,19830],{"__ignoreMap":180},[75,19672,19673,19675,19677,19680,19682,19684,19687,19690],{"class":7383,"line":7384},[75,19674,7388],{"class":7387},[75,19676,7391],{"class":7387},[75,19678,19679],{"class":7394}," BoundedGeneric2",[75,19681,7399],{"class":7398},[75,19683,16256],{"class":7387},[75,19685,19686],{"class":7387}," extends",[75,19688,19689],{"class":7387}," Number",[75,19691,7405],{"class":7398},[75,19693,19694],{"class":7383,"line":181},[75,19695,19696],{"class":7398},"    T data;\n",[75,19698,19699,19701,19703,19705,19708,19711,19713],{"class":7383,"line":7430},[75,19700,7439],{"class":7387},[75,19702,19679],{"class":7394},[75,19704,16580],{"class":7398},[75,19706,19707],{"class":7451},"t",[75,19709,19710],{"class":7398},") { data ",[75,19712,7421],{"class":7387},[75,19714,19715],{"class":7398}," t; }\n",[75,19717,19718,19721,19724],{"class":7383,"line":7436},[75,19719,19720],{"class":7387},"    void",[75,19722,19723],{"class":7394}," display",[75,19725,7495],{"class":7398},[75,19727,19728,19730,19732,19734,19737,19739],{"class":7383,"line":7458},[75,19729,12229],{"class":7398},[75,19731,7767],{"class":7394},[75,19733,7698],{"class":7398},[75,19735,19736],{"class":7589},"\"Value is : \"",[75,19738,7593],{"class":7387},[75,19740,19741],{"class":7398}," data);\n",[75,19743,19744,19746,19748,19750,19753,19755,19758,19761,19763,19765],{"class":7383,"line":7474},[75,19745,12229],{"class":7398},[75,19747,7767],{"class":7394},[75,19749,7698],{"class":7398},[75,19751,19752],{"class":7589},"\" and type is \"",[75,19754,7593],{"class":7387},[75,19756,19757],{"class":7398}," data.",[75,19759,19760],{"class":7394},"getClass",[75,19762,8699],{"class":7398},[75,19764,8649],{"class":7394},[75,19766,8357],{"class":7398},[75,19768,19769],{"class":7383,"line":7480},[75,19770,7477],{"class":7398},[75,19772,19773],{"class":7383,"line":7485},[75,19774,7610],{"class":7398},[75,19776,19777],{"class":7383,"line":7498},[75,19778,7433],{"class":7398},[75,19780,19781],{"class":7383,"line":7516},[75,19782,19783],{"class":7470},"\u002F\u002F Usage:\n",[75,19785,19786,19789,19791,19794,19796,19798,19801,19803,19805],{"class":7383,"line":7521},[75,19787,19788],{"class":7398},"BoundedGeneric2\u003C",[75,19790,9143],{"class":7387},[75,19792,19793],{"class":7398},"> b1 ",[75,19795,7421],{"class":7387},[75,19797,7424],{"class":7387},[75,19799,19800],{"class":7398}," BoundedGeneric2\u003C>(",[75,19802,8830],{"class":8127},[75,19804,7752],{"class":7398},[75,19806,19807],{"class":7470},"\u002F\u002F ✅\n",[75,19809,19810,19812,19814,19817,19819,19821,19823,19826,19828],{"class":7383,"line":7526},[75,19811,19788],{"class":7398},[75,19813,14874],{"class":7387},[75,19815,19816],{"class":7398},">  b2 ",[75,19818,7421],{"class":7387},[75,19820,7424],{"class":7387},[75,19822,19800],{"class":7398},[75,19824,19825],{"class":8127},"3.14",[75,19827,18053],{"class":7398},[75,19829,19807],{"class":7470},[75,19831,19832,19834,19836,19839,19841,19843,19845,19848,19850],{"class":7383,"line":7539},[75,19833,19788],{"class":7398},[75,19835,7674],{"class":7387},[75,19837,19838],{"class":7398},">  b3 ",[75,19840,7421],{"class":7387},[75,19842,7424],{"class":7387},[75,19844,19800],{"class":7398},[75,19846,19847],{"class":7589},"\"Hi\"",[75,19849,18053],{"class":7398},[75,19851,18813],{"class":7470},[7166,19853,19855],{"id":19854},"trivia","Trivia",[12,19857,15781,19858,19861,19862,19864,19865,19868,19869,6959],{},[106,19859,19860],{},"unbounded"," generic ",[6979,19863,7620],{}," is secretly the same as ",[6979,19866,19867],{},"\u003CE extends Object>"," — because every class in Java extends ",[6979,19870,19871],{},"Object",[6964,19873,19875],{"id":19874},"_9-raw-types-and-why-theyre-a-trap","9. Raw types (and why they're a trap)",[12,19877,9407,19878,19881],{},[106,19879,19880],{},"raw type"," is a generic class used without specifying a type parameter:",[6972,19883,19885],{"className":7376,"code":19884,"language":7378,"meta":180,"style":180},"ArrayList list = new ArrayList();   \u002F\u002F raw — no \u003C>\n",[6979,19886,19887],{"__ignoreMap":180},[75,19888,19889,19891,19893,19895,19897,19899],{"class":7383,"line":7384},[75,19890,18173],{"class":7398},[75,19892,7421],{"class":7387},[75,19894,7424],{"class":7387},[75,19896,18180],{"class":7394},[75,19898,9268],{"class":7398},[75,19900,19901],{"class":7470},"\u002F\u002F raw — no \u003C>\n",[12,19903,16745,19904,19907,19908,19911,19912,19915],{},[7289,19905,19906],{},"roughly"," equivalent to ",[6979,19909,19910],{},"ArrayList\u003CObject>",". Java still allows raw types so old pre-2004 code keeps working (this is called ",[106,19913,19914],{},"backward compatibility","). But you should not write new code this way.",[7166,19917,19919],{"id":19918},"why-raw-types-are-unsafe","Why raw types are unsafe",[12,19921,19922],{},"Look at this innocent-looking method:",[6972,19924,19926],{"className":7376,"code":19925,"language":7378,"meta":180,"style":180},"public static Comparable max(Comparable o1, Comparable o2) {\n    if (o1.compareTo(o2) > 0) return o1;\n    else return o2;\n}\n",[6979,19927,19928,19943,19966,19977],{"__ignoreMap":180},[75,19929,19930,19932,19934,19937,19940],{"class":7383,"line":7384},[75,19931,7388],{"class":7387},[75,19933,8155],{"class":7387},[75,19935,19936],{"class":7398}," Comparable ",[75,19938,19939],{"class":7394},"max",[75,19941,19942],{"class":7398},"(Comparable o1, Comparable o2) {\n",[75,19944,19945,19947,19950,19952,19955,19957,19959,19961,19963],{"class":7383,"line":181},[75,19946,10225],{"class":7387},[75,19948,19949],{"class":7398}," (o1.",[75,19951,8522],{"class":7394},[75,19953,19954],{"class":7398},"(o2) ",[75,19956,8279],{"class":7387},[75,19958,8282],{"class":8127},[75,19960,10629],{"class":7398},[75,19962,8636],{"class":7387},[75,19964,19965],{"class":7398}," o1;\n",[75,19967,19968,19971,19974],{"class":7383,"line":7430},[75,19969,19970],{"class":7387},"    else",[75,19972,19973],{"class":7387}," return",[75,19975,19976],{"class":7398}," o2;\n",[75,19978,19979],{"class":7383,"line":7436},[75,19980,7610],{"class":7398},[12,19982,19983],{},"It compiles. It looks fine. Then:",[6972,19985,19987],{"className":7376,"code":19986,"language":7378,"meta":180,"style":180},"Max.max(\"Welcome\", 23);   \u002F\u002F 💥 ClassCastException at runtime\n",[6979,19988,19989],{"__ignoreMap":180},[75,19990,19991,19994,19996,19998,20001,20003,20006,20008],{"class":7383,"line":7384},[75,19992,19993],{"class":7398},"Max.",[75,19995,19939],{"class":7394},[75,19997,7698],{"class":7398},[75,19999,20000],{"class":7589},"\"Welcome\"",[75,20002,7298],{"class":7398},[75,20004,20005],{"class":8127},"23",[75,20007,18053],{"class":7398},[75,20009,20010],{"class":7470},"\u002F\u002F 💥 ClassCastException at runtime\n",[12,20012,20013,20014,20017],{},"The compiler can't catch the bug because ",[6979,20015,20016],{},"Comparable"," is raw. Fix it with generics:",[6972,20019,20021],{"className":7376,"code":20020,"language":7378,"meta":180,"style":180},"public static \u003CE extends Comparable\u003CE>> E max(E o1, E o2) {\n    if (o1.compareTo(o2) > 0) return o1;\n    else return o2;\n}\n \nMax.max(\"Welcome\", 23);   \u002F\u002F ❌ now caught at compile time\n",[6979,20022,20023,20048,20068,20076,20080,20084],{"__ignoreMap":180},[75,20024,20025,20027,20029,20031,20034,20036,20038,20041,20043,20045],{"class":7383,"line":7384},[75,20026,7388],{"class":7387},[75,20028,8155],{"class":7387},[75,20030,19325],{"class":7387},[75,20032,20033],{"class":7398},"E extends Comparable",[75,20035,7399],{"class":7387},[75,20037,7402],{"class":7398},[75,20039,20040],{"class":7387},">>",[75,20042,7490],{"class":7398},[75,20044,19939],{"class":7394},[75,20046,20047],{"class":7398},"(E o1, E o2) {\n",[75,20049,20050,20052,20054,20056,20058,20060,20062,20064,20066],{"class":7383,"line":181},[75,20051,10225],{"class":7387},[75,20053,19949],{"class":7398},[75,20055,8522],{"class":7394},[75,20057,19954],{"class":7398},[75,20059,8279],{"class":7387},[75,20061,8282],{"class":8127},[75,20063,10629],{"class":7398},[75,20065,8636],{"class":7387},[75,20067,19965],{"class":7398},[75,20069,20070,20072,20074],{"class":7383,"line":7430},[75,20071,19970],{"class":7387},[75,20073,19973],{"class":7387},[75,20075,19976],{"class":7398},[75,20077,20078],{"class":7383,"line":7436},[75,20079,7610],{"class":7398},[75,20081,20082],{"class":7383,"line":7458},[75,20083,7433],{"class":7398},[75,20085,20086,20088,20090,20092,20094,20096,20098,20100],{"class":7383,"line":7474},[75,20087,19993],{"class":7398},[75,20089,19939],{"class":7394},[75,20091,7698],{"class":7398},[75,20093,20000],{"class":7589},[75,20095,7298],{"class":7398},[75,20097,20005],{"class":8127},[75,20099,18053],{"class":7398},[75,20101,20102],{"class":7470},"\u002F\u002F ❌ now caught at compile time\n",[6964,20104,20106,20107,7298,20109,7298,20112],{"id":20105},"_10-wildcards-extends-t-super-t","10. Wildcards: ",[6979,20108,8512],{},[6979,20110,20111],{},"? extends T",[6979,20113,20114],{},"? super T",[12,20116,20117],{},"This part trips people up, so go slowly.",[7166,20119,20121],{"id":20120},"why-wildcards-exist","Why wildcards exist",[12,20123,20124,20125,20128,20129,20132],{},"You might think: \"Integer is a subtype of Number, so ",[6979,20126,20127],{},"ArrayList\u003CInteger>"," should be a subtype of ",[6979,20130,20131],{},"ArrayList\u003CNumber>",", right?\"",[12,20134,20135,20138,20139,7042,20141,20143,20144,20147],{},[106,20136,20137],{},"Wrong."," In Java, ",[6979,20140,20127],{},[6979,20142,20131],{}," are ",[7289,20145,20146],{},"unrelated"," types. This code fails to compile:",[6972,20149,20151],{"className":7376,"code":20150,"language":7378,"meta":180,"style":180},"public static void display(ArrayList\u003CNumber> list) { \u002F* ... *\u002F }\n \nArrayList\u003CInteger> list1 = new ArrayList\u003C>();\ndisplay(list1);   \u002F\u002F ❌ compile error\n",[6979,20152,20153,20179,20183,20198],{"__ignoreMap":180},[75,20154,20155,20157,20159,20161,20163,20166,20168,20170,20172,20175,20177],{"class":7383,"line":7384},[75,20156,7388],{"class":7387},[75,20158,8155],{"class":7387},[75,20160,7442],{"class":7387},[75,20162,19723],{"class":7394},[75,20164,20165],{"class":7398},"(ArrayList",[75,20167,7399],{"class":7387},[75,20169,19653],{"class":7398},[75,20171,8279],{"class":7387},[75,20173,20174],{"class":7398}," list) { ",[75,20176,19215],{"class":7470},[75,20178,17884],{"class":7398},[75,20180,20181],{"class":7383,"line":181},[75,20182,7433],{"class":7398},[75,20184,20185,20187,20189,20192,20194,20196],{"class":7383,"line":7430},[75,20186,18091],{"class":7398},[75,20188,9143],{"class":7387},[75,20190,20191],{"class":7398},"> list1 ",[75,20193,7421],{"class":7387},[75,20195,7424],{"class":7387},[75,20197,18270],{"class":7398},[75,20199,20200,20203,20206],{"class":7383,"line":7436},[75,20201,20202],{"class":7394},"display",[75,20204,20205],{"class":7398},"(list1);   ",[75,20207,18813],{"class":7470},[12,20209,20210,20211,20214,20215,20217],{},"The fix: ",[106,20212,20213],{},"wildcards",". Use ",[6979,20216,8512],{}," to say \"I don't care what specific type — just a list of something.\"",[7166,20219,20221],{"id":20220},"the-three-forms","The three forms",[7085,20223,20224,20236],{},[7088,20225,20226],{},[7091,20227,20228,20231,20234],{},[7094,20229,20230],{},"Form",[7094,20232,20233],{},"Name",[7094,20235,18673],{},[7104,20237,20238,20251,20268],{},[7091,20239,20240,20245,20248],{},[7109,20241,20242],{},[6979,20243,20244],{},"\u003C?>",[7109,20246,20247],{},"Unbounded wildcard",[7109,20249,20250],{},"Any type at all",[7091,20252,20253,20258,20261],{},[7109,20254,20255],{},[6979,20256,20257],{},"\u003C? extends T>",[7109,20259,20260],{},"Upper-bounded wildcard",[7109,20262,20263,20264,20267],{},"Some unknown ",[106,20265,20266],{},"subtype"," of T",[7091,20269,20270,20275,20278],{},[7109,20271,20272],{},[6979,20273,20274],{},"\u003C? super T>",[7109,20276,20277],{},"Lower-bounded wildcard",[7109,20279,20263,20280,20267],{},[106,20281,20282],{},"supertype",[12,20284,20285,20287,20288,6959],{},[6979,20286,20244],{}," is exactly the same as ",[6979,20289,20290],{},"\u003C? extends Object>",[7166,20292,20294],{"id":20293},"fixing-the-earlier-example","Fixing the earlier example",[6972,20296,20298],{"className":7376,"code":20297,"language":7378,"meta":180,"style":180},"public static void display(ArrayList\u003C?> list) { \u002F* ... *\u002F }\n\u002F\u002F now display(list1) works for ArrayList\u003CInteger>, ArrayList\u003CDouble>, etc.\n",[6979,20299,20300,20320],{"__ignoreMap":180},[75,20301,20302,20304,20306,20308,20310,20312,20314,20316,20318],{"class":7383,"line":7384},[75,20303,7388],{"class":7387},[75,20305,8155],{"class":7387},[75,20307,7442],{"class":7387},[75,20309,19723],{"class":7394},[75,20311,20165],{"class":7398},[75,20313,20244],{"class":7387},[75,20315,20174],{"class":7398},[75,20317,19215],{"class":7470},[75,20319,17884],{"class":7398},[75,20321,20322],{"class":7383,"line":181},[75,20323,20324],{"class":7470},"\u002F\u002F now display(list1) works for ArrayList\u003CInteger>, ArrayList\u003CDouble>, etc.\n",[6964,20326,20328],{"id":20327},"_11-type-erasure-the-magic-trick-behind-generics","11. Type erasure (the magic trick behind generics)",[12,20330,20331,20332],{},"Here's the surprising part: ",[106,20333,20334],{},"at runtime, generics don't exist.",[12,20336,20337,20338,20341,20342,6959],{},"When the Java compiler is done with your code, it ",[106,20339,20340],{},"erases"," all the type parameters. The compiled bytecode looks like the old, pre-generics code. This is called ",[106,20343,20344],{},"type erasure",[12,20346,20347],{},"What happens:",[23,20349,20350,20357,20365],{},[26,20351,20352,20354,20355],{},[6979,20353,7620],{}," → replaced with ",[6979,20356,19871],{},[26,20358,20359,20354,20362,20364],{},[6979,20360,20361],{},"\u003CE extends Number>",[6979,20363,19653],{}," (the bounded type)",[26,20366,20367],{},"Casts are auto-inserted where needed\nSo this:",[6972,20369,20371],{"className":7376,"code":20370,"language":7378,"meta":180,"style":180},"ArrayList\u003CString> list = new ArrayList\u003C>();\nlist.add(\"Oklahoma\");\nString state = list.get(0);\n",[6979,20372,20373,20387,20399],{"__ignoreMap":180},[75,20374,20375,20377,20379,20381,20383,20385],{"class":7383,"line":7384},[75,20376,18091],{"class":7398},[75,20378,7674],{"class":7387},[75,20380,7418],{"class":7398},[75,20382,7421],{"class":7387},[75,20384,7424],{"class":7387},[75,20386,18270],{"class":7398},[75,20388,20389,20391,20393,20395,20397],{"class":7383,"line":181},[75,20390,18191],{"class":7398},[75,20392,8062],{"class":7394},[75,20394,7698],{"class":7398},[75,20396,8206],{"class":7589},[75,20398,7738],{"class":7398},[75,20400,20401,20404,20406,20408,20410,20412,20414],{"class":7383,"line":7430},[75,20402,20403],{"class":7398},"String state ",[75,20405,7421],{"class":7387},[75,20407,7504],{"class":7398},[75,20409,13819],{"class":7394},[75,20411,7698],{"class":7398},[75,20413,9180],{"class":8127},[75,20415,7738],{"class":7398},[12,20417,20418],{},"...effectively becomes this at runtime:",[6972,20420,20422],{"className":7376,"code":20421,"language":7378,"meta":180,"style":180},"ArrayList list = new ArrayList();\nlist.add(\"Oklahoma\");\nString state = (String) list.get(0);\n",[6979,20423,20424,20436,20448],{"__ignoreMap":180},[75,20425,20426,20428,20430,20432,20434],{"class":7383,"line":7384},[75,20427,18173],{"class":7398},[75,20429,7421],{"class":7387},[75,20431,7424],{"class":7387},[75,20433,18180],{"class":7394},[75,20435,7549],{"class":7398},[75,20437,20438,20440,20442,20444,20446],{"class":7383,"line":181},[75,20439,18191],{"class":7398},[75,20441,8062],{"class":7394},[75,20443,7698],{"class":7398},[75,20445,8206],{"class":7589},[75,20447,7738],{"class":7398},[75,20449,20450,20452,20454,20456,20458,20460,20462],{"class":7383,"line":7430},[75,20451,20403],{"class":7398},[75,20453,7421],{"class":7387},[75,20455,18210],{"class":7398},[75,20457,13819],{"class":7394},[75,20459,7698],{"class":7398},[75,20461,9180],{"class":8127},[75,20463,7738],{"class":7398},[12,20465,20466,20467,20470],{},"Generics are basically ",[106,20468,20469],{},"a compile-time-only feature",". They protect you while you're writing code, then quietly disappear.",[7166,20472,20474],{"id":20473},"a-weird-consequence","A weird consequence",[6972,20476,20478],{"className":7376,"code":20477,"language":7378,"meta":180,"style":180},"ArrayList\u003CString>  list1 = new ArrayList\u003C>();\nArrayList\u003CInteger> list2 = new ArrayList\u003C>();\n \nSystem.out.println(list1 instanceof ArrayList);   \u002F\u002F true\nSystem.out.println(list2 instanceof ArrayList);   \u002F\u002F true\n",[6979,20479,20480,20495,20510,20514,20532],{"__ignoreMap":180},[75,20481,20482,20484,20486,20489,20491,20493],{"class":7383,"line":7384},[75,20483,18091],{"class":7398},[75,20485,7674],{"class":7387},[75,20487,20488],{"class":7398},">  list1 ",[75,20490,7421],{"class":7387},[75,20492,7424],{"class":7387},[75,20494,18270],{"class":7398},[75,20496,20497,20499,20501,20504,20506,20508],{"class":7383,"line":181},[75,20498,18091],{"class":7398},[75,20500,9143],{"class":7387},[75,20502,20503],{"class":7398},"> list2 ",[75,20505,7421],{"class":7387},[75,20507,7424],{"class":7387},[75,20509,18270],{"class":7398},[75,20511,20512],{"class":7383,"line":7430},[75,20513,7433],{"class":7398},[75,20515,20516,20518,20520,20523,20526,20529],{"class":7383,"line":7436},[75,20517,7764],{"class":7398},[75,20519,7767],{"class":7394},[75,20521,20522],{"class":7398},"(list1 ",[75,20524,20525],{"class":7387},"instanceof",[75,20527,20528],{"class":7398}," ArrayList);   ",[75,20530,20531],{"class":7470},"\u002F\u002F true\n",[75,20533,20534,20536,20538,20541,20543,20545],{"class":7383,"line":7458},[75,20535,7764],{"class":7398},[75,20537,7767],{"class":7394},[75,20539,20540],{"class":7398},"(list2 ",[75,20542,20525],{"class":7387},[75,20544,20528],{"class":7398},[75,20546,20531],{"class":7470},[12,20548,20549,20550,7042,20553,20555,20556,7206,20559,7206,20563,16253,20566,7042,20569,20571],{},"Even though ",[6979,20551,20552],{},"ArrayList\u003CString>",[6979,20554,20127],{}," look like two different types, ",[106,20557,20558],{},"only one",[106,20560,20561],{},[6979,20562,7193],{},[106,20564,20565],{},"class is actually loaded into the JVM",[6979,20567,20568],{},"\u003CString>",[6979,20570,19584],{}," parts vanish.",[6964,20573,20575],{"id":20574},"_12-the-four-restrictions-on-generics","12. The four restrictions on generics",[12,20577,20578,20579,20582],{},"Because of type erasure, there are things you simply ",[7289,20580,20581],{},"cannot"," do. Memorize these — they show up on every Java generics exam.",[7166,20584,20586,20587],{"id":20585},"restriction-1-cant-create-an-instance-of-e","❌ Restriction 1: Can't create an instance of ",[6979,20588,7402],{},[6972,20590,20592],{"className":7376,"code":20591,"language":7378,"meta":180,"style":180},"E object = new E();   \u002F\u002F compile error\n",[6979,20593,20594],{"__ignoreMap":180},[75,20595,20596,20599,20601,20603,20606,20608],{"class":7383,"line":7384},[75,20597,20598],{"class":7398},"E object ",[75,20600,7421],{"class":7387},[75,20602,7424],{"class":7387},[75,20604,20605],{"class":7394}," E",[75,20607,9268],{"class":7398},[75,20609,20610],{"class":7470},"\u002F\u002F compile error\n",[12,20612,20613,20614,20617,20618,20620],{},"Why? ",[6979,20615,20616],{},"new E()"," would run at runtime, but by then ",[6979,20619,7402],{}," has been erased. The JVM has no idea what type to actually instantiate.",[7166,20622,20624],{"id":20623},"restriction-2-cant-create-a-generic-array","❌ Restriction 2: Can't create a generic array",[6972,20626,20628],{"className":7376,"code":20627,"language":7378,"meta":180,"style":180},"E[] elements = new E[capacity];   \u002F\u002F compile error\n",[6979,20629,20630],{"__ignoreMap":180},[75,20631,20632,20634,20637,20639,20641,20643,20646],{"class":7383,"line":7384},[75,20633,7402],{"class":7387},[75,20635,20636],{"class":7398},"[] elements ",[75,20638,7421],{"class":7387},[75,20640,7424],{"class":7387},[75,20642,20605],{"class":7387},[75,20644,20645],{"class":7398},"[capacity];   ",[75,20647,20610],{"class":7470},[12,20649,20650],{},"Workaround:",[6972,20652,20654],{"className":7376,"code":20653,"language":7378,"meta":180,"style":180},"E[] elements = (E[]) new Object[capacity];   \u002F\u002F ⚠️ unchecked warning\n",[6979,20655,20656],{"__ignoreMap":180},[75,20657,20658,20660,20662,20664,20666,20668,20671,20673,20676,20678],{"class":7383,"line":7384},[75,20659,7402],{"class":7387},[75,20661,20636],{"class":7398},[75,20663,7421],{"class":7387},[75,20665,10476],{"class":7398},[75,20667,7402],{"class":7387},[75,20669,20670],{"class":7398},"[]) ",[75,20672,8823],{"class":7387},[75,20674,20675],{"class":7387}," Object",[75,20677,20645],{"class":7398},[75,20679,20680],{"class":7470},"\u002F\u002F ⚠️ unchecked warning\n",[12,20682,20683],{},"This compiles but the compiler can't promise the cast will always succeed at runtime. Use carefully.",[7166,20685,20687],{"id":20686},"restriction-3-generic-type-cant-be-used-in-a-static-context","❌ Restriction 3: Generic type can't be used in a static context",[6972,20689,20691],{"className":7376,"code":20690,"language":7378,"meta":180,"style":180},"public class Test\u003CE> {\n    public static void m(E o1) { }    \u002F\u002F ❌ illegal\n    public static E o1;               \u002F\u002F ❌ illegal\n    static { E o2; }                  \u002F\u002F ❌ illegal\n}\n",[6979,20692,20693,20708,20730,20741,20751],{"__ignoreMap":180},[75,20694,20695,20697,20699,20702,20704,20706],{"class":7383,"line":7384},[75,20696,7388],{"class":7387},[75,20698,7391],{"class":7387},[75,20700,20701],{"class":7394}," Test",[75,20703,7399],{"class":7398},[75,20705,7402],{"class":7387},[75,20707,7405],{"class":7398},[75,20709,20710,20712,20714,20716,20719,20721,20724,20727],{"class":7383,"line":181},[75,20711,7439],{"class":7387},[75,20713,8155],{"class":7387},[75,20715,7442],{"class":7387},[75,20717,20718],{"class":7394}," m",[75,20720,7448],{"class":7398},[75,20722,20723],{"class":7451},"o1",[75,20725,20726],{"class":7398},") { }    ",[75,20728,20729],{"class":7470},"\u002F\u002F ❌ illegal\n",[75,20731,20732,20734,20736,20739],{"class":7383,"line":7430},[75,20733,7439],{"class":7387},[75,20735,8155],{"class":7387},[75,20737,20738],{"class":7398}," E o1;               ",[75,20740,20729],{"class":7470},[75,20742,20743,20746,20749],{"class":7383,"line":7436},[75,20744,20745],{"class":7387},"    static",[75,20747,20748],{"class":7398}," { E o2; }                  ",[75,20750,20729],{"class":7470},[75,20752,20753],{"class":7383,"line":7458},[75,20754,7610],{"class":7398},[12,20756,20757,20758,20761,20762,20764],{},"Why? Static members belong to the ",[7289,20759,20760],{},"class itself",", not any specific instance. But ",[6979,20763,7402],{}," is decided per-instance, so it's meaningless in a static context.",[7166,20766,20768],{"id":20767},"restriction-4-exception-classes-cant-be-generic","❌ Restriction 4: Exception classes can't be generic",[6972,20770,20772],{"className":7376,"code":20771,"language":7378,"meta":180,"style":180},"public class MyException\u003CT> extends Exception { }   \u002F\u002F ❌ illegal\n",[6979,20773,20774],{"__ignoreMap":180},[75,20775,20776,20778,20780,20783,20785,20787,20789,20791,20794,20797],{"class":7383,"line":7384},[75,20777,7388],{"class":7387},[75,20779,7391],{"class":7387},[75,20781,20782],{"class":7394}," MyException",[75,20784,7399],{"class":7398},[75,20786,16256],{"class":7387},[75,20788,11390],{"class":7398},[75,20790,19609],{"class":7387},[75,20792,20793],{"class":7394}," Exception",[75,20795,20796],{"class":7398}," { }   ",[75,20798,20729],{"class":7470},[12,20800,20613,20801,20803,20804,20807,20808,20811,20812,6959],{},[6979,20802,15177],{}," blocks have to inspect types at runtime — but generics are erased before runtime. So you can't write ",[6979,20805,20806],{},"catch (MyException\u003CString> ex)"," because the JVM literally can't tell ",[6979,20809,20810],{},"MyException\u003CString>"," apart from ",[6979,20813,20814],{},"MyException\u003CInteger>",[6964,20816,20818],{"id":20817},"_13-quick-cheat-sheet","13. Quick cheat sheet",[6972,20820,20822],{"className":7376,"code":20821,"language":7378,"meta":180,"style":180},"\u002F\u002F Generic class\npublic class Box\u003CT> { T item; }\n \n\u002F\u002F Multiple type parameters\npublic class Pair\u003CK, V> { K key; V value; }\n \n\u002F\u002F Generic interface\npublic interface Comparable\u003CE> { int compareTo(E o); }\n \n\u002F\u002F Generic method\npublic static \u003CE> void print(E[] list) { \u002F* ... *\u002F }\n \n\u002F\u002F Bounded generic\npublic class NumberBox\u003CT extends Number> { T data; }\n \n\u002F\u002F Wildcard - any type\npublic static void show(ArrayList\u003C?> list) { \u002F* ... *\u002F }\n \n\u002F\u002F Wildcard - upper bounded\npublic static void sum(ArrayList\u003C? extends Number> list) { \u002F* ... *\u002F }\n \n\u002F\u002F Wildcard - lower bounded\npublic static void fill(ArrayList\u003C? super Integer> list) { \u002F* ... *\u002F }\n",[6979,20823,20824,20829,20845,20849,20854,20874,20878,20883,20908,20912,20917,20944,20948,20953,20973,20977,20982,21003,21007,21012,21039,21043,21048],{"__ignoreMap":180},[75,20825,20826],{"class":7383,"line":7384},[75,20827,20828],{"class":7470},"\u002F\u002F Generic class\n",[75,20830,20831,20833,20835,20838,20840,20842],{"class":7383,"line":181},[75,20832,7388],{"class":7387},[75,20834,7391],{"class":7387},[75,20836,20837],{"class":7394}," Box",[75,20839,7399],{"class":7398},[75,20841,16256],{"class":7387},[75,20843,20844],{"class":7398},"> { T item; }\n",[75,20846,20847],{"class":7383,"line":7430},[75,20848,7433],{"class":7398},[75,20850,20851],{"class":7383,"line":7436},[75,20852,20853],{"class":7470},"\u002F\u002F Multiple type parameters\n",[75,20855,20856,20858,20860,20863,20865,20867,20869,20871],{"class":7383,"line":7458},[75,20857,7388],{"class":7387},[75,20859,7391],{"class":7387},[75,20861,20862],{"class":7394}," Pair",[75,20864,7399],{"class":7398},[75,20866,18696],{"class":7387},[75,20868,7298],{"class":7398},[75,20870,18711],{"class":7387},[75,20872,20873],{"class":7398},"> { K key; V value; }\n",[75,20875,20876],{"class":7383,"line":7474},[75,20877,7433],{"class":7398},[75,20879,20880],{"class":7383,"line":7480},[75,20881,20882],{"class":7470},"\u002F\u002F Generic interface\n",[75,20884,20885,20887,20889,20891,20893,20895,20897,20899,20901,20903,20905],{"class":7383,"line":7485},[75,20886,7388],{"class":7387},[75,20888,16505],{"class":7387},[75,20890,8542],{"class":7394},[75,20892,7399],{"class":7398},[75,20894,7402],{"class":7387},[75,20896,19212],{"class":7398},[75,20898,8020],{"class":7387},[75,20900,8676],{"class":7394},[75,20902,7448],{"class":7398},[75,20904,9709],{"class":7451},[75,20906,20907],{"class":7398},"); }\n",[75,20909,20910],{"class":7383,"line":7498},[75,20911,7433],{"class":7398},[75,20913,20914],{"class":7383,"line":7516},[75,20915,20916],{"class":7470},"\u002F\u002F Generic method\n",[75,20918,20919,20921,20923,20925,20927,20929,20931,20933,20935,20937,20940,20942],{"class":7383,"line":7521},[75,20920,7388],{"class":7387},[75,20922,8155],{"class":7387},[75,20924,19325],{"class":7387},[75,20926,7402],{"class":7398},[75,20928,8279],{"class":7387},[75,20930,7442],{"class":7387},[75,20932,19369],{"class":7394},[75,20934,7698],{"class":7398},[75,20936,7402],{"class":7387},[75,20938,20939],{"class":7398},"[] list) { ",[75,20941,19215],{"class":7470},[75,20943,17884],{"class":7398},[75,20945,20946],{"class":7383,"line":7526},[75,20947,7433],{"class":7398},[75,20949,20950],{"class":7383,"line":7539},[75,20951,20952],{"class":7470},"\u002F\u002F Bounded generic\n",[75,20954,20955,20957,20959,20962,20964,20966,20968,20970],{"class":7383,"line":7552},[75,20956,7388],{"class":7387},[75,20958,7391],{"class":7387},[75,20960,20961],{"class":7394}," NumberBox",[75,20963,7399],{"class":7398},[75,20965,16256],{"class":7387},[75,20967,19686],{"class":7387},[75,20969,19689],{"class":7387},[75,20971,20972],{"class":7398},"> { T data; 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What even is a generic?",{"id":18159,"depth":181,"text":18160},{"id":18332,"depth":181,"text":21086,"children":21087},"3. Meeting ArrayList properly",[21088,21089],{"id":18427,"depth":7430,"text":18428},{"id":18546,"depth":7430,"text":18547},{"id":18653,"depth":181,"text":18654},{"id":18754,"depth":181,"text":18755},{"id":18845,"depth":181,"text":21093,"children":21094},"6. Writing your own generic class",[21095],{"id":19190,"depth":7430,"text":19191},{"id":19298,"depth":181,"text":19299},{"id":19588,"depth":181,"text":21098,"children":21099},"8. Bounded generics — \"I want any type, but only certain ones\"",[21100],{"id":19854,"depth":7430,"text":19855},{"id":19874,"depth":181,"text":19875,"children":21102},[21103],{"id":19918,"depth":7430,"text":19919},{"id":20105,"depth":181,"text":21105,"children":21106},"10. Wildcards: ?, ? extends T, ? super T",[21107,21108,21109],{"id":20120,"depth":7430,"text":20121},{"id":20220,"depth":7430,"text":20221},{"id":20293,"depth":7430,"text":20294},{"id":20327,"depth":181,"text":20328,"children":21111},[21112],{"id":20473,"depth":7430,"text":20474},{"id":20574,"depth":181,"text":20575,"children":21114},[21115,21117,21118,21119],{"id":20585,"depth":7430,"text":21116},"❌ Restriction 1: Can't create an instance of E",{"id":20623,"depth":7430,"text":20624},{"id":20686,"depth":7430,"text":20687},{"id":20767,"depth":7430,"text":20768},{"id":20817,"depth":181,"text":20818},{},"\u002Fblog\u002FJava-Generics",{"title":17981,"description":180},{"loc":21122},"blog\u002FJava-Generics","1NAqPd2ioUIVkURA3LlHu8eZauFjE5dfQsUM0-QxAls",{"id":21128,"title":21129,"author":7,"body":21130,"date":17971,"description":21249,"draft":185,"edited_at":13169,"extension":186,"featured_image":21250,"meta":21251,"navigation":189,"path":21252,"pinned":185,"seo":21253,"sitemap":21254,"stem":21255,"tags":194,"__hash__":21256},"blog\u002Fblog\u002FSCI（三）受傷機制--ii.md","SCI（三）受傷機制--ii",{"type":9,"value":21131,"toc":21247},[21132,21149,21171,21173,21182,21205,21207,21210,21213,21224,21233,21236],[12,21133,21134,21135,21137,21138,21140,21141,21143,21144,21148],{},"接著來講創傷性胸椎損傷，胸椎雖然比起頸椎損傷發生少見，但假如胸椎損傷了，基本上都是完全性損傷",[16,21136],{},"\n而T1到T10因為結構很牢固穩定，假如要造成傷害，那個力度一定十分大才能造成",[16,21139],{},"\n假如下胸椎損傷，可能會傷及AKA(Adamkiewicz,亞當氏動脈)，像前面提到的會累及至T4",[16,21142],{},"\n原因多為槍傷，撞車或高處墜下；常發生在",[75,21145,21147],{"className":21146},[78,79],"T12到L1","（這裡應該是說交界處吧）",[23,21150,21151,21159,21165,21168],{},[26,21152,21153,21154],{},"先來講屈曲型損傷",[23,21155,21156],{},[26,21157,21158],{},"楔型骨折：胸椎比較常見的是這種類型骨折，因為胸椎正常是後凹，會將垂直力轉變回屈曲力導致；不過在嚴重個案，也會伴隨後方韌帶受損",[26,21160,21161,21162,21164],{},"接下來是屈曲合併旋轉型損傷，不僅有前方擠壓，後方拉開，還加上了剪力，這三個力合併在一起會破壞三個柱（三柱模式的那個），導致不穩定",[16,21163],{},"\n而神經性損傷也是最常見的，通常因為脊椎脫位加上骨折碎片進入脊髓腔導致的",[26,21166,21167],{},"然後是垂直擠壓型損傷，通常是被掉落物砸到，高空且胸椎臀部或雙著腳地導致；常發生在T10 T11 T12，屬爆裂性骨折，與頸椎的爆裂性骨折類似",[26,21169,21170],{},"最後是伸展或側向彎曲損傷，他們發生率是最少的，假如有出現的話通常會是前縱韌帶斷裂（伸展）或側向楔型骨折（側向彎曲損傷）",[36,21172],{},[12,21174,21175,21176,21178,21179,21181],{},"最後是創傷性的腰椎損傷，活動度來看高於胸椎但低於頸椎",[16,21177],{},"\n常見受傷會因為高處墜下，車禍，槍傷，受重物直接撞擊等",[16,21180],{},"\n好發在胸腰椎交界處；通常是不完全性損傷，因為血流供應較好，椎管較大，且L1 L2以下沒有脊髓主體（脊髓圓錐在L1 L2）",[23,21183,21184,21187,21190,21196,21202],{},[26,21185,21186],{},"屈曲型損傷（對，又是楔型骨折）：通常沒有神經系統損傷，好發在T1，與胸椎機制類似",[26,21188,21189],{},"屈曲合併旋轉型損傷：一樣與胸椎機制類似，造成脫位，並使骨折碎片進入椎管",[26,21191,21192,21193,21195],{},"屈曲合併牽拉型損傷：通常是戴橫向安全帶（戴在薦椎那種的）同時沒有配戴肩帶所造成的安全帶損傷；15%到27%的案例全造成神經損傷",[16,21194],{},"\n造成這個傷害的力是水平的；損傷部位包括椎體後方、纖維環、椎弓、後縱韌帶等；前方可能受壓；",[26,21197,21198,21199,21201],{},"剪力型損傷：雖然一樣是水平力，但上一個是著重被拉開，這一個是兩節脊椎被相反拉開；成因有跌落在不平整地面，或者受到從背後的撞擊導致剪力發生",[16,21200],{},"\n可能是韌帶完全斷裂，繼而椎體脫位，並造成嚴重的神經性損傷",[26,21203,21204],{},"垂直擠壓型損傷：一樣是垂直作用的爆裂性骨折；通常是跌倒引起；有50%累及神經性損傷，通常是不完全性損傷",[36,21206],{},[12,21208,21209],{},"現在來說非創傷性，機制包括疾病進程，血管功能受損，先天性神經管缺陷",[12,21211,21212],{},"先來說疾病類型",[23,21214,21215,21218,21221],{},[26,21216,21217],{},"橫斷性脊髓炎：就是說某一節斷的兩側發生發炎",[26,21219,21220],{},"肌萎縮側索硬化症(ALS)：一種進展迅速跟致命的神經系統疾病，會攻擊運動神經元",[26,21222,21223],{},"多發性硬化症(MS)：免疫系統攻擊覆蓋在神經纖維的髓鞘，導致身體與大腦之間的訊息傳遞出問題",[12,21225,21226,21227,21229,21230,21232],{},"然後是血管功能受損，脊髓循環受損導致缺血進而累及該level及下方區域的神經損傷；原因可能是血栓、腫脹、壓迫、血管畸型與功能性障礙",[16,21228],{},"\n這一頁有配圖的，圖上的病是前脊髓動脈症候群，我就說明一下可能會連累到的部位",[16,21231],{},"\n前皮質脊髓徑，前角，外側脊髓丘腦徑（前外側系統），外側皮質脊髓徑",[12,21234,21235],{},"最後是先天性神經管缺陷，最常見就脊柱裂，跟神經管閉合和後椎弓缺陷有關，常見有這兩類",[23,21237,21238,21241,21244],{},[26,21239,21240],{},"隱性脊柱裂：一個或多個椎弓未能融合，脊髓跟腦脊膜(meninges)沒有影響，留在椎管內；跟神經或肌肉功能障礙無關；但在該部位表面會有一叢毛髮生長",[26,21242,21243],{},"腦脊膜膨出：是比較少見的類型；僅有腦脊膜(meninges)和腦脊髓液進到了囊腫內，脊髓還留在椎管內，但有可能表現出異常",[26,21245,21246],{},"脊髓腦脊膜膨出：比較常見而且還比較危險的類型；脊髓跟腦脊膜全都跑到囊腫內；發病的區域及以下會有不同程度的感覺跟運動功能異常",{"title":180,"searchDepth":181,"depth":181,"links":21248},[],"胸腰椎到非完全性損傷","\u002Fimages\u002Fuploads\u002F1778659464649-74635514_p0.webp",{},"\u002Fblog\u002FSCI（三）受傷機制--ii",{"title":21129,"description":21249},{"loc":21252},"blog\u002FSCI（三）受傷機制--ii","z0s36tN6Ravyr8Z8guj6jzUkYr277j93KuJ0RFs11NA",{"id":21258,"title":21259,"author":6946,"body":21260,"date":24033,"description":180,"draft":185,"edited_at":24033,"extension":186,"featured_image":194,"meta":24034,"navigation":189,"path":24035,"pinned":185,"seo":24036,"sitemap":24037,"stem":24038,"tags":194,"__hash__":24039},"blog\u002Fblog\u002FData-Structure-Concept-Revision.md","Data Structure Concept Revision",{"type":9,"value":21261,"toc":23995},[21262,21266,21270,21277,21280,21314,21318,21324,21327,21337,21349,21355,21358,21491,21494,21525,21528,21534,21537,21567,21571,21574,21592,21597,21639,21645,21668,21672,21678,21705,21709,21712,21749,21755,21758,21862,21869,21878,21884,21887,21903,21965,21969,21972,21997,22018,22022,22026,22032,22038,22050,22057,22063,22066,22070,22074,22077,22099,22103,22183,22200,22203,22252,22256,22262,22311,22324,22342,22345,22368,22381,22385,22388,22398,22483,22497,22581,22590,22594,22605,22644,22667,22671,22683,22700,22703,22928,22930,22936,22957,22969,22973,22981,23062,23073,23077,23081,23100,23106,23110,23123,23192,23198,23202,23353,23356,23365,23391,23395,23432,23437,23492,23501,23505,23508,23542,23589,23592,23625,23629,23646,23745,23751,23755,23836,23839,23874,23878,23881,23987,23989,23992],[6964,21263,21265],{"id":21264},"part-1-objects-and-classes","Part 1: Objects and Classes",[7166,21267,21269],{"id":21268},"what-is-an-object","What is an \"object\"?",[12,21271,21272,21273,21276],{},"Forget code for a second. Look around you — a chair, a phone, a cup of coffee. Each of those is an ",[106,21274,21275],{},"object",". In programming, an object is also a \"thing\", just a digital one. It represents some entity we want to model in our program.",[12,21278,21279],{},"Every object has three properties:",[23,21281,21282,21288,21294,21300,21307],{},[26,21283,21284,21287],{},[106,21285,21286],{},"A unique identity"," — it's distinguishable from other objects, even similar ones. Two identical-looking mugs are still two separate mugs.",[26,21289,21290,21293],{},[106,21291,21292],{},"State"," — the data that describes it. A mug's state might include its color, its volume, and whether it's currently full.",[26,21295,21296,21299],{},[106,21297,21298],{},"Behavior"," — the things it can do. A mug can be filled, emptied, or washed.\nIn Java terms:",[26,21301,21302,21303,21306],{},"State is stored in ",[106,21304,21305],{},"data fields"," (also called variables, properties, or attributes).",[26,21308,21309,21310,21313],{},"Behavior is implemented as ",[106,21311,21312],{},"methods"," (functions that belong to the object).",[7166,21315,21317],{"id":21316},"what-is-a-class","What is a \"class\"?",[12,21319,21320,21321,21323],{},"If an object is a real thing, a ",[106,21322,9664],{}," is the blueprint that describes what that thing looks like.",[12,21325,21326],{},"Think of a class like a cookie cutter, and objects like the cookies. The cutter (class) defines the shape, but you can stamp out as many cookies (objects) as you like, each one its own individual piece.",[12,21328,21329,21330,7067,21333,21336],{},"Or in more formal terms: a class is a ",[106,21331,21332],{},"template",[106,21334,21335],{},"contract"," that defines what data fields and methods every object of that type will have.",[7351,21338,21339],{},[12,21340,21341,21344,21345,21348],{},[106,21342,21343],{},"Key idea:"," An object is an ",[7289,21346,21347],{},"instance"," of a class. You can create many objects from one class.",[7166,21350,21352,21353,7391],{"id":21351},"a-first-example-the-circle-class","A first example: the ",[6979,21354,18131],{},[12,21356,21357],{},"Let's model circles. Every circle has a radius and can compute its area.",[6972,21359,21361],{"className":7376,"code":21360,"language":7378,"meta":180,"style":180},"class Circle {\n    \u002F** The radius of this circle *\u002F\n    double radius = 1.0;\n \n    \u002F** Construct a circle object with default radius *\u002F\n    Circle() {\n    }\n \n    \u002F** Construct a circle object with a given radius *\u002F\n    Circle(double newRadius) {\n        radius = newRadius;\n    }\n \n    \u002F** Return the area of this circle *\u002F\n    double getArea() {\n        return radius * radius * 3.14159;\n    }\n}\n",[6979,21362,21363,21372,21377,21392,21396,21401,21408,21412,21416,21421,21434,21444,21448,21452,21457,21466,21483,21487],{"__ignoreMap":180},[75,21364,21365,21367,21370],{"class":7383,"line":7384},[75,21366,9664],{"class":7387},[75,21368,21369],{"class":7394}," Circle",[75,21371,8147],{"class":7398},[75,21373,21374],{"class":7383,"line":181},[75,21375,21376],{"class":7470},"    \u002F** The radius of this circle *\u002F\n",[75,21378,21379,21382,21385,21387,21390],{"class":7383,"line":7430},[75,21380,21381],{"class":7387},"    double",[75,21383,21384],{"class":7398}," radius ",[75,21386,7421],{"class":7387},[75,21388,21389],{"class":8127}," 1.0",[75,21391,8131],{"class":7398},[75,21393,21394],{"class":7383,"line":7436},[75,21395,7433],{"class":7398},[75,21397,21398],{"class":7383,"line":7458},[75,21399,21400],{"class":7470},"    \u002F** Construct a circle object with default radius *\u002F\n",[75,21402,21403,21406],{"class":7383,"line":7474},[75,21404,21405],{"class":7394},"    Circle",[75,21407,7495],{"class":7398},[75,21409,21410],{"class":7383,"line":7480},[75,21411,7477],{"class":7398},[75,21413,21414],{"class":7383,"line":7485},[75,21415,7433],{"class":7398},[75,21417,21418],{"class":7383,"line":7498},[75,21419,21420],{"class":7470},"    \u002F** Construct a circle object with a given radius *\u002F\n",[75,21422,21423,21425,21427,21429,21432],{"class":7383,"line":7516},[75,21424,21405],{"class":7394},[75,21426,7698],{"class":7398},[75,21428,18825],{"class":7387},[75,21430,21431],{"class":7451}," newRadius",[75,21433,7455],{"class":7398},[75,21435,21436,21439,21441],{"class":7383,"line":7521},[75,21437,21438],{"class":7398},"        radius ",[75,21440,7421],{"class":7387},[75,21442,21443],{"class":7398}," newRadius;\n",[75,21445,21446],{"class":7383,"line":7526},[75,21447,7477],{"class":7398},[75,21449,21450],{"class":7383,"line":7539},[75,21451,7433],{"class":7398},[75,21453,21454],{"class":7383,"line":7552},[75,21455,21456],{"class":7470},"    \u002F** Return the area of this circle *\u002F\n",[75,21458,21459,21461,21464],{"class":7383,"line":7557},[75,21460,21381],{"class":7387},[75,21462,21463],{"class":7394}," getArea",[75,21465,7495],{"class":7398},[75,21467,21468,21470,21472,21474,21476,21478,21481],{"class":7383,"line":7562},[75,21469,7501],{"class":7387},[75,21471,21384],{"class":7398},[75,21473,8128],{"class":7387},[75,21475,21384],{"class":7398},[75,21477,8128],{"class":7387},[75,21479,21480],{"class":8127}," 3.14159",[75,21482,8131],{"class":7398},[75,21484,21485],{"class":7383,"line":7571},[75,21486,7477],{"class":7398},[75,21488,21489],{"class":7383,"line":7584},[75,21490,7610],{"class":7398},[12,21492,21493],{},"Three pieces here:",[23,21495,21496,21505,21514],{},[26,21497,21498,18365,21501,21504],{},[6979,21499,21500],{},"radius",[106,21502,21503],{},"data field"," (the state).",[26,21506,21507,7621,21510,21513],{},[6979,21508,21509],{},"getArea()",[106,21511,21512],{},"method"," (the behavior).",[26,21515,21516,7042,21519,20143,21522,6959],{},[6979,21517,21518],{},"Circle()",[6979,21520,21521],{},"Circle(double newRadius)",[106,21523,21524],{},"constructors",[7166,21526,21527],{"id":21524},"Constructors",[12,21529,9407,21530,21533],{},[106,21531,21532],{},"constructor"," is a special method that runs when you create a new object. It \"constructs\" the object — sets it up, gives it initial values.",[12,21535,21536],{},"Three important rules:",[3603,21538,21539,21545,21554],{},[26,21540,21541,21542,6959],{},"The constructor's name ",[106,21543,21544],{},"must match the class name exactly",[26,21546,21547,21548,21551,21552,6959],{},"It has ",[106,21549,21550],{},"no return type"," — not even ",[6979,21553,8035],{},[26,21555,21556,21557,21559,21560,21563,21564,21566],{},"It's invoked using the ",[6979,21558,8823],{}," keyword.\nA constructor with no parameters is called a ",[106,21561,21562],{},"no-arg constructor",". If you write a class without ",[7289,21565,19592],{}," constructor at all, Java quietly gives you a free default no-arg constructor with an empty body. But the moment you write even one constructor yourself, that freebie disappears — Java won't make one for you anymore.",[7166,21568,21570],{"id":21569},"creating-objects","Creating objects",[12,21572,21573],{},"The pattern looks like this:",[6972,21575,21577],{"className":7376,"code":21576,"language":7378,"meta":180,"style":180},"ClassName variableName = new ClassName();\n",[6979,21578,21579],{"__ignoreMap":180},[75,21580,21581,21584,21586,21588,21590],{"class":7383,"line":7384},[75,21582,21583],{"class":7398},"ClassName variableName ",[75,21585,7421],{"class":7387},[75,21587,7424],{"class":7387},[75,21589,18870],{"class":7394},[75,21591,7549],{"class":7398},[12,21593,21594,21595,7024],{},"So for our ",[6979,21596,18131],{},[6972,21598,21600],{"className":7376,"code":21599,"language":7378,"meta":180,"style":180},"Circle myCircle = new Circle();           \u002F\u002F uses no-arg constructor, radius = 1.0\nCircle bigCircle = new Circle(25);        \u002F\u002F uses the other constructor, radius = 25\n",[6979,21601,21602,21619],{"__ignoreMap":180},[75,21603,21604,21607,21609,21611,21613,21616],{"class":7383,"line":7384},[75,21605,21606],{"class":7398},"Circle myCircle ",[75,21608,7421],{"class":7387},[75,21610,7424],{"class":7387},[75,21612,21369],{"class":7394},[75,21614,21615],{"class":7398},"();           ",[75,21617,21618],{"class":7470},"\u002F\u002F uses no-arg constructor, radius = 1.0\n",[75,21620,21621,21624,21626,21628,21630,21632,21634,21636],{"class":7383,"line":181},[75,21622,21623],{"class":7398},"Circle bigCircle ",[75,21625,7421],{"class":7387},[75,21627,7424],{"class":7387},[75,21629,21369],{"class":7394},[75,21631,7698],{"class":7398},[75,21633,17409],{"class":8127},[75,21635,7721],{"class":7398},[75,21637,21638],{"class":7470},"\u002F\u002F uses the other constructor, radius = 25\n",[12,21640,21641,21644],{},[6979,21642,21643],{},"new Circle()"," actually does two things:",[3603,21646,21647,21650],{},[26,21648,21649],{},"Creates a new circle object in memory.",[26,21651,21652,21653,21655,21656,21659,21660,21663,21664,21667],{},"Returns a ",[7289,21654,9653],{}," to it.\nThat reference is what gets stored in ",[6979,21657,21658],{},"myCircle",". The variable doesn't ",[7289,21661,21662],{},"contain"," the object — it ",[7289,21665,21666],{},"points to"," it. This distinction matters, as we'll see in a moment.",[7166,21669,21671],{"id":21670},"accessing-an-objects-members","Accessing an object's members",[12,21673,21674,21675,21677],{},"Once you have an object, use the dot (",[6979,21676,6959],{},") operator:",[6972,21679,21681],{"className":7376,"code":21680,"language":7378,"meta":180,"style":180},"myCircle.radius          \u002F\u002F read the data field\nmyCircle.getArea()       \u002F\u002F invoke a method\n",[6979,21682,21683,21691],{"__ignoreMap":180},[75,21684,21685,21688],{"class":7383,"line":7384},[75,21686,21687],{"class":7398},"myCircle.radius          ",[75,21689,21690],{"class":7470},"\u002F\u002F read the data field\n",[75,21692,21693,21696,21699,21702],{"class":7383,"line":181},[75,21694,21695],{"class":7398},"myCircle.",[75,21697,21698],{"class":7394},"getArea",[75,21700,21701],{"class":7398},"()       ",[75,21703,21704],{"class":7470},"\u002F\u002F invoke a method\n",[7166,21706,21708],{"id":21707},"primitive-types-vs-object-types","Primitive types vs. object types",[12,21710,21711],{},"Java has two flavors of variables:",[6972,21713,21715],{"className":7376,"code":21714,"language":7378,"meta":180,"style":180},"int i = 1;               \u002F\u002F primitive: i directly holds the value 1\nCircle c = new Circle(); \u002F\u002F object: c holds a reference pointing to the Circle\n",[6979,21716,21717,21733],{"__ignoreMap":180},[75,21718,21719,21721,21723,21725,21727,21730],{"class":7383,"line":7384},[75,21720,8020],{"class":7387},[75,21722,10481],{"class":7398},[75,21724,7421],{"class":7387},[75,21726,10486],{"class":8127},[75,21728,21729],{"class":7398},";               ",[75,21731,21732],{"class":7470},"\u002F\u002F primitive: i directly holds the value 1\n",[75,21734,21735,21738,21740,21742,21744,21746],{"class":7383,"line":181},[75,21736,21737],{"class":7398},"Circle c ",[75,21739,7421],{"class":7387},[75,21741,7424],{"class":7387},[75,21743,21369],{"class":7394},[75,21745,7510],{"class":7398},[75,21747,21748],{"class":7470},"\u002F\u002F object: c holds a reference pointing to the Circle\n",[12,21750,21751,21752,21754],{},"For primitives, the variable ",[7289,21753,17990],{}," the value. For objects, the variable is a \"pointer\" to the object stored somewhere else in memory.",[12,21756,21757],{},"This becomes important when you assign one variable to another:",[6972,21759,21761],{"className":7376,"code":21760,"language":7378,"meta":180,"style":180},"\u002F\u002F Primitives — copies the value\nint i = 1, j = 2;\ni = j;   \u002F\u002F now i = 2, j = 2. Two independent copies.\n \n\u002F\u002F Objects — copies the reference, not the object\nCircle c1 = new Circle(5);\nCircle c2 = new Circle(9);\nc1 = c2;  \u002F\u002F c1 now points to the same object as c2 (radius 9)\n          \u002F\u002F The old radius-5 circle is now unreachable — garbage.\n",[6979,21762,21763,21768,21787,21800,21804,21809,21826,21844,21857],{"__ignoreMap":180},[75,21764,21765],{"class":7383,"line":7384},[75,21766,21767],{"class":7470},"\u002F\u002F Primitives — copies the value\n",[75,21769,21770,21772,21774,21776,21778,21781,21783,21785],{"class":7383,"line":181},[75,21771,8020],{"class":7387},[75,21773,10481],{"class":7398},[75,21775,7421],{"class":7387},[75,21777,10486],{"class":8127},[75,21779,21780],{"class":7398},", j ",[75,21782,7421],{"class":7387},[75,21784,10860],{"class":8127},[75,21786,8131],{"class":7398},[75,21788,21789,21792,21794,21797],{"class":7383,"line":7430},[75,21790,21791],{"class":7398},"i ",[75,21793,7421],{"class":7387},[75,21795,21796],{"class":7398}," j;   ",[75,21798,21799],{"class":7470},"\u002F\u002F now i = 2, j = 2. Two independent copies.\n",[75,21801,21802],{"class":7383,"line":7436},[75,21803,7433],{"class":7398},[75,21805,21806],{"class":7383,"line":7458},[75,21807,21808],{"class":7470},"\u002F\u002F Objects — copies the reference, not the object\n",[75,21810,21811,21814,21816,21818,21820,21822,21824],{"class":7383,"line":7474},[75,21812,21813],{"class":7398},"Circle c1 ",[75,21815,7421],{"class":7387},[75,21817,7424],{"class":7387},[75,21819,21369],{"class":7394},[75,21821,7698],{"class":7398},[75,21823,15531],{"class":8127},[75,21825,7738],{"class":7398},[75,21827,21828,21831,21833,21835,21837,21839,21842],{"class":7383,"line":7480},[75,21829,21830],{"class":7398},"Circle c2 ",[75,21832,7421],{"class":7387},[75,21834,7424],{"class":7387},[75,21836,21369],{"class":7394},[75,21838,7698],{"class":7398},[75,21840,21841],{"class":8127},"9",[75,21843,7738],{"class":7398},[75,21845,21846,21849,21851,21854],{"class":7383,"line":7485},[75,21847,21848],{"class":7398},"c1 ",[75,21850,7421],{"class":7387},[75,21852,21853],{"class":7398}," c2;  ",[75,21855,21856],{"class":7470},"\u002F\u002F c1 now points to the same object as c2 (radius 9)\n",[75,21858,21859],{"class":7383,"line":7498},[75,21860,21861],{"class":7470},"          \u002F\u002F The old radius-5 circle is now unreachable — garbage.\n",[12,21863,21864,21865,21868],{},"Java will eventually clean up that \"garbage\" object automatically. That's the ",[106,21866,21867],{},"garbage collector"," doing its job.",[7166,21870,21872,21873,21875,21876],{"id":21871},"visibility-modifiers-public-vs-private","Visibility modifiers: ",[6979,21874,7388],{}," vs. ",[6979,21877,7656],{},[12,21879,21880,21881,6959],{},"By default, a class member (a field or method) is accessible to any class in the ",[106,21882,21883],{},"same package",[12,21885,21886],{},"Two important modifiers change this:",[23,21888,21889,21896],{},[26,21890,21891,21895],{},[106,21892,21893],{},[6979,21894,7388],{}," — visible everywhere. Any class, any package.",[26,21897,21898,21902],{},[106,21899,21900],{},[6979,21901,7656],{}," — visible only inside the class that declares it.",[6972,21904,21906],{"className":7376,"code":21905,"language":7378,"meta":180,"style":180},"public class Account {\n    private double balance;   \u002F\u002F hidden from the outside\n \n    public double getBalance() {   \u002F\u002F controlled access\n        return balance;\n    }\n}\n",[6979,21907,21908,21919,21931,21935,21950,21957,21961],{"__ignoreMap":180},[75,21909,21910,21912,21914,21917],{"class":7383,"line":7384},[75,21911,7388],{"class":7387},[75,21913,7391],{"class":7387},[75,21915,21916],{"class":7394}," Account",[75,21918,8147],{"class":7398},[75,21920,21921,21923,21925,21928],{"class":7383,"line":181},[75,21922,7410],{"class":7387},[75,21924,14858],{"class":7387},[75,21926,21927],{"class":7398}," balance;   ",[75,21929,21930],{"class":7470},"\u002F\u002F hidden from the outside\n",[75,21932,21933],{"class":7383,"line":7430},[75,21934,7433],{"class":7398},[75,21936,21937,21939,21941,21944,21947],{"class":7383,"line":7436},[75,21938,7439],{"class":7387},[75,21940,14858],{"class":7387},[75,21942,21943],{"class":7394}," getBalance",[75,21945,21946],{"class":7398},"() {   ",[75,21948,21949],{"class":7470},"\u002F\u002F controlled access\n",[75,21951,21952,21954],{"class":7383,"line":7458},[75,21953,7501],{"class":7387},[75,21955,21956],{"class":7398}," balance;\n",[75,21958,21959],{"class":7383,"line":7474},[75,21960,7477],{"class":7398},[75,21962,21963],{"class":7383,"line":7480},[75,21964,7610],{"class":7398},[7166,21966,21968],{"id":21967},"why-should-data-fields-be-private","Why should data fields be private?",[12,21970,21971],{},"This is one of the most important habits in OOP. Two reasons:",[3603,21973,21974,21984],{},[26,21975,21976,21979,21980,21983],{},[106,21977,21978],{},"To protect data."," If ",[6979,21981,21982],{},"balance"," were public, any code anywhere could set it to a negative billion. Making it private forces callers to go through methods you control — methods that can validate, log, or refuse the change.",[26,21985,21986,21989,21990,7042,21993,21996],{},[106,21987,21988],{},"To make the class easy to maintain."," If you later decide to store the balance differently (say, in cents instead of dollars), only the inside of the class changes. Code that uses the class via ",[6979,21991,21992],{},"getBalance()",[6979,21994,21995],{},"setBalance()"," doesn't have to be rewritten.\nThese controlled-access methods have standard names:",[23,21998,21999,22008],{},[26,22000,22001,22004,22005,22007],{},[106,22002,22003],{},"Accessor"," (getter): ",[6979,22006,21992],{}," — reads a private field.",[26,22009,22010,22013,22014,22017],{},[106,22011,22012],{},"Mutator"," (setter): ",[6979,22015,22016],{},"setBalance(double value)"," — modifies a private field.",[6964,22019,22021],{"id":22020},"part-2-thinking-in-objects","Part 2: Thinking in Objects",[7166,22023,22025],{"id":22024},"abstraction-and-encapsulation","Abstraction and encapsulation",[12,22027,22028,22029],{},"These are the two big ideas behind \"thinking in objects\". They sound abstract (no pun intended), but they boil down to one principle: ",[106,22030,22031],{},"hide the messy details, expose a clean interface.",[12,22033,22034,22037],{},[106,22035,22036],{},"Encapsulation"," means bundling data and the methods that operate on it inside a single unit (a class), and hiding the internal state behind private fields. The outside world only talks to the class through its public methods.",[12,22039,22040,22043,22044,22046,22047,22049],{},[106,22041,22042],{},"Abstraction"," means separating ",[7289,22045,9441],{}," a class does from ",[7289,22048,9518],{}," it does it. As the user of a class, you only need to know its public contract — the method signatures. You don't need to read the implementation.",[12,22051,22052,22053,22056],{},"Think of a TV remote. You press the volume button. You don't care whether the remote uses infrared, Bluetooth, or magic — you just care that the volume changes. The remote ",[7289,22054,22055],{},"abstracts away"," the complexity.",[6972,22058,22061],{"className":22059,"code":22060,"language":6977},[6975],"[ Class implementation ]  \u003C--->  [ Class Contract ]  \u003C--->  [ Clients ]\n   (hidden, black box)        (public methods\u002Fconstants)   (the users)\n",[6979,22062,22060],{"__ignoreMap":180},[12,22064,22065],{},"When you design a class, ask: \"What's the smallest, cleanest interface I can expose that still lets users do what they need?\" Everything else stays private.",[6964,22067,22069],{"id":22068},"part-3-inheritance-and-polymorphism","Part 3: Inheritance and Polymorphism",[7166,22071,22073],{"id":22072},"the-motivation-avoid-repeating-yourself","The motivation: avoid repeating yourself",[12,22075,22076],{},"Suppose I want to model circles, rectangles, and triangles. All three share things: a color, whether they're filled, the date they were created. If I write that information into each class separately, I'll have three copies of the same code. If I later want to add an \"outline thickness\" property, I'd have to update all three.",[12,22078,22079,22082,22083,22086,22087,7298,22089,7153,22092,7206,22095,22098],{},[106,22080,22081],{},"Inheritance"," solves this. We define a general class first — say, ",[6979,22084,22085],{},"GeometricObject"," — that holds the shared stuff. Then ",[6979,22088,18131],{},[6979,22090,22091],{},"Rectangle",[6979,22093,22094],{},"Triangle",[7289,22096,22097],{},"extend"," it, inheriting everything it has.",[7166,22100,22102],{"id":22101},"superclass-and-subclass","Superclass and subclass",[6972,22104,22106],{"className":7376,"code":22105,"language":7378,"meta":180,"style":180},"class GeometricObject {\n    private String color = \"white\";\n    private boolean filled;\n    \u002F\u002F ...constructors, getters, setters...\n}\n \nclass Circle extends GeometricObject {\n    private double radius;\n    \u002F\u002F ...inherits color, filled, etc. for free...\n}\n",[6979,22107,22108,22117,22131,22140,22145,22149,22153,22165,22174,22179],{"__ignoreMap":180},[75,22109,22110,22112,22115],{"class":7383,"line":7384},[75,22111,9664],{"class":7387},[75,22113,22114],{"class":7394}," GeometricObject",[75,22116,8147],{"class":7398},[75,22118,22119,22121,22124,22126,22129],{"class":7383,"line":181},[75,22120,7410],{"class":7387},[75,22122,22123],{"class":7398}," String color ",[75,22125,7421],{"class":7387},[75,22127,22128],{"class":7589}," \"white\"",[75,22130,8131],{"class":7398},[75,22132,22133,22135,22137],{"class":7383,"line":7430},[75,22134,7410],{"class":7387},[75,22136,9278],{"class":7387},[75,22138,22139],{"class":7398}," filled;\n",[75,22141,22142],{"class":7383,"line":7436},[75,22143,22144],{"class":7470},"    \u002F\u002F ...constructors, getters, setters...\n",[75,22146,22147],{"class":7383,"line":7458},[75,22148,7610],{"class":7398},[75,22150,22151],{"class":7383,"line":7474},[75,22152,7433],{"class":7398},[75,22154,22155,22157,22159,22161,22163],{"class":7383,"line":7480},[75,22156,9664],{"class":7387},[75,22158,21369],{"class":7394},[75,22160,19686],{"class":7387},[75,22162,22114],{"class":7394},[75,22164,8147],{"class":7398},[75,22166,22167,22169,22171],{"class":7383,"line":7485},[75,22168,7410],{"class":7387},[75,22170,14858],{"class":7387},[75,22172,22173],{"class":7398}," radius;\n",[75,22175,22176],{"class":7383,"line":7498},[75,22177,22178],{"class":7470},"    \u002F\u002F ...inherits color, filled, etc. for free...\n",[75,22180,22181],{"class":7383,"line":7516},[75,22182,7610],{"class":7398},[12,22184,22185,18365,22187,22190,22191,18365,22193,22196,22197,22199],{},[6979,22186,22085],{},[106,22188,22189],{},"superclass"," (parent), and ",[6979,22192,18131],{},[106,22194,22195],{},"subclass"," (child). The ",[6979,22198,19609],{}," keyword sets up the relationship.",[12,22201,22202],{},"A few important facts:",[23,22204,22205,22216,22223,22230,22241],{},[26,22206,22207,22208,22211,22212,22215],{},"A subclass is ",[106,22209,22210],{},"not"," simply a subset of a superclass. In fact, a subclass usually has ",[7289,22213,22214],{},"more"," than its parent — it adds its own fields and methods on top.",[26,22217,22218,22219,22222],{},"Private members of the superclass are ",[106,22220,22221],{},"not directly accessible"," in the subclass. You have to go through public getters\u002Fsetters.",[26,22224,22225,22226,22229],{},"Classes are ",[106,22227,22228],{},"extensible"," — you can keep extending subclasses to make new ones.",[26,22231,22232,22233,22236,22237,22240],{},"Java does ",[106,22234,22235],{},"not allow multiple inheritance"," for classes. A class can only extend ",[7289,22238,22239],{},"one"," superclass. (Interfaces are the workaround — more on that later.)",[26,22242,22243,22244,22247,22248,22251],{},"Superclass ",[106,22245,22246],{},"constructors are not inherited",", but they ",[7289,22249,22250],{},"are"," automatically invoked.",[7166,22253,22255],{"id":22254},"calling-the-superclass-constructor","Calling the superclass constructor",[12,22257,22258,22259,7024],{},"To explicitly invoke the parent's constructor, use ",[6979,22260,22261],{},"super",[6972,22263,22265],{"className":7376,"code":22264,"language":7378,"meta":180,"style":180},"public Circle(double radius, String color, boolean filled) {\n    super(color, filled);   \u002F\u002F calls GeometricObject's constructor\n    this.radius = radius;\n}\n",[6979,22266,22267,22285,22296,22307],{"__ignoreMap":180},[75,22268,22269,22271,22273,22275,22277,22280,22282],{"class":7383,"line":7384},[75,22270,7388],{"class":7387},[75,22272,21369],{"class":7394},[75,22274,7698],{"class":7398},[75,22276,18825],{"class":7387},[75,22278,22279],{"class":7398}," radius, String color, ",[75,22281,7944],{"class":7387},[75,22283,22284],{"class":7398}," filled) {\n",[75,22286,22287,22290,22293],{"class":7383,"line":181},[75,22288,22289],{"class":8127},"    super",[75,22291,22292],{"class":7398},"(color, filled);   ",[75,22294,22295],{"class":7470},"\u002F\u002F calls GeometricObject's constructor\n",[75,22297,22298,22300,22303,22305],{"class":7383,"line":7430},[75,22299,8592],{"class":8127},[75,22301,22302],{"class":7398},".radius ",[75,22304,7421],{"class":7387},[75,22306,22173],{"class":7398},[75,22308,22309],{"class":7383,"line":7436},[75,22310,7610],{"class":7398},[12,22312,22313,22316,22317,22320,22321,22323],{},[6979,22314,22315],{},"super()"," must be the ",[7289,22318,22319],{},"first statement"," in the subclass constructor. If you don't write it explicitly, Java inserts an invisible ",[6979,22322,22315],{}," for you. That's why this:",[6972,22325,22327],{"className":7376,"code":22326,"language":7378,"meta":180,"style":180},"public A() {\n}\n",[6979,22328,22329,22338],{"__ignoreMap":180},[75,22330,22331,22333,22336],{"class":7383,"line":7384},[75,22332,7388],{"class":7387},[75,22334,22335],{"class":7394}," A",[75,22337,7495],{"class":7398},[75,22339,22340],{"class":7383,"line":181},[75,22341,7610],{"class":7398},[12,22343,22344],{},"is actually equivalent to:",[6972,22346,22348],{"className":7376,"code":22347,"language":7378,"meta":180,"style":180},"public A() {\n    super();\n}\n",[6979,22349,22350,22358,22364],{"__ignoreMap":180},[75,22351,22352,22354,22356],{"class":7383,"line":7384},[75,22353,7388],{"class":7387},[75,22355,22335],{"class":7394},[75,22357,7495],{"class":7398},[75,22359,22360,22362],{"class":7383,"line":181},[75,22361,22289],{"class":8127},[75,22363,7549],{"class":7398},[75,22365,22366],{"class":7383,"line":7430},[75,22367,7610],{"class":7398},[12,22369,22370,22373,22374,22377,22378,22380],{},[106,22371,22372],{},"Watch out:"," if your superclass only has a constructor that takes arguments (no no-arg constructor), and your subclass constructor doesn't explicitly call ",[6979,22375,22376],{},"super(...)"," with the right arguments, you'll get a compile error. The compiler tries to insert ",[6979,22379,22315],{}," and finds nothing matching.",[7166,22382,22384],{"id":22383},"overriding-vs-overloading","Overriding vs. Overloading",[12,22386,22387],{},"These two words sound similar and trip up everyone at first.",[12,22389,22390,22393,22394,22397],{},[106,22391,22392],{},"Overloading"," — defining multiple methods with the ",[106,22395,22396],{},"same name but different parameter lists"," in the same class (or across an inheritance chain).",[6972,22399,22401],{"className":7376,"code":22400,"language":7378,"meta":180,"style":180},"class Calculator {\n    int add(int a, int b)          { return a + b; }\n    double add(double a, double b) { return a + b; }   \u002F\u002F overload\n}\n",[6979,22402,22403,22412,22446,22479],{"__ignoreMap":180},[75,22404,22405,22407,22410],{"class":7383,"line":7384},[75,22406,9664],{"class":7387},[75,22408,22409],{"class":7394}," Calculator",[75,22411,8147],{"class":7398},[75,22413,22414,22417,22419,22421,22423,22426,22428,22430,22433,22436,22438,22441,22443],{"class":7383,"line":181},[75,22415,22416],{"class":7387},"    int",[75,22418,10395],{"class":7394},[75,22420,7698],{"class":7398},[75,22422,8020],{"class":7387},[75,22424,22425],{"class":7451}," a",[75,22427,7298],{"class":7398},[75,22429,8020],{"class":7387},[75,22431,22432],{"class":7451}," b",[75,22434,22435],{"class":7398},")          { ",[75,22437,8636],{"class":7387},[75,22439,22440],{"class":7398}," a ",[75,22442,8302],{"class":7387},[75,22444,22445],{"class":7398}," b; }\n",[75,22447,22448,22450,22452,22454,22456,22458,22460,22462,22464,22467,22469,22471,22473,22476],{"class":7383,"line":7430},[75,22449,21381],{"class":7387},[75,22451,10395],{"class":7394},[75,22453,7698],{"class":7398},[75,22455,18825],{"class":7387},[75,22457,22425],{"class":7451},[75,22459,7298],{"class":7398},[75,22461,18825],{"class":7387},[75,22463,22432],{"class":7451},[75,22465,22466],{"class":7398},") { ",[75,22468,8636],{"class":7387},[75,22470,22440],{"class":7398},[75,22472,8302],{"class":7387},[75,22474,22475],{"class":7398}," b; }   ",[75,22477,22478],{"class":7470},"\u002F\u002F overload\n",[75,22480,22481],{"class":7383,"line":7436},[75,22482,7610],{"class":7398},[12,22484,22485,22488,22489,22492,22493,22496],{},[106,22486,22487],{},"Overriding"," — a subclass provides a ",[106,22490,22491],{},"new implementation"," of a method that already exists in the superclass. Same name, ",[106,22494,22495],{},"same parameter list",", same return type.",[6972,22498,22500],{"className":7376,"code":22499,"language":7378,"meta":180,"style":180},"class Animal {\n    public String sound() { return \"Some sound\"; }\n}\n \nclass Dog extends Animal {\n    @Override\n    public String sound() { return \"Woof\"; }   \u002F\u002F override\n}\n",[6979,22501,22502,22511,22529,22533,22537,22550,22556,22577],{"__ignoreMap":180},[75,22503,22504,22506,22509],{"class":7383,"line":7384},[75,22505,9664],{"class":7387},[75,22507,22508],{"class":7394}," Animal",[75,22510,8147],{"class":7398},[75,22512,22513,22515,22517,22520,22522,22524,22527],{"class":7383,"line":181},[75,22514,7439],{"class":7387},[75,22516,7576],{"class":7398},[75,22518,22519],{"class":7394},"sound",[75,22521,8633],{"class":7398},[75,22523,8636],{"class":7387},[75,22525,22526],{"class":7589}," \"Some sound\"",[75,22528,18948],{"class":7398},[75,22530,22531],{"class":7383,"line":7430},[75,22532,7610],{"class":7398},[75,22534,22535],{"class":7383,"line":7436},[75,22536,7433],{"class":7398},[75,22538,22539,22541,22544,22546,22548],{"class":7383,"line":7458},[75,22540,9664],{"class":7387},[75,22542,22543],{"class":7394}," Dog",[75,22545,19686],{"class":7387},[75,22547,22508],{"class":7394},[75,22549,8147],{"class":7398},[75,22551,22552,22554],{"class":7383,"line":7474},[75,22553,7565],{"class":7398},[75,22555,7568],{"class":7387},[75,22557,22558,22560,22562,22564,22566,22568,22571,22574],{"class":7383,"line":7480},[75,22559,7439],{"class":7387},[75,22561,7576],{"class":7398},[75,22563,22519],{"class":7394},[75,22565,8633],{"class":7398},[75,22567,8636],{"class":7387},[75,22569,22570],{"class":7589}," \"Woof\"",[75,22572,22573],{"class":7398},"; }   ",[75,22575,22576],{"class":7470},"\u002F\u002F override\n",[75,22578,22579],{"class":7383,"line":7485},[75,22580,7610],{"class":7398},[12,22582,9455,22583,22586,22587,6959],{},[6979,22584,22585],{},"@Override"," annotation isn't strictly required, but it's a good habit — the compiler will check that you really are overriding something, catching typos like ",[6979,22588,22589],{},"sond()",[7166,22591,22593],{"id":22592},"polymorphism","Polymorphism",[12,22595,22596,22597,22599,22600,22602,22603,6959],{},"Here's where it gets interesting. ",[106,22598,22593],{}," is the feature that lets a variable of a ",[7289,22601,20282],{}," refer to an object of a ",[7289,22604,20266],{},[6972,22606,22608],{"className":7376,"code":22607,"language":7378,"meta":180,"style":180},"GeometricObject geoObj = new Circle();   \u002F\u002F legal!\nFruit f = new Apple();                    \u002F\u002F also legal\n",[6979,22609,22610,22626],{"__ignoreMap":180},[75,22611,22612,22615,22617,22619,22621,22623],{"class":7383,"line":7384},[75,22613,22614],{"class":7398},"GeometricObject geoObj ",[75,22616,7421],{"class":7387},[75,22618,7424],{"class":7387},[75,22620,21369],{"class":7394},[75,22622,9268],{"class":7398},[75,22624,22625],{"class":7470},"\u002F\u002F legal!\n",[75,22627,22628,22631,22633,22635,22638,22641],{"class":7383,"line":181},[75,22629,22630],{"class":7398},"Fruit f ",[75,22632,7421],{"class":7387},[75,22634,7424],{"class":7387},[75,22636,22637],{"class":7394}," Apple",[75,22639,22640],{"class":7398},"();                    ",[75,22642,22643],{"class":7470},"\u002F\u002F also legal\n",[12,22645,22646,22647,7653,22650,22652,22653,22656,22657,22659,22660,7206,22662,7206,22665,6959],{},"The variable's ",[7289,22648,22649],{},"declared type",[6979,22651,22085],{},", but the ",[7289,22654,22655],{},"actual object"," it points to is a ",[6979,22658,18131],{},". This is fine because every ",[6979,22661,18131],{},[7289,22663,22664],{},"is a",[6979,22666,22085],{},[7166,22668,22670],{"id":22669},"dynamic-binding","Dynamic binding",[12,22672,22673,22674,22677,22678,22680,22681,8512],{},"So if we call ",[6979,22675,22676],{},"geoObj.toString()",", which method actually runs — the one in ",[6979,22679,22085],{}," or the one in ",[6979,22682,18131],{},[12,22684,22685,7206,22688,22692,22693,22696,22697,6959],{},[106,22686,22687],{},"The one in",[106,22689,22690,6959],{},[6979,22691,18131],{}," Java looks at the ",[7289,22694,22695],{},"actual object's type"," at runtime, not the declared variable type, and picks the most specific version of the method. This is called ",[106,22698,22699],{},"dynamic binding",[12,22701,22702],{},"Example:",[6972,22704,22706],{"className":7376,"code":22705,"language":7378,"meta":180,"style":180},"public class PolymorphismDemo {\n    public static void main(String[] args) {\n        m(new GraduateStudent());\n        m(new Student());\n        m(new Person());\n        m(new Object());\n    }\n \n    public static void m(Object x) {\n        System.out.println(x.toString());\n    }\n}\n \nclass GraduateStudent extends Student { }\n \nclass Student extends Person {\n    public String toString() { return \"Student\"; }\n}\n \nclass Person extends Object {\n    public String toString() { return \"Person\"; }\n}\n",[6979,22707,22708,22719,22739,22753,22766,22779,22791,22795,22799,22816,22829,22833,22837,22841,22854,22858,22870,22887,22891,22895,22907,22924],{"__ignoreMap":180},[75,22709,22710,22712,22714,22717],{"class":7383,"line":7384},[75,22711,7388],{"class":7387},[75,22713,7391],{"class":7387},[75,22715,22716],{"class":7394}," PolymorphismDemo",[75,22718,8147],{"class":7398},[75,22720,22721,22723,22725,22727,22729,22731,22733,22735,22737],{"class":7383,"line":181},[75,22722,7439],{"class":7387},[75,22724,8155],{"class":7387},[75,22726,7442],{"class":7387},[75,22728,8160],{"class":7394},[75,22730,7698],{"class":7398},[75,22732,7674],{"class":7387},[75,22734,8167],{"class":7398},[75,22736,8170],{"class":7451},[75,22738,7455],{"class":7398},[75,22740,22741,22744,22746,22748,22751],{"class":7383,"line":7430},[75,22742,22743],{"class":7394},"        m",[75,22745,7698],{"class":7398},[75,22747,8823],{"class":7387},[75,22749,22750],{"class":7394}," GraduateStudent",[75,22752,8357],{"class":7398},[75,22754,22755,22757,22759,22761,22764],{"class":7383,"line":7436},[75,22756,22743],{"class":7394},[75,22758,7698],{"class":7398},[75,22760,8823],{"class":7387},[75,22762,22763],{"class":7394}," Student",[75,22765,8357],{"class":7398},[75,22767,22768,22770,22772,22774,22777],{"class":7383,"line":7458},[75,22769,22743],{"class":7394},[75,22771,7698],{"class":7398},[75,22773,8823],{"class":7387},[75,22775,22776],{"class":7394}," Person",[75,22778,8357],{"class":7398},[75,22780,22781,22783,22785,22787,22789],{"class":7383,"line":7474},[75,22782,22743],{"class":7394},[75,22784,7698],{"class":7398},[75,22786,8823],{"class":7387},[75,22788,20675],{"class":7394},[75,22790,8357],{"class":7398},[75,22792,22793],{"class":7383,"line":7480},[75,22794,7477],{"class":7398},[75,22796,22797],{"class":7383,"line":7485},[75,22798,7433],{"class":7398},[75,22800,22801,22803,22805,22807,22809,22812,22814],{"class":7383,"line":7498},[75,22802,7439],{"class":7387},[75,22804,8155],{"class":7387},[75,22806,7442],{"class":7387},[75,22808,20718],{"class":7394},[75,22810,22811],{"class":7398},"(Object ",[75,22813,7119],{"class":7451},[75,22815,7455],{"class":7398},[75,22817,22818,22820,22822,22825,22827],{"class":7383,"line":7516},[75,22819,12229],{"class":7398},[75,22821,7767],{"class":7394},[75,22823,22824],{"class":7398},"(x.",[75,22826,7579],{"class":7394},[75,22828,8357],{"class":7398},[75,22830,22831],{"class":7383,"line":7521},[75,22832,7477],{"class":7398},[75,22834,22835],{"class":7383,"line":7526},[75,22836,7610],{"class":7398},[75,22838,22839],{"class":7383,"line":7539},[75,22840,7433],{"class":7398},[75,22842,22843,22845,22847,22849,22851],{"class":7383,"line":7552},[75,22844,9664],{"class":7387},[75,22846,22750],{"class":7394},[75,22848,19686],{"class":7387},[75,22850,22763],{"class":7394},[75,22852,22853],{"class":7398}," { }\n",[75,22855,22856],{"class":7383,"line":7557},[75,22857,7433],{"class":7398},[75,22859,22860,22862,22864,22866,22868],{"class":7383,"line":7562},[75,22861,9664],{"class":7387},[75,22863,22763],{"class":7394},[75,22865,19686],{"class":7387},[75,22867,22776],{"class":7394},[75,22869,8147],{"class":7398},[75,22871,22872,22874,22876,22878,22880,22882,22885],{"class":7383,"line":7571},[75,22873,7439],{"class":7387},[75,22875,7576],{"class":7398},[75,22877,7579],{"class":7394},[75,22879,8633],{"class":7398},[75,22881,8636],{"class":7387},[75,22883,22884],{"class":7589}," \"Student\"",[75,22886,18948],{"class":7398},[75,22888,22889],{"class":7383,"line":7584},[75,22890,7610],{"class":7398},[75,22892,22893],{"class":7383,"line":7602},[75,22894,7433],{"class":7398},[75,22896,22897,22899,22901,22903,22905],{"class":7383,"line":7607},[75,22898,9664],{"class":7387},[75,22900,22776],{"class":7394},[75,22902,19686],{"class":7387},[75,22904,20675],{"class":7394},[75,22906,8147],{"class":7398},[75,22908,22909,22911,22913,22915,22917,22919,22922],{"class":7383,"line":8360},[75,22910,7439],{"class":7387},[75,22912,7576],{"class":7398},[75,22914,7579],{"class":7394},[75,22916,8633],{"class":7398},[75,22918,8636],{"class":7387},[75,22920,22921],{"class":7589}," \"Person\"",[75,22923,18948],{"class":7398},[75,22925,22926],{"class":7383,"line":8374},[75,22927,7610],{"class":7398},[12,22929,9006],{},[6972,22931,22934],{"className":22932,"code":22933,"language":6977},[6975],"Student\nStudent\nPerson\njava.lang.Object@39274\n",[6979,22935,22933],{"__ignoreMap":180},[12,22937,22938,22939,9019,22942,22945,22946,22949,22950,22952,22953,22956],{},"Notice that even though the method signature says ",[6979,22940,22941],{},"m(Object x)",[6979,22943,22944],{},"toString()"," that runs depends on the actual object passed in. ",[6979,22947,22948],{},"GraduateStudent"," doesn't define ",[6979,22951,22944],{}," itself, so Java walks up the inheritance chain and finds ",[6979,22954,22955],{},"Student","'s version. That's dynamic binding in action.",[12,22958,22959,22960,22963,22964,7206,22966,22968],{},"This makes code wonderfully generic — ",[6979,22961,22962],{},"m"," works for ",[7289,22965,19592],{},[6979,22967,19871],{},", but does the right thing for each subtype.",[7166,22970,22972],{"id":22971},"the-full-visibility-modifier-table","The full visibility modifier table",[12,22974,22975,22976,7042,22978,22980],{},"I mentioned ",[6979,22977,7388],{},[6979,22979,7656],{}," earlier. There are actually four levels in Java:",[7085,22982,22983,23002],{},[7088,22984,22985],{},[7091,22986,22987,22990,22993,22996,22999],{},[7094,22988,22989],{},"Modifier",[7094,22991,22992],{},"Same class",[7094,22994,22995],{},"Same package",[7094,22997,22998],{},"Subclass",[7094,23000,23001],{},"Different package",[7104,23003,23004,23019,23035,23048],{},[7091,23005,23006,23010,23013,23015,23017],{},[7109,23007,23008],{},[6979,23009,7388],{},[7109,23011,23012],{},"✓",[7109,23014,23012],{},[7109,23016,23012],{},[7109,23018,23012],{},[7091,23020,23021,23026,23028,23030,23032],{},[7109,23022,23023],{},[6979,23024,23025],{},"protected",[7109,23027,23012],{},[7109,23029,23012],{},[7109,23031,23012],{},[7109,23033,23034],{},"✗",[7091,23036,23037,23040,23042,23044,23046],{},[7109,23038,23039],{},"(default)",[7109,23041,23012],{},[7109,23043,23012],{},[7109,23045,23034],{},[7109,23047,23034],{},[7091,23049,23050,23054,23056,23058,23060],{},[7109,23051,23052],{},[6979,23053,7656],{},[7109,23055,23012],{},[7109,23057,23034],{},[7109,23059,23034],{},[7109,23061,23034],{},[12,23063,23064,23068,23069,23072],{},[106,23065,23066],{},[6979,23067,23025],{}," is the new one. It's like the default, but it ",[7289,23070,23071],{},"also"," lets subclasses see the member, even if the subclass is in a different package. It's the right choice when you want to share something with subclasses but keep it hidden from the wider world.",[6964,23074,23076],{"id":23075},"part-4-abstract-classes-and-interfaces","Part 4: Abstract Classes and Interfaces",[7166,23078,23080],{"id":23079},"why-abstract","Why abstract?",[12,23082,23083,23084,23087,23088,23090,23091,23093,23094,23096,23097,23099],{},"Sometimes you want a superclass that exists ",[7289,23085,23086],{},"only"," to be inherited from — never to be instantiated directly. Think of ",[6979,23089,22085],{},". The very idea of a \"generic geometric object with no specific shape\" doesn't really make sense as a thing you'd create. A ",[6979,23092,18131],{}," makes sense. A ",[6979,23095,22091],{}," makes sense. But a plain ",[6979,23098,22085],{},"? Not so much.",[12,23101,23102,23103,23105],{},"That's where the ",[6979,23104,9514],{}," keyword comes in.",[7166,23107,23109],{"id":23108},"abstract-classes-and-abstract-methods","Abstract classes and abstract methods",[12,23111,15781,23112,23115,23116,23119,23120,23122],{},[106,23113,23114],{},"abstract method"," is a method declared without a body. It's a promise: \"every concrete subclass must provide this.\" An ",[106,23117,23118],{},"abstract class"," is a class that's marked ",[6979,23121,9514],{},", and may contain abstract methods.",[6972,23124,23126],{"className":7376,"code":23125,"language":7378,"meta":180,"style":180},"public abstract class GeometricObject {\n    private String color;\n    \u002F\u002F ...fields and concrete methods...\n \n    public abstract double getArea();        \u002F\u002F no body!\n    public abstract double getPerimeter();   \u002F\u002F no body!\n}\n",[6979,23127,23128,23141,23148,23153,23157,23173,23188],{"__ignoreMap":180},[75,23129,23130,23132,23135,23137,23139],{"class":7383,"line":7384},[75,23131,7388],{"class":7387},[75,23133,23134],{"class":7387}," abstract",[75,23136,7391],{"class":7387},[75,23138,22114],{"class":7394},[75,23140,8147],{"class":7398},[75,23142,23143,23145],{"class":7383,"line":181},[75,23144,7410],{"class":7387},[75,23146,23147],{"class":7398}," String color;\n",[75,23149,23150],{"class":7383,"line":7430},[75,23151,23152],{"class":7470},"    \u002F\u002F ...fields and concrete methods...\n",[75,23154,23155],{"class":7383,"line":7436},[75,23156,7433],{"class":7398},[75,23158,23159,23161,23163,23165,23167,23170],{"class":7383,"line":7458},[75,23160,7439],{"class":7387},[75,23162,23134],{"class":7387},[75,23164,14858],{"class":7387},[75,23166,21463],{"class":7394},[75,23168,23169],{"class":7398},"();        ",[75,23171,23172],{"class":7470},"\u002F\u002F no body!\n",[75,23174,23175,23177,23179,23181,23184,23186],{"class":7383,"line":7474},[75,23176,7439],{"class":7387},[75,23178,23134],{"class":7387},[75,23180,14858],{"class":7387},[75,23182,23183],{"class":7394}," getPerimeter",[75,23185,9268],{"class":7398},[75,23187,23172],{"class":7470},[75,23189,23190],{"class":7383,"line":7480},[75,23191,7610],{"class":7398},[12,23193,23194,23195,6959],{},"In UML diagrams, abstract classes and abstract methods are written in ",[7289,23196,23197],{},"italics",[7166,23199,23201],{"id":23200},"six-rules-about-abstract-classes-the-gotchas","Six rules about abstract classes (the gotchas)",[3603,23203,23204,23217,23256,23265,23277,23283],{},[26,23205,23206,23209,23210,23212,23213,23216],{},[106,23207,23208],{},"An abstract method cannot exist in a non-abstract class."," If you have even one abstract method, the class itself must be declared ",[6979,23211,9514],{},". And if a subclass of an abstract class doesn't implement ",[7289,23214,23215],{},"all"," the inherited abstract methods, that subclass must also be declared abstract.",[26,23218,23219,23222,23223,6959,23225,23246,23248,23249,23252,23253,23255],{},[106,23220,23221],{},"You cannot instantiate an abstract class"," with ",[6979,23224,8823],{},[6972,23226,23228],{"className":7376,"code":23227,"language":7378,"meta":180,"style":180},"GeometricObject g = new GeometricObject();  \u002F\u002F ❌ compile error\n",[6979,23229,23230],{"__ignoreMap":180},[75,23231,23232,23235,23237,23239,23241,23244],{"class":7383,"line":7384},[75,23233,23234],{"class":7398},"GeometricObject g ",[75,23236,7421],{"class":7387},[75,23238,7424],{"class":7387},[75,23240,22114],{"class":7394},[75,23242,23243],{"class":7398},"();  ",[75,23245,18813],{"class":7470},[16,23247],{},"But you ",[7289,23250,23251],{},"can"," still define constructors for it — they get invoked when subclasses are constructed (via the ",[6979,23254,22315],{}," chain).",[26,23257,23258,23261,23262,23264],{},[106,23259,23260],{},"An abstract class doesn't need to have any abstract methods."," It can be 100% concrete methods and still be marked ",[6979,23263,9514],{},". The point is just to prevent instantiation — typically because the class is meant only as a base for other classes.",[26,23266,23267,23270,23271,23273,23274,23276],{},[106,23268,23269],{},"A superclass of an abstract class may be concrete."," For example, ",[6979,23272,19871],{}," (the root of all Java classes) is concrete, but a class like ",[6979,23275,22085],{}," that extends it can still be abstract.",[26,23278,23279,23282],{},[106,23280,23281],{},"A concrete method can be overridden as abstract."," Rare, but useful when an inherited implementation no longer makes sense for a subclass. The subclass would then have to be declared abstract.",[26,23284,23285,23288,23289,23350,23352],{},[106,23286,23287],{},"An abstract class can be used as a data type."," You can't create instances of it directly, but you can declare variables and arrays of that type, then store subclass instances:",[6972,23290,23292],{"className":7376,"code":23291,"language":7378,"meta":180,"style":180},"GeometricObject[] shapes = new GeometricObject[10];\nshapes[0] = new Circle();\nshapes[1] = new Rectangle();\n",[6979,23293,23294,23315,23333],{"__ignoreMap":180},[75,23295,23296,23298,23301,23303,23305,23307,23310,23312],{"class":7383,"line":7384},[75,23297,22085],{"class":7387},[75,23299,23300],{"class":7398},"[] shapes ",[75,23302,7421],{"class":7387},[75,23304,7424],{"class":7387},[75,23306,22114],{"class":7387},[75,23308,23309],{"class":7398},"[",[75,23311,17383],{"class":8127},[75,23313,23314],{"class":7398},"];\n",[75,23316,23317,23320,23322,23325,23327,23329,23331],{"class":7383,"line":181},[75,23318,23319],{"class":7398},"shapes[",[75,23321,9180],{"class":8127},[75,23323,23324],{"class":7398},"] ",[75,23326,7421],{"class":7387},[75,23328,7424],{"class":7387},[75,23330,21369],{"class":7394},[75,23332,7549],{"class":7398},[75,23334,23335,23337,23339,23341,23343,23345,23348],{"class":7383,"line":7430},[75,23336,23319],{"class":7398},[75,23338,8855],{"class":8127},[75,23340,23324],{"class":7398},[75,23342,7421],{"class":7387},[75,23344,7424],{"class":7387},[75,23346,23347],{"class":7394}," Rectangle",[75,23349,7549],{"class":7398},[16,23351],{},"This is polymorphism at work.",[7166,23354,23355],{"id":9437},"Interfaces",[12,23357,15781,23358,23360,23361,23364],{},[106,23359,7058],{}," is a class-like construct that contains ",[106,23362,23363],{},"only constants and abstract methods"," — no real implementation, no constructors, no state. (Modern Java has loosened this slightly with default methods, but the original spirit is what matters.)",[12,23366,23367,23368,23371,23372,7298,23374,7298,23377,23380,23381,23384,23385,23388,23389,6959],{},"The intent of an interface is to specify ",[106,23369,23370],{},"common behavior"," that classes from totally different parts of the hierarchy might share. For example, ",[6979,23373,20016],{},[6979,23375,23376],{},"Edible",[6979,23378,23379],{},"Cloneable",". A ",[6979,23382,23383],{},"Chicken"," and a ",[6979,23386,23387],{},"Fruit"," don't share a common animal-or-plant superclass, but both can be ",[6979,23390,23376],{},[7166,23392,23394],{"id":23393},"defining-an-interface","Defining an interface",[6972,23396,23398],{"className":7376,"code":23397,"language":7378,"meta":180,"style":180},"public interface Edible {\n    \u002F** Describe how to eat *\u002F\n    public abstract String howToEat();\n}\n",[6979,23399,23400,23410,23415,23428],{"__ignoreMap":180},[75,23401,23402,23404,23406,23408],{"class":7383,"line":7384},[75,23403,7388],{"class":7387},[75,23405,16505],{"class":7387},[75,23407,19253],{"class":7394},[75,23409,8147],{"class":7398},[75,23411,23412],{"class":7383,"line":181},[75,23413,23414],{"class":7470},"    \u002F** Describe how to eat *\u002F\n",[75,23416,23417,23419,23421,23423,23426],{"class":7383,"line":7430},[75,23418,7439],{"class":7387},[75,23420,23134],{"class":7387},[75,23422,7576],{"class":7398},[75,23424,23425],{"class":7394},"howToEat",[75,23427,7549],{"class":7398},[75,23429,23430],{"class":7383,"line":7436},[75,23431,7610],{"class":7398},[12,23433,23434,23435,19610],{},"A class signs the contract using the ",[6979,23436,17838],{},[6972,23438,23440],{"className":7376,"code":23439,"language":7378,"meta":180,"style":180},"class Apple extends Fruit implements Edible {\n    @Override\n    public String howToEat() {\n        return \"Apple: Make apple cider\";\n    }\n}\n",[6979,23441,23442,23459,23465,23475,23484,23488],{"__ignoreMap":180},[75,23443,23444,23446,23448,23450,23453,23455,23457],{"class":7383,"line":7384},[75,23445,9664],{"class":7387},[75,23447,22637],{"class":7394},[75,23449,19686],{"class":7387},[75,23451,23452],{"class":7394}," Fruit",[75,23454,8539],{"class":7387},[75,23456,19253],{"class":7394},[75,23458,8147],{"class":7398},[75,23460,23461,23463],{"class":7383,"line":181},[75,23462,7565],{"class":7398},[75,23464,7568],{"class":7387},[75,23466,23467,23469,23471,23473],{"class":7383,"line":7430},[75,23468,7439],{"class":7387},[75,23470,7576],{"class":7398},[75,23472,23425],{"class":7394},[75,23474,7495],{"class":7398},[75,23476,23477,23479,23482],{"class":7383,"line":7436},[75,23478,7501],{"class":7387},[75,23480,23481],{"class":7589}," \"Apple: Make apple cider\"",[75,23483,8131],{"class":7398},[75,23485,23486],{"class":7383,"line":7458},[75,23487,7477],{"class":7398},[75,23489,23490],{"class":7383,"line":7474},[75,23491,7610],{"class":7398},[12,23493,23494,23495,23497,23498,23500],{},"Any class that ",[6979,23496,17838],{}," an interface must provide an implementation for ",[106,23499,7291],{}," method in that interface — or it must itself be declared abstract.",[7166,23502,23504],{"id":23503},"extends-vs-implements-the-key-rules","Extends vs. implements — the key rules",[12,23506,23507],{},"This trips a lot of people up:",[23,23509,23510,23519,23527,23534],{},[26,23511,23512,23513,7206,23515,23518],{},"A class can ",[6979,23514,19609],{},[106,23516,23517],{},"at most one"," class (single inheritance).",[26,23520,23512,23521,7206,23523,23526],{},[6979,23522,17838],{},[106,23524,23525],{},"any number"," of interfaces (multiple).",[26,23528,23529,23530,7206,23532,23526],{},"An interface can ",[6979,23531,19609],{},[106,23533,23525],{},[26,23535,23536,23537,7206,23539,23541],{},"An interface ",[106,23538,20581],{},[6979,23540,17838],{}," an interface.\nPutting it together:",[6972,23543,23545],{"className":7376,"code":23544,"language":7378,"meta":180,"style":180},"public class NewClass extends BaseClass\n        implements Interface1, Interface2, InterfaceN {\n    \u002F\u002F ...\n}\n",[6979,23546,23547,23561,23581,23585],{"__ignoreMap":180},[75,23548,23549,23551,23553,23556,23558],{"class":7383,"line":7384},[75,23550,7388],{"class":7387},[75,23552,7391],{"class":7387},[75,23554,23555],{"class":7394}," NewClass",[75,23557,19686],{"class":7387},[75,23559,23560],{"class":7394}," BaseClass\n",[75,23562,23563,23566,23569,23571,23574,23576,23579],{"class":7383,"line":181},[75,23564,23565],{"class":7387},"        implements",[75,23567,23568],{"class":7394}," Interface1",[75,23570,7298],{"class":7398},[75,23572,23573],{"class":7394},"Interface2",[75,23575,7298],{"class":7398},[75,23577,23578],{"class":7394},"InterfaceN",[75,23580,8147],{"class":7398},[75,23582,23583],{"class":7383,"line":7430},[75,23584,17867],{"class":7470},[75,23586,23587],{"class":7383,"line":7436},[75,23588,7610],{"class":7398},[12,23590,23591],{},"And for interfaces extending other interfaces:",[6972,23593,23595],{"className":7376,"code":23594,"language":7378,"meta":180,"style":180},"public interface NewInterface extends Interface1, Interface2 {\n    \u002F\u002F constants and abstract methods\n}\n",[6979,23596,23597,23616,23621],{"__ignoreMap":180},[75,23598,23599,23601,23603,23606,23608,23610,23612,23614],{"class":7383,"line":7384},[75,23600,7388],{"class":7387},[75,23602,16505],{"class":7387},[75,23604,23605],{"class":7394}," NewInterface",[75,23607,19686],{"class":7387},[75,23609,23568],{"class":7394},[75,23611,7298],{"class":7398},[75,23613,23573],{"class":7394},[75,23615,8147],{"class":7398},[75,23617,23618],{"class":7383,"line":181},[75,23619,23620],{"class":7470},"    \u002F\u002F constants and abstract methods\n",[75,23622,23623],{"class":7383,"line":7430},[75,23624,7610],{"class":7398},[7166,23626,23628],{"id":23627},"omitting-modifiers-in-interfaces","Omitting modifiers in interfaces",[12,23630,23631,23632,23634,23635,23638,23639,23641,23642,23645],{},"In an interface, ",[106,23633,7291],{}," field is implicitly ",[6979,23636,23637],{},"public static final"," (i.e., a constant), and ",[106,23640,7291],{}," method is implicitly ",[6979,23643,23644],{},"public abstract",". So you can leave those modifiers off:",[6972,23647,23649],{"className":7376,"code":23648,"language":7378,"meta":180,"style":180},"public interface T1 {\n    public static final int K = 1;\n    public abstract void p();\n}\n \n\u002F\u002F is exactly equivalent to:\n \npublic interface T1 {\n    int K = 1;\n    void p();\n}\n",[6979,23650,23651,23662,23681,23694,23698,23702,23707,23711,23721,23733,23741],{"__ignoreMap":180},[75,23652,23653,23655,23657,23660],{"class":7383,"line":7384},[75,23654,7388],{"class":7387},[75,23656,16505],{"class":7387},[75,23658,23659],{"class":7394}," T1",[75,23661,8147],{"class":7398},[75,23663,23664,23666,23668,23670,23672,23675,23677,23679],{"class":7383,"line":181},[75,23665,7439],{"class":7387},[75,23667,8155],{"class":7387},[75,23669,17824],{"class":7387},[75,23671,7531],{"class":7387},[75,23673,23674],{"class":7398}," K ",[75,23676,7421],{"class":7387},[75,23678,10486],{"class":8127},[75,23680,8131],{"class":7398},[75,23682,23683,23685,23687,23689,23692],{"class":7383,"line":7430},[75,23684,7439],{"class":7387},[75,23686,23134],{"class":7387},[75,23688,7442],{"class":7387},[75,23690,23691],{"class":7394}," p",[75,23693,7549],{"class":7398},[75,23695,23696],{"class":7383,"line":7436},[75,23697,7610],{"class":7398},[75,23699,23700],{"class":7383,"line":7458},[75,23701,7433],{"class":7398},[75,23703,23704],{"class":7383,"line":7474},[75,23705,23706],{"class":7470},"\u002F\u002F is exactly equivalent to:\n",[75,23708,23709],{"class":7383,"line":7480},[75,23710,7433],{"class":7398},[75,23712,23713,23715,23717,23719],{"class":7383,"line":7485},[75,23714,7388],{"class":7387},[75,23716,16505],{"class":7387},[75,23718,23659],{"class":7394},[75,23720,8147],{"class":7398},[75,23722,23723,23725,23727,23729,23731],{"class":7383,"line":7498},[75,23724,22416],{"class":7387},[75,23726,23674],{"class":7398},[75,23728,7421],{"class":7387},[75,23730,10486],{"class":8127},[75,23732,8131],{"class":7398},[75,23734,23735,23737,23739],{"class":7383,"line":7516},[75,23736,19720],{"class":7387},[75,23738,23691],{"class":7394},[75,23740,7549],{"class":7398},[75,23742,23743],{"class":7383,"line":7521},[75,23744,7610],{"class":7398},[12,23746,23747,23748,6959],{},"You access an interface's constant via the interface name: ",[6979,23749,23750],{},"T1.K",[7166,23752,23754],{"id":23753},"interface-vs-abstract-class-when-to-use-which","Interface vs. Abstract Class — when to use which?",[7085,23756,23757,23775],{},[7088,23758,23759],{},[7091,23760,23761,23765,23770],{},[7094,23762,23763],{},[16,23764],{},[7094,23766,23767],{},[106,23768,23769],{},"Abstract Class",[7094,23771,23772],{},[106,23773,23774],{},"Interface",[7104,23776,23777,23791,23806,23819],{},[7091,23778,23779,23782,23785],{},[7109,23780,23781],{},"Variables",[7109,23783,23784],{},"No restrictions",[7109,23786,23787,23788,23790],{},"Must be ",[6979,23789,23637],{}," (constants only)",[7091,23792,23793,23795,23803],{},[7109,23794,21527],{},[7109,23796,23797,23798,23800,23801,6959],{},"Yes (called by subclasses via ",[6979,23799,22315],{},"). Cannot instantiate with ",[6979,23802,8823],{},[7109,23804,23805],{},"No constructors. Cannot instantiate.",[7091,23807,23808,23811,23814],{},[7109,23809,23810],{},"Methods",[7109,23812,23813],{},"No restrictions (abstract or concrete)",[7109,23815,23816,23817],{},"All methods are implicitly ",[6979,23818,23644],{},[7091,23820,23821,23823,23829],{},[7109,23822,22081],{},[7109,23824,23825,23826,23828],{},"A class can extend ",[106,23827,22239],{}," abstract class",[7109,23830,23831,23832,23835],{},"A class can implement ",[106,23833,23834],{},"many"," interfaces",[12,23837,23838],{},"A rough rule of thumb:",[23,23840,23841,23850],{},[26,23842,23843,23846,23847,23849],{},[106,23844,23845],{},"Use an abstract class"," when subclasses share a common identity and a chunk of common implementation (e.g., all kinds of ",[6979,23848,22085],{}," share color and filled state).",[26,23851,23852,23855,23856,23859,23860,7298,23862,7298,23864,23867,23868,23871,23872,6959],{},[106,23853,23854],{},"Use an interface"," when classes from different parts of the type hierarchy need to share a ",[7289,23857,23858],{},"capability"," (e.g., ",[6979,23861,20016],{},[6979,23863,23376],{},[6979,23865,23866],{},"Runnable","). Interfaces describe what something ",[7289,23869,23870],{},"can do",", regardless of what it ",[7289,23873,17990],{},[6964,23875,23877],{"id":23876},"quick-recap-aka-my-cheat-sheet","Quick Recap (a.k.a. my cheat sheet)",[12,23879,23880],{},"A quick reference I'll come back to before exams:",[23,23882,23883,23892,23895,23900,23907,23910,23925,23932,23942,23947,23955,23962,23973,23979,23984],{},[26,23884,23885,23888,23889,23891],{},[106,23886,23887],{},"Class"," = blueprint. ",[106,23890,19871],{}," = instance of a class.",[26,23893,23894],{},"An object has identity, state (fields), and behavior (methods).",[26,23896,23897,23898,6959],{},"Constructors share the class name, have no return type, are called with ",[6979,23899,8823],{},[26,23901,23902,23903,23906],{},"A default no-arg constructor only exists if you write ",[7289,23904,23905],{},"no"," constructors yourself.",[26,23908,23909],{},"Object variables hold references, not values. Assigning one to another shares the same object.",[26,23911,23912,23913,23915,23916,13084,23918,23921,23922,6959],{},"Make fields ",[6979,23914,7656],{}," and expose them through ",[6979,23917,13819],{},[6979,23919,23920],{},"set"," methods. This is ",[106,23923,23924],{},"encapsulation",[26,23926,23927,10476,23929,23931],{},[106,23928,22081],{},[6979,23930,19609],{},") lets a subclass reuse and extend a superclass.",[26,23933,23934,23935,23938,23939,6959],{},"Java has ",[106,23936,23937],{},"single inheritance for classes"," but ",[106,23940,23941],{},"multiple inheritance for interfaces",[26,23943,23944,23946],{},[6979,23945,22315],{}," calls the parent constructor. It's auto-inserted if you omit it (provided a no-arg parent constructor exists).",[26,23948,23949,23951,23952,23954],{},[106,23950,22392],{}," = same name, different parameters. ",[106,23953,22487],{}," = subclass replaces a parent method (same name, same parameters, same return type).",[26,23956,23957,23959,23960,6959],{},[106,23958,22593],{}," lets a supertype variable hold a subtype object. The actual method that runs is decided at runtime — ",[106,23961,22699],{},[26,23963,23964,23965,23967,23968,23970,23971,6959],{},"Visibility ladder, most to least open: ",[6979,23966,7388],{}," > ",[6979,23969,23025],{}," > (default) > ",[6979,23972,7656],{},[26,23974,23975,23978],{},[106,23976,23977],{},"Abstract class"," — cannot be instantiated, may have abstract methods, can have state and constructors.",[26,23980,23981,23983],{},[106,23982,23774],{}," — only constants and abstract method signatures (classically); a pure contract.",[26,23985,23986],{},"Use abstract classes for \"is-a\" with shared implementation. Use interfaces for \"can-do\" capabilities.",[36,23988],{},[12,23990,23991],{},"That's the whole OOP foundation. Master these and the rest of Data Structures starts to feel a lot less mysterious — because every data structure we'll meet later (linked lists, stacks, queues, trees) is just a class with thoughtfully chosen fields and methods, often built using inheritance or interfaces.",[9332,23993,23994],{},"html pre.shiki code .szBVR, html code.shiki .szBVR{--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .sScJk, html code.shiki .sScJk{--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sVt8B, html code.shiki .sVt8B{--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sJ8bj, html code.shiki .sJ8bj{--shiki-default:#6A737D;--shiki-dark:#6A737D}html pre.shiki code .sj4cs, html code.shiki .sj4cs{--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .s4XuR, html code.shiki .s4XuR{--shiki-default:#E36209;--shiki-dark:#FFAB70}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: 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.sZZnC{--shiki-default:#032F62;--shiki-dark:#9ECBFF}",{"title":180,"searchDepth":181,"depth":181,"links":23996},[23997,24010,24013,24022,24032],{"id":21264,"depth":181,"text":21265,"children":23998},[23999,24000,24001,24003,24004,24005,24006,24007,24009],{"id":21268,"depth":7430,"text":21269},{"id":21316,"depth":7430,"text":21317},{"id":21351,"depth":7430,"text":24002},"A first example: the Circle class",{"id":21524,"depth":7430,"text":21527},{"id":21569,"depth":7430,"text":21570},{"id":21670,"depth":7430,"text":21671},{"id":21707,"depth":7430,"text":21708},{"id":21871,"depth":7430,"text":24008},"Visibility modifiers: public vs. private",{"id":21967,"depth":7430,"text":21968},{"id":22020,"depth":181,"text":22021,"children":24011},[24012],{"id":22024,"depth":7430,"text":22025},{"id":22068,"depth":181,"text":22069,"children":24014},[24015,24016,24017,24018,24019,24020,24021],{"id":22072,"depth":7430,"text":22073},{"id":22101,"depth":7430,"text":22102},{"id":22254,"depth":7430,"text":22255},{"id":22383,"depth":7430,"text":22384},{"id":22592,"depth":7430,"text":22593},{"id":22669,"depth":7430,"text":22670},{"id":22971,"depth":7430,"text":22972},{"id":23075,"depth":181,"text":23076,"children":24023},[24024,24025,24026,24027,24028,24029,24030,24031],{"id":23079,"depth":7430,"text":23080},{"id":23108,"depth":7430,"text":23109},{"id":23200,"depth":7430,"text":23201},{"id":9437,"depth":7430,"text":23355},{"id":23393,"depth":7430,"text":23394},{"id":23503,"depth":7430,"text":23504},{"id":23627,"depth":7430,"text":23628},{"id":23753,"depth":7430,"text":23754},{"id":23876,"depth":181,"text":23877},"2026-05-11",{},"\u002Fblog\u002FData-Structure-Concept-Revision",{"title":21259,"description":180},{"loc":24035},"blog\u002FData-Structure-Concept-Revision","O07n9DHto-cnoQjzvGL87VHiantvLf4hix0EYQVIajo",{"id":24041,"title":24042,"author":7,"body":24043,"date":24182,"description":24183,"draft":185,"edited_at":13169,"extension":186,"featured_image":24184,"meta":24185,"navigation":189,"path":24186,"pinned":185,"seo":24187,"sitemap":24188,"stem":24189,"tags":194,"__hash__":24190},"blog\u002Fblog\u002FSCI（二）受傷機制——創傷性（頸椎）.md","SCI（二）受傷機制--i",{"type":9,"value":24044,"toc":24180},[24045,24048,24051,24087,24089,24106],[12,24046,24047],{},"接下來講受傷機制，首先是分類，分成了創傷性跟非創傷性，先說創傷性",[12,24049,24050],{},"創傷性機制（最常見的SCI類型，如車禍、跌倒、暴力事件、運動傷害）",[23,24052,24053,24056,24059,24073],{},[26,24054,24055],{},"從受力方式上分成了屈曲（或加上旋轉）、過度伸展、壓迫（compression）三種類形",[26,24057,24058],{},"物理成因有，壓迫、牽拉、橫斷、血流供應中斷",[26,24060,24061,24062],{},"損傷模式分成",[23,24063,24064,24067,24070],{},[26,24065,24066],{},"單一（只有一塊椎骨）\u002F連續（附帶上鄰近的椎骨）：最常見",[26,24068,24069],{},"不連續（不連續的受傷節斷，中間由正常的節斷分開）：3到5%",[26,24071,24072],{},"SCIWORA（無影像異常之SCI）：假如是成人的話，多數是退化性病變或椎管狹窄；兒童的話，因為靈活性跟彈性過高導致（已拉傷脊髓）",[26,24074,24075,24076],{},"那從Level來看又分成了頸椎胸椎腰椎的損傷",[23,24077,24078,24081,24084],{},[26,24079,24080],{},"頸椎常見的受力方式：屈曲（或合併旋轉），垂直擠壓，伸展",[26,24082,24083],{},"胸椎的常見受力方式：屈曲（或合併旋轉），垂直擠壓，伸展或側向伸展",[26,24085,24086],{},"腰椎的常見受力方式：屈曲（或合併旋轉，或合併牽拉），剪力，垂直擠壓",[36,24088],{},[12,24090,24091,24092,24094,24095,24097,24098,24102,24103,24105],{},"那頸椎的受傷佔SCI受傷的52%，因為頸椎靈活度高，穩定度較差",[16,24093],{},"\n其中下頸椎的SCI傷害又更容易存活；",[16,24096],{},"\n假如是",[75,24099,24101],{"className":24100},[78,5387],"C1 C2骨折的患者較少會出現神經功能缺失","：因為在顱頸交界椎管空間較大，脊髓僅佔50%",[16,24104],{},"\n不過假如是高位損傷通常難以存活，除非趕緊做心肺復甦",[23,24107,24108,24142,24159],{},[26,24109,24110,24111,24113,24114],{},"那先從屈曲傷害來看，也是頸椎最常見的傷害，以神經性損傷來看頸椎傷害也是最高發生率的",[16,24112],{},"\n而最常見的凹曲傷害是快速減速時，例如追撞前車，會導致前方被擠壓，後方被拉開",[23,24115,24116,24123,24126,24134],{},[26,24117,24118,24119],{},"楔形骨折（穩定型）：就是脊椎前面受嚴重擠壓導致前半部塌陷，一般",[75,24120,24122],{"className":24121},[78,173],"沒有神經損傷問題",[26,24124,24125],{},"屈曲形淚滴狀骨折（不穩定型）：從前方裂開；通常是矢狀骨折，並且會跑到後方，進入椎管；需立即手術處置，嚴重的SCI基本都是這類型",[26,24127,24128,24129,24133],{},"雙側小面關節脫位（極度不穩定型）：因過度屈曲導致",[75,24130,24132],{"className":24131},[78,173],"全部韌帶斷裂","；需立即手術處置，基本會有嚴重的神經性問題",[26,24135,24136,24137],{},"屈曲合併旋轉型",[23,24138,24139],{},[26,24140,24141],{},"單側小面關節脫位（穩定型，除非有顯著韌帶傷害）：小面關節囊跟後方韌帶斷裂；有30%的神經損傷，例如布朗塞加爾症候群或者神經根傷害",[26,24143,24144,24145,24149],{},"接著是垂直擠壓傷害（軸向負荷），是直立（或帶有些微屈）的垂直力導致爆裂性骨折；常發生在跳水進入淺水區頭著地，",[75,24146,24148],{"className":24147},[78,5387],"C4 C5最常受傷伴有完全性四肢癱瘓",[23,24150,24151],{},[26,24152,24153,24154,24158],{},"爆裂性骨折：屬",[75,24155,24157],{"className":24156},[78,173],"粉碎性骨折","，骨頭碎片也會被推到後方進入椎管，導致神經性傷害",[26,24160,24161,24162],{},"最後來看伸直傷害，跌倒時下巴或額頭著地，或後方被追撞導致；常見於C4 C5；與屈曲相反，是前方被拉開，後方被擠壓",[23,24163,24164,24177],{},[26,24165,24166,24167,24171,24172,24176],{},"伸展型淚滴狀骨折（",[75,24168,24170],{"className":24169},[78,173],"伸展時不穩定型","）：椎體前方會有一小塊脫落，伴有",[75,24173,24175],{"className":24174},[78,173],"前縱韌帶和椎間盤連接處","斷裂；",[26,24178,24179],{},"過度伸展型脫位，扭傷或拉傷：前縱韌帶嚴重斷裂，上方的椎骨會跑到後面；前側受到椎體，椎間盤碎片和骨刺的壓迫，後側受黃韌帶壓迫",{"title":180,"searchDepth":181,"depth":181,"links":24181},[],"2026-05-09","說到了頸椎，下一章再寫胸腰椎","\u002Fimages\u002Fuploads\u002F1778470525801-119694510_p0_master1200.webp",{},"\u002Fblog\u002FSCI（二）受傷機制——創傷性（頸椎）",{"title":24042,"description":24183},{"loc":24186},"blog\u002FSCI（二）受傷機制——創傷性（頸椎）","YoNuiIPw6qMFHGtt2UYpH1M_xOD00aFkDaEmbe2NG70",{"id":24192,"title":24193,"author":7,"body":24194,"date":24304,"description":24305,"draft":185,"edited_at":24033,"extension":186,"featured_image":24306,"meta":24307,"navigation":189,"path":24308,"pinned":185,"seo":24309,"sitemap":24310,"stem":24311,"tags":194,"__hash__":24312},"blog\u002Fblog\u002FCVA（十二）輔助用品.md","CVA（十二）輔助用品",{"type":9,"value":24195,"toc":24302},[24196,24199,24202,24223,24226,24273,24284,24296],[12,24197,24198],{},"最後來講到輔助用品的部份",[12,24200,24201],{},"首先是床邊的用品，毛巾，枕頭，毯子",[12,24203,24204,24205,24207,24208,24210,24211,24213,24214,24216,24217,24219,24220,24222],{},"接下來是輪椅類型，分成以下",[16,24206],{},"\nSolid surface to sit（硬坐墊）",[16,24209],{},"\nSupportive backrest（有支撐的背靠）",[16,24212],{},"\nLower seat height and removable and rests（低椅可以讓健側腳也參與移動，還有可以拆下來方便側向轉移位的扶手）",[16,24215],{},"\nOne armed drive: two hand rims on one wheel（可以單手控制兩個輪子來移動）",[16,24218],{},"\nLapboard（膝上桌板）",[16,24221],{},"\nHalf-lapboard or arm trough（半身膝上板或手臂支撐槽）",[12,24224,24225],{},"然後是吊帶",[3603,24227,24228,24236,24244,24251,24259,24267,24270],{},[26,24229,24230,24231,24235],{},"Shoulder immobilizer sling",[75,24232,24234],{"className":24233},[1149],"（不推薦）","，因為會誘發Flexor tone",[26,24237,24238,24239,24243],{},"Clavicle support",[75,24240,24242],{"className":24241},[5387],"（急性期用）","，Supine extend, scapula adduct",[26,24245,24246,24247,24250],{},"Universal cuff and collar sling",[75,24248,24234],{"className":24249},[1149],"，類似1，肱骨抬起",[26,24252,24253,24254,24258],{},"Humeral cuff",[75,24255,24257],{"className":24256},[6760],"（復健期用）","，手臂上抬",[26,24260,24261,24262,24266],{},"Unilateral shoulder orthosis，",[75,24263,24265],{"className":24264},[6760],"萊卡材質","，可以拉伸後回塑，之後能緊緊包覆提供支撐",[26,24268,24269],{},"Shoulder-saddle sling",[26,24271,24272],{},"GivMohr Sling，有在手上打八字，有可能會誘發手指握緊",[12,24274,24275,24276,24278,24279,24283],{},"來到拐杖，注意的點是，拐杖作用是給予更大的BOS，而不是用來輔助承重的",[16,24277],{},"\n在",[75,24280,24282],{"className":24281},[78,79],"上半身控制訓練完，以及下肢開始學習移動時","使用",[12,24285,24286,24287,24289,24290,24292,24293,24295],{},"然後是矯具",[16,24288],{},"\n首先是AFO，確保站立初期可以腳跟著地，擺盪期不會腳尖拖地，嚴重無力病患的早期訓練可以給予踝關節遠端支持，可以控制因為無法背屈而導致的knee過伸",[16,24291],{},"\n繃帶也能起到同樣效果，繃帶更好的一點就是它能用包在鞋子外面",[16,24294],{},"\n最後Arthrofix R ankle and tarsus orthosis這個玩意，他能夠提供踝部和後足穩定的同時，給前足活動度，讓推進更加自然，穿戴也僅需單手即可",[12,24297,24298,24299,24301],{},"接著是一些可移動的器材",[16,24300],{},"\n包括彈力球，彈玻泡棉滾筒，Castered adjustable stools（有滾輪的椅子）",{"title":180,"searchDepth":181,"depth":181,"links":24303},[],"2026-05-07","Assistive Equipment","\u002Fimages\u002Fuploads\u002F1778470849108-118136560_p0_master1200.webp",{},"\u002Fblog\u002FCVA（十二）輔助用品",{"title":24193,"description":24305},{"loc":24308},"blog\u002FCVA（十二）輔助用品","H4hRX6NTOYNNOe_LD2hNNvD1yqVxZAET3JcYnbJbWzM",{"id":24314,"title":24315,"author":7,"body":24316,"date":24467,"description":24468,"draft":185,"edited_at":24304,"extension":186,"featured_image":24469,"meta":24470,"navigation":189,"path":24471,"pinned":185,"seo":24472,"sitemap":24473,"stem":24474,"tags":194,"__hash__":24475},"blog\u002Fblog\u002FCVA（十一）功能性走路ii.md","CVA（十一）功能性走路ii",{"type":9,"value":24317,"toc":24465},[24318,24321,24324,24330,24341,24344,24355,24361,24403,24405],[12,24319,24320],{},"偏癱的步態：下肢符合伸直協同動作，異常的手臂擺動（上肢協同動作），步速慢（步長變短，支撐期增加還有患側畫圈）",[12,24322,24323],{},"代償會變成：Hip hiking，好邊承重偏好，沒有Heel Strike，膝關節過度伸直",[12,24325,24326,24327,24329],{},"接著來看Brunnstrom的步態評估",[16,24328],{},"\n先看站立期",[23,24331,24332,24335,24338],{},[26,24333,24334],{},"Hip：Trendelenburg(臀中肌步態), Trunk forward(Hip不伸直), 穩定\u002F接近正常",[26,24336,24337],{},"Knee：Knee buckles(無法伸直), Hyperextension(mild輕度, moderate中度, servere嚴重), stable of slight flexion微屈下穩定, Near normal",[26,24339,24340],{},"Ankle：Entire sole down（全腳掌著地）, Toe first（腳趾先著地）,內翻足（著地前期，全程內翻）,站立中期外翻，腳趾腳跟接近正常步態",[12,24342,24343],{},"接著是擺盪期",[23,24345,24346,24349,24352],{},[26,24347,24348],{},"Hip:：Circumduction（畫圈步態）,pelvis hike, adduction(剪刀腳), 接近正常, Exaggerated flexion（過度屈曲）, External rotation",[26,24350,24351],{},"Knee：Stiff（無法伸直）, Moderately stiff, 接近正常, Exaggerated flexion（過度屈曲）",[26,24353,24354],{},"Ankle：Toe dragging（腳趾拖地）, Inversion, Exaggerated dorsiflexion（過度背屈）, Whip, Eversion, 接近正常",[12,24356,24357,24358,24360],{},"然後李老師有在這裡給補充原理，那我也寫一下好了，在moodle是叫0510a的pdf",[16,24359],{},"\n一樣是先站立期",[23,24362,24363,24366,24369,24386],{},[26,24364,24365],{},"Trunk and Hip：對側本體覺不佳，軀幹前傾（Hip extension無力，Flexion contracture）",[26,24367,24368],{},"Hip：位置不佳（本體覺不佳），臀中肌步態（外展肌無力），剪刀腳（內收肌痙攣）",[26,24370,24371,24372],{},"Knee：",[23,24373,24374,24380],{},[26,24375,24376,24377,24379],{},"前進過程中Knee Flexion：Flexion Contracture，Hip knee無力，本體覺不佳，Ankle背屈範圍超過正中位，伸直模式或選擇性動作無力",[16,24378],{},"\n伸直肌無力或踝蹠屈",[26,24381,24382,24383,24385],{},"前進過程中Knee Hyperextension：蹠屈痙攣超過90度，本體覺不佳（膝蓋搖晃或者突然回彈+後翻），股四頭肌痙攣",[16,24384],{},"\n膝伸直肌無力（代償將膝蓋鎖在過伸位置）",[26,24387,24388,24389],{},"Ankle：",[23,24390,24391,24394,24397,24400],{},[26,24392,24393],{},"馬蹄足步態：腓腸或和比目魚肌痙攣或攣縮",[26,24395,24396],{},"足內翻（足外側承重）：脛前肌，脛後肌，屈趾肌，和比目魚肌過度活躍或痙攣",[26,24398,24399],{},"Step Length（步長）不等：屈趾肌痙攣導致錘狀趾，因為痛或重量壓在屈曲的腳趾上，使患者無法跨步",[26,24401,24402],{},"患側缺乏背屈：需10度背屈",[12,24404,24343],{},[23,24406,24407,24410,24427,24446],{},[26,24408,24409],{},"Hip and Pelvis：骨盤前轉不足（腹直無力），向健側彎使患側足部離地（屈肌群無力）",[26,24411,24412,24413],{},"Hip：",[23,24414,24415,24418,24421],{},[26,24416,24417],{},"Flex不足：Hip屈肌無力，本體覺不佳，肌四頭肌痙攣，腹肌無力（提髖肌），對側Hip abductor無力，",[26,24419,24420],{},"異常代償：畫圈步態，外旋\u002F內收，軀幹後傾\u002F拖足，慣性作用\u002F無法控制的擺盪",[26,24422,24423],{},[75,24424,24426],{"className":24425},[1149],"過度Hip Flexion：Flexor 協同動作過強",[26,24428,24371,24429],{},[23,24430,24431,24434,24440,24443],{},[26,24432,24433],{},"Knee屈曲不足：Hip屈曲不足且足部離地困，肌四頭肌痙攣",[26,24435,24436],{},[75,24437,24439],{"className":24438},[1149],"Knee過屈且延遲：Flexor協同動作過強",[26,24441,24442],{},"承重時Knee Extend不足：腿後肌痙攣，持續flexor pattern",[26,24444,24445],{},"膝伸直肌無力或本體覺不佳",[26,24447,24388,24448],{},[23,24449,24450,24453,24456,24459],{},[26,24451,24452],{},"持續性馬蹄足和\u002F或內翻足：蹠屈肌攣縮或痙攣，背屈肌無力，背屈肌收縮延遲，擺盪中期拖足",[26,24454,24455],{},"內翻：脛前肌痙攣，腓骨肌與伸趾肌無力",[26,24457,24458],{},"馬蹄內翻足：脛後肌和\u002F或腓腸肌與比目魚肌痙攣",[26,24460,24461],{},[75,24462,24464],{"className":24463},[1149],"過度背屈：Flexor 協同動作過大",{"title":180,"searchDepth":181,"depth":181,"links":24466},[],"2026-05-06","病理步態","\u002Fimages\u002Fuploads\u002F1778138058109-EeQJUMiVoAkcx6-.webp",{},"\u002Fblog\u002FCVA（十一）功能性走路ii",{"title":24315,"description":24468},{"loc":24471},"blog\u002FCVA（十一）功能性走路ii","2N6kx5xh_TSXVyEu9f3VvRTlCrgCw6EcbUvSJdclL9A",{"id":24477,"title":24478,"author":7,"body":24479,"date":24585,"description":24586,"draft":185,"edited_at":24304,"extension":186,"featured_image":24587,"meta":24588,"navigation":189,"path":24589,"pinned":185,"seo":24590,"sitemap":24591,"stem":24592,"tags":194,"__hash__":24593},"blog\u002Fblog\u002FCVA（十）功能性走路.md","CVA（十）功能性走路",{"type":9,"value":24480,"toc":24583},[24481,24484,24490,24502,24525,24528,24551,24554,24556,24577],[12,24482,24483],{},"Locomotion是移動，Gait是直立移動的形態，Walking是其中一種Gait形態，Ambulation跟Walking同義但多為醫用語",[12,24485,24486,24487,24489],{},"Step to follow要求較高，走路必須為Functional Walking（省力的），所以追求的不止是能走，是更接近正常人走",[16,24488],{},"\nStep Length是左腳跟到右腳跟的距離，Stride Length是左腳到下一次左腳的距離",[12,24491,24492,24493,24495,24496,24498,24499,24501],{},"那李老師說，步態周期是七期，Initial contact只是一個事件，用來區分開Terminal swing跟Loading response的",[16,24494],{},"\nMid-stance跟Terminal Stance兩期剛好40%，對應對側腳在做Swing phase全部",[16,24497],{},"\n所以是分成以下周期跟區分開來的事件",[16,24500],{},"\n步態周期：",[3603,24503,24504,24507,24510,24513,24516,24519,24522],{},[26,24505,24506],{},"Loading response 10%",[26,24508,24509],{},"Mid-stance 20%",[26,24511,24512],{},"Terminal stance 20%",[26,24514,24515],{},"Pre-swing 10%",[26,24517,24518],{},"Initial swing 13%",[26,24520,24521],{},"Mid-swing 14%",[26,24523,24524],{},"Terminal swing 13%",[12,24526,24527],{},"區分的事件為",[3603,24529,24530,24533,24536,24539,24542,24545,24548],{},[26,24531,24532],{},"Initial contact——區分Terminal swing和Loading Response",[26,24534,24535],{},"Opposite toe-off——區分Loading Response和Mid-stance",[26,24537,24538],{},"Heel rise——區分Mid-stance和Terminal stance",[26,24540,24541],{},"Opposite initial contact——區分Terminal stance和Pre-swing",[26,24543,24544],{},"Toe-off——區分Preswing和Initial swing",[26,24546,24547],{},"Heel adjacent——區分Initial swing和Mid-swing",[26,24549,24550],{},"Tibia vertical——區分Mid-swing和Terminal swing",[12,24552,24553],{},"病理步態我下一篇再寫",[36,24555],{},[12,24557,24558,24559,24561,24562,24564,24565,24567,24568,24570,24571,24573,24574,24576],{},"接下來講甚麼時候訓練病人開始功能性步態",[16,24560],{},"\n一般病人可以自行站直不需要協助，可以移動下肢，手支撐是最小程度",[16,24563],{},"\n壞腳能承重，不需要拐杖也能跨出好腳",[16,24566],{},"\n壞腳承重時，膝不會hyperextension，ankle不會planterflexion",[16,24569],{},"\n壞腳跨步時，軀幹不會扭曲，PT也不需要協助壞腳做跨步",[16,24572],{},"\n適當協助下，pt能走出正常pattern，且下肢張力不會太強",[16,24575],{},"\n不必強迫pt走以免讓其感到恐懼，以免影響日後訓練",[12,24578,24579,24580,24582],{},"功能性走路的目的有",[16,24581],{},"\n安全，相對省力（功能性步態本身就是省力的），速度夠快，走路外觀相對自然，盡可能獨立行走（無需拐杖），自動化（不用看地板，不用想）",{"title":180,"searchDepth":181,"depth":181,"links":24584},[],"2026-05-05","定義和周期","\u002Fimages\u002Fuploads\u002F1778044035773-Eg_o2kCWsAQCpjp.webp",{},"\u002Fblog\u002FCVA（十）功能性走路",{"title":24478,"description":24586},{"loc":24589},"blog\u002FCVA（十）功能性走路","lcPEwi-cJ5du6GsTqEm7B7UW4TLUVD65NLqm2Uv5aI0",{"id":24595,"title":24596,"author":7,"body":24597,"date":24879,"description":24880,"draft":185,"edited_at":17971,"extension":186,"featured_image":24881,"meta":24882,"navigation":189,"path":24883,"pinned":185,"seo":24884,"sitemap":24885,"stem":24886,"tags":194,"__hash__":24887},"blog\u002Fblog\u002F脊髓損傷SCI（一）.md","脊髓損傷SCI（一）",{"type":9,"value":24598,"toc":24877},[24599,24602,24604,24613,24622,24625,24631,24633,24636,24639,24642,24653,24656,24662,24668,24674,24683,24689,24691,24697,24703,24706,24714,24717,24760,24763,24818,24823,24838,24843,24846,24857,24860],[12,24600,24601],{},"期中考考完了各位，要開新課了，還有兩次考試能夠追分阿，加油",[36,24603],{},[12,24605,24606,24607,24609,24610,24612],{},"SCI從受傷部位區分區的話，分為",[16,24608],{},"\n傷及胸椎或以下的——下半身癱瘓（Paraplegia）",[16,24611],{},"\n傷及頸椎的——四肢癱瘓（Quadplegia\u002FTetraplegia）",[12,24614,24615,24616,24618,24619,24621],{},"而美國來看，平均年齡會落在44歲，多為男性",[16,24617],{},"\n佔比為：機車事故>跌倒>暴力(多指槍傷)>運動>娛樂活動>手術\u002F醫療",[16,24620],{},"\n那癱瘓類型比例最高是不完全性四肢癱瘓",[12,24623,24624],{},"WHO指出，SCI男女比為2：1，且九成都是外傷導致的",[12,24626,24627,24628,24630],{},"在台灣男女比是4：1，年齡層在20~29歲；最多也是交通事故，其次是高處摔下，職業傷害，病變，運動傷害等",[16,24629],{},"\n而癱瘓類型來看，下半身癱瘓佔比最高；假如從部位來看，腰椎受傷佔42%，頸椎受傷佔40%，且頸椎受傷佔比逐年上升",[36,24632],{},[12,24634,24635],{},"脊椎分成，頸椎7節、胸椎12節、腰椎5節、薦椎5節、尾椎4節，一共33節",[12,24637,24638],{},"那前縱韌帶跟後縱韌帶都是從枕骨下到了薦椎，",[12,24640,24641],{},"柱身還分成了三個部份（三柱模式）——假如以下兩柱或以上受傷，才會有不穩定情況",[3603,24643,24644,24647,24650],{},[26,24645,24646],{},"前柱——椎體前部，前蹤韌帶，纖維環前部",[26,24648,24649],{},"中柱——椎體後壁，後蹤韌帶，纖維環後部",[26,24651,24652],{},"後柱——椎弓，棘上韌帶，棘間韌帶，黃韌帶，關節囊",[12,24654,24655],{},"頸椎的後蹤韌帶骨化，黃韌帶退化這兩者常常是老人跌倒造成SCI的原因",[12,24657,24658,24659,24661],{},"從神經節段來分，分成了頸神經8節，胸神經12節，腰神經5節，薦神經5節，尾神經1節",[16,24660],{},"\n其中只有C1神經從枕骨下方出來，C2神經從C1脊椎下方發出，直到C8神經從C7脊椎下方發出，往後的都從他們對應的脊椎下方發出",[12,24663,24664,24665,24667],{},"而這些脊神經的發出，固定會有一組單一的背根和腹根組成",[16,24666],{},"\n背根接收感覺訊號，腹根發出運動訊號",[12,24669,24670,24671,24673],{},"脊髓(spinal cord)的起始位置從延腦尾端，剛好在枕骨大孔上方",[16,24672],{},"\n嬰兒會長到L3，成年人會終止在L2",[12,24675,24676,24677,24679,24680,24682],{},"脊髓圓錐(conus medulllaris)是指脊髓末端最尖端的部份，通常在L1L2左右",[16,24678],{},"\n這裡還包含了從S2到S5以及尾神經出去的神經節段",[16,24681],{},"\n他上方連接的脊髓上圓錐epiconus（從L4到S1出來的節段）",[12,24684,24685,24686,24688],{},"最後講的是馬尾的部份，雖然脊髓只到L2附近",[16,24687],{},"\n但L3以下對應的脊神經還是會從對應的脊椎穿出，而L3以下的這些就叫馬尾",[36,24690],{},[12,24692,24693,24694,24696],{},"從橫截面來看，灰質呈現H形狀",[16,24695],{},"\n後角主管感覺，前角主角運動",[12,24698,24699],{},[75,24700,24702],{"className":24701},[5387],"（這邊要記得背了 會出大概兩三題吧 沒記錯的話 後面病症也會用到這裡旳概念）",[12,24704,24705],{},"白質向上傳感覺，向下傳運動訊息",[23,24707,24708,24711],{},[26,24709,24710],{},"向上傳遞的區域：後柱，背側脊髓小腦徑，腹側脊髓小腦徑，前外側系統ALS（脊髓丘腦徑、脊髓網狀徑，脊髓四疊板徑）",[26,24712,24713],{},"向下傳遞的區域：外側皮質脊髓徑，紅核脊髓徑，外側網狀脊髓徑，內側網狀脊髓徑，前庭脊髓徑，腹側皮質脊髓徑",[12,24715,24716],{},"向下的路分成",[23,24718,24719,24733],{},[26,24720,24721,24722],{},"從運動皮質向下傳的：",[23,24723,24724,24727,24730],{},[26,24725,24726],{},"前側皮質脊髓徑：不在延腦交叉，自發性動作，控制精細動作，特別是近端動作；因為路徑較細，臨床意義較小",[26,24728,24729],{},"外側皮質脊髓徑：在延腦交叉，自發性動作，控制精細動作，特別是遠端動作",[26,24731,24732],{},"皮質延腦徑",[26,24734,24735,24736],{},"從腦幹向下傳的：",[23,24737,24738,24746],{},[26,24739,24740,24741,24745],{},"背外側——紅核脊髓徑：中腦交叉，上肢自發性動作，特別是近端；",[75,24742,24744],{"className":24743},[5387],"促進屈肌收縮","，功能上與外質皮質脊髓徑相關",[26,24747,24748,24749],{},"腹內側",[23,24750,24751,24754,24757],{},[26,24752,24753],{},"四疊板脊髓徑",[26,24755,24756],{},"前庭脊髓徑——平衡和姿勢有關",[26,24758,24759],{},"網狀脊髓徑——平衡、姿勢、調節脊髓反射、中軸控制和近端肢體動作",[12,24761,24762],{},"接著是向上傳的路徑",[23,24764,24765,24807],{},[26,24766,24767,24768],{},"傳到皮質的",[23,24769,24770,24781,24791],{},[26,24771,24772,24776,24777],{},[75,24773,24775],{"className":24774},[6760],"外側皮質丘腦徑（前外側系統ALS）","——剛進脊髓就交叉，",[75,24778,24780],{"className":24779},[79],"管痛覺，溫度覺",[26,24782,24783,24776,24787],{},[75,24784,24786],{"className":24785},[6760],"前側脊髓丘腦徑（前外側系統ALS）",[75,24788,24790],{"className":24789},[79],"管輕觸覺，壓力覺",[26,24792,24793,24797,24798,24802,24803],{},[75,24794,24796],{"className":24795},[5387],"後柱內側蹄系路徑（DMLC）","——在延腦交叉，",[75,24799,24801],{"className":24800},[78,6760],"管本體覺，震動覺，兩點辨別覺","，",[75,24804,24806],{"className":24805},[5387,78],"well-localized touch（明確定位被觸碰的能力）",[26,24808,24809,24810],{},"傳到小腦的",[23,24811,24812,24815],{},[26,24813,24814],{},"背側脊髓小腦徑——不交叉，接收來自下肢跟軀幹的本體覺",[26,24816,24817],{},"腹側脊髓小腦徑——剛進去交叉，在橋腦又交叉回去，接收來自上肢、下肢和軀幹的本體覺",[12,24819,24820],{},[1930,24821],{"alt":1932,"src":24822},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1778232618751-Screenshot_2026-05-08_173003.png",[12,24824,24825,24826,24828,24829,24831,24832,24834,24835,24837],{},"這張圖我就從左到右說",[16,24827],{},"\n首先是皮質脊髓徑，從大腦下來，在延腦（腦幹）交叉，進入脊髓後負責控制肌肉動作(UMN to LMN)",[16,24830],{},"\n第二是DCML，不交叉，到延腦（腦幹）才交叉，管本體覺",[16,24833],{},"\n第三是脊髓丘腦徑，進去直接交叉，後往上傳，管痛覺溫度覺",[16,24836],{},"\n最後是脊髓小腦徑，不交叉，直接往上傳到小腦，管平衡",[12,24839,24840],{},[1930,24841],{"alt":1932,"src":24842},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1778234292566-Screenshot_2026-05-08_175553.png",[12,24844,24845],{},"接下來講血流供應，從基底動脈分出脊椎動脈，再從脊椎動脈分出",[23,24847,24848,24851,24854],{},[26,24849,24850],{},"前脊髓動脈——供應脊髓前2\u002F3",[26,24852,24853],{},"後脊髓動脈——供應脊髓後1\u002F3",[26,24855,24856],{},"分節脊髓動脈",[12,24858,24859],{},"最後提到三種供血系統",[23,24861,24862,24865,24868],{},[26,24863,24864],{},"離心系統，血是從前脊髓動脈來的，供應中心區域，主要有灰質、附近白質的上下行路徑",[26,24866,24867],{},"向心系統，血是從前後脊髓動脈來的，供應周邊區域，主要為白質",[26,24869,24870,24871,24873,24874,24876],{},"AKA(Adamkiewicz)，也叫做前根動脈(the great anterior radicular artery)，那他從T8到L2（脊髓圓錐）都是由他供應前2\u002F3",[16,24872],{},"\n而上段胸椎血流也相對較差，會依賴AKA動脈的側支來供應",[16,24875],{},"\n所以假如AKA受損，代表不只T8以下，T8以上也會被波及，有時候還會上升到T4 level",{"title":180,"searchDepth":181,"depth":181,"links":24878},[],"2026-04-29","最開始到解剖","\u002Fimages\u002Fuploads\u002F1778235826591-0078.webp",{},"\u002Fblog\u002F脊髓損傷SCI（一）",{"title":24596,"description":24880},{"loc":24883},"blog\u002F脊髓損傷SCI（一）","J6f0oeiBSDvTZB-l-TWz4KbnRe2xtBHwusuXBLkqMYw",{"id":24889,"title":24890,"author":6946,"body":24891,"date":31992,"description":180,"draft":185,"edited_at":31993,"extension":186,"featured_image":194,"meta":31994,"navigation":189,"path":31995,"pinned":185,"seo":31996,"sitemap":31997,"stem":31998,"tags":194,"__hash__":31999},"blog\u002Fblog\u002FDecision-Trees.md","Decision Trees",{"type":9,"value":24892,"toc":31967},[24893,24897,24900,24903,24908,24911,24918,25559,25562,25568,25572,25575,25611,25615,25618,25624,25630,25839,25843,25846,25850,25855,26145,26212,26215,26229,27029,27057,27293,27296,27299,27305,27737,27843,27850,29949,29951,29954,30027,30151,30156,30431,30436,30746,30751,31043,31046,31050,31057,31094,31098,31101,31263,31439,31442,31462,31469,31472,31478,31481,31504,31508,31511,31587,31646,31649,31653,31656,31660,31663,31669,31675,31679,31682,31702,31706,31720,31724,31842,31884,31888,31892,31909,31913,31930,31934,31937,31957,31959,31964],[6964,24894,24896],{"id":24895},"what-is-a-decision-tree","What Is a Decision Tree?",[12,24898,24899],{},"A decision tree is exactly what it sounds like — a tree-shaped model that makes decisions. Each internal node asks a question about a feature, each branch represents a possible answer, and each leaf gives you a final classification.",[12,24901,24902],{},"Here's a classic example: should you play tennis today?",[12,24904,24905],{},[1930,24906],{"alt":1932,"src":24907},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1777359100919-1776069090146-Screenshot_2026-04-13_at_4.avif",[12,24909,24910],{},"The tree checks the weather outlook first. If it's overcast, just go play — always yes. If it's sunny, check the humidity. If it's rainy, check the wind. Simple and readable.",[12,24912,24913,24914,24917],{},"We can also express this tree as a set of logical rules (called ",[106,24915,24916],{},"disjunctive normal form","):",[23,24919,24920,25217,25373],{},[26,24921,24922,7206,25191,7206,25215],{},[24923,24924,24928],"mjx-container",{"className":24925,"jax":24927},[24926],"MathJax","SVG",[24929,24930,24937,25021],"svg",{"style":24931,"xmlns":24932,"width":24933,"height":24934,"role":1930,"focusable":8058,"viewBox":24935,"xmlnsXLink":24936},"vertical-align: -0.566ex;","http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","43.069ex","2.262ex","0 -750 19036.6 1000","http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxlink",[24938,24939,24940,24945,24949,24953,24957,24961,24965,24969,24973,24977,24981,24985,24989,24993,24997,25001,25005,25009,25013,25017],"defs",{},[24941,24942],"path",{"id":24943,"d":24944},"MJX-1-TEX-N-28","M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 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is one of the best things about decision trees, you can always convert them into human-readable if-then rules.",[12,25563,25564],{},[1930,25565],{"alt":25566,"src":25567},"0.98","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1777359225311-a-describes-the-components-of-a-decision-tree-the-Nodes-represent-the-possible.webp",[6964,25569,25571],{"id":25570},"why-decision-trees","Why Decision Trees?",[12,25573,25574],{},"Decision trees have several properties that make them a solid starting point for classification problems:",[23,25576,25577,25583,25589,25595,25601],{},[26,25578,25579,25582],{},[106,25580,25581],{},"They handle both discrete and continuous features."," Discrete features like color (red, blue, green) work naturally. For continuous features like temperature, the tree can split on a threshold (e.g., temperature \u003C 30°C vs. ≥ 30°C).",[26,25584,25585,25588],{},[106,25586,25587],{},"They naturally represent disjunctive (OR) concepts."," Some classification boundaries are complex unions of conditions, but decision trees handle this natively.",[26,25590,25591,25594],{},[106,25592,25593],{},"They're robust to noise."," Both mislabeled examples and noisy feature values can be tolerated.",[26,25596,25597,25600],{},[106,25598,25599],{},"They handle missing values."," If a data point doesn't have a value for some feature, the algorithm can still work with it.",[26,25602,25603,25606,25607,25610],{},[106,25604,25605],{},"Classification trees"," produce discrete class labels at the leaves, while ",[106,25608,25609],{},"regression trees"," output continuous real-valued predictions.",[6964,25612,25614],{"id":25613},"how-do-we-build-a-decision-tree","How Do We Build a Decision Tree?",[12,25616,25617],{},"The core algorithm is a top-down, recursive, divide-and-conquer approach. Here's the pseudocode:",[6972,25619,25622],{"className":25620,"code":25621,"language":6977},[6975],"DTree(examples, features):\n    If all examples belong to one class:\n        return a leaf with that class label\n \n    If no features are left:\n        return a leaf with the majority class\n \n    Pick the \"best\" feature F\n    Create a node for F\n    For each value v of F:\n        Let examples_v = subset of examples where F = v\n        If examples_v is empty:\n            attach a leaf with the majority class of examples\n        Else:\n            attach DTree(examples_v, features - {F})\n \n    Return the tree rooted at this node\n",[6979,25623,25621],{"__ignoreMap":180},[12,25625,25626,25627],{},"The key question is: ",[106,25628,25629],{},"how do we pick the \"best\" feature?",[25631,25632,25634,25638],"details",{"className":25633},[13441,13442],[25635,25636,25637],"summary",{},"Pseudocode Explaination",[13438,25639,25642,25647,25650,25689,25694,25697,25732,25737,25740,25799,25804],{"className":25640},[25641],"info-box-content",[12,25643,25644],{},[106,25645,25646],{},"1. Check the Stopping Rules (Base Cases)",[12,25648,25649],{},"Before trying to split the data, the algorithm checks if it should stop growing the tree:",[23,25651,25652,25666],{},[26,25653,25654,7206,25657,25660,25661],{},[106,25655,25656],{},"Pure Data:",[6979,25658,25659],{},"If all examples belong to one class:"," If every item in the current dataset belongs to the exact same category (e.g., they are all \"Spam\"), it stops and creates a leaf node predicting that exact category.\n",[23,25662,25663],{},[26,25664,25665],{},"> Are all the fruits in the basket apples? No, it’s a mix.",[26,25667,25668,7206,25671,25674,25675,25678,25679],{},[106,25669,25670],{},"Out of Features:",[6979,25672,25673],{},"If no features are left:"," If the algorithm has used up all the available features to split on, but there are still mixed classes in the dataset, it stops. It creates a leaf node predicting the ",[106,25676,25677],{},"majority class"," (whichever category is most common in that remaining subset).\n",[23,25680,25681],{},[26,25682,25683,25684,7042,25686,6959],{},"Have we run out of questions to ask? No, we still have ",[7289,25685,18612],{},[7289,25687,25688],{},"Color",[12,25690,25691],{},[106,25692,25693],{},"2. Choose the Split (The Divide Step)",[12,25695,25696],{},"If the algorithm hasn't stopped, it needs to figure out how to divide the data:",[23,25698,25699,25718],{},[26,25700,25701,25704,25705],{},[6979,25702,25703],{},"Pick the \"best\" feature F:"," It evaluates all available features and chooses the one that best separates the classes.\n",[23,25706,25707],{},[26,25708,25709,25710,7042,25712,25714,25715,25717],{},"The algorithm evaluates ",[7289,25711,18612],{},[7289,25713,25688],{}," to see which question separates the fruits best. Let's say it determines that asking about ",[106,25716,18612],{}," is the best first move, because it easily separates the grapes from the larger fruits.",[26,25719,25720,25723,25724],{},[6979,25721,25722],{},"Create a node for F:"," It makes this chosen feature a decision point (a node) in the tree.\n",[23,25725,25726],{},[26,25727,25728,25729],{},"It creates the very first decision point (the root node) in our tree: ",[106,25730,25731],{},"\"What is the Size?\"",[12,25733,25734],{},[106,25735,25736],{},"3. Split the Data and Repeat (The Conquer Step)",[12,25738,25739],{},"Once the best feature is chosen, the algorithm branches out:",[23,25741,25742,25748,25759,25768],{},[26,25743,25744,25747],{},[6979,25745,25746],{},"For each value v of F:"," It creates a branch for every possible answer to the feature's question (e.g., if the feature is \"Weather\", the branches might be \"Sunny\", \"Rainy\", and \"Cloudy\").",[26,25749,25750,25753,25754],{},[6979,25751,25752],{},"Let examples_v = subset...:"," It divides the current dataset, sending the relevant examples down their respective branches.\n",[23,25755,25756],{},[26,25757,25758],{},"It gathers all the \"Small\" fruits into a new subset.",[26,25760,25761,7206,25764,25767],{},[106,25762,25763],{},"Handle Empty Branches:",[6979,25765,25766],{},"If examples_v is empty:"," If a branch ends up with absolutely no data, the algorithm doesn't crash; it simply caps that branch with a leaf node predicting the majority class of the parent data.",[26,25769,25770,7206,25773,25776,25777,25780,25781,25784,25785],{},[106,25771,25772],{},"Recursion:",[6979,25774,25775],{},"Else: attach DTree(...)"," If the branch ",[7289,25778,25779],{},"does"," have data, the algorithm calls itself to repeat the exact same process on this new, smaller subset of data, using only the features that haven't been used yet (",[6979,25782,25783],{},"features - {F}",").\n",[23,25786,25787,25790],{},[26,25788,25789],{},"It runs the algorithm again on just the small fruits.",[26,25791,25792,25795,25796,6959],{},[7289,25793,25794],{},"Recursion kicks in:"," When looking at this \"Small\" subset, it realizes every single fruit here is a Grape! It hits the first stopping rule (If all examples belong to one class) and instantly creates a ",[106,25797,25798],{},"Leaf: Grape",[12,25800,25801],{},[106,25802,25803],{},"4. Return the Result",[23,25805,25806],{},[26,25807,25808,25811,25812],{},[6979,25809,25810],{},"Return the tree rooted at this node:"," Finally, it passes the fully assembled sub-tree back up the chain until the entire decision tree is built.\n",[23,25813,25814],{},[26,25815,25816,25817],{},"The algorithm stitches all these sub-trees and leaves together. It passes the finished flowchart back to you:\n",[23,25818,25819,25824,25829,25834],{},[26,25820,25821],{},[7289,25822,25823],{},"First, check Size.",[26,25825,25826],{},[7289,25827,25828],{},"If Small ➔ It's a Grape.",[26,25830,25831],{},[7289,25832,25833],{},"If Big ➔ Check Color.",[26,25835,25836],{},[7289,25837,25838],{},"If Red\u002FGreen ➔ It's an Apple. If Yellow ➔ It's a Lemon.",[6964,25840,25842],{"id":25841},"entropy-and-information-gain","Entropy and Information Gain",[12,25844,25845],{},"This is where information theory comes in. We want to choose the feature that best separates the data — the one that brings us closest to pure (single-class) subsets.",[7166,25847,25849],{"id":25848},"entropy","Entropy",[12,25851,25852,25854],{},[106,25853,25849],{}," measures the impurity or disorder of a set. For a binary classification:",[12,25856,25857],{},[24923,25858,25860],{"className":25859,"jax":24927},[24926],[24929,25861,25864,25936],{"style":24931,"xmlns":24932,"width":25862,"height":24934,"role":1930,"focusable":8058,"viewBox":25863,"xmlnsXLink":24936},"39.565ex","0 -750 17487.7 1000",[24938,25865,25866,25870,25874,25878,25882,25886,25890,25894,25897,25901,25904,25907,25911,25915,25918,25921,25925,25929,25932],{},[24941,25867],{"id":25868,"d":25869},"MJX-6-TEX-I-1D438","M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z",[24941,25871],{"id":25872,"d":25873},"MJX-6-TEX-I-1D45B","M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z",[24941,25875],{"id":25876,"d":25877},"MJX-6-TEX-I-1D461","M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z",[24941,25879],{"id":25880,"d":25881},"MJX-6-TEX-I-1D45F","M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z",[24941,25883],{"id":25884,"d":25885},"MJX-6-TEX-I-1D45C","M201 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393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z",[24941,25891],{"id":25892,"d":25893},"MJX-6-TEX-I-1D466","M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z",[24941,25895],{"id":25896,"d":24944},"MJX-6-TEX-N-28",[24941,25898],{"id":25899,"d":25900},"MJX-6-TEX-I-1D446","M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z",[24941,25902],{"id":25903,"d":24988},"MJX-6-TEX-N-29",[24941,25905],{"id":25906,"d":24972},"MJX-6-TEX-N-3D",[24941,25908],{"id":25909,"d":25910},"MJX-6-TEX-N-2212","M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z",[24941,25912],{"id":25913,"d":25914},"MJX-6-TEX-N-31","M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",[24941,25916],{"id":25917,"d":24960},"MJX-6-TEX-N-6C",[24941,25919],{"id":25920,"d":24964},"MJX-6-TEX-N-6F",[24941,25922],{"id":25923,"d":25924},"MJX-6-TEX-N-67","M329 409Q373 453 429 453Q459 453 472 434T485 396Q485 382 476 371T449 360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z",[24941,25926],{"id":25927,"d":25928},"MJX-6-TEX-N-32","M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z",[24941,25930],{"id":25931,"d":180},"MJX-6-TEX-N-2061",[24941,25933],{"id":25934,"d":25935},"MJX-6-TEX-N-30","M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z",[25022,25937,25938],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,25939,25940,25947,25954,25961,25968,25975,25982,25989,25995,26002,26008,26014,26021,26037,26060,26067,26071,26082,26087,26092,26105,26120,26125,26129,26140],{"dataMmlNode":25028},[25022,25941,25943],{"dataMmlNode":25942},"mi",[25033,25944],{"dataC":25945,"xLinkHref":25946},"1D438","#MJX-6-TEX-I-1D438",[25022,25948,25950],{"dataMmlNode":25942,"transform":25949},"translate(764,0)",[25033,25951],{"dataC":25952,"xLinkHref":25953},"1D45B","#MJX-6-TEX-I-1D45B",[25022,25955,25957],{"dataMmlNode":25942,"transform":25956},"translate(1364,0)",[25033,25958],{"dataC":25959,"xLinkHref":25960},"1D461","#MJX-6-TEX-I-1D461",[25022,25962,25964],{"dataMmlNode":25942,"transform":25963},"translate(1725,0)",[25033,25965],{"dataC":25966,"xLinkHref":25967},"1D45F","#MJX-6-TEX-I-1D45F",[25022,25969,25971],{"dataMmlNode":25942,"transform":25970},"translate(2176,0)",[25033,25972],{"dataC":25973,"xLinkHref":25974},"1D45C","#MJX-6-TEX-I-1D45C",[25022,25976,25978],{"dataMmlNode":25942,"transform":25977},"translate(2661,0)",[25033,25979],{"dataC":25980,"xLinkHref":25981},"1D45D","#MJX-6-TEX-I-1D45D",[25022,25983,25985],{"dataMmlNode":25942,"transform":25984},"translate(3164,0)",[25033,25986],{"dataC":25987,"xLinkHref":25988},"1D466","#MJX-6-TEX-I-1D466",[25022,25990,25992],{"dataMmlNode":25031,"transform":25991},"translate(3654,0)",[25033,25993],{"dataC":25035,"xLinkHref":25994},"#MJX-6-TEX-N-28",[25022,25996,25998],{"dataMmlNode":25942,"transform":25997},"translate(4043,0)",[25033,25999],{"dataC":26000,"xLinkHref":26001},"1D446","#MJX-6-TEX-I-1D446",[25022,26003,26005],{"dataMmlNode":25031,"transform":26004},"translate(4688,0)",[25033,26006],{"dataC":25108,"xLinkHref":26007},"#MJX-6-TEX-N-29",[25022,26009,26011],{"dataMmlNode":25031,"transform":26010},"translate(5354.8,0)",[25033,26012],{"dataC":25078,"xLinkHref":26013},"#MJX-6-TEX-N-3D",[25022,26015,26017],{"dataMmlNode":25031,"transform":26016},"translate(6410.6,0)",[25033,26018],{"dataC":26019,"xLinkHref":26020},"2212","#MJX-6-TEX-N-2212",[25022,26022,26025,26029],{"dataMmlNode":26023,"transform":26024},"msub","translate(7188.6,0)",[25022,26026,26027],{"dataMmlNode":25942},[25033,26028],{"dataC":25980,"xLinkHref":25981},[25022,26030,26033],{"dataMmlNode":26031,"transform":26032},"mn","translate(536,-150) scale(0.707)",[25033,26034],{"dataC":26035,"xLinkHref":26036},"31","#MJX-6-TEX-N-31",[25022,26038,26040,26053],{"dataMmlNode":26023,"transform":26039},"translate(8294.8,0)",[25022,26041,26042,26045,26049],{"dataMmlNode":25942},[25033,26043],{"dataC":25057,"xLinkHref":26044},"#MJX-6-TEX-N-6C",[25033,26046],{"dataC":25062,"xLinkHref":26047,"transform":26048},"#MJX-6-TEX-N-6F","translate(278,0)",[25033,26050],{"dataC":26051,"xLinkHref":26052,"transform":25049},"67","#MJX-6-TEX-N-67",[25022,26054,26056],{"dataMmlNode":26031,"transform":26055},"translate(1311,-241.4) scale(0.707)",[25033,26057],{"dataC":26058,"xLinkHref":26059},"32","#MJX-6-TEX-N-32",[25022,26061,26063],{"dataMmlNode":25031,"transform":26062},"translate(10009.3,0)",[25033,26064],{"dataC":26065,"xLinkHref":26066},"2061","#MJX-6-TEX-N-2061",[25022,26068,26069],{"dataMmlNode":25031,"transform":26062},[25033,26070],{"dataC":25035,"xLinkHref":25994},[25022,26072,26074,26078],{"dataMmlNode":26023,"transform":26073},"translate(10398.3,0)",[25022,26075,26076],{"dataMmlNode":25942},[25033,26077],{"dataC":25980,"xLinkHref":25981},[25022,26079,26080],{"dataMmlNode":26031,"transform":26032},[25033,26081],{"dataC":26035,"xLinkHref":26036},[25022,26083,26085],{"dataMmlNode":25031,"transform":26084},"translate(11337.9,0)",[25033,26086],{"dataC":25108,"xLinkHref":26007},[25022,26088,26090],{"dataMmlNode":25031,"transform":26089},"translate(11949.1,0)",[25033,26091],{"dataC":26019,"xLinkHref":26020},[25022,26093,26095,26099],{"dataMmlNode":26023,"transform":26094},"translate(12949.3,0)",[25022,26096,26097],{"dataMmlNode":25942},[25033,26098],{"dataC":25980,"xLinkHref":25981},[25022,26100,26101],{"dataMmlNode":26031,"transform":26032},[25033,26102],{"dataC":26103,"xLinkHref":26104},"30","#MJX-6-TEX-N-30",[25022,26106,26108,26116],{"dataMmlNode":26023,"transform":26107},"translate(14055.5,0)",[25022,26109,26110,26112,26114],{"dataMmlNode":25942},[25033,26111],{"dataC":25057,"xLinkHref":26044},[25033,26113],{"dataC":25062,"xLinkHref":26047,"transform":26048},[25033,26115],{"dataC":26051,"xLinkHref":26052,"transform":25049},[25022,26117,26118],{"dataMmlNode":26031,"transform":26055},[25033,26119],{"dataC":26058,"xLinkHref":26059},[25022,26121,26123],{"dataMmlNode":25031,"transform":26122},"translate(15770.1,0)",[25033,26124],{"dataC":26065,"xLinkHref":26066},[25022,26126,26127],{"dataMmlNode":25031,"transform":26122},[25033,26128],{"dataC":25035,"xLinkHref":25994},[25022,26130,26132,26136],{"dataMmlNode":26023,"transform":26131},"translate(16159.1,0)",[25022,26133,26134],{"dataMmlNode":25942},[25033,26135],{"dataC":25980,"xLinkHref":25981},[25022,26137,26138],{"dataMmlNode":26031,"transform":26032},[25033,26139],{"dataC":26103,"xLinkHref":26104},[25022,26141,26143],{"dataMmlNode":25031,"transform":26142},"translate(17098.7,0)",[25033,26144],{"dataC":25108,"xLinkHref":26007},[12,26146,26147,26148,26181,26182,26211],{},"where ",[24923,26149,26151],{"className":26150,"jax":24927},[24926],[24929,26152,26157,26165],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},"vertical-align: -0.439ex;","2.126ex","1.439ex","0 -442 939.6 636",[24938,26158,26159,26162],{},[24941,26160],{"id":26161,"d":25889},"MJX-7-TEX-I-1D45D",[24941,26163],{"id":26164,"d":25914},"MJX-7-TEX-N-31",[25022,26166,26167],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,26168,26169],{"dataMmlNode":25028},[25022,26170,26171,26176],{"dataMmlNode":26023},[25022,26172,26173],{"dataMmlNode":25942},[25033,26174],{"dataC":25980,"xLinkHref":26175},"#MJX-7-TEX-I-1D45D",[25022,26177,26178],{"dataMmlNode":26031,"transform":26032},[25033,26179],{"dataC":26035,"xLinkHref":26180},"#MJX-7-TEX-N-31"," is the fraction of positive examples and ",[24923,26183,26185],{"className":26184,"jax":24927},[24926],[24929,26186,26187,26195],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,26188,26189,26192],{},[24941,26190],{"id":26191,"d":25889},"MJX-8-TEX-I-1D45D",[24941,26193],{"id":26194,"d":25935},"MJX-8-TEX-N-30",[25022,26196,26197],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,26198,26199],{"dataMmlNode":25028},[25022,26200,26201,26206],{"dataMmlNode":26023},[25022,26202,26203],{"dataMmlNode":25942},[25033,26204],{"dataC":25980,"xLinkHref":26205},"#MJX-8-TEX-I-1D45D",[25022,26207,26208],{"dataMmlNode":26031,"transform":26032},[25033,26209],{"dataC":26103,"xLinkHref":26210},"#MJX-8-TEX-N-30"," is the fraction of negatives.",[12,26213,26214],{},"Some intuition:",[23,26216,26217,26223],{},[26,26218,26219,26220,26222],{},"If all examples are the same class → entropy = ",[106,26221,9180],{}," (perfectly pure, no uncertainty)",[26,26224,26225,26226,26228],{},"If examples are evenly split (50\u002F50) → entropy = ",[106,26227,8855],{}," (maximum uncertainty)",[13438,26230,26232,26235,26298,26560,26609,26615,26620,26859,26886,27000,27002,27006,27015],{"className":26231},[13441,13442],[12,26233,26234],{},"Let's break down this equation piece by piece.",[26236,26237,26239,26240,7042,26269,7963],"h4",{"id":26238},"_1-the-fractions-and","1. The Fractions (",[24923,26241,26243],{"className":26242,"jax":24927},[24926],[24929,26244,26245,26253],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,26246,26247,26250],{},[24941,26248],{"id":26249,"d":25889},"MJX-9-TEX-I-1D45D",[24941,26251],{"id":26252,"d":25914},"MJX-9-TEX-N-31",[25022,26254,26255],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,26256,26257],{"dataMmlNode":25028},[25022,26258,26259,26264],{"dataMmlNode":26023},[25022,26260,26261],{"dataMmlNode":25942},[25033,26262],{"dataC":25980,"xLinkHref":26263},"#MJX-9-TEX-I-1D45D",[25022,26265,26266],{"dataMmlNode":26031,"transform":26032},[25033,26267],{"dataC":26035,"xLinkHref":26268},"#MJX-9-TEX-N-31",[24923,26270,26272],{"className":26271,"jax":24927},[24926],[24929,26273,26274,26282],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,26275,26276,26279],{},[24941,26277],{"id":26278,"d":25889},"MJX-10-TEX-I-1D45D",[24941,26280],{"id":26281,"d":25935},"MJX-10-TEX-N-30",[25022,26283,26284],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,26285,26286],{"dataMmlNode":25028},[25022,26287,26288,26293],{"dataMmlNode":26023},[25022,26289,26290],{"dataMmlNode":25942},[25033,26291],{"dataC":25980,"xLinkHref":26292},"#MJX-10-TEX-I-1D45D",[25022,26294,26295],{"dataMmlNode":26031,"transform":26032},[25033,26296],{"dataC":26103,"xLinkHref":26297},"#MJX-10-TEX-N-30",[23,26299,26300,26399],{},[26,26301,26302,26331,26332,6959],{},[24923,26303,26305],{"className":26304,"jax":24927},[24926],[24929,26306,26307,26315],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,26308,26309,26312],{},[24941,26310],{"id":26311,"d":25889},"MJX-11-TEX-I-1D45D",[24941,26313],{"id":26314,"d":25914},"MJX-11-TEX-N-31",[25022,26316,26317],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,26318,26319],{"dataMmlNode":25028},[25022,26320,26321,26326],{"dataMmlNode":26023},[25022,26322,26323],{"dataMmlNode":25942},[25033,26324],{"dataC":25980,"xLinkHref":26325},"#MJX-11-TEX-I-1D45D",[25022,26327,26328],{"dataMmlNode":26031,"transform":26032},[25033,26329],{"dataC":26035,"xLinkHref":26330},"#MJX-11-TEX-N-31"," is the fraction of the basket that is Apples. If you have 10 fruits and 7 are Apples, ",[24923,26333,26335],{"className":26334,"jax":24927},[24926],[24929,26336,26340,26362],{"style":26153,"xmlns":24932,"width":26337,"height":26338,"role":1930,"focusable":8058,"viewBox":26339,"xmlnsXLink":24936},"8.034ex","1.968ex","0 -676 3551.1 870",[24938,26341,26342,26345,26348,26351,26354,26358],{},[24941,26343],{"id":26344,"d":25889},"MJX-12-TEX-I-1D45D",[24941,26346],{"id":26347,"d":25914},"MJX-12-TEX-N-31",[24941,26349],{"id":26350,"d":24972},"MJX-12-TEX-N-3D",[24941,26352],{"id":26353,"d":25935},"MJX-12-TEX-N-30",[24941,26355],{"id":26356,"d":26357},"MJX-12-TEX-N-2E","M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z",[24941,26359],{"id":26360,"d":26361},"MJX-12-TEX-N-37","M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z",[25022,26363,26364],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,26365,26366,26378,26384],{"dataMmlNode":25028},[25022,26367,26368,26373],{"dataMmlNode":26023},[25022,26369,26370],{"dataMmlNode":25942},[25033,26371],{"dataC":25980,"xLinkHref":26372},"#MJX-12-TEX-I-1D45D",[25022,26374,26375],{"dataMmlNode":26031,"transform":26032},[25033,26376],{"dataC":26035,"xLinkHref":26377},"#MJX-12-TEX-N-31",[25022,26379,26381],{"dataMmlNode":25031,"transform":26380},"translate(1217.3,0)",[25033,26382],{"dataC":25078,"xLinkHref":26383},"#MJX-12-TEX-N-3D",[25022,26385,26387,26390,26395],{"dataMmlNode":26031,"transform":26386},"translate(2273.1,0)",[25033,26388],{"dataC":26103,"xLinkHref":26389},"#MJX-12-TEX-N-30",[25033,26391],{"dataC":26392,"xLinkHref":26393,"transform":26394},"2E","#MJX-12-TEX-N-2E","translate(500,0)",[25033,26396],{"dataC":26397,"xLinkHref":26398,"transform":25049},"37","#MJX-12-TEX-N-37",[26,26400,26401,26430,26431,26488,26489,7206,26492,7206,26523,7206,26526,7206,26557],{},[24923,26402,26404],{"className":26403,"jax":24927},[24926],[24929,26405,26406,26414],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,26407,26408,26411],{},[24941,26409],{"id":26410,"d":25889},"MJX-13-TEX-I-1D45D",[24941,26412],{"id":26413,"d":25935},"MJX-13-TEX-N-30",[25022,26415,26416],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,26417,26418],{"dataMmlNode":25028},[25022,26419,26420,26425],{"dataMmlNode":26023},[25022,26421,26422],{"dataMmlNode":25942},[25033,26423],{"dataC":25980,"xLinkHref":26424},"#MJX-13-TEX-I-1D45D",[25022,26426,26427],{"dataMmlNode":26031,"transform":26032},[25033,26428],{"dataC":26103,"xLinkHref":26429},"#MJX-13-TEX-N-30"," is the fraction that is Lemons. In that same basket, 3 are Lemons, so ",[24923,26432,26434],{"className":26433,"jax":24927},[24926],[24929,26435,26438,26456],{"style":26153,"xmlns":24932,"width":26337,"height":26436,"role":1930,"focusable":8058,"viewBox":26437,"xmlnsXLink":24936},"1.946ex","0 -666 3551.1 860",[24938,26439,26440,26443,26446,26449,26452],{},[24941,26441],{"id":26442,"d":25889},"MJX-14-TEX-I-1D45D",[24941,26444],{"id":26445,"d":25935},"MJX-14-TEX-N-30",[24941,26447],{"id":26448,"d":24972},"MJX-14-TEX-N-3D",[24941,26450],{"id":26451,"d":26357},"MJX-14-TEX-N-2E",[24941,26453],{"id":26454,"d":26455},"MJX-14-TEX-N-33","M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z",[25022,26457,26458],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,26459,26460,26472,26477],{"dataMmlNode":25028},[25022,26461,26462,26467],{"dataMmlNode":26023},[25022,26463,26464],{"dataMmlNode":25942},[25033,26465],{"dataC":25980,"xLinkHref":26466},"#MJX-14-TEX-I-1D45D",[25022,26468,26469],{"dataMmlNode":26031,"transform":26032},[25033,26470],{"dataC":26103,"xLinkHref":26471},"#MJX-14-TEX-N-30",[25022,26473,26474],{"dataMmlNode":25031,"transform":26380},[25033,26475],{"dataC":25078,"xLinkHref":26476},"#MJX-14-TEX-N-3D",[25022,26478,26479,26481,26484],{"dataMmlNode":26031,"transform":26386},[25033,26480],{"dataC":26103,"xLinkHref":26471},[25033,26482],{"dataC":26392,"xLinkHref":26483,"transform":26394},"#MJX-14-TEX-N-2E",[25033,26485],{"dataC":26486,"xLinkHref":26487,"transform":25049},"33","#MJX-14-TEX-N-33",".\n",[7289,26490,26491],{},"(Note:",[7289,26493,26494],{},[24923,26495,26497],{"className":26496,"jax":24927},[24926],[24929,26498,26499,26507],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,26500,26501,26504],{},[24941,26502],{"id":26503,"d":25889},"MJX-15-TEX-I-1D45D",[24941,26505],{"id":26506,"d":25914},"MJX-15-TEX-N-31",[25022,26508,26509],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,26510,26511],{"dataMmlNode":25028},[25022,26512,26513,26518],{"dataMmlNode":26023},[25022,26514,26515],{"dataMmlNode":25942},[25033,26516],{"dataC":25980,"xLinkHref":26517},"#MJX-15-TEX-I-1D45D",[25022,26519,26520],{"dataMmlNode":26031,"transform":26032},[25033,26521],{"dataC":26035,"xLinkHref":26522},"#MJX-15-TEX-N-31",[7289,26524,26525],{},"and",[7289,26527,26528],{},[24923,26529,26531],{"className":26530,"jax":24927},[24926],[24929,26532,26533,26541],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,26534,26535,26538],{},[24941,26536],{"id":26537,"d":25889},"MJX-16-TEX-I-1D45D",[24941,26539],{"id":26540,"d":25935},"MJX-16-TEX-N-30",[25022,26542,26543],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,26544,26545],{"dataMmlNode":25028},[25022,26546,26547,26552],{"dataMmlNode":26023},[25022,26548,26549],{"dataMmlNode":25942},[25033,26550],{"dataC":25980,"xLinkHref":26551},"#MJX-16-TEX-I-1D45D",[25022,26553,26554],{"dataMmlNode":26031,"transform":26032},[25033,26555],{"dataC":26103,"xLinkHref":26556},"#MJX-16-TEX-N-30",[7289,26558,26559],{},"must always add up to 1, representing 100% of the basket).",[26236,26561,26563,26564,7963],{"id":26562},"_2-the-logarithm","2. The Logarithm (",[24923,26565,26567],{"className":26566,"jax":24927},[24926],[24929,26568,26573,26587],{"style":26569,"xmlns":24932,"width":26570,"height":26571,"role":1930,"focusable":8058,"viewBox":26572,"xmlnsXLink":24936},"vertical-align: -0.546ex;","3.879ex","2.116ex","0 -694 1714.6 935.4",[24938,26574,26575,26578,26581,26584],{},[24941,26576],{"id":26577,"d":24960},"MJX-17-TEX-N-6C",[24941,26579],{"id":26580,"d":24964},"MJX-17-TEX-N-6F",[24941,26582],{"id":26583,"d":25924},"MJX-17-TEX-N-67",[24941,26585],{"id":26586,"d":25928},"MJX-17-TEX-N-32",[25022,26588,26589],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,26590,26591],{"dataMmlNode":25028},[25022,26592,26593,26604],{"dataMmlNode":26023},[25022,26594,26595,26598,26601],{"dataMmlNode":25942},[25033,26596],{"dataC":25057,"xLinkHref":26597},"#MJX-17-TEX-N-6C",[25033,26599],{"dataC":25062,"xLinkHref":26600,"transform":26048},"#MJX-17-TEX-N-6F",[25033,26602],{"dataC":26051,"xLinkHref":26603,"transform":25049},"#MJX-17-TEX-N-67",[25022,26605,26606],{"dataMmlNode":26031,"transform":26055},[25033,26607],{"dataC":26058,"xLinkHref":26608},"#MJX-17-TEX-N-32",[12,26610,26611,26612],{},"Why use a base-2 logarithm? It measures \"bits\" of information. Think of it as asking: ",[7289,26613,26614],{},"\"How many yes\u002Fno questions (bits) would I need to ask to figure out the identity of a randomly picked fruit?\"",[12,26616,26617],{},[106,26618,26619],{},"The log of a fraction (any number between 0 and 1) is always a negative number.",[23,26621,26622,26743],{},[26,26623,26624],{},[24923,26625,26627],{"className":26626,"jax":24927},[24926],[24929,26628,26631,26673],{"style":24931,"xmlns":24932,"width":26629,"height":24934,"role":1930,"focusable":8058,"viewBox":26630,"xmlnsXLink":24936},"14.439ex","0 -750 6382.1 1000",[24938,26632,26633,26636,26639,26642,26645,26648,26651,26654,26657,26661,26664,26667,26670],{},[24941,26634],{"id":26635,"d":24960},"MJX-18-TEX-N-6C",[24941,26637],{"id":26638,"d":24964},"MJX-18-TEX-N-6F",[24941,26640],{"id":26641,"d":25924},"MJX-18-TEX-N-67",[24941,26643],{"id":26644,"d":25928},"MJX-18-TEX-N-32",[24941,26646],{"id":26647,"d":180},"MJX-18-TEX-N-2061",[24941,26649],{"id":26650,"d":24944},"MJX-18-TEX-N-28",[24941,26652],{"id":26653,"d":25935},"MJX-18-TEX-N-30",[24941,26655],{"id":26656,"d":26357},"MJX-18-TEX-N-2E",[24941,26658],{"id":26659,"d":26660},"MJX-18-TEX-N-35","M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 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It calculates exactly ",[7289,28943,28944],{},"how good"," a specific feature (",[24923,28947,28949],{"className":28948,"jax":24927},[24926],[24929,28950,28951,28956],{"style":27802,"xmlns":24932,"width":27803,"height":27804,"role":1930,"focusable":8058,"viewBox":27805,"xmlnsXLink":24936},[24938,28952,28953],{},[24941,28954],{"id":28955,"d":27343},"MJX-37-TEX-I-1D439",[25022,28957,28958],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,28959,28960],{"dataMmlNode":25028},[25022,28961,28962],{"dataMmlNode":25942},[25033,28963],{"dataC":27463,"xLinkHref":28964},"#MJX-37-TEX-I-1D439",") is at organizing the data.",[23,28967,28968,29089,29120,29276,29385],{},[26,28969,28970,29065,29066,29068,29069,29088],{},[106,28971,28972,7024],{},[24923,28973,28975],{"className":28974,"jax":24927},[24926],[24929,28976,28979,29011],{"style":24931,"xmlns":24932,"width":28977,"height":24934,"role":1930,"focusable":8058,"viewBox":28978,"xmlnsXLink":24936},"11.486ex","0 -750 5077 1000",[24938,28980,28981,28984,28987,28990,28993,28996,28999,29002,29005,29008],{},[24941,28982],{"id":28983,"d":25869},"MJX-38-TEX-I-1D438",[24941,28985],{"id":28986,"d":25873},"MJX-38-TEX-I-1D45B",[24941,28988],{"id":28989,"d":25877},"MJX-38-TEX-I-1D461",[24941,28991],{"id":28992,"d":25881},"MJX-38-TEX-I-1D45F",[24941,28994],{"id":28995,"d":25885},"MJX-38-TEX-I-1D45C",[24941,28997],{"id":28998,"d":25889},"MJX-38-TEX-I-1D45D",[24941,29000],{"id":29001,"d":25893},"MJX-38-TEX-I-1D466",[24941,29003],{"id":29004,"d":24944},"MJX-38-TEX-N-28",[24941,29006],{"id":29007,"d":25900},"MJX-38-TEX-I-1D446",[24941,29009],{"id":29010,"d":24988},"MJX-38-TEX-N-29",[25022,29012,29013],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29014,29015,29020,29025,29030,29035,29040,29045,29050,29055,29060],{"dataMmlNode":25028},[25022,29016,29017],{"dataMmlNode":25942},[25033,29018],{"dataC":25945,"xLinkHref":29019},"#MJX-38-TEX-I-1D438",[25022,29021,29022],{"dataMmlNode":25942,"transform":25949},[25033,29023],{"dataC":25952,"xLinkHref":29024},"#MJX-38-TEX-I-1D45B",[25022,29026,29027],{"dataMmlNode":25942,"transform":25956},[25033,29028],{"dataC":25959,"xLinkHref":29029},"#MJX-38-TEX-I-1D461",[25022,29031,29032],{"dataMmlNode":25942,"transform":25963},[25033,29033],{"dataC":25966,"xLinkHref":29034},"#MJX-38-TEX-I-1D45F",[25022,29036,29037],{"dataMmlNode":25942,"transform":25970},[25033,29038],{"dataC":25973,"xLinkHref":29039},"#MJX-38-TEX-I-1D45C",[25022,29041,29042],{"dataMmlNode":25942,"transform":25977},[25033,29043],{"dataC":25980,"xLinkHref":29044},"#MJX-38-TEX-I-1D45D",[25022,29046,29047],{"dataMmlNode":25942,"transform":25984},[25033,29048],{"dataC":25987,"xLinkHref":29049},"#MJX-38-TEX-I-1D466",[25022,29051,29052],{"dataMmlNode":25031,"transform":25991},[25033,29053],{"dataC":25035,"xLinkHref":29054},"#MJX-38-TEX-N-28",[25022,29056,29057],{"dataMmlNode":25942,"transform":25997},[25033,29058],{"dataC":26000,"xLinkHref":29059},"#MJX-38-TEX-I-1D446",[25022,29061,29062],{"dataMmlNode":25031,"transform":26004},[25033,29063],{"dataC":25108,"xLinkHref":29064},"#MJX-38-TEX-N-29"," This is the messiness of your data ",[7289,29067,10260],{}," you split it. (",[24923,29070,29072],{"className":29071,"jax":24927},[24926],[24929,29073,29074,29079],{"style":27778,"xmlns":24932,"width":27779,"height":27780,"role":1930,"focusable":8058,"viewBox":27781,"xmlnsXLink":24936},[24938,29075,29076],{},[24941,29077],{"id":29078,"d":25900},"MJX-39-TEX-I-1D446",[25022,29080,29081],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29082,29083],{"dataMmlNode":25028},[25022,29084,29085],{"dataMmlNode":25942},[25033,29086],{"dataC":26000,"xLinkHref":29087},"#MJX-39-TEX-I-1D446"," stands for the original Set of data).",[26,29090,29091,29113,29114,29116,29117,29119],{},[106,29092,29093,29094,7963],{},"The Subtraction (",[24923,29095,29097],{"className":29096,"jax":24927},[24926],[24929,29098,29099,29104],{"style":26868,"xmlns":24932,"width":26869,"height":26870,"role":1930,"focusable":8058,"viewBox":26871,"xmlnsXLink":24936},[24938,29100,29101],{},[24941,29102],{"id":29103,"d":25910},"MJX-40-TEX-N-2212",[25022,29105,29106],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29107,29108],{"dataMmlNode":25028},[25022,29109,29110],{"dataMmlNode":25031},[25033,29111],{"dataC":26019,"xLinkHref":29112},"#MJX-40-TEX-N-2212",": We are taking the messiness ",[7289,29115,10260],{}," the split, and subtracting the messiness ",[7289,29118,14466],{}," the split. If the number goes down a lot, our \"Gain\" is high.",[26,29121,29122,29235,29236,29255,29256,29275],{},[106,29123,29124,7024],{},[24923,29125,29127],{"className":29126,"jax":24927},[24926],[24929,29128,29131,29166],{"style":24931,"xmlns":24932,"width":29129,"height":24934,"role":1930,"focusable":8058,"viewBox":29130,"xmlnsXLink":24936},"14.338ex","0 -750 6337.6 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This just means \"for every branch we create.\" If the feature ",[24923,29237,29239],{"className":29238,"jax":24927},[24926],[24929,29240,29241,29246],{"style":27802,"xmlns":24932,"width":27803,"height":27804,"role":1930,"focusable":8058,"viewBox":27805,"xmlnsXLink":24936},[24938,29242,29243],{},[24941,29244],{"id":29245,"d":27343},"MJX-42-TEX-I-1D439",[25022,29247,29248],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29249,29250],{"dataMmlNode":25028},[25022,29251,29252],{"dataMmlNode":25942},[25033,29253],{"dataC":27463,"xLinkHref":29254},"#MJX-42-TEX-I-1D439"," is \"Color\", then ",[24923,29257,29259],{"className":29258,"jax":24927},[24926],[24929,29260,29261,29266],{"style":25196,"xmlns":24932,"width":27826,"height":27827,"role":1930,"focusable":8058,"viewBox":27828,"xmlnsXLink":24936},[24938,29262,29263],{},[24941,29264],{"id":29265,"d":27377},"MJX-43-TEX-I-1D463",[25022,29267,29268],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29269,29270],{"dataMmlNode":25028},[25022,29271,29272],{"dataMmlNode":25942},[25033,29273],{"dataC":27556,"xLinkHref":29274},"#MJX-43-TEX-I-1D463"," represents \"Red\", \"Green\", and \"Yellow\".",[26,29277,29278,29384],{},[106,29279,29280,7024],{},[24923,29281,29283],{"className":29282,"jax":24927},[24926],[24929,29284,29287,29322],{"style":24931,"xmlns":24932,"width":29285,"height":24934,"role":1930,"focusable":8058,"viewBox":29286,"xmlnsXLink":24936},"12.378ex","0 -750 5470.9 1000",[24938,29288,29289,29292,29295,29298,29301,29304,29307,29310,29313,29316,29319],{},[24941,29290],{"id":29291,"d":25869},"MJX-44-TEX-I-1D438",[24941,29293],{"id":29294,"d":25873},"MJX-44-TEX-I-1D45B",[24941,29296],{"id":29297,"d":25877},"MJX-44-TEX-I-1D461",[24941,29299],{"id":29300,"d":25881},"MJX-44-TEX-I-1D45F",[24941,29302],{"id":29303,"d":25885},"MJX-44-TEX-I-1D45C",[24941,29305],{"id":29306,"d":25889},"MJX-44-TEX-I-1D45D",[24941,29308],{"id":29309,"d":25893},"MJX-44-TEX-I-1D466",[24941,29311],{"id":29312,"d":24944},"MJX-44-TEX-N-28",[24941,29314],{"id":29315,"d":25900},"MJX-44-TEX-I-1D446",[24941,29317],{"id":29318,"d":27377},"MJX-44-TEX-I-1D463",[24941,29320],{"id":29321,"d":24988},"MJX-44-TEX-N-29",[25022,29323,29324],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29325,29326,29331,29336,29341,29346,29351,29356,29361,29366,29378],{"dataMmlNode":25028},[25022,29327,29328],{"dataMmlNode":25942},[25033,29329],{"dataC":25945,"xLinkHref":29330},"#MJX-44-TEX-I-1D438",[25022,29332,29333],{"dataMmlNode":25942,"transform":25949},[25033,29334],{"dataC":25952,"xLinkHref":29335},"#MJX-44-TEX-I-1D45B",[25022,29337,29338],{"dataMmlNode":25942,"transform":25956},[25033,29339],{"dataC":25959,"xLinkHref":29340},"#MJX-44-TEX-I-1D461",[25022,29342,29343],{"dataMmlNode":25942,"transform":25963},[25033,29344],{"dataC":25966,"xLinkHref":29345},"#MJX-44-TEX-I-1D45F",[25022,29347,29348],{"dataMmlNode":25942,"transform":25970},[25033,29349],{"dataC":25973,"xLinkHref":29350},"#MJX-44-TEX-I-1D45C",[25022,29352,29353],{"dataMmlNode":25942,"transform":25977},[25033,29354],{"dataC":25980,"xLinkHref":29355},"#MJX-44-TEX-I-1D45D",[25022,29357,29358],{"dataMmlNode":25942,"transform":25984},[25033,29359],{"dataC":25987,"xLinkHref":29360},"#MJX-44-TEX-I-1D466",[25022,29362,29363],{"dataMmlNode":25031,"transform":25991},[25033,29364],{"dataC":25035,"xLinkHref":29365},"#MJX-44-TEX-N-28",[25022,29367,29368,29373],{"dataMmlNode":26023,"transform":25997},[25022,29369,29370],{"dataMmlNode":25942},[25033,29371],{"dataC":26000,"xLinkHref":29372},"#MJX-44-TEX-I-1D446",[25022,29374,29375],{"dataMmlNode":25942,"transform":27643},[25033,29376],{"dataC":27556,"xLinkHref":29377},"#MJX-44-TEX-I-1D463",[25022,29379,29381],{"dataMmlNode":25031,"transform":29380},"translate(5081.9,0)",[25033,29382],{"dataC":25108,"xLinkHref":29383},"#MJX-44-TEX-N-29"," This is the messiness of the data inside one specific branch (e.g., how messy is the \"Red\" pile?).",[26,29386,29387,7206,29452,29455,29456,29484,29485,29528,29529,29562,29563,16419],{},[106,29388,29389],{},[24923,29390,29392],{"className":29391,"jax":24927},[24926],[24929,29393,29396,29407],{"style":28580,"xmlns":24932,"width":29394,"height":28582,"role":1930,"focusable":8058,"viewBox":29395,"xmlnsXLink":24936},"3.547ex","0 -1046.4 1567.8 1592.8",[24938,29397,29398,29401,29404],{},[24941,29399],{"id":29400,"d":27405},"MJX-45-TEX-N-7C",[24941,29402],{"id":29403,"d":25900},"MJX-45-TEX-I-1D446",[24941,29405],{"id":29406,"d":27377},"MJX-45-TEX-I-1D463",[25022,29408,29409],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29410,29411],{"dataMmlNode":25028},[25022,29412,29413,29436,29450],{"dataMmlNode":27622},[25022,29414,29415,29420,29432],{"dataMmlNode":27626,"transform":28837},[25022,29416,29417],{"dataMmlNode":25031,"transform":27630},[25033,29418],{"dataC":27633,"xLinkHref":29419},"#MJX-45-TEX-N-7C",[25022,29421,29422,29427],{"dataMmlNode":26023,"transform":26048},[25022,29423,29424],{"dataMmlNode":25942},[25033,29425],{"dataC":26000,"xLinkHref":29426},"#MJX-45-TEX-I-1D446",[25022,29428,29429],{"dataMmlNode":25942,"transform":27643},[25033,29430],{"dataC":27556,"xLinkHref":29431},"#MJX-45-TEX-I-1D463",[25022,29433,29434],{"dataMmlNode":25031,"transform":27648},[25033,29435],{"dataC":27633,"xLinkHref":29419},[25022,29437,29438,29442,29446],{"dataMmlNode":27626,"transform":28859},[25022,29439,29440],{"dataMmlNode":25031,"transform":27630},[25033,29441],{"dataC":27633,"xLinkHref":29419},[25022,29443,29444],{"dataMmlNode":25942,"transform":26048},[25033,29445],{"dataC":26000,"xLinkHref":29426},[25022,29447,29448],{"dataMmlNode":25031,"transform":27664},[25033,29449],{"dataC":27633,"xLinkHref":29419},[27668,29451],{"width":28874,"height":27671,"x":27672,"y":27673},[106,29453,29454],{},"(The Weight):"," The straight bars ",[24923,29457,29459],{"className":29458,"jax":24927},[24926],[24929,29460,29465,29470],{"style":29461,"xmlns":24932,"width":29462,"height":29463,"role":1930,"focusable":8058,"viewBox":29464,"xmlnsXLink":24936},"vertical-align: -0.564ex;","1.258ex","2.26ex","0 -749.5 556 999",[24938,29466,29467],{},[24941,29468],{"id":29469,"d":27405},"MJX-46-TEX-N-7C",[25022,29471,29472],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29473,29474,29479],{"dataMmlNode":25028},[25022,29475,29476],{"dataMmlNode":25031,"transform":27630},[25033,29477],{"dataC":27633,"xLinkHref":29478},"#MJX-46-TEX-N-7C",[25022,29480,29482],{"dataMmlNode":25031,"transform":29481},"translate(278,0) translate(0 -0.5)",[25033,29483],{"dataC":27633,"xLinkHref":29478}," mean \"count\". ",[24923,29486,29488],{"className":29487,"jax":24927},[24926],[24929,29489,29492,29503],{"style":29461,"xmlns":24932,"width":29490,"height":29463,"role":1930,"focusable":8058,"viewBox":29491,"xmlnsXLink":24936},"3.608ex","0 -749.5 1594.9 999",[24938,29493,29494,29497,29500],{},[24941,29495],{"id":29496,"d":27405},"MJX-47-TEX-N-7C",[24941,29498],{"id":29499,"d":25900},"MJX-47-TEX-I-1D446",[24941,29501],{"id":29502,"d":27377},"MJX-47-TEX-I-1D463",[25022,29504,29505],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29506,29507,29512,29524],{"dataMmlNode":25028},[25022,29508,29509],{"dataMmlNode":25031,"transform":27630},[25033,29510],{"dataC":27633,"xLinkHref":29511},"#MJX-47-TEX-N-7C",[25022,29513,29514,29519],{"dataMmlNode":26023,"transform":26048},[25022,29515,29516],{"dataMmlNode":25942},[25033,29517],{"dataC":26000,"xLinkHref":29518},"#MJX-47-TEX-I-1D446",[25022,29520,29521],{"dataMmlNode":25942,"transform":27643},[25033,29522],{"dataC":27556,"xLinkHref":29523},"#MJX-47-TEX-I-1D463",[25022,29525,29526],{"dataMmlNode":25031,"transform":27648},[25033,29527],{"dataC":27633,"xLinkHref":29511}," is the number of items in this specific branch, and ",[24923,29530,29532],{"className":29531,"jax":24927},[24926],[24929,29533,29536,29544],{"style":29461,"xmlns":24932,"width":29534,"height":29463,"role":1930,"focusable":8058,"viewBox":29535,"xmlnsXLink":24936},"2.717ex","0 -749.5 1201 999",[24938,29537,29538,29541],{},[24941,29539],{"id":29540,"d":27405},"MJX-48-TEX-N-7C",[24941,29542],{"id":29543,"d":25900},"MJX-48-TEX-I-1D446",[25022,29545,29546],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29547,29548,29553,29558],{"dataMmlNode":25028},[25022,29549,29550],{"dataMmlNode":25031,"transform":27630},[25033,29551],{"dataC":27633,"xLinkHref":29552},"#MJX-48-TEX-N-7C",[25022,29554,29555],{"dataMmlNode":25942,"transform":26048},[25033,29556],{"dataC":26000,"xLinkHref":29557},"#MJX-48-TEX-I-1D446",[25022,29559,29560],{"dataMmlNode":25031,"transform":27664},[25033,29561],{"dataC":27633,"xLinkHref":29552}," is the total number of items overall. This creates a fraction (e.g., 4 items out of 10 = ",[24923,29564,29566],{"className":29565,"jax":24927},[24926],[24929,29567,29571,29583],{"style":27778,"xmlns":24932,"width":29568,"height":29569,"role":1930,"focusable":8058,"viewBox":29570,"xmlnsXLink":24936},"2.891ex","1.581ex","0 -677 1278 699",[24938,29572,29573,29576,29579],{},[24941,29574],{"id":29575,"d":25935},"MJX-49-TEX-N-30",[24941,29577],{"id":29578,"d":26357},"MJX-49-TEX-N-2E",[24941,29580],{"id":29581,"d":29582},"MJX-49-TEX-N-34","M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z",[25022,29584,29585],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29586,29587],{"dataMmlNode":25028},[25022,29588,29589,29592,29595],{"dataMmlNode":26031},[25033,29590],{"dataC":26103,"xLinkHref":29591},"#MJX-49-TEX-N-30",[25033,29593],{"dataC":26392,"xLinkHref":29594,"transform":26394},"#MJX-49-TEX-N-2E",[25033,29596],{"dataC":29597,"xLinkHref":29598,"transform":25049},"34","#MJX-49-TEX-N-34",[12,29600,29601,29604,29605],{},[106,29602,29603],{},"Why do we need the weight?:","\nImagine you have 100 fruits. You ask a terrible question: ",[7289,29606,29607],{},"\"Does the fruit have exactly 341 tiny dimples on its skin?\"",[23,29609,29610,29616],{},[26,29611,29612,29615],{},[106,29613,29614],{},"Branch 1 (Yes):"," Only 1 fruit goes here. It happens to be an Apple. The entropy of this branch is 0 (perfectly pure!).",[26,29617,29618,29621],{},[106,29619,29620],{},"Branch 2 (No):"," 99 fruits go here. It's a complete, jumbled mess of Apples, Lemons, and Grapes. The entropy is very high.",[12,29623,29624,29625,29686],{},"If we just averaged the two entropies together, the 1-fruit branch would artificially make the split look really good. By multiplying by ",[24923,29626,29628],{"className":29627,"jax":24927},[24926],[24929,29629,29630,29641],{"style":28580,"xmlns":24932,"width":29394,"height":28582,"role":1930,"focusable":8058,"viewBox":29395,"xmlnsXLink":24936},[24938,29631,29632,29635,29638],{},[24941,29633],{"id":29634,"d":27405},"MJX-50-TEX-N-7C",[24941,29636],{"id":29637,"d":25900},"MJX-50-TEX-I-1D446",[24941,29639],{"id":29640,"d":27377},"MJX-50-TEX-I-1D463",[25022,29642,29643],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29644,29645],{"dataMmlNode":25028},[25022,29646,29647,29670,29684],{"dataMmlNode":27622},[25022,29648,29649,29654,29666],{"dataMmlNode":27626,"transform":28837},[25022,29650,29651],{"dataMmlNode":25031,"transform":27630},[25033,29652],{"dataC":27633,"xLinkHref":29653},"#MJX-50-TEX-N-7C",[25022,29655,29656,29661],{"dataMmlNode":26023,"transform":26048},[25022,29657,29658],{"dataMmlNode":25942},[25033,29659],{"dataC":26000,"xLinkHref":29660},"#MJX-50-TEX-I-1D446",[25022,29662,29663],{"dataMmlNode":25942,"transform":27643},[25033,29664],{"dataC":27556,"xLinkHref":29665},"#MJX-50-TEX-I-1D463",[25022,29667,29668],{"dataMmlNode":25031,"transform":27648},[25033,29669],{"dataC":27633,"xLinkHref":29653},[25022,29671,29672,29676,29680],{"dataMmlNode":27626,"transform":28859},[25022,29673,29674],{"dataMmlNode":25031,"transform":27630},[25033,29675],{"dataC":27633,"xLinkHref":29653},[25022,29677,29678],{"dataMmlNode":25942,"transform":26048},[25033,29679],{"dataC":26000,"xLinkHref":29660},[25022,29681,29682],{"dataMmlNode":25031,"transform":27664},[25033,29683],{"dataC":27633,"xLinkHref":29653},[27668,29685],{"width":28874,"height":27671,"x":27672,"y":27673},", we \"weight\" the branches by their size.",[23,29688,29689,29739],{},[26,29690,29691,29692,29738],{},"Branch 1 gets a weight of ",[24923,29693,29695],{"className":29694,"jax":24927},[24926],[24929,29696,29699,29711],{"style":24931,"xmlns":24932,"width":29697,"height":24934,"role":1930,"focusable":8058,"viewBox":29698,"xmlnsXLink":24936},"5.656ex","0 -750 2500 1000",[24938,29700,29701,29704,29708],{},[24941,29702],{"id":29703,"d":25914},"MJX-51-TEX-N-31",[24941,29705],{"id":29706,"d":29707},"MJX-51-TEX-N-2F","M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z",[24941,29709],{"id":29710,"d":25935},"MJX-51-TEX-N-30",[25022,29712,29713],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29714,29715,29720,29728],{"dataMmlNode":25028},[25022,29716,29717],{"dataMmlNode":26031},[25033,29718],{"dataC":26035,"xLinkHref":29719},"#MJX-51-TEX-N-31",[25022,29721,29722],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":26394},[25022,29723,29724],{"dataMmlNode":25031},[25033,29725],{"dataC":29726,"xLinkHref":29727},"2F","#MJX-51-TEX-N-2F",[25022,29729,29731,29733,29736],{"dataMmlNode":26031,"transform":29730},"translate(1000,0)",[25033,29732],{"dataC":26035,"xLinkHref":29719},[25033,29734],{"dataC":26103,"xLinkHref":29735,"transform":26394},"#MJX-51-TEX-N-30",[25033,29737],{"dataC":26103,"xLinkHref":29735,"transform":29730}," (barely counts).",[26,29740,29741,29742,29794],{},"Branch 2 gets a weight of ",[24923,29743,29745],{"className":29744,"jax":24927},[24926],[24929,29746,29749,29764],{"style":24931,"xmlns":24932,"width":29747,"height":24934,"role":1930,"focusable":8058,"viewBox":29748,"xmlnsXLink":24936},"6.787ex","0 -750 3000 1000",[24938,29750,29751,29755,29758,29761],{},[24941,29752],{"id":29753,"d":29754},"MJX-52-TEX-N-39","M352 287Q304 211 232 211Q154 211 104 270T44 396Q42 412 42 436V444Q42 537 111 606Q171 666 243 666Q245 666 249 666T257 665H261Q273 665 286 663T323 651T370 619T413 560Q456 472 456 334Q456 194 396 97Q361 41 312 10T208 -22Q147 -22 108 7T68 93T121 149Q143 149 158 135T173 96Q173 78 164 65T148 49T135 44L131 43Q131 41 138 37T164 27T206 22H212Q272 22 313 86Q352 142 352 280V287ZM244 248Q292 248 321 297T351 430Q351 508 343 542Q341 552 337 562T323 588T293 615T246 625Q208 625 181 598Q160 576 154 546T147 441Q147 358 152 329T172 282Q197 248 244 248Z",[24941,29756],{"id":29757,"d":29707},"MJX-52-TEX-N-2F",[24941,29759],{"id":29760,"d":25914},"MJX-52-TEX-N-31",[24941,29762],{"id":29763,"d":25935},"MJX-52-TEX-N-30",[25022,29765,29766],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29767,29768,29776,29783],{"dataMmlNode":25028},[25022,29769,29770,29774],{"dataMmlNode":26031},[25033,29771],{"dataC":29772,"xLinkHref":29773},"39","#MJX-52-TEX-N-39",[25033,29775],{"dataC":29772,"xLinkHref":29773,"transform":26394},[25022,29777,29778],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":29730},[25022,29779,29780],{"dataMmlNode":25031},[25033,29781],{"dataC":29726,"xLinkHref":29782},"#MJX-52-TEX-N-2F",[25022,29784,29786,29789,29792],{"dataMmlNode":26031,"transform":29785},"translate(1500,0)",[25033,29787],{"dataC":26035,"xLinkHref":29788},"#MJX-52-TEX-N-31",[25033,29790],{"dataC":26103,"xLinkHref":29791,"transform":26394},"#MJX-52-TEX-N-30",[25033,29793],{"dataC":26103,"xLinkHref":29791,"transform":29730}," (counts heavily).",[12,29796,29797,29800,29932,29933,29936,29937,6959],{},[106,29798,29799],{},"Bringing it all together:",[24923,29801,29803],{"className":29802,"jax":24927},[24926],[24929,29804,29809,29843],{"style":29805,"xmlns":24932,"width":29806,"height":29807,"role":1930,"focusable":8058,"viewBox":29808,"xmlnsXLink":24936},"vertical-align: -0.464ex;","18.208ex","2.059ex","0 -705 8048 910",[24938,29810,29811,29815,29818,29822,29825,29828,29831,29834,29837,29840],{},[24941,29812],{"id":29813,"d":29814},"MJX-53-TEX-I-1D43C","M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z",[24941,29816],{"id":29817,"d":25873},"MJX-53-TEX-I-1D45B",[24941,29819],{"id":29820,"d":29821},"MJX-53-TEX-I-1D453","M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z",[24941,29823],{"id":29824,"d":25885},"MJX-53-TEX-I-1D45C",[24941,29826],{"id":29827,"d":25881},"MJX-53-TEX-I-1D45F",[24941,29829],{"id":29830,"d":28310},"MJX-53-TEX-I-1D45A",[24941,29832],{"id":29833,"d":27323},"MJX-53-TEX-I-1D44E",[24941,29835],{"id":29836,"d":25877},"MJX-53-TEX-I-1D461",[24941,29838],{"id":29839,"d":27108},"MJX-53-TEX-I-1D456",[24941,29841],{"id":29842,"d":27319},"MJX-53-TEX-I-1D43A",[25022,29844,29845],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,29846,29847,29853,29859,29866,29872,29877,29883,29889,29895,29901,29906,29911,29917,29922,29927],{"dataMmlNode":25028},[25022,29848,29849],{"dataMmlNode":25942},[25033,29850],{"dataC":29851,"xLinkHref":29852},"1D43C","#MJX-53-TEX-I-1D43C",[25022,29854,29856],{"dataMmlNode":25942,"transform":29855},"translate(504,0)",[25033,29857],{"dataC":25952,"xLinkHref":29858},"#MJX-53-TEX-I-1D45B",[25022,29860,29862],{"dataMmlNode":25942,"transform":29861},"translate(1104,0)",[25033,29863],{"dataC":29864,"xLinkHref":29865},"1D453","#MJX-53-TEX-I-1D453",[25022,29867,29869],{"dataMmlNode":25942,"transform":29868},"translate(1654,0)",[25033,29870],{"dataC":25973,"xLinkHref":29871},"#MJX-53-TEX-I-1D45C",[25022,29873,29874],{"dataMmlNode":25942,"transform":25141},[25033,29875],{"dataC":25966,"xLinkHref":29876},"#MJX-53-TEX-I-1D45F",[25022,29878,29880],{"dataMmlNode":25942,"transform":29879},"translate(2590,0)",[25033,29881],{"dataC":28479,"xLinkHref":29882},"#MJX-53-TEX-I-1D45A",[25022,29884,29886],{"dataMmlNode":25942,"transform":29885},"translate(3468,0)",[25033,29887],{"dataC":27425,"xLinkHref":29888},"#MJX-53-TEX-I-1D44E",[25022,29890,29892],{"dataMmlNode":25942,"transform":29891},"translate(3997,0)",[25033,29893],{"dataC":25959,"xLinkHref":29894},"#MJX-53-TEX-I-1D461",[25022,29896,29898],{"dataMmlNode":25942,"transform":29897},"translate(4358,0)",[25033,29899],{"dataC":27210,"xLinkHref":29900},"#MJX-53-TEX-I-1D456",[25022,29902,29904],{"dataMmlNode":25942,"transform":29903},"translate(4703,0)",[25033,29905],{"dataC":25973,"xLinkHref":29871},[25022,29907,29909],{"dataMmlNode":25942,"transform":29908},"translate(5188,0)",[25033,29910],{"dataC":25952,"xLinkHref":29858},[25022,29912,29914],{"dataMmlNode":25942,"transform":29913},"translate(5788,0)",[25033,29915],{"dataC":27418,"xLinkHref":29916},"#MJX-53-TEX-I-1D43A",[25022,29918,29920],{"dataMmlNode":25942,"transform":29919},"translate(6574,0)",[25033,29921],{"dataC":27425,"xLinkHref":29888},[25022,29923,29925],{"dataMmlNode":25942,"transform":29924},"translate(7103,0)",[25033,29926],{"dataC":27210,"xLinkHref":29900},[25022,29928,29930],{"dataMmlNode":25942,"transform":29929},"translate(7448,0)",[25033,29931],{"dataC":25952,"xLinkHref":29858}," = ",[6979,29934,29935],{},"(Messiness of Parent)"," MINUS ",[6979,29938,29939],{},"(Weighted Average Messiness of the Children)",[12,29941,29942,29943,29945,29946,29948],{},"When the decision tree is sitting at Step 2 of the pseudocode (\"Pick the best feature\"), it calculates this exact Information Gain equation for ",[7289,29944,18612],{},", and then for ",[7289,29947,25688],{},". Whichever feature gives the highest number (meaning it removed the most messiness) is the winner, and becomes the next node in the tree!",[7166,29950,13610],{"id":13609},[12,29952,29953],{},"Consider four examples with features (size, color, shape) and a binary class 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negatives, so ",[24923,30031,30033],{"className":30032,"jax":24927},[24926],[24929,30034,30037,30081],{"style":24931,"xmlns":24932,"width":30035,"height":24934,"role":1930,"focusable":8058,"viewBox":30036,"xmlnsXLink":24936},"17.395ex","0 -750 7688.6 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1000",[24938,30555,30556,30559,30562,30565,30568,30571,30574,30577,30580,30583,30586,30589,30592,30595,30598,30601,30604,30607,30610],{},[24941,30557],{"id":30558,"d":27319},"MJX-60-TEX-I-1D43A",[24941,30560],{"id":30561,"d":27323},"MJX-60-TEX-I-1D44E",[24941,30563],{"id":30564,"d":27108},"MJX-60-TEX-I-1D456",[24941,30566],{"id":30567,"d":25873},"MJX-60-TEX-I-1D45B",[24941,30569],{"id":30570,"d":24972},"MJX-60-TEX-N-3D",[24941,30572],{"id":30573,"d":25914},"MJX-60-TEX-N-31",[24941,30575],{"id":30576,"d":25910},"MJX-60-TEX-N-2212",[24941,30578],{"id":30579,"d":24944},"MJX-60-TEX-N-28",[24941,30581],{"id":30582,"d":25935},"MJX-60-TEX-N-30",[24941,30584],{"id":30585,"d":26357},"MJX-60-TEX-N-2E",[24941,30587],{"id":30588,"d":26361},"MJX-60-TEX-N-37",[24941,30590],{"id":30591,"d":26660},"MJX-60-TEX-N-35",[24941,30593],{"id":30594,"d":26907},"MJX-60-TEX-N-D7",[24941,30596],{"id":30597,"d":29754},"MJX-60-TEX-N-39",[24941,30599],{"id":30600,"d":30471},"MJX-60-TEX-N-38",[24941,30602],{"id":30603,"d":30313},"MJX-60-TEX-N-2B",[24941,30605],{"id":30606,"d":25928},"MJX-60-TEX-N-32",[24941,30608],{"id":30609,"d":24988},"MJX-60-TEX-N-29",[24941,30611],{"id":30612,"d":26455},"MJX-60-TEX-N-33",[25022,30614,30615],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,30616,30617,30622,30627,30632,30637,30642,30647,30652,30657,30671,30677,30692,30698,30710,30715,30720,30726,30731],{"dataMmlNode":25028},[25022,30618,30619],{"dataMmlNode":25942},[25033,30620],{"dataC":27418,"xLinkHref":30621},"#MJX-60-TEX-I-1D43A",[25022,30623,30624],{"dataMmlNode":25942,"transform":27422},[25033,30625],{"dataC":27425,"xLinkHref":30626},"#MJX-60-TEX-I-1D44E",[25022,30628,30629],{"dataMmlNode":25942,"transform":27429},[25033,30630],{"dataC":27210,"xLinkHref":30631},"#MJX-60-TEX-I-1D456",[25022,30633,30634],{"dataMmlNode":25942,"transform":27435},[25033,30635],{"dataC":25952,"xLinkHref":30636},"#MJX-60-TEX-I-1D45B",[25022,30638,30639],{"dataMmlNode":25031,"transform":30343},[25033,30640],{"dataC":25078,"xLinkHref":30641},"#MJX-60-TEX-N-3D",[25022,30643,30644],{"dataMmlNode":26031,"transform":30349},[25033,30645],{"dataC":26035,"xLinkHref":30646},"#MJX-60-TEX-N-31",[25022,30648,30649],{"dataMmlNode":25031,"transform":30355},[25033,30650],{"dataC":26019,"xLinkHref":30651},"#MJX-60-TEX-N-2212",[25022,30653,30654],{"dataMmlNode":25031,"transform":30361},[25033,30655],{"dataC":25035,"xLinkHref":30656},"#MJX-60-TEX-N-28",[25022,30658,30659,30662,30665,30668],{"dataMmlNode":26031,"transform":30367},[25033,30660],{"dataC":26103,"xLinkHref":30661},"#MJX-60-TEX-N-30",[25033,30663],{"dataC":26392,"xLinkHref":30664,"transform":26394},"#MJX-60-TEX-N-2E",[25033,30666],{"dataC":26397,"xLinkHref":30667,"transform":25049},"#MJX-60-TEX-N-37",[25033,30669],{"dataC":26717,"xLinkHref":30670,"transform":26835},"#MJX-60-TEX-N-35",[25022,30672,30674],{"dataMmlNode":25031,"transform":30673},"translate(7705.2,0)",[25033,30675],{"dataC":26950,"xLinkHref":30676},"#MJX-60-TEX-N-D7",[25022,30678,30680,30682,30684,30687,30689],{"dataMmlNode":26031,"transform":30679},"translate(8705.4,0)",[25033,30681],{"dataC":26103,"xLinkHref":30661},[25033,30683],{"dataC":26392,"xLinkHref":30664,"transform":26394},[25033,30685],{"dataC":29772,"xLinkHref":30686,"transform":25049},"#MJX-60-TEX-N-39",[25033,30688],{"dataC":26035,"xLinkHref":30646,"transform":26835},[25033,30690],{"dataC":30502,"xLinkHref":30691,"transform":30504},"#MJX-60-TEX-N-38",[25022,30693,30695],{"dataMmlNode":25031,"transform":30694},"translate(11205.7,0)",[25033,30696],{"dataC":30393,"xLinkHref":30697},"#MJX-60-TEX-N-2B",[25022,30699,30701,30703,30705,30708],{"dataMmlNode":26031,"transform":30700},"translate(12205.9,0)",[25033,30702],{"dataC":26103,"xLinkHref":30661},[25033,30704],{"dataC":26392,"xLinkHref":30664,"transform":26394},[25033,30706],{"dataC":26058,"xLinkHref":30707,"transform":25049},"#MJX-60-TEX-N-32",[25033,30709],{"dataC":26717,"xLinkHref":30670,"transform":26835},[25022,30711,30713],{"dataMmlNode":25031,"transform":30712},"translate(14206.1,0)",[25033,30714],{"dataC":26950,"xLinkHref":30676},[25022,30716,30718],{"dataMmlNode":26031,"transform":30717},"translate(15206.3,0)",[25033,30719],{"dataC":26103,"xLinkHref":30661},[25022,30721,30723],{"dataMmlNode":25031,"transform":30722},"translate(15706.3,0)",[25033,30724],{"dataC":25108,"xLinkHref":30725},"#MJX-60-TEX-N-29",[25022,30727,30729],{"dataMmlNode":25031,"transform":30728},"translate(16373.1,0)",[25033,30730],{"dataC":25078,"xLinkHref":30641},[25022,30732,30734,30736,30738,30741,30743],{"dataMmlNode":26031,"transform":30733},"translate(17428.9,0)",[25033,30735],{"dataC":26103,"xLinkHref":30661},[25033,30737],{"dataC":26392,"xLinkHref":30664,"transform":26394},[25033,30739],{"dataC":26486,"xLinkHref":30740,"transform":25049},"#MJX-60-TEX-N-33",[25033,30742],{"dataC":26035,"xLinkHref":30646,"transform":26835},[25033,30744],{"dataC":26035,"xLinkHref":30646,"transform":30504}," — much better.",[12,30747,30748],{},[106,30749,30750],{},"Splitting on Shape:",[23,30752,30753,30815,30853],{},[26,30754,30755,30756],{},"circle → 2+, 1− → ",[24923,30757,30759],{"className":30758,"jax":24927},[24926],[24929,30760,30761,30784],{"style":26868,"xmlns":24932,"width":30446,"height":30167,"role":1930,"focusable":8058,"viewBox":30447,"xmlnsXLink":24936},[24938,30762,30763,30766,30769,30772,30775,30778,30781],{},[24941,30764],{"id":30765,"d":25869},"MJX-61-TEX-I-1D438",[24941,30767],{"id":30768,"d":24972},"MJX-61-TEX-N-3D",[24941,30770],{"id":30771,"d":25935},"MJX-61-TEX-N-30",[24941,30773],{"id":30774,"d":26357},"MJX-61-TEX-N-2E",[24941,30776],{"id":30777,"d":29754},"MJX-61-TEX-N-39",[24941,30779],{"id":30780,"d":25914},"MJX-61-TEX-N-31",[24941,30782],{"id":30783,"d":30471},"MJX-61-TEX-N-38",[25022,30785,30786],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,30787,30788,30793,30798],{"dataMmlNode":25028},[25022,30789,30790],{"dataMmlNode":25942},[25033,30791],{"dataC":25945,"xLinkHref":30792},"#MJX-61-TEX-I-1D438",[25022,30794,30795],{"dataMmlNode":25031,"transform":30197},[25033,30796],{"dataC":25078,"xLinkHref":30797},"#MJX-61-TEX-N-3D",[25022,30799,30800,30803,30806,30809,30812],{"dataMmlNode":26031,"transform":30203},[25033,30801],{"dataC":26103,"xLinkHref":30802},"#MJX-61-TEX-N-30",[25033,30804],{"dataC":26392,"xLinkHref":30805,"transform":26394},"#MJX-61-TEX-N-2E",[25033,30807],{"dataC":29772,"xLinkHref":30808,"transform":25049},"#MJX-61-TEX-N-39",[25033,30810],{"dataC":26035,"xLinkHref":30811,"transform":26835},"#MJX-61-TEX-N-31",[25033,30813],{"dataC":30502,"xLinkHref":30814,"transform":30504},"#MJX-61-TEX-N-38",[26,30816,30817,30818],{},"square → 0+, 1− → ",[24923,30819,30821],{"className":30820,"jax":24927},[24926],[24929,30822,30823,30834],{"style":26868,"xmlns":24932,"width":30513,"height":30167,"role":1930,"focusable":8058,"viewBox":30514,"xmlnsXLink":24936},[24938,30824,30825,30828,30831],{},[24941,30826],{"id":30827,"d":25869},"MJX-62-TEX-I-1D438",[24941,30829],{"id":30830,"d":24972},"MJX-62-TEX-N-3D",[24941,30832],{"id":30833,"d":25935},"MJX-62-TEX-N-30",[25022,30835,30836],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,30837,30838,30843,30848],{"dataMmlNode":25028},[25022,30839,30840],{"dataMmlNode":25942},[25033,30841],{"dataC":25945,"xLinkHref":30842},"#MJX-62-TEX-I-1D438",[25022,30844,30845],{"dataMmlNode":25031,"transform":30197},[25033,30846],{"dataC":25078,"xLinkHref":30847},"#MJX-62-TEX-N-3D",[25022,30849,30850],{"dataMmlNode":26031,"transform":30203},[25033,30851],{"dataC":26103,"xLinkHref":30852},"#MJX-62-TEX-N-30",[26,30854,30855,31042],{},[24923,30856,30858],{"className":30857,"jax":24927},[24926],[24929,30859,30860,30919],{"style":24931,"xmlns":24932,"width":30552,"height":24934,"role":1930,"focusable":8058,"viewBox":30553,"xmlnsXLink":24936},[24938,30861,30862,30865,30868,30871,30874,30877,30880,30883,30886,30889,30892,30895,30898,30901,30904,30907,30910,30913,30916],{},[24941,30863],{"id":30864,"d":27319},"MJX-63-TEX-I-1D43A",[24941,30866],{"id":30867,"d":27323},"MJX-63-TEX-I-1D44E",[24941,30869],{"id":30870,"d":27108},"MJX-63-TEX-I-1D456",[24941,30872],{"id":30873,"d":25873},"MJX-63-TEX-I-1D45B",[24941,30875],{"id":30876,"d":24972},"MJX-63-TEX-N-3D",[24941,30878],{"id":30879,"d":25914},"MJX-63-TEX-N-31",[24941,30881],{"id":30882,"d":25910},"MJX-63-TEX-N-2212",[24941,30884],{"id":30885,"d":24944},"MJX-63-TEX-N-28",[24941,30887],{"id":30888,"d":25935},"MJX-63-TEX-N-30",[24941,30890],{"id":30891,"d":26357},"MJX-63-TEX-N-2E",[24941,30893],{"id":30894,"d":26361},"MJX-63-TEX-N-37",[24941,30896],{"id":30897,"d":26660},"MJX-63-TEX-N-35",[24941,30899],{"id":30900,"d":26907},"MJX-63-TEX-N-D7",[24941,30902],{"id":30903,"d":29754},"MJX-63-TEX-N-39",[24941,30905],{"id":30906,"d":30471},"MJX-63-TEX-N-38",[24941,30908],{"id":30909,"d":30313},"MJX-63-TEX-N-2B",[24941,30911],{"id":30912,"d":25928},"MJX-63-TEX-N-32",[24941,30914],{"id":30915,"d":24988},"MJX-63-TEX-N-29",[24941,30917],{"id":30918,"d":26455},"MJX-63-TEX-N-33",[25022,30920,30921],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,30922,30923,30928,30933,30938,30943,30948,30953,30958,30963,30977,30982,30996,31001,31012,31016,31020,31025,31029],{"dataMmlNode":25028},[25022,30924,30925],{"dataMmlNode":25942},[25033,30926],{"dataC":27418,"xLinkHref":30927},"#MJX-63-TEX-I-1D43A",[25022,30929,30930],{"dataMmlNode":25942,"transform":27422},[25033,30931],{"dataC":27425,"xLinkHref":30932},"#MJX-63-TEX-I-1D44E",[25022,30934,30935],{"dataMmlNode":25942,"transform":27429},[25033,30936],{"dataC":27210,"xLinkHref":30937},"#MJX-63-TEX-I-1D456",[25022,30939,30940],{"dataMmlNode":25942,"transform":27435},[25033,30941],{"dataC":25952,"xLinkHref":30942},"#MJX-63-TEX-I-1D45B",[25022,30944,30945],{"dataMmlNode":25031,"transform":30343},[25033,30946],{"dataC":25078,"xLinkHref":30947},"#MJX-63-TEX-N-3D",[25022,30949,30950],{"dataMmlNode":26031,"transform":30349},[25033,30951],{"dataC":26035,"xLinkHref":30952},"#MJX-63-TEX-N-31",[25022,30954,30955],{"dataMmlNode":25031,"transform":30355},[25033,30956],{"dataC":26019,"xLinkHref":30957},"#MJX-63-TEX-N-2212",[25022,30959,30960],{"dataMmlNode":25031,"transform":30361},[25033,30961],{"dataC":25035,"xLinkHref":30962},"#MJX-63-TEX-N-28",[25022,30964,30965,30968,30971,30974],{"dataMmlNode":26031,"transform":30367},[25033,30966],{"dataC":26103,"xLinkHref":30967},"#MJX-63-TEX-N-30",[25033,30969],{"dataC":26392,"xLinkHref":30970,"transform":26394},"#MJX-63-TEX-N-2E",[25033,30972],{"dataC":26397,"xLinkHref":30973,"transform":25049},"#MJX-63-TEX-N-37",[25033,30975],{"dataC":26717,"xLinkHref":30976,"transform":26835},"#MJX-63-TEX-N-35",[25022,30978,30979],{"dataMmlNode":25031,"transform":30673},[25033,30980],{"dataC":26950,"xLinkHref":30981},"#MJX-63-TEX-N-D7",[25022,30983,30984,30986,30988,30991,30993],{"dataMmlNode":26031,"transform":30679},[25033,30985],{"dataC":26103,"xLinkHref":30967},[25033,30987],{"dataC":26392,"xLinkHref":30970,"transform":26394},[25033,30989],{"dataC":29772,"xLinkHref":30990,"transform":25049},"#MJX-63-TEX-N-39",[25033,30992],{"dataC":26035,"xLinkHref":30952,"transform":26835},[25033,30994],{"dataC":30502,"xLinkHref":30995,"transform":30504},"#MJX-63-TEX-N-38",[25022,30997,30998],{"dataMmlNode":25031,"transform":30694},[25033,30999],{"dataC":30393,"xLinkHref":31000},"#MJX-63-TEX-N-2B",[25022,31002,31003,31005,31007,31010],{"dataMmlNode":26031,"transform":30700},[25033,31004],{"dataC":26103,"xLinkHref":30967},[25033,31006],{"dataC":26392,"xLinkHref":30970,"transform":26394},[25033,31008],{"dataC":26058,"xLinkHref":31009,"transform":25049},"#MJX-63-TEX-N-32",[25033,31011],{"dataC":26717,"xLinkHref":30976,"transform":26835},[25022,31013,31014],{"dataMmlNode":25031,"transform":30712},[25033,31015],{"dataC":26950,"xLinkHref":30981},[25022,31017,31018],{"dataMmlNode":26031,"transform":30717},[25033,31019],{"dataC":26103,"xLinkHref":30967},[25022,31021,31022],{"dataMmlNode":25031,"transform":30722},[25033,31023],{"dataC":25108,"xLinkHref":31024},"#MJX-63-TEX-N-29",[25022,31026,31027],{"dataMmlNode":25031,"transform":30728},[25033,31028],{"dataC":25078,"xLinkHref":30947},[25022,31030,31031,31033,31035,31038,31040],{"dataMmlNode":26031,"transform":30733},[25033,31032],{"dataC":26103,"xLinkHref":30967},[25033,31034],{"dataC":26392,"xLinkHref":30970,"transform":26394},[25033,31036],{"dataC":26486,"xLinkHref":31037,"transform":25049},"#MJX-63-TEX-N-33",[25033,31039],{"dataC":26035,"xLinkHref":30952,"transform":26835},[25033,31041],{"dataC":26035,"xLinkHref":30952,"transform":30504}," — tied with color.",[12,31044,31045],{},"So we'd pick either color or shape first (both have gain = 0.311), and definitely not size (gain = 0).",[6964,31047,31049],{"id":31048},"the-id3-algorithm","The ID3 Algorithm",[12,31051,31052,31053,31056],{},"The algorithm we've been describing is called ",[106,31054,31055],{},"ID3",", invented by J. Ross Quinlan in 1979. It uses Shannon's information theory (1948) to select features via information gain. Key characteristics:",[23,31058,31059,31066,31072,31079],{},[26,31060,31061,31062,31065],{},"Builds the tree ",[106,31063,31064],{},"top-down"," with no backtracking (the algorithm is stubbornly committed to its choices, greedy algorithm)",[26,31067,31068,31069,31071],{},"Selects the feature with the ",[106,31070,27848],{}," at each step",[26,31073,31074,31075,31078],{},"Runs ",[106,31076,31077],{},"recursively"," on non-leaf branches until all data is classified",[26,31080,31081,31082,31085,31086,31089,31090,31093],{},"Searches the ",[106,31083,31084],{},"entire dataset"," to construct the tree\nLater improvements led to ",[106,31087,31088],{},"C4.5"," (Quinlan, 1993), which handles continuous attributes, missing values, and pruning. Another parallel development was ",[106,31091,31092],{},"CART"," (Classification and Regression Trees) by Breiman and Friedman.",[6964,31095,31097],{"id":31096},"a-fun-example-classifying-simpsons-characters","A Fun Example: Classifying Simpsons Characters",[12,31099,31100],{},"Let's say we want to classify Simpsons characters as Male or Female using three features: hair length, weight, and age.",[7085,31102,31103,31121],{},[7088,31104,31105],{},[7091,31106,31107,31110,31113,31116,31119],{},[7094,31108,31109],{},"Person",[7094,31111,31112],{},"Hair Length",[7094,31114,31115],{},"Weight",[7094,31117,31118],{},"Age",[7094,31120,23887],{},[7104,31122,31123,31140,31156,31171,31187,31202,31218,31234,31248],{},[7091,31124,31125,31128,31131,31134,31137],{},[7109,31126,31127],{},"Homer",[7109,31129,31130],{},"0\"",[7109,31132,31133],{},"250",[7109,31135,31136],{},"36",[7109,31138,31139],{},"M",[7091,31141,31142,31145,31148,31151,31153],{},[7109,31143,31144],{},"Marge",[7109,31146,31147],{},"10\"",[7109,31149,31150],{},"150",[7109,31152,29597],{},[7109,31154,31155],{},"F",[7091,31157,31158,31161,31164,31167,31169],{},[7109,31159,31160],{},"Bart",[7109,31162,31163],{},"2\"",[7109,31165,31166],{},"90",[7109,31168,17383],{},[7109,31170,31139],{},[7091,31172,31173,31176,31179,31182,31185],{},[7109,31174,31175],{},"Lisa",[7109,31177,31178],{},"6\"",[7109,31180,31181],{},"78",[7109,31183,31184],{},"8",[7109,31186,31155],{},[7091,31188,31189,31192,31195,31198,31200],{},[7109,31190,31191],{},"Maggie",[7109,31193,31194],{},"4\"",[7109,31196,31197],{},"20",[7109,31199,8855],{},[7109,31201,31155],{},[7091,31203,31204,31207,31210,31213,31216],{},[7109,31205,31206],{},"Abe",[7109,31208,31209],{},"1\"",[7109,31211,31212],{},"170",[7109,31214,31215],{},"70",[7109,31217,31139],{},[7091,31219,31220,31223,31226,31229,31232],{},[7109,31221,31222],{},"Selma",[7109,31224,31225],{},"8\"",[7109,31227,31228],{},"160",[7109,31230,31231],{},"41",[7109,31233,31155],{},[7091,31235,31236,31239,31241,31244,31246],{},[7109,31237,31238],{},"Otto",[7109,31240,31147],{},[7109,31242,31243],{},"180",[7109,31245,30502],{},[7109,31247,31139],{},[7091,31249,31250,31253,31255,31258,31261],{},[7109,31251,31252],{},"Krusty",[7109,31254,31178],{},[7109,31256,31257],{},"200",[7109,31259,31260],{},"45",[7109,31262,31139],{},[12,31264,31265,31266,6959],{},"The dataset has 4 females and 5 males, with ",[24923,31267,31269],{"className":31268,"jax":24927},[24926],[24929,31270,31273,31333],{"style":24931,"xmlns":24932,"width":31271,"height":24934,"role":1930,"focusable":8058,"viewBox":31272,"xmlnsXLink":24936},"26.67ex","0 -750 11788.2 1000",[24938,31274,31275,31278,31281,31284,31287,31290,31293,31296,31299,31302,31305,31308,31311,31315,31318,31321,31324,31327,31330],{},[24941,31276],{"id":31277,"d":25869},"MJX-64-TEX-I-1D438",[24941,31279],{"id":31280,"d":25873},"MJX-64-TEX-I-1D45B",[24941,31282],{"id":31283,"d":25877},"MJX-64-TEX-I-1D461",[24941,31285],{"id":31286,"d":25881},"MJX-64-TEX-I-1D45F",[24941,31288],{"id":31289,"d":25885},"MJX-64-TEX-I-1D45C",[24941,31291],{"id":31292,"d":25889},"MJX-64-TEX-I-1D45D",[24941,31294],{"id":31295,"d":25893},"MJX-64-TEX-I-1D466",[24941,31297],{"id":31298,"d":24944},"MJX-64-TEX-N-28",[24941,31300],{"id":31301,"d":29582},"MJX-64-TEX-N-34",[24941,31303],{"id":31304,"d":27343},"MJX-64-TEX-I-1D439",[24941,31306],{"id":31307,"d":27339},"MJX-64-TEX-N-2C",[24941,31309],{"id":31310,"d":26660},"MJX-64-TEX-N-35",[24941,31312],{"id":31313,"d":31314},"MJX-64-TEX-I-1D440","M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",[24941,31316],{"id":31317,"d":24988},"MJX-64-TEX-N-29",[24941,31319],{"id":31320,"d":24972},"MJX-64-TEX-N-3D",[24941,31322],{"id":31323,"d":25935},"MJX-64-TEX-N-30",[24941,31325],{"id":31326,"d":26357},"MJX-64-TEX-N-2E",[24941,31328],{"id":31329,"d":29754},"MJX-64-TEX-N-39",[24941,31331],{"id":31332,"d":25914},"MJX-64-TEX-N-31",[25022,31334,31335],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,31336,31337,31342,31347,31352,31357,31362,31367,31372,31377,31382,31388,31394,31400,31407,31413,31419],{"dataMmlNode":25028},[25022,31338,31339],{"dataMmlNode":25942},[25033,31340],{"dataC":25945,"xLinkHref":31341},"#MJX-64-TEX-I-1D438",[25022,31343,31344],{"dataMmlNode":25942,"transform":25949},[25033,31345],{"dataC":25952,"xLinkHref":31346},"#MJX-64-TEX-I-1D45B",[25022,31348,31349],{"dataMmlNode":25942,"transform":25956},[25033,31350],{"dataC":25959,"xLinkHref":31351},"#MJX-64-TEX-I-1D461",[25022,31353,31354],{"dataMmlNode":25942,"transform":25963},[25033,31355],{"dataC":25966,"xLinkHref":31356},"#MJX-64-TEX-I-1D45F",[25022,31358,31359],{"dataMmlNode":25942,"transform":25970},[25033,31360],{"dataC":25973,"xLinkHref":31361},"#MJX-64-TEX-I-1D45C",[25022,31363,31364],{"dataMmlNode":25942,"transform":25977},[25033,31365],{"dataC":25980,"xLinkHref":31366},"#MJX-64-TEX-I-1D45D",[25022,31368,31369],{"dataMmlNode":25942,"transform":25984},[25033,31370],{"dataC":25987,"xLinkHref":31371},"#MJX-64-TEX-I-1D466",[25022,31373,31374],{"dataMmlNode":25031,"transform":25991},[25033,31375],{"dataC":25035,"xLinkHref":31376},"#MJX-64-TEX-N-28",[25022,31378,31379],{"dataMmlNode":26031,"transform":25997},[25033,31380],{"dataC":29597,"xLinkHref":31381},"#MJX-64-TEX-N-34",[25022,31383,31385],{"dataMmlNode":25942,"transform":31384},"translate(4543,0)",[25033,31386],{"dataC":27463,"xLinkHref":31387},"#MJX-64-TEX-I-1D439",[25022,31389,31391],{"dataMmlNode":25031,"transform":31390},"translate(5292,0)",[25033,31392],{"dataC":27456,"xLinkHref":31393},"#MJX-64-TEX-N-2C",[25022,31395,31397],{"dataMmlNode":26031,"transform":31396},"translate(5736.7,0)",[25033,31398],{"dataC":26717,"xLinkHref":31399},"#MJX-64-TEX-N-35",[25022,31401,31403],{"dataMmlNode":25942,"transform":31402},"translate(6236.7,0)",[25033,31404],{"dataC":31405,"xLinkHref":31406},"1D440","#MJX-64-TEX-I-1D440",[25022,31408,31410],{"dataMmlNode":25031,"transform":31409},"translate(7287.7,0)",[25033,31411],{"dataC":25108,"xLinkHref":31412},"#MJX-64-TEX-N-29",[25022,31414,31416],{"dataMmlNode":25031,"transform":31415},"translate(7954.4,0)",[25033,31417],{"dataC":25078,"xLinkHref":31418},"#MJX-64-TEX-N-3D",[25022,31420,31422,31425,31428,31431,31433,31436],{"dataMmlNode":26031,"transform":31421},"translate(9010.2,0)",[25033,31423],{"dataC":26103,"xLinkHref":31424},"#MJX-64-TEX-N-30",[25033,31426],{"dataC":26392,"xLinkHref":31427,"transform":26394},"#MJX-64-TEX-N-2E",[25033,31429],{"dataC":29772,"xLinkHref":31430,"transform":25049},"#MJX-64-TEX-N-39",[25033,31432],{"dataC":29772,"xLinkHref":31430,"transform":26835},[25033,31434],{"dataC":26035,"xLinkHref":31435,"transform":30504},"#MJX-64-TEX-N-31",[25033,31437],{"dataC":26035,"xLinkHref":31435,"transform":31438},"translate(2278,0)",[12,31440,31441],{},"We compute the information gain for splitting on each feature:",[23,31443,31444,31450,31456],{},[26,31445,31446,31449],{},[106,31447,31448],{},"Hair Length ≤ 5?"," → Gain = 0.0911",[26,31451,31452,31455],{},[106,31453,31454],{},"Weight ≤ 160?"," → Gain = 0.5900 ← Winner!",[26,31457,31458,31461],{},[106,31459,31460],{},"Age ≤ 40?"," → Gain = 0.0183\nWeight gives us the best split. Everyone over 160 lbs turns out to be male (entropy = 0 on that branch — done!). But the ≤160 group still has a mix (4F, 1M), so we recurse.",[12,31463,31464,31465,31468],{},"On the second split, we find that ",[106,31466,31467],{},"Hair Length ≤ 2"," perfectly separates the remaining group: Bart (short hair, male) vs. the females (longer hair).",[12,31470,31471],{},"The final tree:",[6972,31473,31476],{"className":31474,"code":31475,"language":6977},[6975],"            Weight ≤ 160?\n            \u002F           \\\n          yes            no\n          \u002F               \\\n   Hair Length ≤ 2?      Male\n      \u002F        \\\n    yes         no\n    \u002F             \\\n  Male          Female\n",[6979,31477,31475],{"__ignoreMap":180},[12,31479,31480],{},"And as rules:",[23,31482,31483,31489,31494],{},[26,31484,31485,31486],{},"If Weight > 160 → ",[106,31487,31488],{},"Male",[26,31490,31491,31492],{},"Else if Hair Length ≤ 2 → ",[106,31493,31488],{},[26,31495,31496,31497,31500,31501,31503],{},"Else → ",[106,31498,31499],{},"Female","\nNow if Comic Book Guy shows up (Hair: 8\", Weight: 290, Age: 38), the tree says: Weight 290 > 160 → ",[106,31502,31488],{},". Correct!",[6964,31505,31507],{"id":31506},"the-overfitting-problem","The Overfitting Problem",[12,31509,31510],{},"Here's the catch: a tree that perfectly classifies every training example isn't necessarily a good tree.",[12,31512,31513,31516,31517,31541,31542,31578,31579,31582,31583,31586],{},[106,31514,31515],{},"Overfitting"," happens when a model fits the training data too well — including its noise and quirks — and performs poorly on new, unseen data. Formally, hypothesis ",[24923,31518,31520],{"className":31519,"jax":24927},[24926],[24929,31521,31525,31531],{"style":25196,"xmlns":24932,"width":31522,"height":31523,"role":1930,"focusable":8058,"viewBox":31524,"xmlnsXLink":24936},"1.303ex","1.595ex","0 -694 576 705",[24938,31526,31527],{},[24941,31528],{"id":31529,"d":31530},"MJX-65-TEX-I-210E","M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z",[25022,31532,31533],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,31534,31535],{"dataMmlNode":25028},[25022,31536,31537],{"dataMmlNode":25942},[25033,31538],{"dataC":31539,"xLinkHref":31540},"210E","#MJX-65-TEX-I-210E"," overfits if there exists another hypothesis ",[24923,31543,31545],{"className":31544,"jax":24927},[24926],[24929,31546,31550,31559],{"style":25196,"xmlns":24932,"width":31547,"height":31548,"role":1930,"focusable":8058,"viewBox":31549,"xmlnsXLink":24936},"1.931ex","1.742ex","0 -759 853.5 770",[24938,31551,31552,31555],{},[24941,31553],{"id":31554,"d":31530},"MJX-66-TEX-I-210E",[24941,31556],{"id":31557,"d":31558},"MJX-66-TEX-V-2032","M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z",[25022,31560,31561],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,31562,31563],{"dataMmlNode":25028},[25022,31564,31566,31571],{"dataMmlNode":31565},"msup",[25022,31567,31568],{"dataMmlNode":25942},[25033,31569],{"dataC":31539,"xLinkHref":31570},"#MJX-66-TEX-I-210E",[25022,31572,31574],{"dataMmlNode":25031,"transform":31573},"translate(609,363) scale(0.707)",[25033,31575],{"dataC":31576,"xLinkHref":31577},"2032","#MJX-66-TEX-V-2032"," that has ",[7289,31580,31581],{},"higher"," training error but ",[7289,31584,31585],{},"lower"," test error.",[12,31588,31589,31590,31641,31642,31645],{},"Think of it this way: if you measure 10 data points for Ohm's Law (",[24923,31591,31593],{"className":31592,"jax":24927},[24926],[24929,31594,31598,31613],{"style":26868,"xmlns":24932,"width":31595,"height":31596,"role":1930,"focusable":8058,"viewBox":31597,"xmlnsXLink":24936},"7.614ex","1.731ex","0 -683 3365.6 765",[24938,31599,31600,31603,31606,31609],{},[24941,31601],{"id":31602,"d":27385},"MJX-67-TEX-I-1D449",[24941,31604],{"id":31605,"d":24972},"MJX-67-TEX-N-3D",[24941,31607],{"id":31608,"d":29814},"MJX-67-TEX-I-1D43C",[24941,31610],{"id":31611,"d":31612},"MJX-67-TEX-I-1D445","M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z",[25022,31614,31615],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,31616,31617,31622,31628,31634],{"dataMmlNode":25028},[25022,31618,31619],{"dataMmlNode":25942},[25033,31620],{"dataC":27570,"xLinkHref":31621},"#MJX-67-TEX-I-1D449",[25022,31623,31625],{"dataMmlNode":25031,"transform":31624},"translate(1046.8,0)",[25033,31626],{"dataC":25078,"xLinkHref":31627},"#MJX-67-TEX-N-3D",[25022,31629,31631],{"dataMmlNode":25942,"transform":31630},"translate(2102.6,0)",[25033,31632],{"dataC":29851,"xLinkHref":31633},"#MJX-67-TEX-I-1D43C",[25022,31635,31637],{"dataMmlNode":25942,"transform":31636},"translate(2606.6,0)",[25033,31638],{"dataC":31639,"xLinkHref":31640},"1D445","#MJX-67-TEX-I-1D445",") and fit a 9th-degree polynomial, you'll pass through every point perfectly. But that wiggly curve is ",[7289,31643,31644],{},"way"," worse at predicting new measurements than a simple straight line that misses a few points.",[12,31647,31648],{},"The same thing happens with decision trees — noise in the data can cause the algorithm to create extra branches that capture randomness rather than real patterns.",[7166,31650,31652],{"id":31651},"how-noise-causes-overfitting","How Noise Causes Overfitting",[12,31654,31655],{},"Imagine our color\u002Fshape example, and one training instance gets mislabeled. The tree might need to add extra splits on irrelevant features (like size) just to account for that one noisy data point. The result is a bigger, more complex tree that generalizes poorly.",[6964,31657,31659],{"id":31658},"pruning-the-cure-for-overfitting","Pruning: The Cure for Overfitting",[12,31661,31662],{},"There are two main strategies:",[12,31664,31665,31668],{},[106,31666,31667],{},"Prepruning"," — Stop growing the tree early, before it perfectly fits the training data. For example, stop when a node has too few examples to split reliably.",[12,31670,31671,31674],{},[106,31672,31673],{},"Postpruning"," — Grow the full tree first, then cut back branches that don't help generalization. This is generally more effective since it's hard to know when to stop growing.",[7166,31676,31678],{"id":31677},"reduced-error-pruning","Reduced Error Pruning",[12,31680,31681],{},"A common postpruning method:",[3603,31683,31684,31687,31690,31693,31696],{},[26,31685,31686],{},"Split your data into a \"grow\" set and a \"validation\" set",[26,31688,31689],{},"Build a complete tree from the grow set",[26,31691,31692],{},"For each internal node, try replacing its subtree with a leaf (majority class)",[26,31694,31695],{},"If the replacement improves accuracy on the validation set, make it permanent",[26,31697,31698,31699,31701],{},"Repeat until no more pruning helps\nThe downside is that you \"waste\" some training data on validation. An alternative is to run multiple trials with different random splits, average the resulting tree complexity, then grow a final tree from ",[7289,31700,23215],{}," the data up to that average complexity.",[7166,31703,31705],{"id":31704},"other-pruning-approaches","Other Pruning Approaches",[23,31707,31708,31714],{},[26,31709,31710,31713],{},[106,31711,31712],{},"Statistical tests"," — Check whether the pattern at a node is statistically significant or just random chance",[26,31715,31716,31719],{},[106,31717,31718],{},"Minimum Description Length (MDL)"," — Ask whether the extra tree complexity is worth it compared to just memorizing the exceptions",[6964,31721,31723],{"id":31722},"computational-complexity","Computational Complexity",[12,31725,31726,31727,31797,31798,31819,31820,31841],{},"In the worst case (a complete tree testing every feature on every path), building a decision tree takes ",[24923,31728,31730],{"className":31729,"jax":24927},[24926],[24929,31731,31735,31756],{"style":24931,"xmlns":24932,"width":31732,"height":31733,"role":1930,"focusable":8058,"viewBox":31734,"xmlnsXLink":24936},"7.818ex","2.452ex","0 -833.9 3455.6 1083.9",[24938,31736,31737,31741,31744,31747,31750,31753],{},[24941,31738],{"id":31739,"d":31740},"MJX-68-TEX-I-1D442","M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435ZM637 476Q637 565 591 615T476 665Q396 665 322 605Q242 542 200 428T157 216Q157 126 200 73T314 19Q404 19 485 98T608 313Q637 408 637 476Z",[24941,31742],{"id":31743,"d":24944},"MJX-68-TEX-N-28",[24941,31745],{"id":31746,"d":25873},"MJX-68-TEX-I-1D45B",[24941,31748],{"id":31749,"d":28310},"MJX-68-TEX-I-1D45A",[24941,31751],{"id":31752,"d":25928},"MJX-68-TEX-N-32",[24941,31754],{"id":31755,"d":24988},"MJX-68-TEX-N-29",[25022,31757,31758],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,31759,31760,31766,31772,31777,31791],{"dataMmlNode":25028},[25022,31761,31762],{"dataMmlNode":25942},[25033,31763],{"dataC":31764,"xLinkHref":31765},"1D442","#MJX-68-TEX-I-1D442",[25022,31767,31769],{"dataMmlNode":25031,"transform":31768},"translate(763,0)",[25033,31770],{"dataC":25035,"xLinkHref":31771},"#MJX-68-TEX-N-28",[25022,31773,31774],{"dataMmlNode":25942,"transform":27567},[25033,31775],{"dataC":25952,"xLinkHref":31776},"#MJX-68-TEX-I-1D45B",[25022,31778,31780,31785],{"dataMmlNode":31565,"transform":31779},"translate(1752,0)",[25022,31781,31782],{"dataMmlNode":25942},[25033,31783],{"dataC":28479,"xLinkHref":31784},"#MJX-68-TEX-I-1D45A",[25022,31786,31788],{"dataMmlNode":26031,"transform":31787},"translate(911,363) scale(0.707)",[25033,31789],{"dataC":26058,"xLinkHref":31790},"#MJX-68-TEX-N-32",[25022,31792,31794],{"dataMmlNode":25031,"transform":31793},"translate(3066.6,0)",[25033,31795],{"dataC":25108,"xLinkHref":31796},"#MJX-68-TEX-N-29"," time, where ",[24923,31799,31801],{"className":31800,"jax":24927},[24926],[24929,31802,31805,31810],{"style":25196,"xmlns":24932,"width":31803,"height":27038,"role":1930,"focusable":8058,"viewBox":31804,"xmlnsXLink":24936},"1.357ex","0 -442 600 453",[24938,31806,31807],{},[24941,31808],{"id":31809,"d":25873},"MJX-69-TEX-I-1D45B",[25022,31811,31812],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,31813,31814],{"dataMmlNode":25028},[25022,31815,31816],{"dataMmlNode":25942},[25033,31817],{"dataC":25952,"xLinkHref":31818},"#MJX-69-TEX-I-1D45B"," is the number of examples and ",[24923,31821,31823],{"className":31822,"jax":24927},[24926],[24929,31824,31827,31832],{"style":25196,"xmlns":24932,"width":31825,"height":27038,"role":1930,"focusable":8058,"viewBox":31826,"xmlnsXLink":24936},"1.986ex","0 -442 878 453",[24938,31828,31829],{},[24941,31830],{"id":31831,"d":28310},"MJX-70-TEX-I-1D45A",[25022,31833,31834],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,31835,31836],{"dataMmlNode":25028},[25022,31837,31838],{"dataMmlNode":25942},[25033,31839],{"dataC":28479,"xLinkHref":31840},"#MJX-70-TEX-I-1D45A"," is the number of features.",[12,31843,31844,31845,7042,31864,31883],{},"In practice, the tree is rarely complete, and the complexity is roughly linear in both ",[24923,31846,31848],{"className":31847,"jax":24927},[24926],[24929,31849,31850,31855],{"style":25196,"xmlns":24932,"width":31803,"height":27038,"role":1930,"focusable":8058,"viewBox":31804,"xmlnsXLink":24936},[24938,31851,31852],{},[24941,31853],{"id":31854,"d":25873},"MJX-71-TEX-I-1D45B",[25022,31856,31857],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,31858,31859],{"dataMmlNode":25028},[25022,31860,31861],{"dataMmlNode":25942},[25033,31862],{"dataC":25952,"xLinkHref":31863},"#MJX-71-TEX-I-1D45B",[24923,31865,31867],{"className":31866,"jax":24927},[24926],[24929,31868,31869,31874],{"style":25196,"xmlns":24932,"width":31825,"height":27038,"role":1930,"focusable":8058,"viewBox":31826,"xmlnsXLink":24936},[24938,31870,31871],{},[24941,31872],{"id":31873,"d":28310},"MJX-72-TEX-I-1D45A",[25022,31875,31876],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,31877,31878],{"dataMmlNode":25028},[25022,31879,31880],{"dataMmlNode":25942},[25033,31881],{"dataC":28479,"xLinkHref":31882},"#MJX-72-TEX-I-1D45A"," — which is why decision trees scale well to large datasets.",[6964,31885,31887],{"id":31886},"strengths-and-limitations","Strengths and Limitations",[7166,31889,31891],{"id":31890},"strengths","Strengths",[23,31893,31894,31897,31900,31903,31906],{},[26,31895,31896],{},"Produces interpretable, human-readable rules",[26,31898,31899],{},"Fast to build and fast to classify new instances",[26,31901,31902],{},"Handles both categorical and numerical features",[26,31904,31905],{},"Robust to noise and missing data",[26,31907,31908],{},"No need for feature scaling or normalization",[7166,31910,31912],{"id":31911},"limitations","Limitations",[23,31914,31915,31918,31921,31924,31927],{},[26,31916,31917],{},"Prone to overfitting, especially with small datasets",[26,31919,31920],{},"Tests only one feature at a time (axis-aligned splits) — can't easily capture diagonal decision boundaries",[26,31922,31923],{},"Information gain has a bias toward features with many values",[26,31925,31926],{},"Greedy construction doesn't guarantee the globally optimal tree (finding the smallest consistent tree is NP-hard)",[26,31928,31929],{},"Produces a single deterministic hypothesis with no confidence estimates",[6964,31931,31933],{"id":31932},"what-comes-next","What Comes Next?",[12,31935,31936],{},"Decision trees are a foundation for more powerful methods:",[23,31938,31939,31945,31951],{},[26,31940,31941,31944],{},[106,31942,31943],{},"Random Forests"," — build many decision trees on random subsets of features and data, then vote",[26,31946,31947,31950],{},[106,31948,31949],{},"Gradient Boosted Trees"," (XGBoost, LightGBM) — build trees sequentially, each one correcting the errors of the previous",[26,31952,31953,31956],{},[106,31954,31955],{},"C4.5 \u002F C5.0"," — Quinlan's improvements to ID3 with better handling of continuous features, missing values, and pruning\nThese ensemble methods are among the most successful algorithms in practice, especially for structured\u002Ftabular data — and they all build on the decision tree concepts we've covered here.",[36,31958],{},[12,31960,31961],{},[7289,31962,31963],{},"These notes are based on Week 4 of WIA1006\u002FWID3006 Machine Learning at Universiti Malaya (Semester 2, 2025\u002F2026). If you found this helpful, stay tuned for more ML concepts explained from scratch!",[9332,31965,31966],{},"\nmjx-container[jax=\"SVG\"] {\n  direction: ltr;\n}\n\nmjx-container[jax=\"SVG\"] > svg {\n  overflow: visible;\n  min-height: 1px;\n  min-width: 1px;\n}\n\nmjx-container[jax=\"SVG\"] > svg a {\n  fill: blue;\n  stroke: blue;\n}\n\nmjx-container[jax=\"SVG\"][display=\"true\"] {\n  display: block;\n  text-align: center;\n  margin: 1em 0;\n}\n\nmjx-container[jax=\"SVG\"][display=\"true\"][width=\"full\"] {\n  display: flex;\n}\n\nmjx-container[jax=\"SVG\"][justify=\"left\"] {\n  text-align: left;\n}\n\nmjx-container[jax=\"SVG\"][justify=\"right\"] {\n  text-align: right;\n}\n\ng[data-mml-node=\"merror\"] > g {\n  fill: red;\n  stroke: red;\n}\n\ng[data-mml-node=\"merror\"] > rect[data-background] {\n  fill: yellow;\n  stroke: none;\n}\n\ng[data-mml-node=\"mtable\"] > line[data-line], svg[data-table] > g > line[data-line] {\n  stroke-width: 70px;\n  fill: none;\n}\n\ng[data-mml-node=\"mtable\"] > rect[data-frame], svg[data-table] > g > rect[data-frame] {\n  stroke-width: 70px;\n  fill: none;\n}\n\ng[data-mml-node=\"mtable\"] > .mjx-dashed, svg[data-table] > g > .mjx-dashed {\n  stroke-dasharray: 140;\n}\n\ng[data-mml-node=\"mtable\"] > .mjx-dotted, svg[data-table] > g > .mjx-dotted {\n  stroke-linecap: round;\n  stroke-dasharray: 0,140;\n}\n\ng[data-mml-node=\"mtable\"] > g > svg {\n  overflow: visible;\n}\n\n[jax=\"SVG\"] mjx-tool {\n  display: inline-block;\n  position: relative;\n  width: 0;\n  height: 0;\n}\n\n[jax=\"SVG\"] mjx-tool > mjx-tip {\n  position: absolute;\n  top: 0;\n  left: 0;\n}\n\nmjx-tool > mjx-tip {\n  display: inline-block;\n  padding: .2em;\n  border: 1px solid #888;\n  font-size: 70%;\n  background-color: #F8F8F8;\n  color: black;\n  box-shadow: 2px 2px 5px #AAAAAA;\n}\n\ng[data-mml-node=\"maction\"][data-toggle] {\n  cursor: pointer;\n}\n\nmjx-status {\n  display: block;\n  position: fixed;\n  left: 1em;\n  bottom: 1em;\n  min-width: 25%;\n  padding: .2em .4em;\n  border: 1px solid #888;\n  font-size: 90%;\n  background-color: #F8F8F8;\n  color: black;\n}\n\nforeignObject[data-mjx-xml] {\n  font-family: initial;\n  line-height: normal;\n  overflow: visible;\n}\n\nmjx-container[jax=\"SVG\"] path[data-c], mjx-container[jax=\"SVG\"] use[data-c] {\n  stroke-width: 3;\n}\n",{"title":180,"searchDepth":181,"depth":181,"links":31968},[31969,31970,31971,31972,31977,31978,31979,31982,31986,31987,31991],{"id":24895,"depth":181,"text":24896},{"id":25570,"depth":181,"text":25571},{"id":25613,"depth":181,"text":25614},{"id":25841,"depth":181,"text":25842,"children":31973},[31974,31975,31976],{"id":25848,"depth":7430,"text":25849},{"id":27298,"depth":7430,"text":27028},{"id":13609,"depth":7430,"text":13610},{"id":31048,"depth":181,"text":31049},{"id":31096,"depth":181,"text":31097},{"id":31506,"depth":181,"text":31507,"children":31980},[31981],{"id":31651,"depth":7430,"text":31652},{"id":31658,"depth":181,"text":31659,"children":31983},[31984,31985],{"id":31677,"depth":7430,"text":31678},{"id":31704,"depth":7430,"text":31705},{"id":31722,"depth":181,"text":31723},{"id":31886,"depth":181,"text":31887,"children":31988},[31989,31990],{"id":31890,"depth":7430,"text":31891},{"id":31911,"depth":7430,"text":31912},{"id":31932,"depth":181,"text":31933},"2026-04-28","2026-04-30",{},"\u002Fblog\u002FDecision-Trees",{"title":24890,"description":180},{"loc":31995},"blog\u002FDecision-Trees","scI7o6eDpOq-nKdwjALM6YkGpkrtOMrm7xqoLV0lAS4",{"id":32001,"title":32002,"author":6946,"body":32003,"date":42199,"description":180,"draft":185,"edited_at":31992,"extension":186,"featured_image":194,"meta":42200,"navigation":189,"path":42201,"pinned":185,"seo":42202,"sitemap":42203,"stem":42204,"tags":194,"__hash__":42205},"blog\u002Fblog\u002FData-Preprocessing-&-Regression-Analysis.md","Data Preprocessing & Regression Analysis",{"type":9,"value":32004,"toc":42179},[32005,32009,32016,32019,32023,32029,32047,32053,32059,32082,32088,32110,32116,32425,32439,32445,32449,32452,32472,32476,32479,32483,32489,32500,32519,32523,32526,32808,32855,32859,32924,33065,33118,33122,33129,33329,33424,33446,33450,33453,33589,33740,33792,33822,33825,34017,34028,37100,37104,37113,37119,37149,37155,37159,37162,37215,37221,37228,37232,37239,37242,37451,37482,38667,38671,38673,38680,38690,38835,38914,39703,39707,39714,39993,39996,40319,40330,40519,40522,42099,42102,42177],[6964,32006,32008],{"id":32007},"data-preprocessing","Data Preprocessing",[12,32010,32011,32012,32015],{},"Before we can train any Machine Learning model, we need to deal with the messy reality of raw data. ",[106,32013,32014],{},"Data preprocessing"," is the process of transforming raw data into a clean, usable format before feeding it into a model. Think of it like washing and chopping vegetables before cooking. You can't make a good dish with dirty, uncut ingredients.",[12,32017,32018],{},"Raw data typically suffers from problems like outliers, missing values, meaningless combinations, and inconsistent scales. Preprocessing turns that chaos into something a model can actually learn from.",[7166,32020,32022],{"id":32021},"the-7-steps-of-data-preprocessing","The 7 Steps of Data Preprocessing",[12,32024,32025,32028],{},[106,32026,32027],{},"Step 1 — Acquire the dataset."," Gather data from various sources and combine them into a proper format. The structure of your dataset will depend on your domain (business data looks very different from medical data).",[12,32030,32031,32034,32035,32038,32039,32042,32043,32046],{},[106,32032,32033],{},"Step 2 — Import crucial libraries."," In Python, three libraries form the backbone of preprocessing: ",[106,32036,32037],{},"NumPy"," (scientific computing), ",[106,32040,32041],{},"Pandas"," (data manipulation and analysis), and ",[106,32044,32045],{},"Matplotlib"," (2D plotting and visualization).",[12,32048,32049,32052],{},[106,32050,32051],{},"Step 3 — Import the dataset."," Load your data into your working environment. Make sure your working directory is set correctly so your code can find the files.",[12,32054,32055,32058],{},[106,32056,32057],{},"Step 4 — Handle missing values."," This is critical. If you ignore missing data, your model will draw faulty conclusions. Common strategies include filling missing values with the column mean\u002Fmedian, or dropping incomplete rows entirely.",[25631,32060,32062,32065],{"className":32061},[13441,13442],[25635,32063,32064],{},"What's imputation (filling in) and dropping (removing row)?",[13438,32066,32068,32079],{"className":32067},[25641],[12,32069,32070,32071,32074,32075,32078],{},"Imagine a spreadsheet of patient records where some rows have no entry for \"blood pressure.\" If you feed that gap straight into a model, it either crashes or silently learns something wrong. You have two main options: ",[7289,32072,32073],{},"imputation"," (filling in a reasonable substitute) or ",[7289,32076,32077],{},"deletion"," (removing the incomplete row).",[12,32080,32081],{},"For imputation, the most common approach is replacing the blank with the column's mean or median. Mean works well for normally distributed data, while median is safer when you have outliers — for instance, if most blood pressures are around 120 but one reading is 300, the mean gets pulled up, while the median stays stable. Deletion is simpler but risky: if 30% of your rows are missing that field, you're throwing away a lot of data.",[12,32083,32084,32087],{},[106,32085,32086],{},"Step 5 — Encode categorical data."," ML models run on math — they need numbers, not labels like \"Red\" or \"Male.\" Encoding methods include dummy variables, label encoding (assigning integers to categories), and one-hot encoding (creating binary columns for each category).",[25631,32089,32091,32094],{"className":32090},[13441,13442],[25635,32092,32093],{},"Label encoding vs one-hot encoding",[13438,32095,32097,32104],{"className":32096},[25641],[12,32098,32099,32100,32103],{},"Models do math, so a column like \"Color\" with values Red, Blue, Green is meaningless to them as text. ",[7289,32101,32102],{},"Label encoding"," assigns each category an integer (Red=0, Blue=1, Green=2), which is quick but introduces a problem: the model may interpret Blue as \"between\" Red and Green, or assume Green > Red, when there's no such ordering.",[12,32105,32106,32109],{},[7289,32107,32108],{},"One-hot encoding"," avoids this by creating a separate binary column for each category, so you'd get three columns (is_Red, is_Blue, is_Green), each containing 0 or 1. The trade-off is that one-hot encoding adds more columns, which can be expensive with high-cardinality features (imagine a \"City\" column with 500 unique values).",[12,32111,32112,32115],{},[106,32113,32114],{},"Step 6 — Feature scaling."," When your features have wildly different ranges (e.g., age 0–100 vs. salary 0–1,000,000), some algorithms get confused. Feature scaling standardizes variables to a common range. Two popular methods are Min-Max scaling and Standardisation (z-score).",[25631,32117,32119,32122],{"className":32118},[13441,13442],[25635,32120,32121],{},"Feature scaling",[13438,32123,32125,32128,32422],{"className":32124},[25641],[12,32126,32127],{},"Say you have two features: age (ranging 18–65) and income (ranging 20,000–200,000). A distance-based algorithm like k-nearest neighbours calculates how \"close\" two data points are. Without scaling, income dominates that distance calculation simply because its numbers are bigger — age differences become basically invisible. Scaling fixes this by putting both features on a comparable range.",[12,32129,32130,32133,32134,32266,32267,32270,32271,32421],{},[7289,32131,32132],{},"Min-Max scaling"," squeezes everything into 0–1 using the formula ",[24923,32135,32137],{"className":32136,"jax":24927},[24926],[24929,32138,32141,32169],{"style":24931,"xmlns":24932,"width":32139,"height":24934,"role":1930,"focusable":8058,"viewBox":32140,"xmlnsXLink":24936},"24.203ex","0 -750 10697.9 1000",[24938,32142,32143,32145,32149,32152,32155,32158,32161,32163,32166],{},[24941,32144],{"id":24943,"d":24944},[24941,32146],{"id":32147,"d":32148},"MJX-1-TEX-I-1D465","M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",[24941,32150],{"id":32151,"d":25910},"MJX-1-TEX-N-2212",[24941,32153],{"id":32154,"d":28310},"MJX-1-TEX-I-1D45A",[24941,32156],{"id":32157,"d":27108},"MJX-1-TEX-I-1D456",[24941,32159],{"id":32160,"d":25873},"MJX-1-TEX-I-1D45B",[24941,32162],{"id":24987,"d":24988},[24941,32164],{"id":32165,"d":29707},"MJX-1-TEX-N-2F",[24941,32167],{"id":32168,"d":27323},"MJX-1-TEX-I-1D44E",[25022,32170,32171],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,32172,32173,32177,32183,32189,32195,32201,32207,32212,32220,32225,32230,32236,32241,32246,32251,32256,32261],{"dataMmlNode":25028},[25022,32174,32175],{"dataMmlNode":25031},[25033,32176],{"dataC":25035,"xLinkHref":25036},[25022,32178,32179],{"dataMmlNode":25942,"transform":25040},[25033,32180],{"dataC":32181,"xLinkHref":32182},"1D465","#MJX-1-TEX-I-1D465",[25022,32184,32186],{"dataMmlNode":25031,"transform":32185},"translate(1183.2,0)",[25033,32187],{"dataC":26019,"xLinkHref":32188},"#MJX-1-TEX-N-2212",[25022,32190,32192],{"dataMmlNode":25942,"transform":32191},"translate(2183.4,0)",[25033,32193],{"dataC":28479,"xLinkHref":32194},"#MJX-1-TEX-I-1D45A",[25022,32196,32198],{"dataMmlNode":25942,"transform":32197},"translate(3061.4,0)",[25033,32199],{"dataC":27210,"xLinkHref":32200},"#MJX-1-TEX-I-1D456",[25022,32202,32204],{"dataMmlNode":25942,"transform":32203},"translate(3406.4,0)",[25033,32205],{"dataC":25952,"xLinkHref":32206},"#MJX-1-TEX-I-1D45B",[25022,32208,32210],{"dataMmlNode":25031,"transform":32209},"translate(4006.4,0)",[25033,32211],{"dataC":25108,"xLinkHref":25109},[25022,32213,32215],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":32214},"translate(4395.4,0)",[25022,32216,32217],{"dataMmlNode":25031},[25033,32218],{"dataC":29726,"xLinkHref":32219},"#MJX-1-TEX-N-2F",[25022,32221,32223],{"dataMmlNode":25031,"transform":32222},"translate(4895.4,0)",[25033,32224],{"dataC":25035,"xLinkHref":25036},[25022,32226,32228],{"dataMmlNode":25942,"transform":32227},"translate(5284.4,0)",[25033,32229],{"dataC":28479,"xLinkHref":32194},[25022,32231,32233],{"dataMmlNode":25942,"transform":32232},"translate(6162.4,0)",[25033,32234],{"dataC":27425,"xLinkHref":32235},"#MJX-1-TEX-I-1D44E",[25022,32237,32239],{"dataMmlNode":25942,"transform":32238},"translate(6691.4,0)",[25033,32240],{"dataC":32181,"xLinkHref":32182},[25022,32242,32244],{"dataMmlNode":25031,"transform":32243},"translate(7485.7,0)",[25033,32245],{"dataC":26019,"xLinkHref":32188},[25022,32247,32249],{"dataMmlNode":25942,"transform":32248},"translate(8485.9,0)",[25033,32250],{"dataC":28479,"xLinkHref":32194},[25022,32252,32254],{"dataMmlNode":25942,"transform":32253},"translate(9363.9,0)",[25033,32255],{"dataC":27210,"xLinkHref":32200},[25022,32257,32259],{"dataMmlNode":25942,"transform":32258},"translate(9708.9,0)",[25033,32260],{"dataC":25952,"xLinkHref":32206},[25022,32262,32264],{"dataMmlNode":25031,"transform":32263},"translate(10308.9,0)",[25033,32265],{"dataC":25108,"xLinkHref":25109},". So an age of 18 becomes 0, an age of 65 becomes 1, and everything else falls between. ",[7289,32268,32269],{},"Standardisation (z-score)"," instead centres the data around 0 with a standard deviation of 1, using ",[24923,32272,32274],{"className":32273,"jax":24927},[24926],[24929,32275,32278,32324],{"style":24931,"xmlns":24932,"width":32276,"height":24934,"role":1930,"focusable":8058,"viewBox":32277,"xmlnsXLink":24936},"20.06ex","0 -750 8866.4 1000",[24938,32279,32280,32283,32286,32289,32292,32295,32298,32301,32304,32307,32310,32313,32317,32321],{},[24941,32281],{"id":32282,"d":24944},"MJX-2-TEX-N-28",[24941,32284],{"id":32285,"d":32148},"MJX-2-TEX-I-1D465",[24941,32287],{"id":32288,"d":25910},"MJX-2-TEX-N-2212",[24941,32290],{"id":32291,"d":28310},"MJX-2-TEX-I-1D45A",[24941,32293],{"id":32294,"d":27397},"MJX-2-TEX-I-1D452",[24941,32296],{"id":32297,"d":27323},"MJX-2-TEX-I-1D44E",[24941,32299],{"id":32300,"d":25873},"MJX-2-TEX-I-1D45B",[24941,32302],{"id":32303,"d":24988},"MJX-2-TEX-N-29",[24941,32305],{"id":32306,"d":29707},"MJX-2-TEX-N-2F",[24941,32308],{"id":32309,"d":27401},"MJX-2-TEX-I-1D460",[24941,32311],{"id":32312,"d":25877},"MJX-2-TEX-I-1D461",[24941,32314],{"id":32315,"d":32316},"MJX-2-TEX-I-1D451","M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z",[24941,32318],{"id":32319,"d":32320},"MJX-2-TEX-N-5F","M0 -62V-25H499V-62H0Z",[24941,32322],{"id":32323,"d":27377},"MJX-2-TEX-I-1D463",[25022,32325,32326],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,32327,32328,32333,32338,32343,32348,32353,32359,32365,32371,32379,32385,32391,32398,32405,32410,32415],{"dataMmlNode":25028},[25022,32329,32330],{"dataMmlNode":25031},[25033,32331],{"dataC":25035,"xLinkHref":32332},"#MJX-2-TEX-N-28",[25022,32334,32335],{"dataMmlNode":25942,"transform":25040},[25033,32336],{"dataC":32181,"xLinkHref":32337},"#MJX-2-TEX-I-1D465",[25022,32339,32340],{"dataMmlNode":25031,"transform":32185},[25033,32341],{"dataC":26019,"xLinkHref":32342},"#MJX-2-TEX-N-2212",[25022,32344,32345],{"dataMmlNode":25942,"transform":32191},[25033,32346],{"dataC":28479,"xLinkHref":32347},"#MJX-2-TEX-I-1D45A",[25022,32349,32350],{"dataMmlNode":25942,"transform":32197},[25033,32351],{"dataC":27596,"xLinkHref":32352},"#MJX-2-TEX-I-1D452",[25022,32354,32356],{"dataMmlNode":25942,"transform":32355},"translate(3527.4,0)",[25033,32357],{"dataC":27425,"xLinkHref":32358},"#MJX-2-TEX-I-1D44E",[25022,32360,32362],{"dataMmlNode":25942,"transform":32361},"translate(4056.4,0)",[25033,32363],{"dataC":25952,"xLinkHref":32364},"#MJX-2-TEX-I-1D45B",[25022,32366,32368],{"dataMmlNode":25031,"transform":32367},"translate(4656.4,0)",[25033,32369],{"dataC":25108,"xLinkHref":32370},"#MJX-2-TEX-N-29",[25022,32372,32374],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":32373},"translate(5045.4,0)",[25022,32375,32376],{"dataMmlNode":25031},[25033,32377],{"dataC":29726,"xLinkHref":32378},"#MJX-2-TEX-N-2F",[25022,32380,32382],{"dataMmlNode":25942,"transform":32381},"translate(5545.4,0)",[25033,32383],{"dataC":27603,"xLinkHref":32384},"#MJX-2-TEX-I-1D460",[25022,32386,32388],{"dataMmlNode":25942,"transform":32387},"translate(6014.4,0)",[25033,32389],{"dataC":25959,"xLinkHref":32390},"#MJX-2-TEX-I-1D461",[25022,32392,32394],{"dataMmlNode":25942,"transform":32393},"translate(6375.4,0)",[25033,32395],{"dataC":32396,"xLinkHref":32397},"1D451","#MJX-2-TEX-I-1D451",[25022,32399,32401],{"dataMmlNode":25942,"transform":32400},"translate(6895.4,0)",[25033,32402],{"dataC":32403,"xLinkHref":32404},"5F","#MJX-2-TEX-N-5F",[25022,32406,32408],{"dataMmlNode":25942,"transform":32407},"translate(7395.4,0)",[25033,32409],{"dataC":32396,"xLinkHref":32397},[25022,32411,32413],{"dataMmlNode":25942,"transform":32412},"translate(7915.4,0)",[25033,32414],{"dataC":27596,"xLinkHref":32352},[25022,32416,32418],{"dataMmlNode":25942,"transform":32417},"translate(8381.4,0)",[25033,32419],{"dataC":27556,"xLinkHref":32420},"#MJX-2-TEX-I-1D463",". Min-Max is great when you know your data has fixed bounds and no extreme outliers. Standardisation is more robust to outliers and is generally preferred for algorithms like logistic regression or SVMs.",[12,32423,32424],{},"Not all algorithms need feature scaling. Tree-based models (like decision trees and random forests) split on thresholds and don't care about magnitude, so scaling doesn't affect them. It matters most for distance-based and gradient-based algorithms.",[12,32426,32427,32430,32431,32434,32435,32438],{},[106,32428,32429],{},"Step 7 — Split the dataset."," Divide your data into a ",[106,32432,32433],{},"training set"," (for learning) and a ",[106,32436,32437],{},"test set"," (for validation). Common splits follow the Pareto principle: 80\u002F20, 70\u002F30, or 60\u002F40.",[12,32440,32441],{},[1930,32442],{"alt":32443,"src":32444},"0.63","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1777275466503-Screenshot_2026-04-27_at_3.37.33_PM.png",[7166,32446,32448],{"id":32447},"other-preprocessing-techniques","Other Preprocessing Techniques",[12,32450,32451],{},"Beyond the core 7 steps, you may also encounter:",[23,32453,32454,32460,32466],{},[26,32455,32456,32459],{},[106,32457,32458],{},"Data Integration"," — combining data from multiple source systems into a unified set.",[26,32461,32462,32465],{},[106,32463,32464],{},"Data Transformation"," — converting data from one format to another (e.g., source system format → destination format).",[26,32467,32468,32471],{},[106,32469,32470],{},"Data Discretization"," — turning continuous values into discrete intervals (e.g., converting exact ages into age groups like \"18–25\", \"26–35\").",[6964,32473,32475],{"id":32474},"regression-analysis","Regression Analysis",[12,32477,32478],{},"Now that our data is clean, let's learn how to make predictions. We'll start with a classic example.",[7166,32480,32482],{"id":32481},"the-housing-price-problem","The Housing Price Problem",[12,32484,32485,32486],{},"Imagine we have a dataset of 47 houses in SS17, Petaling Jaya, with their living areas (in square feet) and prices (in RM thousands). The question is simple: ",[106,32487,32488],{},"given the size of a house, can we predict its price?",[12,32490,32491,32492,32495,32496,32499],{},"This is a ",[106,32493,32494],{},"supervised learning"," problem. We have input-output pairs and want to learn a mapping between them. Since the output (price) is a continuous value, this is specifically called a ",[106,32497,32498],{},"regression"," problem.",[13438,32501,32503,32506],{"className":32502},[13441,13442],[12,32504,32505],{},"Regression is a method for finding the relationship between variables so you can predict a continuous outcome from one or more inputs.",[12,32507,32508],{},[32509,32510,32511,32515],"figure",{},[1930,32512],{"alt":32513,"src":32514},"0.96","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1777279629723-1N1-K-A43_98pYZ27fnupDA-2048x1418.jpeg",[32516,32517,32518],"figcaption",{},"Linear Regression, Image by Dan White 1000 on Shutterstock",[7166,32520,32522],{"id":32521},"notation","Notation",[12,32524,32525],{},"Before diving in, let's set up our notation:",[23,32527,32528,32581,32637,32731,32755],{},[26,32529,32530,32580],{},[106,32531,32532],{},[24923,32533,32535],{"className":32534,"jax":24927},[24926],[24929,32536,32540,32552],{"style":25196,"xmlns":24932,"width":32537,"height":32538,"role":1930,"focusable":8058,"viewBox":32539,"xmlnsXLink":24936},"3.278ex","2.046ex","0 -893.3 1449.1 904.3",[24938,32541,32542,32545,32547,32550],{},[24941,32543],{"id":32544,"d":32148},"MJX-3-TEX-I-1D465",[24941,32546],{"id":25230,"d":24944},[24941,32548],{"id":32549,"d":27108},"MJX-3-TEX-I-1D456",[24941,32551],{"id":25276,"d":24988},[25022,32553,32554],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,32555,32556],{"dataMmlNode":25028},[25022,32557,32558,32563],{"dataMmlNode":31565},[25022,32559,32560],{"dataMmlNode":25942},[25033,32561],{"dataC":32181,"xLinkHref":32562},"#MJX-3-TEX-I-1D465",[25022,32564,32566,32570,32575],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},"translate(605,363) scale(0.707)",[25022,32567,32568],{"dataMmlNode":25031},[25033,32569],{"dataC":25035,"xLinkHref":25285},[25022,32571,32572],{"dataMmlNode":25942,"transform":25040},[25033,32573],{"dataC":27210,"xLinkHref":32574},"#MJX-3-TEX-I-1D456",[25022,32576,32578],{"dataMmlNode":25031,"transform":32577},"translate(734,0)",[25033,32579],{"dataC":25108,"xLinkHref":25351}," = input features (e.g., living area of the i-th house)",[26,32582,32583,32636],{},[106,32584,32585],{},[24923,32586,32588],{"className":32587,"jax":24927},[24926],[24929,32589,32593,32607],{"style":29805,"xmlns":24932,"width":32590,"height":32591,"role":1930,"focusable":8058,"viewBox":32592,"xmlnsXLink":24936},"3.093ex","2.485ex","0 -893.3 1367.1 1098.3",[24938,32594,32595,32598,32601,32604],{},[24941,32596],{"id":32597,"d":25893},"MJX-4-TEX-I-1D466",[24941,32599],{"id":32600,"d":24944},"MJX-4-TEX-N-28",[24941,32602],{"id":32603,"d":27108},"MJX-4-TEX-I-1D456",[24941,32605],{"id":32606,"d":24988},"MJX-4-TEX-N-29",[25022,32608,32609],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,32610,32611],{"dataMmlNode":25028},[25022,32612,32613,32618],{"dataMmlNode":31565},[25022,32614,32615],{"dataMmlNode":25942},[25033,32616],{"dataC":25987,"xLinkHref":32617},"#MJX-4-TEX-I-1D466",[25022,32619,32621,32626,32631],{"dataMmlNode":27203,"transform":32620,"dataMjxTexclass":27205},"translate(523,363) scale(0.707)",[25022,32622,32623],{"dataMmlNode":25031},[25033,32624],{"dataC":25035,"xLinkHref":32625},"#MJX-4-TEX-N-28",[25022,32627,32628],{"dataMmlNode":25942,"transform":25040},[25033,32629],{"dataC":27210,"xLinkHref":32630},"#MJX-4-TEX-I-1D456",[25022,32632,32633],{"dataMmlNode":25031,"transform":32577},[25033,32634],{"dataC":25108,"xLinkHref":32635},"#MJX-4-TEX-N-29"," = output\u002Ftarget variable (e.g., price of the i-th house)",[26,32638,32639,32730],{},[106,32640,32641],{},[24923,32642,32644],{"className":32643,"jax":24927},[24926],[24929,32645,32649,32667],{"style":24931,"xmlns":24932,"width":32646,"height":32647,"role":1930,"focusable":8058,"viewBox":32648,"xmlnsXLink":24936},"9.138ex","2.587ex","0 -893.3 4038.8 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= a single training example",[26,32732,32733,32754],{},[106,32734,32735],{},[24923,32736,32738],{"className":32737,"jax":24927},[24926],[24929,32739,32740,32745],{"style":25196,"xmlns":24932,"width":31825,"height":27038,"role":1930,"focusable":8058,"viewBox":31826,"xmlnsXLink":24936},[24938,32741,32742],{},[24941,32743],{"id":32744,"d":28310},"MJX-6-TEX-I-1D45A",[25022,32746,32747],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,32748,32749],{"dataMmlNode":25028},[25022,32750,32751],{"dataMmlNode":25942},[25033,32752],{"dataC":28479,"xLinkHref":32753},"#MJX-6-TEX-I-1D45A"," = total number of training examples",[26,32756,32757,32807],{},[106,32758,32759],{},[24923,32760,32762],{"className":32761,"jax":24927},[24926],[24929,32763,32766,32780],{"style":24931,"xmlns":24932,"width":32764,"height":24934,"role":1930,"focusable":8058,"viewBox":32765,"xmlnsXLink":24936},"4.357ex","0 -750 1926 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predictor we want to learn",[12,32809,32810,32811,32854],{},"The goal: learn a function h such that 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scale(0.707)",[25022,33306,33307],{"dataMmlNode":25031},[25033,33308],{"dataC":25035,"xLinkHref":33194},[25022,33310,33311],{"dataMmlNode":25942,"transform":25040},[25033,33312],{"dataC":27210,"xLinkHref":33243},[25022,33314,33315],{"dataMmlNode":25031,"transform":32577},[25033,33316],{"dataC":25108,"xLinkHref":33206},[25022,33318,33320,33324],{"dataMmlNode":31565,"transform":33319},"translate(11545.8,0)",[25022,33321,33322],{"dataMmlNode":25031},[25033,33323],{"dataC":25108,"xLinkHref":33206},[25022,33325,33327],{"dataMmlNode":26031,"transform":33326},"translate(422,413) scale(0.707)",[25033,33328],{"dataC":26058,"xLinkHref":33226},[12,33330,33331,33332,33377,33378,7206,33381,6959],{},"The smaller ",[24923,33333,33335],{"className":33334,"jax":24927},[24926],[24929,33336,33339,33353],{"style":24931,"xmlns":24932,"width":33337,"height":24934,"role":1930,"focusable":8058,"viewBox":33338,"xmlnsXLink":24936},"4.253ex","0 -750 1880 1000",[24938,33340,33341,33344,33347,33350],{},[24941,33342],{"id":33343,"d":33143},"MJX-15-TEX-I-1D43D",[24941,33345],{"id":33346,"d":24944},"MJX-15-TEX-N-28",[24941,33348],{"id":33349,"d":32951},"MJX-15-TEX-I-1D703",[24941,33351],{"id":33352,"d":24988},"MJX-15-TEX-N-29",[25022,33354,33355],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,33356,33357,33362,33367,33372],{"dataMmlNode":25028},[25022,33358,33359],{"dataMmlNode":25942},[25033,33360],{"dataC":33187,"xLinkHref":33361},"#MJX-15-TEX-I-1D43D",[25022,33363,33364],{"dataMmlNode":25031,"transform":33191},[25033,33365],{"dataC":25035,"xLinkHref":33366},"#MJX-15-TEX-N-28",[25022,33368,33369],{"dataMmlNode":25942,"transform":33197},[25033,33370],{"dataC":33000,"xLinkHref":33371},"#MJX-15-TEX-I-1D703",[25022,33373,33374],{"dataMmlNode":25031,"transform":33203},[25033,33375],{"dataC":25108,"xLinkHref":33376},"#MJX-15-TEX-N-29"," is, the better our model fits the data. So we want to ",[106,33379,33380],{},"minimise",[24923,33382,33384],{"className":33383,"jax":24927},[24926],[24929,33385,33386,33400],{"style":24931,"xmlns":24932,"width":33337,"height":24934,"role":1930,"focusable":8058,"viewBox":33338,"xmlnsXLink":24936},[24938,33387,33388,33391,33394,33397],{},[24941,33389],{"id":33390,"d":33143},"MJX-16-TEX-I-1D43D",[24941,33392],{"id":33393,"d":24944},"MJX-16-TEX-N-28",[24941,33395],{"id":33396,"d":32951},"MJX-16-TEX-I-1D703",[24941,33398],{"id":33399,"d":24988},"MJX-16-TEX-N-29",[25022,33401,33402],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,33403,33404,33409,33414,33419],{"dataMmlNode":25028},[25022,33405,33406],{"dataMmlNode":25942},[25033,33407],{"dataC":33187,"xLinkHref":33408},"#MJX-16-TEX-I-1D43D",[25022,33410,33411],{"dataMmlNode":25031,"transform":33191},[25033,33412],{"dataC":25035,"xLinkHref":33413},"#MJX-16-TEX-N-28",[25022,33415,33416],{"dataMmlNode":25942,"transform":33197},[25033,33417],{"dataC":33000,"xLinkHref":33418},"#MJX-16-TEX-I-1D703",[25022,33420,33421],{"dataMmlNode":25031,"transform":33203},[25033,33422],{"dataC":25108,"xLinkHref":33423},"#MJX-16-TEX-N-29",[25631,33425,33427,33430],{"className":33426},[13441,13442],[25635,33428,33429],{},"Recap Cost Function",[13438,33431,33433],{"className":33432},[25641],[12,33434,33435,33436,33441,33442,33445],{},"You can learn more about cost function ",[2330,33437,33440],{"href":33438,"rel":33439},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002FIntroduction-to-Machine-Learning#the-lms-least-mean-squares-algorithm",[2334],"here",". The core idea remains pretty much the same, only difference is that this formula writes the cost as the sum of all errors, which corresponds to ",[106,33443,33444],{},"batch gradient descent"," where you accumulate the total error across every example before making one update. Batch is more stable per step, but each step is expensive because you have to loop through the entire dataset.",[7166,33447,33449],{"id":33448},"gradient-descent-finding-the-best-parameters","Gradient Descent — Finding the Best Parameters",[12,33451,33452],{},"Gradient descent is an iterative optimization algorithm. Here's the intuition:",[3603,33454,33455,33477,33519,33586],{},[26,33456,33457,33458,6959],{},"Start with a random guess for ",[24923,33459,33461],{"className":33460,"jax":24927},[24926],[24929,33462,33463,33468],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,33464,33465],{},[24941,33466],{"id":33467,"d":32951},"MJX-17-TEX-I-1D703",[25022,33469,33470],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,33471,33472],{"dataMmlNode":25028},[25022,33473,33474],{"dataMmlNode":25942},[25033,33475],{"dataC":33000,"xLinkHref":33476},"#MJX-17-TEX-I-1D703",[26,33478,33479,33480,6959],{},"Compute the cost ",[24923,33481,33483],{"className":33482,"jax":24927},[24926],[24929,33484,33485,33497],{"style":24931,"xmlns":24932,"width":33337,"height":24934,"role":1930,"focusable":8058,"viewBox":33338,"xmlnsXLink":24936},[24938,33486,33487,33490,33492,33495],{},[24941,33488],{"id":33489,"d":33143},"MJX-18-TEX-I-1D43D",[24941,33491],{"id":26650,"d":24944},[24941,33493],{"id":33494,"d":32951},"MJX-18-TEX-I-1D703",[24941,33496],{"id":26663,"d":24988},[25022,33498,33499],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,33500,33501,33506,33510,33515],{"dataMmlNode":25028},[25022,33502,33503],{"dataMmlNode":25942},[25033,33504],{"dataC":33187,"xLinkHref":33505},"#MJX-18-TEX-I-1D43D",[25022,33507,33508],{"dataMmlNode":25031,"transform":33191},[25033,33509],{"dataC":25035,"xLinkHref":26705},[25022,33511,33512],{"dataMmlNode":25942,"transform":33197},[25033,33513],{"dataC":33000,"xLinkHref":33514},"#MJX-18-TEX-I-1D703",[25022,33516,33517],{"dataMmlNode":25031,"transform":33203},[25033,33518],{"dataC":25108,"xLinkHref":26724},[26,33520,33521,33522,33541,33542,33585],{},"Adjust ",[24923,33523,33525],{"className":33524,"jax":24927},[24926],[24929,33526,33527,33532],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,33528,33529],{},[24941,33530],{"id":33531,"d":32951},"MJX-19-TEX-I-1D703",[25022,33533,33534],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,33535,33536],{"dataMmlNode":25028},[25022,33537,33538],{"dataMmlNode":25942},[25033,33539],{"dataC":33000,"xLinkHref":33540},"#MJX-19-TEX-I-1D703"," in the direction that reduces ",[24923,33543,33545],{"className":33544,"jax":24927},[24926],[24929,33546,33547,33561],{"style":24931,"xmlns":24932,"width":33337,"height":24934,"role":1930,"focusable":8058,"viewBox":33338,"xmlnsXLink":24936},[24938,33548,33549,33552,33555,33558],{},[24941,33550],{"id":33551,"d":33143},"MJX-20-TEX-I-1D43D",[24941,33553],{"id":33554,"d":24944},"MJX-20-TEX-N-28",[24941,33556],{"id":33557,"d":32951},"MJX-20-TEX-I-1D703",[24941,33559],{"id":33560,"d":24988},"MJX-20-TEX-N-29",[25022,33562,33563],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,33564,33565,33570,33575,33580],{"dataMmlNode":25028},[25022,33566,33567],{"dataMmlNode":25942},[25033,33568],{"dataC":33187,"xLinkHref":33569},"#MJX-20-TEX-I-1D43D",[25022,33571,33572],{"dataMmlNode":25031,"transform":33191},[25033,33573],{"dataC":25035,"xLinkHref":33574},"#MJX-20-TEX-N-28",[25022,33576,33577],{"dataMmlNode":25942,"transform":33197},[25033,33578],{"dataC":33000,"xLinkHref":33579},"#MJX-20-TEX-I-1D703",[25022,33581,33582],{"dataMmlNode":25031,"transform":33203},[25033,33583],{"dataC":25108,"xLinkHref":33584},"#MJX-20-TEX-N-29"," the most (the direction of steepest descent).",[26,33587,33588],{},"Repeat until convergence.\nThe update rule is:",[24923,33590,33592],{"className":33591,"jax":24927,"display":13594},[24926],[24929,33593,33598,33632],{"style":33594,"xmlns":24932,"width":33595,"height":33596,"role":1930,"focusable":8058,"viewBox":33597,"xmlnsXLink":24936},"vertical-align: -2.218ex;","18.205ex","5.521ex","0 -1460 8046.7 2440.2",[24938,33599,33600,33603,33607,33611,33614,33617,33621,33625,33628,33630],{},[24941,33601],{"id":33602,"d":32951},"MJX-21-TEX-I-1D703",[24941,33604],{"id":33605,"d":33606},"MJX-21-TEX-I-1D457","M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 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306Z",[24941,33626],{"id":33627,"d":33143},"MJX-21-TEX-I-1D43D",[24941,33629],{"id":26925,"d":24944},[24941,33631],{"id":26928,"d":24988},[25022,33633,33634],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,33635,33636,33649,33663,33674,33680,33687],{"dataMmlNode":25028},[25022,33637,33638,33643],{"dataMmlNode":26023},[25022,33639,33640],{"dataMmlNode":25942},[25033,33641],{"dataC":33000,"xLinkHref":33642},"#MJX-21-TEX-I-1D703",[25022,33644,33645],{"dataMmlNode":25942,"transform":33004},[25033,33646],{"dataC":33647,"xLinkHref":33648},"1D457","#MJX-21-TEX-I-1D457",[25022,33650,33652,33658],{"dataMmlNode":25031,"transform":33651},"translate(1121.1,0)",[25022,33653,33654],{"dataMmlNode":6977},[25033,33655],{"dataC":33656,"xLinkHref":33657},"3A","#MJX-21-TEX-N-3A",[25022,33659,33660],{"dataMmlNode":6977,"transform":26048},[25033,33661],{"dataC":25078,"xLinkHref":33662},"#MJX-21-TEX-N-3D",[25022,33664,33666,33670],{"dataMmlNode":26023,"transform":33665},"translate(2454.9,0)",[25022,33667,33668],{"dataMmlNode":25942},[25033,33669],{"dataC":33000,"xLinkHref":33642},[25022,33671,33672],{"dataMmlNode":25942,"transform":33004},[25033,33673],{"dataC":33647,"xLinkHref":33648},[25022,33675,33677],{"dataMmlNode":25031,"transform":33676},"translate(3520.4,0)",[25033,33678],{"dataC":26019,"xLinkHref":33679},"#MJX-21-TEX-N-2212",[25022,33681,33683],{"dataMmlNode":25942,"transform":33682},"translate(4520.7,0)",[25033,33684],{"dataC":33685,"xLinkHref":33686},"1D6FC","#MJX-21-TEX-I-1D6FC",[25022,33688,33690,33720,33737],{"dataMmlNode":27622,"transform":33689},"translate(5160.7,0)",[25022,33691,33693,33699,33705,33710,33715],{"dataMmlNode":27626,"transform":33692},"translate(220,710)",[25022,33694,33695],{"dataMmlNode":25942},[25033,33696],{"dataC":33697,"xLinkHref":33698},"1D715","#MJX-21-TEX-I-1D715",[25022,33700,33702],{"dataMmlNode":25942,"transform":33701},"translate(566,0)",[25033,33703],{"dataC":33187,"xLinkHref":33704},"#MJX-21-TEX-I-1D43D",[25022,33706,33708],{"dataMmlNode":25031,"transform":33707},"translate(1199,0)",[25033,33709],{"dataC":25035,"xLinkHref":26981},[25022,33711,33713],{"dataMmlNode":25942,"transform":33712},"translate(1588,0)",[25033,33714],{"dataC":33000,"xLinkHref":33642},[25022,33716,33718],{"dataMmlNode":25031,"transform":33717},"translate(2057,0)",[25033,33719],{"dataC":25108,"xLinkHref":26998},[25022,33721,33723,33727],{"dataMmlNode":27626,"transform":33722},"translate(738.3,-686)",[25022,33724,33725],{"dataMmlNode":25942},[25033,33726],{"dataC":33697,"xLinkHref":33698},[25022,33728,33729,33733],{"dataMmlNode":26023,"transform":33701},[25022,33730,33731],{"dataMmlNode":25942},[25033,33732],{"dataC":33000,"xLinkHref":33642},[25022,33734,33735],{"dataMmlNode":25942,"transform":33004},[25033,33736],{"dataC":33647,"xLinkHref":33648},[27668,33738],{"width":33739,"height":27671,"x":27672,"y":27673},2646,[25631,33741,33743,33746],{"className":33742},[13441,13442],[25635,33744,33745],{},"Understanding the formula",[13438,33747,33749,33783],{"className":33748},[25641],[12,33750,33751,33782],{},[24923,33752,33754],{"className":33753,"jax":24927},[24926],[24929,33755,33758,33766],{"style":26868,"xmlns":24932,"width":33756,"height":26870,"role":1930,"focusable":8058,"viewBox":33757,"xmlnsXLink":24936},"2.389ex","0 -583 1056 665",[24938,33759,33760,33763],{},[24941,33761],{"id":33762,"d":33610},"MJX-22-TEX-N-3A",[24941,33764],{"id":33765,"d":24972},"MJX-22-TEX-N-3D",[25022,33767,33768],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,33769,33770],{"dataMmlNode":25028},[25022,33771,33772,33777],{"dataMmlNode":25031},[25022,33773,33774],{"dataMmlNode":6977},[25033,33775],{"dataC":33656,"xLinkHref":33776},"#MJX-22-TEX-N-3A",[25022,33778,33779],{"dataMmlNode":6977,"transform":26048},[25033,33780],{"dataC":25078,"xLinkHref":33781},"#MJX-22-TEX-N-3D"," means \"assign.\" It computes the expression on the right using the current values, then updates the left side with the result.",[12,33784,33785,33786,33791],{},"This is gradient descent in its simplest form, adjust the weight by stepping in the opposite direction of the slope, using a small learning rate so we don't overshoot. If you want to learn more, check out ",[2330,33787,33790],{"href":33788,"rel":33789},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002FArtificial-Neural-Networks-and-Backpropagation#updating-output-layer-weights",[2334],"this post",", but just note that it's the same update rule, just applied in a deeper context. Linear regression uses the simplest form of this update, back-propagation extends it to networks with multiple layers.",[12,33793,33794,33795,18365,33818,33821],{},"Here, ",[106,33796,33797],{},[24923,33798,33800],{"className":33799,"jax":24927},[24926],[24929,33801,33804,33809],{"style":25196,"xmlns":24932,"width":33802,"height":27038,"role":1930,"focusable":8058,"viewBox":33803,"xmlnsXLink":24936},"1.448ex","0 -442 640 453",[24938,33805,33806],{},[24941,33807],{"id":33808,"d":33620},"MJX-23-TEX-I-1D6FC",[25022,33810,33811],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,33812,33813],{"dataMmlNode":25028},[25022,33814,33815],{"dataMmlNode":25942},[25033,33816],{"dataC":33685,"xLinkHref":33817},"#MJX-23-TEX-I-1D6FC",[106,33819,33820],{},"learning rate"," — it controls how big each step is. Too large and you overshoot; too small and convergence is painfully slow.",[12,33823,33824],{},"Working out the partial derivative, the update becomes:",[24923,33826,33828],{"className":33827,"jax":24927,"display":13594},[24926],[24929,33829,33834,33869],{"style":33830,"xmlns":24932,"width":33831,"height":33832,"role":1930,"focusable":8058,"viewBox":33833,"xmlnsXLink":24936},"vertical-align: -0.991ex;","28.915ex","3.391ex","0 -1060.7 12780.3 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2583.6",[24938,35854,35855,35858,35861,35864,35867,35870,35873,35876,35879],{},[24941,35856],{"id":35857,"d":33624},"MJX-42-TEX-I-1D715",[24941,35859],{"id":35860,"d":31530},"MJX-42-TEX-I-210E",[24941,35862],{"id":35863,"d":24944},"MJX-42-TEX-N-28",[24941,35865],{"id":35866,"d":32148},"MJX-42-TEX-I-1D465",[24941,35868],{"id":35869,"d":27108},"MJX-42-TEX-I-1D456",[24941,35871],{"id":35872,"d":24988},"MJX-42-TEX-N-29",[24941,35874],{"id":35875,"d":32951},"MJX-42-TEX-I-1D703",[24941,35877],{"id":35878,"d":33606},"MJX-42-TEX-I-1D457",[24941,35880],{"id":35881,"d":24972},"MJX-42-TEX-N-3D",[25022,35883,35884],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,35885,35886,35952,35957],{"dataMmlNode":25028},[25022,35887,35888,35932,35950],{"dataMmlNode":27622},[25022,35889,35890,35895,35900,35905,35928],{"dataMmlNode":27626,"transform":33692},[25022,35891,35892],{"dataMmlNode":25942},[25033,35893],{"dataC":33697,"xLinkHref":35894},"#MJX-42-TEX-I-1D715",[25022,35896,35897],{"dataMmlNode":25942,"transform":33701},[25033,35898],{"dataC":31539,"xLinkHref":35899},"#MJX-42-TEX-I-210E",[25022,35901,35902],{"dataMmlNode":25031,"transform":34835},[25033,35903],{"dataC":25035,"xLinkHref":35904},"#MJX-42-TEX-N-28",[25022,35906,35907,35912],{"dataMmlNode":31565,"transform":34840},[25022,35908,35909],{"dataMmlNode":25942},[25033,35910],{"dataC":32181,"xLinkHref":35911},"#MJX-42-TEX-I-1D465",[25022,35913,35914,35918,35923],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},[25022,35915,35916],{"dataMmlNode":25031},[25033,35917],{"dataC":25035,"xLinkHref":35904},[25022,35919,35920],{"dataMmlNode":25942,"transform":25040},[25033,35921],{"dataC":27210,"xLinkHref":35922},"#MJX-42-TEX-I-1D456",[25022,35924,35925],{"dataMmlNode":25031,"transform":32577},[25033,35926],{"dataC":25108,"xLinkHref":35927},"#MJX-42-TEX-N-29",[25022,35929,35930],{"dataMmlNode":25031,"transform":34863},[25033,35931],{"dataC":25108,"xLinkHref":35927},[25022,35933,35934,35938],{"dataMmlNode":27626,"transform":34918},[25022,35935,35936],{"dataMmlNode":25942},[25033,35937],{"dataC":33697,"xLinkHref":35894},[25022,35939,35940,35945],{"dataMmlNode":26023,"transform":33701},[25022,35941,35942],{"dataMmlNode":25942},[25033,35943],{"dataC":33000,"xLinkHref":35944},"#MJX-42-TEX-I-1D703",[25022,35946,35947],{"dataMmlNode":25942,"transform":33004},[25033,35948],{"dataC":33647,"xLinkHref":35949},"#MJX-42-TEX-I-1D457",[27668,35951],{"width":34868,"height":27671,"x":27672,"y":27673},[25022,35953,35954],{"dataMmlNode":25031,"transform":35336},[25033,35955],{"dataC":25078,"xLinkHref":35956},"#MJX-42-TEX-N-3D",[25022,35958,35959,35963,35977],{"dataMmlNode":33991,"transform":35342},[25022,35960,35961],{"dataMmlNode":25942},[25033,35962],{"dataC":32181,"xLinkHref":35911},[25022,35964,35965,35969,35973],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,35966,35967],{"dataMmlNode":25031},[25033,35968],{"dataC":25035,"xLinkHref":35904},[25022,35970,35971],{"dataMmlNode":25942,"transform":25040},[25033,35972],{"dataC":27210,"xLinkHref":35922},[25022,35974,35975],{"dataMmlNode":25031,"transform":32577},[25033,35976],{"dataC":25108,"xLinkHref":35927},[25022,35978,35979],{"dataMmlNode":25942,"transform":34014},[25033,35980],{"dataC":33647,"xLinkHref":35949},[26236,35982,35984],{"id":35983},"combine-the-chain-rule","Combine 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1799",[24938,36897,36898,36901,36904,36907,36910,36913,36916,36919,36922,36925,36928,36931,36934,36937,36940,36943],{},[24941,36899],{"id":36900,"d":32951},"MJX-48-TEX-I-1D703",[24941,36902],{"id":36903,"d":33606},"MJX-48-TEX-I-1D457",[24941,36905],{"id":36906,"d":33610},"MJX-48-TEX-N-3A",[24941,36908],{"id":36909,"d":24972},"MJX-48-TEX-N-3D",[24941,36911],{"id":36912,"d":30313},"MJX-48-TEX-N-2B",[24941,36914],{"id":36915,"d":33620},"MJX-48-TEX-I-1D6FC",[24941,36917],{"id":36918,"d":34234},"MJX-48-TEX-LO-28",[24941,36920],{"id":36921,"d":25893},"MJX-48-TEX-I-1D466",[24941,36923],{"id":36924,"d":24944},"MJX-48-TEX-N-28",[24941,36926],{"id":36927,"d":27108},"MJX-48-TEX-I-1D456",[24941,36929],{"id":36930,"d":24988},"MJX-48-TEX-N-29",[24941,36932],{"id":36933,"d":25910},"MJX-48-TEX-N-2212",[24941,36935],{"id":36936,"d":31530},"MJX-48-TEX-I-210E",[24941,36938],{"id":36939,"d":32148},"MJX-48-TEX-I-1D465",[24941,36941],{"id":36942,"d":34250},"MJX-48-TEX-LO-29",[24941,36944],{"id":36945,"d":27409},"MJX-48-TEX-N-22C5",[25022,36947,36948],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,36949,36950,36962,36974,36984,36989,36994,37069,37075],{"dataMmlNode":25028},[25022,36951,36952,36957],{"dataMmlNode":26023},[25022,36953,36954],{"dataMmlNode":25942},[25033,36955],{"dataC":33000,"xLinkHref":36956},"#MJX-48-TEX-I-1D703",[25022,36958,36959],{"dataMmlNode":25942,"transform":33004},[25033,36960],{"dataC":33647,"xLinkHref":36961},"#MJX-48-TEX-I-1D457",[25022,36963,36964,36969],{"dataMmlNode":25031,"transform":33651},[25022,36965,36966],{"dataMmlNode":6977},[25033,36967],{"dataC":33656,"xLinkHref":36968},"#MJX-48-TEX-N-3A",[25022,36970,36971],{"dataMmlNode":6977,"transform":26048},[25033,36972],{"dataC":25078,"xLinkHref":36973},"#MJX-48-TEX-N-3D",[25022,36975,36976,36980],{"dataMmlNode":26023,"transform":33665},[25022,36977,36978],{"dataMmlNode":25942},[25033,36979],{"dataC":33000,"xLinkHref":36956},[25022,36981,36982],{"dataMmlNode":25942,"transform":33004},[25033,36983],{"dataC":33647,"xLinkHref":36961},[25022,36985,36986],{"dataMmlNode":25031,"transform":33676},[25033,36987],{"dataC":30393,"xLinkHref":36988},"#MJX-48-TEX-N-2B",[25022,36990,36991],{"dataMmlNode":25942,"transform":33682},[25033,36992],{"dataC":33685,"xLinkHref":36993},"#MJX-48-TEX-I-1D6FC",[25022,36995,36996,37001,37025,37030,37035,37039,37060,37064],{"dataMmlNode":27626,"transform":36360},[25022,36997,36998],{"dataMmlNode":25031,"transform":27630},[25033,36999],{"dataC":25035,"xLinkHref":37000},"#MJX-48-TEX-LO-28",[25022,37002,37003,37008],{"dataMmlNode":31565,"transform":34318},[25022,37004,37005],{"dataMmlNode":25942},[25033,37006],{"dataC":25987,"xLinkHref":37007},"#MJX-48-TEX-I-1D466",[25022,37009,37010,37015,37020],{"dataMmlNode":27203,"transform":33304,"dataMjxTexclass":27205},[25022,37011,37012],{"dataMmlNode":25031},[25033,37013],{"dataC":25035,"xLinkHref":37014},"#MJX-48-TEX-N-28",[25022,37016,37017],{"dataMmlNode":25942,"transform":25040},[25033,37018],{"dataC":27210,"xLinkHref":37019},"#MJX-48-TEX-I-1D456",[25022,37021,37022],{"dataMmlNode":25031,"transform":32577},[25033,37023],{"dataC":25108,"xLinkHref":37024},"#MJX-48-TEX-N-29",[25022,37026,37027],{"dataMmlNode":25031,"transform":36800},[25033,37028],{"dataC":26019,"xLinkHref":37029},"#MJX-48-TEX-N-2212",[25022,37031,37032],{"dataMmlNode":25942,"transform":36806},[25033,37033],{"dataC":31539,"xLinkHref":37034},"#MJX-48-TEX-I-210E",[25022,37036,37037],{"dataMmlNode":25031,"transform":36812},[25033,37038],{"dataC":25035,"xLinkHref":37014},[25022,37040,37041,37046],{"dataMmlNode":31565,"transform":36817},[25022,37042,37043],{"dataMmlNode":25942},[25033,37044],{"dataC":32181,"xLinkHref":37045},"#MJX-48-TEX-I-1D465",[25022,37047,37048,37052,37056],{"dataMmlNode":27203,"transform":33270,"dataMjxTexclass":27205},[25022,37049,37050],{"dataMmlNode":25031},[25033,37051],{"dataC":25035,"xLinkHref":37014},[25022,37053,37054],{"dataMmlNode":25942,"transform":25040},[25033,37055],{"dataC":27210,"xLinkHref":37019},[25022,37057,37058],{"dataMmlNode":25031,"transform":32577},[25033,37059],{"dataC":25108,"xLinkHref":37024},[25022,37061,37062],{"dataMmlNode":25031,"transform":36839},[25033,37063],{"dataC":25108,"xLinkHref":37024},[25022,37065,37066],{"dataMmlNode":25031,"transform":34384},[25033,37067],{"dataC":25108,"xLinkHref":37068},"#MJX-48-TEX-LO-29",[25022,37070,37072],{"dataMmlNode":25031,"transform":37071},"translate(12136.2,0)",[25033,37073],{"dataC":27679,"xLinkHref":37074},"#MJX-48-TEX-N-22C5",[25022,37076,37078,37082,37096],{"dataMmlNode":33991,"transform":37077},"translate(12636.4,0)",[25022,37079,37080],{"dataMmlNode":25942},[25033,37081],{"dataC":32181,"xLinkHref":37045},[25022,37083,37084,37088,37092],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,37085,37086],{"dataMmlNode":25031},[25033,37087],{"dataC":25035,"xLinkHref":37014},[25022,37089,37090],{"dataMmlNode":25942,"transform":25040},[25033,37091],{"dataC":27210,"xLinkHref":37019},[25022,37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vs. Stochastic Gradient Descent",[12,37105,37106,37109,37110,37112],{},[106,37107,37108],{},"Batch Gradient Descent"," sums the gradients over ",[7289,37111,23215],{}," training examples before making a single update. It's precise but can be slow for large datasets.",[12,37114,37115],{},[1930,37116],{"alt":37117,"src":37118},"0.84","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1777294390662-1775987318720-Screenshot_2026-04-12_at_5.webp",[12,37120,37121,37124,37125,37144,37145,37148],{},[106,37122,37123],{},"Stochastic (Incremental) Gradient Descent"," updates ",[24923,37126,37128],{"className":37127,"jax":24927},[24926],[24929,37129,37130,37135],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,37131,37132],{},[24941,37133],{"id":37134,"d":32951},"MJX-49-TEX-I-1D703",[25022,37136,37137],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,37138,37139],{"dataMmlNode":25028},[25022,37140,37141],{"dataMmlNode":25942},[25033,37142],{"dataC":33000,"xLinkHref":37143},"#MJX-49-TEX-I-1D703"," after ",[7289,37146,37147],{},"each"," training example. It's faster and often works well in practice, though the path to convergence is noisier.",[12,37150,37151],{},[1930,37152],{"alt":37153,"src":37154},"0.70","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1777294405734-1775987315179-Screenshot_2026-04-12_at_5.webp",[7166,37156,37158],{"id":37157},"underfitting-vs-overfitting","Underfitting vs. Overfitting",[12,37160,37161],{},"This is one of the most important concepts in ML. Consider fitting different models to our housing data:",[23,37163,37164,37173,37206],{},[26,37165,9407,37166,37169,37170,6959],{},[106,37167,37168],{},"straight line"," (linear fit) may be too simple — it doesn't capture the curve in the data. This is ",[106,37171,37172],{},"underfitting",[26,37174,9407,37175,37178,37179,37205],{},[106,37176,37177],{},"quadratic"," (adding ",[24923,37180,37182],{"className":37181,"jax":24927},[24926],[24929,37183,37184,37189],{"style":32867,"xmlns":24932,"width":32868,"height":32869,"role":1930,"focusable":8058,"viewBox":32870,"xmlnsXLink":24936},[24938,37185,37186],{},[24941,37187],{"id":37188,"d":32148},"MJX-50-TEX-I-1D465",[25022,37190,37191],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,37192,37193,37198],{"dataMmlNode":25028},[25022,37194,37195],{"dataMmlNode":25942},[25033,37196],{"dataC":32181,"xLinkHref":37197},"#MJX-50-TEX-I-1D465",[25022,37199,37200],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":32887},[25022,37201,37202],{"dataMmlNode":25031},[6977,37203,37204],{"dataVariant":32892,"transform":25025,"font-size":32893,"font-family":32894},"²",") fits better, capturing more structure.",[26,37207,9407,37208,37211,37212,6959],{},[106,37209,37210],{},"5th-order polynomial"," passes through every data point perfectly, but it would make terrible predictions on new data. This is ",[106,37213,37214],{},"overfitting",[12,37216,37217],{},[1930,37218],{"alt":37219,"src":37220},"0.90","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1777294461971-1775987112930-Screenshot_2026-04-12_at_5.webp",[12,37222,37223,37224,37227],{},"The key insight: a model that memorizes training data is not necessarily a good model. We want one that ",[106,37225,37226],{},"generalizes"," to unseen data.",[7166,37229,37231],{"id":37230},"locally-weighted-linear-regression","Locally Weighted Linear Regression",[12,37233,37234,37235,37238],{},"Standard linear regression fits one global line to all data. 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scale(0.707)",[25033,37420],{"dataC":26058,"xLinkHref":37421},"#MJX-51-TEX-N-32",[25022,37423,37425,37429],{"dataMmlNode":27626,"transform":37424},"translate(1695.3,-719.9)",[25022,37426,37427],{"dataMmlNode":26031},[25033,37428],{"dataC":26058,"xLinkHref":37421},[25022,37430,37431,37437],{"dataMmlNode":31565,"transform":26394},[25022,37432,37433],{"dataMmlNode":25942},[25033,37434],{"dataC":37435,"xLinkHref":37436},"1D70F","#MJX-51-TEX-I-1D70F",[25022,37438,37440],{"dataMmlNode":26031,"transform":37439},"translate(604,289) scale(0.707)",[25033,37441],{"dataC":26058,"xLinkHref":37421},[27668,37443],{"width":37444,"height":27671,"x":27672,"y":27673},4658.1,[25022,37446,37448],{"dataMmlNode":25031,"transform":37447},"translate(6468.1,0) translate(0 -0.5)",[25033,37449],{"dataC":25108,"xLinkHref":37450},"#MJX-51-TEX-S4-29",[12,37452,37453,37454,37477,37478,37481],{},"The parameter ",[24923,37455,37457],{"className":37456,"jax":24927},[24926],[24929,37458,37463,37468],{"style":37459,"xmlns":24932,"width":37460,"height":37461,"role":1930,"focusable":8058,"viewBox":37462,"xmlnsXLink":24936},"vertical-align: -0.029ex;","1.17ex","1.005ex","0 -431 517 444",[24938,37464,37465],{},[24941,37466],{"id":37467,"d":37296},"MJX-52-TEX-I-1D70F",[25022,37469,37470],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,37471,37472],{"dataMmlNode":25028},[25022,37473,37474],{"dataMmlNode":25942},[25033,37475],{"dataC":37435,"xLinkHref":37476},"#MJX-52-TEX-I-1D70F"," (tau) controls how quickly the weights decay with distance. LWLR is a ",[106,37479,37480],{},"non-parametric"," method — it doesn't commit to a fixed set of parameters but re-fits for each query point.",[25631,37483,37485,37488],{"className":37484},[13441,13442],[25635,37486,37487],{},"The Intuition behind Locally Weighted Linear Regression (LWLR)",[13438,37489,37491,37498,37530,37578,37582,37585,37768,37771,38393,38397,38522],{"className":37490},[25641],[12,37492,37493,37494,37497],{},"In standard linear regression, when you fit a line, ",[106,37495,37496],{},"every single training example has an equal vote",". A house that is 5,000 sq ft has the exact same influence on the line as a house that is 1,000 sq ft.",[12,37499,37500,37501,37504,37505,37526,37527],{},"LWLR changes this. If you want to predict the price of a house that is ",[106,37502,37503],{},"1,500 sq ft"," (your query point, ",[24923,37506,37508],{"className":37507,"jax":24927},[24926],[24929,37509,37512,37517],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},"1.294ex","0 -442 572 453",[24938,37513,37514],{},[24941,37515],{"id":37516,"d":32148},"MJX-53-TEX-I-1D465",[25022,37518,37519],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,37520,37521],{"dataMmlNode":25028},[25022,37522,37523],{"dataMmlNode":25942},[25033,37524],{"dataC":32181,"xLinkHref":37525},"#MJX-53-TEX-I-1D465","), LWLR says: ",[7289,37528,37529],{},"\"I only really care about training examples that are similar in size to 1,500 sq ft. The 5,000 sq ft houses shouldn't influence my prediction right now.\"",[12,37531,37532,37533,6959],{},"To do this, we multiply the error of each training example by a weight, ",[24923,37534,37536],{"className":37535,"jax":24927},[24926],[24929,37537,37539,37551],{"style":25196,"xmlns":24932,"width":28582,"height":32538,"role":1930,"focusable":8058,"viewBox":37538,"xmlnsXLink":24936},"0 -893.3 1593.1 904.3",[24938,37540,37541,37544,37546,37549],{},[24941,37542],{"id":37543,"d":37254},"MJX-54-TEX-I-1D464",[24941,37545],{"id":30062,"d":24944},[24941,37547],{"id":37548,"d":27108},"MJX-54-TEX-I-1D456",[24941,37550],{"id":30068,"d":24988},[25022,37552,37553],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,37554,37555],{"dataMmlNode":25028},[25022,37556,37557,37562],{"dataMmlNode":31565},[25022,37558,37559],{"dataMmlNode":25942},[25033,37560],{"dataC":37310,"xLinkHref":37561},"#MJX-54-TEX-I-1D464",[25022,37563,37565,37569,37574],{"dataMmlNode":27203,"transform":37564,"dataMjxTexclass":27205},"translate(749,363) scale(0.707)",[25022,37566,37567],{"dataMmlNode":25031},[25033,37568],{"dataC":25035,"xLinkHref":30124},[25022,37570,37571],{"dataMmlNode":25942,"transform":25040},[25033,37572],{"dataC":27210,"xLinkHref":37573},"#MJX-54-TEX-I-1D456",[25022,37575,37576],{"dataMmlNode":25031,"transform":32577},[25033,37577],{"dataC":25108,"xLinkHref":30134},[7166,37579,37581],{"id":37580},"deconstructing-the-equation","Deconstructing the Equation",[12,37583,37584],{},"The weight formula is a Gaussian (bell-shaped) curve:",[24923,37586,37588],{"className":37587,"jax":24927,"display":13594},[24926],[24929,37589,37590,37636],{"style":34674,"xmlns":24932,"width":37247,"height":34676,"role":1930,"focusable":8058,"viewBox":37248,"xmlnsXLink":24936},[24938,37591,37592,37595,37598,37601,37604,37606,37609,37612,37615,37618,37621,37624,37627,37630,37633],{},[24941,37593],{"id":37594,"d":37254},"MJX-55-TEX-I-1D464",[24941,37596],{"id":37597,"d":24944},"MJX-55-TEX-N-28",[24941,37599],{"id":37600,"d":27108},"MJX-55-TEX-I-1D456",[24941,37602],{"id":37603,"d":24988},"MJX-55-TEX-N-29",[24941,37605],{"id":30176,"d":24972},[24941,37607],{"id":37608,"d":25259},"MJX-55-TEX-N-65",[24941,37610],{"id":37611,"d":37273},"MJX-55-TEX-N-78",[24941,37613],{"id":37614,"d":37277},"MJX-55-TEX-N-70",[24941,37616],{"id":37617,"d":180},"MJX-55-TEX-N-2061",[24941,37619],{"id":37620,"d":34706},"MJX-55-TEX-S4-28",[24941,37622],{"id":37623,"d":25910},"MJX-55-TEX-N-2212",[24941,37625],{"id":37626,"d":32148},"MJX-55-TEX-I-1D465",[24941,37628],{"id":37629,"d":25928},"MJX-55-TEX-N-32",[24941,37631],{"id":37632,"d":37296},"MJX-55-TEX-I-1D70F",[24941,37634],{"id":37635,"d":34719},"MJX-55-TEX-S4-29",[25022,37637,37638],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,37639,37640,37664,37668,37679,37684],{"dataMmlNode":25028},[25022,37641,37642,37647],{"dataMmlNode":31565},[25022,37643,37644],{"dataMmlNode":25942},[25033,37645],{"dataC":37310,"xLinkHref":37646},"#MJX-55-TEX-I-1D464",[25022,37648,37649,37654,37659],{"dataMmlNode":27203,"transform":37314,"dataMjxTexclass":27205},[25022,37650,37651],{"dataMmlNode":25031},[25033,37652],{"dataC":25035,"xLinkHref":37653},"#MJX-55-TEX-N-28",[25022,37655,37656],{"dataMmlNode":25942,"transform":25040},[25033,37657],{"dataC":27210,"xLinkHref":37658},"#MJX-55-TEX-I-1D456",[25022,37660,37661],{"dataMmlNode":25031,"transform":32577},[25033,37662],{"dataC":25108,"xLinkHref":37663},"#MJX-55-TEX-N-29",[25022,37665,37666],{"dataMmlNode":25031,"transform":37332},[25033,37667],{"dataC":25078,"xLinkHref":30200},[25022,37669,37670,37673,37676],{"dataMmlNode":25942,"transform":37338},[25033,37671],{"dataC":25323,"xLinkHref":37672},"#MJX-55-TEX-N-65",[25033,37674],{"dataC":31181,"xLinkHref":37675,"transform":37345},"#MJX-55-TEX-N-78",[25033,37677],{"dataC":31215,"xLinkHref":37678,"transform":37349},"#MJX-55-TEX-N-70",[25022,37680,37681],{"dataMmlNode":25031,"transform":37352},[25033,37682],{"dataC":26065,"xLinkHref":37683},"#MJX-55-TEX-N-2061",[25022,37685,37686,37691,37696,37763],{"dataMmlNode":27626,"transform":37358},[25022,37687,37688],{"dataMmlNode":25031,"transform":27630},[25033,37689],{"dataC":25035,"xLinkHref":37690},"#MJX-55-TEX-S4-28",[25022,37692,37693],{"dataMmlNode":25031,"transform":34797},[25033,37694],{"dataC":26019,"xLinkHref":37695},"#MJX-55-TEX-N-2212",[25022,37697,37698,37744,37761],{"dataMmlNode":27622,"transform":37371},[25022,37699,37700,37704,37725,37729,37733],{"dataMmlNode":27626,"transform":33692},[25022,37701,37702],{"dataMmlNode":25031},[25033,37703],{"dataC":25035,"xLinkHref":37653},[25022,37705,37706,37711],{"dataMmlNode":31565,"transform":25040},[25022,37707,37708],{"dataMmlNode":25942},[25033,37709],{"dataC":32181,"xLinkHref":37710},"#MJX-55-TEX-I-1D465",[25022,37712,37713,37717,37721],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},[25022,37714,37715],{"dataMmlNode":25031},[25033,37716],{"dataC":25035,"xLinkHref":37653},[25022,37718,37719],{"dataMmlNode":25942,"transform":25040},[25033,37720],{"dataC":27210,"xLinkHref":37658},[25022,37722,37723],{"dataMmlNode":25031,"transform":32577},[25033,37724],{"dataC":25108,"xLinkHref":37663},[25022,37726,37727],{"dataMmlNode":25031,"transform":37401},[25033,37728],{"dataC":26019,"xLinkHref":37695},[25022,37730,37731],{"dataMmlNode":25942,"transform":37406},[25033,37732],{"dataC":32181,"xLinkHref":37710},[25022,37734,37735,37739],{"dataMmlNode":31565,"transform":37411},[25022,37736,37737],{"dataMmlNode":25031},[25033,37738],{"dataC":25108,"xLinkHref":37663},[25022,37740,37741],{"dataMmlNode":26031,"transform":37418},[25033,37742],{"dataC":26058,"xLinkHref":37743},"#MJX-55-TEX-N-32",[25022,37745,37746,37750],{"dataMmlNode":27626,"transform":37424},[25022,37747,37748],{"dataMmlNode":26031},[25033,37749],{"dataC":26058,"xLinkHref":37743},[25022,37751,37752,37757],{"dataMmlNode":31565,"transform":26394},[25022,37753,37754],{"dataMmlNode":25942},[25033,37755],{"dataC":37435,"xLinkHref":37756},"#MJX-55-TEX-I-1D70F",[25022,37758,37759],{"dataMmlNode":26031,"transform":37439},[25033,37760],{"dataC":26058,"xLinkHref":37743},[27668,37762],{"width":37444,"height":27671,"x":27672,"y":27673},[25022,37764,37765],{"dataMmlNode":25031,"transform":37447},[25033,37766],{"dataC":25108,"xLinkHref":37767},"#MJX-55-TEX-S4-29",[12,37769,37770],{},"Here 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is the squared distance between a training example ",[24923,37861,37863],{"className":37862,"jax":24927},[24926],[24929,37864,37865,37876],{"style":25196,"xmlns":24932,"width":32537,"height":32538,"role":1930,"focusable":8058,"viewBox":32539,"xmlnsXLink":24936},[24938,37866,37867,37870,37872,37874],{},[24941,37868],{"id":37869,"d":32148},"MJX-57-TEX-I-1D465",[24941,37871],{"id":30297,"d":24944},[24941,37873],{"id":30282,"d":27108},[24941,37875],{"id":30316,"d":24988},[25022,37877,37878],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,37879,37880],{"dataMmlNode":25028},[25022,37881,37882,37887],{"dataMmlNode":31565},[25022,37883,37884],{"dataMmlNode":25942},[25033,37885],{"dataC":32181,"xLinkHref":37886},"#MJX-57-TEX-I-1D465",[25022,37888,37889,37893,37897],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},[25022,37890,37891],{"dataMmlNode":25031},[25033,37892],{"dataC":25035,"xLinkHref":30364},[25022,37894,37895],{"dataMmlNode":25942,"transform":25040},[25033,37896],{"dataC":27210,"xLinkHref":30335},[25022,37898,37899],{"dataMmlNode":25031,"transform":32577},[25033,37900],{"dataC":25108,"xLinkHref":30419}," and the specific point you are trying to predict ",[24923,37903,37905],{"className":37904,"jax":24927},[24926],[24929,37906,37907,37912],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,37908,37909],{},[24941,37910],{"id":37911,"d":32148},"MJX-58-TEX-I-1D465",[25022,37913,37914],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,37915,37916],{"dataMmlNode":25028},[25022,37917,37918],{"dataMmlNode":25942},[25033,37919],{"dataC":32181,"xLinkHref":37920},"#MJX-58-TEX-I-1D465",[23,37922,37923,37928],{},[26,37924,37925,37926,6959],{},"If the training point is very close to your query point, this value is almost ",[106,37927,9180],{},[26,37929,37930,37931,6959],{},"If the training point is very far away, this value is ",[106,37932,37933],{},"large",[26,37935,37936,7206,37939,38016,38018,38019,38040,38041],{},[106,37937,37938],{},"The Exponential & Negative Sign:",[24923,37940,37942],{"className":37941,"jax":24927},[24926],[24929,37943,37946,37972],{"style":24931,"xmlns":24932,"width":37944,"height":24934,"role":1930,"focusable":8058,"viewBox":37945,"xmlnsXLink":24936},"10.006ex","0 -750 4422.7 1000",[24938,37947,37948,37951,37954,37957,37960,37963,37966,37969],{},[24941,37949],{"id":37950,"d":25259},"MJX-59-TEX-N-65",[24941,37952],{"id":37953,"d":37273},"MJX-59-TEX-N-78",[24941,37955],{"id":37956,"d":37277},"MJX-59-TEX-N-70",[24941,37958],{"id":37959,"d":180},"MJX-59-TEX-N-2061",[24941,37961],{"id":37962,"d":24944},"MJX-59-TEX-N-28",[24941,37964],{"id":37965,"d":25910},"MJX-59-TEX-N-2212",[24941,37967],{"id":37968,"d":34448},"MJX-59-TEX-N-2026",[24941,37970],{"id":37971,"d":24988},"MJX-59-TEX-N-29",[25022,37973,37974],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,37975,37976,37987,37993,37998,38004,38010],{"dataMmlNode":25028},[25022,37977,37978,37981,37984],{"dataMmlNode":25942},[25033,37979],{"dataC":25323,"xLinkHref":37980},"#MJX-59-TEX-N-65",[25033,37982],{"dataC":31181,"xLinkHref":37983,"transform":37345},"#MJX-59-TEX-N-78",[25033,37985],{"dataC":31215,"xLinkHref":37986,"transform":37349},"#MJX-59-TEX-N-70",[25022,37988,37990],{"dataMmlNode":25031,"transform":37989},"translate(1528,0)",[25033,37991],{"dataC":26065,"xLinkHref":37992},"#MJX-59-TEX-N-2061",[25022,37994,37995],{"dataMmlNode":25031,"transform":37989},[25033,37996],{"dataC":25035,"xLinkHref":37997},"#MJX-59-TEX-N-28",[25022,37999,38001],{"dataMmlNode":25031,"transform":38000},"translate(1917,0)",[25033,38002],{"dataC":26019,"xLinkHref":38003},"#MJX-59-TEX-N-2212",[25022,38005,38007],{"dataMmlNode":25031,"transform":38006},"translate(2861.7,0)",[25033,38008],{"dataC":34660,"xLinkHref":38009},"#MJX-59-TEX-N-2026",[25022,38011,38013],{"dataMmlNode":25031,"transform":38012},"translate(4033.7,0)",[25033,38014],{"dataC":25108,"xLinkHref":38015},"#MJX-59-TEX-N-29",[16,38017],{},"Because of the negative sign, we are doing ",[24923,38020,38022],{"className":38021,"jax":24927},[24926],[24929,38023,38026,38031],{"style":25196,"xmlns":24932,"width":38024,"height":27038,"role":1930,"focusable":8058,"viewBox":38025,"xmlnsXLink":24936},"1.054ex","0 -442 466 453",[24938,38027,38028],{},[24941,38029],{"id":38030,"d":27397},"MJX-60-TEX-I-1D452",[25022,38032,38033],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38034,38035],{"dataMmlNode":25028},[25022,38036,38037],{"dataMmlNode":25942},[25033,38038],{"dataC":27596,"xLinkHref":38039},"#MJX-60-TEX-I-1D452"," to the power of a negative number.",[23,38042,38043,38133],{},[26,38044,38045,38046,38063,38064,38111,38112,38129,38130,6959],{},"If the distance is roughly ",[24923,38047,38049],{"className":38048,"jax":24927},[24926],[24929,38050,38051,38055],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,38052,38053],{},[24941,38054],{"id":30771,"d":25935},[25022,38056,38057],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38058,38059],{"dataMmlNode":25028},[25022,38060,38061],{"dataMmlNode":26031},[25033,38062],{"dataC":26103,"xLinkHref":30802}," (the points are close), we get ",[24923,38065,38067],{"className":38066,"jax":24927},[24926],[24929,38068,38072,38084],{"style":26868,"xmlns":24932,"width":38069,"height":38070,"role":1930,"focusable":8058,"viewBox":38071,"xmlnsXLink":24936},"6.19ex","2.072ex","0 -833.9 2736.1 915.9",[24938,38073,38074,38077,38079,38081],{},[24941,38075],{"id":38076,"d":27397},"MJX-62-TEX-I-1D452",[24941,38078],{"id":30833,"d":25935},[24941,38080],{"id":30830,"d":24972},[24941,38082],{"id":38083,"d":25914},"MJX-62-TEX-N-31",[25022,38085,38086],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38087,38088,38100,38105],{"dataMmlNode":25028},[25022,38089,38090,38095],{"dataMmlNode":31565},[25022,38091,38092],{"dataMmlNode":25942},[25033,38093],{"dataC":27596,"xLinkHref":38094},"#MJX-62-TEX-I-1D452",[25022,38096,38098],{"dataMmlNode":26031,"transform":38097},"translate(499,363) scale(0.707)",[25033,38099],{"dataC":26103,"xLinkHref":30852},[25022,38101,38103],{"dataMmlNode":25031,"transform":38102},"translate(1180.3,0)",[25033,38104],{"dataC":25078,"xLinkHref":30847},[25022,38106,38108],{"dataMmlNode":26031,"transform":38107},"translate(2236.1,0)",[25033,38109],{"dataC":26035,"xLinkHref":38110},"#MJX-62-TEX-N-31",". The weight is ",[24923,38113,38115],{"className":38114,"jax":24927},[24926],[24929,38116,38117,38121],{"style":27802,"xmlns":24932,"width":35168,"height":35169,"role":1930,"focusable":8058,"viewBox":35170,"xmlnsXLink":24936},[24938,38118,38119],{},[24941,38120],{"id":30879,"d":25914},[25022,38122,38123],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38124,38125],{"dataMmlNode":25028},[25022,38126,38127],{"dataMmlNode":26031},[25033,38128],{"dataC":26035,"xLinkHref":30952},". The model gives this point ",[106,38131,38132],{},"maximum importance",[26,38134,38135,38136,38204,38205,38207,38208,38227,38228,38231],{},"If the distance is large (the points are far), we get ",[24923,38137,38139],{"className":38138,"jax":24927},[24926],[24929,38140,38144,38167],{"style":25196,"xmlns":24932,"width":38141,"height":38142,"role":1930,"focusable":8058,"viewBox":38143,"xmlnsXLink":24936},"5.869ex","1.956ex","0 -853.7 2594 864.7",[24938,38145,38146,38149,38152,38155,38158,38161,38164],{},[24941,38147],{"id":38148,"d":27397},"MJX-64-TEX-I-1D452",[24941,38150],{"id":38151,"d":25910},"MJX-64-TEX-N-2212",[24941,38153],{"id":38154,"d":24960},"MJX-64-TEX-N-6C",[24941,38156],{"id":38157,"d":25020},"MJX-64-TEX-N-61",[24941,38159],{"id":38160,"d":25016},"MJX-64-TEX-N-72",[24941,38162],{"id":38163,"d":25924},"MJX-64-TEX-N-67",[24941,38165],{"id":38166,"d":25259},"MJX-64-TEX-N-65",[25022,38168,38169],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38170,38171],{"dataMmlNode":25028},[25022,38172,38173,38178],{"dataMmlNode":31565},[25022,38174,38175],{"dataMmlNode":25942},[25033,38176],{"dataC":27596,"xLinkHref":38177},"#MJX-64-TEX-I-1D452",[25022,38179,38180,38185],{"dataMmlNode":27203,"transform":38097,"dataMjxTexclass":27205},[25022,38181,38182],{"dataMmlNode":25031},[25033,38183],{"dataC":26019,"xLinkHref":38184},"#MJX-64-TEX-N-2212",[25022,38186,38187,38190,38193,38196,38200],{"dataMmlNode":25039,"transform":25049},[25033,38188],{"dataC":25057,"xLinkHref":38189},"#MJX-64-TEX-N-6C",[25033,38191],{"dataC":25180,"xLinkHref":38192,"transform":26048},"#MJX-64-TEX-N-61",[25033,38194],{"dataC":25172,"xLinkHref":38195,"transform":25049},"#MJX-64-TEX-N-72",[25033,38197],{"dataC":26051,"xLinkHref":38198,"transform":38199},"#MJX-64-TEX-N-67","translate(1170,0)",[25033,38201],{"dataC":25323,"xLinkHref":38202,"transform":38203},"#MJX-64-TEX-N-65","translate(1670,0)",", which approaches ",[106,38206,9180],{},". The weight becomes ",[24923,38209,38211],{"className":38210,"jax":24927},[24926],[24929,38212,38213,38218],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,38214,38215],{},[24941,38216],{"id":38217,"d":25935},"MJX-65-TEX-N-30",[25022,38219,38220],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38221,38222],{"dataMmlNode":25028},[25022,38223,38224],{"dataMmlNode":26031},[25033,38225],{"dataC":26103,"xLinkHref":38226},"#MJX-65-TEX-N-30",". The model completely ",[106,38229,38230],{},"ignores"," this point.",[26,38233,38234,7206,38237,38256,38257,38259,38278,38279,6959,38298],{},[106,38235,38236],{},"The Bandwidth Parameter:",[24923,38238,38240],{"className":38239,"jax":24927},[24926],[24929,38241,38242,38247],{"style":37459,"xmlns":24932,"width":37460,"height":37461,"role":1930,"focusable":8058,"viewBox":37462,"xmlnsXLink":24936},[24938,38243,38244],{},[24941,38245],{"id":38246,"d":37296},"MJX-66-TEX-I-1D70F",[25022,38248,38249],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38250,38251],{"dataMmlNode":25028},[25022,38252,38253],{"dataMmlNode":25942},[25033,38254],{"dataC":37435,"xLinkHref":38255},"#MJX-66-TEX-I-1D70F"," (Tau)",[16,38258],{},[24923,38260,38262],{"className":38261,"jax":24927},[24926],[24929,38263,38264,38269],{"style":37459,"xmlns":24932,"width":37460,"height":37461,"role":1930,"focusable":8058,"viewBox":37462,"xmlnsXLink":24936},[24938,38265,38266],{},[24941,38267],{"id":38268,"d":37296},"MJX-67-TEX-I-1D70F",[25022,38270,38271],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38272,38273],{"dataMmlNode":25028},[25022,38274,38275],{"dataMmlNode":25942},[25033,38276],{"dataC":37435,"xLinkHref":38277},"#MJX-67-TEX-I-1D70F"," dictates how \"fat\" or \"thin\" our bell curve is. It controls how quickly the weights fall off to zero as you move away from the query point ",[24923,38280,38282],{"className":38281,"jax":24927},[24926],[24929,38283,38284,38289],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,38285,38286],{},[24941,38287],{"id":38288,"d":32148},"MJX-68-TEX-I-1D465",[25022,38290,38291],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38292,38293],{"dataMmlNode":25028},[25022,38294,38295],{"dataMmlNode":25942},[25033,38296],{"dataC":32181,"xLinkHref":38297},"#MJX-68-TEX-I-1D465",[23,38299,38300,38347],{},[26,38301,38302,7206,38305,38326,38327,38346],{},[106,38303,38304],{},"Large",[106,38306,38307,7024],{},[24923,38308,38310],{"className":38309,"jax":24927},[24926],[24929,38311,38312,38317],{"style":37459,"xmlns":24932,"width":37460,"height":37461,"role":1930,"focusable":8058,"viewBox":37462,"xmlnsXLink":24936},[24938,38313,38314],{},[24941,38315],{"id":38316,"d":37296},"MJX-69-TEX-I-1D70F",[25022,38318,38319],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38320,38321],{"dataMmlNode":25028},[25022,38322,38323],{"dataMmlNode":25942},[25033,38324],{"dataC":37435,"xLinkHref":38325},"#MJX-69-TEX-I-1D70F"," The weights fall off very slowly. The model looks at a very wide \"neighborhood\" of points. If ",[24923,38328,38330],{"className":38329,"jax":24927},[24926],[24929,38331,38332,38337],{"style":37459,"xmlns":24932,"width":37460,"height":37461,"role":1930,"focusable":8058,"viewBox":37462,"xmlnsXLink":24936},[24938,38333,38334],{},[24941,38335],{"id":38336,"d":37296},"MJX-70-TEX-I-1D70F",[25022,38338,38339],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38340,38341],{"dataMmlNode":25028},[25022,38342,38343],{"dataMmlNode":25942},[25033,38344],{"dataC":37435,"xLinkHref":38345},"#MJX-70-TEX-I-1D70F"," is infinitely large, LWLR just becomes standard linear regression.",[26,38348,38349,7206,38352,38373,38374,6959],{},[106,38350,38351],{},"Small",[106,38353,38354,7024],{},[24923,38355,38357],{"className":38356,"jax":24927},[24926],[24929,38358,38359,38364],{"style":37459,"xmlns":24932,"width":37460,"height":37461,"role":1930,"focusable":8058,"viewBox":37462,"xmlnsXLink":24936},[24938,38360,38361],{},[24941,38362],{"id":38363,"d":37296},"MJX-71-TEX-I-1D70F",[25022,38365,38366],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38367,38368],{"dataMmlNode":25028},[25022,38369,38370],{"dataMmlNode":25942},[25033,38371],{"dataC":37435,"xLinkHref":38372},"#MJX-71-TEX-I-1D70F"," The weights fall off rapidly. The model only looks at a very tiny, strict neighborhood of points immediately next to ",[24923,38375,38377],{"className":38376,"jax":24927},[24926],[24929,38378,38379,38384],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,38380,38381],{},[24941,38382],{"id":38383,"d":32148},"MJX-72-TEX-I-1D465",[25022,38385,38386],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38387,38388],{"dataMmlNode":25028},[25022,38389,38390],{"dataMmlNode":25942},[25033,38391],{"dataC":32181,"xLinkHref":38392},"#MJX-72-TEX-I-1D465",[7166,38394,38396],{"id":38395},"why-is-it-called-non-parametric","Why is it called \"Non-parametric\"?",[12,38398,38399,38400,38403,38404,38423,38424,38521],{},"Standard linear regression is ",[106,38401,38402],{},"parametric",". You look at the data once, calculate your ",[24923,38405,38407],{"className":38406,"jax":24927},[24926],[24929,38408,38409,38414],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,38410,38411],{},[24941,38412],{"id":38413,"d":32951},"MJX-73-TEX-I-1D703",[25022,38415,38416],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38417,38418],{"dataMmlNode":25028},[25022,38419,38420],{"dataMmlNode":25942},[25033,38421],{"dataC":33000,"xLinkHref":38422},"#MJX-73-TEX-I-1D703"," parameters, and then you can throw the training data away forever. Your formula ",[24923,38425,38427],{"className":38426,"jax":24927},[24926],[24929,38428,38431,38460],{"style":24931,"xmlns":24932,"width":38429,"height":24934,"role":1930,"focusable":8058,"viewBox":38430,"xmlnsXLink":24936},"15.532ex","0 -750 6865.1 1000",[24938,38432,38433,38436,38439,38442,38445,38448,38451,38454,38457],{},[24941,38434],{"id":38435,"d":31530},"MJX-74-TEX-I-210E",[24941,38437],{"id":38438,"d":24944},"MJX-74-TEX-N-28",[24941,38440],{"id":38441,"d":32148},"MJX-74-TEX-I-1D465",[24941,38443],{"id":38444,"d":24988},"MJX-74-TEX-N-29",[24941,38446],{"id":38447,"d":24972},"MJX-74-TEX-N-3D",[24941,38449],{"id":38450,"d":32951},"MJX-74-TEX-I-1D703",[24941,38452],{"id":38453,"d":25935},"MJX-74-TEX-N-30",[24941,38455],{"id":38456,"d":30313},"MJX-74-TEX-N-2B",[24941,38458],{"id":38459,"d":25914},"MJX-74-TEX-N-31",[25022,38461,38462],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38463,38464,38469,38474,38479,38484,38489,38501,38506,38517],{"dataMmlNode":25028},[25022,38465,38466],{"dataMmlNode":25942},[25033,38467],{"dataC":31539,"xLinkHref":38468},"#MJX-74-TEX-I-210E",[25022,38470,38471],{"dataMmlNode":25031,"transform":32791},[25033,38472],{"dataC":25035,"xLinkHref":38473},"#MJX-74-TEX-N-28",[25022,38475,38476],{"dataMmlNode":25942,"transform":32797},[25033,38477],{"dataC":32181,"xLinkHref":38478},"#MJX-74-TEX-I-1D465",[25022,38480,38481],{"dataMmlNode":25031,"transform":32803},[25033,38482],{"dataC":25108,"xLinkHref":38483},"#MJX-74-TEX-N-29",[25022,38485,38486],{"dataMmlNode":25031,"transform":32989},[25033,38487],{"dataC":25078,"xLinkHref":38488},"#MJX-74-TEX-N-3D",[25022,38490,38491,38496],{"dataMmlNode":26023,"transform":32995},[25022,38492,38493],{"dataMmlNode":25942},[25033,38494],{"dataC":33000,"xLinkHref":38495},"#MJX-74-TEX-I-1D703",[25022,38497,38498],{"dataMmlNode":26031,"transform":33004},[25033,38499],{"dataC":26103,"xLinkHref":38500},"#MJX-74-TEX-N-30",[25022,38502,38503],{"dataMmlNode":25031,"transform":33010},[25033,38504],{"dataC":30393,"xLinkHref":38505},"#MJX-74-TEX-N-2B",[25022,38507,38508,38512],{"dataMmlNode":26023,"transform":33016},[25022,38509,38510],{"dataMmlNode":25942},[25033,38511],{"dataC":33000,"xLinkHref":38495},[25022,38513,38514],{"dataMmlNode":26031,"transform":33004},[25033,38515],{"dataC":26035,"xLinkHref":38516},"#MJX-74-TEX-N-31",[25022,38518,38519],{"dataMmlNode":25942,"transform":33027},[25033,38520],{"dataC":32181,"xLinkHref":38478}," does all the work.",[12,38523,38524,38525,38527,38528,38575,38576,38595,38596,7206,38599,7206,38620,38623,38624,38643,38644,38663,38664,6959],{},"LWLR is ",[106,38526,37480],{},". Because the weights ",[24923,38529,38531],{"className":38530,"jax":24927},[24926],[24929,38532,38533,38547],{"style":25196,"xmlns":24932,"width":28582,"height":32538,"role":1930,"focusable":8058,"viewBox":37538,"xmlnsXLink":24936},[24938,38534,38535,38538,38541,38544],{},[24941,38536],{"id":38537,"d":37254},"MJX-75-TEX-I-1D464",[24941,38539],{"id":38540,"d":24944},"MJX-75-TEX-N-28",[24941,38542],{"id":38543,"d":27108},"MJX-75-TEX-I-1D456",[24941,38545],{"id":38546,"d":24988},"MJX-75-TEX-N-29",[25022,38548,38549],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38550,38551],{"dataMmlNode":25028},[25022,38552,38553,38558],{"dataMmlNode":31565},[25022,38554,38555],{"dataMmlNode":25942},[25033,38556],{"dataC":37310,"xLinkHref":38557},"#MJX-75-TEX-I-1D464",[25022,38559,38560,38565,38570],{"dataMmlNode":27203,"transform":37564,"dataMjxTexclass":27205},[25022,38561,38562],{"dataMmlNode":25031},[25033,38563],{"dataC":25035,"xLinkHref":38564},"#MJX-75-TEX-N-28",[25022,38566,38567],{"dataMmlNode":25942,"transform":25040},[25033,38568],{"dataC":27210,"xLinkHref":38569},"#MJX-75-TEX-I-1D456",[25022,38571,38572],{"dataMmlNode":25031,"transform":32577},[25033,38573],{"dataC":25108,"xLinkHref":38574},"#MJX-75-TEX-N-29"," depend on ",[24923,38577,38579],{"className":38578,"jax":24927},[24926],[24929,38580,38581,38586],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,38582,38583],{},[24941,38584],{"id":38585,"d":32148},"MJX-76-TEX-I-1D465",[25022,38587,38588],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38589,38590],{"dataMmlNode":25028},[25022,38591,38592],{"dataMmlNode":25942},[25033,38593],{"dataC":32181,"xLinkHref":38594},"#MJX-76-TEX-I-1D465"," (the specific thing you are trying to predict right now, ",[106,38597,38598],{},"you must recalculate",[106,38600,38601],{},[24923,38602,38604],{"className":38603,"jax":24927},[24926],[24929,38605,38606,38611],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,38607,38608],{},[24941,38609],{"id":38610,"d":32951},"MJX-77-TEX-I-1D703",[25022,38612,38613],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38614,38615],{"dataMmlNode":25028},[25022,38616,38617],{"dataMmlNode":25942},[25033,38618],{"dataC":33000,"xLinkHref":38619},"#MJX-77-TEX-I-1D703",[106,38621,38622],{},"from scratch every single time you want to make a new prediction.","), you cannot pre-calculate your ",[24923,38625,38627],{"className":38626,"jax":24927},[24926],[24929,38628,38629,38634],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,38630,38631],{},[24941,38632],{"id":38633,"d":32951},"MJX-78-TEX-I-1D703",[25022,38635,38636],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38637,38638],{"dataMmlNode":25028},[25022,38639,38640],{"dataMmlNode":25942},[25033,38641],{"dataC":33000,"xLinkHref":38642},"#MJX-78-TEX-I-1D703"," values. You have to keep the entire training dataset in memory and calculate a brand new set of ",[24923,38645,38647],{"className":38646,"jax":24927},[24926],[24929,38648,38649,38654],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,38650,38651],{},[24941,38652],{"id":38653,"d":32951},"MJX-79-TEX-I-1D703",[25022,38655,38656],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38657,38658],{"dataMmlNode":25028},[25022,38659,38660],{"dataMmlNode":25942},[25033,38661],{"dataC":33000,"xLinkHref":38662},"#MJX-79-TEX-I-1D703"," values for ",[106,38665,38666],{},"every single prediction you make",[6964,38668,38670],{"id":38669},"logistic-regression","Logistic Regression",[16,38672],{},[12,38674,38675,38676,38679],{},"Everything above assumes y is continuous. But what if y can only be ",[106,38677,38678],{},"0 or 1"," (e.g., spam vs. not spam, tumor is malignant vs. benign)?",[12,38681,38682,38683,38686,38687,7024],{},"We can't use a plain linear function because it can output values outside ",[75,38684,38685],{},"0, 1",". Instead, we wrap it in the ",[106,38688,38689],{},"sigmoid (logistic) function",[24923,38691,38693],{"className":38692,"jax":24927,"display":13594},[24926],[24929,38694,38699,38735],{"style":38695,"xmlns":24932,"width":38696,"height":38697,"role":1930,"focusable":8058,"viewBox":38698,"xmlnsXLink":24936},"vertical-align: -1.977ex;","18.286ex","5.013ex","0 -1342 8082.6 2215.9",[24938,38700,38701,38704,38707,38710,38713,38716,38719,38722,38725,38728,38731],{},[24941,38702],{"id":38703,"d":31530},"MJX-80-TEX-I-210E",[24941,38705],{"id":38706,"d":32951},"MJX-80-TEX-I-1D703",[24941,38708],{"id":38709,"d":24944},"MJX-80-TEX-N-28",[24941,38711],{"id":38712,"d":32148},"MJX-80-TEX-I-1D465",[24941,38714],{"id":38715,"d":24988},"MJX-80-TEX-N-29",[24941,38717],{"id":38718,"d":24972},"MJX-80-TEX-N-3D",[24941,38720],{"id":38721,"d":25914},"MJX-80-TEX-N-31",[24941,38723],{"id":38724,"d":30313},"MJX-80-TEX-N-2B",[24941,38726],{"id":38727,"d":27397},"MJX-80-TEX-I-1D452",[24941,38729],{"id":38730,"d":25910},"MJX-80-TEX-N-2212",[24941,38732],{"id":38733,"d":38734},"MJX-80-TEX-I-1D447","M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z",[25022,38736,38737],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38738,38739,38752,38758,38764,38770,38776],{"dataMmlNode":25028},[25022,38740,38741,38746],{"dataMmlNode":26023},[25022,38742,38743],{"dataMmlNode":25942},[25033,38744],{"dataC":31539,"xLinkHref":38745},"#MJX-80-TEX-I-210E",[25022,38747,38749],{"dataMmlNode":25942,"transform":38748},"translate(609,-150) scale(0.707)",[25033,38750],{"dataC":33000,"xLinkHref":38751},"#MJX-80-TEX-I-1D703",[25022,38753,38755],{"dataMmlNode":25031,"transform":38754},"translate(990.6,0)",[25033,38756],{"dataC":25035,"xLinkHref":38757},"#MJX-80-TEX-N-28",[25022,38759,38761],{"dataMmlNode":25942,"transform":38760},"translate(1379.6,0)",[25033,38762],{"dataC":32181,"xLinkHref":38763},"#MJX-80-TEX-I-1D465",[25022,38765,38767],{"dataMmlNode":25031,"transform":38766},"translate(1951.6,0)",[25033,38768],{"dataC":25108,"xLinkHref":38769},"#MJX-80-TEX-N-29",[25022,38771,38773],{"dataMmlNode":25031,"transform":38772},"translate(2618.4,0)",[25033,38774],{"dataC":25078,"xLinkHref":38775},"#MJX-80-TEX-N-3D",[25022,38777,38779,38785,38832],{"dataMmlNode":27622,"transform":38778},"translate(3674.2,0)",[25022,38780,38782],{"dataMmlNode":26031,"transform":38781},"translate(1954.2,676)",[25033,38783],{"dataC":26035,"xLinkHref":38784},"#MJX-80-TEX-N-31",[25022,38786,38788,38792,38798],{"dataMmlNode":27626,"transform":38787},"translate(220,-791.9)",[25022,38789,38790],{"dataMmlNode":26031},[25033,38791],{"dataC":26035,"xLinkHref":38784},[25022,38793,38795],{"dataMmlNode":25031,"transform":38794},"translate(722.2,0)",[25033,38796],{"dataC":30393,"xLinkHref":38797},"#MJX-80-TEX-N-2B",[25022,38799,38801,38806],{"dataMmlNode":31565,"transform":38800},"translate(1722.4,0)",[25022,38802,38803],{"dataMmlNode":25942},[25033,38804],{"dataC":27596,"xLinkHref":38805},"#MJX-80-TEX-I-1D452",[25022,38807,38809,38814,38827],{"dataMmlNode":27203,"transform":38808,"dataMjxTexclass":27205},"translate(499,289) scale(0.707)",[25022,38810,38811],{"dataMmlNode":25031},[25033,38812],{"dataC":26019,"xLinkHref":38813},"#MJX-80-TEX-N-2212",[25022,38815,38816,38820],{"dataMmlNode":31565,"transform":25049},[25022,38817,38818],{"dataMmlNode":25942},[25033,38819],{"dataC":33000,"xLinkHref":38751},[25022,38821,38823],{"dataMmlNode":25942,"transform":38822},"translate(502,289) scale(0.707)",[25033,38824],{"dataC":38825,"xLinkHref":38826},"1D447","#MJX-80-TEX-I-1D447",[25022,38828,38830],{"dataMmlNode":25942,"transform":38829},"translate(1827.8,0)",[25033,38831],{"dataC":32181,"xLinkHref":38763},[27668,38833],{"width":38834,"height":27671,"x":27672,"y":27673},4168.4,[12,38836,38837,38838,38913],{},"The sigmoid squashes any real number into the range (0, 1), which we interpret as a probability. If ",[24923,38839,38841],{"className":38840,"jax":24927},[24926],[24929,38842,38845,38872],{"style":24931,"xmlns":24932,"width":38843,"height":24934,"role":1930,"focusable":8058,"viewBox":38844,"xmlnsXLink":24936},"10.266ex","0 -750 4537.6 1000",[24938,38846,38847,38850,38853,38856,38859,38863,38866,38869],{},[24941,38848],{"id":38849,"d":31530},"MJX-81-TEX-I-210E",[24941,38851],{"id":38852,"d":24944},"MJX-81-TEX-N-28",[24941,38854],{"id":38855,"d":32148},"MJX-81-TEX-I-1D465",[24941,38857],{"id":38858,"d":24988},"MJX-81-TEX-N-29",[24941,38860],{"id":38861,"d":38862},"MJX-81-TEX-N-2265","M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z",[24941,38864],{"id":38865,"d":25935},"MJX-81-TEX-N-30",[24941,38867],{"id":38868,"d":26357},"MJX-81-TEX-N-2E",[24941,38870],{"id":38871,"d":26660},"MJX-81-TEX-N-35",[25022,38873,38874],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38875,38876,38881,38886,38891,38896,38902],{"dataMmlNode":25028},[25022,38877,38878],{"dataMmlNode":25942},[25033,38879],{"dataC":31539,"xLinkHref":38880},"#MJX-81-TEX-I-210E",[25022,38882,38883],{"dataMmlNode":25031,"transform":32791},[25033,38884],{"dataC":25035,"xLinkHref":38885},"#MJX-81-TEX-N-28",[25022,38887,38888],{"dataMmlNode":25942,"transform":32797},[25033,38889],{"dataC":32181,"xLinkHref":38890},"#MJX-81-TEX-I-1D465",[25022,38892,38893],{"dataMmlNode":25031,"transform":32803},[25033,38894],{"dataC":25108,"xLinkHref":38895},"#MJX-81-TEX-N-29",[25022,38897,38898],{"dataMmlNode":25031,"transform":32989},[25033,38899],{"dataC":38900,"xLinkHref":38901},"2265","#MJX-81-TEX-N-2265",[25022,38903,38904,38907,38910],{"dataMmlNode":26031,"transform":32995},[25033,38905],{"dataC":26103,"xLinkHref":38906},"#MJX-81-TEX-N-30",[25033,38908],{"dataC":26392,"xLinkHref":38909,"transform":26394},"#MJX-81-TEX-N-2E",[25033,38911],{"dataC":26717,"xLinkHref":38912,"transform":25049},"#MJX-81-TEX-N-35",", we predict y = 1; otherwise, y = 0.",[25631,38915,38917,38920],{"className":38916},[13441,13442],[25635,38918,38919],{},"The Problem with Linear Regression for Classification",[13438,38921,38923,39022,39031,39075,39186,39190,39233,39244,39381,39489,39698],{"className":38922},[25641],[12,38924,38925,38926,38966,38967,39002,39003,16419],{},"Imagine you are a doctor trying to predict if a tumor is Malignant (",[24923,38927,38929],{"className":38928,"jax":24927},[24926],[24929,38930,38934,38945],{"style":29805,"xmlns":24932,"width":38931,"height":38932,"role":1930,"focusable":8058,"viewBox":38933,"xmlnsXLink":24936},"5.257ex","1.971ex","0 -666 2323.6 871",[24938,38935,38936,38939,38942],{},[24941,38937],{"id":38938,"d":25893},"MJX-82-TEX-I-1D466",[24941,38940],{"id":38941,"d":24972},"MJX-82-TEX-N-3D",[24941,38943],{"id":38944,"d":25914},"MJX-82-TEX-N-31",[25022,38946,38947],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38948,38949,38954,38960],{"dataMmlNode":25028},[25022,38950,38951],{"dataMmlNode":25942},[25033,38952],{"dataC":25987,"xLinkHref":38953},"#MJX-82-TEX-I-1D466",[25022,38955,38957],{"dataMmlNode":25031,"transform":38956},"translate(767.8,0)",[25033,38958],{"dataC":25078,"xLinkHref":38959},"#MJX-82-TEX-N-3D",[25022,38961,38963],{"dataMmlNode":26031,"transform":38962},"translate(1823.6,0)",[25033,38964],{"dataC":26035,"xLinkHref":38965},"#MJX-82-TEX-N-31",") or Benign (",[24923,38968,38970],{"className":38969,"jax":24927},[24926],[24929,38971,38972,38983],{"style":29805,"xmlns":24932,"width":38931,"height":38932,"role":1930,"focusable":8058,"viewBox":38933,"xmlnsXLink":24936},[24938,38973,38974,38977,38980],{},[24941,38975],{"id":38976,"d":25893},"MJX-83-TEX-I-1D466",[24941,38978],{"id":38979,"d":24972},"MJX-83-TEX-N-3D",[24941,38981],{"id":38982,"d":25935},"MJX-83-TEX-N-30",[25022,38984,38985],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,38986,38987,38992,38997],{"dataMmlNode":25028},[25022,38988,38989],{"dataMmlNode":25942},[25033,38990],{"dataC":25987,"xLinkHref":38991},"#MJX-83-TEX-I-1D466",[25022,38993,38994],{"dataMmlNode":25031,"transform":38956},[25033,38995],{"dataC":25078,"xLinkHref":38996},"#MJX-83-TEX-N-3D",[25022,38998,38999],{"dataMmlNode":26031,"transform":38962},[25033,39000],{"dataC":26103,"xLinkHref":39001},"#MJX-83-TEX-N-30",") based on its size (",[24923,39004,39006],{"className":39005,"jax":24927},[24926],[24929,39007,39008,39013],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,39009,39010],{},[24941,39011],{"id":39012,"d":32148},"MJX-84-TEX-I-1D465",[25022,39014,39015],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39016,39017],{"dataMmlNode":25028},[25022,39018,39019],{"dataMmlNode":25942},[25033,39020],{"dataC":32181,"xLinkHref":39021},"#MJX-84-TEX-I-1D465",[12,39023,39024,39025,39027,39028],{},"If you use standard ",[106,39026,32858],{},", you fit a straight line through the data. You might say: ",[7289,39029,39030],{},"\"If the line outputs a value greater than 0.5, I will predict 1 (Malignant). If it's less than 0.5, I predict 0 (Benign).\"",[12,39032,39033,39034,39037,39038,39070,39071,39074],{},"This works okay until a patient comes in with a ",[106,39035,39036],{},"massive"," tumor.\nBecause linear regression is a straight line, it will try to accommodate this massive outlier. The line will tilt upwards heavily. Suddenly, your ",[24923,39039,39041],{"className":39040,"jax":24927},[24926],[24929,39042,39044,39055],{"style":27778,"xmlns":24932,"width":29568,"height":35776,"role":1930,"focusable":8058,"viewBox":39043,"xmlnsXLink":24936},"0 -666 1278 688",[24938,39045,39046,39049,39052],{},[24941,39047],{"id":39048,"d":25935},"MJX-85-TEX-N-30",[24941,39050],{"id":39051,"d":26357},"MJX-85-TEX-N-2E",[24941,39053],{"id":39054,"d":26660},"MJX-85-TEX-N-35",[25022,39056,39057],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39058,39059],{"dataMmlNode":25028},[25022,39060,39061,39064,39067],{"dataMmlNode":26031},[25033,39062],{"dataC":26103,"xLinkHref":39063},"#MJX-85-TEX-N-30",[25033,39065],{"dataC":26392,"xLinkHref":39066,"transform":26394},"#MJX-85-TEX-N-2E",[25033,39068],{"dataC":26717,"xLinkHref":39069,"transform":25049},"#MJX-85-TEX-N-35"," threshold shifts, and tumors that you ",[7289,39072,39073],{},"used"," to correctly predict as Malignant are now being predicted as Benign.",[12,39076,39077,39078,7067,39127,39185],{},"Even worse, the straight line might output a value like ",[24923,39079,39081],{"className":39080,"jax":24927},[24926],[24929,39082,39085,39102],{"style":29805,"xmlns":24932,"width":39083,"height":38932,"role":1930,"focusable":8058,"viewBox":39084,"xmlnsXLink":24936},"7.017ex","0 -666 3101.6 871",[24938,39086,39087,39090,39093,39096,39099],{},[24941,39088],{"id":39089,"d":25893},"MJX-86-TEX-I-1D466",[24941,39091],{"id":39092,"d":24972},"MJX-86-TEX-N-3D",[24941,39094],{"id":39095,"d":26455},"MJX-86-TEX-N-33",[24941,39097],{"id":39098,"d":26357},"MJX-86-TEX-N-2E",[24941,39100],{"id":39101,"d":26660},"MJX-86-TEX-N-35",[25022,39103,39104],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39105,39106,39111,39116],{"dataMmlNode":25028},[25022,39107,39108],{"dataMmlNode":25942},[25033,39109],{"dataC":25987,"xLinkHref":39110},"#MJX-86-TEX-I-1D466",[25022,39112,39113],{"dataMmlNode":25031,"transform":38956},[25033,39114],{"dataC":25078,"xLinkHref":39115},"#MJX-86-TEX-N-3D",[25022,39117,39118,39121,39124],{"dataMmlNode":26031,"transform":38962},[25033,39119],{"dataC":26486,"xLinkHref":39120},"#MJX-86-TEX-N-33",[25033,39122],{"dataC":26392,"xLinkHref":39123,"transform":26394},"#MJX-86-TEX-N-2E",[25033,39125],{"dataC":26717,"xLinkHref":39126,"transform":25049},"#MJX-86-TEX-N-35",[24923,39128,39130],{"className":39129,"jax":24927},[24926],[24929,39131,39134,39154],{"style":29805,"xmlns":24932,"width":39132,"height":38932,"role":1930,"focusable":8058,"viewBox":39133,"xmlnsXLink":24936},"8.777ex","0 -666 3879.6 871",[24938,39135,39136,39139,39142,39145,39148,39151],{},[24941,39137],{"id":39138,"d":25893},"MJX-87-TEX-I-1D466",[24941,39140],{"id":39141,"d":24972},"MJX-87-TEX-N-3D",[24941,39143],{"id":39144,"d":25910},"MJX-87-TEX-N-2212",[24941,39146],{"id":39147,"d":25914},"MJX-87-TEX-N-31",[24941,39149],{"id":39150,"d":26357},"MJX-87-TEX-N-2E",[24941,39152],{"id":39153,"d":25928},"MJX-87-TEX-N-32",[25022,39155,39156],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39157,39158,39163,39168,39173],{"dataMmlNode":25028},[25022,39159,39160],{"dataMmlNode":25942},[25033,39161],{"dataC":25987,"xLinkHref":39162},"#MJX-87-TEX-I-1D466",[25022,39164,39165],{"dataMmlNode":25031,"transform":38956},[25033,39166],{"dataC":25078,"xLinkHref":39167},"#MJX-87-TEX-N-3D",[25022,39169,39170],{"dataMmlNode":25031,"transform":38962},[25033,39171],{"dataC":26019,"xLinkHref":39172},"#MJX-87-TEX-N-2212",[25022,39174,39176,39179,39182],{"dataMmlNode":26031,"transform":39175},"translate(2601.6,0)",[25033,39177],{"dataC":26035,"xLinkHref":39178},"#MJX-87-TEX-N-31",[25033,39180],{"dataC":26392,"xLinkHref":39181,"transform":26394},"#MJX-87-TEX-N-2E",[25033,39183],{"dataC":26058,"xLinkHref":39184,"transform":25049},"#MJX-87-TEX-N-32",". What does a \"3.5\" or \"-1.2\" mean when the answer can only be 0 or 1? It's nonsense.",[7166,39187,39189],{"id":39188},"the-solution-the-sigmoid-function","The Solution: The Sigmoid Function",[12,39191,39192,39193,39212,39213,39232],{},"Instead of a straight line, we want a curve that flatlines at ",[24923,39194,39196],{"className":39195,"jax":24927},[24926],[24929,39197,39198,39203],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,39199,39200],{},[24941,39201],{"id":39202,"d":25935},"MJX-88-TEX-N-30",[25022,39204,39205],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39206,39207],{"dataMmlNode":25028},[25022,39208,39209],{"dataMmlNode":26031},[25033,39210],{"dataC":26103,"xLinkHref":39211},"#MJX-88-TEX-N-30"," and flatlines at ",[24923,39214,39216],{"className":39215,"jax":24927},[24926],[24929,39217,39218,39223],{"style":27802,"xmlns":24932,"width":35168,"height":35169,"role":1930,"focusable":8058,"viewBox":35170,"xmlnsXLink":24936},[24938,39219,39220],{},[24941,39221],{"id":39222,"d":25914},"MJX-89-TEX-N-31",[25022,39224,39225],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39226,39227],{"dataMmlNode":25028},[25022,39228,39229],{"dataMmlNode":26031},[25033,39230],{"dataC":26035,"xLinkHref":39231},"#MJX-89-TEX-N-31",". We want the \"S-shape\" curve.",[12,39234,39235],{},[32509,39236,39237,39241],{},[1930,39238],{"alt":39239,"src":39240},"0.78","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1777302135699-Screenshot_2026-04-27_at_11.02.04_PM.png",[32516,39242,39243],{},"Sigmoid Function Graph",[12,39245,39246,39247],{},"This is what the Sigmoid function does:\n",[24923,39248,39250],{"className":39249,"jax":24927},[24926],[24929,39251,39256,39291],{"style":39252,"xmlns":24932,"width":39253,"height":39254,"role":1930,"focusable":8058,"viewBox":39255,"xmlnsXLink":24936},"vertical-align: -1.324ex;","15.179ex","3.281ex","0 -864.9 6709.1 1450.3",[24938,39257,39258,39261,39264,39267,39270,39273,39276,39279,39282,39285,39288],{},[24941,39259],{"id":39260,"d":31530},"MJX-90-TEX-I-210E",[24941,39262],{"id":39263,"d":32951},"MJX-90-TEX-I-1D703",[24941,39265],{"id":39266,"d":24944},"MJX-90-TEX-N-28",[24941,39268],{"id":39269,"d":32148},"MJX-90-TEX-I-1D465",[24941,39271],{"id":39272,"d":24988},"MJX-90-TEX-N-29",[24941,39274],{"id":39275,"d":24972},"MJX-90-TEX-N-3D",[24941,39277],{"id":39278,"d":25914},"MJX-90-TEX-N-31",[24941,39280],{"id":39281,"d":30313},"MJX-90-TEX-N-2B",[24941,39283],{"id":39284,"d":27397},"MJX-90-TEX-I-1D452",[24941,39286],{"id":39287,"d":25910},"MJX-90-TEX-N-2212",[24941,39289],{"id":39290,"d":38734},"MJX-90-TEX-I-1D447",[25022,39292,39293],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39294,39295,39307,39312,39317,39322,39327],{"dataMmlNode":25028},[25022,39296,39297,39302],{"dataMmlNode":26023},[25022,39298,39299],{"dataMmlNode":25942},[25033,39300],{"dataC":31539,"xLinkHref":39301},"#MJX-90-TEX-I-210E",[25022,39303,39304],{"dataMmlNode":25942,"transform":38748},[25033,39305],{"dataC":33000,"xLinkHref":39306},"#MJX-90-TEX-I-1D703",[25022,39308,39309],{"dataMmlNode":25031,"transform":38754},[25033,39310],{"dataC":25035,"xLinkHref":39311},"#MJX-90-TEX-N-28",[25022,39313,39314],{"dataMmlNode":25942,"transform":38760},[25033,39315],{"dataC":32181,"xLinkHref":39316},"#MJX-90-TEX-I-1D465",[25022,39318,39319],{"dataMmlNode":25031,"transform":38766},[25033,39320],{"dataC":25108,"xLinkHref":39321},"#MJX-90-TEX-N-29",[25022,39323,39324],{"dataMmlNode":25031,"transform":38772},[25033,39325],{"dataC":25078,"xLinkHref":39326},"#MJX-90-TEX-N-3D",[25022,39328,39329,39335,39378],{"dataMmlNode":27622,"transform":38778},[25022,39330,39332],{"dataMmlNode":26031,"transform":39331},"translate(1340.7,394) scale(0.707)",[25033,39333],{"dataC":26035,"xLinkHref":39334},"#MJX-90-TEX-N-31",[25022,39336,39338,39342,39347],{"dataMmlNode":27626,"transform":39337},"translate(220,-527.4) scale(0.707)",[25022,39339,39340],{"dataMmlNode":26031},[25033,39341],{"dataC":26035,"xLinkHref":39334},[25022,39343,39344],{"dataMmlNode":25031,"transform":26394},[25033,39345],{"dataC":30393,"xLinkHref":39346},"#MJX-90-TEX-N-2B",[25022,39348,39349,39354],{"dataMmlNode":31565,"transform":26835},[25022,39350,39351],{"dataMmlNode":25942},[25033,39352],{"dataC":27596,"xLinkHref":39353},"#MJX-90-TEX-I-1D452",[25022,39355,39356,39361,39373],{"dataMmlNode":27203,"transform":38808,"dataMjxTexclass":27205},[25022,39357,39358],{"dataMmlNode":25031},[25033,39359],{"dataC":26019,"xLinkHref":39360},"#MJX-90-TEX-N-2212",[25022,39362,39363,39367],{"dataMmlNode":31565,"transform":25049},[25022,39364,39365],{"dataMmlNode":25942},[25033,39366],{"dataC":33000,"xLinkHref":39306},[25022,39368,39370],{"dataMmlNode":25942,"transform":39369},"translate(502,289)",[25033,39371],{"dataC":38825,"xLinkHref":39372},"#MJX-90-TEX-I-1D447",[25022,39374,39376],{"dataMmlNode":25942,"transform":39375},"translate(2034,0)",[25033,39377],{"dataC":32181,"xLinkHref":39316},[27668,39379],{"width":39380,"height":27671,"x":27672,"y":27673},2794.9,[12,39382,39383,39384,39426,39427,39430,39431,10720,39462,16419],{},"Think of ",[24923,39385,39387],{"className":39386,"jax":24927},[24926],[24929,39388,39392,39403],{"style":25196,"xmlns":24932,"width":39389,"height":39390,"role":1930,"focusable":8058,"viewBox":39391,"xmlnsXLink":24936},"3.669ex","1.929ex","0 -841.7 1621.8 852.7",[24938,39393,39394,39397,39400],{},[24941,39395],{"id":39396,"d":32951},"MJX-91-TEX-I-1D703",[24941,39398],{"id":39399,"d":38734},"MJX-91-TEX-I-1D447",[24941,39401],{"id":39402,"d":32148},"MJX-91-TEX-I-1D465",[25022,39404,39405],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39406,39407,39420],{"dataMmlNode":25028},[25022,39408,39409,39414],{"dataMmlNode":31565},[25022,39410,39411],{"dataMmlNode":25942},[25033,39412],{"dataC":33000,"xLinkHref":39413},"#MJX-91-TEX-I-1D703",[25022,39415,39417],{"dataMmlNode":25942,"transform":39416},"translate(502,363) scale(0.707)",[25033,39418],{"dataC":38825,"xLinkHref":39419},"#MJX-91-TEX-I-1D447",[25022,39421,39423],{"dataMmlNode":25942,"transform":39422},"translate(1049.8,0)",[25033,39424],{"dataC":32181,"xLinkHref":39425},"#MJX-91-TEX-I-1D465"," as a ",[106,39428,39429],{},"\"raw confidence score\""," (which can be any number from ",[24923,39432,39434],{"className":39433,"jax":24927},[24926],[24929,39435,39438,39447],{"style":26868,"xmlns":24932,"width":39436,"height":26870,"role":1930,"focusable":8058,"viewBox":39437,"xmlnsXLink":24936},"4.023ex","0 -583 1778 665",[24938,39439,39440,39443],{},[24941,39441],{"id":39442,"d":25910},"MJX-92-TEX-N-2212",[24941,39444],{"id":39445,"d":39446},"MJX-92-TEX-N-221E","M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z",[25022,39448,39449],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39450,39451,39456],{"dataMmlNode":25028},[25022,39452,39453],{"dataMmlNode":25031},[25033,39454],{"dataC":26019,"xLinkHref":39455},"#MJX-92-TEX-N-2212",[25022,39457,39458],{"dataMmlNode":25942,"transform":25049},[25033,39459],{"dataC":39460,"xLinkHref":39461},"221E","#MJX-92-TEX-N-221E",[24923,39463,39465],{"className":39464,"jax":24927},[24926],[24929,39466,39467,39475],{"style":26868,"xmlns":24932,"width":39436,"height":26870,"role":1930,"focusable":8058,"viewBox":39437,"xmlnsXLink":24936},[24938,39468,39469,39472],{},[24941,39470],{"id":39471,"d":30313},"MJX-93-TEX-N-2B",[24941,39473],{"id":39474,"d":39446},"MJX-93-TEX-N-221E",[25022,39476,39477],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39478,39479,39484],{"dataMmlNode":25028},[25022,39480,39481],{"dataMmlNode":25031},[25033,39482],{"dataC":30393,"xLinkHref":39483},"#MJX-93-TEX-N-2B",[25022,39485,39486],{"dataMmlNode":25942,"transform":25049},[25033,39487],{"dataC":39460,"xLinkHref":39488},"#MJX-93-TEX-N-221E",[23,39490,39491,39546,39619],{},[26,39492,39493,39494,39513,39514,39545],{},"If the raw score is ",[24923,39495,39497],{"className":39496,"jax":24927},[24926],[24929,39498,39499,39504],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,39500,39501],{},[24941,39502],{"id":39503,"d":25935},"MJX-94-TEX-N-30",[25022,39505,39506],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39507,39508],{"dataMmlNode":25028},[25022,39509,39510],{"dataMmlNode":26031},[25033,39511],{"dataC":26103,"xLinkHref":39512},"#MJX-94-TEX-N-30",", the sigmoid turns it into exactly ",[24923,39515,39517],{"className":39516,"jax":24927},[24926],[24929,39518,39519,39530],{"style":27778,"xmlns":24932,"width":29568,"height":35776,"role":1930,"focusable":8058,"viewBox":39043,"xmlnsXLink":24936},[24938,39520,39521,39524,39527],{},[24941,39522],{"id":39523,"d":25935},"MJX-95-TEX-N-30",[24941,39525],{"id":39526,"d":26357},"MJX-95-TEX-N-2E",[24941,39528],{"id":39529,"d":26660},"MJX-95-TEX-N-35",[25022,39531,39532],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39533,39534],{"dataMmlNode":25028},[25022,39535,39536,39539,39542],{"dataMmlNode":26031},[25033,39537],{"dataC":26103,"xLinkHref":39538},"#MJX-95-TEX-N-30",[25033,39540],{"dataC":26392,"xLinkHref":39541,"transform":26394},"#MJX-95-TEX-N-2E",[25033,39543],{"dataC":26717,"xLinkHref":39544,"transform":25049},"#MJX-95-TEX-N-35"," (50% probability).",[26,39547,39548,39549,39578,39579,39618],{},"If the raw score is a huge positive number (like 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688",[24938,39587,39588,39591,39594],{},[24941,39589],{"id":39590,"d":25935},"MJX-97-TEX-N-30",[24941,39592],{"id":39593,"d":26357},"MJX-97-TEX-N-2E",[24941,39595],{"id":39596,"d":29754},"MJX-97-TEX-N-39",[25022,39598,39599],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,39600,39601],{"dataMmlNode":25028},[25022,39602,39603,39606,39609,39612,39614,39616],{"dataMmlNode":26031},[25033,39604],{"dataC":26103,"xLinkHref":39605},"#MJX-97-TEX-N-30",[25033,39607],{"dataC":26392,"xLinkHref":39608,"transform":26394},"#MJX-97-TEX-N-2E",[25033,39610],{"dataC":29772,"xLinkHref":39611,"transform":25049},"#MJX-97-TEX-N-39",[25033,39613],{"dataC":29772,"xLinkHref":39611,"transform":26835},[25033,39615],{"dataC":29772,"xLinkHref":39611,"transform":30504},[25033,39617],{"dataC":29772,"xLinkHref":39611,"transform":31438}," (99.99% probability it is Malignant).",[26,39620,39621,39622,39578,39660,39697],{},"If the raw score is a huge negative number (like 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form is identical, but remember that h(x) is now the sigmoid function, not a linear function. This elegant symmetry is one of the beautiful things about these algorithms.",[25631,40523,40525,40528],{"className":40524},[13441,13442],[25635,40526,40527],{},"The Log-Likelihood Cost Function",[13438,40529,40531,40688,40773,40796,40800,40803,40894,40901,40905,41068,41239,41242,41245,41500,41503,42035,42042,42044,42047,42093],{"className":40530},[25641],[12,40532,40533,40534,40687],{},"In Linear Regression, we used the Least-Squares cost function (minimising the squared error). Because the prediction is just a straight line, Least-Squares naturally creates a perfect \"U-shaped\" error graph (a parabola) ",[24923,40535,40537],{"className":40536,"jax":24927},[24926],[24929,40538,40541,40596],{"style":24931,"xmlns":24932,"width":40539,"height":31733,"role":1930,"focusable":8058,"viewBox":40540,"xmlnsXLink":24936},"22.335ex","0 -833.9 9872 1083.9",[24938,40542,40543,40546,40550,40553,40556,40559,40562,40565,40568,40571,40574,40577,40581,40584,40587,40590,40593],{},[24941,40544],{"id":40545,"d":24944},"MJX-104-TEX-N-28",[24941,40547],{"id":40548,"d":40549},"MJX-104-TEX-N-50","M130 622Q123 629 119 631T103 634T60 637H27V683H214Q237 683 276 683T331 684Q419 684 471 671T567 616Q624 563 624 489Q624 421 573 372T451 307Q429 302 328 301H234V181Q234 62 237 58Q245 47 304 46H337V0H326Q305 3 182 3Q47 3 38 0H27V46H60Q102 47 111 49T130 61V622ZM507 488Q507 514 506 528T500 564T483 597T450 620T397 635Q385 637 307 637H286Q237 637 234 628Q231 624 231 483V342H302H339Q390 342 423 349T481 382Q507 411 507 488Z",[24941,40551],{"id":40552,"d":25016},"MJX-104-TEX-N-72",[24941,40554],{"id":40555,"d":25259},"MJX-104-TEX-N-65",[24941,40557],{"id":40558,"d":25008},"MJX-104-TEX-N-64",[24941,40560],{"id":40561,"d":25004},"MJX-104-TEX-N-69",[24941,40563],{"id":40564,"d":25266},"MJX-104-TEX-N-63",[24941,40566],{"id":40567,"d":24956},"MJX-104-TEX-N-74",[24941,40569],{"id":40570,"d":24964},"MJX-104-TEX-N-6F",[24941,40572],{"id":40573,"d":24980},"MJX-104-TEX-N-6E",[24941,40575],{"id":40576,"d":25910},"MJX-104-TEX-N-2212",[24941,40578],{"id":40579,"d":40580},"MJX-104-TEX-N-41","M255 0Q240 3 140 3Q48 3 39 0H32V46H47Q119 49 139 88Q140 91 192 245T295 553T348 708Q351 716 366 716H376Q396 715 400 709Q402 707 508 390L617 67Q624 54 636 51T687 46H717V0H708Q699 3 581 3Q458 3 437 0H427V46H440Q510 46 510 64Q510 66 486 138L462 209H229L209 150Q189 91 189 85Q189 72 209 59T259 46H264V0H255ZM447 255L345 557L244 256Q244 255 345 255H447Z",[24941,40582],{"id":40583,"d":24952},"MJX-104-TEX-N-75",[24941,40585],{"id":40586,"d":25020},"MJX-104-TEX-N-61",[24941,40588],{"id":40589,"d":24960},"MJX-104-TEX-N-6C",[24941,40591],{"id":40592,"d":24988},"MJX-104-TEX-N-29",[24941,40594],{"id":40595,"d":25928},"MJX-104-TEX-N-32",[25022,40597,40598],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,40599,40600,40605,40646,40652,40674],{"dataMmlNode":25028},[25022,40601,40602],{"dataMmlNode":25031},[25033,40603],{"dataC":25035,"xLinkHref":40604},"#MJX-104-TEX-N-28",[25022,40606,40607,40611,40615,40619,40623,40627,40631,40635,40638,40642],{"dataMmlNode":25039,"transform":25040},[25033,40608],{"dataC":40609,"xLinkHref":40610},"50","#MJX-104-TEX-N-50",[25033,40612],{"dataC":25172,"xLinkHref":40613,"transform":40614},"#MJX-104-TEX-N-72","translate(681,0)",[25033,40616],{"dataC":25323,"xLinkHref":40617,"transform":40618},"#MJX-104-TEX-N-65","translate(1073,0)",[25033,40620],{"dataC":25144,"xLinkHref":40621,"transform":40622},"#MJX-104-TEX-N-64","translate(1517,0)",[25033,40624],{"dataC":25139,"xLinkHref":40625,"transform":40626},"#MJX-104-TEX-N-69","translate(2073,0)",[25033,40628],{"dataC":25331,"xLinkHref":40629,"transform":40630},"#MJX-104-TEX-N-63","translate(2351,0)",[25033,40632],{"dataC":25052,"xLinkHref":40633,"transform":40634},"#MJX-104-TEX-N-74","translate(2795,0)",[25033,40636],{"dataC":25139,"xLinkHref":40625,"transform":40637},"translate(3184,0)",[25033,40639],{"dataC":25062,"xLinkHref":40640,"transform":40641},"#MJX-104-TEX-N-6F","translate(3462,0)",[25033,40643],{"dataC":25092,"xLinkHref":40644,"transform":40645},"#MJX-104-TEX-N-6E","translate(3962,0)",[25022,40647,40649],{"dataMmlNode":25031,"transform":40648},"translate(5129.2,0)",[25033,40650],{"dataC":26019,"xLinkHref":40651},"#MJX-104-TEX-N-2212",[25022,40653,40655,40658,40660,40663,40667,40670],{"dataMmlNode":25039,"transform":40654},"translate(6129.4,0)",[25033,40656],{"dataC":31231,"xLinkHref":40657},"#MJX-104-TEX-N-41",[25033,40659],{"dataC":25331,"xLinkHref":40629,"transform":25131},[25033,40661],{"dataC":25052,"xLinkHref":40633,"transform":40662},"translate(1194,0)",[25033,40664],{"dataC":25047,"xLinkHref":40665,"transform":40666},"#MJX-104-TEX-N-75","translate(1583,0)",[25033,40668],{"dataC":25180,"xLinkHref":40669,"transform":25141},"#MJX-104-TEX-N-61",[25033,40671],{"dataC":25057,"xLinkHref":40672,"transform":40673},"#MJX-104-TEX-N-6C","translate(2639,0)",[25022,40675,40677,40682],{"dataMmlNode":31565,"transform":40676},"translate(9046.4,0)",[25022,40678,40679],{"dataMmlNode":25031},[25033,40680],{"dataC":25108,"xLinkHref":40681},"#MJX-104-TEX-N-29",[25022,40683,40684],{"dataMmlNode":26031,"transform":37418},[25033,40685],{"dataC":26058,"xLinkHref":40686},"#MJX-104-TEX-N-32",". Gradient Descent just acts like a marble rolling down the steep sides until it hits the single bottom. It never gets stuck because there are no flat surfaces.",[12,40689,40690,40691,40772],{},"But if we try to use Least-Squares for Logistic Regression, we have to shove the complicated Sigmoid fraction ",[24923,40692,40694],{"className":40693,"jax":24927},[24926],[24929,40695,40700,40720],{"style":40696,"xmlns":24932,"width":40697,"height":40698,"role":1930,"focusable":8058,"viewBox":40699,"xmlnsXLink":24936},"vertical-align: -1.029ex;","5.976ex","2.986ex","0 -864.9 2641.4 1319.8",[24938,40701,40702,40705,40708,40711,40714,40717],{},[24941,40703],{"id":40704,"d":25914},"MJX-105-TEX-N-31",[24941,40706],{"id":40707,"d":30313},"MJX-105-TEX-N-2B",[24941,40709],{"id":40710,"d":27397},"MJX-105-TEX-I-1D452",[24941,40712],{"id":40713,"d":25910},"MJX-105-TEX-N-2212",[24941,40715],{"id":40716,"d":32951},"MJX-105-TEX-I-1D703",[24941,40718],{"id":40719,"d":32148},"MJX-105-TEX-I-1D465",[25022,40721,40722],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,40723,40724],{"dataMmlNode":25028},[25022,40725,40726,40732,40769],{"dataMmlNode":27622},[25022,40727,40729],{"dataMmlNode":26031,"transform":40728},"translate(1143.9,394) scale(0.707)",[25033,40730],{"dataC":26035,"xLinkHref":40731},"#MJX-105-TEX-N-31",[25022,40733,40735,40739,40744],{"dataMmlNode":27626,"transform":40734},"translate(220,-396.9) scale(0.707)",[25022,40736,40737],{"dataMmlNode":26031},[25033,40738],{"dataC":26035,"xLinkHref":40731},[25022,40740,40741],{"dataMmlNode":25031,"transform":26394},[25033,40742],{"dataC":30393,"xLinkHref":40743},"#MJX-105-TEX-N-2B",[25022,40745,40746,40751],{"dataMmlNode":31565,"transform":26835},[25022,40747,40748],{"dataMmlNode":25942},[25033,40749],{"dataC":27596,"xLinkHref":40750},"#MJX-105-TEX-I-1D452",[25022,40752,40753,40758,40763],{"dataMmlNode":27203,"transform":38808,"dataMjxTexclass":27205},[25022,40754,40755],{"dataMmlNode":25031},[25033,40756],{"dataC":26019,"xLinkHref":40757},"#MJX-105-TEX-N-2212",[25022,40759,40760],{"dataMmlNode":25942,"transform":25049},[25033,40761],{"dataC":33000,"xLinkHref":40762},"#MJX-105-TEX-I-1D703",[25022,40764,40766],{"dataMmlNode":25942,"transform":40765},"translate(1247,0)",[25033,40767],{"dataC":32181,"xLinkHref":40768},"#MJX-105-TEX-I-1D465",[27668,40770],{"width":40771,"height":27671,"x":27672,"y":27673},2401.4," inside the squared formula.",[12,40774,40775,40776,40795],{},"The math breaks. The ",[24923,40777,40779],{"className":40778,"jax":24927},[24926],[24929,40780,40781,40786],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,40782,40783],{},[24941,40784],{"id":40785,"d":32951},"MJX-106-TEX-I-1D703",[25022,40787,40788],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,40789,40790],{"dataMmlNode":25028},[25022,40791,40792],{"dataMmlNode":25942},[25033,40793],{"dataC":33000,"xLinkHref":40794},"#MJX-106-TEX-I-1D703"," weight gets trapped in an exponent at the bottom of a fraction. This ruins the U-shape, stretching the arms out until they become completely flat, and crumpling the graph into a landscape with fake valleys (local minima). If the marble rolls onto a flat section (e.g., the model is confidently wrong), the slope becomes zero. Gradient Descent stops, and the model stops learning.",[26236,40797,40799],{"id":40798},"maximum-likelihood-estimation","Maximum Likelihood Estimation",[12,40801,40802],{},"Because we are dealing with 0s and 1s, statisticians stopped measuring \"distance\" (Least Squares) and started measuring \"probability\".",[12,40804,40805,40806,40809,40810,6959],{},"If you want to know the total chance that your model predicted Patient 1 correctly (90% sure) AND Patient 2 correctly (80% sure), you ",[106,40807,40808],{},"multiply"," the probabilities together: ",[24923,40811,40813],{"className":40812,"jax":24927},[24926],[24929,40814,40818,40844],{"style":26868,"xmlns":24932,"width":40815,"height":40816,"role":1930,"focusable":8058,"viewBox":40817,"xmlnsXLink":24936},"15.588ex","1.715ex","0 -676 6890 758",[24938,40819,40820,40823,40826,40829,40832,40835,40838,40841],{},[24941,40821],{"id":40822,"d":25935},"MJX-107-TEX-N-30",[24941,40824],{"id":40825,"d":26357},"MJX-107-TEX-N-2E",[24941,40827],{"id":40828,"d":29754},"MJX-107-TEX-N-39",[24941,40830],{"id":40831,"d":26907},"MJX-107-TEX-N-D7",[24941,40833],{"id":40834,"d":30471},"MJX-107-TEX-N-38",[24941,40836],{"id":40837,"d":24972},"MJX-107-TEX-N-3D",[24941,40839],{"id":40840,"d":26361},"MJX-107-TEX-N-37",[24941,40842],{"id":40843,"d":25928},"MJX-107-TEX-N-32",[25022,40845,40846],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,40847,40848,40859,40865,40875,40881],{"dataMmlNode":25028},[25022,40849,40850,40853,40856],{"dataMmlNode":26031},[25033,40851],{"dataC":26103,"xLinkHref":40852},"#MJX-107-TEX-N-30",[25033,40854],{"dataC":26392,"xLinkHref":40855,"transform":26394},"#MJX-107-TEX-N-2E",[25033,40857],{"dataC":29772,"xLinkHref":40858,"transform":25049},"#MJX-107-TEX-N-39",[25022,40860,40862],{"dataMmlNode":25031,"transform":40861},"translate(1500.2,0)",[25033,40863],{"dataC":26950,"xLinkHref":40864},"#MJX-107-TEX-N-D7",[25022,40866,40868,40870,40872],{"dataMmlNode":26031,"transform":40867},"translate(2500.4,0)",[25033,40869],{"dataC":26103,"xLinkHref":40852},[25033,40871],{"dataC":26392,"xLinkHref":40855,"transform":26394},[25033,40873],{"dataC":30502,"xLinkHref":40874,"transform":25049},"#MJX-107-TEX-N-38",[25022,40876,40878],{"dataMmlNode":25031,"transform":40877},"translate(4056.2,0)",[25033,40879],{"dataC":25078,"xLinkHref":40880},"#MJX-107-TEX-N-3D",[25022,40882,40884,40886,40888,40891],{"dataMmlNode":26031,"transform":40883},"translate(5112,0)",[25033,40885],{"dataC":26103,"xLinkHref":40852},[25033,40887],{"dataC":26392,"xLinkHref":40855,"transform":26394},[25033,40889],{"dataC":26397,"xLinkHref":40890,"transform":25049},"#MJX-107-TEX-N-37",[25033,40892],{"dataC":26058,"xLinkHref":40893,"transform":26835},"#MJX-107-TEX-N-32",[12,40895,40896,40897,40900],{},"This is called ",[106,40898,40899],{},"Maximum Likelihood",". We want the model to adjust its weights to make the final multiplied success rate as high as absolutely possible.",[26236,40902,40904],{"id":40903},"summoning-the-logarithm","Summoning the Logarithm",[12,40906,40907,40908,40986,40987,41047,41048,41067],{},"There is one problem: if you multiply 10,000 probabilities together (",[24923,40909,40911],{"className":40910,"jax":24927},[24926],[24929,40912,40915,40935],{"style":27778,"xmlns":24932,"width":40913,"height":35776,"role":1930,"focusable":8058,"viewBox":40914,"xmlnsXLink":24936},"18.366ex","0 -666 8117.6 688",[24938,40916,40917,40920,40923,40926,40929,40932],{},[24941,40918],{"id":40919,"d":25935},"MJX-108-TEX-N-30",[24941,40921],{"id":40922,"d":26357},"MJX-108-TEX-N-2E",[24941,40924],{"id":40925,"d":29754},"MJX-108-TEX-N-39",[24941,40927],{"id":40928,"d":26907},"MJX-108-TEX-N-D7",[24941,40930],{"id":40931,"d":30471},"MJX-108-TEX-N-38",[24941,40933],{"id":40934,"d":34448},"MJX-108-TEX-N-2026",[25022,40936,40937],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,40938,40939,40950,40955,40964,40969,40980],{"dataMmlNode":25028},[25022,40940,40941,40944,40947],{"dataMmlNode":26031},[25033,40942],{"dataC":26103,"xLinkHref":40943},"#MJX-108-TEX-N-30",[25033,40945],{"dataC":26392,"xLinkHref":40946,"transform":26394},"#MJX-108-TEX-N-2E",[25033,40948],{"dataC":29772,"xLinkHref":40949,"transform":25049},"#MJX-108-TEX-N-39",[25022,40951,40952],{"dataMmlNode":25031,"transform":40861},[25033,40953],{"dataC":26950,"xLinkHref":40954},"#MJX-108-TEX-N-D7",[25022,40956,40957,40959,40961],{"dataMmlNode":26031,"transform":40867},[25033,40958],{"dataC":26103,"xLinkHref":40943},[25033,40960],{"dataC":26392,"xLinkHref":40946,"transform":26394},[25033,40962],{"dataC":30502,"xLinkHref":40963,"transform":25049},"#MJX-108-TEX-N-38",[25022,40965,40967],{"dataMmlNode":25031,"transform":40966},"translate(4000.7,0)",[25033,40968],{"dataC":26950,"xLinkHref":40954},[25022,40970,40972,40974,40976,40978],{"dataMmlNode":26031,"transform":40971},"translate(5000.9,0)",[25033,40973],{"dataC":26103,"xLinkHref":40943},[25033,40975],{"dataC":26392,"xLinkHref":40946,"transform":26394},[25033,40977],{"dataC":29772,"xLinkHref":40949,"transform":25049},[25033,40979],{"dataC":29772,"xLinkHref":40949,"transform":26835},[25022,40981,40983],{"dataMmlNode":25031,"transform":40982},"translate(6945.6,0)",[25033,40984],{"dataC":34660,"xLinkHref":40985},"#MJX-108-TEX-N-2026","), the number becomes incredibly tiny (e.g., ",[24923,40988,40990],{"className":40989,"jax":24927},[24926],[24929,40991,40994,41005],{"style":27778,"xmlns":24932,"width":40992,"height":35776,"role":1930,"focusable":8058,"viewBox":40993,"xmlnsXLink":24936},"14.204ex","0 -666 6278 688",[24938,40995,40996,40999,41002],{},[24941,40997],{"id":40998,"d":25935},"MJX-109-TEX-N-30",[24941,41000],{"id":41001,"d":26357},"MJX-109-TEX-N-2E",[24941,41003],{"id":41004,"d":25914},"MJX-109-TEX-N-31",[25022,41006,41007],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,41008,41009],{"dataMmlNode":25028},[25022,41010,41011,41014,41017,41019,41021,41023,41025,41028,41031,41034,41037,41040,41043],{"dataMmlNode":26031},[25033,41012],{"dataC":26103,"xLinkHref":41013},"#MJX-109-TEX-N-30",[25033,41015],{"dataC":26392,"xLinkHref":41016,"transform":26394},"#MJX-109-TEX-N-2E",[25033,41018],{"dataC":26103,"xLinkHref":41013,"transform":25049},[25033,41020],{"dataC":26103,"xLinkHref":41013,"transform":26835},[25033,41022],{"dataC":26103,"xLinkHref":41013,"transform":30504},[25033,41024],{"dataC":26103,"xLinkHref":41013,"transform":31438},[25033,41026],{"dataC":26103,"xLinkHref":41013,"transform":41027},"translate(2778,0)",[25033,41029],{"dataC":26103,"xLinkHref":41013,"transform":41030},"translate(3278,0)",[25033,41032],{"dataC":26103,"xLinkHref":41013,"transform":41033},"translate(3778,0)",[25033,41035],{"dataC":26103,"xLinkHref":41013,"transform":41036},"translate(4278,0)",[25033,41038],{"dataC":26103,"xLinkHref":41013,"transform":41039},"translate(4778,0)",[25033,41041],{"dataC":26103,"xLinkHref":41013,"transform":41042},"translate(5278,0)",[25033,41044],{"dataC":26035,"xLinkHref":41045,"transform":41046},"#MJX-109-TEX-N-31","translate(5778,0)","). The computer can't handle decimals that small and will just round it to ",[24923,41049,41051],{"className":41050,"jax":24927},[24926],[24929,41052,41053,41058],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,41054,41055],{},[24941,41056],{"id":41057,"d":25935},"MJX-110-TEX-N-30",[25022,41059,41060],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,41061,41062],{"dataMmlNode":25028},[25022,41063,41064],{"dataMmlNode":26031},[25033,41065],{"dataC":26103,"xLinkHref":41066},"#MJX-110-TEX-N-30"," (a glitch called \"underflow\"), breaking the math.",[12,41069,41070,41071,41074,41075,41238],{},"To stop the computer from crashing, we wrap the whole thing in a ",[106,41072,41073],{},"logarithm",". 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becomes ",[24923,41602,41604],{"className":41603,"jax":24927},[24926],[24929,41605,41606,41611],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,41607,41608],{},[24941,41609],{"id":41610,"d":25935},"MJX-115-TEX-N-30",[25022,41612,41613],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,41614,41615],{"dataMmlNode":25028},[25022,41616,41617],{"dataMmlNode":26031},[25033,41618],{"dataC":26103,"xLinkHref":41619},"#MJX-115-TEX-N-30"," and disappears. We are just left with ",[24923,41622,41624],{"className":41623,"jax":24927},[24926],[24929,41625,41628,41654],{"style":24931,"xmlns":24932,"width":41626,"height":24934,"role":1930,"focusable":8058,"viewBox":41627,"xmlnsXLink":24936},"9.009ex","0 -750 3982 1000",[24938,41629,41630,41633,41636,41639,41642,41645,41648,41651],{},[24941,41631],{"id":41632,"d":24960},"MJX-116-TEX-N-6C",[24941,41634],{"id":41635,"d":24964},"MJX-116-TEX-N-6F",[24941,41637],{"id":41638,"d":25924},"MJX-116-TEX-N-67",[24941,41640],{"id":41641,"d":180},"MJX-116-TEX-N-2061",[24941,41643],{"id":41644,"d":24944},"MJX-116-TEX-N-28",[24941,41646],{"id":41647,"d":31530},"MJX-116-TEX-I-210E",[24941,41649],{"id":41650,"d":32148},"MJX-116-TEX-I-1D465",[24941,41652],{"id":41653,"d":24988},"MJX-116-TEX-N-29",[25022,41655,41656],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,41657,41658,41669,41674,41679,41684,41689,41695,41701],{"dataMmlNode":25028},[25022,41659,41660,41663,41666],{"dataMmlNode":25942},[25033,41661],{"dataC":25057,"xLinkHref":41662},"#MJX-116-TEX-N-6C",[25033,41664],{"dataC":25062,"xLinkHref":41665,"transform":26048},"#MJX-116-TEX-N-6F",[25033,41667],{"dataC":26051,"xLinkHref":41668,"transform":25049},"#MJX-116-TEX-N-67",[25022,41670,41671],{"dataMmlNode":25031,"transform":26835},[25033,41672],{"dataC":26065,"xLinkHref":41673},"#MJX-116-TEX-N-2061",[25022,41675,41676],{"dataMmlNode":25031,"transform":26835},[25033,41677],{"dataC":25035,"xLinkHref":41678},"#MJX-116-TEX-N-28",[25022,41680,41681],{"dataMmlNode":25942,"transform":41146},[25033,41682],{"dataC":31539,"xLinkHref":41683},"#MJX-116-TEX-I-210E",[25022,41685,41687],{"dataMmlNode":25031,"transform":41686},"translate(2243,0)",[25033,41688],{"dataC":25035,"xLinkHref":41678},[25022,41690,41692],{"dataMmlNode":25942,"transform":41691},"translate(2632,0)",[25033,41693],{"dataC":32181,"xLinkHref":41694},"#MJX-116-TEX-I-1D465",[25022,41696,41698],{"dataMmlNode":25031,"transform":41697},"translate(3204,0)",[25033,41699],{"dataC":25108,"xLinkHref":41700},"#MJX-116-TEX-N-29",[25022,41702,41704],{"dataMmlNode":25031,"transform":41703},"translate(3593,0)",[25033,41705],{"dataC":25108,"xLinkHref":41700},". We want our prediction ",[24923,41708,41710],{"className":41709,"jax":24927},[24926],[24929,41711,41712,41726],{"style":24931,"xmlns":24932,"width":32764,"height":24934,"role":1930,"focusable":8058,"viewBox":32765,"xmlnsXLink":24936},[24938,41713,41714,41717,41720,41723],{},[24941,41715],{"id":41716,"d":31530},"MJX-117-TEX-I-210E",[24941,41718],{"id":41719,"d":24944},"MJX-117-TEX-N-28",[24941,41721],{"id":41722,"d":32148},"MJX-117-TEX-I-1D465",[24941,41724],{"id":41725,"d":24988},"MJX-117-TEX-N-29",[25022,41727,41728],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,41729,41730,41735,41740,41745],{"dataMmlNode":25028},[25022,41731,41732],{"dataMmlNode":25942},[25033,41733],{"dataC":31539,"xLinkHref":41734},"#MJX-117-TEX-I-210E",[25022,41736,41737],{"dataMmlNode":25031,"transform":32791},[25033,41738],{"dataC":25035,"xLinkHref":41739},"#MJX-117-TEX-N-28",[25022,41741,41742],{"dataMmlNode":25942,"transform":32797},[25033,41743],{"dataC":32181,"xLinkHref":41744},"#MJX-117-TEX-I-1D465",[25022,41746,41747],{"dataMmlNode":25031,"transform":32803},[25033,41748],{"dataC":25108,"xLinkHref":41749},"#MJX-117-TEX-N-29"," to be as close to ",[24923,41752,41754],{"className":41753,"jax":24927},[24926],[24929,41755,41756,41761],{"style":27802,"xmlns":24932,"width":35168,"height":35169,"role":1930,"focusable":8058,"viewBox":35170,"xmlnsXLink":24936},[24938,41757,41758],{},[24941,41759],{"id":41760,"d":25914},"MJX-118-TEX-N-31",[25022,41762,41763],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,41764,41765],{"dataMmlNode":25028},[25022,41766,41767],{"dataMmlNode":26031},[25033,41768],{"dataC":26035,"xLinkHref":41769},"#MJX-118-TEX-N-31"," as possible.",[26,41772,41773,7206,41775,41812,41813,41600,41850,41869,41870,41706,41973,41750,42016,41770],{},[106,41774,41509],{},[106,41776,41777,7024],{},[24923,41778,41780],{"className":41779,"jax":24927},[24926],[24929,41781,41782,41793],{"style":29805,"xmlns":24932,"width":38931,"height":38932,"role":1930,"focusable":8058,"viewBox":38933,"xmlnsXLink":24936},[24938,41783,41784,41787,41790],{},[24941,41785],{"id":41786,"d":25893},"MJX-119-TEX-I-1D466",[24941,41788],{"id":41789,"d":24972},"MJX-119-TEX-N-3D",[24941,41791],{"id":41792,"d":25935},"MJX-119-TEX-N-30",[25022,41794,41795],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,41796,41797,41802,41807],{"dataMmlNode":25028},[25022,41798,41799],{"dataMmlNode":25942},[25033,41800],{"dataC":25987,"xLinkHref":41801},"#MJX-119-TEX-I-1D466",[25022,41803,41804],{"dataMmlNode":25031,"transform":38956},[25033,41805],{"dataC":25078,"xLinkHref":41806},"#MJX-119-TEX-N-3D",[25022,41808,41809],{"dataMmlNode":26031,"transform":38962},[25033,41810],{"dataC":26103,"xLinkHref":41811},"#MJX-119-TEX-N-30"," The first half of the equation ",[24923,41814,41816],{"className":41815,"jax":24927},[24926],[24929,41817,41819,41830],{"style":24931,"xmlns":24932,"width":29568,"height":24934,"role":1930,"focusable":8058,"viewBox":41818,"xmlnsXLink":24936},"0 -750 1278 1000",[24938,41820,41821,41824,41827],{},[24941,41822],{"id":41823,"d":24944},"MJX-120-TEX-N-28",[24941,41825],{"id":41826,"d":25935},"MJX-120-TEX-N-30",[24941,41828],{"id":41829,"d":24988},"MJX-120-TEX-N-29",[25022,41831,41832],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,41833,41834,41839,41844],{"dataMmlNode":25028},[25022,41835,41836],{"dataMmlNode":25031},[25033,41837],{"dataC":25035,"xLinkHref":41838},"#MJX-120-TEX-N-28",[25022,41840,41841],{"dataMmlNode":26031,"transform":25040},[25033,41842],{"dataC":26103,"xLinkHref":41843},"#MJX-120-TEX-N-30",[25022,41845,41847],{"dataMmlNode":25031,"transform":41846},"translate(889,0)",[25033,41848],{"dataC":25108,"xLinkHref":41849},"#MJX-120-TEX-N-29",[24923,41851,41853],{"className":41852,"jax":24927},[24926],[24929,41854,41855,41860],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,41856,41857],{},[24941,41858],{"id":41859,"d":25935},"MJX-121-TEX-N-30",[25022,41861,41862],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,41863,41864],{"dataMmlNode":25028},[25022,41865,41866],{"dataMmlNode":26031},[25033,41867],{"dataC":26103,"xLinkHref":41868},"#MJX-121-TEX-N-30"," and disappears. We are left with ",[24923,41871,41873],{"className":41872,"jax":24927},[24926],[24929,41874,41877,41909],{"style":24931,"xmlns":24932,"width":41875,"height":24934,"role":1930,"focusable":8058,"viewBox":41876,"xmlnsXLink":24936},"12.906ex","0 -750 5704.4 1000",[24938,41878,41879,41882,41885,41888,41891,41894,41897,41900,41903,41906],{},[24941,41880],{"id":41881,"d":24960},"MJX-122-TEX-N-6C",[24941,41883],{"id":41884,"d":24964},"MJX-122-TEX-N-6F",[24941,41886],{"id":41887,"d":25924},"MJX-122-TEX-N-67",[24941,41889],{"id":41890,"d":180},"MJX-122-TEX-N-2061",[24941,41892],{"id":41893,"d":24944},"MJX-122-TEX-N-28",[24941,41895],{"id":41896,"d":25914},"MJX-122-TEX-N-31",[24941,41898],{"id":41899,"d":25910},"MJX-122-TEX-N-2212",[24941,41901],{"id":41902,"d":31530},"MJX-122-TEX-I-210E",[24941,41904],{"id":41905,"d":32148},"MJX-122-TEX-I-1D465",[24941,41907],{"id":41908,"d":24988},"MJX-122-TEX-N-29",[25022,41910,41911],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,41912,41913,41924,41929,41934,41939,41945,41951,41956,41962,41968],{"dataMmlNode":25028},[25022,41914,41915,41918,41921],{"dataMmlNode":25942},[25033,41916],{"dataC":25057,"xLinkHref":41917},"#MJX-122-TEX-N-6C",[25033,41919],{"dataC":25062,"xLinkHref":41920,"transform":26048},"#MJX-122-TEX-N-6F",[25033,41922],{"dataC":26051,"xLinkHref":41923,"transform":25049},"#MJX-122-TEX-N-67",[25022,41925,41926],{"dataMmlNode":25031,"transform":26835},[25033,41927],{"dataC":26065,"xLinkHref":41928},"#MJX-122-TEX-N-2061",[25022,41930,41931],{"dataMmlNode":25031,"transform":26835},[25033,41932],{"dataC":25035,"xLinkHref":41933},"#MJX-122-TEX-N-28",[25022,41935,41936],{"dataMmlNode":26031,"transform":41146},[25033,41937],{"dataC":26035,"xLinkHref":41938},"#MJX-122-TEX-N-31",[25022,41940,41942],{"dataMmlNode":25031,"transform":41941},"translate(2389.2,0)",[25033,41943],{"dataC":26019,"xLinkHref":41944},"#MJX-122-TEX-N-2212",[25022,41946,41948],{"dataMmlNode":25942,"transform":41947},"translate(3389.4,0)",[25033,41949],{"dataC":31539,"xLinkHref":41950},"#MJX-122-TEX-I-210E",[25022,41952,41954],{"dataMmlNode":25031,"transform":41953},"translate(3965.4,0)",[25033,41955],{"dataC":25035,"xLinkHref":41933},[25022,41957,41959],{"dataMmlNode":25942,"transform":41958},"translate(4354.4,0)",[25033,41960],{"dataC":32181,"xLinkHref":41961},"#MJX-122-TEX-I-1D465",[25022,41963,41965],{"dataMmlNode":25031,"transform":41964},"translate(4926.4,0)",[25033,41966],{"dataC":25108,"xLinkHref":41967},"#MJX-122-TEX-N-29",[25022,41969,41971],{"dataMmlNode":25031,"transform":41970},"translate(5315.4,0)",[25033,41972],{"dataC":25108,"xLinkHref":41967},[24923,41974,41976],{"className":41975,"jax":24927},[24926],[24929,41977,41978,41992],{"style":24931,"xmlns":24932,"width":32764,"height":24934,"role":1930,"focusable":8058,"viewBox":32765,"xmlnsXLink":24936},[24938,41979,41980,41983,41986,41989],{},[24941,41981],{"id":41982,"d":31530},"MJX-123-TEX-I-210E",[24941,41984],{"id":41985,"d":24944},"MJX-123-TEX-N-28",[24941,41987],{"id":41988,"d":32148},"MJX-123-TEX-I-1D465",[24941,41990],{"id":41991,"d":24988},"MJX-123-TEX-N-29",[25022,41993,41994],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,41995,41996,42001,42006,42011],{"dataMmlNode":25028},[25022,41997,41998],{"dataMmlNode":25942},[25033,41999],{"dataC":31539,"xLinkHref":42000},"#MJX-123-TEX-I-210E",[25022,42002,42003],{"dataMmlNode":25031,"transform":32791},[25033,42004],{"dataC":25035,"xLinkHref":42005},"#MJX-123-TEX-N-28",[25022,42007,42008],{"dataMmlNode":25942,"transform":32797},[25033,42009],{"dataC":32181,"xLinkHref":42010},"#MJX-123-TEX-I-1D465",[25022,42012,42013],{"dataMmlNode":25031,"transform":32803},[25033,42014],{"dataC":25108,"xLinkHref":42015},"#MJX-123-TEX-N-29",[24923,42017,42019],{"className":42018,"jax":24927},[24926],[24929,42020,42021,42026],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,42022,42023],{},[24941,42024],{"id":42025,"d":25935},"MJX-124-TEX-N-30",[25022,42027,42028],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,42029,42030],{"dataMmlNode":25028},[25022,42031,42032],{"dataMmlNode":26031},[25033,42033],{"dataC":26103,"xLinkHref":42034},"#MJX-124-TEX-N-30",[12,42036,42037,42038,42041],{},"Because of the logarithm, the model is ",[106,42039,42040],{},"heavily punished if it is confidently wrong."," (e.g., If it predicts a 99% probability of a tumor being Malignant, but it's actually Benign, the error slope becomes a massive cliff, screaming at the model to turn around).",[16,42043],{},[12,42045,42046],{},"Statisticians didn't add the log to fix the bumpy error graph, they added it so computers wouldn't crash.",[12,42048,42049,42050,42092],{},"However, because the Sigmoid formula relies on an exponential (",[24923,42051,42053],{"className":42052,"jax":24927},[24926],[24929,42054,42058,42069],{"style":25196,"xmlns":24932,"width":42055,"height":42056,"role":1930,"focusable":8058,"viewBox":42057,"xmlnsXLink":24936},"3.402ex","1.779ex","0 -775.2 1503.6 786.2",[24938,42059,42060,42063,42066],{},[24941,42061],{"id":42062,"d":27397},"MJX-125-TEX-I-1D452",[24941,42064],{"id":42065,"d":25910},"MJX-125-TEX-N-2212",[24941,42067],{"id":42068,"d":32148},"MJX-125-TEX-I-1D465",[25022,42070,42071],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,42072,42073],{"dataMmlNode":25028},[25022,42074,42075,42080],{"dataMmlNode":31565},[25022,42076,42077],{"dataMmlNode":25942},[25033,42078],{"dataC":27596,"xLinkHref":42079},"#MJX-125-TEX-I-1D452",[25022,42081,42082,42087],{"dataMmlNode":27203,"transform":38097,"dataMjxTexclass":27205},[25022,42083,42084],{"dataMmlNode":25031},[25033,42085],{"dataC":26019,"xLinkHref":42086},"#MJX-125-TEX-N-2212",[25022,42088,42089],{"dataMmlNode":25942,"transform":25049},[25033,42090],{"dataC":32181,"xLinkHref":42091},"#MJX-125-TEX-I-1D465","), and logarithms are the mathematical \"undo button\" for exponents... wrapping the Sigmoid in a logarithm perfectly canceled out the messy fraction.",[12,42094,42095,42096],{},"It magically ironed out the flat spots and local minima, turning the error graph ",[106,42097,42098],{},"back into a perfect, smooth U-shape bowl.",[6964,42100,42101],{"id":25635},"Summary",[7085,42103,42104,42114],{},[7088,42105,42106],{},[7091,42107,42108,42111],{},[7094,42109,42110],{},"Topic",[7094,42112,42113],{},"Key Idea",[7104,42115,42116,42123,42130,42138,42146,42154,42162,42170],{},[7091,42117,42118,42120],{},[7109,42119,32008],{},[7109,42121,42122],{},"Clean and transform raw data before training",[7091,42124,42125,42127],{},[7109,42126,32858],{},[7109,42128,42129],{},"Predict continuous values with a linear model",[7091,42131,42132,42135],{},[7109,42133,42134],{},"Cost Function (J)",[7109,42136,42137],{},"Measures how wrong our predictions are",[7091,42139,42140,42143],{},[7109,42141,42142],{},"Gradient Descent",[7109,42144,42145],{},"Iteratively adjust parameters to minimize cost",[7091,42147,42148,42151],{},[7109,42149,42150],{},"Batch vs. Stochastic GD",[7109,42152,42153],{},"Update after all examples vs. after each example",[7091,42155,42156,42159],{},[7109,42157,42158],{},"Underfitting \u002F Overfitting",[7109,42160,42161],{},"Too simple vs. too complex models",[7091,42163,42164,42167],{},[7109,42165,42166],{},"Locally Weighted LR",[7109,42168,42169],{},"Give nearby points more influence",[7091,42171,42172,42174],{},[7109,42173,38670],{},[7109,42175,42176],{},"Classify discrete outcomes using the sigmoid function",[9332,42178,31966],{},{"title":180,"searchDepth":181,"depth":181,"links":42180},[42181,42185,42195,42198],{"id":32007,"depth":181,"text":32008,"children":42182},[42183,42184],{"id":32021,"depth":7430,"text":32022},{"id":32447,"depth":7430,"text":32448},{"id":32474,"depth":181,"text":32475,"children":42186},[42187,42188,42189,42190,42191,42192,42193,42194],{"id":32481,"depth":7430,"text":32482},{"id":32521,"depth":7430,"text":32522},{"id":32857,"depth":7430,"text":32858},{"id":33120,"depth":7430,"text":33121},{"id":33448,"depth":7430,"text":33449},{"id":37102,"depth":7430,"text":37103},{"id":37157,"depth":7430,"text":37158},{"id":37230,"depth":7430,"text":37231},{"id":38669,"depth":181,"text":38670,"children":42196},[42197],{"id":39705,"depth":7430,"text":39706},{"id":25635,"depth":181,"text":42101},"2026-04-27",{},"\u002Fblog\u002FData-Preprocessing-&-Regression-Analysis",{"title":32002,"description":180},{"loc":42201},"blog\u002FData-Preprocessing-&-Regression-Analysis","KB9QC-cZ2vN8WOj-CC7svUlaw43IIXaHJvITMAh6dDQ",{"id":42207,"title":42208,"author":6946,"body":42209,"date":47819,"description":180,"draft":185,"edited_at":47819,"extension":186,"featured_image":194,"meta":47820,"navigation":189,"path":47821,"pinned":185,"seo":47822,"sitemap":47823,"stem":47824,"tags":194,"__hash__":47825},"blog\u002Fblog\u002FIntroduction-to-Machine-Learning.md","Introduction to Machine Learning",{"type":9,"value":42210,"toc":47797},[42211,42215,42222,42229,42232,42246,42250,42253,42256,42286,42292,42296,42299,42305,42311,42317,42323,42327,42331,42334,42366,42370,42373,42387,42391,42394,42414,42475,42481,42485,42488,42514,42518,42521,42525,42578,42886,42892,42896,42899,43294,43297,43625,43688,43692,43702,43706,43709,44048,44100,47069,47072,47083,47089,47262,47457,47461,47464,47490,47505,47658,47662,47665,47671,47677,47683,47689,47692,47696,47699,47705,47711,47714,47718,47721,47759,47763,47795],[6964,42212,42214],{"id":42213},"what-even-is-learning","What Even Is \"Learning\"?",[12,42216,42217,42218,42221],{},"Before we talk about ",[7289,42219,42220],{},"machine"," learning, let's think about what learning means in general.",[12,42223,42224,42225,42228],{},"Herbert Simon, a Nobel Prize-winning scientist, once described learning as ",[106,42226,42227],{},"any process by which a system improves its performance from experience",". That's a beautifully simple definition, and it applies just as well to computers as it does to humans.",[12,42230,42231],{},"When we talk about learning in the context of machines, the \"tasks\" we want them to improve at generally fall into two buckets:",[23,42233,42234,42240],{},[26,42235,42236,42239],{},[106,42237,42238],{},"Classification"," — assigning things to categories (e.g., \"Is this email spam or not?\")",[26,42241,42242,42245],{},[106,42243,42244],{},"Problem solving \u002F Planning \u002F Control"," — taking actions to achieve a goal (e.g., \"How should a robot navigate a maze?\")",[6964,42247,42249],{"id":42248},"where-does-machine-learning-fit-in-the-ai-landscape","Where Does Machine Learning Fit in the AI Landscape?",[12,42251,42252],{},"You've probably heard the terms AI, Machine Learning, Deep Learning, and Data Science thrown around interchangeably. They're related, but they're not the same thing.",[12,42254,42255],{},"Think of it as a set of nested circles:",[23,42257,42258,42264,42270,42276],{},[26,42259,42260,42263],{},[106,42261,42262],{},"Artificial Intelligence (AI)"," is the broadest concept. It's about creating machines that can mimic intelligent human behaviour.",[26,42265,42266,42269],{},[106,42267,42268],{},"Machine Learning (ML)"," is a subset of AI. Instead of being explicitly programmed with rules, ML systems learn patterns from data and use those patterns to make predictions or decisions.",[26,42271,42272,42275],{},[106,42273,42274],{},"Deep Learning (DL)"," is a subset of ML that uses multi-layered neural networks to tackle complex problems like image recognition and natural language processing.",[26,42277,42278,42281,42282,42285],{},[106,42279,42280],{},"Data Science"," overlaps with all of these. It's the broader discipline of extracting insights from data using statistics, scientific methods, and algorithms.\nThe key takeaway: ",[106,42283,42284],{},"ML is about building models from training data to make predictions",", rather than writing rules by hand.",[12,42287,42288],{},[1930,42289],{"alt":42290,"src":42291},"0.83","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1777198853679-1775145963610-Screenshot_2026-04-03_000510.png",[6964,42293,42295],{"id":42294},"the-main-flavours-of-machine-learning","The Main Flavours of Machine Learning",[12,42297,42298],{},"Machine learning approaches can be grouped into several categories based on how the system learns:",[12,42300,42301,42304],{},[106,42302,42303],{},"Supervised Learning"," — The model learns from labeled examples. You give it inputs paired with the correct outputs, and it learns the mapping between them. This includes regression (predicting a continuous value, like house prices) and classification (predicting a category, like spam vs. not spam).",[12,42306,42307,42310],{},[106,42308,42309],{},"Unsupervised Learning"," — The model receives data without labels and must find structure on its own. The most common task here is clustering — grouping similar data points together.",[12,42312,42313,42316],{},[106,42314,42315],{},"Reinforcement Learning"," — The model learns by interacting with an environment. It takes actions, receives rewards or penalties, and gradually figures out the best strategy. Think of it like training a dog with treats.",[12,42318,42319,42322],{},[106,42320,42321],{},"Self-Supervised Learning"," — A newer paradigm where the model generates its own labels from the data itself (for example, masking a word in a sentence and learning to predict it).",[6964,42324,42326],{"id":42325},"real-world-applications","Real-World Applications",[7166,42328,42330],{"id":42329},"classification-examples","Classification Examples",[12,42332,42333],{},"Classification is everywhere in daily life:",[23,42335,42336,42342,42348,42354,42360],{},[26,42337,42338,42341],{},[106,42339,42340],{},"Medical diagnosis"," — Is this X-ray showing signs of pneumonia?",[26,42343,42344,42347],{},[106,42345,42346],{},"Spam filtering"," — Should this email go to your inbox or junk folder?",[26,42349,42350,42353],{},[106,42351,42352],{},"Fraud detection"," — Is this credit card transaction suspicious?",[26,42355,42356,42359],{},[106,42357,42358],{},"Recommendation systems"," — Which movies, books, or songs might you enjoy?",[26,42361,42362,42365],{},[106,42363,42364],{},"Speech and handwriting recognition"," — Converting spoken words or handwritten text into digital text.",[7166,42367,42369],{"id":42368},"problem-solving-planning-control-examples","Problem Solving \u002F Planning \u002F Control Examples",[12,42371,42372],{},"These are tasks where an agent takes actions in an environment to achieve a goal:",[23,42374,42375,42378,42381,42384],{},[26,42376,42377],{},"Playing board games like chess or checkers",[26,42379,42380],{},"Self-driving cars navigating roads",[26,42382,42383],{},"Controlling robots or video game characters",[26,42385,42386],{},"Flying drones or helicopters autonomously",[6964,42388,42390],{"id":42389},"defining-a-learning-task-t-p-e","Defining a Learning Task: T, P, E",[12,42392,42393],{},"One of the most useful frameworks for thinking about ML problems comes from Tom Mitchell's classic definition. Every learning task can be described by three components:",[23,42395,42396,42402,42408],{},[26,42397,42398,42401],{},[106,42399,42400],{},"T (Task)"," — What is the system trying to do?",[26,42403,42404,42407],{},[106,42405,42406],{},"P (Performance)"," — How do we measure success?",[26,42409,42410,42413],{},[106,42411,42412],{},"E (Experience)"," — What data does the system learn from?\nHere are a few examples to make this concrete:",[7085,42415,42416,42429],{},[7088,42417,42418],{},[7091,42419,42420,42423,42426],{},[7094,42421,42422],{},"Task (T)",[7094,42424,42425],{},"Performance (P)",[7094,42427,42428],{},"Experience (E)",[7104,42430,42431,42442,42453,42464],{},[7091,42432,42433,42436,42439],{},[7109,42434,42435],{},"Playing checkers",[7109,42437,42438],{},"% of games won",[7109,42440,42441],{},"Self-play practice games",[7091,42443,42444,42447,42450],{},[7109,42445,42446],{},"Recognizing handwritten words",[7109,42448,42449],{},"% of words correctly classified",[7109,42451,42452],{},"Database of labeled handwriting images",[7091,42454,42455,42458,42461],{},[7109,42456,42457],{},"Driving on highways",[7109,42459,42460],{},"Average distance before a human-judged error",[7109,42462,42463],{},"Recorded images and steering commands from a human driver",[7091,42465,42466,42469,42472],{},[7109,42467,42468],{},"Spam classification",[7109,42470,42471],{},"% of emails correctly classified",[7109,42473,42474],{},"Database of emails with human labels",[12,42476,42477,42478],{},"This framework is great for getting clarity before you start any ML project. Ask yourself: ",[7289,42479,42480],{},"What's my T, P, and E?",[6964,42482,42484],{"id":42483},"designing-a-learning-system","Designing a Learning System",[12,42486,42487],{},"When you set out to build an ML system, there are four key design decisions:",[3603,42489,42490,42496,42502,42508],{},[26,42491,42492,42495],{},[106,42493,42494],{},"Choose the training experience"," — What kind of data will the system learn from? Is it direct (labeled input-output pairs) or indirect (like game outcomes where individual moves aren't labeled)?",[26,42497,42498,42501],{},[106,42499,42500],{},"Choose the target function"," — What exactly should the system learn? For a checkers player, this might be an evaluation function that scores how favorable a board position is.",[26,42503,42504,42507],{},[106,42505,42506],{},"Choose a representation"," — How will the target function be expressed? Options include lookup tables, linear functions, decision trees, neural networks, and many more.",[26,42509,42510,42513],{},[106,42511,42512],{},"Choose a learning algorithm"," — How will the system search for the best function? This could be gradient descent, dynamic programming, evolutionary algorithms, etc.",[6964,42515,42517],{"id":42516},"a-concrete-example-learning-to-play-checkers","A Concrete Example: Learning to Play Checkers",[12,42519,42520],{},"To tie everything together, let's walk through a classic example: Arthur Samuel's checkers-playing program from 1959, one of the earliest ML systems ever built.",[7166,42522,42524],{"id":42523},"the-target-function","The Target Function",[12,42526,42527,42528,7206,42531,42577],{},"We want to learn an ",[106,42529,42530],{},"evaluation function",[24923,42532,42534],{"className":42533,"jax":24927},[24926],[24929,42535,42538,42551],{"style":24931,"xmlns":24932,"width":42536,"height":24934,"role":1930,"focusable":8058,"viewBox":42537,"xmlnsXLink":24936},"4.471ex","0 -750 1976 1000",[24938,42539,42540,42543,42545,42549],{},[24941,42541],{"id":42542,"d":27385},"MJX-1-TEX-I-1D449",[24941,42544],{"id":24943,"d":24944},[24941,42546],{"id":42547,"d":42548},"MJX-1-TEX-I-1D44F","M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z",[24941,42550],{"id":24987,"d":24988},[25022,42552,42553],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,42554,42555,42560,42565,42572],{"dataMmlNode":25028},[25022,42556,42557],{"dataMmlNode":25942},[25033,42558],{"dataC":27570,"xLinkHref":42559},"#MJX-1-TEX-I-1D449",[25022,42561,42563],{"dataMmlNode":25031,"transform":42562},"translate(769,0)",[25033,42564],{"dataC":25035,"xLinkHref":25036},[25022,42566,42568],{"dataMmlNode":25942,"transform":42567},"translate(1158,0)",[25033,42569],{"dataC":42570,"xLinkHref":42571},"1D44F","#MJX-1-TEX-I-1D44F",[25022,42573,42575],{"dataMmlNode":25031,"transform":42574},"translate(1587,0)",[25033,42576],{"dataC":25108,"xLinkHref":25109}," that takes a board state and returns a score indicating how favorable it is. Ideally:",[23,42579,42580,42672,42768,42843],{},[26,42581,42582,42649,42650,42671],{},[24923,42583,42585],{"className":42584,"jax":24927},[24926],[24929,42586,42589,42610],{"style":24931,"xmlns":24932,"width":42587,"height":24934,"role":1930,"focusable":8058,"viewBox":42588,"xmlnsXLink":24936},"10.881ex","0 -750 4809.6 1000",[24938,42590,42591,42594,42596,42599,42601,42604,42607],{},[24941,42592],{"id":42593,"d":27385},"MJX-2-TEX-I-1D449",[24941,42595],{"id":32282,"d":24944},[24941,42597],{"id":42598,"d":42548},"MJX-2-TEX-I-1D44F",[24941,42600],{"id":32303,"d":24988},[24941,42602],{"id":42603,"d":24972},"MJX-2-TEX-N-3D",[24941,42605],{"id":42606,"d":25914},"MJX-2-TEX-N-31",[24941,42608],{"id":42609,"d":25935},"MJX-2-TEX-N-30",[25022,42611,42612],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,42613,42614,42619,42623,42628,42632,42638],{"dataMmlNode":25028},[25022,42615,42616],{"dataMmlNode":25942},[25033,42617],{"dataC":27570,"xLinkHref":42618},"#MJX-2-TEX-I-1D449",[25022,42620,42621],{"dataMmlNode":25031,"transform":42562},[25033,42622],{"dataC":25035,"xLinkHref":32332},[25022,42624,42625],{"dataMmlNode":25942,"transform":42567},[25033,42626],{"dataC":42570,"xLinkHref":42627},"#MJX-2-TEX-I-1D44F",[25022,42629,42630],{"dataMmlNode":25031,"transform":42574},[25033,42631],{"dataC":25108,"xLinkHref":32370},[25022,42633,42635],{"dataMmlNode":25031,"transform":42634},"translate(2253.8,0)",[25033,42636],{"dataC":25078,"xLinkHref":42637},"#MJX-2-TEX-N-3D",[25022,42639,42641,42644,42647],{"dataMmlNode":26031,"transform":42640},"translate(3309.6,0)",[25033,42642],{"dataC":26035,"xLinkHref":42643},"#MJX-2-TEX-N-31",[25033,42645],{"dataC":26103,"xLinkHref":42646,"transform":26394},"#MJX-2-TEX-N-30",[25033,42648],{"dataC":26103,"xLinkHref":42646,"transform":29730}," if ",[24923,42651,42653],{"className":42652,"jax":24927},[24926],[24929,42654,42657,42662],{"style":25196,"xmlns":24932,"width":42655,"height":31523,"role":1930,"focusable":8058,"viewBox":42656,"xmlnsXLink":24936},"0.971ex","0 -694 429 705",[24938,42658,42659],{},[24941,42660],{"id":42661,"d":42548},"MJX-3-TEX-I-1D44F",[25022,42663,42664],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,42665,42666],{"dataMmlNode":25028},[25022,42667,42668],{"dataMmlNode":25942},[25033,42669],{"dataC":42570,"xLinkHref":42670},"#MJX-3-TEX-I-1D44F"," is a winning position",[26,42673,42674,42649,42748,42767],{},[24923,42675,42677],{"className":42676,"jax":24927},[24926],[24929,42678,42681,42705],{"style":24931,"xmlns":24932,"width":42679,"height":24934,"role":1930,"focusable":8058,"viewBox":42680,"xmlnsXLink":24936},"12.642ex","0 -750 5587.6 1000",[24938,42682,42683,42686,42688,42691,42693,42696,42699,42702],{},[24941,42684],{"id":42685,"d":27385},"MJX-4-TEX-I-1D449",[24941,42687],{"id":32600,"d":24944},[24941,42689],{"id":42690,"d":42548},"MJX-4-TEX-I-1D44F",[24941,42692],{"id":32606,"d":24988},[24941,42694],{"id":42695,"d":24972},"MJX-4-TEX-N-3D",[24941,42697],{"id":42698,"d":25910},"MJX-4-TEX-N-2212",[24941,42700],{"id":42701,"d":25914},"MJX-4-TEX-N-31",[24941,42703],{"id":42704,"d":25935},"MJX-4-TEX-N-30",[25022,42706,42707],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,42708,42709,42714,42718,42723,42727,42732,42737],{"dataMmlNode":25028},[25022,42710,42711],{"dataMmlNode":25942},[25033,42712],{"dataC":27570,"xLinkHref":42713},"#MJX-4-TEX-I-1D449",[25022,42715,42716],{"dataMmlNode":25031,"transform":42562},[25033,42717],{"dataC":25035,"xLinkHref":32625},[25022,42719,42720],{"dataMmlNode":25942,"transform":42567},[25033,42721],{"dataC":42570,"xLinkHref":42722},"#MJX-4-TEX-I-1D44F",[25022,42724,42725],{"dataMmlNode":25031,"transform":42574},[25033,42726],{"dataC":25108,"xLinkHref":32635},[25022,42728,42729],{"dataMmlNode":25031,"transform":42634},[25033,42730],{"dataC":25078,"xLinkHref":42731},"#MJX-4-TEX-N-3D",[25022,42733,42734],{"dataMmlNode":25031,"transform":42640},[25033,42735],{"dataC":26019,"xLinkHref":42736},"#MJX-4-TEX-N-2212",[25022,42738,42740,42743,42746],{"dataMmlNode":26031,"transform":42739},"translate(4087.6,0)",[25033,42741],{"dataC":26035,"xLinkHref":42742},"#MJX-4-TEX-N-31",[25033,42744],{"dataC":26103,"xLinkHref":42745,"transform":26394},"#MJX-4-TEX-N-30",[25033,42747],{"dataC":26103,"xLinkHref":42745,"transform":29730},[24923,42749,42751],{"className":42750,"jax":24927},[24926],[24929,42752,42753,42758],{"style":25196,"xmlns":24932,"width":42655,"height":31523,"role":1930,"focusable":8058,"viewBox":42656,"xmlnsXLink":24936},[24938,42754,42755],{},[24941,42756],{"id":42757,"d":42548},"MJX-5-TEX-I-1D44F",[25022,42759,42760],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,42761,42762],{"dataMmlNode":25028},[25022,42763,42764],{"dataMmlNode":25942},[25033,42765],{"dataC":42570,"xLinkHref":42766},"#MJX-5-TEX-I-1D44F"," is a losing position",[26,42769,42770,42649,42823,42842],{},[24923,42771,42773],{"className":42772,"jax":24927},[24926],[24929,42774,42777,42793],{"style":24931,"xmlns":24932,"width":42775,"height":24934,"role":1930,"focusable":8058,"viewBox":42776,"xmlnsXLink":24936},"8.619ex","0 -750 3809.6 1000",[24938,42778,42779,42782,42784,42787,42789,42791],{},[24941,42780],{"id":42781,"d":27385},"MJX-6-TEX-I-1D449",[24941,42783],{"id":25896,"d":24944},[24941,42785],{"id":42786,"d":42548},"MJX-6-TEX-I-1D44F",[24941,42788],{"id":25903,"d":24988},[24941,42790],{"id":25906,"d":24972},[24941,42792],{"id":25934,"d":25935},[25022,42794,42795],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,42796,42797,42802,42806,42811,42815,42819],{"dataMmlNode":25028},[25022,42798,42799],{"dataMmlNode":25942},[25033,42800],{"dataC":27570,"xLinkHref":42801},"#MJX-6-TEX-I-1D449",[25022,42803,42804],{"dataMmlNode":25031,"transform":42562},[25033,42805],{"dataC":25035,"xLinkHref":25994},[25022,42807,42808],{"dataMmlNode":25942,"transform":42567},[25033,42809],{"dataC":42570,"xLinkHref":42810},"#MJX-6-TEX-I-1D44F",[25022,42812,42813],{"dataMmlNode":25031,"transform":42574},[25033,42814],{"dataC":25108,"xLinkHref":26007},[25022,42816,42817],{"dataMmlNode":25031,"transform":42634},[25033,42818],{"dataC":25078,"xLinkHref":26013},[25022,42820,42821],{"dataMmlNode":26031,"transform":42640},[25033,42822],{"dataC":26103,"xLinkHref":26104},[24923,42824,42826],{"className":42825,"jax":24927},[24926],[24929,42827,42828,42833],{"style":25196,"xmlns":24932,"width":42655,"height":31523,"role":1930,"focusable":8058,"viewBox":42656,"xmlnsXLink":24936},[24938,42829,42830],{},[24941,42831],{"id":42832,"d":42548},"MJX-7-TEX-I-1D44F",[25022,42834,42835],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,42836,42837],{"dataMmlNode":25028},[25022,42838,42839],{"dataMmlNode":25942},[25033,42840],{"dataC":42570,"xLinkHref":42841},"#MJX-7-TEX-I-1D44F"," is a draw",[26,42844,42845,42846,42885],{},"Otherwise, ",[24923,42847,42849],{"className":42848,"jax":24927},[24926],[24929,42850,42851,42863],{"style":24931,"xmlns":24932,"width":42536,"height":24934,"role":1930,"focusable":8058,"viewBox":42537,"xmlnsXLink":24936},[24938,42852,42853,42856,42858,42861],{},[24941,42854],{"id":42855,"d":27385},"MJX-8-TEX-I-1D449",[24941,42857],{"id":32823,"d":24944},[24941,42859],{"id":42860,"d":42548},"MJX-8-TEX-I-1D44F",[24941,42862],{"id":32829,"d":24988},[25022,42864,42865],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,42866,42867,42872,42876,42881],{"dataMmlNode":25028},[25022,42868,42869],{"dataMmlNode":25942},[25033,42870],{"dataC":27570,"xLinkHref":42871},"#MJX-8-TEX-I-1D449",[25022,42873,42874],{"dataMmlNode":25031,"transform":42562},[25033,42875],{"dataC":25035,"xLinkHref":32843},[25022,42877,42878],{"dataMmlNode":25942,"transform":42567},[25033,42879],{"dataC":42570,"xLinkHref":42880},"#MJX-8-TEX-I-1D44F",[25022,42882,42883],{"dataMmlNode":25031,"transform":42574},[25033,42884],{"dataC":25108,"xLinkHref":32853}," equals the value of the best reachable final state under optimal play",[12,42887,42888,42889,6959],{},"The problem? Computing this perfectly requires searching the entire game tree, which is astronomically large. 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1000",[24938,43529,43530,43533,43536,43538],{},[24941,43531],{"id":43532,"d":42548},"MJX-14-TEX-I-1D44F",[24941,43534],{"id":43535,"d":25877},"MJX-14-TEX-I-1D461",[24941,43537],{"id":33146,"d":24944},[24941,43539],{"id":33152,"d":24988},[25022,43541,43542],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,43543,43544,43549,43554,43559,43564],{"dataMmlNode":25028},[25022,43545,43546],{"dataMmlNode":25942},[25033,43547],{"dataC":42570,"xLinkHref":43548},"#MJX-14-TEX-I-1D44F",[25022,43550,43551],{"dataMmlNode":25942,"transform":43332},[25033,43552],{"dataC":25959,"xLinkHref":43553},"#MJX-14-TEX-I-1D461",[25022,43555,43557],{"dataMmlNode":25031,"transform":43556},"translate(790,0)",[25033,43558],{"dataC":25035,"xLinkHref":33194},[25022,43560,43562],{"dataMmlNode":25942,"transform":43561},"translate(1179,0)",[25033,43563],{"dataC":42570,"xLinkHref":43548},[25022,43565,43567],{"dataMmlNode":25031,"transform":43566},"translate(1608,0)",[25033,43568],{"dataC":25108,"xLinkHref":33206}," = number of black pieces under threat",[26,43571,43572,43624],{},[24923,43573,43575],{"className":43574,"jax":24927},[24926],[24929,43576,43579,43594],{"style":24931,"xmlns":24932,"width":43577,"height":24934,"role":1930,"focusable":8058,"viewBox":43578,"xmlnsXLink":24936},"4.568ex","0 -750 2019 1000",[24938,43580,43581,43584,43587,43589,43592],{},[24941,43582],{"id":43583,"d":25881},"MJX-15-TEX-I-1D45F",[24941,43585],{"id":43586,"d":25877},"MJX-15-TEX-I-1D461",[24941,43588],{"id":33346,"d":24944},[24941,43590],{"id":43591,"d":42548},"MJX-15-TEX-I-1D44F",[24941,43593],{"id":33352,"d":24988},[25022,43595,43596],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,43597,43598,43603,43608,43613,43619],{"dataMmlNode":25028},[25022,43599,43600],{"dataMmlNode":25942},[25033,43601],{"dataC":25966,"xLinkHref":43602},"#MJX-15-TEX-I-1D45F",[25022,43604,43605],{"dataMmlNode":25942,"transform":43386},[25033,43606],{"dataC":25959,"xLinkHref":43607},"#MJX-15-TEX-I-1D461",[25022,43609,43611],{"dataMmlNode":25031,"transform":43610},"translate(812,0)",[25033,43612],{"dataC":25035,"xLinkHref":33366},[25022,43614,43616],{"dataMmlNode":25942,"transform":43615},"translate(1201,0)",[25033,43617],{"dataC":42570,"xLinkHref":43618},"#MJX-15-TEX-I-1D44F",[25022,43620,43622],{"dataMmlNode":25031,"transform":43621},"translate(1630,0)",[25033,43623],{"dataC":25108,"xLinkHref":33376}," = number of red pieces under threat",[12,43626,43627,43628,43657,43658,43684,43685,6959],{},"The weights (",[24923,43629,43631],{"className":43630,"jax":24927},[24926],[24929,43632,43635,43640],{"style":32867,"xmlns":24932,"width":43633,"height":32869,"role":1930,"focusable":8058,"viewBox":43634,"xmlnsXLink":24936},"2.977ex","0 -750 1316 950",[24938,43636,43637],{},[24941,43638],{"id":43639,"d":37254},"MJX-16-TEX-I-1D464",[25022,43641,43642],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,43643,43644,43649],{"dataMmlNode":25028},[25022,43645,43646],{"dataMmlNode":25942},[25033,43647],{"dataC":37310,"xLinkHref":43648},"#MJX-16-TEX-I-1D464",[25022,43650,43652],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":43651},"translate(716,0)",[25022,43653,43654],{"dataMmlNode":25031},[6977,43655,43656],{"dataVariant":32892,"transform":25025,"font-size":32893,"font-family":32894},"₀"," through 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from self-play, we don't have direct labels for every board position. Instead, we use ",[106,43696,43697],{},"temporal difference learning",": the estimated value of a board position is updated to be closer to the estimated value of the ",[7289,43700,9765],{}," board position in actual play. Over many games, accurate values from end-game positions gradually \"back up\" to earlier positions.",[7166,43703,43705],{"id":43704},"the-lms-least-mean-squares-algorithm","The LMS (Least Mean Squares) Algorithm",[12,43707,43708],{},"To adjust the weights, we use gradient descent. For each training example:",[3603,43710,43711,43888],{},[26,43712,43713,43714],{},"Compute the error: ",[24923,43715,43717],{"className":43716,"jax":24927},[24926],[24929,43718,43721,43758],{"style":24931,"xmlns":24932,"width":43719,"height":24934,"role":1930,"focusable":8058,"viewBox":43720,"xmlnsXLink":24936},"26.092ex","0 -750 11532.4 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scale(0.707)",[25022,43822,43823],{"dataMmlNode":25942},[25033,43824],{"dataC":25959,"xLinkHref":43825},"#MJX-18-TEX-I-1D461",[25022,43827,43829],{"dataMmlNode":25942,"transform":43828},"translate(361,0)",[25033,43830],{"dataC":25966,"xLinkHref":43772},[25022,43832,43833],{"dataMmlNode":25942,"transform":43610},[25033,43834],{"dataC":27425,"xLinkHref":43835},"#MJX-18-TEX-I-1D44E",[25022,43837,43839],{"dataMmlNode":25942,"transform":43838},"translate(1341,0)",[25033,43840],{"dataC":27210,"xLinkHref":43841},"#MJX-18-TEX-I-1D456",[25022,43843,43845],{"dataMmlNode":25942,"transform":43844},"translate(1686,0)",[25033,43846],{"dataC":25952,"xLinkHref":43847},"#MJX-18-TEX-I-1D45B",[25022,43849,43851],{"dataMmlNode":25031,"transform":43850},"translate(7127,0)",[25033,43852],{"dataC":25035,"xLinkHref":26705},[25022,43854,43856],{"dataMmlNode":25942,"transform":43855},"translate(7516,0)",[25033,43857],{"dataC":42570,"xLinkHref":43799},[25022,43859,43861],{"dataMmlNode":25031,"transform":43860},"translate(7945,0)",[25033,43862],{"dataC":25108,"xLinkHref":26724},[25022,43864,43866],{"dataMmlNode":25031,"transform":43865},"translate(8556.2,0)",[25033,43867],{"dataC":26019,"xLinkHref":26736},[25022,43869,43871],{"dataMmlNode":25942,"transform":43870},"translate(9556.4,0)",[25033,43872],{"dataC":27570,"xLinkHref":43817},[25022,43874,43876],{"dataMmlNode":25031,"transform":43875},"translate(10325.4,0)",[25033,43877],{"dataC":25035,"xLinkHref":26705},[25022,43879,43881],{"dataMmlNode":25942,"transform":43880},"translate(10714.4,0)",[25033,43882],{"dataC":42570,"xLinkHref":43799},[25022,43884,43886],{"dataMmlNode":25031,"transform":43885},"translate(11143.4,0)",[25033,43887],{"dataC":25108,"xLinkHref":26724},[26,43889,43890,43891],{},"Update each weight: ",[24923,43892,43894],{"className":43893,"jax":24927},[24926],[24929,43895,43898,43936],{"style":24931,"xmlns":24932,"width":43896,"height":24934,"role":1930,"focusable":8058,"viewBox":43897,"xmlnsXLink":24936},"24.542ex","0 -750 10847.7 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scale(0.707)",[25033,43998],{"dataC":27210,"xLinkHref":43951},[25022,44000,44002],{"dataMmlNode":25031,"transform":44001},"translate(6836.5,0)",[25033,44003],{"dataC":27679,"xLinkHref":43985},[25022,44005,44007],{"dataMmlNode":25942,"transform":44006},"translate(7336.7,0)",[25033,44008],{"dataC":27596,"xLinkHref":44009},"#MJX-19-TEX-I-1D452",[25022,44011,44013],{"dataMmlNode":25942,"transform":44012},"translate(7802.7,0)",[25033,44014],{"dataC":25966,"xLinkHref":44015},"#MJX-19-TEX-I-1D45F",[25022,44017,44019],{"dataMmlNode":25942,"transform":44018},"translate(8253.7,0)",[25033,44020],{"dataC":25966,"xLinkHref":44015},[25022,44022,44024],{"dataMmlNode":25942,"transform":44023},"translate(8704.7,0)",[25033,44025],{"dataC":25973,"xLinkHref":44026},"#MJX-19-TEX-I-1D45C",[25022,44028,44030],{"dataMmlNode":25942,"transform":44029},"translate(9189.7,0)",[25033,44031],{"dataC":25966,"xLinkHref":44015},[25022,44033,44035],{"dataMmlNode":25031,"transform":44034},"translate(9640.7,0)",[25033,44036],{"dataC":25035,"xLinkHref":26821},[25022,44038,44040],{"dataMmlNode":25942,"transform":44039},"translate(10029.7,0)",[25033,44041],{"dataC":42570,"xLinkHref":44042},"#MJX-19-TEX-I-1D44F",[25022,44044,44046],{"dataMmlNode":25031,"transform":44045},"translate(10458.7,0)",[25033,44047],{"dataC":25108,"xLinkHref":26841},[12,44049,33794,44050,44069,44070,44099],{},[24923,44051,44053],{"className":44052,"jax":24927},[24926],[24929,44054,44055,44060],{"style":25196,"xmlns":24932,"width":27037,"height":27038,"role":1930,"focusable":8058,"viewBox":27039,"xmlnsXLink":24936},[24938,44056,44057],{},[24941,44058],{"id":44059,"d":27045},"MJX-20-TEX-I-1D450",[25022,44061,44062],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44063,44064],{"dataMmlNode":25028},[25022,44065,44066],{"dataMmlNode":25942},[25033,44067],{"dataC":27054,"xLinkHref":44068},"#MJX-20-TEX-I-1D450"," is a small learning rate, and ",[24923,44071,44073],{"className":44072,"jax":24927},[24926],[24929,44074,44078,44083],{"style":29805,"xmlns":24932,"width":44075,"height":44076,"role":1930,"focusable":8058,"viewBox":44077,"xmlnsXLink":24936},"2.602ex","2.161ex","0 -750 1150 955",[24938,44079,44080],{},[24941,44081],{"id":44082,"d":29821},"MJX-21-TEX-I-1D453",[25022,44084,44085],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44086,44087,44092],{"dataMmlNode":25028},[25022,44088,44089],{"dataMmlNode":25942},[25033,44090],{"dataC":29864,"xLinkHref":44091},"#MJX-21-TEX-I-1D453",[25022,44093,44095],{"dataMmlNode":25942,"transform":44094},"translate(550,0)",[6977,44096,44098],{"dataVariant":44097,"transform":25025,"font-size":32893,"font-family":32894,"font-style":44097},"italic","ᵢ"," is the value of the i-th feature for board b.",[25631,44101,44103,44106],{"className":44102},[13441,13442],[25635,44104,44105],{},"Deriving LMS",[13438,44107,44109,44158,44380,44450,44474,44611,44614,44617,44792,44795,44832,44835,44943,44988,45050,45121,45130,45133,45263,45546,45549,45752,45755,45923,45929,45952,46079,46082,46243,46246,46402,46404,46496,46591,46594,46670,46672,46678,46698,46701,46705,46775,46781,46916,46921,47063],{"className":44108},[25641],[12,44110,44111,44112,6959,44155,44157],{},"Let's take the guess from our AI, and call it ",[24923,44113,44115],{"className":44114,"jax":24927},[24926],[24929,44116,44117,44131],{"style":24931,"xmlns":24932,"width":42536,"height":24934,"role":1930,"focusable":8058,"viewBox":42537,"xmlnsXLink":24936},[24938,44118,44119,44122,44125,44128],{},[24941,44120],{"id":44121,"d":27385},"MJX-22-TEX-I-1D449",[24941,44123],{"id":44124,"d":24944},"MJX-22-TEX-N-28",[24941,44126],{"id":44127,"d":42548},"MJX-22-TEX-I-1D44F",[24941,44129],{"id":44130,"d":24988},"MJX-22-TEX-N-29",[25022,44132,44133],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44134,44135,44140,44145,44150],{"dataMmlNode":25028},[25022,44136,44137],{"dataMmlNode":25942},[25033,44138],{"dataC":27570,"xLinkHref":44139},"#MJX-22-TEX-I-1D449",[25022,44141,44142],{"dataMmlNode":25031,"transform":42562},[25033,44143],{"dataC":25035,"xLinkHref":44144},"#MJX-22-TEX-N-28",[25022,44146,44147],{"dataMmlNode":25942,"transform":42567},[25033,44148],{"dataC":42570,"xLinkHref":44149},"#MJX-22-TEX-I-1D44F",[25022,44151,44152],{"dataMmlNode":25031,"transform":42574},[25033,44153],{"dataC":25108,"xLinkHref":44154},"#MJX-22-TEX-N-29",[16,44156],{},"\nAI is just taking an approximation here, or just guessing random number, we can't really tell, so we can't rely on it entirely.",[12,44159,44160,44161,44241,44242,44244,44245,6959],{},"So, we define another variable, ",[24923,44162,44164],{"className":44163,"jax":24927},[24926],[24929,44165,44168,44191],{"style":24931,"xmlns":24932,"width":44166,"height":24934,"role":1930,"focusable":8058,"viewBox":44167,"xmlnsXLink":24936},"7.895ex","0 -750 3489.4 1000",[24938,44169,44170,44173,44175,44177,44180,44182,44184,44186,44189],{},[24941,44171],{"id":44172,"d":27385},"MJX-23-TEX-I-1D449",[24941,44174],{"id":27076,"d":25877},[24941,44176],{"id":27079,"d":25881},[24941,44178],{"id":44179,"d":27323},"MJX-23-TEX-I-1D44E",[24941,44181],{"id":27107,"d":27108},[24941,44183],{"id":27073,"d":25873},[24941,44185],{"id":27091,"d":24944},[24941,44187],{"id":44188,"d":42548},"MJX-23-TEX-I-1D44F",[24941,44190],{"id":27097,"d":24988},[25022,44192,44193],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44194,44195,44225,44230,44236],{"dataMmlNode":25028},[25022,44196,44197,44202],{"dataMmlNode":26023},[25022,44198,44199],{"dataMmlNode":25942},[25033,44200],{"dataC":27570,"xLinkHref":44201},"#MJX-23-TEX-I-1D449",[25022,44203,44204,44208,44212,44217,44221],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,44205,44206],{"dataMmlNode":25942},[25033,44207],{"dataC":25959,"xLinkHref":27151},[25022,44209,44210],{"dataMmlNode":25942,"transform":43828},[25033,44211],{"dataC":25966,"xLinkHref":27156},[25022,44213,44214],{"dataMmlNode":25942,"transform":43610},[25033,44215],{"dataC":27425,"xLinkHref":44216},"#MJX-23-TEX-I-1D44E",[25022,44218,44219],{"dataMmlNode":25942,"transform":43838},[25033,44220],{"dataC":27210,"xLinkHref":27211},[25022,44222,44223],{"dataMmlNode":25942,"transform":43844},[25033,44224],{"dataC":25952,"xLinkHref":27146},[25022,44226,44228],{"dataMmlNode":25031,"transform":44227},"translate(2282.4,0)",[25033,44229],{"dataC":25035,"xLinkHref":27176},[25022,44231,44233],{"dataMmlNode":25942,"transform":44232},"translate(2671.4,0)",[25033,44234],{"dataC":42570,"xLinkHref":44235},"#MJX-23-TEX-I-1D44F",[25022,44237,44239],{"dataMmlNode":25031,"transform":44238},"translate(3100.4,0)",[25033,44240],{"dataC":25108,"xLinkHref":27186}," that represents the answer that the AI should have guessed.",[16,44243],{},"\nAnd the raw mistake is just ",[24923,44246,44248],{"className":44247,"jax":24927},[24926],[24929,44249,44250,44279],{"style":24931,"xmlns":24932,"width":43719,"height":24934,"role":1930,"focusable":8058,"viewBox":43720,"xmlnsXLink":24936},[24938,44251,44252,44254,44256,44258,44260,44263,44265,44267,44269,44271,44273,44275,44277],{},[24941,44253],{"id":27396,"d":27397},[24941,44255],{"id":27358,"d":25881},[24941,44257],{"id":27361,"d":25885},[24941,44259],{"id":27332,"d":24944},[24941,44261],{"id":44262,"d":42548},"MJX-24-TEX-I-1D44F",[24941,44264],{"id":27346,"d":24988},[24941,44266],{"id":27349,"d":24972},[24941,44268],{"id":27384,"d":27385},[24941,44270],{"id":27355,"d":25877},[24941,44272],{"id":27322,"d":27323},[24941,44274],{"id":27326,"d":27108},[24941,44276],{"id":27329,"d":25873},[24941,44278],{"id":27370,"d":25910},[25022,44280,44281],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44282,44283,44287,44291,44295,44299,44303,44307,44312,44316,44320,44348,44352,44356,44360,44364,44368,44372,44376],{"dataMmlNode":25028},[25022,44284,44285],{"dataMmlNode":25942},[25033,44286],{"dataC":27596,"xLinkHref":27597},[25022,44288,44289],{"dataMmlNode":25942,"transform":43769},[25033,44290],{"dataC":25966,"xLinkHref":27499},[25022,44292,44293],{"dataMmlNode":25942,"transform":43775},[25033,44294],{"dataC":25966,"xLinkHref":27499},[25022,44296,44297],{"dataMmlNode":25942,"transform":43780},[25033,44298],{"dataC":25973,"xLinkHref":27505},[25022,44300,44301],{"dataMmlNode":25942,"transform":43786},[25033,44302],{"dataC":25966,"xLinkHref":27499},[25022,44304,44305],{"dataMmlNode":25031,"transform":43791},[25033,44306],{"dataC":25035,"xLinkHref":27444},[25022,44308,44309],{"dataMmlNode":25942,"transform":43796},[25033,44310],{"dataC":42570,"xLinkHref":44311},"#MJX-24-TEX-I-1D44F",[25022,44313,44314],{"dataMmlNode":25031,"transform":43802},[25033,44315],{"dataC":25108,"xLinkHref":27470},[25022,44317,44318],{"dataMmlNode":25031,"transform":43807},[25033,44319],{"dataC":25078,"xLinkHref":27476},[25022,44321,44322,44326],{"dataMmlNode":26023,"transform":43812},[25022,44323,44324],{"dataMmlNode":25942},[25033,44325],{"dataC":27570,"xLinkHref":27571},[25022,44327,44328,44332,44336,44340,44344],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,44329,44330],{"dataMmlNode":25942},[25033,44331],{"dataC":25959,"xLinkHref":27493},[25022,44333,44334],{"dataMmlNode":25942,"transform":43828},[25033,44335],{"dataC":25966,"xLinkHref":27499},[25022,44337,44338],{"dataMmlNode":25942,"transform":43610},[25033,44339],{"dataC":27425,"xLinkHref":27426},[25022,44341,44342],{"dataMmlNode":25942,"transform":43838},[25033,44343],{"dataC":27210,"xLinkHref":27432},[25022,44345,44346],{"dataMmlNode":25942,"transform":43844},[25033,44347],{"dataC":25952,"xLinkHref":27438},[25022,44349,44350],{"dataMmlNode":25031,"transform":43850},[25033,44351],{"dataC":25035,"xLinkHref":27444},[25022,44353,44354],{"dataMmlNode":25942,"transform":43855},[25033,44355],{"dataC":42570,"xLinkHref":44311},[25022,44357,44358],{"dataMmlNode":25031,"transform":43860},[25033,44359],{"dataC":25108,"xLinkHref":27470},[25022,44361,44362],{"dataMmlNode":25031,"transform":43865},[25033,44363],{"dataC":26019,"xLinkHref":27538},[25022,44365,44366],{"dataMmlNode":25942,"transform":43870},[25033,44367],{"dataC":27570,"xLinkHref":27571},[25022,44369,44370],{"dataMmlNode":25031,"transform":43875},[25033,44371],{"dataC":25035,"xLinkHref":27444},[25022,44373,44374],{"dataMmlNode":25942,"transform":43880},[25033,44375],{"dataC":42570,"xLinkHref":44311},[25022,44377,44378],{"dataMmlNode":25031,"transform":43885},[25033,44379],{"dataC":25108,"xLinkHref":27470},[12,44381,44382,44383,44449],{},"Let's place ",[24923,44384,44386],{"className":44385,"jax":24927},[24926],[24929,44387,44391,44411],{"style":27744,"xmlns":24932,"width":44388,"height":44389,"role":1930,"focusable":8058,"viewBox":44390,"xmlnsXLink":24936},"5.164ex","1.902ex","0 -683 2282.4 840.8",[24938,44392,44393,44396,44399,44402,44405,44408],{},[24941,44394],{"id":44395,"d":27385},"MJX-25-TEX-I-1D449",[24941,44397],{"id":44398,"d":25877},"MJX-25-TEX-I-1D461",[24941,44400],{"id":44401,"d":25881},"MJX-25-TEX-I-1D45F",[24941,44403],{"id":44404,"d":27323},"MJX-25-TEX-I-1D44E",[24941,44406],{"id":44407,"d":27108},"MJX-25-TEX-I-1D456",[24941,44409],{"id":44410,"d":25873},"MJX-25-TEX-I-1D45B",[25022,44412,44413],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44414,44415],{"dataMmlNode":25028},[25022,44416,44417,44422],{"dataMmlNode":26023},[25022,44418,44419],{"dataMmlNode":25942},[25033,44420],{"dataC":27570,"xLinkHref":44421},"#MJX-25-TEX-I-1D449",[25022,44423,44424,44429,44434,44439,44444],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,44425,44426],{"dataMmlNode":25942},[25033,44427],{"dataC":25959,"xLinkHref":44428},"#MJX-25-TEX-I-1D461",[25022,44430,44431],{"dataMmlNode":25942,"transform":43828},[25033,44432],{"dataC":25966,"xLinkHref":44433},"#MJX-25-TEX-I-1D45F",[25022,44435,44436],{"dataMmlNode":25942,"transform":43610},[25033,44437],{"dataC":27425,"xLinkHref":44438},"#MJX-25-TEX-I-1D44E",[25022,44440,44441],{"dataMmlNode":25942,"transform":43838},[25033,44442],{"dataC":27210,"xLinkHref":44443},"#MJX-25-TEX-I-1D456",[25022,44445,44446],{"dataMmlNode":25942,"transform":43844},[25033,44447],{"dataC":25952,"xLinkHref":44448},"#MJX-25-TEX-I-1D45B"," aside first, and just pretend that it's a magical function that spits out the accurate score every time.",[12,44451,44452,44453,6959],{},"Now, we need a way to tell AI how bad it is doing overall. That's our cost function, let's call it ",[24923,44454,44456],{"className":44455,"jax":24927},[24926],[24929,44457,44460,44465],{"style":27778,"xmlns":24932,"width":44458,"height":31523,"role":1930,"focusable":8058,"viewBox":44459,"xmlnsXLink":24936},"1.432ex","0 -683 633 705",[24938,44461,44462],{},[24941,44463],{"id":44464,"d":33143},"MJX-26-TEX-I-1D43D",[25022,44466,44467],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44468,44469],{"dataMmlNode":25028},[25022,44470,44471],{"dataMmlNode":25942},[25033,44472],{"dataC":33187,"xLinkHref":44473},"#MJX-26-TEX-I-1D43D",[24923,44475,44477],{"className":44476,"jax":24927,"display":13594},[24926],[24929,44478,44481,44513],{"style":24931,"xmlns":24932,"width":44479,"height":24934,"role":1930,"focusable":8058,"viewBox":44480,"xmlnsXLink":24936},"21.34ex","0 -750 9432.4 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this creates a problem: A huge positive error (guessing too low) and a huge negative error (guessing too high) will cancel each other out, making it look like the model is doing a great job when it isn't.",[12,44615,44616],{},"By squaring the error:",[24923,44618,44620],{"className":44619,"jax":24927,"display":13594},[24926],[24929,44621,44626,44667],{"style":44622,"xmlns":24932,"width":44623,"height":44624,"role":1930,"focusable":8058,"viewBox":44625,"xmlnsXLink":24936},"vertical-align: -1.552ex;","26.203ex","4.588ex","0 -1342 11581.6 2028",[24938,44627,44628,44631,44633,44635,44637,44639,44642,44645,44648,44651,44653,44656,44659,44662,44664],{},[24941,44629],{"id":44630,"d":33143},"MJX-28-TEX-I-1D43D",[24941,44632],{"id":34422,"d":24972},[24941,44634],{"id":34434,"d":25914},[24941,44636],{"id":34437,"d":25928},[24941,44638],{"id":34410,"d":24944},[24941,44640],{"id":44641,"d":27385},"MJX-28-TEX-I-1D449",[24941,44643],{"id":44644,"d":25877},"MJX-28-TEX-I-1D461",[24941,44646],{"id":44647,"d":25881},"MJX-28-TEX-I-1D45F",[24941,44649],{"id":44650,"d":27323},"MJX-28-TEX-I-1D44E",[24941,44652],{"id":34416,"d":27108},[24941,44654],{"id":44655,"d":25873},"MJX-28-TEX-I-1D45B",[24941,44657],{"id":44658,"d":180},"MJX-28-TEX-N-200B",[24941,44660],{"id":44661,"d":42548},"MJX-28-TEX-I-1D44F",[24941,44663],{"id":34419,"d":24988},[24941,44665],{"id":44666,"d":25910},"MJX-28-TEX-N-2212",[25022,44668,44669],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44670,44671,44676,44680,44692,44697,44703,44739,44744,44750,44755,44761,44766,44771,44776,44781],{"dataMmlNode":25028},[25022,44672,44673],{"dataMmlNode":25942},[25033,44674],{"dataC":33187,"xLinkHref":44675},"#MJX-28-TEX-I-1D43D",[25022,44677,44678],{"dataMmlNode":25031,"transform":44523},[25033,44679],{"dataC":25078,"xLinkHref":34496},[25022,44681,44682,44686,44690],{"dataMmlNode":27622,"transform":44528},[25022,44683,44684],{"dataMmlNode":26031,"transform":33217},[25033,44685],{"dataC":26035,"xLinkHref":34527},[25022,44687,44688],{"dataMmlNode":26031,"transform":33223},[25033,44689],{"dataC":26058,"xLinkHref":34570},[27668,44691],{"width":33229,"height":27671,"x":27672,"y":27673},[25022,44693,44695],{"dataMmlNode":25031,"transform":44694},"translate(2906.6,0)",[25033,44696],{"dataC":25035,"xLinkHref":34462},[25022,44698,44700],{"dataMmlNode":25942,"transform":44699},"translate(3295.6,0)",[25033,44701],{"dataC":27570,"xLinkHref":44702},"#MJX-28-TEX-I-1D449",[25022,44704,44706,44711,44716,44721,44725,44730],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":44705},"translate(4064.6,0)",[25022,44707,44708],{"dataMmlNode":25942},[25033,44709],{"dataC":25959,"xLinkHref":44710},"#MJX-28-TEX-I-1D461",[25022,44712,44713],{"dataMmlNode":25942,"transform":43828},[25033,44714],{"dataC":25966,"xLinkHref":44715},"#MJX-28-TEX-I-1D45F",[25022,44717,44718],{"dataMmlNode":25942,"transform":43610},[25033,44719],{"dataC":27425,"xLinkHref":44720},"#MJX-28-TEX-I-1D44E",[25022,44722,44723],{"dataMmlNode":25942,"transform":43838},[25033,44724],{"dataC":27210,"xLinkHref":34480},[25022,44726,44727],{"dataMmlNode":25942,"transform":43844},[25033,44728],{"dataC":25952,"xLinkHref":44729},"#MJX-28-TEX-I-1D45B",[25022,44731,44733],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":44732},"translate(2286,0)",[25022,44734,44735],{"dataMmlNode":25031},[25033,44736],{"dataC":44737,"xLinkHref":44738},"200B","#MJX-28-TEX-N-200B",[25022,44740,44742],{"dataMmlNode":25031,"transform":44741},"translate(6350.6,0)",[25033,44743],{"dataC":25035,"xLinkHref":34462},[25022,44745,44747],{"dataMmlNode":25942,"transform":44746},"translate(6739.6,0)",[25033,44748],{"dataC":42570,"xLinkHref":44749},"#MJX-28-TEX-I-1D44F",[25022,44751,44753],{"dataMmlNode":25031,"transform":44752},"translate(7168.6,0)",[25033,44754],{"dataC":25108,"xLinkHref":34485},[25022,44756,44758],{"dataMmlNode":25031,"transform":44757},"translate(7779.8,0)",[25033,44759],{"dataC":26019,"xLinkHref":44760},"#MJX-28-TEX-N-2212",[25022,44762,44764],{"dataMmlNode":25942,"transform":44763},"translate(8780,0)",[25033,44765],{"dataC":27570,"xLinkHref":44702},[25022,44767,44769],{"dataMmlNode":25031,"transform":44768},"translate(9549,0)",[25033,44770],{"dataC":25035,"xLinkHref":34462},[25022,44772,44774],{"dataMmlNode":25942,"transform":44773},"translate(9938,0)",[25033,44775],{"dataC":42570,"xLinkHref":44749},[25022,44777,44779],{"dataMmlNode":25031,"transform":44778},"translate(10367,0)",[25033,44780],{"dataC":25108,"xLinkHref":34485},[25022,44782,44784,44788],{"dataMmlNode":31565,"transform":44783},"translate(10756,0)",[25022,44785,44786],{"dataMmlNode":25031},[25033,44787],{"dataC":25108,"xLinkHref":34485},[25022,44789,44790],{"dataMmlNode":26031,"transform":33326},[25033,44791],{"dataC":26058,"xLinkHref":34570},[12,44793,44794],{},"...we ensure that all errors are positive, and larger errors are heavily penalised.",[12,44796,44797,44798,6959],{},"When you use calculus to find the derivative (gradient) of that squared error equation to figure out how to adjust the weights, the 2 from the exponent drops down, cancels out the ",[24923,44799,44801],{"className":44800,"jax":24927},[24926],[24929,44802,44807,44813],{"style":44803,"xmlns":24932,"width":44804,"height":44805,"role":1930,"focusable":8058,"viewBox":44806,"xmlnsXLink":24936},"vertical-align: -0.781ex;","1.795ex","2.737ex","0 -864.9 793.6 1209.9",[24938,44808,44809,44811],{},[24941,44810],{"id":27920,"d":25914},[24941,44812],{"id":27935,"d":25928},[25022,44814,44815],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44816,44817],{"dataMmlNode":25028},[25022,44818,44819,44824,44829],{"dataMmlNode":27622},[25022,44820,44822],{"dataMmlNode":26031,"transform":44821},"translate(220,394) scale(0.707)",[25033,44823],{"dataC":26035,"xLinkHref":28029},[25022,44825,44827],{"dataMmlNode":26031,"transform":44826},"translate(220,-345) scale(0.707)",[25033,44828],{"dataC":26058,"xLinkHref":28065},[27668,44830],{"width":44831,"height":27671,"x":27672,"y":27673},553.6,[12,44833,44834],{},"Your mission is now to make J as close to zero as possible.",[7351,44836,44837,44872],{},[12,44838,44839,44840,44871],{},"You might ask, why is ",[24923,44841,44843],{"className":44842,"jax":24927},[24926],[24929,44844,44845,44853],{"style":44803,"xmlns":24932,"width":44804,"height":44805,"role":1930,"focusable":8058,"viewBox":44806,"xmlnsXLink":24936},[24938,44846,44847,44850],{},[24941,44848],{"id":44849,"d":25914},"MJX-30-TEX-N-31",[24941,44851],{"id":44852,"d":25928},"MJX-30-TEX-N-32",[25022,44854,44855],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44856,44857],{"dataMmlNode":25028},[25022,44858,44859,44864,44869],{"dataMmlNode":27622},[25022,44860,44861],{"dataMmlNode":26031,"transform":44821},[25033,44862],{"dataC":26035,"xLinkHref":44863},"#MJX-30-TEX-N-31",[25022,44865,44866],{"dataMmlNode":26031,"transform":44826},[25033,44867],{"dataC":26058,"xLinkHref":44868},"#MJX-30-TEX-N-32",[27668,44870],{"width":44831,"height":27671,"x":27672,"y":27673},"  allowed here?",[12,44873,44874,44875,44894,44895,44912,44913,44942],{},"As our aim is to minimise ",[24923,44876,44878],{"className":44877,"jax":24927},[24926],[24929,44879,44880,44885],{"style":27778,"xmlns":24932,"width":44458,"height":31523,"role":1930,"focusable":8058,"viewBox":44459,"xmlnsXLink":24936},[24938,44881,44882],{},[24941,44883],{"id":44884,"d":33143},"MJX-31-TEX-I-1D43D",[25022,44886,44887],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44888,44889],{"dataMmlNode":25028},[25022,44890,44891],{"dataMmlNode":25942},[25033,44892],{"dataC":33187,"xLinkHref":44893},"#MJX-31-TEX-I-1D43D"," (turning it to 0), although we are halving the hill's height and the gradient here, but it doesn't matter as long as we can get to the bottom of the hill. And we also utilise a learning rate ",[24923,44896,44898],{"className":44897,"jax":24927},[24926],[24929,44899,44900,44904],{"style":25196,"xmlns":24932,"width":27037,"height":27038,"role":1930,"focusable":8058,"viewBox":27039,"xmlnsXLink":24936},[24938,44901,44902],{},[24941,44903],{"id":28168,"d":27045},[25022,44905,44906],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44907,44908],{"dataMmlNode":25028},[25022,44909,44910],{"dataMmlNode":25942},[25033,44911],{"dataC":27054,"xLinkHref":28183}," later in the equation too, which can be anything the user wants to be. So this ",[24923,44914,44916],{"className":44915,"jax":24927},[24926],[24929,44917,44918,44925],{"style":44803,"xmlns":24932,"width":44804,"height":44805,"role":1930,"focusable":8058,"viewBox":44806,"xmlnsXLink":24936},[24938,44919,44920,44922],{},[24941,44921],{"id":35175,"d":25914},[24941,44923],{"id":44924,"d":25928},"MJX-33-TEX-N-32",[25022,44926,44927],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44928,44929],{"dataMmlNode":25028},[25022,44930,44931,44935,44940],{"dataMmlNode":27622},[25022,44932,44933],{"dataMmlNode":26031,"transform":44821},[25033,44934],{"dataC":26035,"xLinkHref":35184},[25022,44936,44937],{"dataMmlNode":26031,"transform":44826},[25033,44938],{"dataC":26058,"xLinkHref":44939},"#MJX-33-TEX-N-32",[27668,44941],{"width":44831,"height":27671,"x":27672,"y":27673}," gets absorbed into the learning rate anyways.",[12,44944,44945,44946,44987],{},"To make J smaller, you have to tweak your weights (",[24923,44947,44949],{"className":44948,"jax":24927},[24926],[24929,44950,44954,44964],{"style":27744,"xmlns":24932,"width":44951,"height":44952,"role":1930,"focusable":8058,"viewBox":44953,"xmlnsXLink":24936},"2.36ex","1.359ex","0 -443 1043 600.8",[24938,44955,44956,44959,44961],{},[24941,44957],{"id":44958,"d":37254},"MJX-34-TEX-I-1D464",[24941,44960],{"id":28244,"d":27108},[24941,44962],{"id":44963,"d":180},"MJX-34-TEX-N-200B",[25022,44965,44966],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,44967,44968,44979],{"dataMmlNode":25028},[25022,44969,44970,44975],{"dataMmlNode":26023},[25022,44971,44972],{"dataMmlNode":25942},[25033,44973],{"dataC":37310,"xLinkHref":44974},"#MJX-34-TEX-I-1D464",[25022,44976,44977],{"dataMmlNode":25942,"transform":43003},[25033,44978],{"dataC":27210,"xLinkHref":28253},[25022,44980,44982],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":44981},"translate(1043,0)",[25022,44983,44984],{"dataMmlNode":25031},[25033,44985],{"dataC":44737,"xLinkHref":44986},"#MJX-34-TEX-N-200B","). But do you tweak them up or down?",[12,44989,44990,44991],{},"You ask: ",[7289,44992,44993,44994,45031,45032,45049],{},"\"If I bump up a specific weight (",[24923,44995,44997],{"className":44996,"jax":24927},[24926],[24929,44998,44999,45009],{"style":27744,"xmlns":24932,"width":44951,"height":44952,"role":1930,"focusable":8058,"viewBox":44953,"xmlnsXLink":24936},[24938,45000,45001,45004,45006],{},[24941,45002],{"id":45003,"d":37254},"MJX-35-TEX-I-1D464",[24941,45005],{"id":35237,"d":27108},[24941,45007],{"id":45008,"d":180},"MJX-35-TEX-N-200B",[25022,45010,45011],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,45012,45013,45024],{"dataMmlNode":25028},[25022,45014,45015,45020],{"dataMmlNode":26023},[25022,45016,45017],{"dataMmlNode":25942},[25033,45018],{"dataC":37310,"xLinkHref":45019},"#MJX-35-TEX-I-1D464",[25022,45021,45022],{"dataMmlNode":25942,"transform":43003},[25033,45023],{"dataC":27210,"xLinkHref":35323},[25022,45025,45026],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":44981},[25022,45027,45028],{"dataMmlNode":25031},[25033,45029],{"dataC":44737,"xLinkHref":45030},"#MJX-35-TEX-N-200B",") by a tiny amount, does my total error (",[24923,45033,45035],{"className":45034,"jax":24927},[24926],[24929,45036,45037,45041],{"style":27778,"xmlns":24932,"width":44458,"height":31523,"role":1930,"focusable":8058,"viewBox":44459,"xmlnsXLink":24936},[24938,45038,45039],{},[24941,45040],{"id":35465,"d":33143},[25022,45042,45043],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,45044,45045],{"dataMmlNode":25028},[25022,45046,45047],{"dataMmlNode":25942},[25033,45048],{"dataC":33187,"xLinkHref":35510},") go up or down, and by how much?\"",[12,45051,45052,45053,6959],{},"So, you take the derivative of your Cost function with respect to that specific weight: ",[24923,45054,45056],{"className":45055,"jax":24927},[24926],[24929,45057,45062,45076],{"style":45058,"xmlns":24932,"width":45059,"height":45060,"role":1930,"focusable":8058,"viewBox":45061,"xmlnsXLink":24936},"vertical-align: -1.034ex;","3.569ex","3.07ex","0 -899.6 1577.7 1356.7",[24938,45063,45064,45067,45070,45073],{},[24941,45065],{"id":45066,"d":33624},"MJX-37-TEX-I-1D715",[24941,45068],{"id":45069,"d":33143},"MJX-37-TEX-I-1D43D",[24941,45071],{"id":45072,"d":37254},"MJX-37-TEX-I-1D464",[24941,45074],{"id":45075,"d":27108},"MJX-37-TEX-I-1D456",[25022,45077,45078],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,45079,45080],{"dataMmlNode":25028},[25022,45081,45082,45097,45118],{"dataMmlNode":27622},[25022,45083,45085,45090],{"dataMmlNode":27626,"transform":45084},"translate(364.9,394) scale(0.707)",[25022,45086,45087],{"dataMmlNode":25942},[25033,45088],{"dataC":33697,"xLinkHref":45089},"#MJX-37-TEX-I-1D715",[25022,45091,45092],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":33701},[25022,45093,45094],{"dataMmlNode":25942},[25033,45095],{"dataC":33187,"xLinkHref":45096},"#MJX-37-TEX-I-1D43D",[25022,45098,45100,45104],{"dataMmlNode":27626,"transform":45099},"translate(220,-345.6) scale(0.707)",[25022,45101,45102],{"dataMmlNode":25942},[25033,45103],{"dataC":33697,"xLinkHref":45089},[25022,45105,45106],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":33701},[25022,45107,45108,45113],{"dataMmlNode":26023},[25022,45109,45110],{"dataMmlNode":25942},[25033,45111],{"dataC":37310,"xLinkHref":45112},"#MJX-37-TEX-I-1D464",[25022,45114,45115],{"dataMmlNode":25942,"transform":43003},[25033,45116],{"dataC":27210,"xLinkHref":45117},"#MJX-37-TEX-I-1D456",[27668,45119],{"width":45120,"height":27671,"x":27672,"y":27673},1337.7,[7351,45122,45123],{},[12,45124,45125,45126,6959],{},"Learn more about partial derivatives ",[2330,45127,33440],{"href":45128,"rel":45129},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002FArtificial-Neural-Networks-and-Backpropagation#partial-derivative",[2334],[12,45131,45132],{},"Now we can use chain rule on our cost function.",[12,45134,45135,45136],{},"Outside: ",[24923,45137,45139],{"className":45138,"jax":24927},[24926],[24929,45140,45143,45172],{"style":24931,"xmlns":24932,"width":45141,"height":24934,"role":1930,"focusable":8058,"viewBox":45142,"xmlnsXLink":24936},"16.891ex","0 -750 7465.9 1000",[24938,45144,45145,45147,45150,45152,45154,45157,45159,45161,45164,45167,45169],{},[24941,45146],{"id":29004,"d":24944},[24941,45148],{"id":45149,"d":27385},"MJX-38-TEX-I-1D449",[24941,45151],{"id":28989,"d":25877},[24941,45153],{"id":28992,"d":25881},[24941,45155],{"id":45156,"d":27323},"MJX-38-TEX-I-1D44E",[24941,45158],{"id":35698,"d":27108},[24941,45160],{"id":28986,"d":25873},[24941,45162],{"id":45163,"d":180},"MJX-38-TEX-N-200B",[24941,45165],{"id":45166,"d":42548},"MJX-38-TEX-I-1D44F",[24941,45168],{"id":29010,"d":24988},[24941,45170],{"id":45171,"d":25910},"MJX-38-TEX-N-2212",[25022,45173,45174],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,45175,45176,45180,45217,45221,45227,45232,45238,45243,45248,45253,45258],{"dataMmlNode":25028},[25022,45177,45178],{"dataMmlNode":25031},[25033,45179],{"dataC":25035,"xLinkHref":29054},[25022,45181,45182,45187],{"dataMmlNode":26023,"transform":25040},[25022,45183,45184],{"dataMmlNode":25942},[25033,45185],{"dataC":27570,"xLinkHref":45186},"#MJX-38-TEX-I-1D449",[25022,45188,45189,45193,45197,45202,45206,45210],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,45190,45191],{"dataMmlNode":25942},[25033,45192],{"dataC":25959,"xLinkHref":29029},[25022,45194,45195],{"dataMmlNode":25942,"transform":43828},[25033,45196],{"dataC":25966,"xLinkHref":29034},[25022,45198,45199],{"dataMmlNode":25942,"transform":43610},[25033,45200],{"dataC":27425,"xLinkHref":45201},"#MJX-38-TEX-I-1D44E",[25022,45203,45204],{"dataMmlNode":25942,"transform":43838},[25033,45205],{"dataC":27210,"xLinkHref":35731},[25022,45207,45208],{"dataMmlNode":25942,"transform":43844},[25033,45209],{"dataC":25952,"xLinkHref":29024},[25022,45211,45212],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":44732},[25022,45213,45214],{"dataMmlNode":25031},[25033,45215],{"dataC":44737,"xLinkHref":45216},"#MJX-38-TEX-N-200B",[25022,45218,45219],{"dataMmlNode":25031,"transform":44232},[25033,45220],{"dataC":25035,"xLinkHref":29054},[25022,45222,45224],{"dataMmlNode":25942,"transform":45223},"translate(3060.4,0)",[25033,45225],{"dataC":42570,"xLinkHref":45226},"#MJX-38-TEX-I-1D44F",[25022,45228,45230],{"dataMmlNode":25031,"transform":45229},"translate(3489.4,0)",[25033,45231],{"dataC":25108,"xLinkHref":29064},[25022,45233,45235],{"dataMmlNode":25031,"transform":45234},"translate(4100.7,0)",[25033,45236],{"dataC":26019,"xLinkHref":45237},"#MJX-38-TEX-N-2212",[25022,45239,45241],{"dataMmlNode":25942,"transform":45240},"translate(5100.9,0)",[25033,45242],{"dataC":27570,"xLinkHref":45186},[25022,45244,45246],{"dataMmlNode":25031,"transform":45245},"translate(5869.9,0)",[25033,45247],{"dataC":25035,"xLinkHref":29054},[25022,45249,45251],{"dataMmlNode":25942,"transform":45250},"translate(6258.9,0)",[25033,45252],{"dataC":42570,"xLinkHref":45226},[25022,45254,45256],{"dataMmlNode":25031,"transform":45255},"translate(6687.9,0)",[25033,45257],{"dataC":25108,"xLinkHref":29064},[25022,45259,45261],{"dataMmlNode":25031,"transform":45260},"translate(7076.9,0)",[25033,45262],{"dataC":25108,"xLinkHref":29064},[12,45264,45265,45266,45309,45310,45448,45449,45476,45477,45506,45507,45510,45511,45545],{},"Inside: The guess 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is made of ",[24923,45311,45313],{"className":45312,"jax":24927},[24926],[24929,45314,45317,45342],{"style":29805,"xmlns":24932,"width":45315,"height":29807,"role":1930,"focusable":8058,"viewBox":45316,"xmlnsXLink":24936},"20.575ex","0 -705 9094.3 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etc. If you are taking the derivative for just one specific weight (",[24923,45450,45452],{"className":45451,"jax":24927},[24926],[24929,45453,45454,45461],{"style":27744,"xmlns":24932,"width":44951,"height":44952,"role":1930,"focusable":8058,"viewBox":44953,"xmlnsXLink":24936},[24938,45455,45456,45459],{},[24941,45457],{"id":45458,"d":37254},"MJX-41-TEX-I-1D464",[24941,45460],{"id":35808,"d":27108},[25022,45462,45463],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,45464,45465],{"dataMmlNode":25028},[25022,45466,45467,45472],{"dataMmlNode":26023},[25022,45468,45469],{"dataMmlNode":25942},[25033,45470],{"dataC":37310,"xLinkHref":45471},"#MJX-41-TEX-I-1D464",[25022,45473,45474],{"dataMmlNode":25942,"transform":43003},[25033,45475],{"dataC":27210,"xLinkHref":35835},"​), all the other weights vanish because they are constants. The only thing left is the feature attached to it (",[24923,45478,45480],{"className":45479,"jax":24927},[24926],[24929,45481,45484,45491],{"style":29805,"xmlns":24932,"width":45482,"height":29807,"role":1930,"focusable":8058,"viewBox":45483,"xmlnsXLink":24936},"1.848ex","0 -705 817 910",[24938,45485,45486,45489],{},[24941,45487],{"id":45488,"d":29821},"MJX-42-TEX-I-1D453",[24941,45490],{"id":35869,"d":27108},[25022,45492,45493],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,45494,45495],{"dataMmlNode":25028},[25022,45496,45497,45502],{"dataMmlNode":26023},[25022,45498,45499],{"dataMmlNode":25942},[25033,45500],{"dataC":29864,"xLinkHref":45501},"#MJX-42-TEX-I-1D453",[25022,45503,45504],{"dataMmlNode":25942,"transform":43996},[25033,45505],{"dataC":27210,"xLinkHref":35922},"). Because the guess is being ",[7289,45508,45509],{},"subtracted"," in the raw error, a negative sign pops out: ",[24923,45512,45514],{"className":45513,"jax":24927},[24926],[24929,45515,45517,45526],{"style":29805,"xmlns":24932,"width":29490,"height":29807,"role":1930,"focusable":8058,"viewBox":45516,"xmlnsXLink":24936},"0 -705 1595 910",[24938,45518,45519,45521,45524],{},[24941,45520],{"id":36040,"d":25910},[24941,45522],{"id":45523,"d":29821},"MJX-43-TEX-I-1D453",[24941,45525],{"id":36024,"d":27108},[25022,45527,45528],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,45529,45530,45534],{"dataMmlNode":25028},[25022,45531,45532],{"dataMmlNode":25031},[25033,45533],{"dataC":26019,"xLinkHref":36177},[25022,45535,45536,45541],{"dataMmlNode":26023,"transform":25049},[25022,45537,45538],{"dataMmlNode":25942},[25033,45539],{"dataC":29864,"xLinkHref":45540},"#MJX-43-TEX-I-1D453",[25022,45542,45543],{"dataMmlNode":25942,"transform":43996},[25033,45544],{"dataC":27210,"xLinkHref":36122},"​.",[12,45547,45548],{},"Multiply the outside and the inside together, and you get the gradient:",[24923,45550,45552],{"className":45551,"jax":24927,"display":13594},[24926],[24929,45553,45556,45597],{"style":24931,"xmlns":24932,"width":45554,"height":24934,"role":1930,"focusable":8058,"viewBox":45555,"xmlnsXLink":24936},"34.332ex","0 -750 15174.8 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we can write simply as:",[24923,45756,45758],{"className":45757,"jax":24927,"display":13594},[24926],[24929,45759,45762,45809],{"style":24931,"xmlns":24932,"width":45760,"height":24934,"role":1930,"focusable":8058,"viewBox":45761,"xmlnsXLink":24936},"25.385ex","0 -750 11220 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",[24923,46409,46411],{"className":46410,"jax":24927},[24926],[24929,46412,46413,46442],{"style":24931,"xmlns":24932,"width":44166,"height":24934,"role":1930,"focusable":8058,"viewBox":44167,"xmlnsXLink":24936},[24938,46414,46415,46418,46421,46424,46427,46430,46433,46436,46439],{},[24941,46416],{"id":46417,"d":27385},"MJX-50-TEX-I-1D449",[24941,46419],{"id":46420,"d":25877},"MJX-50-TEX-I-1D461",[24941,46422],{"id":46423,"d":25881},"MJX-50-TEX-I-1D45F",[24941,46425],{"id":46426,"d":27323},"MJX-50-TEX-I-1D44E",[24941,46428],{"id":46429,"d":27108},"MJX-50-TEX-I-1D456",[24941,46431],{"id":46432,"d":25873},"MJX-50-TEX-I-1D45B",[24941,46434],{"id":46435,"d":24944},"MJX-50-TEX-N-28",[24941,46437],{"id":46438,"d":42548},"MJX-50-TEX-I-1D44F",[24941,46440],{"id":46441,"d":24988},"MJX-50-TEX-N-29",[25022,46443,46444],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,46445,46446,46480,46485,46490],{"dataMmlNode":25028},[25022,46447,46448,46453],{"dataMmlNode":26023},[25022,46449,46450],{"dataMmlNode":25942},[25033,46451],{"dataC":27570,"xLinkHref":46452},"#MJX-50-TEX-I-1D449",[25022,46454,46455,46460,46465,46470,46475],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,46456,46457],{"dataMmlNode":25942},[25033,46458],{"dataC":25959,"xLinkHref":46459},"#MJX-50-TEX-I-1D461",[25022,46461,46462],{"dataMmlNode":25942,"transform":43828},[25033,46463],{"dataC":25966,"xLinkHref":46464},"#MJX-50-TEX-I-1D45F",[25022,46466,46467],{"dataMmlNode":25942,"transform":43610},[25033,46468],{"dataC":27425,"xLinkHref":46469},"#MJX-50-TEX-I-1D44E",[25022,46471,46472],{"dataMmlNode":25942,"transform":43838},[25033,46473],{"dataC":27210,"xLinkHref":46474},"#MJX-50-TEX-I-1D456",[25022,46476,46477],{"dataMmlNode":25942,"transform":43844},[25033,46478],{"dataC":25952,"xLinkHref":46479},"#MJX-50-TEX-I-1D45B",[25022,46481,46482],{"dataMmlNode":25031,"transform":44227},[25033,46483],{"dataC":25035,"xLinkHref":46484},"#MJX-50-TEX-N-28",[25022,46486,46487],{"dataMmlNode":25942,"transform":44232},[25033,46488],{"dataC":42570,"xLinkHref":46489},"#MJX-50-TEX-I-1D44F",[25022,46491,46492],{"dataMmlNode":25031,"transform":44238},[25033,46493],{"dataC":25108,"xLinkHref":46494},"#MJX-50-TEX-N-29"," then?",[12,46497,46498,46499,46590],{},"The LMS math works perfectly, but it relies entirely on having 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the \"true\" score.",[12,46592,46593],{},"There is no human sitting there labelling every single move with a perfect 1 to 100 score.",[12,46595,46596,46597,8512],{},"So, how does the AI generate its own ",[24923,46598,46600],{"className":46599,"jax":24927},[24926],[24929,46601,46602,46625],{"style":27744,"xmlns":24932,"width":44388,"height":44389,"role":1930,"focusable":8058,"viewBox":44390,"xmlnsXLink":24936},[24938,46603,46604,46607,46610,46613,46616,46619,46622],{},[24941,46605],{"id":46606,"d":27385},"MJX-52-TEX-I-1D449",[24941,46608],{"id":46609,"d":25877},"MJX-52-TEX-I-1D461",[24941,46611],{"id":46612,"d":25881},"MJX-52-TEX-I-1D45F",[24941,46614],{"id":46615,"d":27323},"MJX-52-TEX-I-1D44E",[24941,46617],{"id":46618,"d":27108},"MJX-52-TEX-I-1D456",[24941,46620],{"id":46621,"d":25873},"MJX-52-TEX-I-1D45B",[24941,46623],{"id":46624,"d":180},"MJX-52-TEX-N-200B",[25022,46626,46627],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,46628,46629],{"dataMmlNode":25028},[25022,46630,46631,46636],{"dataMmlNode":26023},[25022,46632,46633],{"dataMmlNode":25942},[25033,46634],{"dataC":27570,"xLinkHref":46635},"#MJX-52-TEX-I-1D449",[25022,46637,46638,46643,46648,46653,46658,46663],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,46639,46640],{"dataMmlNode":25942},[25033,46641],{"dataC":25959,"xLinkHref":46642},"#MJX-52-TEX-I-1D461",[25022,46644,46645],{"dataMmlNode":25942,"transform":43828},[25033,46646],{"dataC":25966,"xLinkHref":46647},"#MJX-52-TEX-I-1D45F",[25022,46649,46650],{"dataMmlNode":25942,"transform":43610},[25033,46651],{"dataC":27425,"xLinkHref":46652},"#MJX-52-TEX-I-1D44E",[25022,46654,46655],{"dataMmlNode":25942,"transform":43838},[25033,46656],{"dataC":27210,"xLinkHref":46657},"#MJX-52-TEX-I-1D456",[25022,46659,46660],{"dataMmlNode":25942,"transform":43844},[25033,46661],{"dataC":25952,"xLinkHref":46662},"#MJX-52-TEX-I-1D45B",[25022,46664,46665],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":44732},[25022,46666,46667],{"dataMmlNode":25031},[25033,46668],{"dataC":44737,"xLinkHref":46669},"#MJX-52-TEX-N-200B",[16,46671],{},[12,46673,46674,46675,7024],{},"The AI doesn't know the exact value of a mid-game board, but the rules of Checkers provide an absolute, undeniable mathematical truth at the very end of the game. These are called ",[106,46676,46677],{},"Terminal States",[23,46679,46680,46686,46692],{},[26,46681,46682,46685],{},[106,46683,46684],{},"Win"," = 100",[26,46687,46688,46691],{},[106,46689,46690],{},"Loss"," = −100",[26,46693,46694,46697],{},[106,46695,46696],{},"Draw"," = 0",[12,46699,46700],{},"The AI is not allowed to guess the score of a Terminal State. The game environment forces these numbers to be the absolute truth. Everything the AI learns is anchored to these final outcomes.",[26236,46702,46704],{"id":46703},"temporal-difference-td-learning","Temporal Difference (TD) Learning",[12,46706,46707,46708,46771,46772],{},"Since the AI lacks a true ",[24923,46709,46711],{"className":46710,"jax":24927},[24926],[24929,46712,46713,46731],{"style":27744,"xmlns":24932,"width":44388,"height":44389,"role":1930,"focusable":8058,"viewBox":44390,"xmlnsXLink":24936},[24938,46714,46715,46718,46720,46722,46724,46726,46728],{},[24941,46716],{"id":46717,"d":27385},"MJX-53-TEX-I-1D449",[24941,46719],{"id":29836,"d":25877},[24941,46721],{"id":29827,"d":25881},[24941,46723],{"id":29833,"d":27323},[24941,46725],{"id":29839,"d":27108},[24941,46727],{"id":29817,"d":25873},[24941,46729],{"id":46730,"d":180},"MJX-53-TEX-N-200B",[25022,46732,46733],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,46734,46735],{"dataMmlNode":25028},[25022,46736,46737,46742],{"dataMmlNode":26023},[25022,46738,46739],{"dataMmlNode":25942},[25033,46740],{"dataC":27570,"xLinkHref":46741},"#MJX-53-TEX-I-1D449",[25022,46743,46744,46748,46752,46756,46760,46764],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,46745,46746],{"dataMmlNode":25942},[25033,46747],{"dataC":25959,"xLinkHref":29894},[25022,46749,46750],{"dataMmlNode":25942,"transform":43828},[25033,46751],{"dataC":25966,"xLinkHref":29876},[25022,46753,46754],{"dataMmlNode":25942,"transform":43610},[25033,46755],{"dataC":27425,"xLinkHref":29888},[25022,46757,46758],{"dataMmlNode":25942,"transform":43838},[25033,46759],{"dataC":27210,"xLinkHref":29900},[25022,46761,46762],{"dataMmlNode":25942,"transform":43844},[25033,46763],{"dataC":25952,"xLinkHref":29858},[25022,46765,46766],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":44732},[25022,46767,46768],{"dataMmlNode":25031},[25033,46769],{"dataC":44737,"xLinkHref":46770},"#MJX-53-TEX-N-200B"," for intermediate moves (like Move 10), it uses a clever trick: ",[106,46773,46774],{},"It uses its own prediction from Move 11 as the \"truth\" for Move 10.",[12,46776,46777,46778],{},"This is Temporal Difference learning. The AI essentially says: ",[7289,46779,46780],{},"\"I have better, more updated information after making a move than I did before making it. Therefore, my guess for the next state is a better target than my current guess.\"",[24923,46782,46784],{"className":46783,"jax":24927,"display":13594},[24926],[24929,46785,46788,46819],{"style":24931,"xmlns":24932,"width":46786,"height":24934,"role":1930,"focusable":8058,"viewBox":46787,"xmlnsXLink":24936},"18.347ex","0 -750 8109.5 1000",[24938,46789,46790,46793,46795,46797,46800,46802,46804,46806,46809,46811,46813,46816],{},[24941,46791],{"id":46792,"d":27385},"MJX-54-TEX-I-1D449",[24941,46794],{"id":30047,"d":25877},[24941,46796],{"id":30050,"d":25881},[24941,46798],{"id":46799,"d":27323},"MJX-54-TEX-I-1D44E",[24941,46801],{"id":37548,"d":27108},[24941,46803],{"id":30044,"d":25873},[24941,46805],{"id":30062,"d":24944},[24941,46807],{"id":46808,"d":42548},"MJX-54-TEX-I-1D44F",[24941,46810],{"id":30068,"d":24988},[24941,46812],{"id":30071,"d":24972},[24941,46814],{"id":46815,"d":27397},"MJX-54-TEX-I-1D452",[24941,46817],{"id":46818,"d":32148},"MJX-54-TEX-I-1D465",[25022,46820,46821],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,46822,46823,46853,46857,46862,46866,46871,46901,46906,46911],{"dataMmlNode":25028},[25022,46824,46825,46830],{"dataMmlNode":26023},[25022,46826,46827],{"dataMmlNode":25942},[25033,46828],{"dataC":27570,"xLinkHref":46829},"#MJX-54-TEX-I-1D449",[25022,46831,46832,46836,46840,46845,46849],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,46833,46834],{"dataMmlNode":25942},[25033,46835],{"dataC":25959,"xLinkHref":30099},[25022,46837,46838],{"dataMmlNode":25942,"transform":43828},[25033,46839],{"dataC":25966,"xLinkHref":30104},[25022,46841,46842],{"dataMmlNode":25942,"transform":43610},[25033,46843],{"dataC":27425,"xLinkHref":46844},"#MJX-54-TEX-I-1D44E",[25022,46846,46847],{"dataMmlNode":25942,"transform":43838},[25033,46848],{"dataC":27210,"xLinkHref":37573},[25022,46850,46851],{"dataMmlNode":25942,"transform":43844},[25033,46852],{"dataC":25952,"xLinkHref":30094},[25022,46854,46855],{"dataMmlNode":25031,"transform":44227},[25033,46856],{"dataC":25035,"xLinkHref":30124},[25022,46858,46859],{"dataMmlNode":25942,"transform":44232},[25033,46860],{"dataC":42570,"xLinkHref":46861},"#MJX-54-TEX-I-1D44F",[25022,46863,46864],{"dataMmlNode":25031,"transform":44238},[25033,46865],{"dataC":25108,"xLinkHref":30134},[25022,46867,46869],{"dataMmlNode":25031,"transform":46868},"translate(3767.2,0)",[25033,46870],{"dataC":25078,"xLinkHref":30139},[25022,46872,46874,46878],{"dataMmlNode":26023,"transform":46873},"translate(4823,0)",[25022,46875,46876],{"dataMmlNode":25942},[25033,46877],{"dataC":27570,"xLinkHref":46829},[25022,46879,46880,46884,46890,46896],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,46881,46882],{"dataMmlNode":25942},[25033,46883],{"dataC":25952,"xLinkHref":30094},[25022,46885,46887],{"dataMmlNode":25942,"transform":46886},"translate(600,0)",[25033,46888],{"dataC":27596,"xLinkHref":46889},"#MJX-54-TEX-I-1D452",[25022,46891,46893],{"dataMmlNode":25942,"transform":46892},"translate(1066,0)",[25033,46894],{"dataC":32181,"xLinkHref":46895},"#MJX-54-TEX-I-1D465",[25022,46897,46899],{"dataMmlNode":25942,"transform":46898},"translate(1638,0)",[25033,46900],{"dataC":25959,"xLinkHref":30099},[25022,46902,46904],{"dataMmlNode":25031,"transform":46903},"translate(6902.5,0)",[25033,46905],{"dataC":25035,"xLinkHref":30124},[25022,46907,46909],{"dataMmlNode":25942,"transform":46908},"translate(7291.5,0)",[25033,46910],{"dataC":42570,"xLinkHref":46861},[25022,46912,46914],{"dataMmlNode":25031,"transform":46913},"translate(7720.5,0)",[25033,46915],{"dataC":25108,"xLinkHref":30134},[12,46917,46918],{},[106,46919,46920],{},"How it works in practice:",[3603,46922,46923,46929,46935,46941,47019,47043],{},[26,46924,46925,46928],{},[106,46926,46927],{},"The Guess:"," At Step 4, the AI evaluates the board and guesses a score of 90.",[26,46930,46931,46934],{},[106,46932,46933],{},"The Move:"," The AI makes a move and immediately wins the game (Step 5).",[26,46936,46937,46940],{},[106,46938,46939],{},"The Correction:"," The environment declares Step 5 is a Terminal State worth 100.",[26,46942,46943,46946,46947,47018],{},[106,46944,46945],{},"The Update:"," The AI looks back at Step 4. It realizes its guess of 90 was wrong, because Step 4 directly led to a 100. It sets ",[24923,46948,46950],{"className":46949,"jax":24927},[24926],[24929,46951,46952,46974],{"style":27744,"xmlns":24932,"width":44388,"height":44389,"role":1930,"focusable":8058,"viewBox":44390,"xmlnsXLink":24936},[24938,46953,46954,46957,46960,46963,46966,46968,46971],{},[24941,46955],{"id":46956,"d":27385},"MJX-55-TEX-I-1D449",[24941,46958],{"id":46959,"d":25877},"MJX-55-TEX-I-1D461",[24941,46961],{"id":46962,"d":25881},"MJX-55-TEX-I-1D45F",[24941,46964],{"id":46965,"d":27323},"MJX-55-TEX-I-1D44E",[24941,46967],{"id":37600,"d":27108},[24941,46969],{"id":46970,"d":25873},"MJX-55-TEX-I-1D45B",[24941,46972],{"id":46973,"d":180},"MJX-55-TEX-N-200B",[25022,46975,46976],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,46977,46978],{"dataMmlNode":25028},[25022,46979,46980,46985],{"dataMmlNode":26023},[25022,46981,46982],{"dataMmlNode":25942},[25033,46983],{"dataC":27570,"xLinkHref":46984},"#MJX-55-TEX-I-1D449",[25022,46986,46987,46992,46997,47002,47006,47011],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,46988,46989],{"dataMmlNode":25942},[25033,46990],{"dataC":25959,"xLinkHref":46991},"#MJX-55-TEX-I-1D461",[25022,46993,46994],{"dataMmlNode":25942,"transform":43828},[25033,46995],{"dataC":25966,"xLinkHref":46996},"#MJX-55-TEX-I-1D45F",[25022,46998,46999],{"dataMmlNode":25942,"transform":43610},[25033,47000],{"dataC":27425,"xLinkHref":47001},"#MJX-55-TEX-I-1D44E",[25022,47003,47004],{"dataMmlNode":25942,"transform":43838},[25033,47005],{"dataC":27210,"xLinkHref":37658},[25022,47007,47008],{"dataMmlNode":25942,"transform":43844},[25033,47009],{"dataC":25952,"xLinkHref":47010},"#MJX-55-TEX-I-1D45B",[25022,47012,47013],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":44732},[25022,47014,47015],{"dataMmlNode":25031},[25033,47016],{"dataC":44737,"xLinkHref":47017},"#MJX-55-TEX-N-200B"," for Step 4 to 100, and the LMS algorithm kicks in to adjust the weights up.",[26,47020,47021,47024,47025],{},[106,47022,47023],{},"The Win (Pushing Up):"," Let's say the AI evaluates the state at 90. It wins, so the target is 100.\n",[23,47026,47027,47030,47033],{},[26,47028,47029],{},"Error=+10",[26,47031,47032],{},"Adjustment=0.1⋅10=+1",[26,47034,47035,47038,47039,47042],{},[106,47036,47037],{},"New Value = 91"," (It steps ",[7289,47040,47041],{},"towards"," 100, but doesn't jump all the way there).",[26,47044,47045,47048,47049],{},[106,47046,47047],{},"The Loss (Dialling Back):"," The next game, it reaches that same state, confidently guesses 91, but falls into a trap and loses. The target is now −100.\n",[23,47050,47051,47054,47057],{},[26,47052,47053],{},"Error=−100−91=−191",[26,47055,47056],{},"Adjustment=0.1⋅(−191)=−19.1",[26,47058,47059,47062],{},[106,47060,47061],{},"New Value = 71.9"," (It takes a massive hit and dials way back).",[12,47064,47065,47066,47068],{},"Over thousands of games, the absolute certainty of the end-game (100 or -100) slowly ripples backward through the moves. Step 4 learns from the Win, then Step 3 learns from Step 4, and so on. Eventually, the evaluation function perfectly maps out the true probability of winning from ",[7289,47067,19592],{}," starting board state.",[12,47070,47071],{},"The intuition is simple:",[23,47073,47074,47077,47080],{},[26,47075,47076],{},"If the prediction is correct → no change",[26,47078,47079],{},"If the prediction is too high → decrease weights proportionally",[26,47081,47082],{},"If the prediction is too low → increase weights proportionally",[12,47084,47085,47086,7024],{},"Under reasonable conditions, LMS is guaranteed to converge to the weights that minimise the ",[106,47087,47088],{},"Mean Squared Error (MSE)",[24923,47090,47092],{"className":47091,"jax":24927,"display":13594},[24926],[24929,47093,47096,47134],{"style":27062,"xmlns":24932,"width":47094,"height":27064,"role":1930,"focusable":8058,"viewBox":47095,"xmlnsXLink":24936},"24.202ex","0 -1562.5 10697.1 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scale(0.707)",[25022,47201,47202],{"dataMmlNode":25942},[25033,47203],{"dataC":25952,"xLinkHref":47171},[25022,47205,47207],{"dataMmlNode":25031,"transform":47206},"translate(6444.2,0)",[25033,47208],{"dataC":25035,"xLinkHref":37813},[25022,47210,47212,47218],{"dataMmlNode":26023,"transform":47211},"translate(6833.2,0)",[25022,47213,47214],{"dataMmlNode":25942},[25033,47215],{"dataC":47216,"xLinkHref":47217},"1D44C","#MJX-56-TEX-I-1D44C",[25022,47219,47221],{"dataMmlNode":25942,"transform":47220},"translate(614,-150) scale(0.707)",[25033,47222],{"dataC":27210,"xLinkHref":37831},[25022,47224,47226],{"dataMmlNode":25031,"transform":47225},"translate(7963.4,0)",[25033,47227],{"dataC":26019,"xLinkHref":37841},[25022,47229,47231,47247],{"dataMmlNode":26023,"transform":47230},"translate(8963.6,0)",[25022,47232,47233],{"dataMmlNode":27203,"dataMjxTexclass":27205},[25022,47234,47236,47240],{"dataMmlNode":47235},"mover",[25022,47237,47238],{"dataMmlNode":25942},[25033,47239],{"dataC":47216,"xLinkHref":47217},[25022,47241,47243],{"dataMmlNode":25031,"transform":47242},"translate(427,257) translate(-250 0)",[25033,47244],{"dataC":47245,"xLinkHref":47246},"5E","#MJX-56-TEX-N-5E",[25022,47248,47249],{"dataMmlNode":25942,"transform":47220},[25033,47250],{"dataC":27210,"xLinkHref":37831},[25022,47252,47254,47258],{"dataMmlNode":31565,"transform":47253},"translate(9871.6,0)",[25022,47255,47256],{"dataMmlNode":25031},[25033,47257],{"dataC":25108,"xLinkHref":37836},[25022,47259,47260],{"dataMmlNode":26031,"transform":33326},[25033,47261],{"dataC":26058,"xLinkHref":37856},[25631,47263,47265,47267],{"className":47264},[13441,13442],[25635,47266,33745],{},[13438,47268,47270,47273,47321,47374,47454],{"className":47269},[25641],[12,47271,47272],{},"This formula is really just an ultimate scoreboard for AI.",[12,47274,47275,7206,47278,47320],{},[106,47276,47277],{},"Mean:",[24923,47279,47281],{"className":47280,"jax":24927},[24926],[24929,47282,47287,47296],{"style":47283,"xmlns":24932,"width":47284,"height":47285,"role":1930,"focusable":8058,"viewBox":47286,"xmlnsXLink":24936},"vertical-align: -0.798ex;","4.722ex","2.755ex","0 -864.9 2086.9 1217.7",[24938,47288,47289,47291,47293],{},[24941,47290],{"id":30291,"d":25914},[24941,47292],{"id":30285,"d":25873},[24941,47294],{"id":47295,"d":27911},"MJX-57-TEX-SO-2211",[25022,47297,47298],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,47299,47300,47314],{"dataMmlNode":25028},[25022,47301,47302,47307,47311],{"dataMmlNode":27622},[25022,47303,47305],{"dataMmlNode":26031,"transform":47304},"translate(255.4,394) scale(0.707)",[25033,47306],{"dataC":26035,"xLinkHref":30352},[25022,47308,47309],{"dataMmlNode":25942,"transform":44826},[25033,47310],{"dataC":25952,"xLinkHref":30340},[27668,47312],{"width":47313,"height":27671,"x":27672,"y":27673},624.3,[25022,47315,47317],{"dataMmlNode":25031,"transform":47316},"translate(1030.9,0)",[25033,47318],{"dataC":27199,"xLinkHref":47319},"#MJX-57-TEX-SO-2211"," (Add them up and divide by n)",[12,47322,47323,7206,47326],{},[106,47324,47325],{},"Squared:",[24923,47327,47329],{"className":47328,"jax":24927},[24926],[24929,47330,47333,47347],{"style":24931,"xmlns":24932,"width":47331,"height":31733,"role":1930,"focusable":8058,"viewBox":47332,"xmlnsXLink":24936},"5.399ex","0 -833.9 2386.6 1083.9",[24938,47334,47335,47338,47341,47344],{},[24941,47336],{"id":47337,"d":24944},"MJX-58-TEX-N-28",[24941,47339],{"id":47340,"d":34448},"MJX-58-TEX-N-2026",[24941,47342],{"id":47343,"d":24988},"MJX-58-TEX-N-29",[24941,47345],{"id":47346,"d":25928},"MJX-58-TEX-N-32",[25022,47348,47349],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,47350,47351,47356,47361],{"dataMmlNode":25028},[25022,47352,47353],{"dataMmlNode":25031},[25033,47354],{"dataC":25035,"xLinkHref":47355},"#MJX-58-TEX-N-28",[25022,47357,47358],{"dataMmlNode":25031,"transform":25040},[25033,47359],{"dataC":34660,"xLinkHref":47360},"#MJX-58-TEX-N-2026",[25022,47362,47364,47369],{"dataMmlNode":31565,"transform":47363},"translate(1561,0)",[25022,47365,47366],{"dataMmlNode":25031},[25033,47367],{"dataC":25108,"xLinkHref":47368},"#MJX-58-TEX-N-29",[25022,47370,47371],{"dataMmlNode":26031,"transform":37418},[25033,47372],{"dataC":26058,"xLinkHref":47373},"#MJX-58-TEX-N-32",[12,47375,47376,7206,47379],{},[106,47377,47378],{},"Error:",[24923,47380,47382],{"className":47381,"jax":24927},[24926],[24929,47383,47387,47404],{"style":24931,"xmlns":24932,"width":47384,"height":47385,"role":1930,"focusable":8058,"viewBox":47386,"xmlnsXLink":24936},"8.634ex","2.943ex","0 -1051 3816.3 1301",[24938,47388,47389,47391,47394,47397,47399,47402],{},[24941,47390],{"id":37962,"d":24944},[24941,47392],{"id":47393,"d":47123},"MJX-59-TEX-I-1D44C",[24941,47395],{"id":47396,"d":27108},"MJX-59-TEX-I-1D456",[24941,47398],{"id":37965,"d":25910},[24941,47400],{"id":47401,"d":47129},"MJX-59-TEX-N-5E",[24941,47403],{"id":37971,"d":24988},[25022,47405,47406],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,47407,47408,47412,47424,47429,47449],{"dataMmlNode":25028},[25022,47409,47410],{"dataMmlNode":25031},[25033,47411],{"dataC":25035,"xLinkHref":37997},[25022,47413,47414,47419],{"dataMmlNode":26023,"transform":25040},[25022,47415,47416],{"dataMmlNode":25942},[25033,47417],{"dataC":47216,"xLinkHref":47418},"#MJX-59-TEX-I-1D44C",[25022,47420,47421],{"dataMmlNode":25942,"transform":47220},[25033,47422],{"dataC":27210,"xLinkHref":47423},"#MJX-59-TEX-I-1D456",[25022,47425,47427],{"dataMmlNode":25031,"transform":47426},"translate(1519.2,0)",[25033,47428],{"dataC":26019,"xLinkHref":38003},[25022,47430,47432,47445],{"dataMmlNode":26023,"transform":47431},"translate(2519.4,0)",[25022,47433,47434],{"dataMmlNode":27203,"dataMjxTexclass":27205},[25022,47435,47436,47440],{"dataMmlNode":47235},[25022,47437,47438],{"dataMmlNode":25942},[25033,47439],{"dataC":47216,"xLinkHref":47418},[25022,47441,47442],{"dataMmlNode":25031,"transform":47242},[25033,47443],{"dataC":47245,"xLinkHref":47444},"#MJX-59-TEX-N-5E",[25022,47446,47447],{"dataMmlNode":25942,"transform":47220},[25033,47448],{"dataC":27210,"xLinkHref":47423},[25022,47450,47452],{"dataMmlNode":25031,"transform":47451},"translate(3427.3,0)",[25033,47453],{"dataC":25108,"xLinkHref":38015},[12,47455,47456],{},"It adds up all the errors, and find the average error rate so that you can compare it between models.",[6964,47458,47460],{"id":47459},"training-data-where-does-it-come-from","Training Data: Where Does It Come From?",[12,47462,47463],{},"The source and nature of training data matters a lot. A few scenarios:",[23,47465,47466,47472,47478,47484],{},[26,47467,47468,47471],{},[106,47469,47470],{},"Random examples"," provided by the environment (most common in practice)",[26,47473,47474,47477],{},[106,47475,47476],{},"Teacher-selected examples"," chosen to be maximally informative (like \"near-miss\" examples)",[26,47479,47480,47483],{},[106,47481,47482],{},"Active learning"," where the model queries an oracle \u002F human for labels on examples it's unsure about",[26,47485,47486,47489],{},[106,47487,47488],{},"Self-directed experimentation"," where the learner designs its own experiments",[12,47491,47492,47493,47496,47497,47500,47501,47504],{},"A key assumption in most ML is that training and test data are ",[106,47494,47495],{},"independently and identically distributed (IID)"," — drawn from the same underlying distribution. When this assumption breaks down, you need techniques like ",[106,47498,47499],{},"transfer learning"," (different distributions) or ",[106,47502,47503],{},"collective classification"," (non-independent examples).",[25631,47506,47508,47511],{"className":47507},[13441,13442],[25635,47509,47510],{},"What is IID? (Independently and Identically Distributed)",[13438,47512,47514,47517,47522,47525,47539,47544,47547,47565,47571,47573,47577,47582,47585,47591,47606,47609,47611,47615,47620,47623,47626,47637,47652],{"className":47513},[25641],[12,47515,47516],{},"IID is a mathematical assumption that ML models make about the world. It assumes your training data and your real-world test data are perfectly consistent. Let's split it into its two halves:",[12,47518,47519],{},[106,47520,47521],{},"1. \"Independently\"",[12,47523,47524],{},"This means that one piece of data has absolutely no connection to the next piece of data. Drawing one doesn't change the probability of the next.",[23,47526,47527,47533],{},[26,47528,47529,47532],{},[106,47530,47531],{},"IID (Independent):"," Rolling a die. Rolling a 6 doesn't change the odds of rolling a 6 on the next turn. Diagnosing patients in a clinic; Patient A having a cold doesn't magically make Patient B have a broken leg.",[26,47534,47535,47538],{},[106,47536,47537],{},"NOT Independent:"," Predicting the weather. If it is raining on Tuesday, it is highly likely to be raining on Wednesday. The data points are linked in time.",[12,47540,47541],{},[106,47542,47543],{},"2. \"Identically Distributed\"",[12,47545,47546],{},"This means all your data is pulled from the exact same \"world,\" under the exact same rules, conditions, and demographics.",[23,47548,47549,47559],{},[26,47550,47551,47554,47555,47558],{},[106,47552,47553],{},"IID (Identical):"," Training an AI to grade math tests from a specific school, and testing it on different math tests from that ",[7289,47556,47557],{},"same"," school.",[26,47560,47561,47564],{},[106,47562,47563],{},"NOT Identical:"," Training a self-driving car entirely on the sunny, wide streets of Phoenix, Arizona (Training Data), and then dropping it into a blizzard in the narrow streets of Boston (Test Data). The \"distribution\" of the data—the weather, the roads, the driver behavior—has completely changed.",[12,47566,47567,47568],{},"When an AI model is trained, it only learns the exact rules of its training data. ",[106,47569,47570],{},"If the real world is not IID to the training data, the AI will confidently make terrible predictions.",[36,47572],{},[26236,47574,47576],{"id":47575},"what-is-transfer-learning","What is Transfer Learning?",[12,47578,47579],{},[7289,47580,47581],{},"(The fix for when data is NOT Identically Distributed)",[12,47583,47584],{},"Let's say the \"Identically Distributed\" assumption breaks. You want to build a medical AI to detect a very rare disease in X-rays. Because the disease is rare, you only have 100 training images. That isn't enough to train a neural network from scratch.",[12,47586,47587,47590],{},[106,47588,47589],{},"Transfer Learning"," is the process of taking a model trained on a massive, different dataset, and \"transferring\" its foundational knowledge to your new problem.",[3603,47592,47593,47596,47599],{},[26,47594,47595],{},"You take a model that Google already trained on 10 million random internet images (cats, dogs, cars, trees). This model already knows how to detect edges, shapes, textures, and shadows.",[26,47597,47598],{},"You take that pre-trained model and do a little bit of extra training using your 100 rare X-rays.",[26,47600,47601,47602,47605],{},"The model ",[7289,47603,47604],{},"transfers"," its general knowledge of shapes and textures to the specific task of reading X-rays.",[12,47607,47608],{},"It saves massive amounts of time and data when your target environment doesn't match a massive, easily available training environment.",[36,47610],{},[26236,47612,47614],{"id":47613},"what-is-collective-classification","What is Collective Classification?",[12,47616,47617],{},[7289,47618,47619],{},"(The fix for when data is NOT Independent)",[12,47621,47622],{},"Let's say the \"Independent\" assumption breaks. Your data points are heavily connected to one another, usually in a network or a graph.",[12,47624,47625],{},"Imagine you are trying to catch bot accounts on a social media platform. If you look at an account strictly in isolation (its bio, its profile picture, its post frequency), you might not be able to tell if it's a bot.",[12,47627,47628,47629,47632,47633,47636],{},"However, in a social network, users aren't independent. ",[106,47630,47631],{},"Collective Classification"," is a technique that looks at the ",[7289,47634,47635],{},"relationships"," between data points.",[23,47638,47639,47642,47645],{},[26,47640,47641],{},"If Account A follows 500 known bots.",[26,47643,47644],{},"If Account A is followed by 500 known bots.",[26,47646,47647,47648,47651],{},"Collective Classification uses the known labels of the ",[7289,47649,47650],{},"neighbors"," to classify the target. It concludes Account A is almost certainly a bot, too!",[12,47653,47654,47657],{},[106,47655,47656],{},"To sum up your notes:"," IID is the \"perfect laboratory condition\" for an AI. Transfer Learning and Collective Classification are the rescue tools you use when those perfect conditions shatter in the real world.",[6964,47659,47661],{"id":47660},"how-we-represent-target-functions","How We Represent Target Functions",[12,47663,47664],{},"Different representations offer different trade-offs between expressiveness and learnability:",[12,47666,47667,47670],{},[106,47668,47669],{},"Numerical functions"," — linear regression, neural networks, support vector machines. These are great at capturing smooth, continuous patterns.",[12,47672,47673,47676],{},[106,47674,47675],{},"Symbolic functions"," — decision trees, logical rules. These produce human-interpretable models.",[12,47678,47679,47682],{},[106,47680,47681],{},"Instance-based functions"," — nearest-neighbor, case-based reasoning. These make predictions by comparing new examples to stored training examples.",[12,47684,47685,47688],{},[106,47686,47687],{},"Probabilistic graphical models"," — Naïve Bayes, Bayesian networks, Hidden Markov Models. These explicitly model uncertainty and dependencies between variables.",[12,47690,47691],{},"The more expressive a representation, the more complex functions it can capture — but it also needs more data to learn accurately. This is a fundamental trade-off in ML.",[6964,47693,47695],{"id":47694},"evaluating-learning-systems","Evaluating Learning Systems",[12,47697,47698],{},"How do we know if a learning system is actually good? There are two main approaches:",[12,47700,47701,47704],{},[106,47702,47703],{},"Experimental evaluation"," — Run controlled experiments using cross-validation on benchmark datasets. Measure things like test accuracy, training time, and testing time. Use statistical tests to determine whether differences between methods are meaningful.",[12,47706,47707,47710],{},[106,47708,47709],{},"Theoretical analysis"," — Prove mathematical guarantees about algorithms, such as computational complexity, the ability to fit data, and sample complexity (how many examples are needed to learn well).",[12,47712,47713],{},"In practice, most ML work uses a combination of both.",[6964,47715,47717],{"id":47716},"a-brief-history-of-machine-learning","A Brief History of Machine Learning",[12,47719,47720],{},"ML has a rich history spanning several decades:",[23,47722,47723,47729,47735,47741,47747,47753],{},[26,47724,47725,47728],{},[106,47726,47727],{},"1950s"," — Samuel's checkers player at IBM, one of the first programs to learn from experience",[26,47730,47731,47734],{},[106,47732,47733],{},"1960s"," — The Perceptron (an early neural network), and Minsky & Papert's proof of its limitations",[26,47736,47737,47740],{},[106,47738,47739],{},"1970s"," — Symbolic approaches take center stage: decision trees (ID3), expert systems, and scientific discovery programs",[26,47742,47743,47746],{},[106,47744,47745],{},"1980s"," — Neural networks make a comeback with backpropagation; PAC learning theory provides mathematical foundations",[26,47748,47749,47752],{},[106,47750,47751],{},"1990s"," — Data mining, reinforcement learning, ensemble methods (bagging, boosting), and Bayesian approaches",[26,47754,47755,47758],{},[106,47756,47757],{},"2000s"," — Support vector machines, kernel methods, transfer learning, and ML applications in security, robotics, and personalization\nAnd of course, from the 2010s onward, deep learning has transformed the field — but that's a story for later weeks!",[6964,47760,47762],{"id":47761},"key-takeaways","Key Takeaways",[3603,47764,47765,47771,47777,47783,47789],{},[26,47766,47767,47770],{},[106,47768,47769],{},"Machine learning is about learning from experience"," — using data to approximate functions, rather than coding rules by hand.",[26,47772,47773,47776],{},[106,47774,47775],{},"Every ML problem can be framed as T, P, E"," — the task, the performance metric, and the experience.",[26,47778,47779,47782],{},[106,47780,47781],{},"Designing an ML system involves four choices",": training experience, target function, representation, and learning algorithm.",[26,47784,47785,47788],{},[106,47786,47787],{},"There's always a trade-off"," between model expressiveness and the amount of data needed.",[26,47790,47791,47794],{},[106,47792,47793],{},"Evaluation matters"," — a model is only as good as its performance on unseen data.",[9332,47796,31966],{},{"title":180,"searchDepth":181,"depth":181,"links":47798},[47799,47800,47801,47802,47806,47807,47808,47814,47815,47816,47817,47818],{"id":42213,"depth":181,"text":42214},{"id":42248,"depth":181,"text":42249},{"id":42294,"depth":181,"text":42295},{"id":42325,"depth":181,"text":42326,"children":47803},[47804,47805],{"id":42329,"depth":7430,"text":42330},{"id":42368,"depth":7430,"text":42369},{"id":42389,"depth":181,"text":42390},{"id":42483,"depth":181,"text":42484},{"id":42516,"depth":181,"text":42517,"children":47809},[47810,47811,47812,47813],{"id":42523,"depth":7430,"text":42524},{"id":42894,"depth":7430,"text":42895},{"id":43690,"depth":7430,"text":43691},{"id":43704,"depth":7430,"text":43705},{"id":47459,"depth":181,"text":47460},{"id":47660,"depth":181,"text":47661},{"id":47694,"depth":181,"text":47695},{"id":47716,"depth":181,"text":47717},{"id":47761,"depth":181,"text":47762},"2026-04-26",{},"\u002Fblog\u002FIntroduction-to-Machine-Learning",{"title":42208,"description":180},{"loc":47821},"blog\u002FIntroduction-to-Machine-Learning","frvXEvSWd33D6c8NnWst_v81G5mL_ZJ2qVq4O4xGfyw",{"id":47827,"title":47828,"author":6946,"body":47829,"date":73345,"description":180,"draft":185,"edited_at":47819,"extension":186,"featured_image":194,"meta":73346,"navigation":189,"path":73347,"pinned":185,"seo":73348,"sitemap":73349,"stem":73350,"tags":194,"__hash__":73351},"blog\u002Fblog\u002FArtificial-Neural-Networks-and-Backpropagation.md","Artificial Neural Networks and Backpropagation",{"type":9,"value":47830,"toc":73313},[47831,47835,47842,47862,47868,47872,47875,48047,48051,48054,48278,48285,48289,48292,48296,48299,48325,48329,48341,48612,48623,48627,48630,48633,48637,48651,48662,48666,48674,48680,48692,48697,48705,48709,48712,48718,48723,48729,48734,48740,48746,48750,48757,48763,48770,48774,48781,48785,48788,49587,49642,49828,50832,50836,50839,50882,51117,51440,51446,51714,51865,51869,51875,51975,51978,52115,53794,53798,53824,53827,56715,56782,56785,56979,58370,58526,58543,58546,58890,60158,60464,63720,63787,63920,64503,65980,65983,66446,66693,69432,69436,69494,69770,69773,69778,69917,70040,70157,70274,70324,70343,70366,70676,70679,70906,70934,70937,71372,72746,72750,72753,72759,72920,72942,73011,73026,73029,73035,73039,73042,73046,73118,73122,73204,73208,73211,73213,73239,73243,73311],[6964,47832,47834],{"id":47833},"_1-the-paradigm-shift-why-artificial-neural-networks","1. The Paradigm Shift: Why Artificial Neural Networks?",[12,47836,47837,47838,47841],{},"For decades, computing has been dominated by the ",[106,47839,47840],{},"Von Neumann architecture",", a design predicated on sequential processing, explicit memory addressing, and rigid algorithmic logic. While highly effective for arithmetic calculations and strictly defined logical operations, the Von Neumann model struggles profoundly with tasks that humans find trivial, such as:",[23,47843,47844,47850,47856],{},[26,47845,47846,47849],{},[106,47847,47848],{},"Pattern Recognition:"," Identifying faces, deciphering handwritten characters, or processing natural speech.",[26,47851,47852,47855],{},[106,47853,47854],{},"Content-Addressable Recall:"," Retrieving complex memories based on partial cues rather than explicit memory addresses.",[26,47857,47858,47861],{},[106,47859,47860],{},"Approximate Reasoning:"," Making common-sense decisions in ambiguous, ill-defined environments (e.g., driving a car, playing a sport).",[12,47863,47864,47865,6959],{},"These tasks are difficult to program algorithmically because they rely on experience, adaptation, and the ability to tolerate noise, rather than rigid mathematical logic. To solve these problems, computer science draws inspiration from the human brain, leading to the development of ",[106,47866,47867],{},"Artificial Neural Networks (ANNs)",[7166,47869,47871],{"id":47870},"biological-neural-networks-bnn-vs-von-neumann-machines","Biological Neural Networks (BNN) vs. Von Neumann Machines",[12,47873,47874],{},"The fundamental difference between biological brains and traditional computers lies in their architecture and processing methods.",[7085,47876,47877,47891],{},[7088,47878,47879],{},[7091,47880,47881,47885,47888],{},[7094,47882,47884],{"align":47883},"left","Feature",[7094,47886,47887],{"align":47883},"Von Neumann Machine",[7094,47889,47890],{"align":47883},"Human Brain (Biological Neural Network)",[7104,47892,47893,47946,47959,48008,48021,48034],{},[7091,47894,47895,47900,47903],{},[7109,47896,47897],{"align":47883},[106,47898,47899],{},"Processors",[7109,47901,47902],{"align":47883},"One or a few high-speed processors (nanosecond operations).",[7109,47904,47905,47906,47945],{"align":47883},"Billions (",[24923,47907,47909],{"className":47908,"jax":24927},[24926],[24929,47910,47914,47922],{"style":27778,"xmlns":24932,"width":47911,"height":47912,"role":1930,"focusable":8058,"viewBox":47913,"xmlnsXLink":24936},"4.05ex","2.005ex","0 -864 1790.1 886",[24938,47915,47916,47919],{},[24941,47917],{"id":47918,"d":25914},"MJX-1-TEX-N-31",[24941,47920],{"id":47921,"d":25935},"MJX-1-TEX-N-30",[25022,47923,47924],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,47925,47926],{"dataMmlNode":25028},[25022,47927,47928,47936],{"dataMmlNode":31565},[25022,47929,47930,47933],{"dataMmlNode":26031},[25033,47931],{"dataC":26035,"xLinkHref":47932},"#MJX-1-TEX-N-31",[25033,47934],{"dataC":26103,"xLinkHref":47935,"transform":26394},"#MJX-1-TEX-N-30",[25022,47937,47939],{"dataMmlNode":27203,"transform":47938,"dataMjxTexclass":27205},"translate(1033,393.1) scale(0.707)",[25022,47940,47941,47943],{"dataMmlNode":26031},[25033,47942],{"dataC":26035,"xLinkHref":47932},[25033,47944],{"dataC":26035,"xLinkHref":47932,"transform":26394},") of low-speed processors (millisecond operations).",[7091,47947,47948,47953,47956],{},[7109,47949,47950],{"align":47883},[106,47951,47952],{},"Computing Power",[7109,47954,47955],{"align":47883},"Massive power concentrated in localized CPU units.",[7109,47957,47958],{"align":47883},"Limited individual power, relying on massive collective parallelization.",[7091,47960,47961,47966,47969],{},[7109,47962,47963],{"align":47883},[106,47964,47965],{},"Connectivity",[7109,47967,47968],{"align":47883},"Shared, high-speed buses routing data sequentially.",[7109,47970,47971,47972,48007],{"align":47883},"Massive interconnections (",[24923,47973,47975],{"className":47974,"jax":24927},[24926],[24929,47976,47977,47986],{"style":27778,"xmlns":24932,"width":47911,"height":47912,"role":1930,"focusable":8058,"viewBox":47913,"xmlnsXLink":24936},[24938,47978,47979,47981,47983],{},[24941,47980],{"id":42606,"d":25914},[24941,47982],{"id":42609,"d":25935},[24941,47984],{"id":47985,"d":26660},"MJX-2-TEX-N-35",[25022,47987,47988],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,47989,47990],{"dataMmlNode":25028},[25022,47991,47992,47998],{"dataMmlNode":31565},[25022,47993,47994,47996],{"dataMmlNode":26031},[25033,47995],{"dataC":26035,"xLinkHref":42643},[25033,47997],{"dataC":26103,"xLinkHref":42646,"transform":26394},[25022,47999,48000],{"dataMmlNode":27203,"transform":47938,"dataMjxTexclass":27205},[25022,48001,48002,48004],{"dataMmlNode":26031},[25033,48003],{"dataC":26035,"xLinkHref":42643},[25033,48005],{"dataC":26717,"xLinkHref":48006,"transform":26394},"#MJX-2-TEX-N-35"," synapses) operating concurrently.",[7091,48009,48010,48015,48018],{},[7109,48011,48012],{"align":47883},[106,48013,48014],{},"Memory Access",[7109,48016,48017],{"align":47883},"Sequential access via explicit physical addresses.",[7109,48019,48020],{"align":47883},"Content-addressable recall (retrieval by association).",[7091,48022,48023,48028,48031],{},[7109,48024,48025],{"align":47883},[106,48026,48027],{},"Knowledge Storage",[7109,48029,48030],{"align":47883},"Knowledge and problem-solving logic are explicitly separated from the computing hardware.",[7109,48032,48033],{"align":47883},"Knowledge is distributed and physically resides in the synaptic connectivity between neurons.",[7091,48035,48036,48041,48044],{},[7109,48037,48038],{"align":47883},[106,48039,48040],{},"Adaptability",[7109,48042,48043],{"align":47883},"Hard-coded and rigid; highly susceptible to catastrophic failure if hardware is damaged.",[7109,48045,48046],{"align":47883},"Highly adaptive; learns by altering network connectivity. Exhibits graceful degradation if partially damaged.",[7166,48048,48050],{"id":48049},"bnn-vs-ann","BNN vs. ANN",[12,48052,48053],{},"While ANNs are inspired by biological brains, the practical engineered version is a tiny, brittle imitation of the real thing. The slides include this distinct comparison:",[7085,48055,48056,48068],{},[7088,48057,48058],{},[7091,48059,48060,48062,48065],{},[7094,48061,47884],{"align":47883},[7094,48063,48064],{"align":47883},"Biological Neural Network (BNN)",[7094,48066,48067],{"align":47883},"Artificial Neural Network (ANN)",[7104,48069,48070,48083,48239,48252,48265],{},[7091,48071,48072,48077,48080],{},[7109,48073,48074],{"align":47883},[106,48075,48076],{},"Parallelism",[7109,48078,48079],{"align":47883},"Massively parallel, slow per neuron, but superior overall",[7109,48081,48082],{"align":47883},"Massively parallel, fast per node, but inferior overall",[7091,48084,48085,48090,48165],{},[7109,48086,48087],{"align":47883},[106,48088,48089],{},"Scale",[7109,48091,48092,48093,48128,48129,48164],{"align":47883},"~",[24923,48094,48096],{"className":48095,"jax":24927},[24926],[24929,48097,48098,48106],{"style":27778,"xmlns":24932,"width":47911,"height":47912,"role":1930,"focusable":8058,"viewBox":47913,"xmlnsXLink":24936},[24938,48099,48100,48103],{},[24941,48101],{"id":48102,"d":25914},"MJX-3-TEX-N-31",[24941,48104],{"id":48105,"d":25935},"MJX-3-TEX-N-30",[25022,48107,48108],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48109,48110],{"dataMmlNode":25028},[25022,48111,48112,48120],{"dataMmlNode":31565},[25022,48113,48114,48117],{"dataMmlNode":26031},[25033,48115],{"dataC":26035,"xLinkHref":48116},"#MJX-3-TEX-N-31",[25033,48118],{"dataC":26103,"xLinkHref":48119,"transform":26394},"#MJX-3-TEX-N-30",[25022,48121,48122],{"dataMmlNode":27203,"transform":47938,"dataMjxTexclass":27205},[25022,48123,48124,48126],{"dataMmlNode":26031},[25033,48125],{"dataC":26035,"xLinkHref":48116},[25033,48127],{"dataC":26035,"xLinkHref":48116,"transform":26394}," neurons, ~",[24923,48130,48132],{"className":48131,"jax":24927},[24926],[24929,48133,48134,48143],{"style":27778,"xmlns":24932,"width":47911,"height":47912,"role":1930,"focusable":8058,"viewBox":47913,"xmlnsXLink":24936},[24938,48135,48136,48138,48140],{},[24941,48137],{"id":42701,"d":25914},[24941,48139],{"id":42704,"d":25935},[24941,48141],{"id":48142,"d":26660},"MJX-4-TEX-N-35",[25022,48144,48145],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48146,48147],{"dataMmlNode":25028},[25022,48148,48149,48155],{"dataMmlNode":31565},[25022,48150,48151,48153],{"dataMmlNode":26031},[25033,48152],{"dataC":26035,"xLinkHref":42742},[25033,48154],{"dataC":26103,"xLinkHref":42745,"transform":26394},[25022,48156,48157],{"dataMmlNode":27203,"transform":47938,"dataMjxTexclass":27205},[25022,48158,48159,48161],{"dataMmlNode":26031},[25033,48160],{"dataC":26035,"xLinkHref":42742},[25033,48162],{"dataC":26717,"xLinkHref":48163,"transform":26394},"#MJX-4-TEX-N-35"," interconnections",[7109,48166,48167,48168,10720,48205,48238],{"align":47883},"Typically ",[24923,48169,48171],{"className":48170,"jax":24927},[24926],[24929,48172,48175,48186],{"style":27778,"xmlns":24932,"width":48173,"height":47912,"role":1930,"focusable":8058,"viewBox":48174,"xmlnsXLink":24936},"3.25ex","0 -864 1436.6 886",[24938,48176,48177,48180,48183],{},[24941,48178],{"id":48179,"d":25914},"MJX-5-TEX-N-31",[24941,48181],{"id":48182,"d":25935},"MJX-5-TEX-N-30",[24941,48184],{"id":48185,"d":25928},"MJX-5-TEX-N-32",[25022,48187,48188],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48189,48190],{"dataMmlNode":25028},[25022,48191,48192,48200],{"dataMmlNode":31565},[25022,48193,48194,48197],{"dataMmlNode":26031},[25033,48195],{"dataC":26035,"xLinkHref":48196},"#MJX-5-TEX-N-31",[25033,48198],{"dataC":26103,"xLinkHref":48199,"transform":26394},"#MJX-5-TEX-N-30",[25022,48201,48202],{"dataMmlNode":26031,"transform":47938},[25033,48203],{"dataC":26058,"xLinkHref":48204},"#MJX-5-TEX-N-32",[24923,48206,48208],{"className":48207,"jax":24927},[24926],[24929,48209,48212,48221],{"style":27778,"xmlns":24932,"width":48173,"height":48210,"role":1930,"focusable":8058,"viewBox":48211,"xmlnsXLink":24936},"2.022ex","0 -871.8 1436.6 893.8",[24938,48213,48214,48216,48218],{},[24941,48215],{"id":25913,"d":25914},[24941,48217],{"id":25934,"d":25935},[24941,48219],{"id":48220,"d":29582},"MJX-6-TEX-N-34",[25022,48222,48223],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48224,48225],{"dataMmlNode":25028},[25022,48226,48227,48233],{"dataMmlNode":31565},[25022,48228,48229,48231],{"dataMmlNode":26031},[25033,48230],{"dataC":26035,"xLinkHref":26036},[25033,48232],{"dataC":26103,"xLinkHref":26104,"transform":26394},[25022,48234,48235],{"dataMmlNode":26031,"transform":47938},[25033,48236],{"dataC":29597,"xLinkHref":48237},"#MJX-6-TEX-N-34"," nodes; depends on application and design",[7091,48240,48241,48246,48249],{},[7109,48242,48243],{"align":47883},[106,48244,48245],{},"Ambiguity",[7109,48247,48248],{"align":47883},"Tolerates ambiguity natively",[7109,48250,48251],{"align":47883},"Requires precise, structured, formatted data to handle ambiguity",[7091,48253,48254,48259,48262],{},[7109,48255,48256],{"align":47883},[106,48257,48258],{},"Fault tolerance",[7109,48260,48261],{"align":47883},"Performance degrades gracefully under partial damage",[7109,48263,48264],{"align":47883},"Robust performance is possible but not automatic",[7091,48266,48267,48272,48275],{},[7109,48268,48269],{"align":47883},[106,48270,48271],{},"Information storage",[7109,48273,48274],{"align":47883},"Stored in the synapses (the connections themselves)",[7109,48276,48277],{"align":47883},"Stored in continuous memory locations",[12,48279,48280,48281,48284],{},"The takeaway: ANNs borrow the ",[7289,48282,48283],{},"idea"," of distributed parallel computation, but they're orders of magnitude smaller and far less robust than the biological systems that inspired them.",[6964,48286,48288],{"id":48287},"_2-from-biological-to-artificial-neurons","2. From Biological to Artificial Neurons",[12,48290,48291],{},"To engineer an ANN, we must abstract the biological mechanisms of the brain into mathematical models.",[7166,48293,48295],{"id":48294},"the-biological-neuron","The Biological Neuron",[12,48297,48298],{},"A biological neuron consists of four primary components:",[3603,48300,48301,48307,48313,48319],{},[26,48302,48303,48306],{},[106,48304,48305],{},"Dendrites:"," Branching structures that receive incoming signals from other neurons.",[26,48308,48309,48312],{},[106,48310,48311],{},"Soma (Cell Body):"," Accumulates the incoming signals.",[26,48314,48315,48318],{},[106,48316,48317],{},"Axon:"," A long fiber that transmits the signal outward if the accumulated stimulus exceeds a certain threshold (action potential).",[26,48320,48321,48324],{},[106,48322,48323],{},"Synapse:"," The microscopic gap between the axon of one neuron and the dendrite of another. The efficiency (strength) of the signal exchange across this gap determines how strongly one neuron influences another.",[7166,48326,48328],{"id":48327},"the-artificial-neuron","The Artificial Neuron",[12,48330,48331,48332,7298,48334,7632,48337,48340],{},"The artificial neuron (often called a ",[106,48333,9609],{},[106,48335,48336],{},"unit",[106,48338,48339],{},"perceptron",") mirrors this biological structure mathematically:",[3603,48342,48343,48440,48538,48563],{},[26,48344,48345,48439],{},[106,48346,48347,48348,24917],{},"Inputs (",[24923,48349,48351],{"className":48350,"jax":24927},[24926],[24929,48352,48356,48376],{"style":26153,"xmlns":24932,"width":48353,"height":48354,"role":1930,"focusable":8058,"viewBox":48355,"xmlnsXLink":24936},"13.212ex","1.984ex","0 -683 5839.6 877",[24938,48357,48358,48362,48364,48367,48370,48373],{},[24941,48359],{"id":48360,"d":48361},"MJX-7-TEX-I-1D44B","M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z",[24941,48363],{"id":26164,"d":25914},[24941,48365],{"id":48366,"d":27339},"MJX-7-TEX-N-2C",[24941,48368],{"id":48369,"d":25928},"MJX-7-TEX-N-32",[24941,48371],{"id":48372,"d":26357},"MJX-7-TEX-N-2E",[24941,48374],{"id":48375,"d":28310},"MJX-7-TEX-I-1D45A",[25022,48377,48378],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48379,48380,48393,48399,48411,48417,48422,48427],{"dataMmlNode":25028},[25022,48381,48382,48388],{"dataMmlNode":26023},[25022,48383,48384],{"dataMmlNode":25942},[25033,48385],{"dataC":48386,"xLinkHref":48387},"1D44B","#MJX-7-TEX-I-1D44B",[25022,48389,48391],{"dataMmlNode":26031,"transform":48390},"translate(861,-150) scale(0.707)",[25033,48392],{"dataC":26035,"xLinkHref":26180},[25022,48394,48396],{"dataMmlNode":25031,"transform":48395},"translate(1264.6,0)",[25033,48397],{"dataC":27456,"xLinkHref":48398},"#MJX-7-TEX-N-2C",[25022,48400,48402,48406],{"dataMmlNode":26023,"transform":48401},"translate(1709.2,0)",[25022,48403,48404],{"dataMmlNode":25942},[25033,48405],{"dataC":48386,"xLinkHref":48387},[25022,48407,48408],{"dataMmlNode":26031,"transform":48390},[25033,48409],{"dataC":26058,"xLinkHref":48410},"#MJX-7-TEX-N-32",[25022,48412,48414],{"dataMmlNode":25031,"transform":48413},"translate(2973.8,0)",[25033,48415],{"dataC":26392,"xLinkHref":48416},"#MJX-7-TEX-N-2E",[25022,48418,48420],{"dataMmlNode":25031,"transform":48419},"translate(3418.4,0)",[25033,48421],{"dataC":26392,"xLinkHref":48416},[25022,48423,48425],{"dataMmlNode":25031,"transform":48424},"translate(3863.1,0)",[25033,48426],{"dataC":26392,"xLinkHref":48416},[25022,48428,48430,48434],{"dataMmlNode":26023,"transform":48429},"translate(4307.8,0)",[25022,48431,48432],{"dataMmlNode":25942},[25033,48433],{"dataC":48386,"xLinkHref":48387},[25022,48435,48436],{"dataMmlNode":25942,"transform":48390},[25033,48437],{"dataC":28479,"xLinkHref":48438},"#MJX-7-TEX-I-1D45A"," Represent the incoming signals (dendrites).",[26,48441,48442,48537],{},[106,48443,48444,48445,24917],{},"Weights (",[24923,48446,48448],{"className":48447,"jax":24927},[24926],[24929,48449,48452,48473],{"style":26153,"xmlns":24932,"width":48450,"height":48354,"role":1930,"focusable":8058,"viewBox":48451,"xmlnsXLink":24936},"13.999ex","0 -683 6187.6 877",[24938,48453,48454,48458,48461,48464,48467,48470],{},[24941,48455],{"id":48456,"d":48457},"MJX-8-TEX-I-1D44A","M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 628T831 637Q817 637 817 647Q817 650 819 660Q823 676 825 679T839 682Q842 682 856 682T895 682T949 681Q1015 681 1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 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683Z",[24941,48459],{"id":48460,"d":25914},"MJX-8-TEX-N-31",[24941,48462],{"id":48463,"d":27339},"MJX-8-TEX-N-2C",[24941,48465],{"id":48466,"d":25928},"MJX-8-TEX-N-32",[24941,48468],{"id":48469,"d":26357},"MJX-8-TEX-N-2E",[24941,48471],{"id":48472,"d":28310},"MJX-8-TEX-I-1D45A",[25022,48474,48475],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48476,48477,48491,48497,48509,48515,48520,48525],{"dataMmlNode":25028},[25022,48478,48479,48485],{"dataMmlNode":26023},[25022,48480,48481],{"dataMmlNode":25942},[25033,48482],{"dataC":48483,"xLinkHref":48484},"1D44A","#MJX-8-TEX-I-1D44A",[25022,48486,48488],{"dataMmlNode":26031,"transform":48487},"translate(977,-150) scale(0.707)",[25033,48489],{"dataC":26035,"xLinkHref":48490},"#MJX-8-TEX-N-31",[25022,48492,48494],{"dataMmlNode":25031,"transform":48493},"translate(1380.6,0)",[25033,48495],{"dataC":27456,"xLinkHref":48496},"#MJX-8-TEX-N-2C",[25022,48498,48500,48504],{"dataMmlNode":26023,"transform":48499},"translate(1825.2,0)",[25022,48501,48502],{"dataMmlNode":25942},[25033,48503],{"dataC":48483,"xLinkHref":48484},[25022,48505,48506],{"dataMmlNode":26031,"transform":48487},[25033,48507],{"dataC":26058,"xLinkHref":48508},"#MJX-8-TEX-N-32",[25022,48510,48512],{"dataMmlNode":25031,"transform":48511},"translate(3205.8,0)",[25033,48513],{"dataC":26392,"xLinkHref":48514},"#MJX-8-TEX-N-2E",[25022,48516,48518],{"dataMmlNode":25031,"transform":48517},"translate(3650.4,0)",[25033,48519],{"dataC":26392,"xLinkHref":48514},[25022,48521,48523],{"dataMmlNode":25031,"transform":48522},"translate(4095.1,0)",[25033,48524],{"dataC":26392,"xLinkHref":48514},[25022,48526,48528,48532],{"dataMmlNode":26023,"transform":48527},"translate(4539.8,0)",[25022,48529,48530],{"dataMmlNode":25942},[25033,48531],{"dataC":48483,"xLinkHref":48484},[25022,48533,48534],{"dataMmlNode":25942,"transform":48487},[25033,48535],{"dataC":28479,"xLinkHref":48536},"#MJX-8-TEX-I-1D45A"," Represent the synaptic strength. A high weight amplifies an input; a low weight diminishes it.",[26,48539,48540,48562],{},[106,48541,48542,48543,24917],{},"Summation Function (",[24923,48544,48546],{"className":48545,"jax":24927},[24926],[24929,48547,48548,48553],{"style":27802,"xmlns":24932,"width":28112,"height":28113,"role":1930,"focusable":8058,"viewBox":28114,"xmlnsXLink":24936},[24938,48549,48550],{},[24941,48551],{"id":48552,"d":28120},"MJX-9-TEX-N-3A3",[25022,48554,48555],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48556,48557],{"dataMmlNode":25028},[25022,48558,48559],{"dataMmlNode":25942},[25033,48560],{"dataC":28129,"xLinkHref":48561},"#MJX-9-TEX-N-3A3"," Acts as the soma, computing the weighted sum of all incoming signals.",[26,48564,48565,48589,48590,48611],{},[106,48566,48567,48568,24917],{},"Activation Function (",[24923,48569,48571],{"className":48570,"jax":24927},[24926],[24929,48572,48575,48580],{"style":29805,"xmlns":24932,"width":48573,"height":29807,"role":1930,"focusable":8058,"viewBox":48574,"xmlnsXLink":24936},"1.244ex","0 -705 550 910",[24938,48576,48577],{},[24941,48578],{"id":48579,"d":29821},"MJX-10-TEX-I-1D453",[25022,48581,48582],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48583,48584],{"dataMmlNode":25028},[25022,48585,48586],{"dataMmlNode":25942},[25033,48587],{"dataC":29864,"xLinkHref":48588},"#MJX-10-TEX-I-1D453"," Acts as the axon, determining the final output (",[24923,48591,48593],{"className":48592,"jax":24927},[24926],[24929,48594,48597,48602],{"style":27802,"xmlns":24932,"width":48595,"height":28113,"role":1930,"focusable":8058,"viewBox":48596,"xmlnsXLink":24936},"1.726ex","0 -683 763 683",[24938,48598,48599],{},[24941,48600],{"id":48601,"d":47123},"MJX-11-TEX-I-1D44C",[25022,48603,48604],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48605,48606],{"dataMmlNode":25028},[25022,48607,48608],{"dataMmlNode":25942},[25033,48609],{"dataC":47216,"xLinkHref":48610},"#MJX-11-TEX-I-1D44C",") based on the aggregated signal. It evaluates whether the \"neuron\" should fire.",[12,48613,48614],{},[32509,48615,48616,48620],{},[1930,48617],{"alt":48618,"src":48619},"0.68","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776932426798-Comparison-between-biological-neuron-and-artificial-neuron-40.webp",[32516,48621,48622],{},"Biological neuron (a) vs. Artificial neuron (b)",[6964,48624,48626],{"id":48625},"_3-neural-network-architecture-and-topology","3. Neural Network Architecture and Topology",[12,48628,48629],{},"An Artificial Neural Network is essentially a collection of these artificial neurons connected by weighted links. What a network can compute is primarily determined by its architecture and the values of its weights.",[12,48631,48632],{},"Networks are classified by several attributes:",[7166,48634,48636],{"id":48635},"_1-connection-type","1. Connection Type",[23,48638,48639,48645],{},[26,48640,48641,48644],{},[106,48642,48643],{},"Static (Feedforward):"," Signals travel in strictly one direction, from input to output. There are no loops.",[26,48646,48647,48650],{},[106,48648,48649],{},"Dynamic (Recurrent\u002FFeedback):"," Network connections form directed cycles. Outputs of nodes are fed back as inputs to themselves or previous layers, allowing the network to maintain an internal state or \"memory\" of past inputs.",[12,48652,48653],{},[32509,48654,48655,48659],{},[1930,48656],{"alt":48657,"src":48658},"0.86","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776961525934-a-Static-neuron-b-Dynamic-neuron.png",[32516,48660,48661],{},"Static neural network (a) vs. Dynamic neural network (b)",[7166,48663,48665],{"id":48664},"_2-topology","2. Topology",[23,48667,48668],{},[26,48669,48670,48673],{},[106,48671,48672],{},"Single-layer:"," Inputs map directly to a single layer of output nodes.",[12,48675,48676],{},[1930,48677],{"alt":48678,"src":48679},"0.47","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776961715501-Single-layer-neural-network-22.webp",[23,48681,48682],{},[26,48683,48684,48687,48688,48691],{},[106,48685,48686],{},"Multi-layer:"," Contains one or more ",[106,48689,48690],{},"hidden layers"," between the input and output layers. This dramatically increases the representational power of the network, allowing it to learn non-linear decision boundaries.",[12,48693,48694],{},[1930,48695],{"alt":32443,"src":48696},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776961728478-Multi-layer-neural-network-22.png",[23,48698,48699],{},[26,48700,48701,48704],{},[106,48702,48703],{},"Self-organized:"," Networks that autonomously organize their topology based on data patterns (often used in unsupervised learning).",[6964,48706,48708],{"id":48707},"_4-learning-paradigms","4. Learning Paradigms",[12,48710,48711],{},"For a neural network to be useful, it must \"learn.\" Learning is the process of adjusting the synaptic weights to minimize the difference between the network's current output and the desired outcome.",[12,48713,48714,48717],{},[106,48715,48716],{},"Supervised Learning:"," The network is provided with a training dataset consisting of paired inputs and explicit desired outputs (labels). An algorithm calculates the error between the network's prediction and the true label, adjusting the weights to correct the error.",[12,48719,48720],{},[1930,48721],{"alt":32443,"src":48722},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776933654458-1520193384546.jpg",[12,48724,48725,48728],{},[106,48726,48727],{},"Unsupervised Learning:"," The network receives input data without any explicit labels. The algorithm's goal is to discover hidden structures, correlations, or natural clusters within the raw data.",[12,48730,48731],{},[1930,48732],{"alt":32443,"src":48733},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776933608367-1520209274175.jpg",[12,48735,48736,48739],{},[106,48737,48738],{},"Reinforcement Learning:"," The network (acting as an agent) interacts with an environment. It receives positive rewards for correct actions and negative rewards for incorrect ones, adjusting its weights to maximize total cumulative reward over time.",[12,48741,48742],{},[1930,48743],{"alt":48744,"src":48745},"0.62","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776933710552-1520209208509.jpg",[6964,48747,48749],{"id":48748},"_5-the-multilayer-perceptron-mlp-and-backpropagation","5. The Multilayer Perceptron (MLP) and Backpropagation",[12,48751,48752,48753,48756],{},"The most fundamental architecture for supervised learning is the ",[106,48754,48755],{},"Multilayer Perceptron (MLP)",". An MLP consists of an input layer, at least one hidden layer, and an output layer.",[12,48758,48759],{},[1930,48760],{"alt":48761,"src":48762},"0.50","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776933907355-Screenshot_2026-04-23_at_4.44.52_PM.png",[12,48764,48765,48766,48769],{},"To train an MLP, we use the ",[106,48767,48768],{},"Backpropagation (BP)"," algorithm. Backpropagation is a gradient descent optimization technique that propagates the output error backward through the network to update the weights of the hidden layers.",[7166,48771,48773],{"id":48772},"the-challenge-of-hidden-layers","The Challenge of Hidden Layers",[12,48775,48776,48777,48780],{},"In a single-layer network, updating weights is straightforward (using the Delta Rule) because we know the explicit target value for the output node. However, in a multi-layer network, we ",[7289,48778,48779],{},"do not know"," the target output for the nodes in the hidden layers. Backpropagation solves this by calculating how much each hidden node contributed to the final output error, assigning \"blame\" proportionately using the chain rule of calculus.",[7166,48782,48784],{"id":48783},"mathematical-notation-for-backpropagation","Mathematical Notation for Backpropagation",[12,48786,48787],{},"Let us define the parameters for a network with an input layer (Layer 0), a hidden layer (Layer 1), and an output layer (Layer 2):",[23,48789,48790,49043,49226,49264,49302,49374],{},[26,48791,48792,48795],{},[106,48793,48794],{},"Weights:",[23,48796,48797,48983],{},[26,48798,48799,48856,48857,48941,48942,48959,48960,6959],{},[24923,48800,48802],{"className":48801,"jax":24927},[24926],[24929,48803,48806,48822],{"style":25196,"xmlns":24932,"width":48804,"height":32538,"role":1930,"focusable":8058,"viewBox":48805,"xmlnsXLink":24936},"5.097ex","0 -893.3 2252.8 904.3",[24938,48807,48808,48811,48813,48815,48818,48820],{},[24941,48809],{"id":48810,"d":37254},"MJX-12-TEX-I-1D464",[24941,48812],{"id":43425,"d":24944},[24941,48814],{"id":26347,"d":25914},[24941,48816],{"id":48817,"d":27339},"MJX-12-TEX-N-2C",[24941,48819],{"id":26353,"d":25935},[24941,48821],{"id":43428,"d":24988},[25022,48823,48824],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48825,48826],{"dataMmlNode":25028},[25022,48827,48828,48833],{"dataMmlNode":31565},[25022,48829,48830],{"dataMmlNode":25942},[25033,48831],{"dataC":37310,"xLinkHref":48832},"#MJX-12-TEX-I-1D464",[25022,48834,48835,48839,48843,48848,48852],{"dataMmlNode":27203,"transform":37564,"dataMjxTexclass":27205},[25022,48836,48837],{"dataMmlNode":25031},[25033,48838],{"dataC":25035,"xLinkHref":43448},[25022,48840,48841],{"dataMmlNode":26031,"transform":25040},[25033,48842],{"dataC":26035,"xLinkHref":26377},[25022,48844,48845],{"dataMmlNode":25031,"transform":41846},[25033,48846],{"dataC":27456,"xLinkHref":48847},"#MJX-12-TEX-N-2C",[25022,48849,48850],{"dataMmlNode":26031,"transform":28333},[25033,48851],{"dataC":26103,"xLinkHref":26389},[25022,48853,48854],{"dataMmlNode":25031,"transform":41146},[25033,48855],{"dataC":25108,"xLinkHref":43459},": Weight matrix from the input layer to the hidden layer. Specifically, ",[24923,48858,48860],{"className":48859,"jax":24927},[24926],[24929,48861,48863,48886],{"style":33830,"xmlns":24932,"width":48804,"height":33832,"role":1930,"focusable":8058,"viewBox":48862,"xmlnsXLink":24936},"0 -1060.7 2252.8 1498.8",[24938,48864,48865,48868,48870,48873,48876,48878,48880,48883],{},[24941,48866],{"id":48867,"d":37254},"MJX-13-TEX-I-1D464",[24941,48869],{"id":43480,"d":24944},[24941,48871],{"id":48872,"d":25914},"MJX-13-TEX-N-31",[24941,48874],{"id":48875,"d":27339},"MJX-13-TEX-N-2C",[24941,48877],{"id":26413,"d":25935},[24941,48879],{"id":43486,"d":24988},[24941,48881],{"id":48882,"d":33606},"MJX-13-TEX-I-1D457",[24941,48884],{"id":48885,"d":27108},"MJX-13-TEX-I-1D456",[25022,48887,48888],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48889,48890],{"dataMmlNode":25028},[25022,48891,48892,48897,48922],{"dataMmlNode":33991},[25022,48893,48894],{"dataMmlNode":25942},[25033,48895],{"dataC":37310,"xLinkHref":48896},"#MJX-13-TEX-I-1D464",[25022,48898,48900,48904,48909,48914,48918],{"dataMmlNode":27203,"transform":48899,"dataMjxTexclass":27205},"translate(749,530.4) scale(0.707)",[25022,48901,48902],{"dataMmlNode":25031},[25033,48903],{"dataC":25035,"xLinkHref":43505},[25022,48905,48906],{"dataMmlNode":26031,"transform":25040},[25033,48907],{"dataC":26035,"xLinkHref":48908},"#MJX-13-TEX-N-31",[25022,48910,48911],{"dataMmlNode":25031,"transform":41846},[25033,48912],{"dataC":27456,"xLinkHref":48913},"#MJX-13-TEX-N-2C",[25022,48915,48916],{"dataMmlNode":26031,"transform":28333},[25033,48917],{"dataC":26103,"xLinkHref":26429},[25022,48919,48920],{"dataMmlNode":25031,"transform":41146},[25033,48921],{"dataC":25108,"xLinkHref":43517},[25022,48923,48925,48930,48935],{"dataMmlNode":27203,"transform":48924,"dataMjxTexclass":27205},"translate(749,-293.8) scale(0.707)",[25022,48926,48927],{"dataMmlNode":25942},[25033,48928],{"dataC":33647,"xLinkHref":48929},"#MJX-13-TEX-I-1D457",[25022,48931,48933],{"dataMmlNode":25031,"transform":48932},"translate(412,0)",[25033,48934],{"dataC":27456,"xLinkHref":48913},[25022,48936,48938],{"dataMmlNode":25942,"transform":48937},"translate(690,0)",[25033,48939],{"dataC":27210,"xLinkHref":48940},"#MJX-13-TEX-I-1D456"," is the weight connecting input node ",[24923,48943,48945],{"className":48944,"jax":24927},[24926],[24929,48946,48947,48951],{"style":25196,"xmlns":24932,"width":28237,"height":28238,"role":1930,"focusable":8058,"viewBox":28239,"xmlnsXLink":24936},[24938,48948,48949],{},[24941,48950],{"id":33166,"d":27108},[25022,48952,48953],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48954,48955],{"dataMmlNode":25028},[25022,48956,48957],{"dataMmlNode":25942},[25033,48958],{"dataC":27210,"xLinkHref":33243}," to hidden node ",[24923,48961,48963],{"className":48962,"jax":24927},[24926],[24929,48964,48969,48974],{"style":48965,"xmlns":24932,"width":48966,"height":48967,"role":1930,"focusable":8058,"viewBox":48968,"xmlnsXLink":24936},"vertical-align: -0.462ex;","0.932ex","1.957ex","0 -661 412 865",[24938,48970,48971],{},[24941,48972],{"id":48973,"d":33606},"MJX-15-TEX-I-1D457",[25022,48975,48976],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,48977,48978],{"dataMmlNode":25028},[25022,48979,48980],{"dataMmlNode":25942},[25033,48981],{"dataC":33647,"xLinkHref":48982},"#MJX-15-TEX-I-1D457",[26,48984,48985,49042],{},[24923,48986,48988],{"className":48987,"jax":24927},[24926],[24929,48989,48990,49007],{"style":25196,"xmlns":24932,"width":48804,"height":32538,"role":1930,"focusable":8058,"viewBox":48805,"xmlnsXLink":24936},[24938,48991,48992,48994,48996,48999,49002,49005],{},[24941,48993],{"id":43639,"d":37254},[24941,48995],{"id":33393,"d":24944},[24941,48997],{"id":48998,"d":25928},"MJX-16-TEX-N-32",[24941,49000],{"id":49001,"d":27339},"MJX-16-TEX-N-2C",[24941,49003],{"id":49004,"d":25914},"MJX-16-TEX-N-31",[24941,49006],{"id":33399,"d":24988},[25022,49008,49009],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,49010,49011],{"dataMmlNode":25028},[25022,49012,49013,49017],{"dataMmlNode":31565},[25022,49014,49015],{"dataMmlNode":25942},[25033,49016],{"dataC":37310,"xLinkHref":43648},[25022,49018,49019,49023,49028,49033,49038],{"dataMmlNode":27203,"transform":37564,"dataMjxTexclass":27205},[25022,49020,49021],{"dataMmlNode":25031},[25033,49022],{"dataC":25035,"xLinkHref":33413},[25022,49024,49025],{"dataMmlNode":26031,"transform":25040},[25033,49026],{"dataC":26058,"xLinkHref":49027},"#MJX-16-TEX-N-32",[25022,49029,49030],{"dataMmlNode":25031,"transform":41846},[25033,49031],{"dataC":27456,"xLinkHref":49032},"#MJX-16-TEX-N-2C",[25022,49034,49035],{"dataMmlNode":26031,"transform":28333},[25033,49036],{"dataC":26035,"xLinkHref":49037},"#MJX-16-TEX-N-31",[25022,49039,49040],{"dataMmlNode":25031,"transform":41146},[25033,49041],{"dataC":25108,"xLinkHref":33423},": Weight matrix from the hidden layer to the output layer.",[26,49044,49045,49048,49049,49070,49071,6959],{},[106,49046,49047],{},"Training Samples:"," A set of ",[24923,49050,49052],{"className":49051,"jax":24927},[24926],[24929,49053,49056,49061],{"style":27802,"xmlns":24932,"width":49054,"height":28113,"role":1930,"focusable":8058,"viewBox":49055,"xmlnsXLink":24936},"1.699ex","0 -683 751 683",[24938,49057,49058],{},[24941,49059],{"id":49060,"d":39730},"MJX-17-TEX-I-1D443",[25022,49062,49063],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,49064,49065],{"dataMmlNode":25028},[25022,49066,49067],{"dataMmlNode":25942},[25033,49068],{"dataC":39770,"xLinkHref":49069},"#MJX-17-TEX-I-1D443"," samples, denoted as ",[24923,49072,49074],{"className":49073,"jax":24927},[24926],[24929,49075,49078,49116],{"style":28262,"xmlns":24932,"width":49076,"height":28264,"role":1930,"focusable":8058,"viewBox":49077,"xmlnsXLink":24936},"23.386ex","0 -750 10336.5 1037.2",[24938,49079,49080,49084,49086,49089,49092,49095,49098,49100,49103,49105,49107,49109,49112],{},[24941,49081],{"id":49082,"d":49083},"MJX-18-TEX-N-7B","M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 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scale(0.707)",[25022,49741,49742],{"dataMmlNode":25942},[25033,49743],{"dataC":43135,"xLinkHref":49744},"#MJX-29-TEX-I-1D458",[25022,49746,49748],{"dataMmlNode":25031,"transform":49747},"translate(521,0)",[25033,49749],{"dataC":25078,"xLinkHref":27997},[25022,49751,49753],{"dataMmlNode":26031,"transform":49752},"translate(1299,0)",[25033,49754],{"dataC":26035,"xLinkHref":28029},[25022,49756,49758],{"dataMmlNode":27203,"transform":49757,"dataMjxTexclass":27205},"translate(407.7,1150) scale(0.707)",[25022,49759,49760],{"dataMmlNode":25942},[25033,49761],{"dataC":49639,"xLinkHref":49762},"#MJX-29-TEX-I-1D43E",[25022,49764,49766],{"dataMmlNode":25031,"transform":49765},"translate(5152.2,0)",[25033,49767],{"dataC":25035,"xLinkHref":27982},[25022,49769,49771,49776],{"dataMmlNode":26023,"transform":49770},"translate(5541.2,0)",[25022,49772,49773],{"dataMmlNode":25942},[25033,49774],{"dataC":32396,"xLinkHref":49775},"#MJX-29-TEX-I-1D451",[25022,49777,49778,49782,49787],{"dataMmlNode":27203,"transform":49158,"dataMjxTexclass":27205},[25022,49779,49780],{"dataMmlNode":25942},[25033,49781],{"dataC":25980,"xLinkHref":27972},[25022,49783,49784],{"dataMmlNode":25031,"transform":49420},[25033,49785],{"dataC":27456,"xLinkHref":49786},"#MJX-29-TEX-N-2C",[25022,49788,49789],{"dataMmlNode":25942,"transform":49426},[25033,49790],{"dataC":43135,"xLinkHref":49744},[25022,49792,49794],{"dataMmlNode":25031,"transform":49793},"translate(7287.1,0)",[25033,49795],{"dataC":26019,"xLinkHref":28035},[25022,49797,49799,49803],{"dataMmlNode":26023,"transform":49798},"translate(8287.3,0)",[25022,49800,49801],{"dataMmlNode":25942},[25033,49802],{"dataC":25973,"xLinkHref":27967},[25022,49804,49805,49809,49813],{"dataMmlNode":27203,"transform":49369,"dataMjxTexclass":27205},[25022,49806,49807],{"dataMmlNode":25942},[25033,49808],{"dataC":25980,"xLinkHref":27972},[25022,49810,49811],{"dataMmlNode":25031,"transform":49420},[25033,49812],{"dataC":27456,"xLinkHref":49786},[25022,49814,49815],{"dataMmlNode":25942,"transform":49426},[25033,49816],{"dataC":43135,"xLinkHref":49744},[25022,49818,49820,49824],{"dataMmlNode":31565,"transform":49819},"translate(9776,0)",[25022,49821,49822],{"dataMmlNode":25031},[25033,49823],{"dataC":25108,"xLinkHref":27992},[25022,49825,49826],{"dataMmlNode":26031,"transform":33326},[25033,49827],{"dataC":26058,"xLinkHref":28065},[25631,49829,49831,49834],{"className":49830},[13441,13442],[25635,49832,49833],{},"Making sense of the formulas",[13438,49835,49837,49847,49850,49883,49890,49894,50005,50076,50148,50234,50238,50423,50787],{"className":49836},[25641],[12,49838,9407,49839,49842,49843,49846],{},[106,49840,49841],{},"vector"," is simply a ",[106,49844,49845],{},"list or array of numbers"," arranged in a specific order.",[12,49848,49849],{},"When we train a neural network, we feed it data (inputs) and tell it what the correct answer should be (targets).",[12,49851,49852,49853,49856,49857,16419,49876,49878,49879,49882],{},"The input here is the list of features you are feeding into the network for ",[7289,49854,49855],{},"one specific example"," (sample ",[24923,49858,49860],{"className":49859,"jax":24927},[24926],[24929,49861,49862,49867],{"style":26153,"xmlns":24932,"width":49454,"height":26155,"role":1930,"focusable":8058,"viewBox":49455,"xmlnsXLink":24936},[24938,49863,49864],{},[24941,49865],{"id":49866,"d":25889},"MJX-30-TEX-I-1D45D",[25022,49868,49869],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,49870,49871],{"dataMmlNode":25028},[25022,49872,49873],{"dataMmlNode":25942},[25033,49874],{"dataC":25980,"xLinkHref":49875},"#MJX-30-TEX-I-1D45D",[16,49877],{},"\nLet's say you are training an AI to predict a house's price. Your input vector for one specific house might be: ",[6979,49880,49881],{},"[3, 2000, 15]",". This vector represents 3 bedrooms, 2000 square feet, and 15 years old. The network receives these three numbers at its input layer.",[12,49884,49885,49886,49889],{},"The output is a vector because the output might be more than one number. For example, an AI that's trained to classify animals as Dog, Cat, or Bird, the output layer will contain 3 nodes, ",[6979,49887,49888],{},"[0, 1, 0]"," (0% Dog, 100% Cat, 0% Bird).",[26236,49891,49893],{"id":49892},"the-simple-error-for-one-node","The Simple Error for One 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988.2",[24938,50015,50016,50019,50022,50025],{},[24941,50017],{"id":50018,"d":32316},"MJX-32-TEX-I-1D451",[24941,50020],{"id":50021,"d":25889},"MJX-32-TEX-I-1D45D",[24941,50023],{"id":50024,"d":27339},"MJX-32-TEX-N-2C",[24941,50026],{"id":50027,"d":33606},"MJX-32-TEX-I-1D457",[25022,50029,50030],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50031,50032],{"dataMmlNode":25028},[25022,50033,50034,50039],{"dataMmlNode":26023},[25022,50035,50036],{"dataMmlNode":25942},[25033,50037],{"dataC":32396,"xLinkHref":50038},"#MJX-32-TEX-I-1D451",[25022,50040,50041,50046,50051],{"dataMmlNode":27203,"transform":49158,"dataMjxTexclass":27205},[25022,50042,50043],{"dataMmlNode":25942},[25033,50044],{"dataC":25980,"xLinkHref":50045},"#MJX-32-TEX-I-1D45D",[25022,50047,50048],{"dataMmlNode":25031,"transform":49420},[25033,50049],{"dataC":27456,"xLinkHref":50050},"#MJX-32-TEX-N-2C",[25022,50052,50053],{"dataMmlNode":25942,"transform":49426},[25033,50054],{"dataC":33647,"xLinkHref":50055},"#MJX-32-TEX-I-1D457"," is your Desired target for node ",[24923,50058,50060],{"className":50059,"jax":24927},[24926],[24929,50061,50062,50067],{"style":48965,"xmlns":24932,"width":48966,"height":48967,"role":1930,"focusable":8058,"viewBox":48968,"xmlnsXLink":24936},[24938,50063,50064],{},[24941,50065],{"id":50066,"d":33606},"MJX-33-TEX-I-1D457",[25022,50068,50069],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50070,50071],{"dataMmlNode":25028},[25022,50072,50073],{"dataMmlNode":25942},[25033,50074],{"dataC":33647,"xLinkHref":50075},"#MJX-33-TEX-I-1D457",[12,50077,50078,50128,50129,6959],{},[24923,50079,50081],{"className":50080,"jax":24927},[24926],[24929,50082,50086,50100],{"style":35650,"xmlns":24932,"width":50083,"height":50084,"role":1930,"focusable":8058,"viewBox":50085,"xmlnsXLink":24936},"3.205ex","1.666ex","0 -442 1416.6 736.2",[24938,50087,50088,50091,50094,50097],{},[24941,50089],{"id":50090,"d":25893},"MJX-34-TEX-I-1D466",[24941,50092],{"id":50093,"d":25889},"MJX-34-TEX-I-1D45D",[24941,50095],{"id":50096,"d":27339},"MJX-34-TEX-N-2C",[24941,50098],{"id":50099,"d":33606},"MJX-34-TEX-I-1D457",[25022,50101,50102],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50103,50104],{"dataMmlNode":25028},[25022,50105,50106,50111],{"dataMmlNode":26023},[25022,50107,50108],{"dataMmlNode":25942},[25033,50109],{"dataC":25987,"xLinkHref":50110},"#MJX-34-TEX-I-1D466",[25022,50112,50113,50118,50123],{"dataMmlNode":27203,"transform":43996,"dataMjxTexclass":27205},[25022,50114,50115],{"dataMmlNode":25942},[25033,50116],{"dataC":25980,"xLinkHref":50117},"#MJX-34-TEX-I-1D45D",[25022,50119,50120],{"dataMmlNode":25031,"transform":49420},[25033,50121],{"dataC":27456,"xLinkHref":50122},"#MJX-34-TEX-N-2C",[25022,50124,50125],{"dataMmlNode":25942,"transform":49426},[25033,50126],{"dataC":33647,"xLinkHref":50127},"#MJX-34-TEX-I-1D457"," is your actual output (Y) for node ",[24923,50130,50132],{"className":50131,"jax":24927},[24926],[24929,50133,50134,50139],{"style":48965,"xmlns":24932,"width":48966,"height":48967,"role":1930,"focusable":8058,"viewBox":48968,"xmlnsXLink":24936},[24938,50135,50136],{},[24941,50137],{"id":50138,"d":33606},"MJX-35-TEX-I-1D457",[25022,50140,50141],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50142,50143],{"dataMmlNode":25028},[25022,50144,50145],{"dataMmlNode":25942},[25033,50146],{"dataC":33647,"xLinkHref":50147},"#MJX-35-TEX-I-1D457",[12,50149,50150,50151,50170,50171,50202,50203,6959],{},"This formula just says: Error = (What we wanted) - (What the network actually guessed). If the target was ",[24923,50152,50154],{"className":50153,"jax":24927},[24926],[24929,50155,50156,50161],{"style":27802,"xmlns":24932,"width":35168,"height":35169,"role":1930,"focusable":8058,"viewBox":35170,"xmlnsXLink":24936},[24938,50157,50158],{},[24941,50159],{"id":50160,"d":25914},"MJX-36-TEX-N-31",[25022,50162,50163],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50164,50165],{"dataMmlNode":25028},[25022,50166,50167],{"dataMmlNode":26031},[25033,50168],{"dataC":26035,"xLinkHref":50169},"#MJX-36-TEX-N-31"," and the network guessed 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",[24923,50204,50206],{"className":50205,"jax":24927},[24926],[24929,50207,50208,50219],{"style":27778,"xmlns":24932,"width":29568,"height":35776,"role":1930,"focusable":8058,"viewBox":39043,"xmlnsXLink":24936},[24938,50209,50210,50213,50216],{},[24941,50211],{"id":50212,"d":25935},"MJX-38-TEX-N-30",[24941,50214],{"id":50215,"d":26357},"MJX-38-TEX-N-2E",[24941,50217],{"id":50218,"d":25928},"MJX-38-TEX-N-32",[25022,50220,50221],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50222,50223],{"dataMmlNode":25028},[25022,50224,50225,50228,50231],{"dataMmlNode":26031},[25033,50226],{"dataC":26103,"xLinkHref":50227},"#MJX-38-TEX-N-30",[25033,50229],{"dataC":26392,"xLinkHref":50230,"transform":26394},"#MJX-38-TEX-N-2E",[25033,50232],{"dataC":26058,"xLinkHref":50233,"transform":25049},"#MJX-38-TEX-N-32",[26236,50235,50237],{"id":50236},"the-big-objective-function-sum-squared-error","The Big Objective Function (Sum Squared 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First, calculate the error (Desired minus Output) for a specific output node. Then, square it.",[23,50536,50537],{},[26,50538,50539,50540,50581,50582,50621],{},"Why square it? 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Second, it heavily penalizes large errors (an error of 4 becomes 16, screaming at the network \"FIX THIS!\").",[26,50623,50624,50682,50683,50704],{},[24923,50625,50627],{"className":50626,"jax":24927},[24926],[24929,50628,50632,50648],{"style":27869,"xmlns":24932,"width":50629,"height":50630,"role":1930,"focusable":8058,"viewBox":50631,"xmlnsXLink":24936},"5.455ex","2.949ex","0 -960 2411.1 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Add up those squared errors for all the output nodes (",[24923,50684,50686],{"className":50685,"jax":24927},[24926],[24929,50687,50690,50695],{"style":25196,"xmlns":24932,"width":50688,"height":31523,"role":1930,"focusable":8058,"viewBox":50689,"xmlnsXLink":24936},"1.179ex","0 -694 521 705",[24938,50691,50692],{},[24941,50693],{"id":50694,"d":42951},"MJX-44-TEX-I-1D458",[25022,50696,50697],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50698,50699],{"dataMmlNode":25028},[25022,50700,50701],{"dataMmlNode":25942},[25033,50702],{"dataC":43135,"xLinkHref":50703},"#MJX-44-TEX-I-1D458",") in the network for that one specific sample.",[26,50706,50707,50766,50767,50786],{},[24923,50708,50710],{"className":50709,"jax":24927},[24926],[24929,50711,50716,50732],{"style":50712,"xmlns":24932,"width":50713,"height":50714,"role":1930,"focusable":8058,"viewBox":50715,"xmlnsXLink":24936},"vertical-align: -0.956ex;","5.426ex","3.128ex","0 -960 2398.4 1382.5",[24938,50717,50718,50721,50724,50727,50729],{},[24941,50719],{"id":50720,"d":27911},"MJX-45-TEX-SO-2211",[24941,50722],{"id":50723,"d":39730},"MJX-45-TEX-I-1D443",[24941,50725],{"id":50726,"d":25889},"MJX-45-TEX-I-1D45D",[24941,50728],{"id":45790,"d":24972},[24941,50730],{"id":50731,"d":25914},"MJX-45-TEX-N-31",[25022,50733,50734],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50735,50736],{"dataMmlNode":25028},[25022,50737,50738,50743,50750],{"dataMmlNode":27194},[25022,50739,50740],{"dataMmlNode":25031},[25033,50741],{"dataC":27199,"xLinkHref":50742},"#MJX-45-TEX-SO-2211",[25022,50744,50745],{"dataMmlNode":27203,"transform":28007,"dataMjxTexclass":27205},[25022,50746,50747],{"dataMmlNode":25942},[25033,50748],{"dataC":39770,"xLinkHref":50749},"#MJX-45-TEX-I-1D443",[25022,50751,50752,50757,50761],{"dataMmlNode":27203,"transform":28015,"dataMjxTexclass":27205},[25022,50753,50754],{"dataMmlNode":25942},[25033,50755],{"dataC":25980,"xLinkHref":50756},"#MJX-45-TEX-I-1D45D",[25022,50758,50759],{"dataMmlNode":25031,"transform":49420},[25033,50760],{"dataC":25078,"xLinkHref":45857},[25022,50762,50763],{"dataMmlNode":26031,"transform":49719},[25033,50764],{"dataC":26035,"xLinkHref":50765},"#MJX-45-TEX-N-31",": Do that for every single training sample (",[24923,50768,50770],{"className":50769,"jax":24927},[24926],[24929,50771,50772,50777],{"style":26153,"xmlns":24932,"width":49454,"height":26155,"role":1930,"focusable":8058,"viewBox":49455,"xmlnsXLink":24936},[24938,50773,50774],{},[24941,50775],{"id":50776,"d":25889},"MJX-46-TEX-I-1D45D",[25022,50778,50779],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50780,50781],{"dataMmlNode":25028},[25022,50782,50783],{"dataMmlNode":25942},[25033,50784],{"dataC":25980,"xLinkHref":50785},"#MJX-46-TEX-I-1D45D",") in your entire dataset, and add them all together into one giant number.",[12,50788,50789,50790,50811,50812,50831],{},"In plain English: The Objective Function (",[24923,50791,50793],{"className":50792,"jax":24927},[24926],[24929,50794,50797,50802],{"style":27802,"xmlns":24932,"width":50795,"height":27804,"role":1930,"focusable":8058,"viewBox":50796,"xmlnsXLink":24936},"1.729ex","0 -680 764 680",[24938,50798,50799],{},[24941,50800],{"id":50801,"d":25869},"MJX-47-TEX-I-1D438",[25022,50803,50804],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50805,50806],{"dataMmlNode":25028},[25022,50807,50808],{"dataMmlNode":25942},[25033,50809],{"dataC":25945,"xLinkHref":50810},"#MJX-47-TEX-I-1D438",") gives you a single, massive number that represents the total amount of wrongness across your entire dataset. The whole goal of Backpropagation is to tweak the network's weights to make that giant number ",[24923,50813,50815],{"className":50814,"jax":24927},[24926],[24929,50816,50817,50822],{"style":27802,"xmlns":24932,"width":50795,"height":27804,"role":1930,"focusable":8058,"viewBox":50796,"xmlnsXLink":24936},[24938,50818,50819],{},[24941,50820],{"id":50821,"d":25869},"MJX-48-TEX-I-1D438",[25022,50823,50824],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50825,50826],{"dataMmlNode":25028},[25022,50827,50828],{"dataMmlNode":25942},[25033,50829],{"dataC":25945,"xLinkHref":50830},"#MJX-48-TEX-I-1D438"," as close to zero as possible!",[7166,50833,50835],{"id":50834},"step-1-forward-computing-the-forward-pass","Step 1: Forward Computing (The Forward Pass)",[12,50837,50838],{},"Data propagates forward from inputs to outputs.",[12,50840,50841,50844,50845,50864,50865,6959],{},[106,50842,50843],{},"Hidden Layer Calculation:"," The input vector ",[24923,50846,50848],{"className":50847,"jax":24927},[24926],[24929,50849,50850,50855],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,50851,50852],{},[24941,50853],{"id":50854,"d":32148},"MJX-49-TEX-I-1D465",[25022,50856,50857],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50858,50859],{"dataMmlNode":25028},[25022,50860,50861],{"dataMmlNode":25942},[25033,50862],{"dataC":32181,"xLinkHref":50863},"#MJX-49-TEX-I-1D465"," is multiplied by the input-to-hidden weights to generate a net input, which is then passed through an activation function ",[24923,50866,50868],{"className":50867,"jax":24927},[24926],[24929,50869,50870,50874],{"style":27778,"xmlns":24932,"width":27779,"height":27780,"role":1930,"focusable":8058,"viewBox":27781,"xmlnsXLink":24936},[24938,50871,50872],{},[24941,50873],{"id":29637,"d":25900},[25022,50875,50876],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50877,50878],{"dataMmlNode":25028},[25022,50879,50880],{"dataMmlNode":25942},[25033,50881],{"dataC":26000,"xLinkHref":29660},[24923,50883,50885],{"className":50884,"jax":24927,"display":13594},[24926],[24929,50886,50889,50936],{"style":34674,"xmlns":24932,"width":50887,"height":34676,"role":1930,"focusable":8058,"viewBox":50888,"xmlnsXLink":24936},"35.754ex","0 -1749.5 15803.2 2999",[24938,50890,50891,50893,50895,50897,50899,50902,50904,50907,50910,50912,50915,50917,50920,50922,50925,50927,50929,50932,50934],{},[24941,50892],{"id":37289,"d":32148},[24941,50894],{"id":37257,"d":24944},[24941,50896],{"id":29703,"d":25914},[24941,50898],{"id":37263,"d":24988},[24941,50900],{"id":50901,"d":33606},"MJX-51-TEX-I-1D457",[24941,50903],{"id":37266,"d":24972},[24941,50905],{"id":50906,"d":25900},"MJX-51-TEX-I-1D446",[24941,50908],{"id":50909,"d":34234},"MJX-51-TEX-LO-28",[24941,50911],{"id":46522,"d":25873},[24941,50913],{"id":50914,"d":27397},"MJX-51-TEX-I-1D452",[24941,50916],{"id":46511,"d":25877},[24941,50918],{"id":50919,"d":34250},"MJX-51-TEX-LO-29",[24941,50921],{"id":37283,"d":34706},[24941,50923],{"id":50924,"d":27104},"MJX-51-TEX-LO-2211",[24941,50926],{"id":37260,"d":27108},[24941,50928],{"id":37253,"d":37254},[24941,50930],{"id":50931,"d":27339},"MJX-51-TEX-N-2C",[24941,50933],{"id":29710,"d":25935},[24941,50935],{"id":37299,"d":34719},[25022,50937,50938],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,50939,50940,50965,50970,50976,51027,51032,51037],{"dataMmlNode":25028},[25022,50941,50942,50946,50960],{"dataMmlNode":33991},[25022,50943,50944],{"dataMmlNode":25942},[25033,50945],{"dataC":32181,"xLinkHref":37384},[25022,50947,50948,50952,50956],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,50949,50950],{"dataMmlNode":25031},[25033,50951],{"dataC":25035,"xLinkHref":37319},[25022,50953,50954],{"dataMmlNode":26031,"transform":25040},[25033,50955],{"dataC":26035,"xLinkHref":29719},[25022,50957,50958],{"dataMmlNode":25031,"transform":41846},[25033,50959],{"dataC":25108,"xLinkHref":37329},[25022,50961,50962],{"dataMmlNode":25942,"transform":34014},[25033,50963],{"dataC":33647,"xLinkHref":50964},"#MJX-51-TEX-I-1D457",[25022,50966,50968],{"dataMmlNode":25031,"transform":50967},"translate(1836.5,0)",[25033,50969],{"dataC":25078,"xLinkHref":37335},[25022,50971,50973],{"dataMmlNode":25942,"transform":50972},"translate(2892.2,0)",[25033,50974],{"dataC":26000,"xLinkHref":50975},"#MJX-51-TEX-I-1D446",[25022,50977,50979,50984,50988,50994,51021],{"dataMmlNode":27626,"transform":50978},"translate(3703.9,0)",[25022,50980,50981],{"dataMmlNode":25031,"transform":27630},[25033,50982],{"dataC":25035,"xLinkHref":50983},"#MJX-51-TEX-LO-28",[25022,50985,50986],{"dataMmlNode":25942,"transform":34318},[25033,50987],{"dataC":25952,"xLinkHref":46569},[25022,50989,50991],{"dataMmlNode":25942,"transform":50990},"translate(1197,0)",[25033,50992],{"dataC":27596,"xLinkHref":50993},"#MJX-51-TEX-I-1D452",[25022,50995,50997,51001,51016],{"dataMmlNode":33991,"transform":50996},"translate(1663,0)",[25022,50998,50999],{"dataMmlNode":25942},[25033,51000],{"dataC":25959,"xLinkHref":46550},[25022,51002,51004,51008,51012],{"dataMmlNode":27203,"transform":51003,"dataMjxTexclass":27205},"translate(394,530.4) scale(0.707)",[25022,51005,51006],{"dataMmlNode":25031},[25033,51007],{"dataC":25035,"xLinkHref":37319},[25022,51009,51010],{"dataMmlNode":26031,"transform":25040},[25033,51011],{"dataC":26035,"xLinkHref":29719},[25022,51013,51014],{"dataMmlNode":25031,"transform":41846},[25033,51015],{"dataC":25108,"xLinkHref":37329},[25022,51017,51019],{"dataMmlNode":25942,"transform":51018},"translate(394,-293.8) scale(0.707)",[25033,51020],{"dataC":33647,"xLinkHref":50964},[25022,51022,51024],{"dataMmlNode":25031,"transform":51023},"translate(3010.7,0) translate(0 -0.5)",[25033,51025],{"dataC":25108,"xLinkHref":51026},"#MJX-51-TEX-LO-29",[25022,51028,51030],{"dataMmlNode":25031,"transform":51029},"translate(7589.4,0)",[25033,51031],{"dataC":25078,"xLinkHref":37335},[25022,51033,51035],{"dataMmlNode":25942,"transform":51034},"translate(8645.1,0)",[25033,51036],{"dataC":26000,"xLinkHref":50975},[25022,51038,51040,51044,51057,51101,51112],{"dataMmlNode":27626,"transform":51039},"translate(9456.8,0)",[25022,51041,51042],{"dataMmlNode":25031,"transform":27630},[25033,51043],{"dataC":25035,"xLinkHref":37363},[25022,51045,51046,51051],{"dataMmlNode":27541,"transform":34797},[25022,51047,51048],{"dataMmlNode":25031},[25033,51049],{"dataC":27199,"xLinkHref":51050},"#MJX-51-TEX-LO-2211",[25022,51052,51053],{"dataMmlNode":27203,"transform":33240,"dataMjxTexclass":27205},[25022,51054,51055],{"dataMmlNode":25942},[25033,51056],{"dataC":27210,"xLinkHref":37324},[25022,51058,51060,51064,51087],{"dataMmlNode":33991,"transform":51059},"translate(2402.7,0)",[25022,51061,51062],{"dataMmlNode":25942},[25033,51063],{"dataC":37310,"xLinkHref":37311},[25022,51065,51066,51070,51074,51079,51083],{"dataMmlNode":27203,"transform":48899,"dataMjxTexclass":27205},[25022,51067,51068],{"dataMmlNode":25031},[25033,51069],{"dataC":25035,"xLinkHref":37319},[25022,51071,51072],{"dataMmlNode":26031,"transform":25040},[25033,51073],{"dataC":26035,"xLinkHref":29719},[25022,51075,51076],{"dataMmlNode":25031,"transform":41846},[25033,51077],{"dataC":27456,"xLinkHref":51078},"#MJX-51-TEX-N-2C",[25022,51080,51081],{"dataMmlNode":26031,"transform":28333},[25033,51082],{"dataC":26103,"xLinkHref":29735},[25022,51084,51085],{"dataMmlNode":25031,"transform":41146},[25033,51086],{"dataC":25108,"xLinkHref":37329},[25022,51088,51089,51093,51097],{"dataMmlNode":27203,"transform":48924,"dataMjxTexclass":27205},[25022,51090,51091],{"dataMmlNode":25942},[25033,51092],{"dataC":33647,"xLinkHref":50964},[25022,51094,51095],{"dataMmlNode":25031,"transform":48932},[25033,51096],{"dataC":27456,"xLinkHref":51078},[25022,51098,51099],{"dataMmlNode":25942,"transform":48937},[25033,51100],{"dataC":27210,"xLinkHref":37324},[25022,51102,51104,51108],{"dataMmlNode":26023,"transform":51103},"translate(4655.5,0)",[25022,51105,51106],{"dataMmlNode":25942},[25033,51107],{"dataC":32181,"xLinkHref":37384},[25022,51109,51110],{"dataMmlNode":25942,"transform":33034},[25033,51111],{"dataC":27210,"xLinkHref":37324},[25022,51113,51115],{"dataMmlNode":25031,"transform":51114},"translate(5554.4,0) translate(0 -0.5)",[25033,51116],{"dataC":25108,"xLinkHref":37450},[25631,51118,51120,51123],{"className":51119},[13441,13442],[25635,51121,51122],{},"Formula explanation",[13438,51124,51126,51161,51215,51328],{"className":51125},[25641],[12,51127,51128,10476,51131,51160],{},[106,51129,51130],{},"The Inputs",[24923,51132,51134],{"className":51133,"jax":24927},[24926],[24929,51135,51138,51145],{"style":27744,"xmlns":24932,"width":51136,"height":31803,"role":1930,"focusable":8058,"viewBox":51137,"xmlnsXLink":24936},"2.034ex","0 -442 899 599.8",[24938,51139,51140,51143],{},[24941,51141],{"id":51142,"d":32148},"MJX-52-TEX-I-1D465",[24941,51144],{"id":46618,"d":27108},[25022,51146,51147],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,51148,51149],{"dataMmlNode":25028},[25022,51150,51151,51156],{"dataMmlNode":26023},[25022,51152,51153],{"dataMmlNode":25942},[25033,51154],{"dataC":32181,"xLinkHref":51155},"#MJX-52-TEX-I-1D465",[25022,51157,51158],{"dataMmlNode":25942,"transform":33034},[25033,51159],{"dataC":27210,"xLinkHref":46657},"): These are the numbers you are feeding into the network (like the house size or number of bedrooms we talked about earlier).",[12,51162,51163,10476,51166,51214],{},[106,51164,51165],{},"The Weights",[24923,51167,51169],{"className":51168,"jax":24927},[24926],[24929,51170,51174,51187],{"style":35650,"xmlns":24932,"width":51171,"height":51172,"role":1930,"focusable":8058,"viewBox":51173,"xmlnsXLink":24936},"3.463ex","1.668ex","0 -443 1530.9 737.2",[24938,51175,51176,51179,51182,51185],{},[24941,51177],{"id":51178,"d":37254},"MJX-53-TEX-I-1D464",[24941,51180],{"id":51181,"d":33606},"MJX-53-TEX-I-1D457",[24941,51183],{"id":51184,"d":27339},"MJX-53-TEX-N-2C",[24941,51186],{"id":29839,"d":27108},[25022,51188,51189],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,51190,51191],{"dataMmlNode":25028},[25022,51192,51193,51198],{"dataMmlNode":26023},[25022,51194,51195],{"dataMmlNode":25942},[25033,51196],{"dataC":37310,"xLinkHref":51197},"#MJX-53-TEX-I-1D464",[25022,51199,51200,51205,51210],{"dataMmlNode":27203,"transform":43003,"dataMjxTexclass":27205},[25022,51201,51202],{"dataMmlNode":25942},[25033,51203],{"dataC":33647,"xLinkHref":51204},"#MJX-53-TEX-I-1D457",[25022,51206,51207],{"dataMmlNode":25031,"transform":48932},[25033,51208],{"dataC":27456,"xLinkHref":51209},"#MJX-53-TEX-N-2C",[25022,51211,51212],{"dataMmlNode":25942,"transform":48937},[25033,51213],{"dataC":27210,"xLinkHref":29900},"): Think of weights as \"importance dials.\" The network multiplies each input by its specific weight to decide how much that input matters.",[12,51216,51217,10476,51220,51264,51265,16419],{},[106,51218,51219],{},"The Multiply and Add",[24923,51221,51223],{"className":51222,"jax":24927},[24926],[24929,51224,51227,51239],{"style":24931,"xmlns":24932,"width":51225,"height":24934,"role":1930,"focusable":8058,"viewBox":51226,"xmlnsXLink":24936},"7.315ex","0 -750 3233.1 1000",[24938,51228,51229,51232,51234,51237],{},[24941,51230],{"id":51231,"d":27911},"MJX-54-TEX-SO-2211",[24941,51233],{"id":37543,"d":37254},[24941,51235],{"id":51236,"d":27409},"MJX-54-TEX-N-22C5",[24941,51238],{"id":46818,"d":32148},[25022,51240,51241],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,51242,51243,51248,51253,51259],{"dataMmlNode":25028},[25022,51244,51245],{"dataMmlNode":25031},[25033,51246],{"dataC":27199,"xLinkHref":51247},"#MJX-54-TEX-SO-2211",[25022,51249,51251],{"dataMmlNode":25942,"transform":51250},"translate(1222.7,0)",[25033,51252],{"dataC":37310,"xLinkHref":37561},[25022,51254,51256],{"dataMmlNode":25031,"transform":51255},"translate(2160.9,0)",[25033,51257],{"dataC":27679,"xLinkHref":51258},"#MJX-54-TEX-N-22C5",[25022,51260,51262],{"dataMmlNode":25942,"transform":51261},"translate(2661.1,0)",[25033,51263],{"dataC":32181,"xLinkHref":46895},"): The network takes Input 1, multiplies it by Weight 1. Then Input 2 times Weight 2. It does this for all inputs and adds them all together into one single number. This is called the \"net input\" (",[24923,51266,51268],{"className":51267,"jax":24927},[24926],[24929,51269,51272,51290],{"style":33830,"xmlns":24932,"width":51270,"height":33832,"role":1930,"focusable":8058,"viewBox":51271,"xmlnsXLink":24936},"5.461ex","0 -1060.7 2413.7 1498.8",[24938,51273,51274,51276,51279,51281,51283,51285,51287],{},[24941,51275],{"id":46970,"d":25873},[24941,51277],{"id":51278,"d":27397},"MJX-55-TEX-I-1D452",[24941,51280],{"id":46959,"d":25877},[24941,51282],{"id":37597,"d":24944},[24941,51284],{"id":30179,"d":25914},[24941,51286],{"id":37603,"d":24988},[24941,51288],{"id":51289,"d":33606},"MJX-55-TEX-I-1D457",[25022,51291,51292],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,51293,51294,51298,51303],{"dataMmlNode":25028},[25022,51295,51296],{"dataMmlNode":25942},[25033,51297],{"dataC":25952,"xLinkHref":47010},[25022,51299,51300],{"dataMmlNode":25942,"transform":46886},[25033,51301],{"dataC":27596,"xLinkHref":51302},"#MJX-55-TEX-I-1D452",[25022,51304,51305,51309,51323],{"dataMmlNode":33991,"transform":46892},[25022,51306,51307],{"dataMmlNode":25942},[25033,51308],{"dataC":25959,"xLinkHref":46991},[25022,51310,51311,51315,51319],{"dataMmlNode":27203,"transform":51003,"dataMjxTexclass":27205},[25022,51312,51313],{"dataMmlNode":25031},[25033,51314],{"dataC":25035,"xLinkHref":37653},[25022,51316,51317],{"dataMmlNode":26031,"transform":25040},[25033,51318],{"dataC":26035,"xLinkHref":30206},[25022,51320,51321],{"dataMmlNode":25031,"transform":41846},[25033,51322],{"dataC":25108,"xLinkHref":37663},[25022,51324,51325],{"dataMmlNode":25942,"transform":51018},[25033,51326],{"dataC":33647,"xLinkHref":51327},"#MJX-55-TEX-I-1D457",[12,51329,51330,10476,51333,51350,51351,51370,51371,51420,51421,6959],{},[106,51331,51332],{},"The Activation Function",[24923,51334,51336],{"className":51335,"jax":24927},[24926],[24929,51337,51338,51342],{"style":27778,"xmlns":24932,"width":27779,"height":27780,"role":1930,"focusable":8058,"viewBox":27781,"xmlnsXLink":24936},[24938,51339,51340],{},[24941,51341],{"id":47103,"d":25900},[25022,51343,51344],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,51345,51346],{"dataMmlNode":25028},[25022,51347,51348],{"dataMmlNode":25942},[25033,51349],{"dataC":26000,"xLinkHref":47148},"): The network doesn't just pass that raw sum to the next layer. It passes it through a mathematical filter called an Activation Function (represented by the ",[24923,51352,51354],{"className":51353,"jax":24927},[24926],[24929,51355,51356,51361],{"style":27778,"xmlns":24932,"width":27779,"height":27780,"role":1930,"focusable":8058,"viewBox":27781,"xmlnsXLink":24936},[24938,51357,51358],{},[24941,51359],{"id":51360,"d":25900},"MJX-57-TEX-I-1D446",[25022,51362,51363],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,51364,51365],{"dataMmlNode":25028},[25022,51366,51367],{"dataMmlNode":25942},[25033,51368],{"dataC":26000,"xLinkHref":51369},"#MJX-57-TEX-I-1D446","). This function \"squishes\" the number into a specific range (often between 0 and 1, or -1 and 1). This is crucial because it allows the network to learn complex, non-linear patterns rather than just drawing straight lines.The Result (",[24923,51372,51374],{"className":51373,"jax":24927},[24926],[24929,51375,51378,51391],{"style":33830,"xmlns":24932,"width":51376,"height":33832,"role":1930,"focusable":8058,"viewBox":51377,"xmlnsXLink":24936},"3.526ex","0 -1060.7 1558.7 1498.8",[24938,51379,51380,51382,51384,51386,51388],{},[24941,51381],{"id":37911,"d":32148},[24941,51383],{"id":47337,"d":24944},[24941,51385],{"id":30467,"d":25914},[24941,51387],{"id":47343,"d":24988},[24941,51389],{"id":51390,"d":33606},"MJX-58-TEX-I-1D457",[25022,51392,51393],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,51394,51395],{"dataMmlNode":25028},[25022,51396,51397,51401,51415],{"dataMmlNode":33991},[25022,51398,51399],{"dataMmlNode":25942},[25033,51400],{"dataC":32181,"xLinkHref":37920},[25022,51402,51403,51407,51411],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,51404,51405],{"dataMmlNode":25031},[25033,51406],{"dataC":25035,"xLinkHref":47355},[25022,51408,51409],{"dataMmlNode":26031,"transform":25040},[25033,51410],{"dataC":26035,"xLinkHref":30499},[25022,51412,51413],{"dataMmlNode":25031,"transform":41846},[25033,51414],{"dataC":25108,"xLinkHref":47368},[25022,51416,51417],{"dataMmlNode":25942,"transform":34014},[25033,51418],{"dataC":33647,"xLinkHref":51419},"#MJX-58-TEX-I-1D457","): This is the final, squished number that spits out of hidden node ",[24923,51422,51424],{"className":51423,"jax":24927},[24926],[24929,51425,51426,51431],{"style":48965,"xmlns":24932,"width":48966,"height":48967,"role":1930,"focusable":8058,"viewBox":48968,"xmlnsXLink":24936},[24938,51427,51428],{},[24941,51429],{"id":51430,"d":33606},"MJX-59-TEX-I-1D457",[25022,51432,51433],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,51434,51435],{"dataMmlNode":25028},[25022,51436,51437],{"dataMmlNode":25942},[25033,51438],{"dataC":33647,"xLinkHref":51439},"#MJX-59-TEX-I-1D457",[12,51441,51442,51445],{},[106,51443,51444],{},"Output Layer Calculation:"," The hidden layer's output becomes the input for the output layer.",[24923,51447,51449],{"className":51448,"jax":24927,"display":13594},[24926],[24929,51450,51455,51513],{"style":51451,"xmlns":24932,"width":51452,"height":51453,"role":1930,"focusable":8058,"viewBox":51454,"xmlnsXLink":24936},"vertical-align: 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scale(0.707)",[25022,51671,51672],{"dataMmlNode":25942},[25033,51673],{"dataC":43135,"xLinkHref":51528},[25022,51675,51676],{"dataMmlNode":25031,"transform":49747},[25033,51677],{"dataC":27456,"xLinkHref":51658},[25022,51679,51681],{"dataMmlNode":25942,"transform":51680},"translate(799,0)",[25033,51682],{"dataC":33647,"xLinkHref":51636},[25022,51684,51685,51690,51704],{"dataMmlNode":33991,"transform":51103},[25022,51686,51687],{"dataMmlNode":25942},[25033,51688],{"dataC":32181,"xLinkHref":51689},"#MJX-60-TEX-I-1D465",[25022,51691,51692,51696,51700],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,51693,51694],{"dataMmlNode":25031},[25033,51695],{"dataC":25035,"xLinkHref":30656},[25022,51697,51698],{"dataMmlNode":26031,"transform":25040},[25033,51699],{"dataC":26035,"xLinkHref":30646},[25022,51701,51702],{"dataMmlNode":25031,"transform":41846},[25033,51703],{"dataC":25108,"xLinkHref":30725},[25022,51705,51706],{"dataMmlNode":25942,"transform":34014},[25033,51707],{"dataC":33647,"xLinkHref":51636},[25022,51709,51711],{"dataMmlNode":25031,"transform":51710},"translate(6214.2,0) translate(0 -0.5)",[25033,51712],{"dataC":25108,"xLinkHref":51713},"#MJX-60-TEX-S4-29",[25631,51715,51717,51719],{"className":51716},[13441,13442],[25635,51718,51122],{},[13438,51720,51722],{"className":51721},[25641],[12,51723,51724,51725,51754,51755,51808,51809,51864],{},"This is the exact same formula as the first one. The only thing that has changed is where the data is coming from: Instead of using the raw data (",[24923,51726,51728],{"className":51727,"jax":24927},[24926],[24929,51729,51730,51738],{"style":27744,"xmlns":24932,"width":51136,"height":31803,"role":1930,"focusable":8058,"viewBox":51137,"xmlnsXLink":24936},[24938,51731,51732,51735],{},[24941,51733],{"id":51734,"d":32148},"MJX-61-TEX-I-1D465",[24941,51736],{"id":51737,"d":27108},"MJX-61-TEX-I-1D456",[25022,51739,51740],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,51741,51742],{"dataMmlNode":25028},[25022,51743,51744,51749],{"dataMmlNode":26023},[25022,51745,51746],{"dataMmlNode":25942},[25033,51747],{"dataC":32181,"xLinkHref":51748},"#MJX-61-TEX-I-1D465",[25022,51750,51751],{"dataMmlNode":25942,"transform":33034},[25033,51752],{"dataC":27210,"xLinkHref":51753},"#MJX-61-TEX-I-1D456",") as the input, it uses the output from the hidden layer (",[24923,51756,51758],{"className":51757,"jax":24927},[24926],[24929,51759,51760,51776],{"style":33830,"xmlns":24932,"width":51376,"height":33832,"role":1930,"focusable":8058,"viewBox":51377,"xmlnsXLink":24936},[24938,51761,51762,51765,51768,51770,51773],{},[24941,51763],{"id":51764,"d":32148},"MJX-62-TEX-I-1D465",[24941,51766],{"id":51767,"d":24944},"MJX-62-TEX-N-28",[24941,51769],{"id":38083,"d":25914},[24941,51771],{"id":51772,"d":24988},"MJX-62-TEX-N-29",[24941,51774],{"id":51775,"d":33606},"MJX-62-TEX-I-1D457",[25022,51777,51778],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,51779,51780],{"dataMmlNode":25028},[25022,51781,51782,51787,51803],{"dataMmlNode":33991},[25022,51783,51784],{"dataMmlNode":25942},[25033,51785],{"dataC":32181,"xLinkHref":51786},"#MJX-62-TEX-I-1D465",[25022,51788,51789,51794,51798],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,51790,51791],{"dataMmlNode":25031},[25033,51792],{"dataC":25035,"xLinkHref":51793},"#MJX-62-TEX-N-28",[25022,51795,51796],{"dataMmlNode":26031,"transform":25040},[25033,51797],{"dataC":26035,"xLinkHref":38110},[25022,51799,51800],{"dataMmlNode":25031,"transform":41846},[25033,51801],{"dataC":25108,"xLinkHref":51802},"#MJX-62-TEX-N-29",[25022,51804,51805],{"dataMmlNode":25942,"transform":34014},[25033,51806],{"dataC":33647,"xLinkHref":51807},"#MJX-62-TEX-I-1D457",") that we just calculated above, and tt uses a new set of weights (",[24923,51810,51812],{"className":51811,"jax":24927},[24926],[24929,51813,51814,51830],{"style":25196,"xmlns":24932,"width":48804,"height":32538,"role":1930,"focusable":8058,"viewBox":48805,"xmlnsXLink":24936},[24938,51815,51816,51819,51821,51823,51826,51828],{},[24941,51817],{"id":51818,"d":37254},"MJX-63-TEX-I-1D464",[24941,51820],{"id":30885,"d":24944},[24941,51822],{"id":30912,"d":25928},[24941,51824],{"id":51825,"d":27339},"MJX-63-TEX-N-2C",[24941,51827],{"id":30879,"d":25914},[24941,51829],{"id":30915,"d":24988},[25022,51831,51832],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,51833,51834],{"dataMmlNode":25028},[25022,51835,51836,51841],{"dataMmlNode":31565},[25022,51837,51838],{"dataMmlNode":25942},[25033,51839],{"dataC":37310,"xLinkHref":51840},"#MJX-63-TEX-I-1D464",[25022,51842,51843,51847,51851,51856,51860],{"dataMmlNode":27203,"transform":37564,"dataMjxTexclass":27205},[25022,51844,51845],{"dataMmlNode":25031},[25033,51846],{"dataC":25035,"xLinkHref":30962},[25022,51848,51849],{"dataMmlNode":26031,"transform":25040},[25033,51850],{"dataC":26058,"xLinkHref":31009},[25022,51852,51853],{"dataMmlNode":25031,"transform":41846},[25033,51854],{"dataC":27456,"xLinkHref":51855},"#MJX-63-TEX-N-2C",[25022,51857,51858],{"dataMmlNode":26031,"transform":28333},[25033,51859],{"dataC":26035,"xLinkHref":30952},[25022,51861,51862],{"dataMmlNode":25031,"transform":41146},[25033,51863],{"dataC":25108,"xLinkHref":31024},") connecting the hidden layer to the output layer.",[7166,51866,51868],{"id":51867},"step-2-the-activation-function-sigmoid","Step 2: The Activation Function (Sigmoid)",[12,51870,51871,51872,7024],{},"For backpropagation to work, the activation function must be non-linear and differentiable. 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no exponentials needed.",[26236,53735,53737],{"id":53736},"note-on-saturation","Note on Saturation",[12,53739,53740,53741,53793],{},"If the net input ",[24923,53742,53744],{"className":53743,"jax":24927},[24926],[24929,53745,53748,53762],{"style":29461,"xmlns":24932,"width":53746,"height":29463,"role":1930,"focusable":8058,"viewBox":53747,"xmlnsXLink":24936},"4.486ex","0 -749.5 1983 999",[24938,53749,53750,53753,53756,53759],{},[24941,53751],{"id":53752,"d":27405},"MJX-82-TEX-N-7C",[24941,53754],{"id":53755,"d":25873},"MJX-82-TEX-I-1D45B",[24941,53757],{"id":53758,"d":27397},"MJX-82-TEX-I-1D452",[24941,53760],{"id":53761,"d":25877},"MJX-82-TEX-I-1D461",[25022,53763,53764],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,53765,53766,53771,53776,53782,53788],{"dataMmlNode":25028},[25022,53767,53768],{"dataMmlNode":25031,"transform":27630},[25033,53769],{"dataC":27633,"xLinkHref":53770},"#MJX-82-TEX-N-7C",[25022,53772,53773],{"dataMmlNode":25942,"transform":26048},[25033,53774],{"dataC":25952,"xLinkHref":53775},"#MJX-82-TEX-I-1D45B",[25022,53777,53779],{"dataMmlNode":25942,"transform":53778},"translate(878,0)",[25033,53780],{"dataC":27596,"xLinkHref":53781},"#MJX-82-TEX-I-1D452",[25022,53783,53785],{"dataMmlNode":25942,"transform":53784},"translate(1344,0)",[25033,53786],{"dataC":25959,"xLinkHref":53787},"#MJX-82-TEX-I-1D461",[25022,53789,53791],{"dataMmlNode":25031,"transform":53790},"translate(1705,0) translate(0 -0.5)",[25033,53792],{"dataC":27633,"xLinkHref":53770}," becomes exceptionally large (positive or negative), the sigmoid function enters a \"saturation region\" where the curve flattens out. In these regions, the derivative approaches zero, behaving like a hard step function. This can cause a problem known as \"vanishing gradients,\" where the network stops learning.",[7166,53795,53797],{"id":53796},"step-3-backpropagating-the-error-the-backward-pass","Step 3: Backpropagating the Error (The Backward Pass)",[12,53799,53800,53801,53820,53821,53823],{},"We must update the weights to minimize the error ",[24923,53802,53804],{"className":53803,"jax":24927},[24926],[24929,53805,53806,53811],{"style":27802,"xmlns":24932,"width":50795,"height":27804,"role":1930,"focusable":8058,"viewBox":50796,"xmlnsXLink":24936},[24938,53807,53808],{},[24941,53809],{"id":53810,"d":25869},"MJX-83-TEX-I-1D438",[25022,53812,53813],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,53814,53815],{"dataMmlNode":25028},[25022,53816,53817],{"dataMmlNode":25942},[25033,53818],{"dataC":25945,"xLinkHref":53819},"#MJX-83-TEX-I-1D438",". We do this using ",[106,53822,42142],{},", updating each weight by an amount proportional to the negative gradient of the error with respect to that weight.",[12,53825,53826],{},"Before diving into the neural network math, let's do a quick recap on the calculus tools we need: Partial Derivatives and the Chain Rule. You can freely jump to the next section if you are already familiar with them.",[25631,53828,53831,53834],{"className":53829},[13441,53830],"info-box-warning",[25635,53832,53833],{},"Prerequisite: Partial Derivatives & The Chain Rule",[13438,53835,53837,53863,53866,53902,54147,54150,54258,54268,54271,54274,54351,54416,54511,54558,54665,54740,54826,54849,54954,55026,55029,55122,55195,55257,55439,55536,55646,55728,55803,55825,55927,55999,56051,56203,56290,56293,56360,56421,56424,56426,56430,56437,56529,56659],{"className":53836},[25641],[7166,53838,53840,53841,7963],{"id":53839},"partial-derivative","Partial Derivative (",[24923,53842,53844],{"className":53843,"jax":24927},[24926],[24929,53845,53849,53854],{"style":27778,"xmlns":24932,"width":53846,"height":53847,"role":1930,"focusable":8058,"viewBox":53848,"xmlnsXLink":24936},"1.281ex","1.667ex","0 -715 566 737",[24938,53850,53851],{},[24941,53852],{"id":53853,"d":33624},"MJX-84-TEX-I-1D715",[25022,53855,53856],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,53857,53858],{"dataMmlNode":25028},[25022,53859,53860],{"dataMmlNode":25942},[25033,53861],{"dataC":33697,"xLinkHref":53862},"#MJX-84-TEX-I-1D715",[12,53864,53865],{},"Before anything else, I think it is worth understanding how partial derivative works. We will skip the boring analogy and jump right into the math.",[12,53867,53868,53869,7963],{},"Let's first look at a standard derivative (",[24923,53870,53872],{"className":53871,"jax":24927},[24926],[24929,53873,53877,53885],{"style":29805,"xmlns":24932,"width":53874,"height":53875,"role":1930,"focusable":8058,"viewBox":53876,"xmlnsXLink":24936},"1.992ex","2.181ex","0 -759 880.5 964",[24938,53878,53879,53882],{},[24941,53880],{"id":53881,"d":29821},"MJX-85-TEX-I-1D453",[24941,53883],{"id":53884,"d":31558},"MJX-85-TEX-V-2032",[25022,53886,53887],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,53888,53889],{"dataMmlNode":25028},[25022,53890,53891,53896],{"dataMmlNode":31565},[25022,53892,53893],{"dataMmlNode":25942},[25033,53894],{"dataC":29864,"xLinkHref":53895},"#MJX-85-TEX-I-1D453",[25022,53897,53899],{"dataMmlNode":25031,"transform":53898},"translate(636,363) scale(0.707)",[25033,53900],{"dataC":31576,"xLinkHref":53901},"#MJX-85-TEX-V-2032",[12,53903,53904,53905,53974,53975,54034,54035,54065,54066,54093,54094,54127,54128,6959],{},"A standard derivative tells you the slope of a given equation. Take ",[24923,53906,53908],{"className":53907,"jax":24927},[24926],[24929,53909,53912,53931],{"style":24931,"xmlns":24932,"width":53910,"height":31733,"role":1930,"focusable":8058,"viewBox":53911,"xmlnsXLink":24936},"9.598ex","0 -833.9 4242.1 1083.9",[24938,53913,53914,53917,53920,53923,53926,53928],{},[24941,53915],{"id":53916,"d":29821},"MJX-86-TEX-I-1D453",[24941,53918],{"id":53919,"d":24944},"MJX-86-TEX-N-28",[24941,53921],{"id":53922,"d":32148},"MJX-86-TEX-I-1D465",[24941,53924],{"id":53925,"d":24988},"MJX-86-TEX-N-29",[24941,53927],{"id":39092,"d":24972},[24941,53929],{"id":53930,"d":25928},"MJX-86-TEX-N-32",[25022,53932,53933],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,53934,53935,53940,53945,53951,53957,53962],{"dataMmlNode":25028},[25022,53936,53937],{"dataMmlNode":25942},[25033,53938],{"dataC":29864,"xLinkHref":53939},"#MJX-86-TEX-I-1D453",[25022,53941,53942],{"dataMmlNode":25031,"transform":44094},[25033,53943],{"dataC":25035,"xLinkHref":53944},"#MJX-86-TEX-N-28",[25022,53946,53948],{"dataMmlNode":25942,"transform":53947},"translate(939,0)",[25033,53949],{"dataC":32181,"xLinkHref":53950},"#MJX-86-TEX-I-1D465",[25022,53952,53954],{"dataMmlNode":25031,"transform":53953},"translate(1511,0)",[25033,53955],{"dataC":25108,"xLinkHref":53956},"#MJX-86-TEX-N-29",[25022,53958,53960],{"dataMmlNode":25031,"transform":53959},"translate(2177.8,0)",[25033,53961],{"dataC":25078,"xLinkHref":39115},[25022,53963,53965,53969],{"dataMmlNode":31565,"transform":53964},"translate(3233.6,0)",[25022,53966,53967],{"dataMmlNode":25942},[25033,53968],{"dataC":32181,"xLinkHref":53950},[25022,53970,53971],{"dataMmlNode":26031,"transform":32565},[25033,53972],{"dataC":26058,"xLinkHref":53973},"#MJX-86-TEX-N-32"," as example, ",[24923,53976,53978],{"className":53977,"jax":24927},[24926],[24929,53979,53983,54000],{"style":24931,"xmlns":24932,"width":53980,"height":53981,"role":1930,"focusable":8058,"viewBox":53982,"xmlnsXLink":24936},"5.046ex","2.283ex","0 -759 2230.5 1009",[24938,53984,53985,53988,53991,53994,53997],{},[24941,53986],{"id":53987,"d":29821},"MJX-87-TEX-I-1D453",[24941,53989],{"id":53990,"d":31558},"MJX-87-TEX-V-2032",[24941,53992],{"id":53993,"d":24944},"MJX-87-TEX-N-28",[24941,53995],{"id":53996,"d":32148},"MJX-87-TEX-I-1D465",[24941,53998],{"id":53999,"d":24988},"MJX-87-TEX-N-29",[25022,54001,54002],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54003,54004,54016,54022,54028],{"dataMmlNode":25028},[25022,54005,54006,54011],{"dataMmlNode":31565},[25022,54007,54008],{"dataMmlNode":25942},[25033,54009],{"dataC":29864,"xLinkHref":54010},"#MJX-87-TEX-I-1D453",[25022,54012,54013],{"dataMmlNode":25031,"transform":53898},[25033,54014],{"dataC":31576,"xLinkHref":54015},"#MJX-87-TEX-V-2032",[25022,54017,54019],{"dataMmlNode":25031,"transform":54018},"translate(880.5,0)",[25033,54020],{"dataC":25035,"xLinkHref":54021},"#MJX-87-TEX-N-28",[25022,54023,54025],{"dataMmlNode":25942,"transform":54024},"translate(1269.5,0)",[25033,54026],{"dataC":32181,"xLinkHref":54027},"#MJX-87-TEX-I-1D465",[25022,54029,54031],{"dataMmlNode":25031,"transform":54030},"translate(1841.5,0)",[25033,54032],{"dataC":25108,"xLinkHref":54033},"#MJX-87-TEX-N-29","is ",[24923,54036,54038],{"className":54037,"jax":24927},[24926],[24929,54039,54043,54051],{"style":25196,"xmlns":24932,"width":54040,"height":54041,"role":1930,"focusable":8058,"viewBox":54042,"xmlnsXLink":24936},"2.425ex","1.532ex","0 -666 1072 677",[24938,54044,54045,54048],{},[24941,54046],{"id":54047,"d":25928},"MJX-88-TEX-N-32",[24941,54049],{"id":54050,"d":32148},"MJX-88-TEX-I-1D465",[25022,54052,54053],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54054,54055,54060],{"dataMmlNode":25028},[25022,54056,54057],{"dataMmlNode":26031},[25033,54058],{"dataC":26058,"xLinkHref":54059},"#MJX-88-TEX-N-32",[25022,54061,54062],{"dataMmlNode":25942,"transform":26394},[25033,54063],{"dataC":32181,"xLinkHref":54064},"#MJX-88-TEX-I-1D465",", this means at any given point in the graph, the slope of the graph is ",[24923,54067,54069],{"className":54068,"jax":24927},[24926],[24929,54070,54071,54079],{"style":25196,"xmlns":24932,"width":54040,"height":54041,"role":1930,"focusable":8058,"viewBox":54042,"xmlnsXLink":24936},[24938,54072,54073,54076],{},[24941,54074],{"id":54075,"d":25928},"MJX-89-TEX-N-32",[24941,54077],{"id":54078,"d":32148},"MJX-89-TEX-I-1D465",[25022,54080,54081],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54082,54083,54088],{"dataMmlNode":25028},[25022,54084,54085],{"dataMmlNode":26031},[25033,54086],{"dataC":26058,"xLinkHref":54087},"#MJX-89-TEX-N-32",[25022,54089,54090],{"dataMmlNode":25942,"transform":26394},[25033,54091],{"dataC":32181,"xLinkHref":54092},"#MJX-89-TEX-I-1D465",". If you are standing at ",[24923,54095,54097],{"className":54096,"jax":24927},[24926],[24929,54098,54101,54110],{"style":26868,"xmlns":24932,"width":54099,"height":28162,"role":1930,"focusable":8058,"viewBox":54100,"xmlnsXLink":24936},"5.442ex","0 -665 2405.6 747",[24938,54102,54103,54105,54107],{},[24941,54104],{"id":39269,"d":32148},[24941,54106],{"id":39275,"d":24972},[24941,54108],{"id":54109,"d":26455},"MJX-90-TEX-N-33",[25022,54111,54112],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54113,54114,54118,54122],{"dataMmlNode":25028},[25022,54115,54116],{"dataMmlNode":25942},[25033,54117],{"dataC":32181,"xLinkHref":39316},[25022,54119,54120],{"dataMmlNode":25031,"transform":35003},[25033,54121],{"dataC":25078,"xLinkHref":39326},[25022,54123,54124],{"dataMmlNode":26031,"transform":35009},[25033,54125],{"dataC":26486,"xLinkHref":54126},"#MJX-90-TEX-N-33",", the slope is exactly ",[24923,54129,54131],{"className":54130,"jax":24927},[24926],[24929,54132,54133,54138],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,54134,54135],{},[24941,54136],{"id":54137,"d":42964},"MJX-91-TEX-N-36",[25022,54139,54140],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54141,54142],{"dataMmlNode":25028},[25022,54143,54144],{"dataMmlNode":26031},[25033,54145],{"dataC":31136,"xLinkHref":54146},"#MJX-91-TEX-N-36",[12,54148,54149],{},"But what if you have more than 1 variable? Take this example:",[24923,54151,54153],{"className":54152,"jax":24927,"display":13594},[24926],[24929,54154,54158,54187],{"style":24931,"xmlns":24932,"width":54155,"height":54156,"role":1930,"focusable":8058,"viewBox":54157,"xmlnsXLink":24936},"16.574ex","2.565ex","0 -883.9 7325.8 1133.9",[24938,54159,54160,54163,54166,54169,54172,54175,54178,54181,54184],{},[24941,54161],{"id":54162,"d":29821},"MJX-92-TEX-I-1D453",[24941,54164],{"id":54165,"d":24944},"MJX-92-TEX-N-28",[24941,54167],{"id":54168,"d":32148},"MJX-92-TEX-I-1D465",[24941,54170],{"id":54171,"d":27339},"MJX-92-TEX-N-2C",[24941,54173],{"id":54174,"d":25893},"MJX-92-TEX-I-1D466",[24941,54176],{"id":54177,"d":24988},"MJX-92-TEX-N-29",[24941,54179],{"id":54180,"d":24972},"MJX-92-TEX-N-3D",[24941,54182],{"id":54183,"d":25928},"MJX-92-TEX-N-32",[24941,54185],{"id":54186,"d":30313},"MJX-92-TEX-N-2B",[25022,54188,54189],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54190,54191,54196,54201,54206,54211,54217,54223,54229,54241,54247],{"dataMmlNode":25028},[25022,54192,54193],{"dataMmlNode":25942},[25033,54194],{"dataC":29864,"xLinkHref":54195},"#MJX-92-TEX-I-1D453",[25022,54197,54198],{"dataMmlNode":25031,"transform":44094},[25033,54199],{"dataC":25035,"xLinkHref":54200},"#MJX-92-TEX-N-28",[25022,54202,54203],{"dataMmlNode":25942,"transform":53947},[25033,54204],{"dataC":32181,"xLinkHref":54205},"#MJX-92-TEX-I-1D465",[25022,54207,54208],{"dataMmlNode":25031,"transform":53953},[25033,54209],{"dataC":27456,"xLinkHref":54210},"#MJX-92-TEX-N-2C",[25022,54212,54214],{"dataMmlNode":25942,"transform":54213},"translate(1955.7,0)",[25033,54215],{"dataC":25987,"xLinkHref":54216},"#MJX-92-TEX-I-1D466",[25022,54218,54220],{"dataMmlNode":25031,"transform":54219},"translate(2445.7,0)",[25033,54221],{"dataC":25108,"xLinkHref":54222},"#MJX-92-TEX-N-29",[25022,54224,54226],{"dataMmlNode":25031,"transform":54225},"translate(3112.4,0)",[25033,54227],{"dataC":25078,"xLinkHref":54228},"#MJX-92-TEX-N-3D",[25022,54230,54232,54236],{"dataMmlNode":31565,"transform":54231},"translate(4168.2,0)",[25022,54233,54234],{"dataMmlNode":25942},[25033,54235],{"dataC":32181,"xLinkHref":54205},[25022,54237,54238],{"dataMmlNode":26031,"transform":33270},[25033,54239],{"dataC":26058,"xLinkHref":54240},"#MJX-92-TEX-N-32",[25022,54242,54244],{"dataMmlNode":25031,"transform":54243},"translate(5399,0)",[25033,54245],{"dataC":30393,"xLinkHref":54246},"#MJX-92-TEX-N-2B",[25022,54248,54250,54254],{"dataMmlNode":31565,"transform":54249},"translate(6399.2,0)",[25022,54251,54252],{"dataMmlNode":25942},[25033,54253],{"dataC":25987,"xLinkHref":54216},[25022,54255,54256],{"dataMmlNode":26031,"transform":33304},[25033,54257],{"dataC":26058,"xLinkHref":54240},[12,54259,54260],{},[32509,54261,54262,54265],{},[1930,54263],{"alt":48657,"src":54264},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776951804474-Screenshot_2026-04-23_at_9.42.54_PM.png",[32516,54266,54267],{},"Graph formed by the equation.",[12,54269,54270],{},"Its graph will look something like this.",[12,54272,54273],{},"It's hard to find the slope at a given point, as the you can have the slope face any direction. So we can use partial derivative here. We pretend that one of the variable is a boring, normal number (like 10).",[26236,54275,54277,54278,54297,54298,24917],{"id":54276},"find-the-partial-derivative-with-respect-to-written-as","Find the partial derivative with respect to ",[24923,54279,54281],{"className":54280,"jax":24927},[24926],[24929,54282,54283,54288],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,54284,54285],{},[24941,54286],{"id":54287,"d":32148},"MJX-93-TEX-I-1D465",[25022,54289,54290],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54291,54292],{"dataMmlNode":25028},[25022,54293,54294],{"dataMmlNode":25942},[25033,54295],{"dataC":32181,"xLinkHref":54296},"#MJX-93-TEX-I-1D465"," (written as ",[24923,54299,54301],{"className":54300,"jax":24927},[24926],[24929,54302,54307,54318],{"style":54303,"xmlns":24932,"width":54304,"height":54305,"role":1930,"focusable":8058,"viewBox":54306,"xmlnsXLink":24936},"vertical-align: -0.817ex;","2.816ex","3.058ex","0 -990.5 1244.7 1351.7",[24938,54308,54309,54312,54315],{},[24941,54310],{"id":54311,"d":33624},"MJX-94-TEX-I-1D715",[24941,54313],{"id":54314,"d":29821},"MJX-94-TEX-I-1D453",[24941,54316],{"id":54317,"d":32148},"MJX-94-TEX-I-1D465",[25022,54319,54320],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54321,54322],{"dataMmlNode":25028},[25022,54323,54324,54337,54348],{"dataMmlNode":27622},[25022,54325,54327,54332],{"dataMmlNode":27626,"transform":54326},"translate(227.8,485) scale(0.707)",[25022,54328,54329],{"dataMmlNode":25942},[25033,54330],{"dataC":33697,"xLinkHref":54331},"#MJX-94-TEX-I-1D715",[25022,54333,54334],{"dataMmlNode":25942,"transform":33701},[25033,54335],{"dataC":29864,"xLinkHref":54336},"#MJX-94-TEX-I-1D453",[25022,54338,54339,54343],{"dataMmlNode":27626,"transform":45099},[25022,54340,54341],{"dataMmlNode":25942},[25033,54342],{"dataC":33697,"xLinkHref":54331},[25022,54344,54345],{"dataMmlNode":25942,"transform":33701},[25033,54346],{"dataC":32181,"xLinkHref":54347},"#MJX-94-TEX-I-1D465",[27668,54349],{"width":54350,"height":27671,"x":27672,"y":27673},1004.7,[12,54352,54353,54354,54376,54377,16419],{},"We pretend ",[24923,54355,54357],{"className":54356,"jax":24927},[24926],[24929,54358,54362,54367],{"style":29805,"xmlns":24932,"width":54359,"height":54360,"role":1930,"focusable":8058,"viewBox":54361,"xmlnsXLink":24936},"1.109ex","1.464ex","0 -442 490 647",[24938,54363,54364],{},[24941,54365],{"id":54366,"d":25893},"MJX-95-TEX-I-1D466",[25022,54368,54369],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54370,54371],{"dataMmlNode":25028},[25022,54372,54373],{"dataMmlNode":25942},[25033,54374],{"dataC":25987,"xLinkHref":54375},"#MJX-95-TEX-I-1D466"," is just a normal number (let's say ",[24923,54378,54380],{"className":54379,"jax":24927},[24926],[24929,54381,54384,54396],{"style":29805,"xmlns":24932,"width":54382,"height":38932,"role":1930,"focusable":8058,"viewBox":54383,"xmlnsXLink":24936},"6.388ex","0 -666 2823.6 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",[24923,54420,54422],{"className":54421,"jax":24927},[24926],[24929,54423,54424,54437],{"style":29805,"xmlns":24932,"width":54382,"height":38932,"role":1930,"focusable":8058,"viewBox":54383,"xmlnsXLink":24936},[24938,54425,54426,54429,54432,54435],{},[24941,54427],{"id":54428,"d":25893},"MJX-97-TEX-I-1D466",[24941,54430],{"id":54431,"d":24972},"MJX-97-TEX-N-3D",[24941,54433],{"id":54434,"d":25914},"MJX-97-TEX-N-31",[24941,54436],{"id":39590,"d":25935},[25022,54438,54439],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54440,54441,54446,54451],{"dataMmlNode":25028},[25022,54442,54443],{"dataMmlNode":25942},[25033,54444],{"dataC":25987,"xLinkHref":54445},"#MJX-97-TEX-I-1D466",[25022,54447,54448],{"dataMmlNode":25031,"transform":38956},[25033,54449],{"dataC":25078,"xLinkHref":54450},"#MJX-97-TEX-N-3D",[25022,54452,54453,54456],{"dataMmlNode":26031,"transform":38962},[25033,54454],{"dataC":26035,"xLinkHref":54455},"#MJX-97-TEX-N-31",[25033,54457],{"dataC":26103,"xLinkHref":39605,"transform":26394},[24923,54459,54461],{"className":54460,"jax":24927},[24926],[24929,54462,54465,54480],{"style":29805,"xmlns":24932,"width":54463,"height":27745,"role":1930,"focusable":8058,"viewBox":54464,"xmlnsXLink":24936},"8.507ex","0 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1038.9",[24938,54466,54467,54470,54473,54476,54478],{},[24941,54468],{"id":54469,"d":25893},"MJX-98-TEX-I-1D466",[24941,54471],{"id":54472,"d":25928},"MJX-98-TEX-N-32",[24941,54474],{"id":54475,"d":24972},"MJX-98-TEX-N-3D",[24941,54477],{"id":39637,"d":25914},[24941,54479],{"id":39640,"d":25935},[25022,54481,54482],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54483,54484,54496,54502],{"dataMmlNode":25028},[25022,54485,54486,54491],{"dataMmlNode":31565},[25022,54487,54488],{"dataMmlNode":25942},[25033,54489],{"dataC":25987,"xLinkHref":54490},"#MJX-98-TEX-I-1D466",[25022,54492,54493],{"dataMmlNode":26031,"transform":32620},[25033,54494],{"dataC":26058,"xLinkHref":54495},"#MJX-98-TEX-N-32",[25022,54497,54499],{"dataMmlNode":25031,"transform":54498},"translate(1204.3,0)",[25033,54500],{"dataC":25078,"xLinkHref":54501},"#MJX-98-TEX-N-3D",[25022,54503,54505,54507,54509],{"dataMmlNode":26031,"transform":54504},"translate(2260.1,0)",[25033,54506],{"dataC":26035,"xLinkHref":39654},[25033,54508],{"dataC":26103,"xLinkHref":39657,"transform":26394},[25033,54510],{"dataC":26103,"xLinkHref":39657,"transform":29730},[12,54512,54513,54514,54537,54538,54557],{},"What 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It's just ",[24923,54539,54541],{"className":54540,"jax":24927},[24926],[24929,54542,54543,54548],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,54544,54545],{},[24941,54546],{"id":54547,"d":25935},"MJX-100-TEX-N-30",[25022,54549,54550],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54551,54552],{"dataMmlNode":25028},[25022,54553,54554],{"dataMmlNode":26031},[25033,54555],{"dataC":26103,"xLinkHref":54556},"#MJX-100-TEX-N-30",". 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844.9",[24938,54570,54571,54573],{},[24941,54572],{"id":39884,"d":32148},[24941,54574],{"id":54575,"d":25928},"MJX-101-TEX-N-32",[25022,54577,54578],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54579,54580],{"dataMmlNode":25028},[25022,54581,54582,54586],{"dataMmlNode":31565},[25022,54583,54584],{"dataMmlNode":25942},[25033,54585],{"dataC":32181,"xLinkHref":39941},[25022,54587,54588],{"dataMmlNode":26031,"transform":32565},[25033,54589],{"dataC":26058,"xLinkHref":54590},"#MJX-101-TEX-N-32",[24923,54592,54594],{"className":54593,"jax":24927},[24926],[24929,54595,54596,54603],{"style":25196,"xmlns":24932,"width":54040,"height":54041,"role":1930,"focusable":8058,"viewBox":54042,"xmlnsXLink":24936},[24938,54597,54598,54601],{},[24941,54599],{"id":54600,"d":25928},"MJX-102-TEX-N-32",[24941,54602],{"id":40050,"d":32148},[25022,54604,54605],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54606,54607,54612],{"dataMmlNode":25028},[25022,54608,54609],{"dataMmlNode":26031},[25033,54610],{"dataC":26058,"xLinkHref":54611},"#MJX-102-TEX-N-32",[25022,54613,54614],{"dataMmlNode":25942,"transform":26394},[25033,54615],{"dataC":32181,"xLinkHref":40175},", and the derivative of ",[24923,54618,54620],{"className":54619,"jax":24927},[24926],[24929,54621,54624,54631],{"style":29805,"xmlns":24932,"width":54622,"height":27745,"role":1930,"focusable":8058,"viewBox":54623,"xmlnsXLink":24936},"2.096ex","0 -833.9 926.6 1038.9",[24938,54625,54626,54628],{},[24941,54627],{"id":40360,"d":25893},[24941,54629],{"id":54630,"d":25928},"MJX-103-TEX-N-32",[25022,54632,54633],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54634,54635],{"dataMmlNode":25028},[25022,54636,54637,54641],{"dataMmlNode":31565},[25022,54638,54639],{"dataMmlNode":25942},[25033,54640],{"dataC":25987,"xLinkHref":40435},[25022,54642,54643],{"dataMmlNode":26031,"transform":32620},[25033,54644],{"dataC":26058,"xLinkHref":54645},"#MJX-103-TEX-N-32",[24923,54647,54649],{"className":54648,"jax":24927},[24926],[24929,54650,54651,54656],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,54652,54653],{},[24941,54654],{"id":54655,"d":25935},"MJX-104-TEX-N-30",[25022,54657,54658],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54659,54660],{"dataMmlNode":25028},[25022,54661,54662],{"dataMmlNode":26031},[25033,54663],{"dataC":26103,"xLinkHref":54664},"#MJX-104-TEX-N-30",[12,54666,54667,7206,54670],{},[106,54668,54669],{},"Answer:",[24923,54671,54673],{"className":54672,"jax":24927},[24926],[24929,54674,54677,54693],{"style":54303,"xmlns":24932,"width":54675,"height":54305,"role":1930,"focusable":8058,"viewBox":54676,"xmlnsXLink":24936},"8.258ex","0 -990.5 3650.2 1351.7",[24938,54678,54679,54682,54685,54687,54690],{},[24941,54680],{"id":54681,"d":33624},"MJX-105-TEX-I-1D715",[24941,54683],{"id":54684,"d":29821},"MJX-105-TEX-I-1D453",[24941,54686],{"id":40719,"d":32148},[24941,54688],{"id":54689,"d":24972},"MJX-105-TEX-N-3D",[24941,54691],{"id":54692,"d":25928},"MJX-105-TEX-N-32",[25022,54694,54695],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54696,54697,54723,54729,54735],{"dataMmlNode":25028},[25022,54698,54699,54711,54721],{"dataMmlNode":27622},[25022,54700,54701,54706],{"dataMmlNode":27626,"transform":54326},[25022,54702,54703],{"dataMmlNode":25942},[25033,54704],{"dataC":33697,"xLinkHref":54705},"#MJX-105-TEX-I-1D715",[25022,54707,54708],{"dataMmlNode":25942,"transform":33701},[25033,54709],{"dataC":29864,"xLinkHref":54710},"#MJX-105-TEX-I-1D453",[25022,54712,54713,54717],{"dataMmlNode":27626,"transform":45099},[25022,54714,54715],{"dataMmlNode":25942},[25033,54716],{"dataC":33697,"xLinkHref":54705},[25022,54718,54719],{"dataMmlNode":25942,"transform":33701},[25033,54720],{"dataC":32181,"xLinkHref":40768},[27668,54722],{"width":54350,"height":27671,"x":27672,"y":27673},[25022,54724,54726],{"dataMmlNode":25031,"transform":54725},"translate(1522.5,0)",[25033,54727],{"dataC":25078,"xLinkHref":54728},"#MJX-105-TEX-N-3D",[25022,54730,54732],{"dataMmlNode":26031,"transform":54731},"translate(2578.2,0)",[25033,54733],{"dataC":26058,"xLinkHref":54734},"#MJX-105-TEX-N-32",[25022,54736,54738],{"dataMmlNode":25942,"transform":54737},"translate(3078.2,0)",[25033,54739],{"dataC":32181,"xLinkHref":40768},[26236,54741,54743,7206,54746,7206,54767,7206,54770],{"id":54742},"find-the-partial-derivative-with-respect-to-written-as-1",[106,54744,54745],{},"Find 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1481.1",[24938,54782,54783,54786,54789],{},[24941,54784],{"id":54785,"d":33624},"MJX-107-TEX-I-1D715",[24941,54787],{"id":54788,"d":29821},"MJX-107-TEX-I-1D453",[24941,54790],{"id":54791,"d":25893},"MJX-107-TEX-I-1D466",[25022,54793,54794],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54795,54796],{"dataMmlNode":25028},[25022,54797,54798,54811,54823],{"dataMmlNode":27622},[25022,54799,54801,54806],{"dataMmlNode":27626,"transform":54800},"translate(220,485) scale(0.707)",[25022,54802,54803],{"dataMmlNode":25942},[25033,54804],{"dataC":33697,"xLinkHref":54805},"#MJX-107-TEX-I-1D715",[25022,54807,54808],{"dataMmlNode":25942,"transform":33701},[25033,54809],{"dataC":29864,"xLinkHref":54810},"#MJX-107-TEX-I-1D453",[25022,54812,54814,54818],{"dataMmlNode":27626,"transform":54813},"translate(241.2,-345.6) scale(0.707)",[25022,54815,54816],{"dataMmlNode":25942},[25033,54817],{"dataC":33697,"xLinkHref":54805},[25022,54819,54820],{"dataMmlNode":25942,"transform":33701},[25033,54821],{"dataC":25987,"xLinkHref":54822},"#MJX-107-TEX-I-1D466",[27668,54824],{"width":54825,"height":27671,"x":27672,"y":27673},989.1,[12,54827,54828,54829,54848],{},"Now pretend ",[24923,54830,54832],{"className":54831,"jax":24927},[24926],[24929,54833,54834,54839],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,54835,54836],{},[24941,54837],{"id":54838,"d":32148},"MJX-108-TEX-I-1D465",[25022,54840,54841],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54842,54843],{"dataMmlNode":25028},[25022,54844,54845],{"dataMmlNode":25942},[25033,54846],{"dataC":32181,"xLinkHref":54847},"#MJX-108-TEX-I-1D465"," is the frozen number.",[12,54850,52466,54851,41600,54880,54897,54898,7653,54927,6959],{},[24923,54852,54854],{"className":54853,"jax":24927},[24926],[24929,54855,54856,54864],{"style":25196,"xmlns":24932,"width":54566,"height":54567,"role":1930,"focusable":8058,"viewBox":54568,"xmlnsXLink":24936},[24938,54857,54858,54861],{},[24941,54859],{"id":54860,"d":32148},"MJX-109-TEX-I-1D465",[24941,54862],{"id":54863,"d":25928},"MJX-109-TEX-N-32",[25022,54865,54866],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54867,54868],{"dataMmlNode":25028},[25022,54869,54870,54875],{"dataMmlNode":31565},[25022,54871,54872],{"dataMmlNode":25942},[25033,54873],{"dataC":32181,"xLinkHref":54874},"#MJX-109-TEX-I-1D465",[25022,54876,54877],{"dataMmlNode":26031,"transform":32565},[25033,54878],{"dataC":26058,"xLinkHref":54879},"#MJX-109-TEX-N-32",[24923,54881,54883],{"className":54882,"jax":24927},[24926],[24929,54884,54885,54889],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,54886,54887],{},[24941,54888],{"id":41057,"d":25935},[25022,54890,54891],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,54892,54893],{"dataMmlNode":25028},[25022,54894,54895],{"dataMmlNode":26031},[25033,54896],{"dataC":26103,"xLinkHref":41066},". 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Let's pretend ",[24923,55219,55221],{"className":55220,"jax":24927},[24926],[24929,55222,55223,55236],{"style":29805,"xmlns":24932,"width":54382,"height":38932,"role":1930,"focusable":8058,"viewBox":54383,"xmlnsXLink":24936},[24938,55224,55225,55228,55231,55233],{},[24941,55226],{"id":55227,"d":25893},"MJX-118-TEX-I-1D466",[24941,55229],{"id":55230,"d":24972},"MJX-118-TEX-N-3D",[24941,55232],{"id":41760,"d":25914},[24941,55234],{"id":55235,"d":25935},"MJX-118-TEX-N-30",[25022,55237,55238],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55239,55240,55245,55250],{"dataMmlNode":25028},[25022,55241,55242],{"dataMmlNode":25942},[25033,55243],{"dataC":25987,"xLinkHref":55244},"#MJX-118-TEX-I-1D466",[25022,55246,55247],{"dataMmlNode":25031,"transform":38956},[25033,55248],{"dataC":25078,"xLinkHref":55249},"#MJX-118-TEX-N-3D",[25022,55251,55252,55254],{"dataMmlNode":26031,"transform":38962},[25033,55253],{"dataC":26035,"xLinkHref":41769},[25033,55255],{"dataC":26103,"xLinkHref":55256,"transform":26394},"#MJX-118-TEX-N-30",[12,55258,54418,55259,55294,55295,55394,55395,6959],{},[24923,55260,55262],{"className":55261,"jax":24927},[24926],[24929,55263,55264,55275],{"style":29805,"xmlns":24932,"width":54382,"height":38932,"role":1930,"focusable":8058,"viewBox":54383,"xmlnsXLink":24936},[24938,55265,55266,55268,55270,55273],{},[24941,55267],{"id":41786,"d":25893},[24941,55269],{"id":41789,"d":24972},[24941,55271],{"id":55272,"d":25914},"MJX-119-TEX-N-31",[24941,55274],{"id":41792,"d":25935},[25022,55276,55277],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55278,55279,55283,55287],{"dataMmlNode":25028},[25022,55280,55281],{"dataMmlNode":25942},[25033,55282],{"dataC":25987,"xLinkHref":41801},[25022,55284,55285],{"dataMmlNode":25031,"transform":38956},[25033,55286],{"dataC":25078,"xLinkHref":41806},[25022,55288,55289,55292],{"dataMmlNode":26031,"transform":38962},[25033,55290],{"dataC":26035,"xLinkHref":55291},"#MJX-119-TEX-N-31",[25033,55293],{"dataC":26103,"xLinkHref":41811,"transform":26394},", your equation looks like this: ",[24923,55296,55298],{"className":55297,"jax":24927},[24926],[24929,55299,55302,55331],{"style":24931,"xmlns":24932,"width":55300,"height":31733,"role":1930,"focusable":8058,"viewBox":55301,"xmlnsXLink":24936},"16.26ex","0 -833.9 7187 1083.9",[24938,55303,55304,55307,55309,55312,55314,55317,55320,55323,55326,55329],{},[24941,55305],{"id":55306,"d":29821},"MJX-120-TEX-I-1D453",[24941,55308],{"id":41823,"d":24944},[24941,55310],{"id":55311,"d":32148},"MJX-120-TEX-I-1D465",[24941,55313],{"id":41829,"d":24988},[24941,55315],{"id":55316,"d":24972},"MJX-120-TEX-N-3D",[24941,55318],{"id":55319,"d":26455},"MJX-120-TEX-N-33",[24941,55321],{"id":55322,"d":27409},"MJX-120-TEX-N-22C5",[24941,55324],{"id":55325,"d":25928},"MJX-120-TEX-N-32",[24941,55327],{"id":55328,"d":25914},"MJX-120-TEX-N-31",[24941,55330],{"id":41826,"d":25935},[25022,55332,55333],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55334,55335,55340,55344,55349,55353,55358,55363,55369,55381,55386],{"dataMmlNode":25028},[25022,55336,55337],{"dataMmlNode":25942},[25033,55338],{"dataC":29864,"xLinkHref":55339},"#MJX-120-TEX-I-1D453",[25022,55341,55342],{"dataMmlNode":25031,"transform":44094},[25033,55343],{"dataC":25035,"xLinkHref":41838},[25022,55345,55346],{"dataMmlNode":25942,"transform":53947},[25033,55347],{"dataC":32181,"xLinkHref":55348},"#MJX-120-TEX-I-1D465",[25022,55350,55351],{"dataMmlNode":25031,"transform":53953},[25033,55352],{"dataC":25108,"xLinkHref":41849},[25022,55354,55355],{"dataMmlNode":25031,"transform":53959},[25033,55356],{"dataC":25078,"xLinkHref":55357},"#MJX-120-TEX-N-3D",[25022,55359,55360],{"dataMmlNode":26031,"transform":53964},[25033,55361],{"dataC":26486,"xLinkHref":55362},"#MJX-120-TEX-N-33",[25022,55364,55366],{"dataMmlNode":25031,"transform":55365},"translate(3955.8,0)",[25033,55367],{"dataC":27679,"xLinkHref":55368},"#MJX-120-TEX-N-22C5",[25022,55370,55372,55376],{"dataMmlNode":31565,"transform":55371},"translate(4456,0)",[25022,55373,55374],{"dataMmlNode":25942},[25033,55375],{"dataC":32181,"xLinkHref":55348},[25022,55377,55378],{"dataMmlNode":26031,"transform":32565},[25033,55379],{"dataC":26058,"xLinkHref":55380},"#MJX-120-TEX-N-32",[25022,55382,55384],{"dataMmlNode":25031,"transform":55383},"translate(5686.8,0)",[25033,55385],{"dataC":27679,"xLinkHref":55368},[25022,55387,55389,55392],{"dataMmlNode":26031,"transform":55388},"translate(6187,0)",[25033,55390],{"dataC":26035,"xLinkHref":55391},"#MJX-120-TEX-N-31",[25033,55393],{"dataC":26103,"xLinkHref":41843,"transform":26394},", which simplifies to ",[24923,55396,55398],{"className":55397,"jax":24927},[24926],[24929,55399,55403,55416],{"style":27778,"xmlns":24932,"width":55400,"height":55401,"role":1930,"focusable":8058,"viewBox":55402,"xmlnsXLink":24936},"4.544ex","1.937ex","0 -833.9 2008.6 855.9",[24938,55404,55405,55408,55410,55413],{},[24941,55406],{"id":55407,"d":26455},"MJX-121-TEX-N-33",[24941,55409],{"id":41859,"d":25935},[24941,55411],{"id":55412,"d":32148},"MJX-121-TEX-I-1D465",[24941,55414],{"id":55415,"d":25928},"MJX-121-TEX-N-32",[25022,55417,55418],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55419,55420,55427],{"dataMmlNode":25028},[25022,55421,55422,55425],{"dataMmlNode":26031},[25033,55423],{"dataC":26486,"xLinkHref":55424},"#MJX-121-TEX-N-33",[25033,55426],{"dataC":26103,"xLinkHref":41868,"transform":26394},[25022,55428,55429,55434],{"dataMmlNode":31565,"transform":29730},[25022,55430,55431],{"dataMmlNode":25942},[25033,55432],{"dataC":32181,"xLinkHref":55433},"#MJX-121-TEX-I-1D465",[25022,55435,55436],{"dataMmlNode":26031,"transform":32565},[25033,55437],{"dataC":26058,"xLinkHref":55438},"#MJX-121-TEX-N-32",[12,55440,52466,55441,55482,55483,55502,55503,6959],{},[24923,55442,55444],{"className":55443,"jax":24927},[24926],[24929,55445,55446,55459],{"style":27778,"xmlns":24932,"width":55400,"height":55401,"role":1930,"focusable":8058,"viewBox":55402,"xmlnsXLink":24936},[24938,55447,55448,55451,55454,55456],{},[24941,55449],{"id":55450,"d":26455},"MJX-122-TEX-N-33",[24941,55452],{"id":55453,"d":25935},"MJX-122-TEX-N-30",[24941,55455],{"id":41905,"d":32148},[24941,55457],{"id":55458,"d":25928},"MJX-122-TEX-N-32",[25022,55460,55461],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55462,55463,55471],{"dataMmlNode":25028},[25022,55464,55465,55468],{"dataMmlNode":26031},[25033,55466],{"dataC":26486,"xLinkHref":55467},"#MJX-122-TEX-N-33",[25033,55469],{"dataC":26103,"xLinkHref":55470,"transform":26394},"#MJX-122-TEX-N-30",[25022,55472,55473,55477],{"dataMmlNode":31565,"transform":29730},[25022,55474,55475],{"dataMmlNode":25942},[25033,55476],{"dataC":32181,"xLinkHref":41961},[25022,55478,55479],{"dataMmlNode":26031,"transform":32565},[25033,55480],{"dataC":26058,"xLinkHref":55481},"#MJX-122-TEX-N-32"," is easy: you bring down the ",[24923,55484,55486],{"className":55485,"jax":24927},[24926],[24929,55487,55488,55493],{"style":27802,"xmlns":24932,"width":35168,"height":35169,"role":1930,"focusable":8058,"viewBox":35170,"xmlnsXLink":24936},[24938,55489,55490],{},[24941,55491],{"id":55492,"d":25928},"MJX-123-TEX-N-32",[25022,55494,55495],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55496,55497],{"dataMmlNode":25028},[25022,55498,55499],{"dataMmlNode":26031},[25033,55500],{"dataC":26058,"xLinkHref":55501},"#MJX-123-TEX-N-32"," and multiply it, giving you ",[24923,55504,55506],{"className":55505,"jax":24927},[24926],[24929,55507,55510,55520],{"style":27778,"xmlns":24932,"width":55508,"height":35776,"role":1930,"focusable":8058,"viewBox":55509,"xmlnsXLink":24936},"3.557ex","0 -666 1572 688",[24938,55511,55512,55515,55517],{},[24941,55513],{"id":55514,"d":42964},"MJX-124-TEX-N-36",[24941,55516],{"id":42025,"d":25935},[24941,55518],{"id":55519,"d":32148},"MJX-124-TEX-I-1D465",[25022,55521,55522],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55523,55524,55531],{"dataMmlNode":25028},[25022,55525,55526,55529],{"dataMmlNode":26031},[25033,55527],{"dataC":31136,"xLinkHref":55528},"#MJX-124-TEX-N-36",[25033,55530],{"dataC":26103,"xLinkHref":42034,"transform":26394},[25022,55532,55533],{"dataMmlNode":25942,"transform":29730},[25033,55534],{"dataC":32181,"xLinkHref":55535},"#MJX-124-TEX-I-1D465",[12,55537,55538,55539,55565,55566,16253,55585,55606,55607,55626,55627,6959],{},"Now, let's swap the ",[24923,55540,55542],{"className":55541,"jax":24927},[24926],[24929,55543,55545,55553],{"style":27778,"xmlns":24932,"width":24934,"height":35776,"role":1930,"focusable":8058,"viewBox":55544,"xmlnsXLink":24936},"0 -666 1000 688",[24938,55546,55547,55550],{},[24941,55548],{"id":55549,"d":25914},"MJX-125-TEX-N-31",[24941,55551],{"id":55552,"d":25935},"MJX-125-TEX-N-30",[25022,55554,55555],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55556,55557],{"dataMmlNode":25028},[25022,55558,55559,55562],{"dataMmlNode":26031},[25033,55560],{"dataC":26035,"xLinkHref":55561},"#MJX-125-TEX-N-31",[25033,55563],{"dataC":26103,"xLinkHref":55564,"transform":26394},"#MJX-125-TEX-N-30"," back out for the ",[24923,55567,55569],{"className":55568,"jax":24927},[24926],[24929,55570,55571,55576],{"style":29805,"xmlns":24932,"width":54359,"height":54360,"role":1930,"focusable":8058,"viewBox":54361,"xmlnsXLink":24936},[24938,55572,55573],{},[24941,55574],{"id":55575,"d":25893},"MJX-126-TEX-I-1D466",[25022,55577,55578],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55579,55580],{"dataMmlNode":25028},[25022,55581,55582],{"dataMmlNode":25942},[25033,55583],{"dataC":25987,"xLinkHref":55584},"#MJX-126-TEX-I-1D466",[24923,55586,55588],{"className":55587,"jax":24927},[24926],[24929,55589,55592,55597],{"style":27778,"xmlns":24932,"width":35168,"height":55590,"role":1930,"focusable":8058,"viewBox":55591,"xmlnsXLink":24936},"1.554ex","0 -665 500 687",[24938,55593,55594],{},[24941,55595],{"id":55596,"d":26455},"MJX-127-TEX-N-33",[25022,55598,55599],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55600,55601],{"dataMmlNode":25028},[25022,55602,55603],{"dataMmlNode":26031},[25033,55604],{"dataC":26486,"xLinkHref":55605},"#MJX-127-TEX-N-33"," and the ",[24923,55608,55610],{"className":55609,"jax":24927},[24926],[24929,55611,55612,55617],{"style":29805,"xmlns":24932,"width":54359,"height":54360,"role":1930,"focusable":8058,"viewBox":54361,"xmlnsXLink":24936},[24938,55613,55614],{},[24941,55615],{"id":55616,"d":25893},"MJX-128-TEX-I-1D466",[25022,55618,55619],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55620,55621],{"dataMmlNode":25028},[25022,55622,55623],{"dataMmlNode":25942},[25033,55624],{"dataC":25987,"xLinkHref":55625},"#MJX-128-TEX-I-1D466"," just act as \"sticky constants\" that stay attached to the math you are doing on ",[24923,55628,55630],{"className":55629,"jax":24927},[24926],[24929,55631,55632,55637],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,55633,55634],{},[24941,55635],{"id":55636,"d":32148},"MJX-129-TEX-I-1D465",[25022,55638,55639],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55640,55641],{"dataMmlNode":25028},[25022,55642,55643],{"dataMmlNode":25942},[25033,55644],{"dataC":32181,"xLinkHref":55645},"#MJX-129-TEX-I-1D465",[12,55647,55648,7206,55650],{},[106,55649,54669],{},[24923,55651,55653],{"className":55652,"jax":24927},[24926],[24929,55654,55657,55677],{"style":54303,"xmlns":24932,"width":55655,"height":54305,"role":1930,"focusable":8058,"viewBox":55656,"xmlnsXLink":24936},"9.367ex","0 -990.5 4140.2 1351.7",[24938,55658,55659,55662,55665,55668,55671,55674],{},[24941,55660],{"id":55661,"d":33624},"MJX-130-TEX-I-1D715",[24941,55663],{"id":55664,"d":29821},"MJX-130-TEX-I-1D453",[24941,55666],{"id":55667,"d":32148},"MJX-130-TEX-I-1D465",[24941,55669],{"id":55670,"d":24972},"MJX-130-TEX-N-3D",[24941,55672],{"id":55673,"d":42964},"MJX-130-TEX-N-36",[24941,55675],{"id":55676,"d":25893},"MJX-130-TEX-I-1D466",[25022,55678,55679],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55680,55681,55708,55713,55718,55722],{"dataMmlNode":25028},[25022,55682,55683,55695,55706],{"dataMmlNode":27622},[25022,55684,55685,55690],{"dataMmlNode":27626,"transform":54326},[25022,55686,55687],{"dataMmlNode":25942},[25033,55688],{"dataC":33697,"xLinkHref":55689},"#MJX-130-TEX-I-1D715",[25022,55691,55692],{"dataMmlNode":25942,"transform":33701},[25033,55693],{"dataC":29864,"xLinkHref":55694},"#MJX-130-TEX-I-1D453",[25022,55696,55697,55701],{"dataMmlNode":27626,"transform":45099},[25022,55698,55699],{"dataMmlNode":25942},[25033,55700],{"dataC":33697,"xLinkHref":55689},[25022,55702,55703],{"dataMmlNode":25942,"transform":33701},[25033,55704],{"dataC":32181,"xLinkHref":55705},"#MJX-130-TEX-I-1D465",[27668,55707],{"width":54350,"height":27671,"x":27672,"y":27673},[25022,55709,55710],{"dataMmlNode":25031,"transform":54725},[25033,55711],{"dataC":25078,"xLinkHref":55712},"#MJX-130-TEX-N-3D",[25022,55714,55715],{"dataMmlNode":26031,"transform":54731},[25033,55716],{"dataC":31136,"xLinkHref":55717},"#MJX-130-TEX-N-36",[25022,55719,55720],{"dataMmlNode":25942,"transform":54737},[25033,55721],{"dataC":32181,"xLinkHref":55705},[25022,55723,55725],{"dataMmlNode":25942,"transform":55724},"translate(3650.2,0)",[25033,55726],{"dataC":25987,"xLinkHref":55727},"#MJX-130-TEX-I-1D466",[26236,55729,55731,7206,55733,7206,55754],{"id":55730},"find-the-partial-derivative-with-respect-to-1",[106,55732,54745],{},[106,55734,55735],{},[24923,55736,55738],{"className":55737,"jax":24927},[24926],[24929,55739,55740,55745],{"style":29805,"xmlns":24932,"width":54359,"height":54360,"role":1930,"focusable":8058,"viewBox":54361,"xmlnsXLink":24936},[24938,55741,55742],{},[24941,55743],{"id":55744,"d":25893},"MJX-131-TEX-I-1D466",[25022,55746,55747],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55748,55749],{"dataMmlNode":25028},[25022,55750,55751],{"dataMmlNode":25942},[25033,55752],{"dataC":25987,"xLinkHref":55753},"#MJX-131-TEX-I-1D466",[106,55755,7698,55756,24917],{},[24923,55757,55759],{"className":55758,"jax":24927},[24926],[24929,55760,55761,55772],{"style":54777,"xmlns":24932,"width":54778,"height":54779,"role":1930,"focusable":8058,"viewBox":54780,"xmlnsXLink":24936},[24938,55762,55763,55766,55769],{},[24941,55764],{"id":55765,"d":33624},"MJX-132-TEX-I-1D715",[24941,55767],{"id":55768,"d":29821},"MJX-132-TEX-I-1D453",[24941,55770],{"id":55771,"d":25893},"MJX-132-TEX-I-1D466",[25022,55773,55774],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55775,55776],{"dataMmlNode":25028},[25022,55777,55778,55790,55801],{"dataMmlNode":27622},[25022,55779,55780,55785],{"dataMmlNode":27626,"transform":54800},[25022,55781,55782],{"dataMmlNode":25942},[25033,55783],{"dataC":33697,"xLinkHref":55784},"#MJX-132-TEX-I-1D715",[25022,55786,55787],{"dataMmlNode":25942,"transform":33701},[25033,55788],{"dataC":29864,"xLinkHref":55789},"#MJX-132-TEX-I-1D453",[25022,55791,55792,55796],{"dataMmlNode":27626,"transform":54813},[25022,55793,55794],{"dataMmlNode":25942},[25033,55795],{"dataC":33697,"xLinkHref":55784},[25022,55797,55798],{"dataMmlNode":25942,"transform":33701},[25033,55799],{"dataC":25987,"xLinkHref":55800},"#MJX-132-TEX-I-1D466",[27668,55802],{"width":54825,"height":27671,"x":27672,"y":27673},[12,55804,55197,55805,55824],{},[24923,55806,55808],{"className":55807,"jax":24927},[24926],[24929,55809,55810,55815],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,55811,55812],{},[24941,55813],{"id":55814,"d":32148},"MJX-133-TEX-I-1D465",[25022,55816,55817],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55818,55819],{"dataMmlNode":25028},[25022,55820,55821],{"dataMmlNode":25942},[25033,55822],{"dataC":32181,"xLinkHref":55823},"#MJX-133-TEX-I-1D465"," is the constant.",[12,55826,55827,55828,55867,55868,55905,55906,6959],{},"This means that entire chunk of ",[24923,55829,55831],{"className":55830,"jax":24927},[24926],[24929,55832,55835,55846],{"style":27778,"xmlns":24932,"width":55833,"height":55401,"role":1930,"focusable":8058,"viewBox":55834,"xmlnsXLink":24936},"3.413ex","0 -833.9 1508.6 855.9",[24938,55836,55837,55840,55843],{},[24941,55838],{"id":55839,"d":26455},"MJX-134-TEX-N-33",[24941,55841],{"id":55842,"d":32148},"MJX-134-TEX-I-1D465",[24941,55844],{"id":55845,"d":25928},"MJX-134-TEX-N-32",[25022,55847,55848],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55849,55850,55855],{"dataMmlNode":25028},[25022,55851,55852],{"dataMmlNode":26031},[25033,55853],{"dataC":26486,"xLinkHref":55854},"#MJX-134-TEX-N-33",[25022,55856,55857,55862],{"dataMmlNode":31565,"transform":26394},[25022,55858,55859],{"dataMmlNode":25942},[25033,55860],{"dataC":32181,"xLinkHref":55861},"#MJX-134-TEX-I-1D465",[25022,55863,55864],{"dataMmlNode":26031,"transform":32565},[25033,55865],{"dataC":26058,"xLinkHref":55866},"#MJX-134-TEX-N-32"," is just one big, boring number. Let's imagine ",[24923,55869,55871],{"className":55870,"jax":24927},[24926],[24929,55872,55873,55884],{"style":27778,"xmlns":24932,"width":55833,"height":55401,"role":1930,"focusable":8058,"viewBox":55834,"xmlnsXLink":24936},[24938,55874,55875,55878,55881],{},[24941,55876],{"id":55877,"d":26455},"MJX-135-TEX-N-33",[24941,55879],{"id":55880,"d":32148},"MJX-135-TEX-I-1D465",[24941,55882],{"id":55883,"d":25928},"MJX-135-TEX-N-32",[25022,55885,55886],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55887,55888,55893],{"dataMmlNode":25028},[25022,55889,55890],{"dataMmlNode":26031},[25033,55891],{"dataC":26486,"xLinkHref":55892},"#MJX-135-TEX-N-33",[25022,55894,55895,55900],{"dataMmlNode":31565,"transform":26394},[25022,55896,55897],{"dataMmlNode":25942},[25033,55898],{"dataC":32181,"xLinkHref":55899},"#MJX-135-TEX-I-1D465",[25022,55901,55902],{"dataMmlNode":26031,"transform":32565},[25033,55903],{"dataC":26058,"xLinkHref":55904},"#MJX-135-TEX-N-32"," evaluates to ",[24923,55907,55909],{"className":55908,"jax":24927},[24926],[24929,55910,55913,55918],{"style":27778,"xmlns":24932,"width":35168,"height":55911,"role":1930,"focusable":8058,"viewBox":55912,"xmlnsXLink":24936},"1.579ex","0 -676 500 698",[24938,55914,55915],{},[24941,55916],{"id":55917,"d":26361},"MJX-136-TEX-N-37",[25022,55919,55920],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55921,55922],{"dataMmlNode":25028},[25022,55923,55924],{"dataMmlNode":26031},[25033,55925],{"dataC":26397,"xLinkHref":55926},"#MJX-136-TEX-N-37",[12,55928,55929,55930,6959],{},"Your equation would look like this: ",[24923,55931,55933],{"className":55932,"jax":24927},[24926],[24929,55934,55937,55957],{"style":24931,"xmlns":24932,"width":55935,"height":24934,"role":1930,"focusable":8058,"viewBox":55936,"xmlnsXLink":24936},"9.37ex","0 -750 4141.6 1000",[24938,55938,55939,55942,55945,55948,55951,55954],{},[24941,55940],{"id":55941,"d":29821},"MJX-137-TEX-I-1D453",[24941,55943],{"id":55944,"d":24944},"MJX-137-TEX-N-28",[24941,55946],{"id":55947,"d":25893},"MJX-137-TEX-I-1D466",[24941,55949],{"id":55950,"d":24988},"MJX-137-TEX-N-29",[24941,55952],{"id":55953,"d":24972},"MJX-137-TEX-N-3D",[24941,55955],{"id":55956,"d":26361},"MJX-137-TEX-N-37",[25022,55958,55959],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,55960,55961,55966,55971,55976,55982,55988,55994],{"dataMmlNode":25028},[25022,55962,55963],{"dataMmlNode":25942},[25033,55964],{"dataC":29864,"xLinkHref":55965},"#MJX-137-TEX-I-1D453",[25022,55967,55968],{"dataMmlNode":25031,"transform":44094},[25033,55969],{"dataC":25035,"xLinkHref":55970},"#MJX-137-TEX-N-28",[25022,55972,55973],{"dataMmlNode":25942,"transform":53947},[25033,55974],{"dataC":25987,"xLinkHref":55975},"#MJX-137-TEX-I-1D466",[25022,55977,55979],{"dataMmlNode":25031,"transform":55978},"translate(1429,0)",[25033,55980],{"dataC":25108,"xLinkHref":55981},"#MJX-137-TEX-N-29",[25022,55983,55985],{"dataMmlNode":25031,"transform":55984},"translate(2095.8,0)",[25033,55986],{"dataC":25078,"xLinkHref":55987},"#MJX-137-TEX-N-3D",[25022,55989,55991],{"dataMmlNode":26031,"transform":55990},"translate(3151.6,0)",[25033,55992],{"dataC":26397,"xLinkHref":55993},"#MJX-137-TEX-N-37",[25022,55995,55997],{"dataMmlNode":25942,"transform":55996},"translate(3651.6,0)",[25033,55998],{"dataC":25987,"xLinkHref":55975},[12,56000,56001,56002,54537,56031,56050],{},"What is the derivative of ",[24923,56003,56005],{"className":56004,"jax":24927},[24926],[24929,56006,56009,56017],{"style":29805,"xmlns":24932,"width":54932,"height":56007,"role":1930,"focusable":8058,"viewBox":56008,"xmlnsXLink":24936},"1.993ex","0 -676 990 881",[24938,56010,56011,56014],{},[24941,56012],{"id":56013,"d":26361},"MJX-138-TEX-N-37",[24941,56015],{"id":56016,"d":25893},"MJX-138-TEX-I-1D466",[25022,56018,56019],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56020,56021,56026],{"dataMmlNode":25028},[25022,56022,56023],{"dataMmlNode":26031},[25033,56024],{"dataC":26397,"xLinkHref":56025},"#MJX-138-TEX-N-37",[25022,56027,56028],{"dataMmlNode":25942,"transform":26394},[25033,56029],{"dataC":25987,"xLinkHref":56030},"#MJX-138-TEX-I-1D466",[24923,56032,56034],{"className":56033,"jax":24927},[24926],[24929,56035,56036,56041],{"style":27778,"xmlns":24932,"width":35168,"height":55911,"role":1930,"focusable":8058,"viewBox":55912,"xmlnsXLink":24936},[24938,56037,56038],{},[24941,56039],{"id":56040,"d":26361},"MJX-139-TEX-N-37",[25022,56042,56043],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56044,56045],{"dataMmlNode":25028},[25022,56046,56047],{"dataMmlNode":26031},[25033,56048],{"dataC":26397,"xLinkHref":56049},"#MJX-139-TEX-N-37"," (the slope of a straight line is just the number attached to it).",[12,56052,56053,56054,7653,56081,56100,56101,35162,56166,6959],{},"So, if the derivative of ",[24923,56055,56057],{"className":56056,"jax":24927},[24926],[24929,56058,56059,56067],{"style":29805,"xmlns":24932,"width":54932,"height":56007,"role":1930,"focusable":8058,"viewBox":56008,"xmlnsXLink":24936},[24938,56060,56061,56064],{},[24941,56062],{"id":56063,"d":26361},"MJX-140-TEX-N-37",[24941,56065],{"id":56066,"d":25893},"MJX-140-TEX-I-1D466",[25022,56068,56069],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56070,56071,56076],{"dataMmlNode":25028},[25022,56072,56073],{"dataMmlNode":26031},[25033,56074],{"dataC":26397,"xLinkHref":56075},"#MJX-140-TEX-N-37",[25022,56077,56078],{"dataMmlNode":25942,"transform":26394},[25033,56079],{"dataC":25987,"xLinkHref":56080},"#MJX-140-TEX-I-1D466",[24923,56082,56084],{"className":56083,"jax":24927},[24926],[24929,56085,56086,56091],{"style":27778,"xmlns":24932,"width":35168,"height":55911,"role":1930,"focusable":8058,"viewBox":55912,"xmlnsXLink":24936},[24938,56087,56088],{},[24941,56089],{"id":56090,"d":26361},"MJX-141-TEX-N-37",[25022,56092,56093],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56094,56095],{"dataMmlNode":25028},[25022,56096,56097],{"dataMmlNode":26031},[25033,56098],{"dataC":26397,"xLinkHref":56099},"#MJX-141-TEX-N-37",", then the derivative of ",[24923,56102,56104],{"className":56103,"jax":24927},[24926],[24929,56105,56108,56128],{"style":24931,"xmlns":24932,"width":56106,"height":31733,"role":1930,"focusable":8058,"viewBox":56107,"xmlnsXLink":24936},"6.282ex","0 -833.9 2776.6 1083.9",[24938,56109,56110,56113,56116,56119,56122,56125],{},[24941,56111],{"id":56112,"d":24944},"MJX-142-TEX-N-28",[24941,56114],{"id":56115,"d":26455},"MJX-142-TEX-N-33",[24941,56117],{"id":56118,"d":32148},"MJX-142-TEX-I-1D465",[24941,56120],{"id":56121,"d":25928},"MJX-142-TEX-N-32",[24941,56123],{"id":56124,"d":24988},"MJX-142-TEX-N-29",[24941,56126],{"id":56127,"d":25893},"MJX-142-TEX-I-1D466",[25022,56129,56130],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56131,56132,56137,56142,56154,56160],{"dataMmlNode":25028},[25022,56133,56134],{"dataMmlNode":25031},[25033,56135],{"dataC":25035,"xLinkHref":56136},"#MJX-142-TEX-N-28",[25022,56138,56139],{"dataMmlNode":26031,"transform":25040},[25033,56140],{"dataC":26486,"xLinkHref":56141},"#MJX-142-TEX-N-33",[25022,56143,56144,56149],{"dataMmlNode":31565,"transform":41846},[25022,56145,56146],{"dataMmlNode":25942},[25033,56147],{"dataC":32181,"xLinkHref":56148},"#MJX-142-TEX-I-1D465",[25022,56150,56151],{"dataMmlNode":26031,"transform":32565},[25033,56152],{"dataC":26058,"xLinkHref":56153},"#MJX-142-TEX-N-32",[25022,56155,56157],{"dataMmlNode":25031,"transform":56156},"translate(1897.6,0)",[25033,56158],{"dataC":25108,"xLinkHref":56159},"#MJX-142-TEX-N-29",[25022,56161,56163],{"dataMmlNode":25942,"transform":56162},"translate(2286.6,0)",[25033,56164],{"dataC":25987,"xLinkHref":56165},"#MJX-142-TEX-I-1D466",[24923,56167,56169],{"className":56168,"jax":24927},[24926],[24929,56170,56171,56182],{"style":27778,"xmlns":24932,"width":55833,"height":55401,"role":1930,"focusable":8058,"viewBox":55834,"xmlnsXLink":24936},[24938,56172,56173,56176,56179],{},[24941,56174],{"id":56175,"d":26455},"MJX-143-TEX-N-33",[24941,56177],{"id":56178,"d":32148},"MJX-143-TEX-I-1D465",[24941,56180],{"id":56181,"d":25928},"MJX-143-TEX-N-32",[25022,56183,56184],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56185,56186,56191],{"dataMmlNode":25028},[25022,56187,56188],{"dataMmlNode":26031},[25033,56189],{"dataC":26486,"xLinkHref":56190},"#MJX-143-TEX-N-33",[25022,56192,56193,56198],{"dataMmlNode":31565,"transform":26394},[25022,56194,56195],{"dataMmlNode":25942},[25033,56196],{"dataC":32181,"xLinkHref":56197},"#MJX-143-TEX-I-1D465",[25022,56199,56200],{"dataMmlNode":26031,"transform":32565},[25033,56201],{"dataC":26058,"xLinkHref":56202},"#MJX-143-TEX-N-32",[12,56204,56205,7206,56207],{},[106,56206,54669],{},[24923,56208,56210],{"className":56209,"jax":24927},[24926],[24929,56211,56214,56237],{"style":54777,"xmlns":24932,"width":56212,"height":54779,"role":1930,"focusable":8058,"viewBox":56213,"xmlnsXLink":24936},"9.211ex","0 -990.5 4071.2 1481.1",[24938,56215,56216,56219,56222,56225,56228,56231,56234],{},[24941,56217],{"id":56218,"d":33624},"MJX-144-TEX-I-1D715",[24941,56220],{"id":56221,"d":29821},"MJX-144-TEX-I-1D453",[24941,56223],{"id":56224,"d":25893},"MJX-144-TEX-I-1D466",[24941,56226],{"id":56227,"d":24972},"MJX-144-TEX-N-3D",[24941,56229],{"id":56230,"d":26455},"MJX-144-TEX-N-33",[24941,56232],{"id":56233,"d":32148},"MJX-144-TEX-I-1D465",[24941,56235],{"id":56236,"d":25928},"MJX-144-TEX-N-32",[25022,56238,56239],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56240,56241,56268,56273,56278],{"dataMmlNode":25028},[25022,56242,56243,56255,56266],{"dataMmlNode":27622},[25022,56244,56245,56250],{"dataMmlNode":27626,"transform":54800},[25022,56246,56247],{"dataMmlNode":25942},[25033,56248],{"dataC":33697,"xLinkHref":56249},"#MJX-144-TEX-I-1D715",[25022,56251,56252],{"dataMmlNode":25942,"transform":33701},[25033,56253],{"dataC":29864,"xLinkHref":56254},"#MJX-144-TEX-I-1D453",[25022,56256,56257,56261],{"dataMmlNode":27626,"transform":54813},[25022,56258,56259],{"dataMmlNode":25942},[25033,56260],{"dataC":33697,"xLinkHref":56249},[25022,56262,56263],{"dataMmlNode":25942,"transform":33701},[25033,56264],{"dataC":25987,"xLinkHref":56265},"#MJX-144-TEX-I-1D466",[27668,56267],{"width":54825,"height":27671,"x":27672,"y":27673},[25022,56269,56270],{"dataMmlNode":25031,"transform":55012},[25033,56271],{"dataC":25078,"xLinkHref":56272},"#MJX-144-TEX-N-3D",[25022,56274,56275],{"dataMmlNode":26031,"transform":55017},[25033,56276],{"dataC":26486,"xLinkHref":56277},"#MJX-144-TEX-N-33",[25022,56279,56280,56285],{"dataMmlNode":31565,"transform":55023},[25022,56281,56282],{"dataMmlNode":25942},[25033,56283],{"dataC":32181,"xLinkHref":56284},"#MJX-144-TEX-I-1D465",[25022,56286,56287],{"dataMmlNode":26031,"transform":32565},[25033,56288],{"dataC":26058,"xLinkHref":56289},"#MJX-144-TEX-N-32",[12,56291,56292],{},"So the partial derivative essentially allows you to \"lock\" one of the axis by freezing all other variables. We can take a complex 3D graph, slice it into 2D cross-section, and do regular, easy standard derivative on that slice. A neural network can consists of billions of variables. Trying to comprehend a mathematical shape with a billion dimensions is literally impossible for a human brain.",[12,56294,56295,56296,56359],{},"Just as an example, in a network, ",[24923,56297,56299],{"className":56298,"jax":24927},[24926],[24929,56300,56305,56319],{"style":56301,"xmlns":24932,"width":56302,"height":56303,"role":1930,"focusable":8058,"viewBox":56304,"xmlnsXLink":24936},"vertical-align: -1.022ex;","3.745ex","3.057ex","0 -899.6 1655.2 1351.2",[24938,56306,56307,56310,56313,56316],{},[24941,56308],{"id":56309,"d":33624},"MJX-145-TEX-I-1D715",[24941,56311],{"id":56312,"d":25869},"MJX-145-TEX-I-1D438",[24941,56314],{"id":56315,"d":37254},"MJX-145-TEX-I-1D464",[24941,56317],{"id":56318,"d":25914},"MJX-145-TEX-N-31",[25022,56320,56321],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56322,56323],{"dataMmlNode":25028},[25022,56324,56325,56338,56356],{"dataMmlNode":27622},[25022,56326,56328,56333],{"dataMmlNode":27626,"transform":56327},"translate(357.4,394) scale(0.707)",[25022,56329,56330],{"dataMmlNode":25942},[25033,56331],{"dataC":33697,"xLinkHref":56332},"#MJX-145-TEX-I-1D715",[25022,56334,56335],{"dataMmlNode":25942,"transform":33701},[25033,56336],{"dataC":25945,"xLinkHref":56337},"#MJX-145-TEX-I-1D438",[25022,56339,56340,56344],{"dataMmlNode":27626,"transform":45099},[25022,56341,56342],{"dataMmlNode":25942},[25033,56343],{"dataC":33697,"xLinkHref":56332},[25022,56345,56346,56351],{"dataMmlNode":26023,"transform":33701},[25022,56347,56348],{"dataMmlNode":25942},[25033,56349],{"dataC":37310,"xLinkHref":56350},"#MJX-145-TEX-I-1D464",[25022,56352,56353],{"dataMmlNode":26031,"transform":43003},[25033,56354],{"dataC":26035,"xLinkHref":56355},"#MJX-145-TEX-N-31",[27668,56357],{"width":56358,"height":27671,"x":27672,"y":27673},1415.2," means: \"Freeze weights 2 through 1,000,000. Only nudge weight 1. How much did the error change?\"",[12,56361,56362,56363,56420],{},"Then the network calculates ",[24923,56364,56366],{"className":56365,"jax":24927},[24926],[24929,56367,56368,56382],{"style":56301,"xmlns":24932,"width":56302,"height":56303,"role":1930,"focusable":8058,"viewBox":56304,"xmlnsXLink":24936},[24938,56369,56370,56373,56376,56379],{},[24941,56371],{"id":56372,"d":33624},"MJX-146-TEX-I-1D715",[24941,56374],{"id":56375,"d":25869},"MJX-146-TEX-I-1D438",[24941,56377],{"id":56378,"d":37254},"MJX-146-TEX-I-1D464",[24941,56380],{"id":56381,"d":25928},"MJX-146-TEX-N-32",[25022,56383,56384],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56385,56386],{"dataMmlNode":25028},[25022,56387,56388,56400,56418],{"dataMmlNode":27622},[25022,56389,56390,56395],{"dataMmlNode":27626,"transform":56327},[25022,56391,56392],{"dataMmlNode":25942},[25033,56393],{"dataC":33697,"xLinkHref":56394},"#MJX-146-TEX-I-1D715",[25022,56396,56397],{"dataMmlNode":25942,"transform":33701},[25033,56398],{"dataC":25945,"xLinkHref":56399},"#MJX-146-TEX-I-1D438",[25022,56401,56402,56406],{"dataMmlNode":27626,"transform":45099},[25022,56403,56404],{"dataMmlNode":25942},[25033,56405],{"dataC":33697,"xLinkHref":56394},[25022,56407,56408,56413],{"dataMmlNode":26023,"transform":33701},[25022,56409,56410],{"dataMmlNode":25942},[25033,56411],{"dataC":37310,"xLinkHref":56412},"#MJX-146-TEX-I-1D464",[25022,56414,56415],{"dataMmlNode":26031,"transform":43003},[25033,56416],{"dataC":26058,"xLinkHref":56417},"#MJX-146-TEX-N-32",[27668,56419],{"width":56358,"height":27671,"x":27672,"y":27673},": \"Okay, freeze weight 1, and freeze weights 3 through 1,000,000. Only nudge weight 2. How much did the error change?\"",[12,56422,56423],{},"It does this for every single weight. Once it knows the \"partial\" blame for every individual weight, it adjusts them all at once, and the network learns!",[36,56425],{},[7166,56427,56429],{"id":56428},"the-chain-rule","The Chain rule",[12,56431,56432,56433,56436],{},"In calculus, the Chain Rule is a formula used to find the derivative of ",[106,56434,56435],{},"nested functions",", meaning a function that is sitting inside another function.",[12,56438,56439,56440],{},"A nested function looks like this: ",[106,56441,56442],{},[24923,56443,56445],{"className":56444,"jax":24927},[24926],[24929,56446,56449,56470],{"style":24931,"xmlns":24932,"width":56447,"height":24934,"role":1930,"focusable":8058,"viewBox":56448,"xmlnsXLink":24936},"11.708ex","0 -750 5175 1000",[24938,56450,56451,56454,56457,56460,56464,56467],{},[24941,56452],{"id":56453,"d":41102},"MJX-147-TEX-I-1D434",[24941,56455],{"id":56456,"d":24944},"MJX-147-TEX-N-28",[24941,56458],{"id":56459,"d":41109},"MJX-147-TEX-I-1D435",[24941,56461],{"id":56462,"d":56463},"MJX-147-TEX-I-1D436","M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z",[24941,56465],{"id":56466,"d":32148},"MJX-147-TEX-I-1D465",[24941,56468],{"id":56469,"d":24988},"MJX-147-TEX-N-29",[25022,56471,56472],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56473,56474,56479,56484,56490,56495,56502,56507,56513,56519,56524],{"dataMmlNode":25028},[25022,56475,56476],{"dataMmlNode":25942},[25033,56477],{"dataC":41149,"xLinkHref":56478},"#MJX-147-TEX-I-1D434",[25022,56480,56481],{"dataMmlNode":25031,"transform":25131},[25033,56482],{"dataC":25035,"xLinkHref":56483},"#MJX-147-TEX-N-28",[25022,56485,56487],{"dataMmlNode":25942,"transform":56486},"translate(1139,0)",[25033,56488],{"dataC":41162,"xLinkHref":56489},"#MJX-147-TEX-I-1D435",[25022,56491,56493],{"dataMmlNode":25031,"transform":56492},"translate(1898,0)",[25033,56494],{"dataC":25035,"xLinkHref":56483},[25022,56496,56498],{"dataMmlNode":25942,"transform":56497},"translate(2287,0)",[25033,56499],{"dataC":56500,"xLinkHref":56501},"1D436","#MJX-147-TEX-I-1D436",[25022,56503,56505],{"dataMmlNode":25031,"transform":56504},"translate(3047,0)",[25033,56506],{"dataC":25035,"xLinkHref":56483},[25022,56508,56510],{"dataMmlNode":25942,"transform":56509},"translate(3436,0)",[25033,56511],{"dataC":32181,"xLinkHref":56512},"#MJX-147-TEX-I-1D465",[25022,56514,56516],{"dataMmlNode":25031,"transform":56515},"translate(4008,0)",[25033,56517],{"dataC":25108,"xLinkHref":56518},"#MJX-147-TEX-N-29",[25022,56520,56522],{"dataMmlNode":25031,"transform":56521},"translate(4397,0)",[25033,56523],{"dataC":25108,"xLinkHref":56518},[25022,56525,56527],{"dataMmlNode":25031,"transform":56526},"translate(4786,0)",[25033,56528],{"dataC":25108,"xLinkHref":56518},[23,56530,56531,56574,56616],{},[26,56532,56533,56552,56553],{},[24923,56534,56536],{"className":56535,"jax":24927},[24926],[24929,56537,56538,56543],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,56539,56540],{},[24941,56541],{"id":56542,"d":32148},"MJX-148-TEX-I-1D465",[25022,56544,56545],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56546,56547],{"dataMmlNode":25028},[25022,56548,56549],{"dataMmlNode":25942},[25033,56550],{"dataC":32181,"xLinkHref":56551},"#MJX-148-TEX-I-1D465"," changes ",[24923,56554,56556],{"className":56555,"jax":24927},[24926],[24929,56557,56560,56565],{"style":27778,"xmlns":24932,"width":56558,"height":27780,"role":1930,"focusable":8058,"viewBox":56559,"xmlnsXLink":24936},"1.719ex","0 -705 760 727",[24938,56561,56562],{},[24941,56563],{"id":56564,"d":56463},"MJX-149-TEX-I-1D436",[25022,56566,56567],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56568,56569],{"dataMmlNode":25028},[25022,56570,56571],{"dataMmlNode":25942},[25033,56572],{"dataC":56500,"xLinkHref":56573},"#MJX-149-TEX-I-1D436",[26,56575,56576,56552,56595],{},[24923,56577,56579],{"className":56578,"jax":24927},[24926],[24929,56580,56581,56586],{"style":27778,"xmlns":24932,"width":56558,"height":27780,"role":1930,"focusable":8058,"viewBox":56559,"xmlnsXLink":24936},[24938,56582,56583],{},[24941,56584],{"id":56585,"d":56463},"MJX-150-TEX-I-1D436",[25022,56587,56588],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56589,56590],{"dataMmlNode":25028},[25022,56591,56592],{"dataMmlNode":25942},[25033,56593],{"dataC":56500,"xLinkHref":56594},"#MJX-150-TEX-I-1D436",[24923,56596,56598],{"className":56597,"jax":24927},[24926],[24929,56599,56602,56607],{"style":27802,"xmlns":24932,"width":56600,"height":28113,"role":1930,"focusable":8058,"viewBox":56601,"xmlnsXLink":24936},"1.717ex","0 -683 759 683",[24938,56603,56604],{},[24941,56605],{"id":56606,"d":41109},"MJX-151-TEX-I-1D435",[25022,56608,56609],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56610,56611],{"dataMmlNode":25028},[25022,56612,56613],{"dataMmlNode":25942},[25033,56614],{"dataC":41162,"xLinkHref":56615},"#MJX-151-TEX-I-1D435",[26,56617,56618,56552,56637],{},[24923,56619,56621],{"className":56620,"jax":24927},[24926],[24929,56622,56623,56628],{"style":27802,"xmlns":24932,"width":56600,"height":28113,"role":1930,"focusable":8058,"viewBox":56601,"xmlnsXLink":24936},[24938,56624,56625],{},[24941,56626],{"id":56627,"d":41109},"MJX-152-TEX-I-1D435",[25022,56629,56630],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56631,56632],{"dataMmlNode":25028},[25022,56633,56634],{"dataMmlNode":25942},[25033,56635],{"dataC":41162,"xLinkHref":56636},"#MJX-152-TEX-I-1D435",[24923,56638,56640],{"className":56639,"jax":24927},[24926],[24929,56641,56645,56650],{"style":27802,"xmlns":24932,"width":56642,"height":56643,"role":1930,"focusable":8058,"viewBox":56644,"xmlnsXLink":24936},"1.697ex","1.62ex","0 -716 750 716",[24938,56646,56647],{},[24941,56648],{"id":56649,"d":41102},"MJX-153-TEX-I-1D434",[25022,56651,56652],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56653,56654],{"dataMmlNode":25028},[25022,56655,56656],{"dataMmlNode":25942},[25033,56657],{"dataC":41149,"xLinkHref":56658},"#MJX-153-TEX-I-1D434",[12,56660,56661,56662,7206,56665,7206,56686,7206,56689,56711,56712],{},"If you want to know ",[106,56663,56664],{},"\"How much does",[106,56666,56667],{},[24923,56668,56670],{"className":56669,"jax":24927},[24926],[24929,56671,56672,56677],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,56673,56674],{},[24941,56675],{"id":56676,"d":32148},"MJX-154-TEX-I-1D465",[25022,56678,56679],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56680,56681],{"dataMmlNode":25028},[25022,56682,56683],{"dataMmlNode":25942},[25033,56684],{"dataC":32181,"xLinkHref":56685},"#MJX-154-TEX-I-1D465",[106,56687,56688],{},"change",[106,56690,56691,56710],{},[24923,56692,56694],{"className":56693,"jax":24927},[24926],[24929,56695,56696,56701],{"style":27802,"xmlns":24932,"width":56642,"height":56643,"role":1930,"focusable":8058,"viewBox":56644,"xmlnsXLink":24936},[24938,56697,56698],{},[24941,56699],{"id":56700,"d":41102},"MJX-155-TEX-I-1D434",[25022,56702,56703],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56704,56705],{"dataMmlNode":25028},[25022,56706,56707],{"dataMmlNode":25942},[25033,56708],{"dataC":41149,"xLinkHref":56709},"#MJX-155-TEX-I-1D434","?\"",", the Chain Rule says you just calculate the rate of change for each nested function individually, and then ",[106,56713,56714],{},"multiply them all together.",[26236,56716,56718,56719,7963],{"id":56717},"updating-output-layer-weights","Updating Output Layer Weights 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update a specific weight, we use this formula:",[24923,56786,56788],{"className":56787,"jax":24927,"display":13594},[24926],[24929,56789,56794,56837],{"style":56790,"xmlns":24932,"width":56791,"height":56792,"role":1930,"focusable":8058,"viewBox":56793,"xmlnsXLink":24936},"vertical-align: -3.353ex;","19.132ex","6.5ex","0 -1391 8456.2 2873.1",[24938,56795,56796,56800,56803,56806,56809,56812,56815,56818,56821,56824,56828,56831,56834],{},[24941,56797],{"id":56798,"d":56799},"MJX-157-TEX-N-394","M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z",[24941,56801],{"id":56802,"d":37254},"MJX-157-TEX-I-1D464",[24941,56804],{"id":56805,"d":24944},"MJX-157-TEX-N-28",[24941,56807],{"id":56808,"d":25928},"MJX-157-TEX-N-32",[24941,56810],{"id":56811,"d":27339},"MJX-157-TEX-N-2C",[24941,56813],{"id":56814,"d":25914},"MJX-157-TEX-N-31",[24941,56816],{"id":56817,"d":24988},"MJX-157-TEX-N-29",[24941,56819],{"id":56820,"d":42951},"MJX-157-TEX-I-1D458",[24941,56822],{"id":56823,"d":33606},"MJX-157-TEX-I-1D457",[24941,56825],{"id":56826,"d":56827},"MJX-157-TEX-N-221D","M56 124T56 216T107 375T238 442Q260 442 280 438T319 425T352 407T382 385T406 361T427 336T442 315T455 297T462 285L469 297Q555 442 679 442Q687 442 722 437V398H718Q710 400 694 400Q657 400 623 383T567 343T527 294T503 253T495 235Q495 231 520 192T554 143Q625 44 696 44Q717 44 719 46H722V-5Q695 -11 678 -11Q552 -11 457 141Q455 145 454 146L447 134Q362 -11 235 -11Q157 -11 107 56ZM93 213Q93 143 126 87T220 31Q258 31 292 48T349 88T389 137T413 178T421 196Q421 200 396 239T362 288Q322 345 288 366T213 387Q163 387 128 337T93 213Z",[24941,56829],{"id":56830,"d":25910},"MJX-157-TEX-N-2212",[24941,56832],{"id":56833,"d":33624},"MJX-157-TEX-I-1D715",[24941,56835],{"id":56836,"d":25869},"MJX-157-TEX-I-1D438",[25022,56838,56839],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,56840,56841,56847,56898,56905,56911],{"dataMmlNode":25028},[25022,56842,56843],{"dataMmlNode":25942},[25033,56844],{"dataC":56845,"xLinkHref":56846},"394","#MJX-157-TEX-N-394",[25022,56848,56850,56855,56882],{"dataMmlNode":33991,"transform":56849},"translate(833,0)",[25022,56851,56852],{"dataMmlNode":25942},[25033,56853],{"dataC":37310,"xLinkHref":56854},"#MJX-157-TEX-I-1D464",[25022,56856,56857,56862,56867,56872,56877],{"dataMmlNode":27203,"transform":48899,"dataMjxTexclass":27205},[25022,56858,56859],{"dataMmlNode":25031},[25033,56860],{"dataC":25035,"xLinkHref":56861},"#MJX-157-TEX-N-28",[25022,56863,56864],{"dataMmlNode":26031,"transform":25040},[25033,56865],{"dataC":26058,"xLinkHref":56866},"#MJX-157-TEX-N-32",[25022,56868,56869],{"dataMmlNode":25031,"transform":41846},[25033,56870],{"dataC":27456,"xLinkHref":56871},"#MJX-157-TEX-N-2C",[25022,56873,56874],{"dataMmlNode":26031,"transform":28333},[25033,56875],{"dataC":26035,"xLinkHref":56876},"#MJX-157-TEX-N-31",[25022,56878,56879],{"dataMmlNode":25031,"transform":41146},[25033,56880],{"dataC":25108,"xLinkHref":56881},"#MJX-157-TEX-N-29",[25022,56883,56884,56889,56893],{"dataMmlNode":27203,"transform":51669,"dataMjxTexclass":27205},[25022,56885,56886],{"dataMmlNode":25942},[25033,56887],{"dataC":43135,"xLinkHref":56888},"#MJX-157-TEX-I-1D458",[25022,56890,56891],{"dataMmlNode":25031,"transform":49747},[25033,56892],{"dataC":27456,"xLinkHref":56871},[25022,56894,56895],{"dataMmlNode":25942,"transform":51680},[25033,56896],{"dataC":33647,"xLinkHref":56897},"#MJX-157-TEX-I-1D457",[25022,56899,56901],{"dataMmlNode":25031,"transform":56900},"translate(3363.6,0)",[25033,56902],{"dataC":56903,"xLinkHref":56904},"221D","#MJX-157-TEX-N-221D",[25022,56906,56908],{"dataMmlNode":25031,"transform":56907},"translate(4419.4,0)",[25033,56909],{"dataC":26019,"xLinkHref":56910},"#MJX-157-TEX-N-2212",[25022,56912,56914,56927,56976],{"dataMmlNode":27622,"transform":56913},"translate(5197.4,0)",[25022,56915,56917,56922],{"dataMmlNode":27626,"transform":56916},"translate(964.4,676)",[25022,56918,56919],{"dataMmlNode":25942},[25033,56920],{"dataC":33697,"xLinkHref":56921},"#MJX-157-TEX-I-1D715",[25022,56923,56924],{"dataMmlNode":25942,"transform":33701},[25033,56925],{"dataC":25945,"xLinkHref":56926},"#MJX-157-TEX-I-1D438",[25022,56928,56930,56934],{"dataMmlNode":27626,"transform":56929},"translate(220,-1020.7)",[25022,56931,56932],{"dataMmlNode":25942},[25033,56933],{"dataC":33697,"xLinkHref":56921},[25022,56935,56936,56940,56962],{"dataMmlNode":33991,"transform":33701},[25022,56937,56938],{"dataMmlNode":25942},[25033,56939],{"dataC":37310,"xLinkHref":56854},[25022,56941,56942,56946,56950,56954,56958],{"dataMmlNode":27203,"transform":48899,"dataMjxTexclass":27205},[25022,56943,56944],{"dataMmlNode":25031},[25033,56945],{"dataC":25035,"xLinkHref":56861},[25022,56947,56948],{"dataMmlNode":26031,"transform":25040},[25033,56949],{"dataC":26058,"xLinkHref":56866},[25022,56951,56952],{"dataMmlNode":25031,"transform":41846},[25033,56953],{"dataC":27456,"xLinkHref":56871},[25022,56955,56956],{"dataMmlNode":26031,"transform":28333},[25033,56957],{"dataC":26035,"xLinkHref":56876},[25022,56959,56960],{"dataMmlNode":25031,"transform":41146},[25033,56961],{"dataC":25108,"xLinkHref":56881},[25022,56963,56964,56968,56972],{"dataMmlNode":27203,"transform":51669,"dataMjxTexclass":27205},[25022,56965,56966],{"dataMmlNode":25942},[25033,56967],{"dataC":43135,"xLinkHref":56888},[25022,56969,56970],{"dataMmlNode":25031,"transform":49747},[25033,56971],{"dataC":27456,"xLinkHref":56871},[25022,56973,56974],{"dataMmlNode":25942,"transform":51680},[25033,56975],{"dataC":33647,"xLinkHref":56897},[27668,56977],{"width":56978,"height":27671,"x":27672,"y":27673},3018.8,[25631,56980,56982,56985],{"className":56981},[13441,13442],[25635,56983,56984],{},"The Learning Rate η",[13438,56986,56988,57103,57202,57224,57238,57241,57468,57620,57624,57627,57728,57900,57924,57955,58087,58090],{"className":56987},[25641],[12,56989,56990,56991,57074,57075,57100,57101,7024],{},"The proportionality ",[24923,56992,56994],{"className":56993,"jax":24927},[24926],[24929,56995,56999,57019],{"style":54303,"xmlns":24932,"width":56996,"height":56997,"role":1930,"focusable":8058,"viewBox":56998,"xmlnsXLink":24936},"11.405ex","2.852ex","0 -899.6 5041 1260.7",[24938,57000,57001,57004,57007,57010,57013,57016],{},[24941,57002],{"id":57003,"d":56799},"MJX-158-TEX-N-394",[24941,57005],{"id":57006,"d":37254},"MJX-158-TEX-I-1D464",[24941,57008],{"id":57009,"d":56827},"MJX-158-TEX-N-221D",[24941,57011],{"id":57012,"d":25910},"MJX-158-TEX-N-2212",[24941,57014],{"id":57015,"d":33624},"MJX-158-TEX-I-1D715",[24941,57017],{"id":57018,"d":25869},"MJX-158-TEX-I-1D438",[25022,57020,57021],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57022,57023,57028,57033,57039,57045],{"dataMmlNode":25028},[25022,57024,57025],{"dataMmlNode":25942},[25033,57026],{"dataC":56845,"xLinkHref":57027},"#MJX-158-TEX-N-394",[25022,57029,57030],{"dataMmlNode":25942,"transform":56849},[25033,57031],{"dataC":37310,"xLinkHref":57032},"#MJX-158-TEX-I-1D464",[25022,57034,57036],{"dataMmlNode":25031,"transform":57035},"translate(1826.8,0)",[25033,57037],{"dataC":56903,"xLinkHref":57038},"#MJX-158-TEX-N-221D",[25022,57040,57042],{"dataMmlNode":25031,"transform":57041},"translate(2882.6,0)",[25033,57043],{"dataC":26019,"xLinkHref":57044},"#MJX-158-TEX-N-2212",[25022,57046,57048,57060,57071],{"dataMmlNode":27622,"transform":57047},"translate(3660.6,0)",[25022,57049,57050,57055],{"dataMmlNode":27626,"transform":44821},[25022,57051,57052],{"dataMmlNode":25942},[25033,57053],{"dataC":33697,"xLinkHref":57054},"#MJX-158-TEX-I-1D715",[25022,57056,57057],{"dataMmlNode":25942,"transform":33701},[25033,57058],{"dataC":25945,"xLinkHref":57059},"#MJX-158-TEX-I-1D438",[25022,57061,57063,57067],{"dataMmlNode":27626,"transform":57062},"translate(237,-345.6) scale(0.707)",[25022,57064,57065],{"dataMmlNode":25942},[25033,57066],{"dataC":33697,"xLinkHref":57054},[25022,57068,57069],{"dataMmlNode":25942,"transform":33701},[25033,57070],{"dataC":37310,"xLinkHref":57032},[27668,57072],{"width":57073,"height":27671,"x":27672,"y":27673},1140.5," becomes a usable equation by introducing a small constant ",[24923,57076,57078],{"className":57077,"jax":24927},[24926],[24929,57079,57084,57090],{"style":57080,"xmlns":24932,"width":57081,"height":57082,"role":1930,"focusable":8058,"viewBox":57083,"xmlnsXLink":24936},"vertical-align: -0.489ex;","1.124ex","1.489ex","0 -442 497 658",[24938,57085,57086],{},[24941,57087],{"id":57088,"d":57089},"MJX-159-TEX-I-1D702","M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z",[25022,57091,57092],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57093,57094],{"dataMmlNode":25028},[25022,57095,57096],{"dataMmlNode":25942},[25033,57097],{"dataC":57098,"xLinkHref":57099},"1D702","#MJX-159-TEX-I-1D702"," (eta), called the ",[106,57102,33820],{},[24923,57104,57106],{"className":57105,"jax":24927,"display":13594},[24926],[24929,57107,57112,57138],{"style":57108,"xmlns":24932,"width":57109,"height":57110,"role":1930,"focusable":8058,"viewBox":57111,"xmlnsXLink":24936},"vertical-align: -1.602ex;","15.045ex","4.749ex","0 -1391 6650 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is typically a small positive number (0.001 to 0.5). 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scale(0.707)",[25033,57412],{"dataC":33647,"xLinkHref":57342},[25022,57414,57416],{"dataMmlNode":25031,"transform":57415},"translate(9368.2,0)",[25033,57417],{"dataC":27679,"xLinkHref":57398},[25022,57419,57421,57426,57431],{"dataMmlNode":33991,"transform":57420},"translate(9868.4,0)",[25022,57422,57423],{"dataMmlNode":25942},[25033,57424],{"dataC":29864,"xLinkHref":57425},"#MJX-162-TEX-I-1D453",[25022,57427,57429],{"dataMmlNode":25031,"transform":57428},"translate(636,413) scale(0.707)",[25033,57430],{"dataC":31576,"xLinkHref":57310},[25022,57432,57434],{"dataMmlNode":25942,"transform":57433},"translate(523,-247) scale(0.707)",[25033,57435],{"dataC":33647,"xLinkHref":57342},[25022,57437,57439],{"dataMmlNode":25031,"transform":57438},"translate(10748.9,0)",[25033,57440],{"dataC":25035,"xLinkHref":57318},[25022,57442,57444],{"dataMmlNode":25942,"transform":57443},"translate(11137.9,0)",[25033,57445],{"dataC":27596,"xLinkHref":57446},"#MJX-162-TEX-I-1D452",[25022,57448,57450],{"dataMmlNode":25031,"transform":57449},"translate(11603.9,0)",[25033,57451],{"dataC":25108,"xLinkHref":57336},[25022,57453,57455],{"dataMmlNode":25031,"transform":57454},"translate(12215.1,0)",[25033,57456],{"dataC":27679,"xLinkHref":57398},[25022,57458,57460,57464],{"dataMmlNode":26023,"transform":57459},"translate(12715.3,0)",[25022,57461,57462],{"dataMmlNode":25942},[25033,57463],{"dataC":32181,"xLinkHref":57325},[25022,57465,57466],{"dataMmlNode":25942,"transform":33034},[25033,57467],{"dataC":27210,"xLinkHref":57330},[12,57469,26147,57470,57502,57503,7298,57522,57589,57590,57619],{},[24923,57471,57473],{"className":57472,"jax":24927},[24926],[24929,57474,57478,57486],{"style":35650,"xmlns":24932,"width":57475,"height":57476,"role":1930,"focusable":8058,"viewBox":57477,"xmlnsXLink":24936},"1.851ex","2.288ex","0 -717 818.3 1011.2",[24938,57479,57480,57483],{},[24941,57481],{"id":57482,"d":57288},"MJX-163-TEX-I-1D6FF",[24941,57484],{"id":57485,"d":33606},"MJX-163-TEX-I-1D457",[25022,57487,57488],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57489,57490],{"dataMmlNode":25028},[25022,57491,57492,57497],{"dataMmlNode":26023},[25022,57493,57494],{"dataMmlNode":25942},[25033,57495],{"dataC":57406,"xLinkHref":57496},"#MJX-163-TEX-I-1D6FF",[25022,57498,57499],{"dataMmlNode":25942,"transform":57410},[25033,57500],{"dataC":33647,"xLinkHref":57501},"#MJX-163-TEX-I-1D457"," is the local error signal at node 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1157.2",[24938,57532,57533,57536,57539,57542,57545,57548],{},[24941,57534],{"id":57535,"d":29821},"MJX-165-TEX-I-1D453",[24941,57537],{"id":57538,"d":31558},"MJX-165-TEX-V-2032",[24941,57540],{"id":57541,"d":33606},"MJX-165-TEX-I-1D457",[24941,57543],{"id":57544,"d":24944},"MJX-165-TEX-N-28",[24941,57546],{"id":57547,"d":27397},"MJX-165-TEX-I-1D452",[24941,57549],{"id":57550,"d":24988},"MJX-165-TEX-N-29",[25022,57552,57553],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57554,57555,57573,57578,57583],{"dataMmlNode":25028},[25022,57556,57557,57562,57567],{"dataMmlNode":33991},[25022,57558,57559],{"dataMmlNode":25942},[25033,57560],{"dataC":29864,"xLinkHref":57561},"#MJX-165-TEX-I-1D453",[25022,57563,57564],{"dataMmlNode":25031,"transform":53898},[25033,57565],{"dataC":31576,"xLinkHref":57566},"#MJX-165-TEX-V-2032",[25022,57568,57570],{"dataMmlNode":25942,"transform":57569},"translate(523,-254) scale(0.707)",[25033,57571],{"dataC":33647,"xLinkHref":57572},"#MJX-165-TEX-I-1D457",[25022,57574,57575],{"dataMmlNode":25031,"transform":54018},[25033,57576],{"dataC":25035,"xLinkHref":57577},"#MJX-165-TEX-N-28",[25022,57579,57580],{"dataMmlNode":25942,"transform":54024},[25033,57581],{"dataC":27596,"xLinkHref":57582},"#MJX-165-TEX-I-1D452",[25022,57584,57586],{"dataMmlNode":25031,"transform":57585},"translate(1735.5,0)",[25033,57587],{"dataC":25108,"xLinkHref":57588},"#MJX-165-TEX-N-29"," is the derivative of that node's activation function, and ",[24923,57591,57593],{"className":57592,"jax":24927},[24926],[24929,57594,57595,57603],{"style":27744,"xmlns":24932,"width":51136,"height":31803,"role":1930,"focusable":8058,"viewBox":51137,"xmlnsXLink":24936},[24938,57596,57597,57600],{},[24941,57598],{"id":57599,"d":32148},"MJX-166-TEX-I-1D465",[24941,57601],{"id":57602,"d":27108},"MJX-166-TEX-I-1D456",[25022,57604,57605],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57606,57607],{"dataMmlNode":25028},[25022,57608,57609,57614],{"dataMmlNode":26023},[25022,57610,57611],{"dataMmlNode":25942},[25033,57612],{"dataC":32181,"xLinkHref":57613},"#MJX-166-TEX-I-1D465",[25022,57615,57616],{"dataMmlNode":25942,"transform":33034},[25033,57617],{"dataC":27210,"xLinkHref":57618},"#MJX-166-TEX-I-1D456"," is the incoming signal on the wire being updated.",[26236,57621,57623],{"id":57622},"breaking-down-the-update-rule","Breaking down the Update Rule",[12,57625,57626],{},"Since I'm having trouble understanding the formula, so here's a very very detailed explanation:",[26236,57628,57630,57631],{"id":57629},"the-left-side","The Left Side: ",[24923,57632,57634],{"className":57633,"jax":24927},[24926],[24929,57635,57640,57669],{"style":57636,"xmlns":24932,"width":57637,"height":57638,"role":1930,"focusable":8058,"viewBox":57639,"xmlnsXLink":24936},"vertical-align: -1.044ex;","6.981ex","3.444ex","0 -1060.7 3085.8 1522.1",[24938,57641,57642,57645,57648,57651,57654,57657,57660,57663,57666],{},[24941,57643],{"id":57644,"d":56799},"MJX-167-TEX-N-394",[24941,57646],{"id":57647,"d":37254},"MJX-167-TEX-I-1D464",[24941,57649],{"id":57650,"d":24944},"MJX-167-TEX-N-28",[24941,57652],{"id":57653,"d":25928},"MJX-167-TEX-N-32",[24941,57655],{"id":57656,"d":27339},"MJX-167-TEX-N-2C",[24941,57658],{"id":57659,"d":25914},"MJX-167-TEX-N-31",[24941,57661],{"id":57662,"d":24988},"MJX-167-TEX-N-29",[24941,57664],{"id":57665,"d":42951},"MJX-167-TEX-I-1D458",[24941,57667],{"id":57668,"d":33606},"MJX-167-TEX-I-1D457",[25022,57670,57671],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57672,57673,57678],{"dataMmlNode":25028},[25022,57674,57675],{"dataMmlNode":25942},[25033,57676],{"dataC":56845,"xLinkHref":57677},"#MJX-167-TEX-N-394",[25022,57679,57680,57685,57712],{"dataMmlNode":33991,"transform":56849},[25022,57681,57682],{"dataMmlNode":25942},[25033,57683],{"dataC":37310,"xLinkHref":57684},"#MJX-167-TEX-I-1D464",[25022,57686,57687,57692,57697,57702,57707],{"dataMmlNode":27203,"transform":48899,"dataMjxTexclass":27205},[25022,57688,57689],{"dataMmlNode":25031},[25033,57690],{"dataC":25035,"xLinkHref":57691},"#MJX-167-TEX-N-28",[25022,57693,57694],{"dataMmlNode":26031,"transform":25040},[25033,57695],{"dataC":26058,"xLinkHref":57696},"#MJX-167-TEX-N-32",[25022,57698,57699],{"dataMmlNode":25031,"transform":41846},[25033,57700],{"dataC":27456,"xLinkHref":57701},"#MJX-167-TEX-N-2C",[25022,57703,57704],{"dataMmlNode":26031,"transform":28333},[25033,57705],{"dataC":26035,"xLinkHref":57706},"#MJX-167-TEX-N-31",[25022,57708,57709],{"dataMmlNode":25031,"transform":41146},[25033,57710],{"dataC":25108,"xLinkHref":57711},"#MJX-167-TEX-N-29",[25022,57713,57714,57719,57723],{"dataMmlNode":27203,"transform":51669,"dataMjxTexclass":27205},[25022,57715,57716],{"dataMmlNode":25942},[25033,57717],{"dataC":43135,"xLinkHref":57718},"#MJX-167-TEX-I-1D458",[25022,57720,57721],{"dataMmlNode":25031,"transform":49747},[25033,57722],{"dataC":27456,"xLinkHref":57701},[25022,57724,57725],{"dataMmlNode":25942,"transform":51680},[25033,57726],{"dataC":33647,"xLinkHref":57727},"#MJX-167-TEX-I-1D457",[23,57729,57730,57760,57891],{},[26,57731,57732,57756,57757,6959],{},[106,57733,57734,57735,24917],{},"The Triangle (",[24923,57736,57738],{"className":57737,"jax":24927},[24926],[24929,57739,57742,57747],{"style":27802,"xmlns":24932,"width":57740,"height":56643,"role":1930,"focusable":8058,"viewBox":57741,"xmlnsXLink":24936},"1.885ex","0 -716 833 716",[24938,57743,57744],{},[24941,57745],{"id":57746,"d":56799},"MJX-168-TEX-N-394",[25022,57748,57749],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57750,57751],{"dataMmlNode":25028},[25022,57752,57753],{"dataMmlNode":25942},[25033,57754],{"dataC":56845,"xLinkHref":57755},"#MJX-168-TEX-N-394"," This is the Greek letter Delta, and in math, it simply means ",[106,57758,57759],{},"\"Change\"",[26,57761,57762,57851,57852,57871,57872,16419],{},[106,57763,57764,57765,24917],{},"The Weight (",[24923,57766,57768],{"className":57767,"jax":24927},[24926],[24929,57769,57771,57797],{"style":57636,"xmlns":24932,"width":48804,"height":57638,"role":1930,"focusable":8058,"viewBox":57770,"xmlnsXLink":24936},"0 -1060.7 2252.8 1522.1",[24938,57772,57773,57776,57779,57782,57785,57788,57791,57794],{},[24941,57774],{"id":57775,"d":37254},"MJX-169-TEX-I-1D464",[24941,57777],{"id":57778,"d":24944},"MJX-169-TEX-N-28",[24941,57780],{"id":57781,"d":25928},"MJX-169-TEX-N-32",[24941,57783],{"id":57784,"d":27339},"MJX-169-TEX-N-2C",[24941,57786],{"id":57787,"d":25914},"MJX-169-TEX-N-31",[24941,57789],{"id":57790,"d":24988},"MJX-169-TEX-N-29",[24941,57792],{"id":57793,"d":42951},"MJX-169-TEX-I-1D458",[24941,57795],{"id":57796,"d":33606},"MJX-169-TEX-I-1D457",[25022,57798,57799],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57800,57801],{"dataMmlNode":25028},[25022,57802,57803,57808,57835],{"dataMmlNode":33991},[25022,57804,57805],{"dataMmlNode":25942},[25033,57806],{"dataC":37310,"xLinkHref":57807},"#MJX-169-TEX-I-1D464",[25022,57809,57810,57815,57820,57825,57830],{"dataMmlNode":27203,"transform":48899,"dataMjxTexclass":27205},[25022,57811,57812],{"dataMmlNode":25031},[25033,57813],{"dataC":25035,"xLinkHref":57814},"#MJX-169-TEX-N-28",[25022,57816,57817],{"dataMmlNode":26031,"transform":25040},[25033,57818],{"dataC":26058,"xLinkHref":57819},"#MJX-169-TEX-N-32",[25022,57821,57822],{"dataMmlNode":25031,"transform":41846},[25033,57823],{"dataC":27456,"xLinkHref":57824},"#MJX-169-TEX-N-2C",[25022,57826,57827],{"dataMmlNode":26031,"transform":28333},[25033,57828],{"dataC":26035,"xLinkHref":57829},"#MJX-169-TEX-N-31",[25022,57831,57832],{"dataMmlNode":25031,"transform":41146},[25033,57833],{"dataC":25108,"xLinkHref":57834},"#MJX-169-TEX-N-29",[25022,57836,57837,57842,57846],{"dataMmlNode":27203,"transform":51669,"dataMjxTexclass":27205},[25022,57838,57839],{"dataMmlNode":25942},[25033,57840],{"dataC":43135,"xLinkHref":57841},"#MJX-169-TEX-I-1D458",[25022,57843,57844],{"dataMmlNode":25031,"transform":49747},[25033,57845],{"dataC":27456,"xLinkHref":57824},[25022,57847,57848],{"dataMmlNode":25942,"transform":51680},[25033,57849],{"dataC":33647,"xLinkHref":57850},"#MJX-169-TEX-I-1D457"," This is the specific weight (connection) between a hidden node (",[24923,57853,57855],{"className":57854,"jax":24927},[24926],[24929,57856,57857,57862],{"style":48965,"xmlns":24932,"width":48966,"height":48967,"role":1930,"focusable":8058,"viewBox":48968,"xmlnsXLink":24936},[24938,57858,57859],{},[24941,57860],{"id":57861,"d":33606},"MJX-170-TEX-I-1D457",[25022,57863,57864],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57865,57866],{"dataMmlNode":25028},[25022,57867,57868],{"dataMmlNode":25942},[25033,57869],{"dataC":33647,"xLinkHref":57870},"#MJX-170-TEX-I-1D457",") and an output node (",[24923,57873,57875],{"className":57874,"jax":24927},[24926],[24929,57876,57877,57882],{"style":25196,"xmlns":24932,"width":50688,"height":31523,"role":1930,"focusable":8058,"viewBox":50689,"xmlnsXLink":24936},[24938,57878,57879],{},[24941,57880],{"id":57881,"d":42951},"MJX-171-TEX-I-1D458",[25022,57883,57884],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57885,57886],{"dataMmlNode":25028},[25022,57887,57888],{"dataMmlNode":25942},[25033,57889],{"dataC":43135,"xLinkHref":57890},"#MJX-171-TEX-I-1D458",[26,57892,57893,57896,57897],{},[106,57894,57895],{},"Put it together:"," The entire left side just means: ",[106,57898,57899],{},"\"The exact amount we need to adjust this specific weight.\"",[26236,57901,57903,57904],{"id":57902},"the-middle","The Middle: ",[24923,57905,57907],{"className":57906,"jax":24927},[24926],[24929,57908,57910,57915],{"style":25196,"xmlns":24932,"width":26869,"height":27038,"role":1930,"focusable":8058,"viewBox":57909,"xmlnsXLink":24936},"0 -442 778 453",[24938,57911,57912],{},[24941,57913],{"id":57914,"d":56827},"MJX-172-TEX-N-221D",[25022,57916,57917],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57918,57919],{"dataMmlNode":25028},[25022,57920,57921],{"dataMmlNode":25031},[25033,57922],{"dataC":56903,"xLinkHref":57923},"#MJX-172-TEX-N-221D",[23,57925,57926,57932],{},[26,57927,57928,57929],{},"This symbol means ",[106,57930,57931],{},"\"is proportional to.\"",[26,57933,57934,57935,57954],{},"It basically means the left side scales with the right side (we can replace this symbol with an equals sign ",[24923,57936,57938],{"className":57937,"jax":24927},[24926],[24929,57939,57940,57945],{"style":26868,"xmlns":24932,"width":26869,"height":26870,"role":1930,"focusable":8058,"viewBox":26871,"xmlnsXLink":24936},[24938,57941,57942],{},[24941,57943],{"id":57944,"d":24972},"MJX-173-TEX-N-3D",[25022,57946,57947],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,57948,57949],{"dataMmlNode":25028},[25022,57950,57951],{"dataMmlNode":25031},[25033,57952],{"dataC":25078,"xLinkHref":57953},"#MJX-173-TEX-N-3D"," by multiplying the right side by a small number called a \"learning rate\")",[26236,57956,57958,57959],{"id":57957},"the-right-side","The Right Side: ",[24923,57960,57962],{"className":57961,"jax":24927},[24926],[24929,57963,57968,58003],{"style":57964,"xmlns":24932,"width":57965,"height":57966,"role":1930,"focusable":8058,"viewBox":57967,"xmlnsXLink":24936},"vertical-align: -2.073ex;","7.265ex","4.108ex","0 -899.6 3211.2 1815.9",[24938,57969,57970,57973,57976,57979,57982,57985,57988,57991,57994,57997,58000],{},[24941,57971],{"id":57972,"d":25910},"MJX-174-TEX-N-2212",[24941,57974],{"id":57975,"d":33624},"MJX-174-TEX-I-1D715",[24941,57977],{"id":57978,"d":25869},"MJX-174-TEX-I-1D438",[24941,57980],{"id":57981,"d":37254},"MJX-174-TEX-I-1D464",[24941,57983],{"id":57984,"d":24944},"MJX-174-TEX-N-28",[24941,57986],{"id":57987,"d":25928},"MJX-174-TEX-N-32",[24941,57989],{"id":57990,"d":27339},"MJX-174-TEX-N-2C",[24941,57992],{"id":57993,"d":25914},"MJX-174-TEX-N-31",[24941,57995],{"id":57996,"d":24988},"MJX-174-TEX-N-29",[24941,57998],{"id":57999,"d":42951},"MJX-174-TEX-I-1D458",[24941,58001],{"id":58002,"d":33606},"MJX-174-TEX-I-1D457",[25022,58004,58005],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58006,58007,58012],{"dataMmlNode":25028},[25022,58008,58009],{"dataMmlNode":25031},[25033,58010],{"dataC":26019,"xLinkHref":58011},"#MJX-174-TEX-N-2212",[25022,58013,58014,58027,58084],{"dataMmlNode":27622,"transform":25049},[25022,58015,58017,58022],{"dataMmlNode":27626,"transform":58016},"translate(746.4,394) scale(0.707)",[25022,58018,58019],{"dataMmlNode":25942},[25033,58020],{"dataC":33697,"xLinkHref":58021},"#MJX-174-TEX-I-1D715",[25022,58023,58024],{"dataMmlNode":25942,"transform":33701},[25033,58025],{"dataC":25945,"xLinkHref":58026},"#MJX-174-TEX-I-1D438",[25022,58028,58030,58034],{"dataMmlNode":27626,"transform":58029},"translate(220,-590) scale(0.707)",[25022,58031,58032],{"dataMmlNode":25942},[25033,58033],{"dataC":33697,"xLinkHref":58021},[25022,58035,58036,58041,58068],{"dataMmlNode":33991,"transform":33701},[25022,58037,58038],{"dataMmlNode":25942},[25033,58039],{"dataC":37310,"xLinkHref":58040},"#MJX-174-TEX-I-1D464",[25022,58042,58043,58048,58053,58058,58063],{"dataMmlNode":27203,"transform":48899,"dataMjxTexclass":27205},[25022,58044,58045],{"dataMmlNode":25031},[25033,58046],{"dataC":25035,"xLinkHref":58047},"#MJX-174-TEX-N-28",[25022,58049,58050],{"dataMmlNode":26031,"transform":25040},[25033,58051],{"dataC":26058,"xLinkHref":58052},"#MJX-174-TEX-N-32",[25022,58054,58055],{"dataMmlNode":25031,"transform":41846},[25033,58056],{"dataC":27456,"xLinkHref":58057},"#MJX-174-TEX-N-2C",[25022,58059,58060],{"dataMmlNode":26031,"transform":28333},[25033,58061],{"dataC":26035,"xLinkHref":58062},"#MJX-174-TEX-N-31",[25022,58064,58065],{"dataMmlNode":25031,"transform":41146},[25033,58066],{"dataC":25108,"xLinkHref":58067},"#MJX-174-TEX-N-29",[25022,58069,58070,58075,58079],{"dataMmlNode":27203,"transform":51669,"dataMjxTexclass":27205},[25022,58071,58072],{"dataMmlNode":25942},[25033,58073],{"dataC":43135,"xLinkHref":58074},"#MJX-174-TEX-I-1D458",[25022,58076,58077],{"dataMmlNode":25031,"transform":49747},[25033,58078],{"dataC":27456,"xLinkHref":58057},[25022,58080,58081],{"dataMmlNode":25942,"transform":51680},[25033,58082],{"dataC":33647,"xLinkHref":58083},"#MJX-174-TEX-I-1D457",[27668,58085],{"width":58086,"height":27671,"x":27672,"y":27673},2193.2,[12,58088,58089],{},"This is the most important part, and it has two pieces: the fraction and the negative sign.",[23,58091,58092,58218],{},[26,58093,58094,58146,58147,58150],{},[106,58095,58096,58097,24917],{},"The Fraction (",[24923,58098,58100],{"className":58099,"jax":24927},[24926],[24929,58101,58104,58115],{"style":54303,"xmlns":24932,"width":58102,"height":56997,"role":1930,"focusable":8058,"viewBox":58103,"xmlnsXLink":24936},"3.123ex","0 -899.6 1380.5 1260.7",[24938,58105,58106,58109,58112],{},[24941,58107],{"id":58108,"d":33624},"MJX-175-TEX-I-1D715",[24941,58110],{"id":58111,"d":25869},"MJX-175-TEX-I-1D438",[24941,58113],{"id":58114,"d":37254},"MJX-175-TEX-I-1D464",[25022,58116,58117],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58118,58119],{"dataMmlNode":25028},[25022,58120,58121,58133,58144],{"dataMmlNode":27622},[25022,58122,58123,58128],{"dataMmlNode":27626,"transform":44821},[25022,58124,58125],{"dataMmlNode":25942},[25033,58126],{"dataC":33697,"xLinkHref":58127},"#MJX-175-TEX-I-1D715",[25022,58129,58130],{"dataMmlNode":25942,"transform":33701},[25033,58131],{"dataC":25945,"xLinkHref":58132},"#MJX-175-TEX-I-1D438",[25022,58134,58135,58139],{"dataMmlNode":27626,"transform":57062},[25022,58136,58137],{"dataMmlNode":25942},[25033,58138],{"dataC":33697,"xLinkHref":58127},[25022,58140,58141],{"dataMmlNode":25942,"transform":33701},[25033,58142],{"dataC":37310,"xLinkHref":58143},"#MJX-175-TEX-I-1D464",[27668,58145],{"width":57073,"height":27671,"x":27672,"y":27673}," This is the \"blame.\" It answers the question: ",[7289,58148,58149],{},"\"If I turn this weight UP, what happens to the total Error?\"",[23,58151,58152,58185],{},[26,58153,58154,58155,58182,58183,6959],{},"If the answer is positive (e.g., ",[24923,58156,58158],{"className":58157,"jax":24927},[24926],[24929,58159,58160,58168],{"style":26868,"xmlns":24932,"width":29568,"height":39628,"role":1930,"focusable":8058,"viewBox":52648,"xmlnsXLink":24936},[24938,58161,58162,58165],{},[24941,58163],{"id":58164,"d":30313},"MJX-176-TEX-N-2B",[24941,58166],{"id":58167,"d":26660},"MJX-176-TEX-N-35",[25022,58169,58170],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58171,58172,58177],{"dataMmlNode":25028},[25022,58173,58174],{"dataMmlNode":25031},[25033,58175],{"dataC":30393,"xLinkHref":58176},"#MJX-176-TEX-N-2B",[25022,58178,58179],{"dataMmlNode":26031,"transform":25049},[25033,58180],{"dataC":26717,"xLinkHref":58181},"#MJX-176-TEX-N-35","), it means turning the weight up makes the error ",[106,58184,45928],{},[26,58186,58187,58188,58182,58215,6959],{},"If the answer is negative (e.g., ",[24923,58189,58191],{"className":58190,"jax":24927},[24926],[24929,58192,58193,58201],{"style":26868,"xmlns":24932,"width":29568,"height":39628,"role":1930,"focusable":8058,"viewBox":52648,"xmlnsXLink":24936},[24938,58194,58195,58198],{},[24941,58196],{"id":58197,"d":25910},"MJX-177-TEX-N-2212",[24941,58199],{"id":58200,"d":26660},"MJX-177-TEX-N-35",[25022,58202,58203],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58204,58205,58210],{"dataMmlNode":25028},[25022,58206,58207],{"dataMmlNode":25031},[25033,58208],{"dataC":26019,"xLinkHref":58209},"#MJX-177-TEX-N-2212",[25022,58211,58212],{"dataMmlNode":26031,"transform":25049},[25033,58213],{"dataC":26717,"xLinkHref":58214},"#MJX-177-TEX-N-35",[106,58216,58217],{},"better",[26,58219,58220,58243,58244,58246,58247,58250,58251],{},[106,58221,58222,58223,58242],{},"The Negative Sign (",[24923,58224,58226],{"className":58225,"jax":24927},[24926],[24929,58227,58228,58233],{"style":26868,"xmlns":24932,"width":26869,"height":26870,"role":1930,"focusable":8058,"viewBox":26871,"xmlnsXLink":24936},[24938,58229,58230],{},[24941,58231],{"id":58232,"d":25910},"MJX-178-TEX-N-2212",[25022,58234,58235],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58236,58237],{"dataMmlNode":25028},[25022,58238,58239],{"dataMmlNode":25031},[25033,58240],{"dataC":26019,"xLinkHref":58241},"#MJX-178-TEX-N-2212",") [CRUCIAL!]:"," This is the magic of ",[106,58245,42142],{},". The negative sign tells the network to do the ",[106,58248,58249],{},"exact opposite"," of whatever causes more error.",[23,58252,58253,58312],{},[26,58254,58255,58256,58283,58284,58311],{},"If turning the weight up makes the error worse (",[24923,58257,58259],{"className":58258,"jax":24927},[24926],[24929,58260,58261,58269],{"style":26868,"xmlns":24932,"width":29568,"height":39628,"role":1930,"focusable":8058,"viewBox":52648,"xmlnsXLink":24936},[24938,58262,58263,58266],{},[24941,58264],{"id":58265,"d":30313},"MJX-179-TEX-N-2B",[24941,58267],{"id":58268,"d":26660},"MJX-179-TEX-N-35",[25022,58270,58271],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58272,58273,58278],{"dataMmlNode":25028},[25022,58274,58275],{"dataMmlNode":25031},[25033,58276],{"dataC":30393,"xLinkHref":58277},"#MJX-179-TEX-N-2B",[25022,58279,58280],{"dataMmlNode":26031,"transform":25049},[25033,58281],{"dataC":26717,"xLinkHref":58282},"#MJX-179-TEX-N-35","), the negative sign flips it to ",[24923,58285,58287],{"className":58286,"jax":24927},[24926],[24929,58288,58289,58297],{"style":26868,"xmlns":24932,"width":29568,"height":39628,"role":1930,"focusable":8058,"viewBox":52648,"xmlnsXLink":24936},[24938,58290,58291,58294],{},[24941,58292],{"id":58293,"d":25910},"MJX-180-TEX-N-2212",[24941,58295],{"id":58296,"d":26660},"MJX-180-TEX-N-35",[25022,58298,58299],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58300,58301,58306],{"dataMmlNode":25028},[25022,58302,58303],{"dataMmlNode":25031},[25033,58304],{"dataC":26019,"xLinkHref":58305},"#MJX-180-TEX-N-2212",[25022,58307,58308],{"dataMmlNode":26031,"transform":25049},[25033,58309],{"dataC":26717,"xLinkHref":58310},"#MJX-180-TEX-N-35",". The network says, \"Okay, I will turn the weight DOWN.\"",[26,58313,58314,58315,58283,58342,58369],{},"If turning the weight up makes the error go down (",[24923,58316,58318],{"className":58317,"jax":24927},[24926],[24929,58319,58320,58328],{"style":26868,"xmlns":24932,"width":29568,"height":39628,"role":1930,"focusable":8058,"viewBox":52648,"xmlnsXLink":24936},[24938,58321,58322,58325],{},[24941,58323],{"id":58324,"d":25910},"MJX-181-TEX-N-2212",[24941,58326],{"id":58327,"d":26660},"MJX-181-TEX-N-35",[25022,58329,58330],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58331,58332,58337],{"dataMmlNode":25028},[25022,58333,58334],{"dataMmlNode":25031},[25033,58335],{"dataC":26019,"xLinkHref":58336},"#MJX-181-TEX-N-2212",[25022,58338,58339],{"dataMmlNode":26031,"transform":25049},[25033,58340],{"dataC":26717,"xLinkHref":58341},"#MJX-181-TEX-N-35",[24923,58343,58345],{"className":58344,"jax":24927},[24926],[24929,58346,58347,58355],{"style":26868,"xmlns":24932,"width":29568,"height":39628,"role":1930,"focusable":8058,"viewBox":52648,"xmlnsXLink":24936},[24938,58348,58349,58352],{},[24941,58350],{"id":58351,"d":30313},"MJX-182-TEX-N-2B",[24941,58353],{"id":58354,"d":26660},"MJX-182-TEX-N-35",[25022,58356,58357],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58358,58359,58364],{"dataMmlNode":25028},[25022,58360,58361],{"dataMmlNode":25031},[25033,58362],{"dataC":30393,"xLinkHref":58363},"#MJX-182-TEX-N-2B",[25022,58365,58366],{"dataMmlNode":26031,"transform":25049},[25033,58367],{"dataC":26717,"xLinkHref":58368},"#MJX-182-TEX-N-35",". The network says, \"Great! I will turn the weight UP even more.\"",[12,58371,58372,58373,58420,58421,58504,58505,58525],{},"However, we cannot directly calculate ",[24923,58374,58376],{"className":58375,"jax":24927},[24926],[24929,58377,58378,58389],{"style":54303,"xmlns":24932,"width":58102,"height":56997,"role":1930,"focusable":8058,"viewBox":58103,"xmlnsXLink":24936},[24938,58379,58380,58383,58386],{},[24941,58381],{"id":58382,"d":33624},"MJX-183-TEX-I-1D715",[24941,58384],{"id":58385,"d":25869},"MJX-183-TEX-I-1D438",[24941,58387],{"id":58388,"d":37254},"MJX-183-TEX-I-1D464",[25022,58390,58391],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58392,58393],{"dataMmlNode":25028},[25022,58394,58395,58407,58418],{"dataMmlNode":27622},[25022,58396,58397,58402],{"dataMmlNode":27626,"transform":44821},[25022,58398,58399],{"dataMmlNode":25942},[25033,58400],{"dataC":33697,"xLinkHref":58401},"#MJX-183-TEX-I-1D715",[25022,58403,58404],{"dataMmlNode":25942,"transform":33701},[25033,58405],{"dataC":25945,"xLinkHref":58406},"#MJX-183-TEX-I-1D438",[25022,58408,58409,58413],{"dataMmlNode":27626,"transform":57062},[25022,58410,58411],{"dataMmlNode":25942},[25033,58412],{"dataC":33697,"xLinkHref":58401},[25022,58414,58415],{"dataMmlNode":25942,"transform":33701},[25033,58416],{"dataC":37310,"xLinkHref":58417},"#MJX-183-TEX-I-1D464",[27668,58419],{"width":57073,"height":27671,"x":27672,"y":27673}," because the Error formula ",[24923,58422,58424],{"className":58423,"jax":24927},[24926],[24929,58425,58428,58454],{"style":24931,"xmlns":24932,"width":58426,"height":31733,"role":1930,"focusable":8058,"viewBox":58427,"xmlnsXLink":24936},"12.533ex","0 -833.9 5539.6 1083.9",[24938,58429,58430,58433,58436,58439,58442,58445,58448,58451],{},[24941,58431],{"id":58432,"d":25869},"MJX-184-TEX-I-1D438",[24941,58434],{"id":58435,"d":24972},"MJX-184-TEX-N-3D",[24941,58437],{"id":58438,"d":24944},"MJX-184-TEX-N-28",[24941,58440],{"id":58441,"d":32316},"MJX-184-TEX-I-1D451",[24941,58443],{"id":58444,"d":25910},"MJX-184-TEX-N-2212",[24941,58446],{"id":58447,"d":25885},"MJX-184-TEX-I-1D45C",[24941,58449],{"id":58450,"d":24988},"MJX-184-TEX-N-29",[24941,58452],{"id":58453,"d":25928},"MJX-184-TEX-N-32",[25022,58455,58456],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58457,58458,58463,58468,58473,58479,58485,58491],{"dataMmlNode":25028},[25022,58459,58460],{"dataMmlNode":25942},[25033,58461],{"dataC":25945,"xLinkHref":58462},"#MJX-184-TEX-I-1D438",[25022,58464,58465],{"dataMmlNode":25031,"transform":30197},[25033,58466],{"dataC":25078,"xLinkHref":58467},"#MJX-184-TEX-N-3D",[25022,58469,58470],{"dataMmlNode":25031,"transform":30203},[25033,58471],{"dataC":25035,"xLinkHref":58472},"#MJX-184-TEX-N-28",[25022,58474,58476],{"dataMmlNode":25942,"transform":58475},"translate(2486.6,0)",[25033,58477],{"dataC":32396,"xLinkHref":58478},"#MJX-184-TEX-I-1D451",[25022,58480,58482],{"dataMmlNode":25031,"transform":58481},"translate(3228.8,0)",[25033,58483],{"dataC":26019,"xLinkHref":58484},"#MJX-184-TEX-N-2212",[25022,58486,58488],{"dataMmlNode":25942,"transform":58487},"translate(4229,0)",[25033,58489],{"dataC":25973,"xLinkHref":58490},"#MJX-184-TEX-I-1D45C",[25022,58492,58494,58499],{"dataMmlNode":31565,"transform":58493},"translate(4714,0)",[25022,58495,58496],{"dataMmlNode":25031},[25033,58497],{"dataC":25108,"xLinkHref":58498},"#MJX-184-TEX-N-29",[25022,58500,58501],{"dataMmlNode":26031,"transform":37418},[25033,58502],{"dataC":26058,"xLinkHref":58503},"#MJX-184-TEX-N-32"," does not have a ",[24923,58506,58508],{"className":58507,"jax":24927},[24926],[24929,58509,58511,58516],{"style":25196,"xmlns":24932,"width":56643,"height":27827,"role":1930,"focusable":8058,"viewBox":58510,"xmlnsXLink":24936},"0 -443 716 454",[24938,58512,58513],{},[24941,58514],{"id":58515,"d":37254},"MJX-185-TEX-I-1D464",[25022,58517,58518],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58519,58520],{"dataMmlNode":25028},[25022,58521,58522],{"dataMmlNode":25942},[25033,58523],{"dataC":37310,"xLinkHref":58524},"#MJX-185-TEX-I-1D464"," in it.",[12,58527,58528,58529,58532,58533,58536,58537,58536,58540,6959],{},"To bridge this gap, we use the ",[106,58530,58531],{},"Chain Rule",". We map out the exact sequence of events that happen when a weight is tweaked: a change in the weight alters the ",[106,58534,58535],{},"Net Input",", which alters the ",[106,58538,58539],{},"Output",[106,58541,58542],{},"Error",[12,58544,58545],{},"Mathematically, we chain these three partial derivatives together:",[24923,58547,58549],{"className":58548,"jax":24927,"display":13594},[24926],[24929,58550,58554,58604],{"style":56790,"xmlns":24932,"width":58551,"height":58552,"role":1930,"focusable":8058,"viewBox":58553,"xmlnsXLink":24936},"33.527ex","7.529ex","0 -1845.6 14819 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Mapping out the Chain Rule",[13438,58897,58899,58902,59016,59137,59182,59717,59817,59821,59824,59991],{"className":58898},[25641],[12,58900,58901],{},"Let's imagine we are mathematicians trying to figure out this equation ourselves.",[12,58903,58904,58905,58924,58925,58968,58969,6959],{},"We have a neural network with a huge Error (",[24923,58906,58908],{"className":58907,"jax":24927},[24926],[24929,58909,58910,58915],{"style":27802,"xmlns":24932,"width":50795,"height":27804,"role":1930,"focusable":8058,"viewBox":50796,"xmlnsXLink":24936},[24938,58911,58912],{},[24941,58913],{"id":58914,"d":25869},"MJX-187-TEX-I-1D438",[25022,58916,58917],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58918,58919],{"dataMmlNode":25028},[25022,58920,58921],{"dataMmlNode":25942},[25033,58922],{"dataC":25945,"xLinkHref":58923},"#MJX-187-TEX-I-1D438","). We don't want that. We need to figure out: ",[106,58926,58927,58928,58947,58948,58967],{},"If we tweak a specific weight (",[24923,58929,58931],{"className":58930,"jax":24927},[24926],[24929,58932,58933,58938],{"style":25196,"xmlns":24932,"width":56643,"height":27827,"role":1930,"focusable":8058,"viewBox":58510,"xmlnsXLink":24936},[24938,58934,58935],{},[24941,58936],{"id":58937,"d":37254},"MJX-188-TEX-I-1D464",[25022,58939,58940],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58941,58942],{"dataMmlNode":25028},[25022,58943,58944],{"dataMmlNode":25942},[25033,58945],{"dataC":37310,"xLinkHref":58946},"#MJX-188-TEX-I-1D464","), what happens to the Error (",[24923,58949,58951],{"className":58950,"jax":24927},[24926],[24929,58952,58953,58958],{"style":27802,"xmlns":24932,"width":50795,"height":27804,"role":1930,"focusable":8058,"viewBox":50796,"xmlnsXLink":24936},[24938,58954,58955],{},[24941,58956],{"id":58957,"d":25869},"MJX-189-TEX-I-1D438",[25022,58959,58960],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58961,58962],{"dataMmlNode":25028},[25022,58963,58964],{"dataMmlNode":25942},[25033,58965],{"dataC":25945,"xLinkHref":58966},"#MJX-189-TEX-I-1D438",")?"," In calculus terms, our goal is to find ",[24923,58970,58972],{"className":58971,"jax":24927},[24926],[24929,58973,58974,58985],{"style":54303,"xmlns":24932,"width":58102,"height":56997,"role":1930,"focusable":8058,"viewBox":58103,"xmlnsXLink":24936},[24938,58975,58976,58979,58982],{},[24941,58977],{"id":58978,"d":33624},"MJX-190-TEX-I-1D715",[24941,58980],{"id":58981,"d":25869},"MJX-190-TEX-I-1D438",[24941,58983],{"id":58984,"d":37254},"MJX-190-TEX-I-1D464",[25022,58986,58987],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,58988,58989],{"dataMmlNode":25028},[25022,58990,58991,59003,59014],{"dataMmlNode":27622},[25022,58992,58993,58998],{"dataMmlNode":27626,"transform":44821},[25022,58994,58995],{"dataMmlNode":25942},[25033,58996],{"dataC":33697,"xLinkHref":58997},"#MJX-190-TEX-I-1D715",[25022,58999,59000],{"dataMmlNode":25942,"transform":33701},[25033,59001],{"dataC":25945,"xLinkHref":59002},"#MJX-190-TEX-I-1D438",[25022,59004,59005,59009],{"dataMmlNode":27626,"transform":57062},[25022,59006,59007],{"dataMmlNode":25942},[25033,59008],{"dataC":33697,"xLinkHref":58997},[25022,59010,59011],{"dataMmlNode":25942,"transform":33701},[25033,59012],{"dataC":37310,"xLinkHref":59013},"#MJX-190-TEX-I-1D464",[27668,59015],{"width":57073,"height":27671,"x":27672,"y":27673},[12,59017,59018,59019,59096,59097,59116,59117,59136],{},"However, we can't directly calculate this because the Error formula ",[24923,59020,59022],{"className":59021,"jax":24927},[24926],[24929,59023,59024,59050],{"style":24931,"xmlns":24932,"width":58426,"height":31733,"role":1930,"focusable":8058,"viewBox":58427,"xmlnsXLink":24936},[24938,59025,59026,59029,59032,59035,59038,59041,59044,59047],{},[24941,59027],{"id":59028,"d":25869},"MJX-191-TEX-I-1D438",[24941,59030],{"id":59031,"d":24972},"MJX-191-TEX-N-3D",[24941,59033],{"id":59034,"d":24944},"MJX-191-TEX-N-28",[24941,59036],{"id":59037,"d":32316},"MJX-191-TEX-I-1D451",[24941,59039],{"id":59040,"d":25910},"MJX-191-TEX-N-2212",[24941,59042],{"id":59043,"d":25885},"MJX-191-TEX-I-1D45C",[24941,59045],{"id":59046,"d":24988},"MJX-191-TEX-N-29",[24941,59048],{"id":59049,"d":25928},"MJX-191-TEX-N-32",[25022,59051,59052],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59053,59054,59059,59064,59069,59074,59079,59084],{"dataMmlNode":25028},[25022,59055,59056],{"dataMmlNode":25942},[25033,59057],{"dataC":25945,"xLinkHref":59058},"#MJX-191-TEX-I-1D438",[25022,59060,59061],{"dataMmlNode":25031,"transform":30197},[25033,59062],{"dataC":25078,"xLinkHref":59063},"#MJX-191-TEX-N-3D",[25022,59065,59066],{"dataMmlNode":25031,"transform":30203},[25033,59067],{"dataC":25035,"xLinkHref":59068},"#MJX-191-TEX-N-28",[25022,59070,59071],{"dataMmlNode":25942,"transform":58475},[25033,59072],{"dataC":32396,"xLinkHref":59073},"#MJX-191-TEX-I-1D451",[25022,59075,59076],{"dataMmlNode":25031,"transform":58481},[25033,59077],{"dataC":26019,"xLinkHref":59078},"#MJX-191-TEX-N-2212",[25022,59080,59081],{"dataMmlNode":25942,"transform":58487},[25033,59082],{"dataC":25973,"xLinkHref":59083},"#MJX-191-TEX-I-1D45C",[25022,59085,59086,59091],{"dataMmlNode":31565,"transform":58493},[25022,59087,59088],{"dataMmlNode":25031},[25033,59089],{"dataC":25108,"xLinkHref":59090},"#MJX-191-TEX-N-29",[25022,59092,59093],{"dataMmlNode":26031,"transform":37418},[25033,59094],{"dataC":26058,"xLinkHref":59095},"#MJX-191-TEX-N-32"," doesn't have a ",[24923,59098,59100],{"className":59099,"jax":24927},[24926],[24929,59101,59102,59107],{"style":25196,"xmlns":24932,"width":56643,"height":27827,"role":1930,"focusable":8058,"viewBox":58510,"xmlnsXLink":24936},[24938,59103,59104],{},[24941,59105],{"id":59106,"d":37254},"MJX-192-TEX-I-1D464",[25022,59108,59109],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59110,59111],{"dataMmlNode":25028},[25022,59112,59113],{"dataMmlNode":25942},[25033,59114],{"dataC":37310,"xLinkHref":59115},"#MJX-192-TEX-I-1D464"," in it! There is no ",[24923,59118,59120],{"className":59119,"jax":24927},[24926],[24929,59121,59122,59127],{"style":25196,"xmlns":24932,"width":56643,"height":27827,"role":1930,"focusable":8058,"viewBox":58510,"xmlnsXLink":24936},[24938,59123,59124],{},[24941,59125],{"id":59126,"d":37254},"MJX-193-TEX-I-1D464",[25022,59128,59129],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59130,59131],{"dataMmlNode":25028},[25022,59132,59133],{"dataMmlNode":25942},[25033,59134],{"dataC":37310,"xLinkHref":59135},"#MJX-193-TEX-I-1D464"," to take a derivative of.",[12,59138,59139,59140,10720,59159,59178,59179],{},"To connect ",[24923,59141,59143],{"className":59142,"jax":24927},[24926],[24929,59144,59145,59150],{"style":25196,"xmlns":24932,"width":56643,"height":27827,"role":1930,"focusable":8058,"viewBox":58510,"xmlnsXLink":24936},[24938,59146,59147],{},[24941,59148],{"id":59149,"d":37254},"MJX-194-TEX-I-1D464",[25022,59151,59152],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59153,59154],{"dataMmlNode":25028},[25022,59155,59156],{"dataMmlNode":25942},[25033,59157],{"dataC":37310,"xLinkHref":59158},"#MJX-194-TEX-I-1D464",[24923,59160,59162],{"className":59161,"jax":24927},[24926],[24929,59163,59164,59169],{"style":27802,"xmlns":24932,"width":50795,"height":27804,"role":1930,"focusable":8058,"viewBox":50796,"xmlnsXLink":24936},[24938,59165,59166],{},[24941,59167],{"id":59168,"d":25869},"MJX-195-TEX-I-1D438",[25022,59170,59171],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59172,59173],{"dataMmlNode":25028},[25022,59174,59175],{"dataMmlNode":25942},[25033,59176],{"dataC":25945,"xLinkHref":59177},"#MJX-195-TEX-I-1D438",", we must map out the \"domino effect\" of the network. We ask ourselves: ",[7289,59180,59181],{},"\"If I tweak the weight, what is the very next thing that happens?\"",[23,59183,59184,59374,59579],{},[26,59185,59186,59189,59190,59209,59210,59248,59249,59268,59269,59304,59305],{},[106,59187,59188],{},"Domino 1: The Weight changes the Net Input.","\nThe weight (",[24923,59191,59193],{"className":59192,"jax":24927},[24926],[24929,59194,59195,59200],{"style":25196,"xmlns":24932,"width":56643,"height":27827,"role":1930,"focusable":8058,"viewBox":58510,"xmlnsXLink":24936},[24938,59196,59197],{},[24941,59198],{"id":59199,"d":37254},"MJX-196-TEX-I-1D464",[25022,59201,59202],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59203,59204],{"dataMmlNode":25028},[25022,59205,59206],{"dataMmlNode":25942},[25033,59207],{"dataC":37310,"xLinkHref":59208},"#MJX-196-TEX-I-1D464",") is multiplied by the incoming signal to create the net input (",[24923,59211,59213],{"className":59212,"jax":24927},[24926],[24929,59214,59218,59229],{"style":25196,"xmlns":24932,"width":59215,"height":59216,"role":1930,"focusable":8058,"viewBox":59217,"xmlnsXLink":24936},"3.229ex","1.441ex","0 -626 1427 637",[24938,59219,59220,59223,59226],{},[24941,59221],{"id":59222,"d":25873},"MJX-197-TEX-I-1D45B",[24941,59224],{"id":59225,"d":27397},"MJX-197-TEX-I-1D452",[24941,59227],{"id":59228,"d":25877},"MJX-197-TEX-I-1D461",[25022,59230,59231],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59232,59233,59238,59243],{"dataMmlNode":25028},[25022,59234,59235],{"dataMmlNode":25942},[25033,59236],{"dataC":25952,"xLinkHref":59237},"#MJX-197-TEX-I-1D45B",[25022,59239,59240],{"dataMmlNode":25942,"transform":46886},[25033,59241],{"dataC":27596,"xLinkHref":59242},"#MJX-197-TEX-I-1D452",[25022,59244,59245],{"dataMmlNode":25942,"transform":46892},[25033,59246],{"dataC":25959,"xLinkHref":59247},"#MJX-197-TEX-I-1D461","). Thus, a change in ",[24923,59250,59252],{"className":59251,"jax":24927},[24926],[24929,59253,59254,59259],{"style":25196,"xmlns":24932,"width":56643,"height":27827,"role":1930,"focusable":8058,"viewBox":58510,"xmlnsXLink":24936},[24938,59255,59256],{},[24941,59257],{"id":59258,"d":37254},"MJX-198-TEX-I-1D464",[25022,59260,59261],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59262,59263],{"dataMmlNode":25028},[25022,59264,59265],{"dataMmlNode":25942},[25033,59266],{"dataC":37310,"xLinkHref":59267},"#MJX-198-TEX-I-1D464"," causes a direct change in ",[24923,59270,59272],{"className":59271,"jax":24927},[24926],[24929,59273,59274,59285],{"style":25196,"xmlns":24932,"width":59215,"height":59216,"role":1930,"focusable":8058,"viewBox":59217,"xmlnsXLink":24936},[24938,59275,59276,59279,59282],{},[24941,59277],{"id":59278,"d":25873},"MJX-199-TEX-I-1D45B",[24941,59280],{"id":59281,"d":27397},"MJX-199-TEX-I-1D452",[24941,59283],{"id":59284,"d":25877},"MJX-199-TEX-I-1D461",[25022,59286,59287],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59288,59289,59294,59299],{"dataMmlNode":25028},[25022,59290,59291],{"dataMmlNode":25942},[25033,59292],{"dataC":25952,"xLinkHref":59293},"#MJX-199-TEX-I-1D45B",[25022,59295,59296],{"dataMmlNode":25942,"transform":46886},[25033,59297],{"dataC":27596,"xLinkHref":59298},"#MJX-199-TEX-I-1D452",[25022,59300,59301],{"dataMmlNode":25942,"transform":46892},[25033,59302],{"dataC":25959,"xLinkHref":59303},"#MJX-199-TEX-I-1D461",".\nThis gives us our first piece: ",[106,59306,59307],{},[24923,59308,59310],{"className":59309,"jax":24927},[24926],[24929,59311,59314,59331],{"style":54303,"xmlns":24932,"width":59312,"height":56997,"role":1930,"focusable":8058,"viewBox":59313,"xmlnsXLink":24936},"4.184ex","0 -899.6 1849.3 1260.7",[24938,59315,59316,59319,59322,59325,59328],{},[24941,59317],{"id":59318,"d":33624},"MJX-200-TEX-I-1D715",[24941,59320],{"id":59321,"d":25873},"MJX-200-TEX-I-1D45B",[24941,59323],{"id":59324,"d":27397},"MJX-200-TEX-I-1D452",[24941,59326],{"id":59327,"d":25877},"MJX-200-TEX-I-1D461",[24941,59329],{"id":59330,"d":37254},"MJX-200-TEX-I-1D464",[25022,59332,59333],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59334,59335],{"dataMmlNode":25028},[25022,59336,59337,59359,59371],{"dataMmlNode":27622},[25022,59338,59339,59344,59349,59354],{"dataMmlNode":27626,"transform":44821},[25022,59340,59341],{"dataMmlNode":25942},[25033,59342],{"dataC":33697,"xLinkHref":59343},"#MJX-200-TEX-I-1D715",[25022,59345,59346],{"dataMmlNode":25942,"transform":33701},[25033,59347],{"dataC":25952,"xLinkHref":59348},"#MJX-200-TEX-I-1D45B",[25022,59350,59351],{"dataMmlNode":25942,"transform":58759},[25033,59352],{"dataC":27596,"xLinkHref":59353},"#MJX-200-TEX-I-1D452",[25022,59355,59356],{"dataMmlNode":25942,"transform":58765},[25033,59357],{"dataC":25959,"xLinkHref":59358},"#MJX-200-TEX-I-1D461",[25022,59360,59362,59366],{"dataMmlNode":27626,"transform":59361},"translate(471.4,-345.6) scale(0.707)",[25022,59363,59364],{"dataMmlNode":25942},[25033,59365],{"dataC":33697,"xLinkHref":59343},[25022,59367,59368],{"dataMmlNode":25942,"transform":33701},[25033,59369],{"dataC":37310,"xLinkHref":59370},"#MJX-200-TEX-I-1D464",[27668,59372],{"width":59373,"height":27671,"x":27672,"y":27673},1609.3,[26,59375,59376,59379,59380,59415,59416,59435,59436,59457,59458,59268,59493,59512,59513],{},[106,59377,59378],{},"Domino 2: The Net Input changes the Output.","\nDoes the ",[24923,59381,59383],{"className":59382,"jax":24927},[24926],[24929,59384,59385,59396],{"style":25196,"xmlns":24932,"width":59215,"height":59216,"role":1930,"focusable":8058,"viewBox":59217,"xmlnsXLink":24936},[24938,59386,59387,59390,59393],{},[24941,59388],{"id":59389,"d":25873},"MJX-201-TEX-I-1D45B",[24941,59391],{"id":59392,"d":27397},"MJX-201-TEX-I-1D452",[24941,59394],{"id":59395,"d":25877},"MJX-201-TEX-I-1D461",[25022,59397,59398],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59399,59400,59405,59410],{"dataMmlNode":25028},[25022,59401,59402],{"dataMmlNode":25942},[25033,59403],{"dataC":25952,"xLinkHref":59404},"#MJX-201-TEX-I-1D45B",[25022,59406,59407],{"dataMmlNode":25942,"transform":46886},[25033,59408],{"dataC":27596,"xLinkHref":59409},"#MJX-201-TEX-I-1D452",[25022,59411,59412],{"dataMmlNode":25942,"transform":46892},[25033,59413],{"dataC":25959,"xLinkHref":59414},"#MJX-201-TEX-I-1D461"," directly change the Error? No. First, it has to pass through the activation function (",[24923,59417,59419],{"className":59418,"jax":24927},[24926],[24929,59420,59421,59426],{"style":27778,"xmlns":24932,"width":27779,"height":27780,"role":1930,"focusable":8058,"viewBox":27781,"xmlnsXLink":24936},[24938,59422,59423],{},[24941,59424],{"id":59425,"d":25900},"MJX-202-TEX-I-1D446",[25022,59427,59428],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59429,59430],{"dataMmlNode":25028},[25022,59431,59432],{"dataMmlNode":25942},[25033,59433],{"dataC":26000,"xLinkHref":59434},"#MJX-202-TEX-I-1D446",") to become the final output (",[24923,59437,59439],{"className":59438,"jax":24927},[24926],[24929,59440,59443,59448],{"style":25196,"xmlns":24932,"width":27826,"height":59441,"role":1930,"focusable":8058,"viewBox":59442,"xmlnsXLink":24936},"1.023ex","0 -441 485 452",[24938,59444,59445],{},[24941,59446],{"id":59447,"d":25885},"MJX-203-TEX-I-1D45C",[25022,59449,59450],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59451,59452],{"dataMmlNode":25028},[25022,59453,59454],{"dataMmlNode":25942},[25033,59455],{"dataC":25973,"xLinkHref":59456},"#MJX-203-TEX-I-1D45C","). A change in ",[24923,59459,59461],{"className":59460,"jax":24927},[24926],[24929,59462,59463,59474],{"style":25196,"xmlns":24932,"width":59215,"height":59216,"role":1930,"focusable":8058,"viewBox":59217,"xmlnsXLink":24936},[24938,59464,59465,59468,59471],{},[24941,59466],{"id":59467,"d":25873},"MJX-204-TEX-I-1D45B",[24941,59469],{"id":59470,"d":27397},"MJX-204-TEX-I-1D452",[24941,59472],{"id":59473,"d":25877},"MJX-204-TEX-I-1D461",[25022,59475,59476],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59477,59478,59483,59488],{"dataMmlNode":25028},[25022,59479,59480],{"dataMmlNode":25942},[25033,59481],{"dataC":25952,"xLinkHref":59482},"#MJX-204-TEX-I-1D45B",[25022,59484,59485],{"dataMmlNode":25942,"transform":46886},[25033,59486],{"dataC":27596,"xLinkHref":59487},"#MJX-204-TEX-I-1D452",[25022,59489,59490],{"dataMmlNode":25942,"transform":46892},[25033,59491],{"dataC":25959,"xLinkHref":59492},"#MJX-204-TEX-I-1D461",[24923,59494,59496],{"className":59495,"jax":24927},[24926],[24929,59497,59498,59503],{"style":25196,"xmlns":24932,"width":27826,"height":59441,"role":1930,"focusable":8058,"viewBox":59442,"xmlnsXLink":24936},[24938,59499,59500],{},[24941,59501],{"id":59502,"d":25885},"MJX-205-TEX-I-1D45C",[25022,59504,59505],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59506,59507],{"dataMmlNode":25028},[25022,59508,59509],{"dataMmlNode":25942},[25033,59510],{"dataC":25973,"xLinkHref":59511},"#MJX-205-TEX-I-1D45C",".\nThis gives us our second piece: ",[106,59514,59515],{},[24923,59516,59518],{"className":59517,"jax":24927},[24926],[24929,59519,59520,59537],{"style":54303,"xmlns":24932,"width":59312,"height":56997,"role":1930,"focusable":8058,"viewBox":59313,"xmlnsXLink":24936},[24938,59521,59522,59525,59528,59531,59534],{},[24941,59523],{"id":59524,"d":33624},"MJX-206-TEX-I-1D715",[24941,59526],{"id":59527,"d":25885},"MJX-206-TEX-I-1D45C",[24941,59529],{"id":59530,"d":25873},"MJX-206-TEX-I-1D45B",[24941,59532],{"id":59533,"d":27397},"MJX-206-TEX-I-1D452",[24941,59535],{"id":59536,"d":25877},"MJX-206-TEX-I-1D461",[25022,59538,59539],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59540,59541],{"dataMmlNode":25028},[25022,59542,59543,59556,59577],{"dataMmlNode":27622},[25022,59544,59546,59551],{"dataMmlNode":27626,"transform":59545},"translate(553,394) scale(0.707)",[25022,59547,59548],{"dataMmlNode":25942},[25033,59549],{"dataC":33697,"xLinkHref":59550},"#MJX-206-TEX-I-1D715",[25022,59552,59553],{"dataMmlNode":25942,"transform":33701},[25033,59554],{"dataC":25973,"xLinkHref":59555},"#MJX-206-TEX-I-1D45C",[25022,59557,59558,59562,59567,59572],{"dataMmlNode":27626,"transform":45099},[25022,59559,59560],{"dataMmlNode":25942},[25033,59561],{"dataC":33697,"xLinkHref":59550},[25022,59563,59564],{"dataMmlNode":25942,"transform":33701},[25033,59565],{"dataC":25952,"xLinkHref":59566},"#MJX-206-TEX-I-1D45B",[25022,59568,59569],{"dataMmlNode":25942,"transform":58759},[25033,59570],{"dataC":27596,"xLinkHref":59571},"#MJX-206-TEX-I-1D452",[25022,59573,59574],{"dataMmlNode":25942,"transform":58765},[25033,59575],{"dataC":25959,"xLinkHref":59576},"#MJX-206-TEX-I-1D461",[27668,59578],{"width":59373,"height":27671,"x":27672,"y":27673},[26,59580,59581,59584,59585,59604,59605,59627,59628,59268,59647,59666,59667],{},[106,59582,59583],{},"Domino 3: The Output changes the Error.","\nFinally, the output (",[24923,59586,59588],{"className":59587,"jax":24927},[24926],[24929,59589,59590,59595],{"style":25196,"xmlns":24932,"width":27826,"height":59441,"role":1930,"focusable":8058,"viewBox":59442,"xmlnsXLink":24936},[24938,59591,59592],{},[24941,59593],{"id":59594,"d":25885},"MJX-207-TEX-I-1D45C",[25022,59596,59597],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59598,59599],{"dataMmlNode":25028},[25022,59600,59601],{"dataMmlNode":25942},[25033,59602],{"dataC":25973,"xLinkHref":59603},"#MJX-207-TEX-I-1D45C",") is compared to our dataset target (",[24923,59606,59608],{"className":59607,"jax":24927},[24926],[24929,59609,59613,59618],{"style":33072,"xmlns":24932,"width":59610,"height":59611,"role":1930,"focusable":8058,"viewBox":59612,"xmlnsXLink":24936},"1.176ex","1.593ex","0 -694 520 704",[24938,59614,59615],{},[24941,59616],{"id":59617,"d":32316},"MJX-208-TEX-I-1D451",[25022,59619,59620],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59621,59622],{"dataMmlNode":25028},[25022,59623,59624],{"dataMmlNode":25942},[25033,59625],{"dataC":32396,"xLinkHref":59626},"#MJX-208-TEX-I-1D451",") to calculate the final Error. A change in ",[24923,59629,59631],{"className":59630,"jax":24927},[24926],[24929,59632,59633,59638],{"style":25196,"xmlns":24932,"width":27826,"height":59441,"role":1930,"focusable":8058,"viewBox":59442,"xmlnsXLink":24936},[24938,59634,59635],{},[24941,59636],{"id":59637,"d":25885},"MJX-209-TEX-I-1D45C",[25022,59639,59640],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59641,59642],{"dataMmlNode":25028},[25022,59643,59644],{"dataMmlNode":25942},[25033,59645],{"dataC":25973,"xLinkHref":59646},"#MJX-209-TEX-I-1D45C",[24923,59648,59650],{"className":59649,"jax":24927},[24926],[24929,59651,59652,59657],{"style":27802,"xmlns":24932,"width":50795,"height":27804,"role":1930,"focusable":8058,"viewBox":50796,"xmlnsXLink":24936},[24938,59653,59654],{},[24941,59655],{"id":59656,"d":25869},"MJX-210-TEX-I-1D438",[25022,59658,59659],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59660,59661],{"dataMmlNode":25028},[25022,59662,59663],{"dataMmlNode":25942},[25033,59664],{"dataC":25945,"xLinkHref":59665},"#MJX-210-TEX-I-1D438",".\nThis gives us our final piece: ",[106,59668,59669],{},[24923,59670,59672],{"className":59671,"jax":24927},[24926],[24929,59673,59674,59685],{"style":54303,"xmlns":24932,"width":58102,"height":56997,"role":1930,"focusable":8058,"viewBox":58103,"xmlnsXLink":24936},[24938,59675,59676,59679,59682],{},[24941,59677],{"id":59678,"d":33624},"MJX-211-TEX-I-1D715",[24941,59680],{"id":59681,"d":25869},"MJX-211-TEX-I-1D438",[24941,59683],{"id":59684,"d":25885},"MJX-211-TEX-I-1D45C",[25022,59686,59687],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59688,59689],{"dataMmlNode":25028},[25022,59690,59691,59703,59715],{"dataMmlNode":27622},[25022,59692,59693,59698],{"dataMmlNode":27626,"transform":44821},[25022,59694,59695],{"dataMmlNode":25942},[25033,59696],{"dataC":33697,"xLinkHref":59697},"#MJX-211-TEX-I-1D715",[25022,59699,59700],{"dataMmlNode":25942,"transform":33701},[25033,59701],{"dataC":25945,"xLinkHref":59702},"#MJX-211-TEX-I-1D438",[25022,59704,59706,59710],{"dataMmlNode":27626,"transform":59705},"translate(318.6,-345.6) scale(0.707)",[25022,59707,59708],{"dataMmlNode":25942},[25033,59709],{"dataC":33697,"xLinkHref":59697},[25022,59711,59712],{"dataMmlNode":25942,"transform":33701},[25033,59713],{"dataC":25973,"xLinkHref":59714},"#MJX-211-TEX-I-1D45C",[27668,59716],{"width":57073,"height":27671,"x":27672,"y":27673},[12,59718,59719,59720,59739,59740,59759,59760,59759,59795,49470,59814],{},"We've just mapped out a composite function: ",[24923,59721,59723],{"className":59722,"jax":24927},[24926],[24929,59724,59725,59730],{"style":27802,"xmlns":24932,"width":50795,"height":27804,"role":1930,"focusable":8058,"viewBox":50796,"xmlnsXLink":24936},[24938,59726,59727],{},[24941,59728],{"id":59729,"d":25869},"MJX-212-TEX-I-1D438",[25022,59731,59732],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59733,59734],{"dataMmlNode":25028},[25022,59735,59736],{"dataMmlNode":25942},[25033,59737],{"dataC":25945,"xLinkHref":59738},"#MJX-212-TEX-I-1D438"," depends on ",[24923,59741,59743],{"className":59742,"jax":24927},[24926],[24929,59744,59745,59750],{"style":25196,"xmlns":24932,"width":27826,"height":59441,"role":1930,"focusable":8058,"viewBox":59442,"xmlnsXLink":24936},[24938,59746,59747],{},[24941,59748],{"id":59749,"d":25885},"MJX-213-TEX-I-1D45C",[25022,59751,59752],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59753,59754],{"dataMmlNode":25028},[25022,59755,59756],{"dataMmlNode":25942},[25033,59757],{"dataC":25973,"xLinkHref":59758},"#MJX-213-TEX-I-1D45C",", which depends on ",[24923,59761,59763],{"className":59762,"jax":24927},[24926],[24929,59764,59765,59776],{"style":25196,"xmlns":24932,"width":59215,"height":59216,"role":1930,"focusable":8058,"viewBox":59217,"xmlnsXLink":24936},[24938,59766,59767,59770,59773],{},[24941,59768],{"id":59769,"d":25873},"MJX-214-TEX-I-1D45B",[24941,59771],{"id":59772,"d":27397},"MJX-214-TEX-I-1D452",[24941,59774],{"id":59775,"d":25877},"MJX-214-TEX-I-1D461",[25022,59777,59778],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59779,59780,59785,59790],{"dataMmlNode":25028},[25022,59781,59782],{"dataMmlNode":25942},[25033,59783],{"dataC":25952,"xLinkHref":59784},"#MJX-214-TEX-I-1D45B",[25022,59786,59787],{"dataMmlNode":25942,"transform":46886},[25033,59788],{"dataC":27596,"xLinkHref":59789},"#MJX-214-TEX-I-1D452",[25022,59791,59792],{"dataMmlNode":25942,"transform":46892},[25033,59793],{"dataC":25959,"xLinkHref":59794},"#MJX-214-TEX-I-1D461",[24923,59796,59798],{"className":59797,"jax":24927},[24926],[24929,59799,59800,59805],{"style":25196,"xmlns":24932,"width":56643,"height":27827,"role":1930,"focusable":8058,"viewBox":58510,"xmlnsXLink":24936},[24938,59801,59802],{},[24941,59803],{"id":59804,"d":37254},"MJX-215-TEX-I-1D464",[25022,59806,59807],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59808,59809],{"dataMmlNode":25028},[25022,59810,59811],{"dataMmlNode":25942},[25033,59812],{"dataC":37310,"xLinkHref":59813},"#MJX-215-TEX-I-1D464",[106,59815,59816],{},"Multiply them together, and you get your exact blueprint equation!",[26236,59818,59820],{"id":59819},"decoding-the-labels","Decoding the Labels",[12,59822,59823],{},"The numbers and letters are just coordinate labels to tell you exactly where you are in the network.",[23,59825,59826,59900],{},[26,59827,59828,7206,59831,59888,59889,59892,59893,59896,59897],{},[106,59829,59830],{},"The Superscript",[106,59832,59833,7024],{},[24923,59834,59836],{"className":59835,"jax":24927},[24926],[24929,59837,59840,59857],{"style":24931,"xmlns":24932,"width":59838,"height":24934,"role":1930,"focusable":8058,"viewBox":59839,"xmlnsXLink":24936},"5.029ex","0 -750 2222.7 1000",[24938,59841,59842,59845,59848,59851,59854],{},[24941,59843],{"id":59844,"d":24944},"MJX-216-TEX-N-28",[24941,59846],{"id":59847,"d":25928},"MJX-216-TEX-N-32",[24941,59849],{"id":59850,"d":27339},"MJX-216-TEX-N-2C",[24941,59852],{"id":59853,"d":25914},"MJX-216-TEX-N-31",[24941,59855],{"id":59856,"d":24988},"MJX-216-TEX-N-29",[25022,59858,59859],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59860,59861,59866,59871,59876,59882],{"dataMmlNode":25028},[25022,59862,59863],{"dataMmlNode":25031},[25033,59864],{"dataC":25035,"xLinkHref":59865},"#MJX-216-TEX-N-28",[25022,59867,59868],{"dataMmlNode":26031,"transform":25040},[25033,59869],{"dataC":26058,"xLinkHref":59870},"#MJX-216-TEX-N-32",[25022,59872,59873],{"dataMmlNode":25031,"transform":41846},[25033,59874],{"dataC":27456,"xLinkHref":59875},"#MJX-216-TEX-N-2C",[25022,59877,59879],{"dataMmlNode":26031,"transform":59878},"translate(1333.7,0)",[25033,59880],{"dataC":26035,"xLinkHref":59881},"#MJX-216-TEX-N-31",[25022,59883,59885],{"dataMmlNode":25031,"transform":59884},"translate(1833.7,0)",[25033,59886],{"dataC":25108,"xLinkHref":59887},"#MJX-216-TEX-N-29"," This tells you which layers the weight is bridging. It means this weight connects ",[106,59890,59891],{},"Layer 1"," (hidden) to ",[106,59894,59895],{},"Layer 2"," (output). ",[7289,59898,59899],{},"(Note: It is conventionally written backward as Target, Source).",[26,59901,59902,7206,59905,59946,59947,59968,59969,59990],{},[106,59903,59904],{},"The Subscript",[106,59906,59907,7024],{},[24923,59908,59910],{"className":59909,"jax":24927},[24926],[24929,59911,59915,59926],{"style":48965,"xmlns":24932,"width":59912,"height":59913,"role":1930,"focusable":8058,"viewBox":59914,"xmlnsXLink":24936},"3.117ex","2.032ex","0 -694 1377.7 898",[24938,59916,59917,59920,59923],{},[24941,59918],{"id":59919,"d":42951},"MJX-217-TEX-I-1D458",[24941,59921],{"id":59922,"d":27339},"MJX-217-TEX-N-2C",[24941,59924],{"id":59925,"d":33606},"MJX-217-TEX-I-1D457",[25022,59927,59928],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59929,59930,59935,59940],{"dataMmlNode":25028},[25022,59931,59932],{"dataMmlNode":25942},[25033,59933],{"dataC":43135,"xLinkHref":59934},"#MJX-217-TEX-I-1D458",[25022,59936,59937],{"dataMmlNode":25031,"transform":49747},[25033,59938],{"dataC":27456,"xLinkHref":59939},"#MJX-217-TEX-N-2C",[25022,59941,59943],{"dataMmlNode":25942,"transform":59942},"translate(965.7,0)",[25033,59944],{"dataC":33647,"xLinkHref":59945},"#MJX-217-TEX-I-1D457"," This tells you the specific nodes. It connects node ",[106,59948,59949],{},[24923,59950,59952],{"className":59951,"jax":24927},[24926],[24929,59953,59954,59959],{"style":48965,"xmlns":24932,"width":48966,"height":48967,"role":1930,"focusable":8058,"viewBox":48968,"xmlnsXLink":24936},[24938,59955,59956],{},[24941,59957],{"id":59958,"d":33606},"MJX-218-TEX-I-1D457",[25022,59960,59961],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59962,59963],{"dataMmlNode":25028},[25022,59964,59965],{"dataMmlNode":25942},[25033,59966],{"dataC":33647,"xLinkHref":59967},"#MJX-218-TEX-I-1D457"," (in the hidden layer) to node ",[106,59970,59971],{},[24923,59972,59974],{"className":59973,"jax":24927},[24926],[24929,59975,59976,59981],{"style":25196,"xmlns":24932,"width":50688,"height":31523,"role":1930,"focusable":8058,"viewBox":50689,"xmlnsXLink":24936},[24938,59977,59978],{},[24941,59979],{"id":59980,"d":42951},"MJX-219-TEX-I-1D458",[25022,59982,59983],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,59984,59985],{"dataMmlNode":25028},[25022,59986,59987],{"dataMmlNode":25942},[25033,59988],{"dataC":43135,"xLinkHref":59989},"#MJX-219-TEX-I-1D458"," (in the output layer).",[12,59992,59993,59994,60109,60110,7206,60113,7206,60134,7206,60137],{},"So, ",[24923,59995,59997],{"className":59996,"jax":24927},[24926],[24929,59998,60001,60033],{"style":57964,"xmlns":24932,"width":59999,"height":57966,"role":1930,"focusable":8058,"viewBox":60000,"xmlnsXLink":24936},"5.505ex","0 -899.6 2433.2 1815.9",[24938,60002,60003,60006,60009,60012,60015,60018,60021,60024,60027,60030],{},[24941,60004],{"id":60005,"d":33624},"MJX-220-TEX-I-1D715",[24941,60007],{"id":60008,"d":25869},"MJX-220-TEX-I-1D438",[24941,60010],{"id":60011,"d":37254},"MJX-220-TEX-I-1D464",[24941,60013],{"id":60014,"d":24944},"MJX-220-TEX-N-28",[24941,60016],{"id":60017,"d":25928},"MJX-220-TEX-N-32",[24941,60019],{"id":60020,"d":27339},"MJX-220-TEX-N-2C",[24941,60022],{"id":60023,"d":25914},"MJX-220-TEX-N-31",[24941,60025],{"id":60026,"d":24988},"MJX-220-TEX-N-29",[24941,60028],{"id":60029,"d":42951},"MJX-220-TEX-I-1D458",[24941,60031],{"id":60032,"d":33606},"MJX-220-TEX-I-1D457",[25022,60034,60035],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,60036,60037],{"dataMmlNode":25028},[25022,60038,60039,60051,60107],{"dataMmlNode":27622},[25022,60040,60041,60046],{"dataMmlNode":27626,"transform":58016},[25022,60042,60043],{"dataMmlNode":25942},[25033,60044],{"dataC":33697,"xLinkHref":60045},"#MJX-220-TEX-I-1D715",[25022,60047,60048],{"dataMmlNode":25942,"transform":33701},[25033,60049],{"dataC":25945,"xLinkHref":60050},"#MJX-220-TEX-I-1D438",[25022,60052,60053,60057],{"dataMmlNode":27626,"transform":58029},[25022,60054,60055],{"dataMmlNode":25942},[25033,60056],{"dataC":33697,"xLinkHref":60045},[25022,60058,60059,60064,60091],{"dataMmlNode":33991,"transform":33701},[25022,60060,60061],{"dataMmlNode":25942},[25033,60062],{"dataC":37310,"xLinkHref":60063},"#MJX-220-TEX-I-1D464",[25022,60065,60066,60071,60076,60081,60086],{"dataMmlNode":27203,"transform":48899,"dataMjxTexclass":27205},[25022,60067,60068],{"dataMmlNode":25031},[25033,60069],{"dataC":25035,"xLinkHref":60070},"#MJX-220-TEX-N-28",[25022,60072,60073],{"dataMmlNode":26031,"transform":25040},[25033,60074],{"dataC":26058,"xLinkHref":60075},"#MJX-220-TEX-N-32",[25022,60077,60078],{"dataMmlNode":25031,"transform":41846},[25033,60079],{"dataC":27456,"xLinkHref":60080},"#MJX-220-TEX-N-2C",[25022,60082,60083],{"dataMmlNode":26031,"transform":28333},[25033,60084],{"dataC":26035,"xLinkHref":60085},"#MJX-220-TEX-N-31",[25022,60087,60088],{"dataMmlNode":25031,"transform":41146},[25033,60089],{"dataC":25108,"xLinkHref":60090},"#MJX-220-TEX-N-29",[25022,60092,60093,60098,60102],{"dataMmlNode":27203,"transform":51669,"dataMjxTexclass":27205},[25022,60094,60095],{"dataMmlNode":25942},[25033,60096],{"dataC":43135,"xLinkHref":60097},"#MJX-220-TEX-I-1D458",[25022,60099,60100],{"dataMmlNode":25031,"transform":49747},[25033,60101],{"dataC":27456,"xLinkHref":60080},[25022,60103,60104],{"dataMmlNode":25942,"transform":51680},[25033,60105],{"dataC":33647,"xLinkHref":60106},"#MJX-220-TEX-I-1D457",[27668,60108],{"width":58086,"height":27671,"x":27672,"y":27673}," simply means: ",[7289,60111,60112],{},"\"How much does the total Error change if I tweak this one specific wire connecting hidden node",[7289,60114,60115],{},[24923,60116,60118],{"className":60117,"jax":24927},[24926],[24929,60119,60120,60125],{"style":48965,"xmlns":24932,"width":48966,"height":48967,"role":1930,"focusable":8058,"viewBox":48968,"xmlnsXLink":24936},[24938,60121,60122],{},[24941,60123],{"id":60124,"d":33606},"MJX-221-TEX-I-1D457",[25022,60126,60127],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,60128,60129],{"dataMmlNode":25028},[25022,60130,60131],{"dataMmlNode":25942},[25033,60132],{"dataC":33647,"xLinkHref":60133},"#MJX-221-TEX-I-1D457",[7289,60135,60136],{},"to output 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the other nodes:"," The original Error equation depends on ALL changing outputs. If your network has 3 output nodes, the formula looks like this:\n",[24923,61134,61136],{"className":61135,"jax":24927},[24926],[24929,61137,61139,61174],{"style":24931,"xmlns":24932,"width":25862,"height":31733,"role":1930,"focusable":8058,"viewBox":61138,"xmlnsXLink":24936},"0 -833.9 17487.8 1083.9",[24938,61140,61141,61144,61147,61150,61153,61156,61159,61162,61165,61168,61171],{},[24941,61142],{"id":61143,"d":25869},"MJX-230-TEX-I-1D438",[24941,61145],{"id":61146,"d":24972},"MJX-230-TEX-N-3D",[24941,61148],{"id":61149,"d":24944},"MJX-230-TEX-N-28",[24941,61151],{"id":61152,"d":32316},"MJX-230-TEX-I-1D451",[24941,61154],{"id":61155,"d":25914},"MJX-230-TEX-N-31",[24941,61157],{"id":61158,"d":25910},"MJX-230-TEX-N-2212",[24941,61160],{"id":61161,"d":25885},"MJX-230-TEX-I-1D45C",[24941,61163],{"id":61164,"d":24988},"MJX-230-TEX-N-29",[24941,61166],{"id":61167,"d":25928},"MJX-230-TEX-N-32",[24941,61169],{"id":61170,"d":30313},"MJX-230-TEX-N-2B",[24941,61172],{"id":61173,"d":26455},"MJX-230-TEX-N-33",[25022,61175,61176],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61177,61178,61183,61188,61193,61205,61211,61223,61236,61242,61247,61258,61263,61274,61285,61290,61295,61307,61312,61323],{"dataMmlNode":25028},[25022,61179,61180],{"dataMmlNode":25942},[25033,61181],{"dataC":25945,"xLinkHref":61182},"#MJX-230-TEX-I-1D438",[25022,61184,61185],{"dataMmlNode":25031,"transform":30197},[25033,61186],{"dataC":25078,"xLinkHref":61187},"#MJX-230-TEX-N-3D",[25022,61189,61190],{"dataMmlNode":25031,"transform":30203},[25033,61191],{"dataC":25035,"xLinkHref":61192},"#MJX-230-TEX-N-28",[25022,61194,61195,61200],{"dataMmlNode":26023,"transform":58475},[25022,61196,61197],{"dataMmlNode":25942},[25033,61198],{"dataC":32396,"xLinkHref":61199},"#MJX-230-TEX-I-1D451",[25022,61201,61202],{"dataMmlNode":26031,"transform":49158},[25033,61203],{"dataC":26035,"xLinkHref":61204},"#MJX-230-TEX-N-31",[25022,61206,61208],{"dataMmlNode":25031,"transform":61207},"translate(3665.3,0)",[25033,61209],{"dataC":26019,"xLinkHref":61210},"#MJX-230-TEX-N-2212",[25022,61212,61214,61219],{"dataMmlNode":26023,"transform":61213},"translate(4665.6,0)",[25022,61215,61216],{"dataMmlNode":25942},[25033,61217],{"dataC":25973,"xLinkHref":61218},"#MJX-230-TEX-I-1D45C",[25022,61220,61221],{"dataMmlNode":26031,"transform":49369},[25033,61222],{"dataC":26035,"xLinkHref":61204},[25022,61224,61226,61231],{"dataMmlNode":31565,"transform":61225},"translate(5587.1,0)",[25022,61227,61228],{"dataMmlNode":25031},[25033,61229],{"dataC":25108,"xLinkHref":61230},"#MJX-230-TEX-N-29",[25022,61232,61233],{"dataMmlNode":26031,"transform":37418},[25033,61234],{"dataC":26058,"xLinkHref":61235},"#MJX-230-TEX-N-32",[25022,61237,61239],{"dataMmlNode":25031,"transform":61238},"translate(6634.9,0)",[25033,61240],{"dataC":30393,"xLinkHref":61241},"#MJX-230-TEX-N-2B",[25022,61243,61245],{"dataMmlNode":25031,"transform":61244},"translate(7635.1,0)",[25033,61246],{"dataC":25035,"xLinkHref":61192},[25022,61248,61250,61254],{"dataMmlNode":26023,"transform":61249},"translate(8024.1,0)",[25022,61251,61252],{"dataMmlNode":25942},[25033,61253],{"dataC":32396,"xLinkHref":61199},[25022,61255,61256],{"dataMmlNode":26031,"transform":49158},[25033,61257],{"dataC":26058,"xLinkHref":61235},[25022,61259,61261],{"dataMmlNode":25031,"transform":61260},"translate(9202.9,0)",[25033,61262],{"dataC":26019,"xLinkHref":61210},[25022,61264,61266,61270],{"dataMmlNode":26023,"transform":61265},"translate(10203.1,0)",[25022,61267,61268],{"dataMmlNode":25942},[25033,61269],{"dataC":25973,"xLinkHref":61218},[25022,61271,61272],{"dataMmlNode":26031,"transform":49369},[25033,61273],{"dataC":26058,"xLinkHref":61235},[25022,61275,61277,61281],{"dataMmlNode":31565,"transform":61276},"translate(11124.7,0)",[25022,61278,61279],{"dataMmlNode":25031},[25033,61280],{"dataC":25108,"xLinkHref":61230},[25022,61282,61283],{"dataMmlNode":26031,"transform":37418},[25033,61284],{"dataC":26058,"xLinkHref":61235},[25022,61286,61288],{"dataMmlNode":25031,"transform":61287},"translate(12172.4,0)",[25033,61289],{"dataC":30393,"xLinkHref":61241},[25022,61291,61293],{"dataMmlNode":25031,"transform":61292},"translate(13172.7,0)",[25033,61294],{"dataC":25035,"xLinkHref":61192},[25022,61296,61298,61302],{"dataMmlNode":26023,"transform":61297},"translate(13561.7,0)",[25022,61299,61300],{"dataMmlNode":25942},[25033,61301],{"dataC":32396,"xLinkHref":61199},[25022,61303,61304],{"dataMmlNode":26031,"transform":49158},[25033,61305],{"dataC":26486,"xLinkHref":61306},"#MJX-230-TEX-N-33",[25022,61308,61310],{"dataMmlNode":25031,"transform":61309},"translate(14740.4,0)",[25033,61311],{"dataC":26019,"xLinkHref":61210},[25022,61313,61315,61319],{"dataMmlNode":26023,"transform":61314},"translate(15740.7,0)",[25022,61316,61317],{"dataMmlNode":25942},[25033,61318],{"dataC":25973,"xLinkHref":61218},[25022,61320,61321],{"dataMmlNode":26031,"transform":49369},[25033,61322],{"dataC":26486,"xLinkHref":61306},[25022,61324,61326,61330],{"dataMmlNode":31565,"transform":61325},"translate(16662.2,0)",[25022,61327,61328],{"dataMmlNode":25031},[25033,61329],{"dataC":25108,"xLinkHref":61230},[25022,61331,61332],{"dataMmlNode":26031,"transform":37418},[25033,61333],{"dataC":26058,"xLinkHref":61235},"\nWe only want the partial derivative for the first node (",[24923,61336,61338],{"className":61337,"jax":24927},[24926],[24929,61339,61344,61352],{"style":61340,"xmlns":24932,"width":61341,"height":61342,"role":1930,"focusable":8058,"viewBox":61343,"xmlnsXLink":24936},"vertical-align: -0.339ex;","2.085ex","1.337ex","0 -441 921.6 591",[24938,61345,61346,61349],{},[24941,61347],{"id":61348,"d":25885},"MJX-231-TEX-I-1D45C",[24941,61350],{"id":61351,"d":25914},"MJX-231-TEX-N-31",[25022,61353,61354],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61355,61356],{"dataMmlNode":25028},[25022,61357,61358,61363],{"dataMmlNode":26023},[25022,61359,61360],{"dataMmlNode":25942},[25033,61361],{"dataC":25973,"xLinkHref":61362},"#MJX-231-TEX-I-1D45C",[25022,61364,61365],{"dataMmlNode":26031,"transform":49369},[25033,61366],{"dataC":26035,"xLinkHref":61367},"#MJX-231-TEX-N-31","). So, we lock ",[24923,61370,61372],{"className":61371,"jax":24927},[24926],[24929,61373,61374,61382],{"style":61340,"xmlns":24932,"width":61341,"height":61342,"role":1930,"focusable":8058,"viewBox":61343,"xmlnsXLink":24936},[24938,61375,61376,61379],{},[24941,61377],{"id":61378,"d":25885},"MJX-232-TEX-I-1D45C",[24941,61380],{"id":61381,"d":25928},"MJX-232-TEX-N-32",[25022,61383,61384],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61385,61386],{"dataMmlNode":25028},[25022,61387,61388,61393],{"dataMmlNode":26023},[25022,61389,61390],{"dataMmlNode":25942},[25033,61391],{"dataC":25973,"xLinkHref":61392},"#MJX-232-TEX-I-1D45C",[25022,61394,61395],{"dataMmlNode":26031,"transform":49369},[25033,61396],{"dataC":26058,"xLinkHref":61397},"#MJX-232-TEX-N-32",[24923,61399,61401],{"className":61400,"jax":24927},[24926],[24929,61402,61406,61414],{"style":61403,"xmlns":24932,"width":61341,"height":61404,"role":1930,"focusable":8058,"viewBox":61405,"xmlnsXLink":24936},"vertical-align: -0.375ex;","1.372ex","0 -441 921.6 606.6",[24938,61407,61408,61411],{},[24941,61409],{"id":61410,"d":25885},"MJX-233-TEX-I-1D45C",[24941,61412],{"id":61413,"d":26455},"MJX-233-TEX-N-33",[25022,61415,61416],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61417,61418],{"dataMmlNode":25028},[25022,61419,61420,61425],{"dataMmlNode":26023},[25022,61421,61422],{"dataMmlNode":25942},[25033,61423],{"dataC":25973,"xLinkHref":61424},"#MJX-233-TEX-I-1D45C",[25022,61426,61427],{"dataMmlNode":26031,"transform":49369},[25033,61428],{"dataC":26486,"xLinkHref":61429},"#MJX-233-TEX-N-33"," into place, treating them as constants. This changes the function to ",[24923,61432,61434],{"className":61433,"jax":24927},[24926],[24929,61435,61438,61473],{"style":24931,"xmlns":24932,"width":61436,"height":31733,"role":1930,"focusable":8058,"viewBox":61437,"xmlnsXLink":24936},"22.354ex","0 -833.9 9880.7 1083.9",[24938,61439,61440,61443,61446,61449,61452,61455,61458,61461,61464,61467,61470],{},[24941,61441],{"id":61442,"d":29821},"MJX-234-TEX-I-1D453",[24941,61444],{"id":61445,"d":24944},"MJX-234-TEX-N-28",[24941,61447],{"id":61448,"d":25885},"MJX-234-TEX-I-1D45C",[24941,61450],{"id":61451,"d":25914},"MJX-234-TEX-N-31",[24941,61453],{"id":61454,"d":24988},"MJX-234-TEX-N-29",[24941,61456],{"id":61457,"d":24972},"MJX-234-TEX-N-3D",[24941,61459],{"id":61460,"d":32316},"MJX-234-TEX-I-1D451",[24941,61462],{"id":61463,"d":25910},"MJX-234-TEX-N-2212",[24941,61465],{"id":61466,"d":25928},"MJX-234-TEX-N-32",[24941,61468],{"id":61469,"d":30313},"MJX-234-TEX-N-2B",[24941,61471],{"id":61472,"d":56463},"MJX-234-TEX-I-1D436",[25022,61474,61475],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61476,61477,61482,61487,61499,61505,61511,61516,61528,61534,61545,61557,61563],{"dataMmlNode":25028},[25022,61478,61479],{"dataMmlNode":25942},[25033,61480],{"dataC":29864,"xLinkHref":61481},"#MJX-234-TEX-I-1D453",[25022,61483,61484],{"dataMmlNode":25031,"transform":44094},[25033,61485],{"dataC":25035,"xLinkHref":61486},"#MJX-234-TEX-N-28",[25022,61488,61489,61494],{"dataMmlNode":26023,"transform":53947},[25022,61490,61491],{"dataMmlNode":25942},[25033,61492],{"dataC":25973,"xLinkHref":61493},"#MJX-234-TEX-I-1D45C",[25022,61495,61496],{"dataMmlNode":26031,"transform":49369},[25033,61497],{"dataC":26035,"xLinkHref":61498},"#MJX-234-TEX-N-31",[25022,61500,61502],{"dataMmlNode":25031,"transform":61501},"translate(1860.6,0)",[25033,61503],{"dataC":25108,"xLinkHref":61504},"#MJX-234-TEX-N-29",[25022,61506,61508],{"dataMmlNode":25031,"transform":61507},"translate(2527.3,0)",[25033,61509],{"dataC":25078,"xLinkHref":61510},"#MJX-234-TEX-N-3D",[25022,61512,61514],{"dataMmlNode":25031,"transform":61513},"translate(3583.1,0)",[25033,61515],{"dataC":25035,"xLinkHref":61486},[25022,61517,61519,61524],{"dataMmlNode":26023,"transform":61518},"translate(3972.1,0)",[25022,61520,61521],{"dataMmlNode":25942},[25033,61522],{"dataC":32396,"xLinkHref":61523},"#MJX-234-TEX-I-1D451",[25022,61525,61526],{"dataMmlNode":26031,"transform":49158},[25033,61527],{"dataC":26035,"xLinkHref":61498},[25022,61529,61531],{"dataMmlNode":25031,"transform":61530},"translate(5150.9,0)",[25033,61532],{"dataC":26019,"xLinkHref":61533},"#MJX-234-TEX-N-2212",[25022,61535,61537,61541],{"dataMmlNode":26023,"transform":61536},"translate(6151.1,0)",[25022,61538,61539],{"dataMmlNode":25942},[25033,61540],{"dataC":25973,"xLinkHref":61493},[25022,61542,61543],{"dataMmlNode":26031,"transform":49369},[25033,61544],{"dataC":26035,"xLinkHref":61498},[25022,61546,61548,61552],{"dataMmlNode":31565,"transform":61547},"translate(7072.7,0)",[25022,61549,61550],{"dataMmlNode":25031},[25033,61551],{"dataC":25108,"xLinkHref":61504},[25022,61553,61554],{"dataMmlNode":26031,"transform":37418},[25033,61555],{"dataC":26058,"xLinkHref":61556},"#MJX-234-TEX-N-32",[25022,61558,61560],{"dataMmlNode":25031,"transform":61559},"translate(8120.4,0)",[25033,61561],{"dataC":30393,"xLinkHref":61562},"#MJX-234-TEX-N-2B",[25022,61564,61566],{"dataMmlNode":25942,"transform":61565},"translate(9120.7,0)",[25033,61567],{"dataC":56500,"xLinkHref":61568},"#MJX-234-TEX-I-1D436",[26,61570,61571,61572,61591,61592,61594],{},"Because the derivatives of those frozen constants become ",[24923,61573,61575],{"className":61574,"jax":24927},[24926],[24929,61576,61577,61582],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,61578,61579],{},[24941,61580],{"id":61581,"d":25935},"MJX-235-TEX-N-30",[25022,61583,61584],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61585,61586],{"dataMmlNode":25028},[25022,61587,61588],{"dataMmlNode":26031},[25033,61589],{"dataC":26103,"xLinkHref":61590},"#MJX-235-TEX-N-30",", we are left looking only at:",[16,61593],{},[24923,61595,61597],{"className":61596,"jax":24927},[24926],[24929,61598,61601,61630],{"style":24931,"xmlns":24932,"width":61599,"height":31733,"role":1930,"focusable":8058,"viewBox":61600,"xmlnsXLink":24936},"17.97ex","0 -833.9 7942.8 1083.9",[24938,61602,61603,61606,61609,61612,61615,61618,61621,61624,61627],{},[24941,61604],{"id":61605,"d":29821},"MJX-236-TEX-I-1D453",[24941,61607],{"id":61608,"d":24944},"MJX-236-TEX-N-28",[24941,61610],{"id":61611,"d":25885},"MJX-236-TEX-I-1D45C",[24941,61613],{"id":61614,"d":42951},"MJX-236-TEX-I-1D458",[24941,61616],{"id":61617,"d":24988},"MJX-236-TEX-N-29",[24941,61619],{"id":61620,"d":24972},"MJX-236-TEX-N-3D",[24941,61622],{"id":61623,"d":32316},"MJX-236-TEX-I-1D451",[24941,61625],{"id":61626,"d":25910},"MJX-236-TEX-N-2212",[24941,61628],{"id":61629,"d":25928},"MJX-236-TEX-N-32",[25022,61631,61632],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61633,61634,61639,61644,61656,61662,61668,61673,61685,61691,61702],{"dataMmlNode":25028},[25022,61635,61636],{"dataMmlNode":25942},[25033,61637],{"dataC":29864,"xLinkHref":61638},"#MJX-236-TEX-I-1D453",[25022,61640,61641],{"dataMmlNode":25031,"transform":44094},[25033,61642],{"dataC":25035,"xLinkHref":61643},"#MJX-236-TEX-N-28",[25022,61645,61646,61651],{"dataMmlNode":26023,"transform":53947},[25022,61647,61648],{"dataMmlNode":25942},[25033,61649],{"dataC":25973,"xLinkHref":61650},"#MJX-236-TEX-I-1D45C",[25022,61652,61653],{"dataMmlNode":25942,"transform":49369},[25033,61654],{"dataC":43135,"xLinkHref":61655},"#MJX-236-TEX-I-1D458",[25022,61657,61659],{"dataMmlNode":25031,"transform":61658},"translate(1875.4,0)",[25033,61660],{"dataC":25108,"xLinkHref":61661},"#MJX-236-TEX-N-29",[25022,61663,61665],{"dataMmlNode":25031,"transform":61664},"translate(2542.2,0)",[25033,61666],{"dataC":25078,"xLinkHref":61667},"#MJX-236-TEX-N-3D",[25022,61669,61671],{"dataMmlNode":25031,"transform":61670},"translate(3598,0)",[25033,61672],{"dataC":25035,"xLinkHref":61643},[25022,61674,61676,61681],{"dataMmlNode":26023,"transform":61675},"translate(3987,0)",[25022,61677,61678],{"dataMmlNode":25942},[25033,61679],{"dataC":32396,"xLinkHref":61680},"#MJX-236-TEX-I-1D451",[25022,61682,61683],{"dataMmlNode":25942,"transform":49158},[25033,61684],{"dataC":43135,"xLinkHref":61655},[25022,61686,61688],{"dataMmlNode":25031,"transform":61687},"translate(5180.6,0)",[25033,61689],{"dataC":26019,"xLinkHref":61690},"#MJX-236-TEX-N-2212",[25022,61692,61694,61698],{"dataMmlNode":26023,"transform":61693},"translate(6180.8,0)",[25022,61695,61696],{"dataMmlNode":25942},[25033,61697],{"dataC":25973,"xLinkHref":61650},[25022,61699,61700],{"dataMmlNode":25942,"transform":49369},[25033,61701],{"dataC":43135,"xLinkHref":61655},[25022,61703,61705,61709],{"dataMmlNode":31565,"transform":61704},"translate(7117.2,0)",[25022,61706,61707],{"dataMmlNode":25031},[25033,61708],{"dataC":25108,"xLinkHref":61661},[25022,61710,61711],{"dataMmlNode":26031,"transform":37418},[25033,61712],{"dataC":26058,"xLinkHref":61713},"#MJX-236-TEX-N-32",[26,61715,61716,61719,61720,61739,61740,61742],{},[106,61717,61718],{},"Apply the Power Rule:"," To take the derivative of something squared, we bring the ",[24923,61721,61723],{"className":61722,"jax":24927},[24926],[24929,61724,61725,61730],{"style":27802,"xmlns":24932,"width":35168,"height":35169,"role":1930,"focusable":8058,"viewBox":35170,"xmlnsXLink":24936},[24938,61726,61727],{},[24941,61728],{"id":61729,"d":25928},"MJX-237-TEX-N-32",[25022,61731,61732],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61733,61734],{"dataMmlNode":25028},[25022,61735,61736],{"dataMmlNode":26031},[25033,61737],{"dataC":26058,"xLinkHref":61738},"#MJX-237-TEX-N-32"," down to the front:",[16,61741],{},[24923,61743,61745],{"className":61744,"jax":24927},[24926],[24929,61746,61749,61772],{"style":24931,"xmlns":24932,"width":61747,"height":24934,"role":1930,"focusable":8058,"viewBox":61748,"xmlnsXLink":24936},"9.973ex","0 -750 4408.2 1000",[24938,61750,61751,61754,61757,61760,61763,61766,61769],{},[24941,61752],{"id":61753,"d":25928},"MJX-238-TEX-N-32",[24941,61755],{"id":61756,"d":24944},"MJX-238-TEX-N-28",[24941,61758],{"id":61759,"d":32316},"MJX-238-TEX-I-1D451",[24941,61761],{"id":61762,"d":42951},"MJX-238-TEX-I-1D458",[24941,61764],{"id":61765,"d":25910},"MJX-238-TEX-N-2212",[24941,61767],{"id":61768,"d":25885},"MJX-238-TEX-I-1D45C",[24941,61770],{"id":61771,"d":24988},"MJX-238-TEX-N-29",[25022,61773,61774],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61775,61776,61781,61786,61798,61804,61816],{"dataMmlNode":25028},[25022,61777,61778],{"dataMmlNode":26031},[25033,61779],{"dataC":26058,"xLinkHref":61780},"#MJX-238-TEX-N-32",[25022,61782,61783],{"dataMmlNode":25031,"transform":26394},[25033,61784],{"dataC":25035,"xLinkHref":61785},"#MJX-238-TEX-N-28",[25022,61787,61788,61793],{"dataMmlNode":26023,"transform":41846},[25022,61789,61790],{"dataMmlNode":25942},[25033,61791],{"dataC":32396,"xLinkHref":61792},"#MJX-238-TEX-I-1D451",[25022,61794,61795],{"dataMmlNode":25942,"transform":49158},[25033,61796],{"dataC":43135,"xLinkHref":61797},"#MJX-238-TEX-I-1D458",[25022,61799,61801],{"dataMmlNode":25031,"transform":61800},"translate(2082.6,0)",[25033,61802],{"dataC":26019,"xLinkHref":61803},"#MJX-238-TEX-N-2212",[25022,61805,61807,61812],{"dataMmlNode":26023,"transform":61806},"translate(3082.8,0)",[25022,61808,61809],{"dataMmlNode":25942},[25033,61810],{"dataC":25973,"xLinkHref":61811},"#MJX-238-TEX-I-1D45C",[25022,61813,61814],{"dataMmlNode":25942,"transform":49369},[25033,61815],{"dataC":43135,"xLinkHref":61797},[25022,61817,61819],{"dataMmlNode":25031,"transform":61818},"translate(4019.2,0)",[25033,61820],{"dataC":25108,"xLinkHref":61821},"#MJX-238-TEX-N-29",[26,61823,61824,61827,61828,61860,61861,54897,61880,7653,61919,6959],{},[106,61825,61826],{},"The Inner Chain Rule:"," In calculus, if you take the derivative of the outside (the square), you must multiply it by the derivative of the inside. The derivative of ",[24923,61829,61831],{"className":61830,"jax":24927},[24926],[24929,61832,61836,61844],{"style":27744,"xmlns":24932,"width":61833,"height":61834,"role":1930,"focusable":8058,"viewBox":61835,"xmlnsXLink":24936},"2.198ex","1.927ex","0 -694 971.4 851.8",[24938,61837,61838,61841],{},[24941,61839],{"id":61840,"d":32316},"MJX-239-TEX-I-1D451",[24941,61842],{"id":61843,"d":42951},"MJX-239-TEX-I-1D458",[25022,61845,61846],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61847,61848],{"dataMmlNode":25028},[25022,61849,61850,61855],{"dataMmlNode":26023},[25022,61851,61852],{"dataMmlNode":25942},[25033,61853],{"dataC":32396,"xLinkHref":61854},"#MJX-239-TEX-I-1D451",[25022,61856,61857],{"dataMmlNode":25942,"transform":49158},[25033,61858],{"dataC":43135,"xLinkHref":61859},"#MJX-239-TEX-I-1D458"," (a frozen target number) is ",[24923,61862,61864],{"className":61863,"jax":24927},[24926],[24929,61865,61866,61871],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,61867,61868],{},[24941,61869],{"id":61870,"d":25935},"MJX-240-TEX-N-30",[25022,61872,61873],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61874,61875],{"dataMmlNode":25028},[25022,61876,61877],{"dataMmlNode":26031},[25033,61878],{"dataC":26103,"xLinkHref":61879},"#MJX-240-TEX-N-30",[24923,61881,61883],{"className":61882,"jax":24927},[24926],[24929,61884,61887,61898],{"style":27744,"xmlns":24932,"width":26570,"height":61885,"role":1930,"focusable":8058,"viewBox":61886,"xmlnsXLink":24936},"1.676ex","0 -583 1714.4 740.8",[24938,61888,61889,61892,61895],{},[24941,61890],{"id":61891,"d":25910},"MJX-241-TEX-N-2212",[24941,61893],{"id":61894,"d":25885},"MJX-241-TEX-I-1D45C",[24941,61896],{"id":61897,"d":42951},"MJX-241-TEX-I-1D458",[25022,61899,61900],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61901,61902,61907],{"dataMmlNode":25028},[25022,61903,61904],{"dataMmlNode":25031},[25033,61905],{"dataC":26019,"xLinkHref":61906},"#MJX-241-TEX-N-2212",[25022,61908,61909,61914],{"dataMmlNode":26023,"transform":25049},[25022,61910,61911],{"dataMmlNode":25942},[25033,61912],{"dataC":25973,"xLinkHref":61913},"#MJX-241-TEX-I-1D45C",[25022,61915,61916],{"dataMmlNode":25942,"transform":49369},[25033,61917],{"dataC":43135,"xLinkHref":61918},"#MJX-241-TEX-I-1D458",[24923,61920,61922],{"className":61921,"jax":24927},[24926],[24929,61923,61924,61932],{"style":26868,"xmlns":24932,"width":29568,"height":39628,"role":1930,"focusable":8058,"viewBox":52648,"xmlnsXLink":24936},[24938,61925,61926,61929],{},[24941,61927],{"id":61928,"d":25910},"MJX-242-TEX-N-2212",[24941,61930],{"id":61931,"d":25914},"MJX-242-TEX-N-31",[25022,61933,61934],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61935,61936,61941],{"dataMmlNode":25028},[25022,61937,61938],{"dataMmlNode":25031},[25033,61939],{"dataC":26019,"xLinkHref":61940},"#MJX-242-TEX-N-2212",[25022,61942,61943],{"dataMmlNode":26031,"transform":25049},[25033,61944],{"dataC":26035,"xLinkHref":61945},"#MJX-242-TEX-N-31",[26,61947,61948,61949,7024,61976,61978,7206,61981],{},"Multiply the result by ",[24923,61950,61952],{"className":61951,"jax":24927},[24926],[24929,61953,61954,61962],{"style":26868,"xmlns":24932,"width":29568,"height":39628,"role":1930,"focusable":8058,"viewBox":52648,"xmlnsXLink":24936},[24938,61955,61956,61959],{},[24941,61957],{"id":61958,"d":25910},"MJX-243-TEX-N-2212",[24941,61960],{"id":61961,"d":25914},"MJX-243-TEX-N-31",[25022,61963,61964],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,61965,61966,61971],{"dataMmlNode":25028},[25022,61967,61968],{"dataMmlNode":25031},[25033,61969],{"dataC":26019,"xLinkHref":61970},"#MJX-243-TEX-N-2212",[25022,61972,61973],{"dataMmlNode":26031,"transform":25049},[25033,61974],{"dataC":26035,"xLinkHref":61975},"#MJX-243-TEX-N-31",[16,61977],{},[106,61979,61980],{},"Answer 1:",[24923,61982,61984],{"className":61983,"jax":24927},[24926],[24929,61985,61988,62011],{"style":24931,"xmlns":24932,"width":61986,"height":24934,"role":1930,"focusable":8058,"viewBox":61987,"xmlnsXLink":24936},"11.734ex","0 -750 5186.2 1000",[24938,61989,61990,61993,61996,61999,62002,62005,62008],{},[24941,61991],{"id":61992,"d":25910},"MJX-244-TEX-N-2212",[24941,61994],{"id":61995,"d":25928},"MJX-244-TEX-N-32",[24941,61997],{"id":61998,"d":24944},"MJX-244-TEX-N-28",[24941,62000],{"id":62001,"d":32316},"MJX-244-TEX-I-1D451",[24941,62003],{"id":62004,"d":42951},"MJX-244-TEX-I-1D458",[24941,62006],{"id":62007,"d":25885},"MJX-244-TEX-I-1D45C",[24941,62009],{"id":62010,"d":24988},"MJX-244-TEX-N-29",[25022,62012,62013],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62014,62015,62020,62025,62030,62042,62047,62059],{"dataMmlNode":25028},[25022,62016,62017],{"dataMmlNode":25031},[25033,62018],{"dataC":26019,"xLinkHref":62019},"#MJX-244-TEX-N-2212",[25022,62021,62022],{"dataMmlNode":26031,"transform":25049},[25033,62023],{"dataC":26058,"xLinkHref":62024},"#MJX-244-TEX-N-32",[25022,62026,62027],{"dataMmlNode":25031,"transform":26835},[25033,62028],{"dataC":25035,"xLinkHref":62029},"#MJX-244-TEX-N-28",[25022,62031,62032,62037],{"dataMmlNode":26023,"transform":41146},[25022,62033,62034],{"dataMmlNode":25942},[25033,62035],{"dataC":32396,"xLinkHref":62036},"#MJX-244-TEX-I-1D451",[25022,62038,62039],{"dataMmlNode":25942,"transform":49158},[25033,62040],{"dataC":43135,"xLinkHref":62041},"#MJX-244-TEX-I-1D458",[25022,62043,62045],{"dataMmlNode":25031,"transform":62044},"translate(2860.6,0)",[25033,62046],{"dataC":26019,"xLinkHref":62019},[25022,62048,62050,62055],{"dataMmlNode":26023,"transform":62049},"translate(3860.8,0)",[25022,62051,62052],{"dataMmlNode":25942},[25033,62053],{"dataC":25973,"xLinkHref":62054},"#MJX-244-TEX-I-1D45C",[25022,62056,62057],{"dataMmlNode":25942,"transform":49369},[25033,62058],{"dataC":43135,"xLinkHref":62041},[25022,62060,62062],{"dataMmlNode":25031,"transform":62061},"translate(4797.2,0)",[25033,62063],{"dataC":25108,"xLinkHref":62064},"#MJX-244-TEX-N-29",[12,62066,62067],{},"Some textbooks will drop the 2 in front, as it won't affect the final target (we still want error to be 0), and the 2 is absorbed into the learning rate anyways, which we will talk about later.",[7351,62069,62070,62104,62140,62194,62261],{},[26236,62071,62073,62074,62103],{"id":62072},"why-is-frozen","Why is ",[24923,62075,62077],{"className":62076,"jax":24927},[24926],[24929,62078,62079,62087],{"style":27744,"xmlns":24932,"width":61833,"height":61834,"role":1930,"focusable":8058,"viewBox":61835,"xmlnsXLink":24936},[24938,62080,62081,62084],{},[24941,62082],{"id":62083,"d":32316},"MJX-245-TEX-I-1D451",[24941,62085],{"id":62086,"d":42951},"MJX-245-TEX-I-1D458",[25022,62088,62089],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62090,62091],{"dataMmlNode":25028},[25022,62092,62093,62098],{"dataMmlNode":26023},[25022,62094,62095],{"dataMmlNode":25942},[25033,62096],{"dataC":32396,"xLinkHref":62097},"#MJX-245-TEX-I-1D451",[25022,62099,62100],{"dataMmlNode":25942,"transform":49158},[25033,62101],{"dataC":43135,"xLinkHref":62102},"#MJX-245-TEX-I-1D458"," frozen?",[12,62105,62106,62107,62136,62137,62139],{},"Remember what ",[24923,62108,62110],{"className":62109,"jax":24927},[24926],[24929,62111,62112,62120],{"style":27744,"xmlns":24932,"width":61833,"height":61834,"role":1930,"focusable":8058,"viewBox":61835,"xmlnsXLink":24936},[24938,62113,62114,62117],{},[24941,62115],{"id":62116,"d":32316},"MJX-246-TEX-I-1D451",[24941,62118],{"id":62119,"d":42951},"MJX-246-TEX-I-1D458",[25022,62121,62122],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62123,62124],{"dataMmlNode":25028},[25022,62125,62126,62131],{"dataMmlNode":26023},[25022,62127,62128],{"dataMmlNode":25942},[25033,62129],{"dataC":32396,"xLinkHref":62130},"#MJX-246-TEX-I-1D451",[25022,62132,62133],{"dataMmlNode":25942,"transform":49158},[25033,62134],{"dataC":43135,"xLinkHref":62135},"#MJX-246-TEX-I-1D458"," actually is in the real world: it is your ",[106,62138,13635],{},"esired target. It is the \"ground truth\" label from your dataset.",[12,62141,62142,62143,19180,62145,6959],{},"If you feed the network a picture of a dog, and the label for dog is ",[6979,62144,8855],{},[24923,62146,62148],{"className":62147,"jax":24927},[24926],[24929,62149,62152,62166],{"style":27744,"xmlns":24932,"width":62150,"height":61834,"role":1930,"focusable":8058,"viewBox":62151,"xmlnsXLink":24936},"6.346ex","0 -694 2805 851.8",[24938,62153,62154,62157,62160,62163],{},[24941,62155],{"id":62156,"d":32316},"MJX-247-TEX-I-1D451",[24941,62158],{"id":62159,"d":42951},"MJX-247-TEX-I-1D458",[24941,62161],{"id":62162,"d":24972},"MJX-247-TEX-N-3D",[24941,62164],{"id":62165,"d":25914},"MJX-247-TEX-N-31",[25022,62167,62168],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62169,62170,62182,62188],{"dataMmlNode":25028},[25022,62171,62172,62177],{"dataMmlNode":26023},[25022,62173,62174],{"dataMmlNode":25942},[25033,62175],{"dataC":32396,"xLinkHref":62176},"#MJX-247-TEX-I-1D451",[25022,62178,62179],{"dataMmlNode":25942,"transform":49158},[25033,62180],{"dataC":43135,"xLinkHref":62181},"#MJX-247-TEX-I-1D458",[25022,62183,62185],{"dataMmlNode":25031,"transform":62184},"translate(1249.2,0)",[25033,62186],{"dataC":25078,"xLinkHref":62187},"#MJX-247-TEX-N-3D",[25022,62189,62191],{"dataMmlNode":26031,"transform":62190},"translate(2305,0)",[25033,62192],{"dataC":26035,"xLinkHref":62193},"#MJX-247-TEX-N-31",[12,62195,62196,62197,62226,62227,62230,62231,62260],{},"The output (",[24923,62198,62200],{"className":62199,"jax":24927},[24926],[24929,62201,62202,62210],{"style":27744,"xmlns":24932,"width":60992,"height":60993,"role":1930,"focusable":8058,"viewBox":60994,"xmlnsXLink":24936},[24938,62203,62204,62207],{},[24941,62205],{"id":62206,"d":25885},"MJX-248-TEX-I-1D45C",[24941,62208],{"id":62209,"d":42951},"MJX-248-TEX-I-1D458",[25022,62211,62212],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62213,62214],{"dataMmlNode":25028},[25022,62215,62216,62221],{"dataMmlNode":26023},[25022,62217,62218],{"dataMmlNode":25942},[25033,62219],{"dataC":25973,"xLinkHref":62220},"#MJX-248-TEX-I-1D45C",[25022,62222,62223],{"dataMmlNode":25942,"transform":49369},[25033,62224],{"dataC":43135,"xLinkHref":62225},"#MJX-248-TEX-I-1D458",") is going to change constantly as the network learns and tweaks its weights. But the picture is ",[7289,62228,62229],{},"always"," going to be a dog. The target ",[24923,62232,62234],{"className":62233,"jax":24927},[24926],[24929,62235,62236,62244],{"style":27744,"xmlns":24932,"width":61833,"height":61834,"role":1930,"focusable":8058,"viewBox":61835,"xmlnsXLink":24936},[24938,62237,62238,62241],{},[24941,62239],{"id":62240,"d":32316},"MJX-249-TEX-I-1D451",[24941,62242],{"id":62243,"d":42951},"MJX-249-TEX-I-1D458",[25022,62245,62246],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62247,62248],{"dataMmlNode":25028},[25022,62249,62250,62255],{"dataMmlNode":26023},[25022,62251,62252],{"dataMmlNode":25942},[25033,62253],{"dataC":32396,"xLinkHref":62254},"#MJX-249-TEX-I-1D451",[25022,62256,62257],{"dataMmlNode":25942,"transform":49158},[25033,62258],{"dataC":43135,"xLinkHref":62259},"#MJX-249-TEX-I-1D458"," never changes; it is a permanent, hardcoded number for that specific training sample.",[12,62262,62263,62264,62293,62294,6959],{},"Because ",[24923,62265,62267],{"className":62266,"jax":24927},[24926],[24929,62268,62269,62277],{"style":27744,"xmlns":24932,"width":61833,"height":61834,"role":1930,"focusable":8058,"viewBox":61835,"xmlnsXLink":24936},[24938,62270,62271,62274],{},[24941,62272],{"id":62273,"d":32316},"MJX-250-TEX-I-1D451",[24941,62275],{"id":62276,"d":42951},"MJX-250-TEX-I-1D458",[25022,62278,62279],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62280,62281],{"dataMmlNode":25028},[25022,62282,62283,62288],{"dataMmlNode":26023},[25022,62284,62285],{"dataMmlNode":25942},[25033,62286],{"dataC":32396,"xLinkHref":62287},"#MJX-250-TEX-I-1D451",[25022,62289,62290],{"dataMmlNode":25942,"transform":49158},[25033,62291],{"dataC":43135,"xLinkHref":62292},"#MJX-250-TEX-I-1D458"," is just a plain, unchanging number, the rules of calculus say that its derivative is exactly ",[24923,62295,62297],{"className":62296,"jax":24927},[24926],[24929,62298,62299,62304],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,62300,62301],{},[24941,62302],{"id":62303,"d":25935},"MJX-251-TEX-N-30",[25022,62305,62306],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62307,62308],{"dataMmlNode":25028},[25022,62309,62310],{"dataMmlNode":26031},[25033,62311],{"dataC":26103,"xLinkHref":62312},"#MJX-251-TEX-N-30",[26236,62314,62316,62317],{"id":62315},"step-2-solve","Step 2: Solve ",[24923,62318,62320],{"className":62319,"jax":24927},[24926],[24929,62321,62326,62355],{"style":62322,"xmlns":24932,"width":62323,"height":62324,"role":1930,"focusable":8058,"viewBox":62325,"xmlnsXLink":24936},"vertical-align: -1.855ex;","5.762ex","4.02ex","0 -957.1 2547 1776.9",[24938,62327,62328,62331,62334,62337,62340,62343,62346,62349,62352],{},[24941,62329],{"id":62330,"d":33624},"MJX-252-TEX-I-1D715",[24941,62332],{"id":62333,"d":25885},"MJX-252-TEX-I-1D45C",[24941,62335],{"id":62336,"d":42951},"MJX-252-TEX-I-1D458",[24941,62338],{"id":62339,"d":25873},"MJX-252-TEX-I-1D45B",[24941,62341],{"id":62342,"d":27397},"MJX-252-TEX-I-1D452",[24941,62344],{"id":62345,"d":25877},"MJX-252-TEX-I-1D461",[24941,62347],{"id":62348,"d":24944},"MJX-252-TEX-N-28",[24941,62350],{"id":62351,"d":25928},"MJX-252-TEX-N-32",[24941,62353],{"id":62354,"d":24988},"MJX-252-TEX-N-29",[25022,62356,62357],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62358,62359],{"dataMmlNode":25028},[25022,62360,62361,62381,62425],{"dataMmlNode":27622},[25022,62362,62364,62369],{"dataMmlNode":27626,"transform":62363},"translate(742.3,451.6) scale(0.707)",[25022,62365,62366],{"dataMmlNode":25942},[25033,62367],{"dataC":33697,"xLinkHref":62368},"#MJX-252-TEX-I-1D715",[25022,62370,62371,62376],{"dataMmlNode":26023,"transform":33701},[25022,62372,62373],{"dataMmlNode":25942},[25033,62374],{"dataC":25973,"xLinkHref":62375},"#MJX-252-TEX-I-1D45C",[25022,62377,62378],{"dataMmlNode":25942,"transform":49369},[25033,62379],{"dataC":43135,"xLinkHref":62380},"#MJX-252-TEX-I-1D458",[25022,62382,62383,62387,62392,62397],{"dataMmlNode":27626,"transform":58029},[25022,62384,62385],{"dataMmlNode":25942},[25033,62386],{"dataC":33697,"xLinkHref":62368},[25022,62388,62389],{"dataMmlNode":25942,"transform":33701},[25033,62390],{"dataC":25952,"xLinkHref":62391},"#MJX-252-TEX-I-1D45B",[25022,62393,62394],{"dataMmlNode":25942,"transform":58759},[25033,62395],{"dataC":27596,"xLinkHref":62396},"#MJX-252-TEX-I-1D452",[25022,62398,62399,62404,62421],{"dataMmlNode":33991,"transform":58765},[25022,62400,62401],{"dataMmlNode":25942},[25033,62402],{"dataC":25959,"xLinkHref":62403},"#MJX-252-TEX-I-1D461",[25022,62405,62406,62411,62416],{"dataMmlNode":27203,"transform":51003,"dataMjxTexclass":27205},[25022,62407,62408],{"dataMmlNode":25031},[25033,62409],{"dataC":25035,"xLinkHref":62410},"#MJX-252-TEX-N-28",[25022,62412,62413],{"dataMmlNode":26031,"transform":25040},[25033,62414],{"dataC":26058,"xLinkHref":62415},"#MJX-252-TEX-N-32",[25022,62417,62418],{"dataMmlNode":25031,"transform":41846},[25033,62419],{"dataC":25108,"xLinkHref":62420},"#MJX-252-TEX-N-29",[25022,62422,62423],{"dataMmlNode":25942,"transform":51596},[25033,62424],{"dataC":43135,"xLinkHref":62380},[27668,62426],{"width":62427,"height":27671,"x":27672,"y":27673},2307,[12,62429,62430],{},"We want to find the derivative of the Output with respect to the Net Input.",[12,62432,62433,7206,62435],{},[106,62434,61023],{},[24923,62436,62438],{"className":62437,"jax":24927},[24926],[24929,62439,62440,62472],{"style":60601,"xmlns":24932,"width":60602,"height":60603,"role":1930,"focusable":8058,"viewBox":60604,"xmlnsXLink":24936},[24938,62441,62442,62445,62448,62451,62454,62457,62460,62463,62466,62469],{},[24941,62443],{"id":62444,"d":25885},"MJX-253-TEX-I-1D45C",[24941,62446],{"id":62447,"d":42951},"MJX-253-TEX-I-1D458",[24941,62449],{"id":62450,"d":24972},"MJX-253-TEX-N-3D",[24941,62452],{"id":62453,"d":25900},"MJX-253-TEX-I-1D446",[24941,62455],{"id":62456,"d":24944},"MJX-253-TEX-N-28",[24941,62458],{"id":62459,"d":25873},"MJX-253-TEX-I-1D45B",[24941,62461],{"id":62462,"d":27397},"MJX-253-TEX-I-1D452",[24941,62464],{"id":62465,"d":25877},"MJX-253-TEX-I-1D461",[24941,62467],{"id":62468,"d":25928},"MJX-253-TEX-N-32",[24941,62470],{"id":62471,"d":24988},"MJX-253-TEX-N-29",[25022,62473,62474],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62475,62476,62488,62493,62498,62503,62508,62513,62540],{"dataMmlNode":25028},[25022,62477,62478,62483],{"dataMmlNode":26023},[25022,62479,62480],{"dataMmlNode":25942},[25033,62481],{"dataC":25973,"xLinkHref":62482},"#MJX-253-TEX-I-1D45C",[25022,62484,62485],{"dataMmlNode":25942,"transform":49369},[25033,62486],{"dataC":43135,"xLinkHref":62487},"#MJX-253-TEX-I-1D458",[25022,62489,62490],{"dataMmlNode":25031,"transform":60655},[25033,62491],{"dataC":25078,"xLinkHref":62492},"#MJX-253-TEX-N-3D",[25022,62494,62495],{"dataMmlNode":25942,"transform":60661},[25033,62496],{"dataC":26000,"xLinkHref":62497},"#MJX-253-TEX-I-1D446",[25022,62499,62500],{"dataMmlNode":25031,"transform":60667},[25033,62501],{"dataC":25035,"xLinkHref":62502},"#MJX-253-TEX-N-28",[25022,62504,62505],{"dataMmlNode":25942,"transform":60673},[25033,62506],{"dataC":25952,"xLinkHref":62507},"#MJX-253-TEX-I-1D45B",[25022,62509,62510],{"dataMmlNode":25942,"transform":60679},[25033,62511],{"dataC":27596,"xLinkHref":62512},"#MJX-253-TEX-I-1D452",[25022,62514,62515,62520,62536],{"dataMmlNode":33991,"transform":60685},[25022,62516,62517],{"dataMmlNode":25942},[25033,62518],{"dataC":25959,"xLinkHref":62519},"#MJX-253-TEX-I-1D461",[25022,62521,62522,62526,62531],{"dataMmlNode":27203,"transform":51003,"dataMjxTexclass":27205},[25022,62523,62524],{"dataMmlNode":25031},[25033,62525],{"dataC":25035,"xLinkHref":62502},[25022,62527,62528],{"dataMmlNode":26031,"transform":25040},[25033,62529],{"dataC":26058,"xLinkHref":62530},"#MJX-253-TEX-N-32",[25022,62532,62533],{"dataMmlNode":25031,"transform":41846},[25033,62534],{"dataC":25108,"xLinkHref":62535},"#MJX-253-TEX-N-29",[25022,62537,62538],{"dataMmlNode":25942,"transform":51596},[25033,62539],{"dataC":43135,"xLinkHref":62487},[25022,62541,62542],{"dataMmlNode":25031,"transform":60713},[25033,62543],{"dataC":25108,"xLinkHref":62535},[3603,62545,62546,62569,62625],{},[26,62547,62548,62549,62568],{},"This one requires almost no math! ",[24923,62550,62552],{"className":62551,"jax":24927},[24926],[24929,62553,62554,62559],{"style":27778,"xmlns":24932,"width":27779,"height":27780,"role":1930,"focusable":8058,"viewBox":27781,"xmlnsXLink":24936},[24938,62555,62556],{},[24941,62557],{"id":62558,"d":25900},"MJX-254-TEX-I-1D446",[25022,62560,62561],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62562,62563],{"dataMmlNode":25028},[25022,62564,62565],{"dataMmlNode":25942},[25033,62566],{"dataC":26000,"xLinkHref":62567},"#MJX-254-TEX-I-1D446"," just represents the activation function.",[26,62570,62571,62572,62591,62592,6959],{},"In calculus, the universal shorthand for \"the derivative of function ",[24923,62573,62575],{"className":62574,"jax":24927},[24926],[24929,62576,62577,62582],{"style":27778,"xmlns":24932,"width":27779,"height":27780,"role":1930,"focusable":8058,"viewBox":27781,"xmlnsXLink":24936},[24938,62578,62579],{},[24941,62580],{"id":62581,"d":25900},"MJX-255-TEX-I-1D446",[25022,62583,62584],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62585,62586],{"dataMmlNode":25028},[25022,62587,62588],{"dataMmlNode":25942},[25033,62589],{"dataC":26000,"xLinkHref":62590},"#MJX-255-TEX-I-1D446","\" is simply writing it with a prime symbol: ",[24923,62593,62595],{"className":62594,"jax":24927},[24926],[24929,62596,62600,62608],{"style":27778,"xmlns":24932,"width":62597,"height":62598,"role":1930,"focusable":8058,"viewBox":62599,"xmlnsXLink":24936},"2.204ex","1.767ex","0 -759 974.1 781",[24938,62601,62602,62605],{},[24941,62603],{"id":62604,"d":25900},"MJX-256-TEX-I-1D446",[24941,62606],{"id":62607,"d":31558},"MJX-256-TEX-V-2032",[25022,62609,62610],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,62611,62612],{"dataMmlNode":25028},[25022,62613,62614,62619],{"dataMmlNode":31565},[25022,62615,62616],{"dataMmlNode":25942},[25033,62617],{"dataC":26000,"xLinkHref":62618},"#MJX-256-TEX-I-1D446",[25022,62620,62622],{"dataMmlNode":25031,"transform":62621},"translate(729.6,363) scale(0.707)",[25033,62623],{"dataC":31576,"xLinkHref":62624},"#MJX-256-TEX-V-2032",[26,62626,62627,7206,62630],{},[106,62628,62629],{},"Answer 2:",[24923,62631,62633],{"className":62632,"jax":24927},[24926],[24929,62634,62637,62672],{"style":36892,"xmlns":24932,"width":62635,"height":36894,"role":1930,"focusable":8058,"viewBox":62636,"xmlnsXLink":24936},"10.743ex","0 -1149.5 4748.4 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3: Solve ",[24923,62745,62747],{"className":62746,"jax":24927},[24926],[24929,62748,62751,62789],{"style":57964,"xmlns":24932,"width":62323,"height":62749,"role":1930,"focusable":8058,"viewBox":62750,"xmlnsXLink":24936},"5.059ex","0 -1319.8 2547 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We pretend the input (",[24923,63459,63461],{"className":63460,"jax":24927},[24926],[24929,63462,63463,63480],{"style":33830,"xmlns":24932,"width":51376,"height":33832,"role":1930,"focusable":8058,"viewBox":51377,"xmlnsXLink":24936},[24938,63464,63465,63468,63471,63474,63477],{},[24941,63466],{"id":63467,"d":32148},"MJX-264-TEX-I-1D465",[24941,63469],{"id":63470,"d":24944},"MJX-264-TEX-N-28",[24941,63472],{"id":63473,"d":25914},"MJX-264-TEX-N-31",[24941,63475],{"id":63476,"d":24988},"MJX-264-TEX-N-29",[24941,63478],{"id":63479,"d":33606},"MJX-264-TEX-I-1D457",[25022,63481,63482],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,63483,63484],{"dataMmlNode":25028},[25022,63485,63486,63491,63508],{"dataMmlNode":33991},[25022,63487,63488],{"dataMmlNode":25942},[25033,63489],{"dataC":32181,"xLinkHref":63490},"#MJX-264-TEX-I-1D465",[25022,63492,63493,63498,63503],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,63494,63495],{"dataMmlNode":25031},[25033,63496],{"dataC":25035,"xLinkHref":63497},"#MJX-264-TEX-N-28",[25022,63499,63500],{"dataMmlNode":26031,"transform":25040},[25033,63501],{"dataC":26035,"xLinkHref":63502},"#MJX-264-TEX-N-31",[25022,63504,63505],{"dataMmlNode":25031,"transform":41846},[25033,63506],{"dataC":25108,"xLinkHref":63507},"#MJX-264-TEX-N-29",[25022,63509,63510],{"dataMmlNode":25942,"transform":34014},[25033,63511],{"dataC":33647,"xLinkHref":63512},"#MJX-264-TEX-I-1D457",") is a boring, frozen number like ",[24923,63515,63517],{"className":63516,"jax":24927},[24926],[24929,63518,63519,63527],{"style":27778,"xmlns":24932,"width":24934,"height":35776,"role":1930,"focusable":8058,"viewBox":55544,"xmlnsXLink":24936},[24938,63520,63521,63524],{},[24941,63522],{"id":63523,"d":25914},"MJX-265-TEX-N-31",[24941,63525],{"id":63526,"d":25935},"MJX-265-TEX-N-30",[25022,63528,63529],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,63530,63531],{"dataMmlNode":25028},[25022,63532,63533,63536],{"dataMmlNode":26031},[25033,63534],{"dataC":26035,"xLinkHref":63535},"#MJX-265-TEX-N-31",[25033,63537],{"dataC":26103,"xLinkHref":63538,"transform":26394},"#MJX-265-TEX-N-30",[26,63540,52466,63541,35162,63576,63601,63602,35162,63641,6959,63660,63662,7206,63665],{},[24923,63542,63544],{"className":63543,"jax":24927},[24926],[24929,63545,63548,63559],{"style":27778,"xmlns":24932,"width":63546,"height":35776,"role":1930,"focusable":8058,"viewBox":63547,"xmlnsXLink":24936},"3.882ex","0 -666 1716 688",[24938,63549,63550,63553,63556],{},[24941,63551],{"id":63552,"d":25914},"MJX-266-TEX-N-31",[24941,63554],{"id":63555,"d":25935},"MJX-266-TEX-N-30",[24941,63557],{"id":63558,"d":37254},"MJX-266-TEX-I-1D464",[25022,63560,63561],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,63562,63563,63571],{"dataMmlNode":25028},[25022,63564,63565,63568],{"dataMmlNode":26031},[25033,63566],{"dataC":26035,"xLinkHref":63567},"#MJX-266-TEX-N-31",[25033,63569],{"dataC":26103,"xLinkHref":63570,"transform":26394},"#MJX-266-TEX-N-30",[25022,63572,63573],{"dataMmlNode":25942,"transform":29730},[25033,63574],{"dataC":37310,"xLinkHref":63575},"#MJX-266-TEX-I-1D464",[24923,63577,63579],{"className":63578,"jax":24927},[24926],[24929,63580,63581,63589],{"style":27778,"xmlns":24932,"width":24934,"height":35776,"role":1930,"focusable":8058,"viewBox":55544,"xmlnsXLink":24936},[24938,63582,63583,63586],{},[24941,63584],{"id":63585,"d":25914},"MJX-267-TEX-N-31",[24941,63587],{"id":63588,"d":25935},"MJX-267-TEX-N-30",[25022,63590,63591],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,63592,63593],{"dataMmlNode":25028},[25022,63594,63595,63598],{"dataMmlNode":26031},[25033,63596],{"dataC":26035,"xLinkHref":63597},"#MJX-267-TEX-N-31",[25033,63599],{"dataC":26103,"xLinkHref":63600,"transform":26394},"#MJX-267-TEX-N-30",". Therefore, the derivative of ",[24923,63603,63605],{"className":63604,"jax":24927},[24926],[24929,63606,63609,63620],{"style":25196,"xmlns":24932,"width":63607,"height":27827,"role":1930,"focusable":8058,"viewBox":63608,"xmlnsXLink":24936},"4.549ex","0 -443 2010.4 454",[24938,63610,63611,63614,63617],{},[24941,63612],{"id":63613,"d":32148},"MJX-268-TEX-I-1D465",[24941,63615],{"id":63616,"d":27409},"MJX-268-TEX-N-22C5",[24941,63618],{"id":63619,"d":37254},"MJX-268-TEX-I-1D464",[25022,63621,63622],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,63623,63624,63629,63635],{"dataMmlNode":25028},[25022,63625,63626],{"dataMmlNode":25942},[25033,63627],{"dataC":32181,"xLinkHref":63628},"#MJX-268-TEX-I-1D465",[25022,63630,63632],{"dataMmlNode":25031,"transform":63631},"translate(794.2,0)",[25033,63633],{"dataC":27679,"xLinkHref":63634},"#MJX-268-TEX-N-22C5",[25022,63636,63638],{"dataMmlNode":25942,"transform":63637},"translate(1294.4,0)",[25033,63639],{"dataC":37310,"xLinkHref":63640},"#MJX-268-TEX-I-1D464",[24923,63642,63644],{"className":63643,"jax":24927},[24926],[24929,63645,63646,63651],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,63647,63648],{},[24941,63649],{"id":63650,"d":32148},"MJX-269-TEX-I-1D465",[25022,63652,63653],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,63654,63655],{"dataMmlNode":25028},[25022,63656,63657],{"dataMmlNode":25942},[25033,63658],{"dataC":32181,"xLinkHref":63659},"#MJX-269-TEX-I-1D465",[16,63661],{},[106,63663,63664],{},"Answer 3:",[24923,63666,63668],{"className":63667,"jax":24927},[24926],[24929,63669,63670,63687],{"style":33830,"xmlns":24932,"width":51376,"height":33832,"role":1930,"focusable":8058,"viewBox":51377,"xmlnsXLink":24936},[24938,63671,63672,63675,63678,63681,63684],{},[24941,63673],{"id":63674,"d":32148},"MJX-270-TEX-I-1D465",[24941,63676],{"id":63677,"d":24944},"MJX-270-TEX-N-28",[24941,63679],{"id":63680,"d":25914},"MJX-270-TEX-N-31",[24941,63682],{"id":63683,"d":24988},"MJX-270-TEX-N-29",[24941,63685],{"id":63686,"d":33606},"MJX-270-TEX-I-1D457",[25022,63688,63689],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,63690,63691],{"dataMmlNode":25028},[25022,63692,63693,63698,63715],{"dataMmlNode":33991},[25022,63694,63695],{"dataMmlNode":25942},[25033,63696],{"dataC":32181,"xLinkHref":63697},"#MJX-270-TEX-I-1D465",[25022,63699,63700,63705,63710],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,63701,63702],{"dataMmlNode":25031},[25033,63703],{"dataC":25035,"xLinkHref":63704},"#MJX-270-TEX-N-28",[25022,63706,63707],{"dataMmlNode":26031,"transform":25040},[25033,63708],{"dataC":26035,"xLinkHref":63709},"#MJX-270-TEX-N-31",[25022,63711,63712],{"dataMmlNode":25031,"transform":41846},[25033,63713],{"dataC":25108,"xLinkHref":63714},"#MJX-270-TEX-N-29",[25022,63716,63717],{"dataMmlNode":25942,"transform":34014},[25033,63718],{"dataC":33647,"xLinkHref":63719},"#MJX-270-TEX-I-1D457",[26236,63721,63723,63724,7963],{"id":63722},"updating-hidden-layer-weights","Updating Hidden Layer Weights (",[24923,63725,63727],{"className":63726,"jax":24927},[24926],[24929,63728,63729,63749],{"style":25196,"xmlns":24932,"width":48804,"height":32538,"role":1930,"focusable":8058,"viewBox":48805,"xmlnsXLink":24936},[24938,63730,63731,63734,63737,63740,63743,63746],{},[24941,63732],{"id":63733,"d":37254},"MJX-271-TEX-I-1D464",[24941,63735],{"id":63736,"d":24944},"MJX-271-TEX-N-28",[24941,63738],{"id":63739,"d":25914},"MJX-271-TEX-N-31",[24941,63741],{"id":63742,"d":27339},"MJX-271-TEX-N-2C",[24941,63744],{"id":63745,"d":25935},"MJX-271-TEX-N-30",[24941,63747],{"id":63748,"d":24988},"MJX-271-TEX-N-29",[25022,63750,63751],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,63752,63753],{"dataMmlNode":25028},[25022,63754,63755,63760],{"dataMmlNode":31565},[25022,63756,63757],{"dataMmlNode":25942},[25033,63758],{"dataC":37310,"xLinkHref":63759},"#MJX-271-TEX-I-1D464",[25022,63761,63762,63767,63772,63777,63782],{"dataMmlNode":27203,"transform":37564,"dataMjxTexclass":27205},[25022,63763,63764],{"dataMmlNode":25031},[25033,63765],{"dataC":25035,"xLinkHref":63766},"#MJX-271-TEX-N-28",[25022,63768,63769],{"dataMmlNode":26031,"transform":25040},[25033,63770],{"dataC":26035,"xLinkHref":63771},"#MJX-271-TEX-N-31",[25022,63773,63774],{"dataMmlNode":25031,"transform":41846},[25033,63775],{"dataC":27456,"xLinkHref":63776},"#MJX-271-TEX-N-2C",[25022,63778,63779],{"dataMmlNode":26031,"transform":28333},[25033,63780],{"dataC":26103,"xLinkHref":63781},"#MJX-271-TEX-N-30",[25022,63783,63784],{"dataMmlNode":25031,"transform":41146},[25033,63785],{"dataC":25108,"xLinkHref":63786},"#MJX-271-TEX-N-29",[12,63788,63789,63790,63809,63810,63813,63814,63833,63834,63919],{},"This is the core of backpropagation. A hidden node ",[24923,63791,63793],{"className":63792,"jax":24927},[24926],[24929,63794,63795,63800],{"style":48965,"xmlns":24932,"width":48966,"height":48967,"role":1930,"focusable":8058,"viewBox":48968,"xmlnsXLink":24936},[24938,63796,63797],{},[24941,63798],{"id":63799,"d":33606},"MJX-272-TEX-I-1D457",[25022,63801,63802],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,63803,63804],{"dataMmlNode":25028},[25022,63805,63806],{"dataMmlNode":25942},[25033,63807],{"dataC":33647,"xLinkHref":63808},"#MJX-272-TEX-I-1D457"," connects to ",[7289,63811,63812],{},"multiple"," output nodes ",[24923,63815,63817],{"className":63816,"jax":24927},[24926],[24929,63818,63819,63824],{"style":25196,"xmlns":24932,"width":50688,"height":31523,"role":1930,"focusable":8058,"viewBox":50689,"xmlnsXLink":24936},[24938,63820,63821],{},[24941,63822],{"id":63823,"d":42951},"MJX-273-TEX-I-1D458",[25022,63825,63826],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,63827,63828],{"dataMmlNode":25028},[25022,63829,63830],{"dataMmlNode":25942},[25033,63831],{"dataC":43135,"xLinkHref":63832},"#MJX-273-TEX-I-1D458",". Therefore, the error derivative with respect to a hidden layer weight 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It only calculates the blame for the very last set of weights in the network (the ones touching the final Output). What if we want to calculate the blame for a weight deeper inside the network? Say, between the input layer and a hidden layer. We can apply the same logic, but we will run into two new challanges, longer chains and fork roads.",[26236,64516,64518],{"id":64517},"_1-longer-chains","1. 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If you tweak a deep weight, the ripple effect looks like this:",[3603,64590,64591,64597,64604,64610,64616],{},[26,64592,64593,64594,6959],{},"The Deep Weight changes the ",[106,64595,64596],{},"Deep Net Input",[26,64598,64599,64600,64603],{},"The Deep Net Input changes the ",[106,64601,64602],{},"Deep Output"," (the Hidden Node's signal).",[26,64605,64606,64607,6959],{},"The Hidden Node's signal travels forward and changes the ",[106,64608,64609],{},"Final Net Input",[26,64611,64612,64613,6959],{},"The Final Net Input changes the ",[106,64614,64615],{},"Final Output",[26,64617,64618,64619,6959],{},"The Final Output changes the ",[106,64620,58542],{},[12,64622,64623],{},"If it was just a straight line, your blueprint would just be those 5 derivatives multiplied 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The Fork in the Road (Why we need the ",[24923,65043,65045],{"className":65044,"jax":24927},[24926],[24929,65046,65047,65052],{"style":27802,"xmlns":24932,"width":28112,"height":28113,"role":1930,"focusable":8058,"viewBox":28114,"xmlnsXLink":24936},[24938,65048,65049],{},[24941,65050],{"id":65051,"d":28120},"MJX-278-TEX-N-3A3",[25022,65053,65054],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,65055,65056],{"dataMmlNode":25028},[25022,65057,65058],{"dataMmlNode":25942},[25033,65059],{"dataC":28129,"xLinkHref":65060},"#MJX-278-TEX-N-3A3",[12,65062,65063],{},"But in neural network's context, the chain is almost never a straight line.",[12,65065,65066],{},"If you are trying to trace the Error backward to hidden node's weight, you hit a fork in the road. Which node's Error do you trace?",[12,65068,65069],{},"Of course you have to look at all of them.",[12,65071,65072,65073,65092,65093,65183,65184,65203],{},"If a hidden node ",[24923,65074,65076],{"className":65075,"jax":24927},[24926],[24929,65077,65078,65083],{"style":48965,"xmlns":24932,"width":48966,"height":48967,"role":1930,"focusable":8058,"viewBox":48968,"xmlnsXLink":24936},[24938,65079,65080],{},[24941,65081],{"id":65082,"d":33606},"MJX-279-TEX-I-1D457",[25022,65084,65085],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,65086,65087],{"dataMmlNode":25028},[25022,65088,65089],{"dataMmlNode":25942},[25033,65090],{"dataC":33647,"xLinkHref":65091},"#MJX-279-TEX-I-1D457"," connects to three different outputs (",[24923,65094,65096],{"className":65095,"jax":24927},[24926],[24929,65097,65101,65121],{"style":26153,"xmlns":24932,"width":65098,"height":65099,"role":1930,"focusable":8058,"viewBox":65100,"xmlnsXLink":24936},"17.993ex","2.009ex","0 -694 7953 888",[24938,65102,65103,65106,65109,65112,65115,65118],{},[24941,65104],{"id":65105,"d":42951},"MJX-280-TEX-I-1D458",[24941,65107],{"id":65108,"d":24972},"MJX-280-TEX-N-3D",[24941,65110],{"id":65111,"d":25914},"MJX-280-TEX-N-31",[24941,65113],{"id":65114,"d":27339},"MJX-280-TEX-N-2C",[24941,65116],{"id":65117,"d":25928},"MJX-280-TEX-N-32",[24941,65119],{"id":65120,"d":26455},"MJX-280-TEX-N-33",[25022,65122,65123],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,65124,65125,65130,65136,65142,65148,65153,65157,65163,65168,65173,65178],{"dataMmlNode":25028},[25022,65126,65127],{"dataMmlNode":25942},[25033,65128],{"dataC":43135,"xLinkHref":65129},"#MJX-280-TEX-I-1D458",[25022,65131,65133],{"dataMmlNode":25031,"transform":65132},"translate(798.8,0)",[25033,65134],{"dataC":25078,"xLinkHref":65135},"#MJX-280-TEX-N-3D",[25022,65137,65139],{"dataMmlNode":26031,"transform":65138},"translate(1854.6,0)",[25033,65140],{"dataC":26035,"xLinkHref":65141},"#MJX-280-TEX-N-31",[25022,65143,65145],{"dataMmlNode":25031,"transform":65144},"translate(2354.6,0)",[25033,65146],{"dataC":27456,"xLinkHref":65147},"#MJX-280-TEX-N-2C",[25022,65149,65151],{"dataMmlNode":25942,"transform":65150},"translate(2799.2,0)",[25033,65152],{"dataC":43135,"xLinkHref":65129},[25022,65154,65155],{"dataMmlNode":25031,"transform":61670},[25033,65156],{"dataC":25078,"xLinkHref":65135},[25022,65158,65160],{"dataMmlNode":26031,"transform":65159},"translate(4653.8,0)",[25033,65161],{"dataC":26058,"xLinkHref":65162},"#MJX-280-TEX-N-32",[25022,65164,65166],{"dataMmlNode":25031,"transform":65165},"translate(5153.8,0)",[25033,65167],{"dataC":27456,"xLinkHref":65147},[25022,65169,65171],{"dataMmlNode":25942,"transform":65170},"translate(5598.4,0)",[25033,65172],{"dataC":43135,"xLinkHref":65129},[25022,65174,65176],{"dataMmlNode":25031,"transform":65175},"translate(6397.2,0)",[25033,65177],{"dataC":25078,"xLinkHref":65135},[25022,65179,65180],{"dataMmlNode":26031,"transform":60346},[25033,65181],{"dataC":26486,"xLinkHref":65182},"#MJX-280-TEX-N-33","), then hidden node ",[24923,65185,65187],{"className":65186,"jax":24927},[24926],[24929,65188,65189,65194],{"style":48965,"xmlns":24932,"width":48966,"height":48967,"role":1930,"focusable":8058,"viewBox":48968,"xmlnsXLink":24936},[24938,65190,65191],{},[24941,65192],{"id":65193,"d":33606},"MJX-281-TEX-I-1D457",[25022,65195,65196],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,65197,65198],{"dataMmlNode":25028},[25022,65199,65200],{"dataMmlNode":25942},[25033,65201],{"dataC":33647,"xLinkHref":65202},"#MJX-281-TEX-I-1D457"," is partially responsible for the errors of all three. To find the total blame for the hidden node's weight, you have to:",[3603,65205,65206,65209,65212,65215],{},[26,65207,65208],{},"Trace the 5-step chain back from output node 1.",[26,65210,65211],{},"Trace the 5-step chain back from output node 2.",[26,65213,65214],{},"Trace the 5-step chain back from output node 3.",[26,65216,65217],{},[106,65218,65219],{},"Add them all together.",[12,65221,65222,65223,65242,65243,24917],{},"Now, look at the formula again. It is just the 5-step chain rule we deduced above, wrapped inside a giant Summation (",[24923,65224,65226],{"className":65225,"jax":24927},[24926],[24929,65227,65228,65233],{"style":27802,"xmlns":24932,"width":28112,"height":28113,"role":1930,"focusable":8058,"viewBox":28114,"xmlnsXLink":24936},[24938,65229,65230],{},[24941,65231],{"id":65232,"d":28120},"MJX-282-TEX-N-3A3",[25022,65234,65235],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,65236,65237],{"dataMmlNode":25028},[25022,65238,65239],{"dataMmlNode":25942},[25033,65240],{"dataC":28129,"xLinkHref":65241},"#MJX-282-TEX-N-3A3",") symbol to add up the blame from every node in the next layer 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react if their own Net Input 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This means we must freeze the raw input data (",[24923,69168,69170],{"className":69169,"jax":24927},[24926],[24929,69171,69172,69180],{"style":27744,"xmlns":24932,"width":51136,"height":31803,"role":1930,"focusable":8058,"viewBox":51137,"xmlnsXLink":24936},[24938,69173,69174,69177],{},[24941,69175],{"id":69176,"d":32148},"MJX-320-TEX-I-1D465",[24941,69178],{"id":69179,"d":27108},"MJX-320-TEX-I-1D456",[25022,69181,69182],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,69183,69184],{"dataMmlNode":25028},[25022,69185,69186,69191],{"dataMmlNode":26023},[25022,69187,69188],{"dataMmlNode":25942},[25033,69189],{"dataC":32181,"xLinkHref":69190},"#MJX-320-TEX-I-1D465",[25022,69192,69193],{"dataMmlNode":25942,"transform":33034},[25033,69194],{"dataC":27210,"xLinkHref":69195},"#MJX-320-TEX-I-1D456",[26,69197,69198,69199,69218,69219,69244,69245,69317,69318,69343,69344,35162,69379,6959],{},"Just like we did for the output layer, if we freeze ",[24923,69200,69202],{"className":69201,"jax":24927},[24926],[24929,69203,69204,69209],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,69205,69206],{},[24941,69207],{"id":69208,"d":32148},"MJX-321-TEX-I-1D465",[25022,69210,69211],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,69212,69213],{"dataMmlNode":25028},[25022,69214,69215],{"dataMmlNode":25942},[25033,69216],{"dataC":32181,"xLinkHref":69217},"#MJX-321-TEX-I-1D465"," to be ",[24923,69220,69222],{"className":69221,"jax":24927},[24926],[24929,69223,69224,69232],{"style":27778,"xmlns":24932,"width":24934,"height":35776,"role":1930,"focusable":8058,"viewBox":55544,"xmlnsXLink":24936},[24938,69225,69226,69229],{},[24941,69227],{"id":69228,"d":25914},"MJX-322-TEX-N-31",[24941,69230],{"id":69231,"d":25935},"MJX-322-TEX-N-30",[25022,69233,69234],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,69235,69236],{"dataMmlNode":25028},[25022,69237,69238,69241],{"dataMmlNode":26031},[25033,69239],{"dataC":26035,"xLinkHref":69240},"#MJX-322-TEX-N-31",[25033,69242],{"dataC":26103,"xLinkHref":69243,"transform":26394},"#MJX-322-TEX-N-30",", the equation is ",[24923,69246,69248],{"className":69247,"jax":24927},[24926],[24929,69249,69252,69275],{"style":24931,"xmlns":24932,"width":69250,"height":24934,"role":1930,"focusable":8058,"viewBox":69251,"xmlnsXLink":24936},"11.524ex","0 -750 5093.6 1000",[24938,69253,69254,69257,69260,69263,69266,69269,69272],{},[24941,69255],{"id":69256,"d":29821},"MJX-323-TEX-I-1D453",[24941,69258],{"id":69259,"d":24944},"MJX-323-TEX-N-28",[24941,69261],{"id":69262,"d":37254},"MJX-323-TEX-I-1D464",[24941,69264],{"id":69265,"d":24988},"MJX-323-TEX-N-29",[24941,69267],{"id":69268,"d":24972},"MJX-323-TEX-N-3D",[24941,69270],{"id":69271,"d":25914},"MJX-323-TEX-N-31",[24941,69273],{"id":69274,"d":25935},"MJX-323-TEX-N-30",[25022,69276,69277],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,69278,69279,69284,69289,69294,69299,69304,69312],{"dataMmlNode":25028},[25022,69280,69281],{"dataMmlNode":25942},[25033,69282],{"dataC":29864,"xLinkHref":69283},"#MJX-323-TEX-I-1D453",[25022,69285,69286],{"dataMmlNode":25031,"transform":44094},[25033,69287],{"dataC":25035,"xLinkHref":69288},"#MJX-323-TEX-N-28",[25022,69290,69291],{"dataMmlNode":25942,"transform":53947},[25033,69292],{"dataC":37310,"xLinkHref":69293},"#MJX-323-TEX-I-1D464",[25022,69295,69296],{"dataMmlNode":25031,"transform":63362},[25033,69297],{"dataC":25108,"xLinkHref":69298},"#MJX-323-TEX-N-29",[25022,69300,69301],{"dataMmlNode":25031,"transform":63368},[25033,69302],{"dataC":25078,"xLinkHref":69303},"#MJX-323-TEX-N-3D",[25022,69305,69306,69309],{"dataMmlNode":26031,"transform":63374},[25033,69307],{"dataC":26035,"xLinkHref":69308},"#MJX-323-TEX-N-31",[25033,69310],{"dataC":26103,"xLinkHref":69311,"transform":26394},"#MJX-323-TEX-N-30",[25022,69313,69315],{"dataMmlNode":25942,"transform":69314},"translate(4377.6,0)",[25033,69316],{"dataC":37310,"xLinkHref":69293},". The derivative is ",[24923,69319,69321],{"className":69320,"jax":24927},[24926],[24929,69322,69323,69331],{"style":27778,"xmlns":24932,"width":24934,"height":35776,"role":1930,"focusable":8058,"viewBox":55544,"xmlnsXLink":24936},[24938,69324,69325,69328],{},[24941,69326],{"id":69327,"d":25914},"MJX-324-TEX-N-31",[24941,69329],{"id":69330,"d":25935},"MJX-324-TEX-N-30",[25022,69332,69333],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,69334,69335],{"dataMmlNode":25028},[25022,69336,69337,69340],{"dataMmlNode":26031},[25033,69338],{"dataC":26035,"xLinkHref":69339},"#MJX-324-TEX-N-31",[25033,69341],{"dataC":26103,"xLinkHref":69342,"transform":26394},"#MJX-324-TEX-N-30",". So, the derivative of ",[24923,69345,69347],{"className":69346,"jax":24927},[24926],[24929,69348,69349,69360],{"style":25196,"xmlns":24932,"width":63607,"height":27827,"role":1930,"focusable":8058,"viewBox":63608,"xmlnsXLink":24936},[24938,69350,69351,69354,69357],{},[24941,69352],{"id":69353,"d":32148},"MJX-325-TEX-I-1D465",[24941,69355],{"id":69356,"d":27409},"MJX-325-TEX-N-22C5",[24941,69358],{"id":69359,"d":37254},"MJX-325-TEX-I-1D464",[25022,69361,69362],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,69363,69364,69369,69374],{"dataMmlNode":25028},[25022,69365,69366],{"dataMmlNode":25942},[25033,69367],{"dataC":32181,"xLinkHref":69368},"#MJX-325-TEX-I-1D465",[25022,69370,69371],{"dataMmlNode":25031,"transform":63631},[25033,69372],{"dataC":27679,"xLinkHref":69373},"#MJX-325-TEX-N-22C5",[25022,69375,69376],{"dataMmlNode":25942,"transform":63637},[25033,69377],{"dataC":37310,"xLinkHref":69378},"#MJX-325-TEX-I-1D464",[24923,69380,69382],{"className":69381,"jax":24927},[24926],[24929,69383,69384,69389],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,69385,69386],{},[24941,69387],{"id":69388,"d":32148},"MJX-326-TEX-I-1D465",[25022,69390,69391],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,69392,69393],{"dataMmlNode":25028},[25022,69394,69395],{"dataMmlNode":25942},[25033,69396],{"dataC":32181,"xLinkHref":69397},"#MJX-326-TEX-I-1D465",[26,69399,69400,7206,69403],{},[106,69401,69402],{},"Answer 5:",[24923,69404,69406],{"className":69405,"jax":24927},[24926],[24929,69407,69408,69416],{"style":27744,"xmlns":24932,"width":51136,"height":31803,"role":1930,"focusable":8058,"viewBox":51137,"xmlnsXLink":24936},[24938,69409,69410,69413],{},[24941,69411],{"id":69412,"d":32148},"MJX-327-TEX-I-1D465",[24941,69414],{"id":69415,"d":27108},"MJX-327-TEX-I-1D456",[25022,69417,69418],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,69419,69420],{"dataMmlNode":25028},[25022,69421,69422,69427],{"dataMmlNode":26023},[25022,69423,69424],{"dataMmlNode":25942},[25033,69425],{"dataC":32181,"xLinkHref":69426},"#MJX-327-TEX-I-1D465",[25022,69428,69429],{"dataMmlNode":25942,"transform":33034},[25033,69430],{"dataC":27210,"xLinkHref":69431},"#MJX-327-TEX-I-1D456",[7166,69433,69435],{"id":69434},"a-more-intuitive-view-the-δ-recursion","A More Intuitive View: The δ Recursion",[12,69437,69438,69439,7206,69442,69474,69475,7024],{},"The chain-rule derivations above are mathematically complete, but they hide a beautiful pattern. Once we group terms, backpropagation collapses into a simple recursive rule. Define a ",[106,69440,69441],{},"local error signal",[24923,69443,69445],{"className":69444,"jax":24927},[24926],[24929,69446,69450,69458],{"style":27744,"xmlns":24932,"width":69447,"height":69448,"role":1930,"focusable":8058,"viewBox":69449,"xmlnsXLink":24936},"2.152ex","1.979ex","0 -717 951.3 874.8",[24938,69451,69452,69455],{},[24941,69453],{"id":69454,"d":57288},"MJX-328-TEX-I-1D6FF",[24941,69456],{"id":69457,"d":25873},"MJX-328-TEX-I-1D45B",[25022,69459,69460],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,69461,69462],{"dataMmlNode":25028},[25022,69463,69464,69469],{"dataMmlNode":26023},[25022,69465,69466],{"dataMmlNode":25942},[25033,69467],{"dataC":57406,"xLinkHref":69468},"#MJX-328-TEX-I-1D6FF",[25022,69470,69471],{"dataMmlNode":25942,"transform":57410},[25033,69472],{"dataC":25952,"xLinkHref":69473},"#MJX-328-TEX-I-1D45B"," at every node 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just the \"learning rate,\" a tiny number we use to make sure the network takes small steps instead of giant leaps).",[12,70935,70936],{},"That single line summarizes all of backpropagation. Everything earlier in this document is the proof that this rule actually decreases the error.",[25631,70938,70940,70963],{"className":70939},[13441,13442],[25635,70941,70942,70943,70962],{},"Wait, what is ",[24923,70944,70946],{"className":70945,"jax":24927},[24926],[24929,70947,70948,70953],{"style":33072,"xmlns":24932,"width":37461,"height":27780,"role":1930,"focusable":8058,"viewBox":69623,"xmlnsXLink":24936},[24938,70949,70950],{},[24941,70951],{"id":70952,"d":57288},"MJX-345-TEX-I-1D6FF",[25022,70954,70955],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,70956,70957],{"dataMmlNode":25028},[25022,70958,70959],{"dataMmlNode":25942},[25033,70960],{"dataC":57406,"xLinkHref":70961},"#MJX-345-TEX-I-1D6FF"," actually doing?",[13438,70964,70966,70977,70980,71028,71050,71053,71058,71220,71225,71262,71267,71296,71349],{"className":70965},[25641],[12,70967,70968,70969,70972,70973,70976],{},"The chain-rule math above is the ",[7289,70970,70971],{},"proof"," that backpropagation works, but it is not how a computer actually ",[7289,70974,70975],{},"runs"," the code.",[12,70978,70979],{},"If a neural network is 100 layers deep, calculating the chain-rule equation for the very first layer would require an impossibly long equation. The computer would end up recalculating the exact same math millions of times.",[12,70981,70982,7206,71004,71007,71008,71027],{},[106,70983,70984,70985],{},"The Solution (",[24923,70986,70988],{"className":70987,"jax":24927},[24926],[24929,70989,70990,70995],{"style":33072,"xmlns":24932,"width":37461,"height":27780,"role":1930,"focusable":8058,"viewBox":69623,"xmlnsXLink":24936},[24938,70991,70992],{},[24941,70993],{"id":70994,"d":57288},"MJX-346-TEX-I-1D6FF",[25022,70996,70997],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,70998,70999],{"dataMmlNode":25028},[25022,71000,71001],{"dataMmlNode":25942},[25033,71002],{"dataC":57406,"xLinkHref":71003},"#MJX-346-TEX-I-1D6FF",[106,71005,71006],{},"Recursion):"," Instead of looking at the whole giant equation at once, we chop the \"Blame\" up into a neat little package called ",[24923,71009,71011],{"className":71010,"jax":24927},[24926],[24929,71012,71013,71018],{"style":33072,"xmlns":24932,"width":37461,"height":27780,"role":1930,"focusable":8058,"viewBox":69623,"xmlnsXLink":24936},[24938,71014,71015],{},[24941,71016],{"id":71017,"d":57288},"MJX-347-TEX-I-1D6FF",[25022,71019,71020],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71021,71022],{"dataMmlNode":25028},[25022,71023,71024],{"dataMmlNode":25942},[25033,71025],{"dataC":57406,"xLinkHref":71026},"#MJX-347-TEX-I-1D6FF"," (Delta).",[12,71029,39383,71030,71049],{},[24923,71031,71033],{"className":71032,"jax":24927},[24926],[24929,71034,71035,71040],{"style":33072,"xmlns":24932,"width":37461,"height":27780,"role":1930,"focusable":8058,"viewBox":69623,"xmlnsXLink":24936},[24938,71036,71037],{},[24941,71038],{"id":71039,"d":57288},"MJX-348-TEX-I-1D6FF",[25022,71041,71042],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71043,71044],{"dataMmlNode":25028},[25022,71045,71046],{"dataMmlNode":25942},[25033,71047],{"dataC":57406,"xLinkHref":71048},"#MJX-348-TEX-I-1D6FF"," as a \"Box of Blame.\"\nEvery single node in the network calculates its own personal Box of Blame.",[12,71051,71052],{},"Here is how the algorithm flows:",[12,71054,71055],{},[106,71056,71057],{},"1. The Output Layer (The Final Inspector)",[23,71059,71060,71198],{},[26,71061,71062,71063,6959],{},"The output node calculates its raw error: ",[24923,71064,71066],{"className":71065,"jax":24927},[24926],[24929,71067,71070,71109],{"style":24931,"xmlns":24932,"width":71068,"height":24934,"role":1930,"focusable":8058,"viewBox":71069,"xmlnsXLink":24936},"18.372ex","0 -750 8120.4 1000",[24938,71071,71072,71075,71078,71081,71084,71088,71091,71094,71097,71100,71103,71106],{},[24941,71073],{"id":71074,"d":24944},"MJX-349-TEX-N-28",[24941,71076],{"id":71077,"d":38734},"MJX-349-TEX-I-1D447",[24941,71079],{"id":71080,"d":27323},"MJX-349-TEX-I-1D44E",[24941,71082],{"id":71083,"d":25881},"MJX-349-TEX-I-1D45F",[24941,71085],{"id":71086,"d":71087},"MJX-349-TEX-I-1D454","M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z",[24941,71089],{"id":71090,"d":27397},"MJX-349-TEX-I-1D452",[24941,71092],{"id":71093,"d":25877},"MJX-349-TEX-I-1D461",[24941,71095],{"id":71096,"d":25910},"MJX-349-TEX-N-2212",[24941,71098],{"id":71099,"d":31740},"MJX-349-TEX-I-1D442",[24941,71101],{"id":71102,"d":27393},"MJX-349-TEX-I-1D462",[24941,71104],{"id":71105,"d":25889},"MJX-349-TEX-I-1D45D",[24941,71107],{"id":71108,"d":24988},"MJX-349-TEX-N-29",[25022,71110,71111],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71112,71113,71118,71123,71129,71135,71141,71147,71153,71159,71165,71171,71176,71182,71187,71192],{"dataMmlNode":25028},[25022,71114,71115],{"dataMmlNode":25031},[25033,71116],{"dataC":25035,"xLinkHref":71117},"#MJX-349-TEX-N-28",[25022,71119,71120],{"dataMmlNode":25942,"transform":25040},[25033,71121],{"dataC":38825,"xLinkHref":71122},"#MJX-349-TEX-I-1D447",[25022,71124,71126],{"dataMmlNode":25942,"transform":71125},"translate(1093,0)",[25033,71127],{"dataC":27425,"xLinkHref":71128},"#MJX-349-TEX-I-1D44E",[25022,71130,71132],{"dataMmlNode":25942,"transform":71131},"translate(1622,0)",[25033,71133],{"dataC":25966,"xLinkHref":71134},"#MJX-349-TEX-I-1D45F",[25022,71136,71137],{"dataMmlNode":25942,"transform":40626},[25033,71138],{"dataC":71139,"xLinkHref":71140},"1D454","#MJX-349-TEX-I-1D454",[25022,71142,71144],{"dataMmlNode":25942,"transform":71143},"translate(2550,0)",[25033,71145],{"dataC":27596,"xLinkHref":71146},"#MJX-349-TEX-I-1D452",[25022,71148,71150],{"dataMmlNode":25942,"transform":71149},"translate(3016,0)",[25033,71151],{"dataC":25959,"xLinkHref":71152},"#MJX-349-TEX-I-1D461",[25022,71154,71156],{"dataMmlNode":25031,"transform":71155},"translate(3599.2,0)",[25033,71157],{"dataC":26019,"xLinkHref":71158},"#MJX-349-TEX-N-2212",[25022,71160,71162],{"dataMmlNode":25942,"transform":71161},"translate(4599.4,0)",[25033,71163],{"dataC":31764,"xLinkHref":71164},"#MJX-349-TEX-I-1D442",[25022,71166,71168],{"dataMmlNode":25942,"transform":71167},"translate(5362.4,0)",[25033,71169],{"dataC":27589,"xLinkHref":71170},"#MJX-349-TEX-I-1D462",[25022,71172,71174],{"dataMmlNode":25942,"transform":71173},"translate(5934.4,0)",[25033,71175],{"dataC":25959,"xLinkHref":71152},[25022,71177,71179],{"dataMmlNode":25942,"transform":71178},"translate(6295.4,0)",[25033,71180],{"dataC":25980,"xLinkHref":71181},"#MJX-349-TEX-I-1D45D",[25022,71183,71185],{"dataMmlNode":25942,"transform":71184},"translate(6798.4,0)",[25033,71186],{"dataC":27589,"xLinkHref":71170},[25022,71188,71190],{"dataMmlNode":25942,"transform":71189},"translate(7370.4,0)",[25033,71191],{"dataC":25959,"xLinkHref":71152},[25022,71193,71195],{"dataMmlNode":25031,"transform":71194},"translate(7731.4,0)",[25033,71196],{"dataC":25108,"xLinkHref":71197},"#MJX-349-TEX-N-29",[26,71199,71200,71201,6959],{},"It puts this into its Box of Blame and calls it ",[24923,71202,71204],{"className":71203,"jax":24927},[24926],[24929,71205,71206,71211],{"style":33072,"xmlns":24932,"width":37461,"height":27780,"role":1930,"focusable":8058,"viewBox":69623,"xmlnsXLink":24936},[24938,71207,71208],{},[24941,71209],{"id":71210,"d":57288},"MJX-350-TEX-I-1D6FF",[25022,71212,71213],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71214,71215],{"dataMmlNode":25028},[25022,71216,71217],{"dataMmlNode":25942},[25033,71218],{"dataC":57406,"xLinkHref":71219},"#MJX-350-TEX-I-1D6FF",[12,71221,71222],{},[106,71223,71224],{},"2. The Hidden Layers (Passing the Buck)",[23,71226,71227,71230,71256],{},[26,71228,71229],{},"A hidden node doesn't calculate an error from scratch.",[26,71231,71232,71233],{},"Instead, it just looks at the node in front of it and says: ",[7289,71234,71235,71236,71255],{},"\"Give me your Box of Blame (",[24923,71237,71239],{"className":71238,"jax":24927},[24926],[24929,71240,71241,71246],{"style":33072,"xmlns":24932,"width":37461,"height":27780,"role":1930,"focusable":8058,"viewBox":69623,"xmlnsXLink":24936},[24938,71242,71243],{},[24941,71244],{"id":71245,"d":57288},"MJX-351-TEX-I-1D6FF",[25022,71247,71248],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71249,71250],{"dataMmlNode":25028},[25022,71251,71252],{"dataMmlNode":25942},[25033,71253],{"dataC":57406,"xLinkHref":71254},"#MJX-351-TEX-I-1D6FF","). Multiply it by the Weight connecting us, and hand it backward to me.\"",[26,71257,71258,71259],{},"The hidden node takes that package, adds its own flexibility (activation derivative), and says: ",[7289,71260,71261],{},"\"Great, this is my new Box of Blame!\"",[12,71263,71264],{},[106,71265,71266],{},"3. The Fork in the Road (Summing)",[23,71268,71269],{},[26,71270,71271,71272,71275,71276,71295],{},"If a hidden node sent its signal to ",[7289,71273,71274],{},"three"," different output nodes, it simply collects the Boxes of Blame (",[24923,71277,71279],{"className":71278,"jax":24927},[24926],[24929,71280,71281,71286],{"style":33072,"xmlns":24932,"width":37461,"height":27780,"role":1930,"focusable":8058,"viewBox":69623,"xmlnsXLink":24936},[24938,71282,71283],{},[24941,71284],{"id":71285,"d":57288},"MJX-352-TEX-I-1D6FF",[25022,71287,71288],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71289,71290],{"dataMmlNode":25028},[25022,71291,71292],{"dataMmlNode":25942},[25033,71293],{"dataC":57406,"xLinkHref":71294},"#MJX-352-TEX-I-1D6FF",") from all three of them, multiplies them by their respective weights, and adds them together.",[12,71297,71298,71299,71301,71302,71321,71322,71325,71326,71345,71346,6959],{},"Notice that during the ",[106,71300,70341],{},", the network pushes the raw data (",[24923,71303,71305],{"className":71304,"jax":24927},[24926],[24929,71306,71307,71312],{"style":25196,"xmlns":24932,"width":37510,"height":27038,"role":1930,"focusable":8058,"viewBox":37511,"xmlnsXLink":24936},[24938,71308,71309],{},[24941,71310],{"id":71311,"d":32148},"MJX-353-TEX-I-1D465",[25022,71313,71314],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71315,71316],{"dataMmlNode":25028},[25022,71317,71318],{"dataMmlNode":25942},[25033,71319],{"dataC":32181,"xLinkHref":71320},"#MJX-353-TEX-I-1D465",") left-to-right, multiplying it by the weights.\nDuring the ",[106,71323,71324],{},"Backward Pass",", the network takes the Error packages (",[24923,71327,71329],{"className":71328,"jax":24927},[24926],[24929,71330,71331,71336],{"style":33072,"xmlns":24932,"width":37461,"height":27780,"role":1930,"focusable":8058,"viewBox":69623,"xmlnsXLink":24936},[24938,71332,71333],{},[24941,71334],{"id":71335,"d":57288},"MJX-354-TEX-I-1D6FF",[25022,71337,71338],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71339,71340],{"dataMmlNode":25028},[25022,71341,71342],{"dataMmlNode":25942},[25033,71343],{"dataC":57406,"xLinkHref":71344},"#MJX-354-TEX-I-1D6FF",") and pushes them right-to-left, multiplying them by the ",[7289,71347,71348],{},"exact same weights",[12,71350,71351,71352,71371],{},"The computer never has to calculate a 100-link equation. It just plays a game of hot potato, handing the ",[24923,71353,71355],{"className":71354,"jax":24927},[24926],[24929,71356,71357,71362],{"style":33072,"xmlns":24932,"width":37461,"height":27780,"role":1930,"focusable":8058,"viewBox":69623,"xmlnsXLink":24936},[24938,71358,71359],{},[24941,71360],{"id":71361,"d":57288},"MJX-355-TEX-I-1D6FF",[25022,71363,71364],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71365,71366],{"dataMmlNode":25028},[25022,71367,71368],{"dataMmlNode":25942},[25033,71369],{"dataC":57406,"xLinkHref":71370},"#MJX-355-TEX-I-1D6FF"," packages backward one step at a time!",[25631,71373,71375,71377],{"className":71374},[13441,13442],[25635,71376,66698],{},[13438,71378,71380,71471,71501,71514,71517,71750,71753,71756,71890,71893,72077,72080,72311,72314,72394,72397,72591,72594,72743],{"className":71379},[25641],[12,71381,71382,71383,71402,71403,14448,71450,6959],{},"Remember the partial derivative we calculated earlier? Let's say we did all the chain rule math, and we found out that for one specific weight (",[24923,71384,71386],{"className":71385,"jax":24927},[24926],[24929,71387,71388,71393],{"style":25196,"xmlns":24932,"width":56643,"height":27827,"role":1930,"focusable":8058,"viewBox":58510,"xmlnsXLink":24936},[24938,71389,71390],{},[24941,71391],{"id":71392,"d":37254},"MJX-356-TEX-I-1D464",[25022,71394,71395],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71396,71397],{"dataMmlNode":25028},[25022,71398,71399],{"dataMmlNode":25942},[25033,71400],{"dataC":37310,"xLinkHref":71401},"#MJX-356-TEX-I-1D464","), the slope (the derivative ",[24923,71404,71406],{"className":71405,"jax":24927},[24926],[24929,71407,71408,71419],{"style":54303,"xmlns":24932,"width":58102,"height":56997,"role":1930,"focusable":8058,"viewBox":58103,"xmlnsXLink":24936},[24938,71409,71410,71413,71416],{},[24941,71411],{"id":71412,"d":33624},"MJX-357-TEX-I-1D715",[24941,71414],{"id":71415,"d":25869},"MJX-357-TEX-I-1D438",[24941,71417],{"id":71418,"d":37254},"MJX-357-TEX-I-1D464",[25022,71420,71421],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71422,71423],{"dataMmlNode":25028},[25022,71424,71425,71437,71448],{"dataMmlNode":27622},[25022,71426,71427,71432],{"dataMmlNode":27626,"transform":44821},[25022,71428,71429],{"dataMmlNode":25942},[25033,71430],{"dataC":33697,"xLinkHref":71431},"#MJX-357-TEX-I-1D715",[25022,71433,71434],{"dataMmlNode":25942,"transform":33701},[25033,71435],{"dataC":25945,"xLinkHref":71436},"#MJX-357-TEX-I-1D438",[25022,71438,71439,71443],{"dataMmlNode":27626,"transform":57062},[25022,71440,71441],{"dataMmlNode":25942},[25033,71442],{"dataC":33697,"xLinkHref":71431},[25022,71444,71445],{"dataMmlNode":25942,"transform":33701},[25033,71446],{"dataC":37310,"xLinkHref":71447},"#MJX-357-TEX-I-1D464",[27668,71449],{"width":57073,"height":27671,"x":27672,"y":27673},[106,71451,71452],{},[24923,71453,71455],{"className":71454,"jax":24927},[24926],[24929,71456,71457,71462],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,71458,71459],{},[24941,71460],{"id":71461,"d":26660},"MJX-358-TEX-N-35",[25022,71463,71464],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71465,71466],{"dataMmlNode":25028},[25022,71467,71468],{"dataMmlNode":26031},[25033,71469],{"dataC":26717,"xLinkHref":71470},"#MJX-358-TEX-N-35",[12,71472,71473,71474,71493,71494,71497,71498,71500],{},"What does a positive slope of ",[24923,71475,71477],{"className":71476,"jax":24927},[24926],[24929,71478,71479,71484],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,71480,71481],{},[24941,71482],{"id":71483,"d":26660},"MJX-359-TEX-N-35",[25022,71485,71486],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,71487,71488],{"dataMmlNode":25028},[25022,71489,71490],{"dataMmlNode":26031},[25033,71491],{"dataC":26717,"xLinkHref":71492},"#MJX-359-TEX-N-35"," mean mathematically? It means if we ",[7289,71495,71496],{},"increase"," the weight, the Error will ",[7289,71499,71496],{}," by 5 times as much.",[12,71502,71503,71504,71507,71508,71511,71512,6959],{},"But our goal is to make the Error go ",[106,71505,71506],{},"down"," to zero! If increasing the weight makes the error go up, what should we do? ",[106,71509,71510],{},"We do the exact opposite of the slope."," We need to subtract the slope from our current weight. This is called the ",[106,71513,42142],{},[12,71515,71516],{},"Gradient Descent looks like this:",[24923,71518,71520],{"className":71519,"jax":24927,"display":13594},[24926],[24929,71521,71524,71601],{"style":24931,"xmlns":24932,"width":71522,"height":24934,"role":1930,"focusable":8058,"viewBox":71523,"xmlnsXLink":24936},"36.968ex","0 -750 16339.9 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learning rate is there so that we don't take steps that are too big, to the point that we cross the entire valley.",[12,71754,71755],{},"In math, it looks like this:",[24923,71757,71759],{"className":71758,"jax":24927,"display":13594},[24926],[24929,71760,71763,71798],{"style":57108,"xmlns":24932,"width":71761,"height":57110,"role":1930,"focusable":8058,"viewBox":71762,"xmlnsXLink":24936},"19.462ex","0 -1391 8602.4 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closely at what happens to the negative signs. You are subtracting a negative number. 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1000",[24938,72325,72326,72329,72332,72335,72338],{},[24941,72327],{"id":72328,"d":24944},"MJX-364-TEX-N-28",[24941,72330],{"id":72331,"d":32316},"MJX-364-TEX-I-1D451",[24941,72333],{"id":72334,"d":25910},"MJX-364-TEX-N-2212",[24941,72336],{"id":72337,"d":25885},"MJX-364-TEX-I-1D45C",[24941,72339],{"id":72340,"d":24988},"MJX-364-TEX-N-29",[25022,72342,72343],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,72344,72345,72350,72355,72361,72367],{"dataMmlNode":25028},[25022,72346,72347],{"dataMmlNode":25031},[25033,72348],{"dataC":25035,"xLinkHref":72349},"#MJX-364-TEX-N-28",[25022,72351,72352],{"dataMmlNode":25942,"transform":25040},[25033,72353],{"dataC":32396,"xLinkHref":72354},"#MJX-364-TEX-I-1D451",[25022,72356,72358],{"dataMmlNode":25031,"transform":72357},"translate(1131.2,0)",[25033,72359],{"dataC":26019,"xLinkHref":72360},"#MJX-364-TEX-N-2212",[25022,72362,72364],{"dataMmlNode":25942,"transform":72363},"translate(2131.4,0)",[25033,72365],{"dataC":25973,"xLinkHref":72366},"#MJX-364-TEX-I-1D45C",[25022,72368,72370],{"dataMmlNode":25031,"transform":72369},"translate(2616.4,0)",[25033,72371],{"dataC":25108,"xLinkHref":72372},"#MJX-364-TEX-N-29"," is what we call ",[24923,72375,72377],{"className":72376,"jax":24927},[24926],[24929,72378,72379,72384],{"style":33072,"xmlns":24932,"width":37461,"height":27780,"role":1930,"focusable":8058,"viewBox":69623,"xmlnsXLink":24936},[24938,72380,72381],{},[24941,72382],{"id":72383,"d":57288},"MJX-365-TEX-I-1D6FF",[25022,72385,72386],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,72387,72388],{"dataMmlNode":25028},[25022,72389,72390],{"dataMmlNode":25942},[25033,72391],{"dataC":57406,"xLinkHref":72392},"#MJX-365-TEX-I-1D6FF"," (the blame).",[12,72395,72396],{},"So with some clean up, we get this:",[24923,72398,72400],{"className":72399,"jax":24927,"display":13594},[24926],[24929,72401,72404,72457],{"style":24931,"xmlns":24932,"width":72402,"height":51984,"role":1930,"focusable":8058,"viewBox":72403,"xmlnsXLink":24936},"29.853ex","0 -809 13194.8 1059",[24938,72405,72406,72409,72412,72415,72418,72421,72424,72427,72430,72433,72436,72439,72442,72445,72448,72451,72454],{},[24941,72407],{"id":72408,"d":37254},"MJX-366-TEX-I-1D464",[24941,72410],{"id":72411,"d":25873},"MJX-366-TEX-I-1D45B",[24941,72413],{"id":72414,"d":27397},"MJX-366-TEX-I-1D452",[24941,72416],{"id":72417,"d":24972},"MJX-366-TEX-N-3D",[24941,72419],{"id":72420,"d":25885},"MJX-366-TEX-I-1D45C",[24941,72422],{"id":72423,"d":27389},"MJX-366-TEX-I-1D459",[24941,72425],{"id":72426,"d":32316},"MJX-366-TEX-I-1D451",[24941,72428],{"id":72429,"d":30313},"MJX-366-TEX-N-2B",[24941,72431],{"id":72432,"d":57089},"MJX-366-TEX-I-1D702",[24941,72434],{"id":72435,"d":27409},"MJX-366-TEX-N-22C5",[24941,72437],{"id":72438,"d":57288},"MJX-366-TEX-I-1D6FF",[24941,72440],{"id":72441,"d":25900},"MJX-366-TEX-I-1D446",[24941,72443],{"id":72444,"d":31558},"MJX-366-TEX-V-2032",[24941,72446],{"id":72447,"d":24944},"MJX-366-TEX-N-28",[24941,72449],{"id":72450,"d":25877},"MJX-366-TEX-I-1D461",[24941,72452],{"id":72453,"d":24988},"MJX-366-TEX-N-29",[24941,72455],{"id":72456,"d":32148},"MJX-366-TEX-I-1D465",[25022,72458,72459],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,72460,72461,72484,72489,72512,72517,72522,72528,72534,72539,72552,72558,72563,72568,72574,72580,72585],{"dataMmlNode":25028},[25022,72462,72463,72468],{"dataMmlNode":26023},[25022,72464,72465],{"dataMmlNode":25942},[25033,72466],{"dataC":37310,"xLinkHref":72467},"#MJX-366-TEX-I-1D464",[25022,72469,72470,72475,72480],{"dataMmlNode":27203,"transform":43003,"dataMjxTexclass":27205},[25022,72471,72472],{"dataMmlNode":25942},[25033,72473],{"dataC":25952,"xLinkHref":72474},"#MJX-366-TEX-I-1D45B",[25022,72476,72477],{"dataMmlNode":25942,"transform":46886},[25033,72478],{"dataC":27596,"xLinkHref":72479},"#MJX-366-TEX-I-1D452",[25022,72481,72482],{"dataMmlNode":25942,"transform":46892},[25033,72483],{"dataC":37310,"xLinkHref":72467},[25022,72485,72486],{"dataMmlNode":25031,"transform":71630},[25033,72487],{"dataC":25078,"xLinkHref":72488},"#MJX-366-TEX-N-3D",[25022,72490,72491,72495],{"dataMmlNode":26023,"transform":71636},[25022,72492,72493],{"dataMmlNode":25942},[25033,72494],{"dataC":37310,"xLinkHref":72467},[25022,72496,72497,72502,72507],{"dataMmlNode":27203,"transform":43003,"dataMjxTexclass":27205},[25022,72498,72499],{"dataMmlNode":25942},[25033,72500],{"dataC":25973,"xLinkHref":72501},"#MJX-366-TEX-I-1D45C",[25022,72503,72504],{"dataMmlNode":25942,"transform":27560},[25033,72505],{"dataC":27582,"xLinkHref":72506},"#MJX-366-TEX-I-1D459",[25022,72508,72509],{"dataMmlNode":25942,"transform":71655},[25033,72510],{"dataC":32396,"xLinkHref":72511},"#MJX-366-TEX-I-1D451",[25022,72513,72514],{"dataMmlNode":25031,"transform":71661},[25033,72515],{"dataC":30393,"xLinkHref":72516},"#MJX-366-TEX-N-2B",[25022,72518,72519],{"dataMmlNode":25942,"transform":71667},[25033,72520],{"dataC":57098,"xLinkHref":72521},"#MJX-366-TEX-I-1D702",[25022,72523,72525],{"dataMmlNode":25031,"transform":72524},"translate(7054.6,0)",[25033,72526],{"dataC":27679,"xLinkHref":72527},"#MJX-366-TEX-N-22C5",[25022,72529,72531],{"dataMmlNode":25942,"transform":72530},"translate(7554.9,0)",[25033,72532],{"dataC":57406,"xLinkHref":72533},"#MJX-366-TEX-I-1D6FF",[25022,72535,72537],{"dataMmlNode":25031,"transform":72536},"translate(8221.1,0)",[25033,72538],{"dataC":27679,"xLinkHref":72527},[25022,72540,72542,72547],{"dataMmlNode":31565,"transform":72541},"translate(8721.3,0)",[25022,72543,72544],{"dataMmlNode":25942},[25033,72545],{"dataC":26000,"xLinkHref":72546},"#MJX-366-TEX-I-1D446",[25022,72548,72549],{"dataMmlNode":25031,"transform":52025},[25033,72550],{"dataC":31576,"xLinkHref":72551},"#MJX-366-TEX-V-2032",[25022,72553,72555],{"dataMmlNode":25031,"transform":72554},"translate(9695.4,0)",[25033,72556],{"dataC":25035,"xLinkHref":72557},"#MJX-366-TEX-N-28",[25022,72559,72561],{"dataMmlNode":25942,"transform":72560},"translate(10084.4,0)",[25033,72562],{"dataC":25952,"xLinkHref":72474},[25022,72564,72566],{"dataMmlNode":25942,"transform":72565},"translate(10684.4,0)",[25033,72567],{"dataC":27596,"xLinkHref":72479},[25022,72569,72571],{"dataMmlNode":25942,"transform":72570},"translate(11150.4,0)",[25033,72572],{"dataC":25959,"xLinkHref":72573},"#MJX-366-TEX-I-1D461",[25022,72575,72577],{"dataMmlNode":25031,"transform":72576},"translate(11511.4,0)",[25033,72578],{"dataC":25108,"xLinkHref":72579},"#MJX-366-TEX-N-29",[25022,72581,72583],{"dataMmlNode":25031,"transform":72582},"translate(12122.6,0)",[25033,72584],{"dataC":27679,"xLinkHref":72527},[25022,72586,72588],{"dataMmlNode":25942,"transform":72587},"translate(12622.8,0)",[25033,72589],{"dataC":32181,"xLinkHref":72590},"#MJX-366-TEX-I-1D465",[12,72592,72593],{},"Which, you may have noticed, is our weight update rule.",[12,72595,72596,72670,72671,14448,72700,72719,72720,72739,72740],{},[24923,72597,72599],{"className":72598,"jax":24927},[24926],[24929,72600,72603,72626],{"style":24931,"xmlns":24932,"width":72601,"height":53981,"role":1930,"focusable":8058,"viewBox":72602,"xmlnsXLink":24936},"7.192ex","0 -759 3179.1 1009",[24938,72604,72605,72608,72611,72614,72617,72620,72623],{},[24941,72606],{"id":72607,"d":25900},"MJX-367-TEX-I-1D446",[24941,72609],{"id":72610,"d":31558},"MJX-367-TEX-V-2032",[24941,72612],{"id":72613,"d":24944},"MJX-367-TEX-N-28",[24941,72615],{"id":72616,"d":25873},"MJX-367-TEX-I-1D45B",[24941,72618],{"id":72619,"d":27397},"MJX-367-TEX-I-1D452",[24941,72621],{"id":72622,"d":25877},"MJX-367-TEX-I-1D461",[24941,72624],{"id":72625,"d":24988},"MJX-367-TEX-N-29",[25022,72627,72628],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,72629,72630,72642,72647,72652,72658,72664],{"dataMmlNode":25028},[25022,72631,72632,72637],{"dataMmlNode":31565},[25022,72633,72634],{"dataMmlNode":25942},[25033,72635],{"dataC":26000,"xLinkHref":72636},"#MJX-367-TEX-I-1D446",[25022,72638,72639],{"dataMmlNode":25031,"transform":62621},[25033,72640],{"dataC":31576,"xLinkHref":72641},"#MJX-367-TEX-V-2032",[25022,72643,72644],{"dataMmlNode":25031,"transform":52031},[25033,72645],{"dataC":25035,"xLinkHref":72646},"#MJX-367-TEX-N-28",[25022,72648,72649],{"dataMmlNode":25942,"transform":52037},[25033,72650],{"dataC":25952,"xLinkHref":72651},"#MJX-367-TEX-I-1D45B",[25022,72653,72655],{"dataMmlNode":25942,"transform":72654},"translate(1963.1,0)",[25033,72656],{"dataC":27596,"xLinkHref":72657},"#MJX-367-TEX-I-1D452",[25022,72659,72661],{"dataMmlNode":25942,"transform":72660},"translate(2429.1,0)",[25033,72662],{"dataC":25959,"xLinkHref":72663},"#MJX-367-TEX-I-1D461",[25022,72665,72667],{"dataMmlNode":25031,"transform":72666},"translate(2790.1,0)",[25033,72668],{"dataC":25108,"xLinkHref":72669},"#MJX-367-TEX-N-29"," here acts as a volume dial. If the node is in the flat zone, the little slope (",[24923,72672,72674],{"className":72673,"jax":24927},[24926],[24929,72675,72676,72684],{"style":27778,"xmlns":24932,"width":62597,"height":62598,"role":1930,"focusable":8058,"viewBox":62599,"xmlnsXLink":24936},[24938,72677,72678,72681],{},[24941,72679],{"id":72680,"d":25900},"MJX-368-TEX-I-1D446",[24941,72682],{"id":72683,"d":31558},"MJX-368-TEX-V-2032",[25022,72685,72686],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,72687,72688],{"dataMmlNode":25028},[25022,72689,72690,72695],{"dataMmlNode":31565},[25022,72691,72692],{"dataMmlNode":25942},[25033,72693],{"dataC":26000,"xLinkHref":72694},"#MJX-368-TEX-I-1D446",[25022,72696,72697],{"dataMmlNode":25031,"transform":62621},[25033,72698],{"dataC":31576,"xLinkHref":72699},"#MJX-368-TEX-V-2032",[24923,72701,72703],{"className":72702,"jax":24927},[24926],[24929,72704,72705,72710],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,72706,72707],{},[24941,72708],{"id":72709,"d":25935},"MJX-369-TEX-N-30",[25022,72711,72712],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,72713,72714],{"dataMmlNode":25028},[25022,72715,72716],{"dataMmlNode":26031},[25033,72717],{"dataC":26103,"xLinkHref":72718},"#MJX-369-TEX-N-30",". It multiplies the whole weight update by ",[24923,72721,72723],{"className":72722,"jax":24927},[24926],[24929,72724,72725,72730],{"style":27778,"xmlns":24932,"width":35168,"height":35776,"role":1930,"focusable":8058,"viewBox":35777,"xmlnsXLink":24936},[24938,72726,72727],{},[24941,72728],{"id":72729,"d":25935},"MJX-370-TEX-N-30",[25022,72731,72732],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,72733,72734],{"dataMmlNode":25028},[25022,72735,72736],{"dataMmlNode":26031},[25033,72737],{"dataC":26103,"xLinkHref":72738},"#MJX-370-TEX-N-30",", canceling it out. The network realizes: ",[7289,72741,72742],{},"\"Changing this weight won't affect the error right now, so don't bother moving it.\"",[12,72744,72745],{},"If the node is in the steep middle zone, the little slope ($S'$) is high, and the weight is allowed to update normally.",[7166,72747,72749],{"id":72748},"enhancing-backpropagation","Enhancing Backpropagation",[12,72751,72752],{},"Vanilla gradient descent with a fixed learning rate is slow and gets trapped in shallow valleys of the error surface. Three classical refinements address this:",[12,72754,72755,72758],{},[106,72756,72757],{},"Momentum."," A fraction of the previous weight update is added to the current one:",[24923,72760,72762],{"className":72761,"jax":24927,"display":13594},[24926],[24929,72763,72766,72804],{"style":57108,"xmlns":24932,"width":72764,"height":57110,"role":1930,"focusable":8058,"viewBox":72765,"xmlnsXLink":24936},"27.973ex","0 -1391 12364.1 2099",[24938,72767,72768,72771,72774,72777,72780,72783,72786,72789,72792,72795,72798,72801],{},[24941,72769],{"id":72770,"d":56799},"MJX-371-TEX-N-394",[24941,72772],{"id":72773,"d":37254},"MJX-371-TEX-I-1D464",[24941,72775],{"id":72776,"d":25877},"MJX-371-TEX-I-1D461",[24941,72778],{"id":72779,"d":24972},"MJX-371-TEX-N-3D",[24941,72781],{"id":72782,"d":25910},"MJX-371-TEX-N-2212",[24941,72784],{"id":72785,"d":57089},"MJX-371-TEX-I-1D702",[24941,72787],{"id":72788,"d":27409},"MJX-371-TEX-N-22C5",[24941,72790],{"id":72791,"d":33624},"MJX-371-TEX-I-1D715",[24941,72793],{"id":72794,"d":25869},"MJX-371-TEX-I-1D438",[24941,72796],{"id":72797,"d":30313},"MJX-371-TEX-N-2B",[24941,72799],{"id":72800,"d":33620},"MJX-371-TEX-I-1D6FC",[24941,72802],{"id":72803,"d":25914},"MJX-371-TEX-N-31",[25022,72805,72806],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,72807,72808,72813,72825,72831,72837,72843,72849,72876,72882,72888,72893,72898],{"dataMmlNode":25028},[25022,72809,72810],{"dataMmlNode":25942},[25033,72811],{"dataC":56845,"xLinkHref":72812},"#MJX-371-TEX-N-394",[25022,72814,72815,72820],{"dataMmlNode":26023,"transform":56849},[25022,72816,72817],{"dataMmlNode":25942},[25033,72818],{"dataC":37310,"xLinkHref":72819},"#MJX-371-TEX-I-1D464",[25022,72821,72822],{"dataMmlNode":25942,"transform":43003},[25033,72823],{"dataC":25959,"xLinkHref":72824},"#MJX-371-TEX-I-1D461",[25022,72826,72828],{"dataMmlNode":25031,"transform":72827},"translate(2165,0)",[25033,72829],{"dataC":25078,"xLinkHref":72830},"#MJX-371-TEX-N-3D",[25022,72832,72834],{"dataMmlNode":25031,"transform":72833},"translate(3220.8,0)",[25033,72835],{"dataC":26019,"xLinkHref":72836},"#MJX-371-TEX-N-2212",[25022,72838,72840],{"dataMmlNode":25942,"transform":72839},"translate(3998.8,0)",[25033,72841],{"dataC":57098,"xLinkHref":72842},"#MJX-371-TEX-I-1D702",[25022,72844,72846],{"dataMmlNode":25031,"transform":72845},"translate(4718,0)",[25033,72847],{"dataC":27679,"xLinkHref":72848},"#MJX-371-TEX-N-22C5",[25022,72850,72852,72864,72874],{"dataMmlNode":27622,"transform":72851},"translate(5218.3,0)",[25022,72853,72854,72859],{"dataMmlNode":27626,"transform":33217},[25022,72855,72856],{"dataMmlNode":25942},[25033,72857],{"dataC":33697,"xLinkHref":72858},"#MJX-371-TEX-I-1D715",[25022,72860,72861],{"dataMmlNode":25942,"transform":33701},[25033,72862],{"dataC":25945,"xLinkHref":72863},"#MJX-371-TEX-I-1D438",[25022,72865,72866,72870],{"dataMmlNode":27626,"transform":57190},[25022,72867,72868],{"dataMmlNode":25942},[25033,72869],{"dataC":33697,"xLinkHref":72858},[25022,72871,72872],{"dataMmlNode":25942,"transform":33701},[25033,72873],{"dataC":37310,"xLinkHref":72819},[27668,72875],{"width":57201,"height":27671,"x":27672,"y":27673},[25022,72877,72879],{"dataMmlNode":25031,"transform":72878},"translate(7210.5,0)",[25033,72880],{"dataC":30393,"xLinkHref":72881},"#MJX-371-TEX-N-2B",[25022,72883,72885],{"dataMmlNode":25942,"transform":72884},"translate(8210.7,0)",[25033,72886],{"dataC":33685,"xLinkHref":72887},"#MJX-371-TEX-I-1D6FC",[25022,72889,72891],{"dataMmlNode":25031,"transform":72890},"translate(9072.9,0)",[25033,72892],{"dataC":27679,"xLinkHref":72848},[25022,72894,72896],{"dataMmlNode":25942,"transform":72895},"translate(9573.2,0)",[25033,72897],{"dataC":56845,"xLinkHref":72812},[25022,72899,72901,72905],{"dataMmlNode":26023,"transform":72900},"translate(10406.2,0)",[25022,72902,72903],{"dataMmlNode":25942},[25033,72904],{"dataC":37310,"xLinkHref":72819},[25022,72906,72907,72911,72915],{"dataMmlNode":27203,"transform":43003,"dataMjxTexclass":27205},[25022,72908,72909],{"dataMmlNode":25942},[25033,72910],{"dataC":25959,"xLinkHref":72824},[25022,72912,72913],{"dataMmlNode":25031,"transform":43828},[25033,72914],{"dataC":26019,"xLinkHref":72836},[25022,72916,72917],{"dataMmlNode":26031,"transform":56486},[25033,72918],{"dataC":26035,"xLinkHref":72919},"#MJX-371-TEX-N-31",[12,72921,26147,72922,72941],{},[24923,72923,72925],{"className":72924,"jax":24927},[24926],[24929,72926,72927,72932],{"style":25196,"xmlns":24932,"width":33802,"height":27038,"role":1930,"focusable":8058,"viewBox":33803,"xmlnsXLink":24936},[24938,72928,72929],{},[24941,72930],{"id":72931,"d":33620},"MJX-372-TEX-I-1D6FC",[25022,72933,72934],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,72935,72936],{"dataMmlNode":25028},[25022,72937,72938],{"dataMmlNode":25942},[25033,72939],{"dataC":33685,"xLinkHref":72940},"#MJX-372-TEX-I-1D6FC"," (typically 0.9) is the momentum coefficient. Intuitively, the optimizer behaves like a ball rolling down a hill: it accumulates velocity in directions of consistent descent and damps oscillations in directions where the gradient keeps flipping sign. This both speeds up convergence on flat plateaus and helps escape shallow local minima.",[12,72943,72944,72947,72948,72967,72968,73010],{},[106,72945,72946],{},"Adaptive Learning Rates (Delta-Bar-Delta)."," Rather than picking one ",[24923,72949,72951],{"className":72950,"jax":24927},[24926],[24929,72952,72953,72958],{"style":57080,"xmlns":24932,"width":57081,"height":57082,"role":1930,"focusable":8058,"viewBox":57083,"xmlnsXLink":24936},[24938,72954,72955],{},[24941,72956],{"id":72957,"d":57089},"MJX-373-TEX-I-1D702",[25022,72959,72960],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,72961,72962],{"dataMmlNode":25028},[25022,72963,72964],{"dataMmlNode":25942},[25033,72965],{"dataC":57098,"xLinkHref":72966},"#MJX-373-TEX-I-1D702"," for the whole network, give every weight its own ",[24923,72969,72971],{"className":72970,"jax":24927},[24926],[24929,72972,72975,72986],{"style":35650,"xmlns":24932,"width":72973,"height":50084,"role":1930,"focusable":8058,"viewBox":72974,"xmlnsXLink":24936},"2.523ex","0 -442 1115.3 736.2",[24938,72976,72977,72980,72983],{},[24941,72978],{"id":72979,"d":57089},"MJX-374-TEX-I-1D702",[24941,72981],{"id":72982,"d":27108},"MJX-374-TEX-I-1D456",[24941,72984],{"id":72985,"d":33606},"MJX-374-TEX-I-1D457",[25022,72987,72988],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,72989,72990],{"dataMmlNode":25028},[25022,72991,72992,72997],{"dataMmlNode":26023},[25022,72993,72994],{"dataMmlNode":25942},[25033,72995],{"dataC":57098,"xLinkHref":72996},"#MJX-374-TEX-I-1D702",[25022,72998,73000,73005],{"dataMmlNode":27203,"transform":72999,"dataMjxTexclass":27205},"translate(530,-150) scale(0.707)",[25022,73001,73002],{"dataMmlNode":25942},[25033,73003],{"dataC":27210,"xLinkHref":73004},"#MJX-374-TEX-I-1D456",[25022,73006,73007],{"dataMmlNode":25942,"transform":27214},[25033,73008],{"dataC":33647,"xLinkHref":73009},"#MJX-374-TEX-I-1D457"," that adjusts dynamically:",[23,73012,73013,73019],{},[26,73014,73015,73016,73018],{},"If a weight's gradient direction is consistent across recent updates, ",[7289,73017,71496],{}," its learning rate (we're confident, take bigger steps).",[26,73020,73021,73022,73025],{},"If a weight's gradient keeps oscillating, ",[7289,73023,73024],{},"decrease"," its learning rate (we're overshooting, slow down).",[12,73027,73028],{},"Modern variants of this idea (Adam, RMSProp) are still the default in deep learning today.",[12,73030,73031,73034],{},[106,73032,73033],{},"Quickprop."," A second-order method that approximates the error surface near the current point as a parabola in each weight, then jumps directly to the parabola's minimum. When the assumption holds, Quickprop converges in dramatically fewer epochs than vanilla gradient descent.",[6964,73036,73038],{"id":73037},"_6-applications-of-neural-networks","6. Applications of Neural Networks",[12,73040,73041],{},"Neural networks excel in environments where data is noisy, relationships are non-linear, and traditional algorithmic logic fails.",[7166,73043,73045],{"id":73044},"general-functional-categories","General Functional Categories",[3603,73047,73048,73054,73060,73112],{},[26,73049,73050,73053],{},[106,73051,73052],{},"Clustering (Unsupervised):"," Exploring similarities between data patterns and grouping them. Use cases include data compression and data mining.",[26,73055,73056,73059],{},[106,73057,73058],{},"Classification \u002F Pattern Recognition:"," Assigning an input pattern (e.g., a handwritten symbol or an image) to one of many predefined classes.",[26,73061,73062,73065,73066,73111],{},[106,73063,73064],{},"Function Approximation:"," Finding an estimate of an unknown mathematical function ",[24923,73067,73069],{"className":73068,"jax":24927},[24926],[24929,73070,73073,73087],{"style":24931,"xmlns":24932,"width":73071,"height":24934,"role":1930,"focusable":8058,"viewBox":73072,"xmlnsXLink":24936},"4.299ex","0 -750 1900 1000",[24938,73074,73075,73078,73081,73084],{},[24941,73076],{"id":73077,"d":29821},"MJX-375-TEX-I-1D453",[24941,73079],{"id":73080,"d":24944},"MJX-375-TEX-N-28",[24941,73082],{"id":73083,"d":32148},"MJX-375-TEX-I-1D465",[24941,73085],{"id":73086,"d":24988},"MJX-375-TEX-N-29",[25022,73088,73089],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,73090,73091,73096,73101,73106],{"dataMmlNode":25028},[25022,73092,73093],{"dataMmlNode":25942},[25033,73094],{"dataC":29864,"xLinkHref":73095},"#MJX-375-TEX-I-1D453",[25022,73097,73098],{"dataMmlNode":25031,"transform":44094},[25033,73099],{"dataC":25035,"xLinkHref":73100},"#MJX-375-TEX-N-28",[25022,73102,73103],{"dataMmlNode":25942,"transform":53947},[25033,73104],{"dataC":32181,"xLinkHref":73105},"#MJX-375-TEX-I-1D465",[25022,73107,73108],{"dataMmlNode":25031,"transform":53953},[25033,73109],{"dataC":25108,"xLinkHref":73110},"#MJX-375-TEX-N-29"," subject to noise. This is heavily used in scientific and engineering disciplines.",[26,73113,73114,73117],{},[106,73115,73116],{},"Prediction \u002F Dynamical Systems:"," Forecasting future values of time-sequenced data. Unlike standard function approximation, prediction incorporates the element of time, meaning the system state changes dynamically.",[7166,73119,73121],{"id":73120},"specific-real-world-applications","Specific Real-World Applications",[23,73123,73124,73148,73170,73187],{},[26,73125,73126,73129],{},[106,73127,73128],{},"Medical Diagnosis:",[23,73130,73131,73137,73142],{},[26,73132,73133,73136],{},[7289,73134,73135],{},"Input:"," Patient manifestations (symptoms, lab results, blood tests).",[26,73138,73139,73141],{},[7289,73140,9006],{}," Predicted disease states (e.g., probability of prostate cancer or Hepatitis B).",[26,73143,73144,73147],{},[7289,73145,73146],{},"Advantage:"," Circumvents the need for explicit causal rules, which are often impossible to define in complex human biology.",[26,73149,73150,73153],{},[106,73151,73152],{},"Process Control:",[23,73154,73155,73160,73165],{},[26,73156,73157,73159],{},[7289,73158,73135],{}," Environmental parameters and sensor readings.",[26,73161,73162,73164],{},[7289,73163,9006],{}," Automated control parameters (e.g., adjusting valves, regulating temperature).",[26,73166,73167,73169],{},[7289,73168,73146],{}," Learns ill-structured control functions that are too mathematically complex for standard PID controllers.",[26,73171,73172,73175],{},[106,73173,73174],{},"Financial Forecasting:",[23,73176,73177,73182],{},[26,73178,73179,73181],{},[7289,73180,73135],{}," Macroeconomic factors (CPI, interest rates) and historical stock quotes.",[26,73183,73184,73186],{},[7289,73185,9006],{}," Forecasts of future stock prices or major indices (like the S&P 500).",[26,73188,73189,73192],{},[106,73190,73191],{},"Consumer Credit Evaluation:",[23,73193,73194,73199],{},[26,73195,73196,73198],{},[7289,73197,73135],{}," Personal financial metrics (income, debt-to-income ratio, payment history).",[26,73200,73201,73203],{},[7289,73202,9006],{}," Risk assessment or credit rating scores.",[6964,73205,73207],{"id":73206},"_7-evaluation-of-backpropagation-networks","7. Evaluation of Backpropagation Networks",[12,73209,73210],{},"While Multi-Layer Perceptrons trained with Backpropagation are exceptionally powerful, the architecture has both distinct advantages and inherent limitations.",[7166,73212,31891],{"id":31890},[23,73214,73215,73221,73227,73233],{},[26,73216,73217,73220],{},[106,73218,73219],{},"Great Representation Power:"," The inclusion of non-linear hidden layers allows the network to approximate virtually any continuous function.",[26,73222,73223,73226],{},[106,73224,73225],{},"Wide Practical Applicability:"," Easily adapted to classification, regression, and prediction tasks across countless domains.",[26,73228,73229,73232],{},[106,73230,73231],{},"Easy to Implement:"," The calculus behind backpropagation is complex, but the algorithmic implementation is straightforward matrix multiplication.",[26,73234,73235,73238],{},[106,73236,73237],{},"Good Generalization:"," When trained correctly (often utilizing cross-validation testing), ANNs can make highly accurate predictions on entirely unseen data.",[7166,73240,73242],{"id":73241},"limitations-and-problems","Limitations and Problems",[23,73244,73245,73251,73261,73275,73281,73287,73293,73299,73305],{},[26,73246,73247,73250],{},[106,73248,73249],{},"Slow Convergence:"," Gradient descent can require tens of thousands of epochs (passes through the data) to reach an acceptable error rate.",[26,73252,73253,73256,73257,73260],{},[106,73254,73255],{},"The \"Black Box\" Problem:"," It is nearly impossible to inspect the thousands of weights to understand ",[7289,73258,73259],{},"why"," the network made a specific decision.",[26,73262,73263,73266,73267,73270,73271,73274],{},[106,73264,73265],{},"Local Minima:"," Gradient descent only guarantees finding a ",[7289,73268,73269],{},"local"," minimum error, not necessarily the ",[7289,73272,73273],{},"global"," minimum.",[26,73276,73277,73280],{},[106,73278,73279],{},"Representational Limits:"," Not absolutely every conceivable function can be learned easily within standard operational timeframes.",[26,73282,73283,73286],{},[106,73284,73285],{},"Overfitting \u002F Poor Generalization:"," If trained too long, the network may memorize the training data, leading to a situation where the training error is zero, but the network fails entirely on new data. Cross-validation is strictly required to prevent this.",[26,73288,73289,73292],{},[106,73290,73291],{},"Lack of Quality Assessment:"," There is no mathematically well-founded way to assess the absolute quality of the learning aside from empirical testing on a hold-out set.",[26,73294,73295,73298],{},[106,73296,73297],{},"Network Paralysis:"," If weights become too large, activation functions are driven into deep saturation regions. The derivatives become near zero, freezing the network.",[26,73300,73301,73304],{},[106,73302,73303],{},"Trial-and-Error Architecture:"," Selecting the optimal learning rate, momentum, number of hidden layers, and nodes is still largely an empirical art rather than an exact science.",[26,73306,73307,73310],{},[106,73308,73309],{},"Catastrophic Forgetting (Non-incremental Learning):"," Standard BP networks cannot easily learn \"new\" data on the fly. To incorporate new samples without forgetting old patterns, the network usually must be entirely retrained on the combined dataset.",[9332,73312,31966],{},{"title":180,"searchDepth":181,"depth":181,"links":73314},[73315,73319,73323,73327,73328,73337,73341],{"id":47833,"depth":181,"text":47834,"children":73316},[73317,73318],{"id":47870,"depth":7430,"text":47871},{"id":48049,"depth":7430,"text":48050},{"id":48287,"depth":181,"text":48288,"children":73320},[73321,73322],{"id":48294,"depth":7430,"text":48295},{"id":48327,"depth":7430,"text":48328},{"id":48625,"depth":181,"text":48626,"children":73324},[73325,73326],{"id":48635,"depth":7430,"text":48636},{"id":48664,"depth":7430,"text":48665},{"id":48707,"depth":181,"text":48708},{"id":48748,"depth":181,"text":48749,"children":73329},[73330,73331,73332,73333,73334,73335,73336],{"id":48772,"depth":7430,"text":48773},{"id":48783,"depth":7430,"text":48784},{"id":50834,"depth":7430,"text":50835},{"id":51867,"depth":7430,"text":51868},{"id":53796,"depth":7430,"text":53797},{"id":69434,"depth":7430,"text":69435},{"id":72748,"depth":7430,"text":72749},{"id":73037,"depth":181,"text":73038,"children":73338},[73339,73340],{"id":73044,"depth":7430,"text":73045},{"id":73120,"depth":7430,"text":73121},{"id":73206,"depth":181,"text":73207,"children":73342},[73343,73344],{"id":31890,"depth":7430,"text":31891},{"id":73241,"depth":7430,"text":73242},"2026-04-22",{},"\u002Fblog\u002FArtificial-Neural-Networks-and-Backpropagation",{"title":47828,"description":180},{"loc":73347},"blog\u002FArtificial-Neural-Networks-and-Backpropagation","P3HQFKv5VF57HypbedU76N0iPUwqvblOlwnyRpFk1b8",{"id":73353,"title":73354,"author":7,"body":73355,"date":73403,"description":73404,"draft":185,"edited_at":73403,"extension":186,"featured_image":73405,"meta":73406,"navigation":189,"path":73407,"pinned":185,"seo":73408,"sitemap":73409,"stem":73410,"tags":194,"__hash__":73411},"blog\u002Fblog\u002F小腦（七）評估(ii).md","小腦（七）評估(ii)",{"type":9,"value":73356,"toc":73401},[73357,73360,73363],[12,73358,73359],{},"哇 好像寫不完，先跳過中間的寫這個大紅字的（從81~94空掉）",[12,73361,73362],{},"協調功能障礙評估測試表",[23,73364,73365,73368,73371,73374,73377,73380,73383,73386,73389,73392,73395,73398],{},[26,73366,73367],{},"辨距不良——Pointing and past pointing；畫八字，畫圈；腳跟沿著對側脛骨滑動；腳放地板標記上；坐站姿切換",[26,73369,73370],{},"協同不良——指尖碰鼻尖；手指摸治療師手指；輪流做腳跟對膝蓋；腳趾對治療師手指",[26,73372,73373],{},"交替動作障礙——指尖碰鼻尖；手部旋前旋後；膝蓋extend  flex；走路時改變步速方向",[26,73375,73376],{},"低張力——被動動作；DTR",[26,73378,73379],{},"意向顫抖——觀察功能性活動；輪流鼻尖至手指；指對指；手指對治療師手指；腳趾對治療師手指",[26,73381,73382],{},"休息顫抖——休息時看肢體跟下顎有沒有抖；觀察動作時抖有沒有消失",[26,73384,73385],{},"姿勢顫抖——看坐姿站姿的穩定度",[26,73387,73388],{},"肌無力——固定或維持位置；給予阻力判斷維持位置能力",[26,73390,73391],{},"僵硬——被動動作；觀察功能性活動；觀察休息姿勢",[26,73393,73394],{},"動作緩慢——走路看手臂擺動，改變速度或方向，突然要求停止或轉向；觀察功能性動作（計時）",[26,73396,73397],{},"姿勢干擾——固定及維持位置；坐姿站姿給予干擾；站立時改變BOS",[26,73399,73400],{},"步態干擾——沿直線走，側走，倒走，原地踏步，改變速度方向，繞圈",{"title":180,"searchDepth":181,"depth":181,"links":73402},[],"2026-04-21","P95~98","\u002Fimages\u002Fuploads\u002F1776743958700-IMG_20200712_171445.webp",{},"\u002Fblog\u002F小腦（七）評估(ii)",{"title":73354,"description":73404},{"loc":73407},"blog\u002F小腦（七）評估(ii)","1Pz8Gm0CR0iGY43E8woykq0rBXs7Y6RzDQ7bJ4FlGMk",{"id":73413,"title":73414,"author":7,"body":73415,"date":73403,"description":73499,"draft":185,"edited_at":73403,"extension":186,"featured_image":73500,"meta":73501,"navigation":189,"path":73502,"pinned":185,"seo":73503,"sitemap":73504,"stem":73505,"tags":194,"__hash__":73506},"blog\u002Fblog\u002F小腦（八）治療.md","小腦（八）治療",{"type":9,"value":73416,"toc":73497},[73417,73423,73432,73434,73442,73445,73448,73462,73468,73471,73473,73476,73479,73482,73485,73488,73491],[12,73418,73419,73420,73422],{},"主要有兩種手段——吃藥跟自然恢復",[16,73421],{},"\n在自然恢復中有幾點要注意，退化性小腦疾病會惡化、深層核受損預後比皮質白質受損差，合併多部位受損預後較差",[12,73424,73425,73426,73428,73429,73431],{},"診斷從主要運動功能障礙切入；",[16,73427],{},"\n假如預後良好的受傷，那就以回到原來功能性入手",[16,73430],{},"\n假如持續惡化的受傷，那就從代償+體適能著手",[36,73433],{},[12,73435,73436],{},[106,73437,73438],{},[75,73439,73441],{"className":73440},[78],"治療目的跟代償原則",[12,73443,73444],{},"有氧運動跟阻力訓練——因為容易疲勞導致姿勢控制較差，可以從有氧跟次最大運動著手；如游泳，走路，跳舞，固定腳踏車",[12,73446,73447],{},"更密集且長時間介入——因為小腦受損伴有動作學習缺失，所以要大量重複訓練；如每天十小時，練六個月",[12,73449,73450,73451,73453,73454,73456,73457,73459,73461],{},"接著是代償策略——目的是恢復受傷前的活動跟社會參與能力，即使要利用代償；",[16,73452],{},"\n技巧為：緩慢動作，減少移動肢段數量，給予視覺回饋",[16,73455],{},"\n還可以從增加步寬，以及避免分心來讓穩定性增加",[16,73458],{},[16,73460],{},"\n也可以以使用輔具，比較推四拐，穩定，而且有重量，但只有空間問題",[12,73463,73464,73465,73467],{},"另一種做法是在Trunk加重物——可以讓辨距不良和顫抖問題減少，但患者疲勞會增加（我覺得後效也會有影響，因為他們不太會調適，影響動作呈現）",[16,73466],{},"\n可以根據動作改善表現減少重量",[12,73469,73470],{},"其他的做法有——彈性阻力帶，空氣夾板；增加軀體感覺；水療；生物回饋（肌電等）",[36,73472],{},[12,73474,73475],{},"訓練方式",[12,73477,73478],{},"肌肉無力或不穩——可以使用PNF：包括了關節擠壓，慢且反向維持，交替等長收縮",[12,73480,73481],{},"改善協調性——Frenkel’s Exercise：有簡入繁，由穩定到不穩定",[12,73483,73484],{},"坐姿功能及轉移位——利用轉移板去輔助；坐姿功能性像穿衣，梳理，進食等",[12,73486,73487],{},"獨立站立跟行走準備——可以先訓練髖關節控制（共同收縮）；站姿平衡，坐到站站到坐變換，還要從肌力跟心肺功能著手",[12,73489,73490],{},"靜態平衡——踩泡棉（降低本體覺），並攏或heel to toe（降低BOS）；閉眼（關閉視覺）；雙手抱胸",[12,73492,73493,73494,73496],{},"動態平衡——原地踏步；前後左右跨步；360度轉身；站伸取物，向下抓取東西；",[16,73495],{},"\n增加跨步距離；增加速度；雙腳並攏",{"title":180,"searchDepth":181,"depth":181,"links":73498},[],"寫到動態平衡","\u002Fimages\u002Fuploads\u002F1776746304416-HGWqVs5aoAERy2o.webp",{},"\u002Fblog\u002F小腦（八）治療",{"title":73414,"description":73499},{"loc":73502},"blog\u002F小腦（八）治療","RdWQj5QF4qCztWS6cLC9aLPbsHvVVT8DJpOVvlsC2cg",{"id":73508,"title":73509,"author":7,"body":73510,"date":73403,"description":73626,"draft":185,"edited_at":73403,"extension":186,"featured_image":73627,"meta":73628,"navigation":189,"path":73629,"pinned":185,"seo":73630,"sitemap":73631,"stem":73632,"tags":194,"__hash__":73633},"blog\u002Fblog\u002F小腦（六）評估.md","小腦（六）評估",{"type":9,"value":73511,"toc":73624},[73512,73515,73521,73527,73529,73537,73540,73543,73546,73549,73552,73555,73564,73567,73569,73578,73581,73584,73607,73610,73613,73616,73619,73621],[12,73513,73514],{},"評估主要是看ROM，肌肉力量(MMT)，感覺部份(運動覺，震動覺，兩點辨識覺)；還有就是動作有沒有平滑做出",[12,73516,73517,73518,73520],{},"主要還是看功能性動作部份，包括",[16,73519],{},"\n左右翻身，躺到坐坐到躺，維持姿勢平衡，轉移位，坐到站站到坐，維持直立姿勢，行走（基本功能性動作SOP）",[12,73522,73523,73524,73526],{},"接著是觀察，包括",[16,73525],{},"\n技巧，肢體有沒有無關動作或搖晃，姿勢不穩，協調損傷，動作做完的時間，安全程度",[36,73528],{},[12,73530,73531],{},[106,73532,73533],{},[75,73534,73536],{"className":73535},[78],"基本功能動作問題",[12,73538,73539],{},"翻身——普偏不會有轉不過來問題，除非有嚴重的運動失調",[12,73541,73542],{},"躺到坐動作——也是不太會有甚麼問題，除非有嚴重的運動失調",[12,73544,73545],{},"坐姿穩定——有軀幹失調患者會不穩",[12,73547,73548],{},"坐到站——會變寬姿勢，並且十分動作過程緩慢謹慎",[12,73550,73551],{},"穩定站姿——寬站姿，身體搖晃，屈曲姿勢（降低重心），依賴支撐",[12,73553,73554],{},"轉移位——會持續手支撐，一樣採寬姿勢，過程緩慢且謹慎",[12,73556,73557,73558,73560,73561,73563],{},"行走——基本為 拖著走腳 步態",[16,73559],{},"\n脊髓運動失調：腳掌拍打地面；開眼無異常，閉眼會不穩定，甚至不能走路",[16,73562],{},"\n小腦運動失調：開眼閉眼一樣嚴重；步態採寬步姿，不穩定，不規律，無法走直線",[12,73565,73566],{},"爬樓梯——會增加運動失調表現；極度依賴視覺（盯腳），需要扶手，寬步姿",[36,73568],{},[12,73570,73571,73577],{},[106,73572,73573],{},[75,73574,73576],{"className":73575},[78],"協調的評估","——通常分成粗大動作（姿勢，平衡）跟精細動作（小肌群動作）",[12,73579,73580],{},"著重評估在動作才能上——包括交替動作，動作組成與協同，動作準確度，肢體能否被自己固定",[12,73582,73583],{},"接著是動作的四個層面——M>S>MOS>Skill（對，又是小兒那個）",[3603,73585,73586,73589,73592,73595],{},[26,73587,73588],{},"Mobility——主要是指開始動作，控制動作，動作結束；跟肌肉活性和ROM有關",[26,73590,73591],{},"Stability——在載重層面上維持穩定；主要是坐跟站的穩定；所以出問題就會晃",[26,73593,73594],{},"Mobility on Stability——在穩定姿勢下能改變位置或到新位置的能力，主要是重量轉移；那有問題就是無法重量轉移或者晃",[26,73596,73597,73598,73603,73604,73606],{},"Skill——協調動作，與環境互動完成任務的能力；分成分離，連續，串連動作（",[2330,73599,73602],{"href":73600,"rel":73601},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E6%8E%A7%E5%88%B6%E8%87%AA%E5%B7%B1-Be-water-my-friend%EF%BC%88%E4%BA%8C%EF%BC%89?series=Motor%20Control",[2334],"這邊的圖片下第四段開始","）；也分成open\u002Fclose skill（好像都之前說過）",[16,73605],{},"\n可同時做兩種Task",[12,73608,73609],{},"評估時從簡單到難的測——單肢段，雙肢段對稱，雙肢段不對稱，多肢段任務",[12,73611,73612],{},"評估空間需明亮寬敞；兩張標準椅子，墊子，或者治療床；計時器，眼罩",[12,73614,73615],{},"患者需充份休息；評估前治療師需示範動作；因為小腦問題通常伴有疲勞，或因平衡不好的恐懼等，都會影響評估結果",[12,73617,73618],{},"測的時候重點看：動作性質；時間效率；速度影響；適應能力；穩定性；放置動作（定位動作）；視覺依賴程度；各肢段差異；左右差異；耐受性；一致性",[36,73620],{},[12,73622,73623],{},"接著是評估跑台，先跳過，考完再寫（）",{"title":180,"searchDepth":181,"depth":181,"links":73625},[],"評估--跳過跑台先(81~84)","\u002Fimages\u002Fuploads\u002F1776740017768-IMG_20200616_205355.webp",{},"\u002Fblog\u002F小腦（六）評估",{"title":73509,"description":73626},{"loc":73629},"blog\u002F小腦（六）評估","KXOnM2VM0rSaByWY2k53YjC99eLwOHOCpg_ENVlFy7k",{"id":73635,"title":73636,"author":7,"body":73637,"date":73699,"description":73700,"draft":185,"edited_at":73699,"extension":186,"featured_image":73701,"meta":73702,"navigation":189,"path":73703,"pinned":185,"seo":73704,"sitemap":73705,"stem":73706,"tags":194,"__hash__":73707},"blog\u002Fblog\u002F小腦（三）功能.md","小腦（三）功能",{"type":9,"value":73638,"toc":73697},[73639,73642,73654,73660,73666,73669,73680,73682,73685,73688,73691,73694],[12,73640,73641],{},"小腦主要功能有三個",[12,73643,73644,73645,73647,73648,73650,73651,73653],{},"先來講比較器——主要比較內、外回饋",[16,73646],{},"\n他有一個叫錯誤矯正機制：意思是動作偏離意圖，那就去更正，例如搬重物，出的力不夠，他給你改回來",[16,73649],{},"\n那感覺有閉環，依賴感覺回饋；",[16,73652],{},"\n而也有開環，又稱前饋，用於快速、短時間或者原形動作；因為這些動作太快，不依賴回饋，而是有個程式預設",[12,73655,73656,73657,73659],{},"反射代償器——調節肌肉張力，而且動作在做之前也有預期性代償調整反射",[16,73658],{},"\n假如小腦損傷就會有不正常肌張力（通常low tone），不適合的姿勢調整，辨距不良",[12,73661,73662,73663,73665],{},"動作學習——具備適應性前回饋控制：意思是可以根據以前的記憶建立一個內部模組，並將複雜動作自動化",[16,73664],{},"\n心智練習也跟小腦有關，這時齒狀核跟小腦半球會最活躍",[12,73667,73668],{},"那在治療上的考量有三",[23,73670,73671,73674,73677],{},[26,73672,73673],{},"當比較器失能——導致辨距不良；治療師會挑最有用，最生活化的練習，依賴意識參與校正，需代償策略",[26,73675,73676],{},"當反射代償器失能——異常張力（low tone），走路不會擺動手；以誘發反射為主，並引導患者有意識的學習動作",[26,73678,73679],{},"當以學習動作程式失能——動作記憶會消失；要長期為運動行為修正，步速怎麼教都偏慢，所以要說明，是個漫長的治療過程",[36,73681],{},[12,73683,73684],{},"接著講小腦功能的理論",[12,73686,73687],{},"動作協調理論——協調多關節，產生平滑的多關節動作",[12,73689,73690],{},"時間器假說——負責動作時間表現；決定肌肉啟動的次序",[12,73692,73693],{},"內部模型理論——允許預期性動作控制：因為等感覺回饋再決定動作回傳太慢了；是前饋控制",[12,73695,73696],{},"學習機器——有神經塑性；長期降低電位；有人提出學習是透過攀緣纖維誘導來改變柏金氏輸出來",{"title":180,"searchDepth":181,"depth":181,"links":73698},[],"2026-04-20","功能+理論","\u002Fimages\u002Fuploads\u002F1776681905462-20201206_001939.webp",{},"\u002Fblog\u002F小腦（三）功能",{"title":73636,"description":73700},{"loc":73703},"blog\u002F小腦（三）功能","DIIDbXuv0ZGCndRgStJlmrRkuF7ZTDq-uu8TbZHmJRM",{"id":73709,"title":73710,"author":7,"body":73711,"date":73699,"description":73831,"draft":185,"edited_at":73699,"extension":186,"featured_image":73832,"meta":73833,"navigation":189,"path":73834,"pinned":185,"seo":73835,"sitemap":73836,"stem":73837,"tags":194,"__hash__":73838},"blog\u002Fblog\u002F小腦（二）.md","小腦（二）",{"type":9,"value":73712,"toc":73829},[73713,73719,73725,73728,73789,73791,73794,73803,73806,73827],[12,73714,73715,73716,73718],{},"小腦解剖——在大腦後下方；經由三個軸突束連接，讓上中下三個小腦腳與其他區域連結",[16,73717],{},"\n小腦可以分成前後葉跟絨球小結葉；原發性裂分前後葉；後外側裂分後葉絨球小結葉",[12,73720,73721,73722,73724],{},"小腦皮質構造——神經元較其他腦區多；皮質分三層：分子層，柏金氏層，顆粒細胞層",[16,73723],{},"\n白質層：在小腦深層；灰質：在外沿，有頂核（最內側），間核，齒狀核（最外側）",[12,73726,73727],{},"小腦功能位置",[23,73729,73730,73744,73758,73761,73775],{},[26,73731,73732,73733],{},"內區——在中線區，包含蚓狀部，頂核，是脊髓小腦一部份",[23,73734,73735,73738,73741],{},[26,73736,73737],{},"接收——前庭，網狀，背側和腹側脊髓徑",[26,73739,73740],{},"輸出——頂核",[26,73742,73743],{},"功能——肌肉姿勢控制，張力，直立行走，移動，凝視",[26,73745,73746,73747],{},"中間區——小腦半球中間，包含球核，栓形核，脊髓小腦一部份",[23,73748,73749,73752,73755],{},[26,73750,73751],{},"接收——背側和腹側脊髓徑，網狀核，大腦橋腦小腦徑",[26,73753,73754],{},"輸出——大腦皮質，紅核",[26,73756,73757],{},"功能——控制拮抗機跟協同肌的協調",[26,73759,73760],{},"先提到脊髓小腦——因為內區中間區是小腦唯二會接收脊髓徑的部份；確保動作執行，本體覺整合，姿勢控制",[26,73762,73763,73764],{},"外區——小腦最大功能區，兩個小腦半球的外側跟齒狀核，因為跟大腦最多互動又被稱為大腦小腦",[23,73765,73766,73769,73772],{},[26,73767,73768],{},"接收——來自大腦",[26,73770,73771],{},"輸出——由齒狀核發回大腦",[26,73773,73774],{},"功能——隨意或意志動作，動作計劃，誤差評量，認知功能",[26,73776,73777,73778],{},"絨球小結葉——又稱前庭小腦",[23,73779,73780,73783,73786],{},[26,73781,73782],{},"接收——內耳石，半規管",[26,73784,73785],{},"輸出——沒有跟連結到深層小腦核的部份，直接投射到腦幹跟內外側前庭核",[26,73787,73788],{},"功能——眼球運動，平衡，VOR",[36,73790],{},[12,73792,73793],{},"神經路徑——小腦到脊髓未叉交或雙重交叉；小腦到大腦交叉；所以小腦問題通常都同側病",[12,73795,73796,73797,73799,73800,73802],{},"內在迴路——輸入由苔狀纖維及攀緣纖維，最後作用在柏金氏細胞",[16,73798],{},"\n攀緣纖維：他可以連十顆柏金氏細胞，但柏金氏細胞只能接收一條攀緣纖維；影響柏金氏細胞放電",[16,73801],{},"\n柏金氏細胞:：小腦皮質唯一輸出路徑，到深小小腦核；小腦皮質中唯一人釋放抑制信號的地方",[12,73804,73805],{},"輸入路徑——所以都為興奮性信號，由攀狀及苔狀輸入",[12,73807,73808,73809,73811,73812,73814,73816,73817,73819,73821,73822,73824,73826],{},"輸出路徑",[16,73810],{},"\n球—栓—紅核脊髓徑：影響紅核脊髓核",[16,73813],{},[16,73815],{},"\n頂核網狀路徑：影響網狀脊髓徑",[16,73818],{},[16,73820],{},"\n齒狀核視丘路徑：影響大腦脊髓徑",[16,73823],{},[16,73825],{},"\n頂核前庭路徑：影響前庭脊髓徑",[16,73828],{},{"title":180,"searchDepth":181,"depth":181,"links":73830},[],"解剖跟生理","\u002Fimages\u002Fuploads\u002F1776680161767-77717844_p5_master1200.webp",{},"\u002Fblog\u002F小腦（二）",{"title":73710,"description":73831},{"loc":73834},"blog\u002F小腦（二）","3xXbVwzAyAy3-rqEAVt2mbrueGjYf95u5_wl8PV_lM8",{"id":73840,"title":73841,"author":7,"body":73842,"date":73699,"description":74017,"draft":185,"edited_at":73403,"extension":186,"featured_image":74018,"meta":74019,"navigation":189,"path":74020,"pinned":185,"seo":74021,"sitemap":74022,"stem":74023,"tags":194,"__hash__":74024},"blog\u002Fblog\u002F小腦（五）各功能障礙.md","小腦（五）各功能障礙",{"type":9,"value":73843,"toc":74015},[73844,73850,73852,73858,73860,73863,73865,73868,73870,73879,73881,73884,73886,73898,73900,73903,73905,73920,73922,73934,73936,73945,73947,73950,73952,73955,73957,73983,73985,73988,73999,74001,74004],[12,73845,73846,73847,73849],{},"辨距不良——指判斷物體距離範圍有障礙，無法用手精確觸摸目標；分有辨距過度跟辨距不及",[16,73848],{},"\n多關節比單關節動作嚴重；快速動作會使辨距過度，慢速動作會使辨距不及；無法預測交互作用力矩",[16,73851],{},[12,73853,73854,73855,73857],{},"協調不良——指多關節活動的次序出問題，跟辨距不良有關",[16,73856],{},"\n分解——指一個套動作慢慢拆解成不同次序，是一種代償；正常人手指動作為直線，患者側為曲線",[16,73859],{},[12,73861,73862],{},"交替動作困難——無法快速規律動作；為拮抗肌，協同肌啟動停止時間不良有關",[16,73864],{},[12,73866,73867],{},"回彈現象——失去 立即停止 有力主動動作 的能力；急停會打到自己",[16,73869],{},[12,73871,73872,73873,73875,73876,73878],{},"顫抖——指非意圖搖晃，小腦受損為動作顫抖；",[16,73874],{},"\n意圖顫抖——執行有目的性動作會抖，在動作到結尾最明顯；同一平面動作",[16,73877],{},"\n非意圖顫抖——維持抗動力姿勢會晃",[16,73880],{},[12,73882,73883],{},"肌張力偏低——被動動作的阻力偏低，軟，不結實，DTR偏弱，Extensor tone弱；成人急性期才會這樣，或小腦蚓部發育不全",[16,73885],{},[12,73887,73888,73889,73891,73892,73894,73895,73897],{},"平衡失調——姿勢搖晃增加；過度反應：有干擾時會反應太大反而導致不穩（干擾小也會反應大）",[16,73890],{},"\n單獨小腦前葉受傷：前後晃，高速小幅度晃，可改善",[16,73893],{},"\n前庭小腦受傷：無特定方向，低速大幅度晃，不可改善",[16,73896],{},"\n單獨小腦外葉受傷：基本沒有問題",[16,73899],{},[12,73901,73902],{},"運動失調——一般是感覺跟動作功能合併障礙",[16,73904],{},[12,73906,73907,73908,73910,73911,73913,73914,73916,73917,73919],{},"運動失調步態（醉酒步態）——不穩易跌倒，步寬步長不規律，缺乏手部擺動",[16,73909],{},"\n在擺盪期會hip knee 突然flex，放下，並失控般踩地（大聲踩踏）",[16,73912],{},"\n快走時會無規律轉向，難以轉彎及停止；關節不協調",[16,73915],{},"\n步態程寬步態，手呈防衛型，下肢會突然加速，關於平衡測試如heel to toe會出問題",[16,73918],{},"\n失去平衡是其中的重要因子：假如有平衡問題，通常有運動失調步態；假如無問題，那不太會有運動失調步態",[16,73921],{},[12,73923,73924,73925,73927,73928,73930,73931,73933],{},"動眼陷陷——包括了",[16,73926],{},"\n掃視：眼球迅速移動，有物體定位能力；缺陷就是辨距不良",[16,73929],{},"\n追視：緩緩平滑跟蹤的能力；有問題的話變成掃視型追蹤，先掃視，而且眼的目標的會落在目標後面",[16,73932],{},"\nVOR：假如VOR受傷就是頭眼轉同向（正常是頭眼轉反向，才能讓視角不被帶走）",[16,73935],{},[12,73937,73938,73939,73941,73942,73944],{},"眼球震顫——眼肌肉出現節律型擺動，無法固定視角，而分成",[16,73940],{},"\n凝視誘發眼震——最常見；假如患者注視周邊視角，不管上下左右，都會不自覺得回到中心",[16,73943],{},"\n回彈眼震——注視20秒周邊，要回到中心時，會先跳到反方向短暫移動",[16,73946],{},[12,73948,73949],{},"發音困難——運動型發音困難，說話斷斷續續，音節拖長，模糊，猶豫，音調受損，重音錯位，但語句有意義（次序問題）",[16,73951],{},[12,73953,73954],{},"全身無力——廣泛性肌肉無力；主訴沉重，容易疲乏",[16,73956],{},[12,73958,73959,73960,73962,73963,73965,73966,73968,73969,73971,73972,73974,73975,73977,73979,73980,73982],{},"動作能力受損——是動作行為的關鍵角色；可以利用錯誤調整訊息，改善動作",[16,73961],{},"\n動作調適能力：假如學會一組動作，可以在不同的場合上使用同一組動作，根據環境或需求去做微調",[16,73964],{},"\n後效：例子像是，看到了個大箱子，直覺會覺得很重，所以力度會先預設很大（原始錯誤）",[16,73967],{},"\n但提上手原來沒有很重，就會去調整降低發力（調適）",[16,73970],{},"\n但像是，你以後看到同一個大箱子，也以為它還是這麼輕（移除新指令），就用輕的力去提起，結果重到靠北提不起來（後效就是有儲存參數前提的能力）",[16,73973],{},"\n那缺乏後效就是：假如今天大箱子很重，還是很輕，因為他儲存不了參數前提，所以他都只會用一種參數去提起（調適不能）",[16,73976],{},[16,73978],{},"\n對於復健的影響就十分麻煩了：像剛剛的例子，小腦有自動化微調能力，但小腦患者缺乏這個能力，所以變成需要病人去用意志去調控，而且沒有辦法儲存參數",[16,73981],{},"\n但也不是完全沒救，我們可以利用獎賞機制激活基底核迴路；或者大量重複練習去強化軸突傳導（赫布理論）",[16,73984],{},[12,73986,73987],{},"按功能受損區分",[23,73989,73990,73993,73996],{},[26,73991,73992],{},"絨球小結葉跟中線區——姿勢性張力喪失，直立姿勢與平衡能力喪失，運動失調步態，眼球運動功能受損",[26,73994,73995],{},"中間區——交替動作缺失，動作性震顫，肢體辨距不良",[26,73997,73998],{},"外側區——協調不良，辨距不良，做需要視覺引導的動作障礙",[16,74000],{},[12,74002,74003],{},"按受損區域來分",[23,74005,74006,74009,74012],{},[26,74007,74008],{},"中間區（蚓部，舊小腦）——蹣跚步態，Trunk失調，步態失衡，姿勢性顫抖",[26,74010,74011],{},"半球區（新小腦）——肢體失調，構音困難，low tone",[26,74013,74014],{},"後區（絨球小結葉，原始小腦）——姿勢步態影響，眼球運動障礙",{"title":180,"searchDepth":181,"depth":181,"links":74016},[],"辨距不良——指判斷物體距離範圍有障礙，無法用手精確觸摸目標；分有辨距過度跟辨距不及\n多關節比單關節動作嚴重；快速動作會使辨距過度，慢速動作會使辨距不及；無法預測交互作用力矩","\u002Fimages\u002Fuploads\u002F1776684587198-IMG_20200616_135217.webp",{},"\u002Fblog\u002F小腦（五）各功能障礙",{"title":73841,"description":74017},{"loc":74020},"blog\u002F小腦（五）各功能障礙","JP-V9_cbT8DhVQm4U8u2z5KtkVBeWtVkpJg46pjRMQ4",{"id":74026,"title":74027,"author":7,"body":74028,"date":73699,"description":74080,"draft":185,"edited_at":73699,"extension":186,"featured_image":74081,"meta":74082,"navigation":189,"path":74083,"pinned":185,"seo":74084,"sitemap":74085,"stem":74086,"tags":194,"__hash__":74087},"blog\u002Fblog\u002F小腦（四）損傷.md","小腦（四）損傷",{"type":9,"value":74029,"toc":74078},[74030,74045,74051,74054,74057,74060,74063,74066],[12,74031,74032,74033,74035,74036,74038,74039,74041,74042,74044],{},"先講中風——大概只佔5%；由SCA AICA PICA供應",[16,74034],{},"\nSCA——症狀有同側手臂辨距不良，走路不穩，構音困難，眼球震顫",[16,74037],{},"\nAICA——會導致小腦以外的症；辨距不良，前庭症狀，顏面感覺缺失",[16,74040],{},"\nPICA——暈眩，不穩，運動失調，眼球震顫（多為良性）",[16,74043],{},"\n深小腦核若未受影響，復健程度良好",[12,74046,74047,74048,74050],{},"腫瘤——好發後顱窩；小孩都話多為良性，手術 化療即可；大人的話具侵犯性，預後較差；",[16,74049],{},"\n深小腦核若未受影響，同樣良好",[12,74052,74053],{},"神經性退化疾病——遺傳性，或染色體問題多；偶發性運動失調（罕見）",[12,74055,74056],{},"TBI——為不良預後指標",[12,74058,74059],{},"毒性——酒精或重金屬中毒；前上蚓部：慢性酒精中毒",[12,74061,74062],{},"炎症——多發性硬化症",[12,74064,74065],{},"先天畸型——奇阿里畸型",[12,74067,74068,74069,74071,74072,74074,74075,74077],{},"而預後分成三種類型",[16,74070],{},"\n進行性：餘生都要周期性物理治療維持功能",[16,74073],{},"\n非進行性：較有可能實質性復原",[16,74076],{},"\n更多腦部受牽連：具挑戰",{"title":180,"searchDepth":181,"depth":181,"links":74079},[],"損傷,再給我三小時!","\u002Fimages\u002Fuploads\u002F1776683386464-IMG_20200614_204008.webp",{},"\u002Fblog\u002F小腦（四）損傷",{"title":74027,"description":74080},{"loc":74083},"blog\u002F小腦（四）損傷","DahauWPpfx-18pjfp5blFi0SSV0_9D7suh899_zbwbY",{"id":74089,"title":74090,"author":6946,"body":74091,"date":74911,"description":74912,"draft":185,"edited_at":74911,"extension":186,"featured_image":194,"meta":74913,"navigation":189,"path":74914,"pinned":185,"seo":74915,"sitemap":74916,"stem":74917,"tags":194,"__hash__":74918},"blog\u002Fblog\u002FA-Top-Level-View-of-Computer-Function-and-Interconnection.md","A Top-Level View of Computer Function and Interconnection",{"type":9,"value":74092,"toc":74882},[74093,74100,74103,74107,74114,74146,74150,74153,74176,74179,74193,74197,74204,74208,74234,74238,74241,74267,74271,74282,74302,74313,74330,74333,74337,74344,74353,74357,74410,74414,74424,74442,74445,74449,74452,74482,74486,74496,74500,74503,74513,74518,74528,74533,74578,74582,74585,74596,74600,74607,74611,74617,74627,74637,74643,74646,74650,74657,74661,74668,74692,74696,74766,74770,74776,74782,74796,74800,74807,74822,74825,74849,74851,74854,74869,74879],[12,74094,74095,74096,74099],{},"So you've heard that computers are made of a processor, memory, and I\u002FO devices — but how do they actually ",[7289,74097,74098],{},"work together","? This post walks through the big picture: how a computer fetches and runs instructions, how it deals with interruptions, and how all its components talk to each other through buses and point-to-point links.",[12,74101,74102],{},"If you're new to computer architecture, this is a great starting point. Let's dive in.",[6964,74104,74106],{"id":74105},"_1-the-von-neumann-architecture","1. The Von Neumann Architecture",[12,74108,74109,74110,74113],{},"Almost every modern computer traces its design back to ideas developed by ",[106,74111,74112],{},"John von Neumann"," at the Institute for Advanced Studies in Princeton. The architecture rests on three core principles:",[3603,74115,74116,74122,74132],{},[26,74117,74118,74121],{},[106,74119,74120],{},"Unified memory"," — Both data and instructions live in the same read-write memory. There isn't one memory for programs and another for data; they share the same space.",[26,74123,74124,74127,74128,74131],{},[106,74125,74126],{},"Address-based access"," — Memory contents are referenced by their ",[7289,74129,74130],{},"location"," (address), regardless of whether the content is an integer, a character, or a machine instruction.",[26,74133,74134,74137,74138,74141,74142,74145],{},[106,74135,74136],{},"Sequential execution"," — The processor works through instructions one after another, in order, unless an instruction explicitly tells it to jump somewhere else.\nThis might sound obvious today, but the alternative — a ",[106,74139,74140],{},"hardwired program",", where you physically rewire components to change what the computer does — was once the norm. The von Neumann model gave us the power of ",[7289,74143,74144],{},"software",": change the program in memory, and the same hardware does something completely different.",[7166,74147,74149],{"id":74148},"the-three-core-components","The Three Core Components",[12,74151,74152],{},"At the highest level, a computer is built from three types of modules:",[23,74154,74155,74161,74170],{},[26,74156,74157,74160],{},[106,74158,74159],{},"Processor (CPU)"," — Reads instructions and data, performs computation, writes results, and coordinates everything via control signals. It also receives interrupt signals (more on that soon).",[26,74162,74163,74166,74167,74169],{},[106,74164,74165],{},"Memory"," — A collection of ",[7289,74168,18725],{}," words, each with a unique numerical address (0, 1, …, N−1). The processor can read from or write to any address.",[26,74171,74172,74175],{},[106,74173,74174],{},"I\u002FO Modules"," — From the computer's internal perspective, I\u002FO works a lot like memory — there are read and write operations. A single I\u002FO module may control multiple external devices (keyboard, display, disk, etc.).",[12,74177,74178],{},"Two special registers sit between the processor and memory:",[23,74180,74181,74187],{},[26,74182,74183,74186],{},[106,74184,74185],{},"MAR (Memory Address Register)"," — Holds the address of the memory location the processor wants to access.",[26,74188,74189,74192],{},[106,74190,74191],{},"MBR (Memory Buffer Register)"," — Holds the data being written to or read from that address.",[6964,74194,74196],{"id":74195},"_2-the-instruction-cycle","2. The Instruction Cycle",[12,74198,74199,74200,74203],{},"The fundamental rhythm of a processor is the ",[106,74201,74202],{},"instruction cycle",": fetch an instruction, then execute it. Over and over again. Let's break that down.",[7166,74205,74207],{"id":74206},"the-fetch-phase","The Fetch Phase",[3603,74209,74210,74219,74222,74225,74231],{},[26,74211,9455,74212,74215,74216,74218],{},[106,74213,74214],{},"Program Counter (PC)"," holds the address of the ",[7289,74217,9765],{}," instruction to fetch.",[26,74220,74221],{},"The processor reads the instruction at that address from memory.",[26,74223,74224],{},"The PC is incremented so it points to the following instruction.",[26,74226,74227,74228,6959],{},"The fetched instruction is loaded into the ",[106,74229,74230],{},"Instruction Register (IR)",[26,74232,74233],{},"The processor decodes the instruction and figures out what to do.",[7166,74235,74237],{"id":74236},"the-execute-phase","The Execute Phase",[12,74239,74240],{},"Once the processor knows what the instruction says, the action falls into one of four categories:",[23,74242,74243,74249,74255,74261],{},[26,74244,74245,74248],{},[106,74246,74247],{},"Processor–Memory"," — Transfer data between the CPU and main memory (load\u002Fstore).",[26,74250,74251,74254],{},[106,74252,74253],{},"Processor–I\u002FO"," — Transfer data between the CPU and an I\u002FO module.",[26,74256,74257,74260],{},[106,74258,74259],{},"Data processing"," — Perform arithmetic or logical operations on data.",[26,74262,74263,74266],{},[106,74264,74265],{},"Control"," — Change the sequence of execution (e.g., jump to a different address).",[7166,74268,74270],{"id":74269},"a-simple-example","A Simple Example",[12,74272,74273,74274,74277,74278,74281],{},"Imagine we want to add the contents of memory location ",[106,74275,74276],{},"940"," to the contents of location ",[106,74279,74280],{},"941",". With a simple instruction set, this might take three instruction cycles:",[3603,74283,74284,74290,74296],{},[26,74285,74286,74289],{},[106,74287,74288],{},"LOAD 940"," — Fetch the value at address 940 into the accumulator.",[26,74291,74292,74295],{},[106,74293,74294],{},"ADD 941"," — Fetch the value at address 941 and add it to the accumulator.",[26,74297,74298,74301],{},[106,74299,74300],{},"STORE 941"," — Write the result from the accumulator back to address 941.\nEach of these is one fetch + one execute. Three cycles total.",[12,74303,74304,74305,74308,74309,74312],{},"Now, some processors have more powerful instructions. Consider the PDP-11 instruction ",[6979,74306,74307],{},"ADD B, A",", which does all of this in a ",[7289,74310,74311],{},"single"," instruction cycle — but that single execute phase is more complex:",[3603,74314,74315,74318,74321,74324,74327],{},[26,74316,74317],{},"Fetch the ADD instruction.",[26,74319,74320],{},"Read memory location A into the processor.",[26,74322,74323],{},"Read memory location B into the processor (the CPU needs two internal registers to hold both values).",[26,74325,74326],{},"Add the two values.",[26,74328,74329],{},"Write the result back to location A.",[12,74331,74332],{},"The takeaway: richer instructions can reduce the number of cycles, but each cycle does more work.",[6964,74334,74336],{"id":74335},"_3-interrupts","3. Interrupts",[12,74338,74339,74340,74343],{},"Here's a problem. External devices like printers and disks are ",[7289,74341,74342],{},"much"," slower than the processor. If the CPU sends data to a printer and then just waits for the printer to finish, it wastes thousands of instruction cycles doing nothing. That's terrible for performance.",[12,74345,74346,74349,74350,74352],{},[106,74347,74348],{},"Interrupts"," solve this by letting other modules (I\u002FO controllers, timers, etc.) signal the processor that something needs attention — ",[7289,74351,13580],{}," forcing the CPU to sit idle.",[7166,74354,74356],{"id":74355},"classes-of-interrupts","Classes of Interrupts",[7085,74358,74359,74368],{},[7088,74360,74361],{},[7091,74362,74363,74365],{},[7094,74364,23887],{},[7094,74366,74367],{},"What triggers it",[7104,74369,74370,74380,74390,74400],{},[7091,74371,74372,74377],{},[7109,74373,74374],{},[106,74375,74376],{},"Program",[7109,74378,74379],{},"A condition arising from instruction execution — arithmetic overflow, division by zero, illegal instruction, or an out-of-bounds memory access.",[7091,74381,74382,74387],{},[7109,74383,74384],{},[106,74385,74386],{},"Timer",[7109,74388,74389],{},"A hardware timer inside the processor fires periodically, letting the OS perform housekeeping tasks on a regular schedule.",[7091,74391,74392,74397],{},[7109,74393,74394],{},[106,74395,74396],{},"I\u002FO",[7109,74398,74399],{},"An I\u002FO controller signals that an operation has completed normally, that it needs service, or that an error occurred.",[7091,74401,74402,74407],{},[7109,74403,74404],{},[106,74405,74406],{},"Hardware failure",[7109,74408,74409],{},"Something physical went wrong — power failure, memory parity error, etc.",[7166,74411,74413],{"id":74412},"how-interrupts-improve-efficiency","How Interrupts Improve Efficiency",[12,74415,74416,74419,74420,74423],{},[106,74417,74418],{},"Without interrupts:"," The user program calls WRITE, prepares the I\u002FO data (code segment 4), issues the I\u002FO command, and then ",[7289,74421,74422],{},"waits"," until the device finishes before running the completion code (segment 5). During that wait, the CPU is idle.",[12,74425,74426,74429,74430,74433,74434,74437,74438,74441],{},[106,74427,74428],{},"With interrupts:"," The program issues the I\u002FO command and then ",[7289,74431,74432],{},"keeps executing other instructions",". The I\u002FO device works concurrently in the background. When it finishes, it sends an ",[106,74435,74436],{},"interrupt request signal",". The processor pauses its current work, jumps to an ",[106,74439,74440],{},"interrupt handler"," (a small routine, usually part of the OS, that services the device), and then resumes the original program right where it left off.",[12,74443,74444],{},"This is dramatically more efficient. The CPU stays busy doing useful work instead of waiting.",[7166,74446,74448],{"id":74447},"what-happens-during-an-interrupt","What Happens During an Interrupt",[12,74450,74451],{},"When the processor detects an interrupt (typically checked at the end of each instruction cycle):",[3603,74453,74454,74464,74470,74476],{},[26,74455,74456,74459,74460,74463],{},[106,74457,74458],{},"Suspend"," the current program and ",[106,74461,74462],{},"save its context"," (register contents, PC value, status flags) — usually onto a system stack.",[26,74465,74466,74469],{},[106,74467,74468],{},"Set the PC"," to the starting address of the appropriate interrupt handler.",[26,74471,74472,74475],{},[106,74473,74474],{},"Fetch and execute"," the interrupt handler instructions (determine the interrupt source, perform the needed action).",[26,74477,74478,74481],{},[106,74479,74480],{},"Restore"," the saved context and resume the interrupted program.\nThere is some overhead — the handler must figure out what caused the interrupt and respond — but the time saved by not idling far outweighs it.",[7166,74483,74485],{"id":74484},"short-io-wait-vs-long-io-wait","Short I\u002FO Wait vs. Long I\u002FO Wait",[12,74487,74488,74489,74492,74493,6959],{},"If the I\u002FO operation finishes before the program issues its next WRITE, everything flows smoothly. But if the program reaches a ",[7289,74490,74491],{},"second"," WRITE before the first I\u002FO operation completes, it has to wait — the device is still busy. This is a ",[106,74494,74495],{},"long I\u002FO wait",[7166,74497,74499],{"id":74498},"multiple-interrupts","Multiple Interrupts",[12,74501,74502],{},"What if several devices interrupt at the same time? Two strategies:",[12,74504,74505,74508,74509,74512],{},[106,74506,74507],{},"1. Disabled (sequential) approach:","\nWhile handling one interrupt, the processor ",[7289,74510,74511],{},"disables"," further interrupts. Any new interrupt stays pending. Once the current handler finishes and re-enables interrupts, the processor checks for and services the next pending interrupt.",[23,74514,74515],{},[26,74516,74517],{},"Simple, but it ignores urgency. A time-critical interrupt might have to wait behind a low-priority one.",[12,74519,74520,74523,74524,74527],{},[106,74521,74522],{},"2. Priority-based (nested) approach:","\nEach interrupt source has a ",[106,74525,74526],{},"priority level",". A higher-priority interrupt can preempt (interrupt) a lower-priority handler. Lower-priority interrupts must wait.",[12,74529,74530,74532],{},[106,74531,22702],{}," Suppose we have three devices with these priorities — Printer: 2, Disk: 4, Communications line: 5.",[23,74534,74535,74542,74548,74554,74560,74566,74572],{},[26,74536,74537,74538,74541],{},"At ",[7289,74539,74540],{},"t = 0",", the user program starts.",[26,74543,74537,74544,74547],{},[7289,74545,74546],{},"t = 10",", the printer interrupts. The user program's state is saved; the printer ISR (Interrupt Service Routine) begins.",[26,74549,74537,74550,74553],{},[7289,74551,74552],{},"t = 15",", the communications line interrupts (priority 5 > 2). The printer ISR is paused and its state saved; the comm ISR begins.",[26,74555,74537,74556,74559],{},[7289,74557,74558],{},"t = 20",", the disk interrupts (priority 4 \u003C 5). It must wait — the comm ISR continues.",[26,74561,74537,74562,74565],{},[7289,74563,74564],{},"t = 25",", the comm ISR finishes. The processor restores the printer ISR's state — but before it executes even one instruction, it notices the pending disk interrupt (priority 4 > 2) and services it.",[26,74567,74537,74568,74571],{},[7289,74569,74570],{},"t = 35",", the disk ISR finishes. The printer ISR finally resumes.",[26,74573,74537,74574,74577],{},[7289,74575,74576],{},"t = 40",", the printer ISR finishes. Control returns to the user program.\nThis priority scheme ensures that urgent devices get serviced quickly, even at the cost of delaying less critical ones.",[6964,74579,74581],{"id":74580},"_4-io-function-and-dma","4. I\u002FO Function and DMA",[12,74583,74584],{},"The processor can exchange data with I\u002FO modules directly — reading from or writing to them using special I\u002FO instructions (or memory-mapped I\u002FO, where certain memory addresses correspond to device registers).",[12,74586,74587,74588,74591,74592,74595],{},"But there's a better way for large data transfers: ",[106,74589,74590],{},"Direct Memory Access (DMA)",". With DMA, the processor grants an I\u002FO module the authority to read from or write to main memory ",[7289,74593,74594],{},"on its own",". The I\u002FO module handles the entire transfer, and the processor is free to do other work. The CPU only gets involved at the start (to set up the transfer) and at the end (when the I\u002FO module signals completion via an interrupt).",[6964,74597,74599],{"id":74598},"_5-interconnection-structures","5. Interconnection Structures",[12,74601,74602,74603,74606],{},"So the processor, memory, and I\u002FO modules need to communicate. How? Through an ",[106,74604,74605],{},"interconnection structure",". There are two main approaches.",[7166,74608,74610],{"id":74609},"bus-interconnection","Bus Interconnection",[12,74612,9407,74613,74616],{},[106,74614,74615],{},"bus"," is a shared communication pathway made up of multiple lines (wires). All modules connect to the same bus and share it. A bus has three types of lines:",[12,74618,74619,74622,74623,74626],{},[106,74620,74621],{},"Data Bus:","\nCarries the actual data being transferred. The ",[7289,74624,74625],{},"width"," of the data bus — how many parallel lines it has (e.g., 32, 64, 128 bits) — directly affects how many bits can move at once and is a key factor in overall system performance.",[12,74628,74629,74632,74633,74636],{},[106,74630,74631],{},"Address Bus:","\nSpecifies ",[7289,74634,74635],{},"where"," data should go or come from. The width of the address bus determines the maximum memory the system can address. Higher-order bits select which module (memory chip, I\u002FO port), and lower-order bits select a location within that module.",[12,74638,74639,74642],{},[106,74640,74641],{},"Control Bus:","\nCarries command and timing signals — things like \"this is a read operation,\" \"this is a write,\" and \"the data on the bus is valid now.\" Because the data and address lines are shared by all components, the control bus coordinates who gets to use them and when.",[12,74644,74645],{},"Buses are simple and low-cost, but they become bottlenecks when many high-speed components compete for the same shared pathway.",[7166,74647,74649],{"id":74648},"point-to-point-interconnect","Point-to-Point Interconnect",[12,74651,74652,74653,74656],{},"Modern systems have largely moved away from shared buses toward ",[106,74654,74655],{},"point-to-point interconnects",", where each pair of components has its own dedicated connection. This eliminates the arbitration overhead of shared buses and provides much higher bandwidth.",[6964,74658,74660],{"id":74659},"_6-intel-quickpath-interconnect-qpi","6. Intel QuickPath Interconnect (QPI)",[12,74662,74663,74664,74667],{},"Introduced by Intel in 2008, ",[106,74665,74666],{},"QPI"," is a point-to-point interconnect that replaced the older shared front-side bus. Its key features:",[23,74669,74670,74676,74682],{},[26,74671,74672,74675],{},[106,74673,74674],{},"Multiple direct connections"," — Components are linked in pairs, removing the need for bus arbitration.",[26,74677,74678,74681],{},[106,74679,74680],{},"Layered protocol architecture"," — Rather than simple control signals, QPI uses a structured protocol stack (like a network protocol), which makes it more flexible and robust.",[26,74683,74684,74687,74688,74691],{},[106,74685,74686],{},"Packetized data transfer"," — Data travels in ",[7289,74689,74690],{},"packets",", each containing control headers and error control codes.",[7166,74693,74695],{"id":74694},"qpi-layers","QPI Layers",[7085,74697,74698,74708],{},[7088,74699,74700],{},[7091,74701,74702,74705],{},[7094,74703,74704],{},"Layer",[7094,74706,74707],{},"Role",[7104,74709,74710,74720,74742,74752],{},[7091,74711,74712,74717],{},[7109,74713,74714],{},[106,74715,74716],{},"Physical",[7109,74718,74719],{},"The actual wires and electrical signaling.",[7091,74721,74722,74727],{},[7109,74723,74724],{},[106,74725,74726],{},"Link",[7109,74728,74729,74730,74733,74734,74737,74738,74741],{},"Handles ",[106,74731,74732],{},"flow control"," (preventing a fast sender from overwhelming a slow receiver) and ",[106,74735,74736],{},"error control"," (detecting and recovering from bit errors using CRC codes). Operates on units called ",[7289,74739,74740],{},"flits"," (flow control units): 72 bits of payload + 8 bits of CRC.",[7091,74743,74744,74749],{},[7109,74745,74746],{},[106,74747,74748],{},"Routing",[7109,74750,74751],{},"Determines the path a packet takes through the interconnect network. Routes are defined by firmware.",[7091,74753,74754,74759],{},[7109,74755,74756],{},[106,74757,74758],{},"Protocol",[7109,74760,74761,74762,74765],{},"Defines the packet as the unit of transfer. A critical function here is the ",[106,74763,74764],{},"cache coherency protocol",", which ensures that when multiple caches hold copies of the same memory location, they all stay consistent.",[6964,74767,74769],{"id":74768},"_7-pci-express-pcie","7. PCI Express (PCIe)",[12,74771,74772,74775],{},[106,74773,74774],{},"PCI (Peripheral Component Interconnect)"," was once the dominant I\u002FO bus — a high-bandwidth, processor-independent shared bus. But as device speeds increased, the shared bus became a bottleneck.",[12,74777,74778,74781],{},[106,74779,74780],{},"PCI Express (PCIe)"," replaced it with a point-to-point interconnect scheme. Key motivations:",[23,74783,74784,74790],{},[26,74785,74786,74789],{},[106,74787,74788],{},"High capacity"," for modern I\u002FO devices like Gigabit Ethernet.",[26,74791,74792,74795],{},[106,74793,74794],{},"Support for time-dependent data streams"," (audio, video) that need guaranteed bandwidth.",[7166,74797,74799],{"id":74798},"pcie-transaction-layer-tl","PCIe Transaction Layer (TL)",[12,74801,74802,74803,74806],{},"The Transaction Layer sits at the top of the PCIe protocol stack. It receives read and write requests from software and creates ",[106,74804,74805],{},"request packets"," that travel down through the layers to the destination device.",[12,74808,74809,74810,74813,74814,74817,74818,74821],{},"Most transactions use a ",[106,74811,74812],{},"split transaction"," technique: the source sends a request packet, then waits for a ",[106,74815,74816],{},"completion packet"," in response. Some writes and messages are ",[7289,74819,74820],{},"posted"," (fire-and-forget — no response expected).",[12,74823,74824],{},"The TL supports four address spaces:",[23,74826,74827,74832,74837,74843],{},[26,74828,74829,74831],{},[106,74830,74165],{}," — Maps to system main memory and memory-mapped I\u002FO devices.",[26,74833,74834,74836],{},[106,74835,74396],{}," — For legacy PCI devices with reserved address ranges.",[26,74838,74839,74842],{},[106,74840,74841],{},"Configuration"," — Lets the system read\u002Fwrite configuration registers of PCIe devices.",[26,74844,74845,74848],{},[106,74846,74847],{},"Message"," — For control signals: interrupts, error handling, power management.\nThe TL supports both 32-bit and extended 64-bit memory addressing.",[6964,74850,42101],{"id":25635},[12,74852,74853],{},"Here's the big picture of what we covered:",[12,74855,74856,74857,74860,74861,74864,74865,74868],{},"The computer's fundamental operation is a loop — ",[106,74858,74859],{},"fetch"," an instruction, ",[106,74862,74863],{},"execute"," it, check for ",[106,74866,74867],{},"interrupts",", repeat. Interrupts make this cycle dramatically more efficient by letting the CPU do useful work while slow I\u002FO devices operate in the background. When multiple interrupts compete, a priority system ensures the most urgent ones get serviced first.",[12,74870,74871,74872,74875,74876,74878],{},"All of this requires a way for the processor, memory, and I\u002FO to communicate. Older systems used ",[106,74873,74874],{},"shared buses"," (simple but limited in bandwidth). Modern systems use ",[106,74877,74655],{}," like Intel's QPI and PCIe, which offer dedicated high-speed links, layered protocols, and packetized data transfer.",[12,74880,74881],{},"Understanding this top-level view — the instruction cycle, interrupts, and interconnection — gives you the foundation for everything else in computer architecture. Each of these topics goes much deeper, but now you have the mental map to navigate them.",{"title":180,"searchDepth":181,"depth":181,"links":74883},[74884,74887,74892,74899,74900,74904,74907,74910],{"id":74105,"depth":181,"text":74106,"children":74885},[74886],{"id":74148,"depth":7430,"text":74149},{"id":74195,"depth":181,"text":74196,"children":74888},[74889,74890,74891],{"id":74206,"depth":7430,"text":74207},{"id":74236,"depth":7430,"text":74237},{"id":74269,"depth":7430,"text":74270},{"id":74335,"depth":181,"text":74336,"children":74893},[74894,74895,74896,74897,74898],{"id":74355,"depth":7430,"text":74356},{"id":74412,"depth":7430,"text":74413},{"id":74447,"depth":7430,"text":74448},{"id":74484,"depth":7430,"text":74485},{"id":74498,"depth":7430,"text":74499},{"id":74580,"depth":181,"text":74581},{"id":74598,"depth":181,"text":74599,"children":74901},[74902,74903],{"id":74609,"depth":7430,"text":74610},{"id":74648,"depth":7430,"text":74649},{"id":74659,"depth":181,"text":74660,"children":74905},[74906],{"id":74694,"depth":7430,"text":74695},{"id":74768,"depth":181,"text":74769,"children":74908},[74909],{"id":74798,"depth":7430,"text":74799},{"id":25635,"depth":181,"text":42101},"2026-04-19","So you've heard that computers are made of a processor, memory, and I\u002FO devices — but how do they actually work together? This post walks through the big picture: how a computer fetches and runs instructions, how it deals with interruptions, and how all its components talk to each other through buses and point-to-point links.",{},"\u002Fblog\u002FA-Top-Level-View-of-Computer-Function-and-Interconnection",{"title":74090,"description":74912},{"loc":74914},"blog\u002FA-Top-Level-View-of-Computer-Function-and-Interconnection","UqcUU_uBVGq3CfxbWv0lODHcWxe61h3dxG4b2TSnj3k",{"id":74920,"title":74921,"author":7,"body":74922,"date":74911,"description":74971,"draft":185,"edited_at":74911,"extension":186,"featured_image":74972,"meta":74973,"navigation":189,"path":74974,"pinned":185,"seo":74975,"sitemap":74976,"stem":74977,"tags":194,"__hash__":74978},"blog\u002Fblog\u002FBalance（九）干預.md","Balance（九）干預",{"type":9,"value":74923,"toc":74969},[74924,74927,74930,74964,74966],[12,74925,74926],{},"剛從GMFCS解放回來，還好不難啃",[12,74928,74929],{},"各種前庭問題的干預（以下先省略跑台內容 後面再寫）",[23,74931,74932,74940,74948,74956],{},[26,74933,74934,74935],{},"BPPV——治療目的",[23,74936,74937],{},[26,74938,74939],{},"回復耳石；頭動時暈眩，眼球震顫降低；提升平衡能力；自我治療策略；讓頭部動作可以回到ADL",[26,74941,74942,74943],{},"單側前庭傷害（通常在6~8周後顯著恢復）——治療目的",[23,74944,74945],{},[26,74946,74947],{},"提升頭部動作時的凝視穩定；減敏；促進靜動態姿勢穩定；獨立完成居家運動",[26,74949,74950,74951],{},"雙側前庭傷害——治療目的",[23,74952,74953],{},[26,74954,74955],{},"提升頭部動作時的凝視穩定；增加動靜態平衡；獨立完成居定運動；強化ADL能力",[26,74957,74958,74959],{},"中樞前庭低功能——治療目的",[23,74960,74961],{},[26,74962,74963],{},"確定防跌倒策略和衛教；教育代償性凝視穩定策略；獨立完成居家運動",[36,74965],{},[12,74967,74968],{},"原來我自己傻逼在165頁後面多加了幾頁空白誤導自己還有十多頁PPT，笑死",{"title":180,"searchDepth":181,"depth":181,"links":74970},[],"一樣先跳過跑台描述","\u002Fimages\u002Fuploads\u002F1776583400332-EloH0HNVgAEj6cN.webp",{},"\u002Fblog\u002FBalance（九）干預",{"title":74921,"description":74971},{"loc":74974},"blog\u002FBalance（九）干預","0ADHe9nD4rcMP5GEEAktPphLg5wRxqzoNGOR62-fLtA",{"id":74980,"title":74981,"author":7,"body":74982,"date":74911,"description":75317,"draft":185,"edited_at":74911,"extension":186,"featured_image":75318,"meta":75319,"navigation":189,"path":75320,"pinned":185,"seo":75321,"sitemap":75322,"stem":75323,"tags":194,"__hash__":75324},"blog\u002Fblog\u002FGMFCS.md","GMFCS",{"type":9,"value":74983,"toc":75315},[74984,74987,74990,75007,75142,75145,75171,75173,75176,75193,75195,75198,75218,75220,75223,75243,75245,75248,75264,75266,75269,75286,75288,75291,75310,75312],[12,74985,74986],{},"寫完繼續趕神經 不然我會發神經",[12,74988,74989],{},"GMFCS分級定義",[3603,74991,74992,74995,74998,75001,75004],{},[26,74993,74994],{},"一級——行走不受限",[26,74996,74997],{},"二級——行走受限",[26,74999,75000],{},"三級——使用手持輔具",[26,75002,75003],{},"四級——需要電動輪椅",[26,75005,75006],{},"五級——需要別人手推輪椅",[7085,75008,75009,75032],{},[7088,75010,75011],{},[7091,75012,75013,75017,75020,75023,75026,75029],{},[7094,75014,75015],{"align":47883},[16,75016],{},[7094,75018,75019],{"align":47883},"0-2y",[7094,75021,75022],{"align":47883},"2-4y",[7094,75024,75025],{"align":47883},"4-6y",[7094,75027,75028],{"align":47883},"6-12y",[7094,75030,75031],{"align":47883},"12-18y",[7104,75033,75034,75057,75080,75103,75121],{},[7091,75035,75036,75039,75042,75045,75049,75053],{},[7109,75037,75038],{"align":47883},"I",[7109,75040,75041],{"align":47883},"1. Sitting free on the floor            2. Crawl on hands and knee 3. Pull to stand  4. Takes few step with support",[7109,75043,75044],{"align":47883},"Independent walking without assistance",[7109,75046,75047],{"align":47883},[16,75048],{},[7109,75050,75051],{"align":47883},[16,75052],{},[7109,75054,75055],{"align":47883},[16,75056],{},[7091,75058,75059,75062,75065,75068,75072,75076],{},[7109,75060,75061],{"align":47883},"II",[7109,75063,75064],{"align":47883},"1. Sitting may need support   2. Creep on stomach",[7109,75066,75067],{"align":47883},"but sitting with support",[7109,75069,75070],{"align":47883},[16,75071],{},[7109,75073,75074],{"align":47883},[16,75075],{},[7109,75077,75078],{"align":47883},[16,75079],{},[7091,75081,75082,75085,75088,75091,75095,75099],{},[7109,75083,75084],{"align":47883},"III",[7109,75086,75087],{"align":47883},"1. Low back supported        2. Roll or creep on stomach",[7109,75089,75090],{"align":47883},"1. W-sitting      2. Creep or crawl                3. Walking with walker indoor",[7109,75092,75093],{"align":47883},[16,75094],{},[7109,75096,75097],{"align":47883},[16,75098],{},[7109,75100,75101],{"align":47883},[16,75102],{},[7091,75104,75105,75108,75111,75114,75117,75119],{},[7109,75106,75107],{"align":47883},"IV",[7109,75109,75110],{"align":47883},"1. Head control  2. Roll to supine; may roll to prone",[7109,75112,75113],{"align":47883},"1. Sitting with support (cannot maintain in good alignment) 2. Roll or creep or crawl without reciprocal movements",[7109,75115,75116],{"align":47883},"Self mobility with powered W\u002FC",[7109,75118,75116],{"align":47883},[7109,75120,75116],{"align":47883},[7091,75122,75123,75125,75128,75130,75134,75138],{},[7109,75124,18711],{"align":47883},[7109,75126,75127],{"align":47883},"Unable maintain antigravity head control",[7109,75129,75127],{"align":47883},[7109,75131,75132],{"align":47883},[16,75133],{},[7109,75135,75136],{"align":47883},[16,75137],{},[7109,75139,75140],{"align":47883},[16,75141],{},[12,75143,75144],{},"接著是操作的部份",[23,75146,75147,75150,75153,75156,75159,75162,75165,75168],{},[26,75148,75149],{},"軀幹支持行助行器——可以支撐pelvis trunk",[26,75151,75152],{},"手持式移動輔具——手杖，拐杖，前推或後拉式助行器這一類都的不支撐trunk",[26,75154,75155],{},"肢體協助——需要別人協助移動",[26,75157,75158],{},"電動移動輔助——電動輪椅，代步車，等",[26,75160,75161],{},"手動輪椅自推——自己推",[26,75163,75164],{},"被動運送——被人推",[26,75166,75167],{},"行走——可以帶矯具但不能用輔具",[26,75169,75170],{},"輪式移動工具——包括電動輪椅，手動輪椅還有推車",[36,75172],{},[12,75174,75175],{},"兩歲前",[23,75177,75178,75181,75184,75187,75190],{},[26,75179,75180],{},"一級——能自行轉變姿勢；坐姿能雙手伸出抓物；可以四足爬、扶物站起、扶物跨步；18~24個月獨立行走不用輔具",[26,75182,75183],{},"二級——需用手支撐平衡；腹貼地爬或四足爬行；扶物站起、扶物跨步",[26,75185,75186],{},"三級——下背部被支撐才能坐穩；會翻身、腹貼地爬",[26,75188,75189],{},"四級——有頭控制能力，但坐需要扶住整個身軀；會翻身prone to supine\u002Fsupine to prone但不會爬",[26,75191,75192],{},"五級——因身體功能受損限制自主動作；無法在爬或坐姿控制頭部及軀幹",[36,75194],{},[12,75196,75197],{},"二到四歲",[23,75199,75200,75203,75206,75212,75215],{},[26,75201,75202],{},"一級——坐姿雙手伸出抓物；坐到站\u002F站到坐自行轉換；以行走為移動首選，不需輔助",[26,75204,75205],{},"二級——坐姿雙手伸出抓物會不穩；坐到站\u002F站到坐能自行轉換不需大人，但要拉穩定物體站；以四足爬或輔具作移動首選",[26,75207,75208,75209,75211],{},"三級——以W坐姿維持自行坐，需大人協且進入坐姿；會以腹貼地爬或四足爬（不具備雙腳交替能力）",[16,75210],{},"\n能扶穩定物站，短距離扶物走；可用輔具短距離走，轉向需協助",[26,75213,75214],{},"四級——無支撐會倒，需適應性輔具協具坐和站（九個月沒有站要注意）；以翻滾、腹貼地爬、四足爬坐為移動首選",[26,75216,75217],{},"五級——幾乎不會動",[36,75219],{},[12,75221,75222],{},"四到六歲",[23,75224,75225,75228,75234,75237,75240],{},[26,75226,75227],{},"一級——在椅子坐站切換，且坐穩不需支撐；椅子地板坐站切換不需支撐；室內外自由走，會爬樓梯；開始展現跑跳能力",[26,75229,75230,75231,75233],{},"二級——在椅上能放開雙手，椅子地板坐站切換需支撐穩定表面；",[16,75232],{},"\n室內走不需協助，室外平坦地面短距走；需扶欄杆走樓梯，但無法跑跳",[26,75235,75236],{},"三級——坐姿需被支撐才能雙手活動；移動需輔具；上樓梯需協助",[26,75238,75239],{},"四級——需適應性座椅才能雙手活動；椅子坐站切換需協助；可使用電動輪椅自行移動",[26,75241,75242],{},"五級——幾乎不動",[36,75244],{},[12,75246,75247],{},"六到十二歲",[23,75249,75250,75253,75256,75259,75262],{},[26,75251,75252],{},"一級——能自行室內外獨立行走上下樓梯；有跑跳但速度平衡協調還在進化",[26,75254,75255],{},"二級——大多數環境可以走；需手持或輪式輔具；爬樓梯需扶手或協助；跑跳能力受限",[26,75257,75258],{},"三級——大多數環境需手持輔具行走；坐椅子需骨盤支持帶；長距離，社區，不平整環境需輪式輔具",[26,75260,75261],{},"四級——需他人協助或電動輪椅移動",[26,75263,75242],{},[36,75265],{},[12,75267,75268],{},"十二到十八歲",[23,75270,75271,75274,75277,75280,75283],{},[26,75272,75273],{},"一級——所有環境移動；跑跳與同儕一致，但速度平衡協調可能不同",[26,75275,75276],{},"二級——多數環境移動；不平整或長距需手持或輪式輔具；在校或工作會使用輪式輔助；爬樓梯需扶手",[26,75278,75279],{},"三級——需手持輔具移動；在學校社區需輪式輔具；坐椅子需骨盤支持帶或軀幹支撐才能使用雙手；平整地可短距行走",[26,75281,75282],{},"四級——需輪式輔具移動；需肢體協助設備；常用電動輪椅移動",[26,75284,75285],{},"五級——被動協助移動",[36,75287],{},[12,75289,75290],{},"各分級比較",[23,75292,75293,75296,75299,75307],{},[26,75294,75295],{},"一級VS二級——二級在長距離移動和平衡較差；為平穩多選擇手持扶具；爬樓梯需扶手",[26,75297,75298],{},"二級VS三級——主要為移動形態差異；二級多數環境能獨立；三級在平穩地面手持輔具，社區需輪式",[26,75300,75301,75302],{},"三級VS四級——主要為坐姿及自動移動差異；",[23,75303,75304],{},[26,75305,75306],{},"三級僅需簡單支撐，且具一定獨立行走能力；四級需適應性支撐，且需依靠電動輪椅",[26,75308,75309],{},"四級VS五級——五級在頭部抗重力與軀幹穩定極差；所有動作領域嚴重不足；若要求自行移動需高度改裝電動輪椅",[36,75311],{},[12,75313,75314],{},"還好內容不多也不難啃，嚇死我了",{"title":180,"searchDepth":181,"depth":181,"links":75316},[],"好像是要自己看的補充","\u002Fimages\u002Fuploads\u002F1776579606728-El0Kc5OVMAEdbG5.webp",{},"\u002Fblog\u002FGMFCS",{"title":74981,"description":75317},{"loc":75320},"blog\u002FGMFCS","G29aDr074P62Nd9uZoQMiZkkv7riTkIuliNENvcBRcI",{"id":75326,"title":75327,"author":6946,"body":75328,"date":74911,"description":180,"draft":185,"edited_at":76619,"extension":186,"featured_image":194,"meta":76620,"navigation":189,"path":76621,"pinned":185,"seo":76622,"sitemap":76623,"stem":76624,"tags":194,"__hash__":76625},"blog\u002Fblog\u002FInternal-Memory-How-Your-Computer-Remembers-Things.md","Internal Memory: How Your Computer Remembers Things",{"type":9,"value":75329,"toc":76591},[75330,75334,75340,75347,75350,75361,75372,75376,75383,75409,75436,75439,75443,75446,75568,75579,75643,75647,75652,75674,75683,75687,75703,75714,75719,75741,75746,75749,75798,75802,75813,75827,75834,75852,75856,75862,75867,75872,75883,75887,75893,75899,75919,75923,75926,75932,75940,75949,75959,75973,75977,75984,76001,76044,76049,76068,76125,76142,76181,76185,76188,76249,76253,76263,76274,76277,76280,76291,76295,76298,76312,76322,76347,76351,76354,76358,76369,76376,76380,76390,76397,76471,76482,76486,76489,76496,76500,76507,76521,76525,76528,76538,76552,76556,76563,76566,76570,76573,76579,76586,76589],[6964,75331,75333],{"id":75332},"why-should-you-care-about-memory","Why Should You Care About Memory?",[12,75335,75336,75337],{},"Here's a fact that might surprise you: ",[106,75338,75339],{},"your processor is way faster than your memory.",[12,75341,75342,75343,75346],{},"A 3 GHz processor can execute a simple \"add\" operation in about 0.33 nanoseconds. But fetching data from main memory? That takes over 33 nanoseconds — roughly ",[106,75344,75345],{},"100 times slower",". That means if your system naively accessed memory every time it needed data, loads and stores would bottleneck everything else.",[12,75348,75349],{},"So engineers face an impossible wish list:",[23,75351,75352,75355,75358],{},[26,75353,75354],{},"Memory that runs at processor speed",[26,75356,75357],{},"Memory large enough for all running programs",[26,75359,75360],{},"Memory that's cheap",[12,75362,75363,75364,75367,75368,75371],{},"You can't have all three at once. That tension is the reason we have a ",[7289,75365,75366],{},"memory hierarchy"," — different types of memory layered together, each making a different trade-off between speed, size, and cost. In this post, we'll focus on ",[106,75369,75370],{},"internal memory",": the semiconductor-based memory that lives on or near the processor, as opposed to external storage like hard drives.",[6964,75373,75375],{"id":75374},"the-memory-cell-the-smallest-unit","The Memory Cell: The Smallest Unit",[12,75377,75378,75379,75382],{},"Every piece of memory, at its most basic level, is built from ",[106,75380,75381],{},"memory cells",". A single memory cell:",[23,75384,75385,75396,75403],{},[26,75386,75387,75388,75391,75392,7194,75394],{},"Has ",[106,75389,75390],{},"two stable states",": representing either a ",[6979,75393,9180],{},[6979,75395,8855],{},[26,75397,75398,75399,75402],{},"Can be ",[106,75400,75401],{},"written to"," — setting its state",[26,75404,75398,75405,75408],{},[106,75406,75407],{},"read from"," — sensing its current state",[12,75410,75411,75412,75415,75416,75419,75420,75423,75424,75427,75428,75431,75432,75435],{},"When you ",[7289,75413,75414],{},"write"," to a cell, a ",[106,75417,75418],{},"select"," signal activates it, a ",[106,75421,75422],{},"control"," signal tells it \"this is a write operation,\" and ",[106,75425,75426],{},"data in"," provides the value. When you ",[7289,75429,75430],{},"read",", the select and control signals activate the cell, and a ",[106,75433,75434],{},"sense"," line outputs the stored value.",[12,75437,75438],{},"Think of it like a light switch with a lock — you can flip it on (1) or off (0), lock it in place, and later check which position it's in.",[6964,75440,75442],{"id":75441},"the-big-picture-semiconductor-memory-types","The Big Picture: Semiconductor Memory Types",[12,75444,75445],{},"Before we dive deep, here's a roadmap of the main memory types and how they relate to each other:",[7085,75447,75448,75467],{},[7088,75449,75450],{},[7091,75451,75452,75455,75458,75461,75464],{},[7094,75453,75454],{},"Memory Type",[7094,75456,75457],{},"Category",[7094,75459,75460],{},"Erasure",[7094,75462,75463],{},"Write Mechanism",[7094,75465,75466],{},"Volatile?",[7104,75468,75469,75487,75504,75520,75537,75552],{},[7091,75470,75471,75476,75479,75482,75485],{},[7109,75472,75473],{},[106,75474,75475],{},"RAM",[7109,75477,75478],{},"Read-write",[7109,75480,75481],{},"Electrically, byte-level",[7109,75483,75484],{},"Electrically",[7109,75486,16122],{},[7091,75488,75489,75493,75496,75499,75502],{},[7109,75490,75491],{},[106,75492,1612],{},[7109,75494,75495],{},"Read-only",[7109,75497,75498],{},"Not possible",[7109,75500,75501],{},"Masks (at factory)",[7109,75503,16113],{},[7091,75505,75506,75511,75513,75515,75518],{},[7109,75507,75508],{},[106,75509,75510],{},"PROM",[7109,75512,75495],{},[7109,75514,75498],{},[7109,75516,75517],{},"Electrically (once)",[7109,75519,16113],{},[7091,75521,75522,75527,75530,75533,75535],{},[7109,75523,75524],{},[106,75525,75526],{},"EPROM",[7109,75528,75529],{},"Read-mostly",[7109,75531,75532],{},"UV light, chip-level",[7109,75534,75484],{},[7109,75536,16113],{},[7091,75538,75539,75544,75546,75548,75550],{},[7109,75540,75541],{},[106,75542,75543],{},"EEPROM",[7109,75545,75529],{},[7109,75547,75481],{},[7109,75549,75484],{},[7109,75551,16113],{},[7091,75553,75554,75559,75561,75564,75566],{},[7109,75555,75556],{},[106,75557,75558],{},"Flash",[7109,75560,75529],{},[7109,75562,75563],{},"Electrically, block-level",[7109,75565,75484],{},[7109,75567,16113],{},[12,75569,75570,75571,75574,75575,75578],{},"The two big branches are ",[106,75572,75573],{},"volatile"," memory (loses data without power) and ",[106,75576,75577],{},"nonvolatile"," memory (retains data even when powered off). Let's explore each.",[25631,75580,75582,75585],{"className":75581},[13441,13442],[25635,75583,75584],{},"Explaination",[13438,75586,75588,75633,75636],{"className":75587},[25641],[23,75589,75590,75596,75602,75611,75617,75623],{},[26,75591,75592,75595],{},[106,75593,75594],{},"RAM (Random Access Memory):"," The defining example of volatile memory. It is fast, allows byte-level read\u002Fwrites, and is wiped when power is lost.",[26,75597,75598,75601],{},[106,75599,75600],{},"ROM (Read-Only Memory):"," Hardwired at the factory. Cannot be changed.",[26,75603,75604,75607,75608,75610],{},[106,75605,75606],{},"PROM (Programmable ROM):"," Blank from the factory, but can only be written to ",[7289,75609,18152],{}," (like burning a CD-R).",[26,75612,75613,75616],{},[106,75614,75615],{},"EPROM (Erasable Programmable ROM):"," A massive leap forward, but it required pulling the chip out of the computer and exposing a little quartz window on it to strong UV light to erase it.",[26,75618,75619,75622],{},[106,75620,75621],{},"EEPROM (Electrically Erasable Programmable ROM):"," Allowed memory to be erased electrically without removing the chip, but only one byte at a time (too slow for bulk storage).",[26,75624,75625,75628,75629,75632],{},[106,75626,75627],{},"Flash:"," The defining example of modern non-volatile memory (used in SSDs, USB drives, and smartphones). You need to know that it is electrically erased at the ",[106,75630,75631],{},"block-level"," (which is why writing data to a nearly full SSD can slow down).",[12,75634,75635],{},"Byte-Level means the computer can target a single byte of data (usually 8 bits) and change it without affecting the data sitting right next to it. It is incredibly convenient and fast for making small changes, but building the microscopic circuitry required to target every individual byte makes the physical chip more complex and expensive to manufacture.",[12,75637,75638,75639,75642],{},"Block-level means the memory is divided into larger chunks called \"blocks\" (often thousands of bytes large). To change even a single byte of data, the computer must erase the ",[7289,75640,75641],{},"entire"," block first, and then rewrite the whole block with the new change included. By giving up byte-level precision, engineers were able to drastically simplify the wiring inside the chip. This is exactly why Flash memory became so cheap and can hold massive amounts of data. However, it requires clever software controllers to manage the constant copying and erasing of blocks so the drive does not wear out too quickly.",[6964,75644,75646],{"id":75645},"ram-the-workhorse","RAM: The Workhorse",[12,75648,75649,75651],{},[106,75650,75475],{}," (Random Access Memory) is the memory your computer uses for active work. It's called \"random access\" because any byte can be read or written in roughly the same amount of time, regardless of its location. The key characteristics:",[23,75653,75654,75661,75667],{},[26,75655,75656,75657,75660],{},"Data is read and written using ",[106,75658,75659],{},"electrical signals"," — fast and easy",[26,75662,75663,75664,75666],{},"It's ",[106,75665,75573],{}," — turn off the power, and everything disappears",[26,75668,75669,75670,75673],{},"It serves as ",[106,75671,75672],{},"temporary storage"," for programs and data currently in use",[12,75675,75676,75677,7042,75680,6959],{},"There are two major flavors of RAM: ",[106,75678,75679],{},"DRAM",[106,75681,75682],{},"SRAM",[7166,75684,75686],{"id":75685},"dram-dynamic-ram-the-forgetful-one","DRAM (Dynamic RAM) — The Forgetful One",[12,75688,75689,75690,75693,75694,75696,75697,75699,75700,6959],{},"DRAM stores each bit of data as a ",[106,75691,75692],{},"charge on a tiny capacitor",". A charged capacitor represents a ",[6979,75695,8855],{},"; a discharged one represents ",[6979,75698,9180],{},". The circuit for a single DRAM cell is remarkably simple — just ",[106,75701,75702],{},"one transistor and one capacitor",[12,75704,75705,75706,75709,75710,75713],{},"But there's a catch: capacitors ",[7289,75707,75708],{},"leak",". The charge slowly drains away over time, so the data would be lost if left alone. That's why DRAM needs ",[106,75711,75712],{},"periodic refreshing"," — the memory controller must regularly read every row and write it back to restore the charges. This is what makes it \"dynamic.\"",[12,75715,75716],{},[106,75717,75718],{},"How DRAM reads and writes:",[23,75720,75721,75727],{},[26,75722,75723,75726],{},[106,75724,75725],{},"Writing:"," A voltage is applied to the bit line (high for 1, low for 0). Then the address line is activated, which turns on the transistor, allowing the charge to transfer to the capacitor.",[26,75728,75729,75732,75733,75736,75737,75740],{},[106,75730,75731],{},"Reading:"," The address line is activated, turning on the transistor. The charge stored in the capacitor flows out through the bit line to a ",[106,75734,75735],{},"sense amplifier",", which compares it against a reference voltage to determine whether it's a 0 or 1. Here's the tricky part — reading is ",[106,75738,75739],{},"destructive",". The act of reading drains the capacitor, so the charge must be written back after every read.",[12,75742,75743],{},[106,75744,75745],{},"Refreshing in practice:",[12,75747,75748],{},"A dedicated refresh circuit is built into the chip. It temporarily disables normal access, steps through each row, reads the data, and writes it back. This takes time and slightly reduces the apparent performance of the chip. Every few milliseconds, the entire memory array must be refreshed.",[25631,75750,75752,75755],{"className":75751},[13441,13442],[25635,75753,75754],{},"Additional Explaination",[13438,75756,75758,75761,75767,75773,75787],{"className":75757},[25641],[12,75759,75760],{},"DRAM is arranged in a massive grid, much like a spreadsheet or a city map. To read or write data, the memory controller needs a way to target specific \"coordinates\" on that grid.",[12,75762,75763,75766],{},[106,75764,75765],{},"The Address Line"," is the horizontal wire running across a row of memory cells. Think of the transistor in the DRAM cell as a door blocking access to the capacitor (where the data lives). The address line controls that door. When the memory controller sends a high voltage down the address line, it \"opens the door\" (turns on the transistor) for every single cell in that specific row, connecting their capacitors to the rest of the circuit.",[12,75768,75769,75772],{},[106,75770,75771],{},"The Bit Line"," is the vertical wire running down a column of memory cells. If the address line opens the door, the bit line is the hallway the data travels through.",[23,75774,75775,75781],{},[26,75776,75777,75780],{},[106,75778,75779],{},"During a Write:"," The memory controller forces a high voltage (a 1) or low voltage (a 0) down the bit line. Because the address line opened the transistor door, that voltage flows from the bit line into the capacitor, charging or discharging it.",[26,75782,75783,75786],{},[106,75784,75785],{},"During a Read:"," The transistor door opens, and whatever tiny bit of charge is stored in the capacitor spills out onto the bit line to be read.",[12,75788,75789,75790,75793,75794,75797],{},"The Sense Amplifier is a highly sensitive measuring circuit situated at the end of each bit line. The capacitor inside a DRAM cell is microscopic. When it dumps its charge onto the much larger bit line during a \"read\" operation, the resulting voltage change is incredibly weak—barely a whisper of a signal. The sense amplifier detects that tiny voltage shift and instantly ",[106,75791,75792],{},"amplifies"," it into a strong, clear digital 1 or 0 that the computer's processor can actually understand. Furthermore, because reading the capacitor drained it (destructive read), the sense amplifier immediately pushes that newly amplified strong signal ",[7289,75795,75796],{},"back"," up the bit line to recharge the capacitor, saving the data from being lost forever.",[7166,75799,75801],{"id":75800},"sram-static-ram-the-fast-one","SRAM (Static RAM) — The Fast One",[12,75803,75804,75805,75808,75809,75812],{},"SRAM takes a completely different approach. Instead of a capacitor, it stores each bit using a ",[106,75806,75807],{},"flip-flop"," circuit — a configuration of ",[106,75810,75811],{},"six transistors"," (typically labeled T₁ through T₆). Two pairs of transistors form cross-coupled inverters (T₁\u002FT₃ and T₂\u002FT₄), creating two stable states. The other two transistors (T₅ and T₆) connect the cell to the bit lines and are controlled by the address line.",[23,75814,75815,75821],{},[26,75816,75817,75820],{},[106,75818,75819],{},"State 1:"," Point C₁ is high, C₂ is low. Transistors T₁ and T₄ are off, T₂ and T₃ are on.",[26,75822,75823,75826],{},[106,75824,75825],{},"State 0:"," Point C₂ is high, C₁ is low. Transistors T₂ and T₃ are off, T₁ and T₄ are on.",[12,75828,75829,75830,75833],{},"The beauty of SRAM is that ",[106,75831,75832],{},"as long as power is supplied, the flip-flop holds its state indefinitely"," — no refresh needed. To write, you apply the desired value to bit line B and its complement to B̄, then activate the address line. To read, the value simply appears on bit line B when the address line is activated.",[25631,75835,75837,75840],{"className":75836},[13441,13442],[25635,75838,75839],{},"Visualisation",[13438,75841,75843,75846,75849],{"className":75842},[25641],[7026,75844],{"code":75845},"\u003Ch2 class=\"sr-only\">Interactive SRAM cell showing how cross-coupled inverters store a bit, with step-through for idle, write, and read operations.\u003C\u002Fh2>\n\u003Cstyle>\n  \u002F* Base wrapper styles *\u002F\n  .sram-wrap { \n    max-width: 660px; \n    margin: 0 auto; \n    padding: 0.5rem 0; \n    font-family: var(--font-sans); \n  }\n  .sram-nav { display: flex; gap: 6px; margin-bottom: 14px; flex-wrap: wrap; }\n  .sram-btn { \n    padding: 6px 14px; \n    border-radius: var(--border-radius-md); \n    border: 1px solid var(--color-border-tertiary); \n    background: transparent; \n    color: var(--color-text-primary); \n    font-size: 13px; \n    cursor: pointer; \n    transition: all .2s; \n  }\n  .sram-btn:hover {\n    background: var(--color-border-tertiary);\n  }\n  .sram-btn.active { \n    background: #534AB7; \n    color: #fff; \n    border-color: #534AB7; \n  }\n  .sram-desc { \n    font-size: 14px; \n    line-height: 1.65; \n    color: var(--color-text-primary); \n    margin-bottom: 12px; \n    min-height: 60px; \n  }\n\n  .wire { transition: stroke .4s, opacity .4s; }\n  .node-fill { transition: fill .4s; }\n  .txt-anim { transition: fill .4s; }\n  \n  \u002F* --- SVG Text Colors: Adapt to your site's Dark\u002FLight mode! --- *\u002F\n  svg text {\n    font-family: var(--font-sans);\n    font-size: 12px;\n  }\n  .th { \n    font-weight: bold; \n    fill: currentColor; \n  }\n  .ts { \n    \u002F* Use a solid secondary color! 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lines\u003C\u002Fdesc>\n    \u003Cdefs>\n      \u003Cmarker id=\"arrow\" viewBox=\"0 0 10 10\" refX=\"8\" refY=\"5\" markerWidth=\"6\" markerHeight=\"6\" orient=\"auto-start-reverse\">\n        \u003Cpath d=\"M2 1L8 5L2 9\" fill=\"none\" stroke=\"context-stroke\" stroke-width=\"1.5\" stroke-linecap=\"round\" stroke-linejoin=\"round\"\u002F>\n      \u003C\u002Fmarker>\n    \u003C\u002Fdefs>\n\n    \u003Ctext class=\"ts\" x=\"340\" y=\"24\" text-anchor=\"middle\" id=\"addr-label\">Address line\u003C\u002Ftext>\n    \u003Cline id=\"addr-line\" x1=\"40\" y1=\"40\" x2=\"640\" y2=\"40\" stroke-width=\"2.5\" stroke-linecap=\"round\" class=\"wire\"\u002F>\n\n    \u003Ctext class=\"ts\" x=\"108\" y=\"18\" text-anchor=\"middle\">Bit line B\u003C\u002Ftext>\n    \u003Cline id=\"bl-top\" x1=\"108\" y1=\"28\" x2=\"108\" y2=\"72\" stroke-width=\"2\" stroke-linecap=\"round\" class=\"wire\"\u002F>\n    \u003Cline id=\"bl-bot\" x1=\"108\" y1=\"120\" x2=\"108\" y2=\"400\" stroke-width=\"2\" stroke-linecap=\"round\" class=\"wire\"\u002F>\n\n    \u003Ctext class=\"ts\" x=\"572\" y=\"18\" text-anchor=\"middle\">Bit line B-bar\u003C\u002Ftext>\n    \u003Cline id=\"br-top\" x1=\"572\" y1=\"28\" x2=\"572\" y2=\"72\" stroke-width=\"2\" stroke-linecap=\"round\" class=\"wire\"\u002F>\n    \u003Cline id=\"br-bot\" x1=\"572\" y1=\"120\" x2=\"572\" y2=\"400\" stroke-width=\"2\" stroke-linecap=\"round\" class=\"wire\"\u002F>\n\n    \u003Crect id=\"t5-body\" x=\"80\" y=\"72\" width=\"56\" height=\"48\" rx=\"6\" stroke-width=\"0.5\" class=\"node-fill\"\u002F>\n    \u003Ctext class=\"ts txt-anim\" x=\"108\" y=\"92\" text-anchor=\"middle\" dominant-baseline=\"central\" id=\"t5-txt\">T5\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"108\" y=\"108\" text-anchor=\"middle\" dominant-baseline=\"central\" id=\"t5-state\">\u003C\u002Ftext>\n\n    \u003Crect id=\"t6-body\" x=\"544\" y=\"72\" width=\"56\" height=\"48\" rx=\"6\" stroke-width=\"0.5\" class=\"node-fill\"\u002F>\n    \u003Ctext 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class=\"th txt-anim\" x=\"250\" y=\"195\" text-anchor=\"middle\" dominant-baseline=\"central\" id=\"inv1-title\">Inverter A\u003C\u002Ftext>\n    \u003Ctext class=\"ts txt-anim\" x=\"250\" y=\"215\" text-anchor=\"middle\" dominant-baseline=\"central\" id=\"inv1-sub\">(T1 + T3)\u003C\u002Ftext>\n    \u003Ctext class=\"th txt-anim\" x=\"250\" y=\"245\" text-anchor=\"middle\" dominant-baseline=\"central\" id=\"inv1-out\">\u003C\u002Ftext>\n\n    \u003Crect id=\"inv2-body\" x=\"370\" y=\"160\" width=\"120\" height=\"100\" rx=\"10\" stroke-width=\"0.5\" class=\"node-fill\"\u002F>\n    \u003Ctext class=\"th txt-anim\" x=\"430\" y=\"195\" text-anchor=\"middle\" dominant-baseline=\"central\" id=\"inv2-title\">Inverter B\u003C\u002Ftext>\n    \u003Ctext class=\"ts txt-anim\" x=\"430\" y=\"215\" text-anchor=\"middle\" dominant-baseline=\"central\" id=\"inv2-sub\">(T2 + T4)\u003C\u002Ftext>\n    \u003Ctext class=\"th txt-anim\" x=\"430\" y=\"245\" text-anchor=\"middle\" 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Inverter A outputs HIGH (C1 = 1), which feeds into Inverter B. Inverter B does its job — inverts it — and outputs LOW (C2 = 0). That LOW feeds back into Inverter A, which inverts it to HIGH. The loop sustains itself forever, no refresh needed. The access transistors (T5, T6) are closed — the address line is off, so the cell is isolated from the bit lines.\",\n    addr: G, addrW: 1.5,\n    bl: G, br: G,\n    t5: { fill: GRAY_FILL, stroke: GRAY_STROKE, txt: GRAY_TXT, label: 'T5', state: 'closed' },\n    t6: { fill: GRAY_FILL, stroke: GRAY_STROKE, txt: GRAY_TXT, label: 'T6', state: 'closed' },\n    inv1: { fill: AMBER_FILL, stroke: AMBER_STROKE, txt: AMBER_TXT, out: 'OUT: 1 (HIGH)' },\n    inv2: { fill: BLUE_FILL, stroke: BLUE_STROKE, txt: BLUE_TXT, out: 'OUT: 0 (LOW)' },\n    cross1: H, cross2: L,\n    t5conn: G, t6conn: G,\n    sense: { fill: GRAY_FILL, stroke: GRAY_STROKE, txt: GRAY_TXT, label: 'Sense amplifier (idle)' }\n  },\n  {\n    title: \"Storing a 0\",\n    desc: \"Now the cell holds a \u003Cstrong>0\u003C\u002Fstrong>. Everything flips: Inverter A outputs LOW (C1 = 0), Inverter B outputs HIGH (C2 = 1). The feedback loop holds this state just as stubbornly. Same circuit, opposite stable state. Still no refresh needed — the cross-coupled inverters actively hold the value.\",\n    addr: G, addrW: 1.5,\n    bl: G, br: G,\n    t5: { fill: GRAY_FILL, stroke: GRAY_STROKE, txt: GRAY_TXT, label: 'T5', state: 'closed' },\n    t6: { fill: GRAY_FILL, stroke: GRAY_STROKE, txt: GRAY_TXT, label: 'T6', state: 'closed' },\n    inv1: { fill: BLUE_FILL, stroke: BLUE_STROKE, txt: BLUE_TXT, out: 'OUT: 0 (LOW)' },\n    inv2: { fill: AMBER_FILL, stroke: AMBER_STROKE, txt: AMBER_TXT, out: 'OUT: 1 (HIGH)' },\n    cross1: L, cross2: H,\n    t5conn: G, t6conn: G,\n    sense: { fill: GRAY_FILL, stroke: GRAY_STROKE, txt: GRAY_TXT, label: 'Sense amplifier (idle)' }\n  },\n  {\n    title: \"Writing a 1\",\n    desc: \"To write a \u003Cstrong>1\u003C\u002Fstrong>, the controller forces bit line B to HIGH and B-bar to LOW, then activates the address line. This opens both doors (T5 and T6). The strong external signals flood into the inverters and overpower whatever state they were in, forcing C1 = HIGH and C2 = LOW. When the address line deactivates, the doors close — and the loop now holds the new value on its own.\",\n    addr: PURPLE_STROKE, addrW: 3,\n    bl: H, br: L,\n    t5: { fill: PURPLE_FILL, stroke: PURPLE_STROKE, txt: PURPLE_TXT, label: 'T5', state: 'OPEN' },\n    t6: { fill: PURPLE_FILL, stroke: PURPLE_STROKE, txt: PURPLE_TXT, label: 'T6', state: 'OPEN' },\n    inv1: { fill: AMBER_FILL, stroke: AMBER_STROKE, txt: AMBER_TXT, out: 'OUT: 1 (HIGH)' },\n    inv2: { fill: BLUE_FILL, stroke: BLUE_STROKE, txt: BLUE_TXT, out: 'OUT: 0 (LOW)' },\n    cross1: H, cross2: L,\n    t5conn: H, t6conn: L,\n    sense: { fill: GRAY_FILL, stroke: GRAY_STROKE, txt: GRAY_TXT, label: 'Sense amplifier (idle)' }\n  },\n  {\n    title: \"Reading\",\n    desc: \"To read, both bit lines are pre-charged to a neutral voltage. Then the address line activates, opening T5 and T6. The powered flip-flop pushes its internal voltages outward — B gets pulled slightly toward C1's value, B-bar toward C2's. The sense amplifier detects which line went up and which went down. \u003Cstrong>Reading is non-destructive\u003C\u002Fstrong> — the flip-flop keeps driving its state the whole time, unlike DRAM where reading drains the capacitor.\",\n    addr: PURPLE_STROKE, addrW: 3,\n    bl: H, br: L,\n    t5: { fill: PURPLE_FILL, stroke: PURPLE_STROKE, txt: PURPLE_TXT, label: 'T5', state: 'OPEN' },\n    t6: { fill: PURPLE_FILL, stroke: PURPLE_STROKE, txt: PURPLE_TXT, label: 'T6', state: 'OPEN' },\n    inv1: { fill: AMBER_FILL, stroke: AMBER_STROKE, txt: AMBER_TXT, out: 'OUT: 1 (HIGH)' },\n    inv2: { fill: BLUE_FILL, stroke: BLUE_STROKE, txt: BLUE_TXT, out: 'OUT: 0 (LOW)' },\n    cross1: H, cross2: L,\n    t5conn: H, t6conn: L,\n    sense: { fill: TEAL_FILL, stroke: TEAL_STROKE, txt: TEAL_TXT, label: 'Detected: 1' }\n  }\n];\n\nconst nav = document.getElementById('nav');\nconst desc = document.getElementById('desc');\n\nsteps.forEach((s, i) => {\n  const b = document.createElement('button');\n  b.className = 'sram-btn';\n  b.textContent = s.title;\n  b.onclick = () => show(i);\n  nav.appendChild(b);\n});\n\nfunction setFill(id, fill, stroke) {\n  const el = document.getElementById(id);\n  el.style.fill = fill;\n  el.style.stroke = stroke;\n}\nfunction setStroke(id, color) {\n  document.getElementById(id).style.stroke = color;\n}\nfunction setTxt(id, val, color) {\n  const el = document.getElementById(id);\n  if (val) el.textContent = val;\n  if (color) el.style.fill = color;\n}\n\nfunction show(i) {\n  const s = steps[i];\n  nav.querySelectorAll('.sram-btn').forEach((b,j) => b.classList.toggle('active', j===i));\n  desc.innerHTML = s.desc;\n\n  setStroke('addr-line', s.addr);\n  document.getElementById('addr-line').style.strokeWidth = s.addrW + 'px';\n\n  setStroke('bl-top', s.bl); setStroke('bl-bot', s.bl);\n  setStroke('br-top', s.br); setStroke('br-bot', s.br);\n\n  setFill('t5-body', s.t5.fill, s.t5.stroke);\n  setTxt('t5-txt', s.t5.label, s.t5.txt);\n  setTxt('t5-state', s.t5.state, s.t5.txt);\n  setFill('t6-body', s.t6.fill, s.t6.stroke);\n  setTxt('t6-txt', s.t6.label, s.t6.txt);\n  setTxt('t6-state', s.t6.state, s.t6.txt);\n\n  setFill('inv1-body', s.inv1.fill, s.inv1.stroke);\n  setTxt('inv1-title', 'Inverter A', s.inv1.txt);\n  \n  \u002F\u002F Dynamically color the subtitles so they pop off their backgrounds!\n  setTxt('inv1-sub', '(T1 + T3)', s.inv1.txt);\n  document.getElementById('inv1-sub').style.opacity = '0.65'; \n  setTxt('inv1-out', s.inv1.out, s.inv1.txt);\n  \n  setFill('inv2-body', s.inv2.fill, s.inv2.stroke);\n  setTxt('inv2-title', 'Inverter B', s.inv2.txt);\n  \n  \u002F\u002F Dynamically color the subtitles so they pop off their backgrounds!\n  setTxt('inv2-sub', '(T2 + T4)', s.inv2.txt);\n  document.getElementById('inv2-sub').style.opacity = '0.65';\n  setTxt('inv2-out', s.inv2.out, s.inv2.txt);\n\n  setStroke('cross1', s.cross1);\n  setStroke('cross2', s.cross2);\n  setStroke('t5-conn', s.t5conn);\n  setStroke('t5-to-inv1', s.t5conn);\n  setStroke('t6-conn', s.t6conn);\n  setStroke('t6-to-inv2', s.t6conn);\n\n  setFill('sense-body', s.sense.fill, s.sense.stroke);\n  setTxt('sense-txt', s.sense.label, s.sense.txt);\n}\n\nshow(0);\n\u003C\u002Fscript>",[12,75847,75848],{},"The core insight is that the two inverters create a self-sustaining loop. Each one's output justifies the other's output, so the state holds itself in place with zero outside help. That's the entire reason SRAM doesn't need refresh. DRAM stores a charge on a passive capacitor that slowly leaks. SRAM stores a logical state in an active circuit that continuously reinforces itself.",[12,75850,75851],{},"The price you pay is those 6 transistors per bit (versus DRAM's 1 transistor + 1 capacitor). That's why SRAM is faster but far more expensive per bit — and why it's reserved for small, speed-critical caches while DRAM handles the bulk storage.",[7166,75853,75855],{"id":75854},"sram-vs-dram-a-quick-comparison","SRAM vs. DRAM — A Quick Comparison",[12,75857,75858,75859,75861],{},"Both SRAM and DRAM are ",[106,75860,75573],{}," — they both need power to hold their data. But beyond that, they differ in almost every way:",[12,75863,75864,75866],{},[106,75865,75679],{}," is simpler to build (1 transistor + 1 capacitor per cell), smaller per bit, higher density, and cheaper. But it needs constant refreshing.",[12,75868,75869,75871],{},[106,75870,75682],{}," is faster — no refresh delays, no destructive reads. But it uses 6 transistors per cell, so it's larger, less dense, and more expensive.",[12,75873,75874,75875,75878,75879,75882],{},"This is why ",[106,75876,75877],{},"DRAM is used for main memory"," (where you need lots of gigabytes at a reasonable price) and ",[106,75880,75881],{},"SRAM is used for cache"," (where you need blazing speed and don't mind the higher cost for smaller amounts).",[6964,75884,75886],{"id":75885},"rom-memory-that-doesnt-forget","ROM: Memory That Doesn't Forget",[12,75888,75889,75890,75892],{},"While RAM is great for active computation, sometimes you need memory that survives a power cycle. That's where ",[106,75891,1612],{}," (Read Only Memory) comes in.",[12,75894,75895,75896,75898],{},"ROM is ",[106,75897,75577],{}," — no power source is needed to maintain the stored data. It's used for things like:",[23,75900,75901,75907,75913],{},[26,75902,75903,75906],{},[106,75904,75905],{},"BIOS\u002Ffirmware"," — the code that boots your computer before the OS loads",[26,75908,75909,75912],{},[106,75910,75911],{},"Library subroutines"," — frequently used functions baked into hardware",[26,75914,75915,75918],{},[106,75916,75917],{},"Function tables"," — lookup tables for common calculations",[7166,75920,75922],{"id":75921},"the-rom-family","The ROM Family",[12,75924,75925],{},"ROM comes in several varieties, each with different trade-offs between flexibility and cost:",[12,75927,75928,75931],{},[106,75929,75930],{},"ROM (Mask ROM):"," The data is literally wired in during the manufacturing process using photographic masks. It's permanent and cannot be changed. This makes it extremely cheap at scale, but there's absolutely no room for error — if the data is wrong, you throw away the whole chip. Best for mass-produced devices where the firmware will never change.",[12,75933,75934,75936,75937,75939],{},[106,75935,75606],{}," Like ROM, but the user can write data to it ",[106,75938,18152],{}," after manufacture, using special programming equipment. It's non-erasable, so mistakes are permanent. PROM is ideal for small production runs or prototyping where mask ROM's setup costs aren't justified.",[12,75941,75942,75944,75945,75948],{},[106,75943,75615],{}," A step up — EPROM can be erased by exposing the chip to ",[106,75946,75947],{},"ultraviolet light"," through a small quartz window on the chip package. UV exposure erases the entire chip (you can't selectively erase), and the process takes a relatively long time (minutes to hours). Once erased, it can be reprogrammed electrically.",[12,75950,75951,75954,75955,75958],{},[106,75952,75953],{},"EEPROM (Electrically Erasable PROM):"," No UV lamp needed — EEPROM can be erased and reprogrammed ",[106,75956,75957],{},"electrically at the byte level",". You can update individual bytes without erasing the whole chip. The downsides: writing takes much longer than reading, and EEPROM is more expensive and less dense than EPROM.",[12,75960,75961,75964,75965,75968,75969,75972],{},[106,75962,75963],{},"Flash Memory:"," Flash is the sweet spot between EPROM and EEPROM. It erases ",[106,75966,75967],{},"electrically"," (no UV light needed) but at the ",[106,75970,75971],{},"block level"," rather than byte level. A section of memory cells are erased in a single fast action — that's where the name \"flash\" comes from. It uses only one transistor per bit (like EPROM), achieving high density, while being much faster to program than EEPROM.",[6964,75974,75976],{"id":75975},"inside-a-dram-chip-organisation","Inside a DRAM Chip: Organisation",[12,75978,75979,75980,75983],{},"Let's look at how a real DRAM chip is organised internally, using a ",[106,75981,75982],{},"16 Mbit DRAM (4M × 4)"," as an example. This chip stores 16 megabits of data, arranged as 4 million locations, each storing 4 bits.",[13438,75985,75987],{"className":75986},[13441,13442],[12,75988,75989,75992,75993,75996,75997,76000],{},[106,75990,75991],{},"\"4M × 4\""," means there are ",[106,75994,75995],{},"4 million individual lockers"," (locations), and every time you open one locker, you put in or take out exactly ",[106,75998,75999],{},"4 bits"," of data at once.",[12,76002,76003,76004,76007,76008,76043],{},"The memory array is physically laid out as a ",[106,76005,76006],{},"2048 × 2048 × 4"," grid. But 2048 rows × 2048 columns × 4 bits = 16,777,216 bits = 16 Mbit. To address 2048 rows, you need 11 address lines (since ",[24923,76009,76011],{"className":76010,"jax":24927},[24926],[24929,76012,76015,76020],{"style":32867,"xmlns":24932,"width":76013,"height":32869,"role":1930,"focusable":8058,"viewBox":76014,"xmlnsXLink":24936},"3.846ex","0 -750 1700 950",[24938,76016,76017],{},[24941,76018],{"id":76019,"d":25928},"MJX-1-TEX-N-32",[25022,76021,76022],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,76023,76024,76029,76036],{"dataMmlNode":25028},[25022,76025,76026],{"dataMmlNode":26031},[25033,76027],{"dataC":26058,"xLinkHref":76028},"#MJX-1-TEX-N-32",[25022,76030,76031],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":26394},[25022,76032,76033],{"dataMmlNode":25031},[6977,76034,76035],{"dataVariant":32892,"transform":25025,"font-size":32893,"font-family":32894},"¹",[25022,76037,76039],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":76038},"translate(1100,0)",[25022,76040,76041],{"dataMmlNode":25031},[6977,76042,76035],{"dataVariant":32892,"transform":25025,"font-size":32893,"font-family":32894}," = 2048).",[12,76045,76046],{},[106,76047,76048],{},"The clever trick — address multiplexing:",[12,76050,76051,76052,76055,76056,76059,76060,76063,76064,76067],{},"Instead of using 22 address pins (11 for rows + 11 for columns), DRAM chips ",[106,76053,76054],{},"multiplex"," the address: they send the row address and column address ",[7289,76057,76058],{},"over the same pins at different times",". First, the row address is sent and latched with the ",[106,76061,76062],{},"RAS (Row Address Strobe)"," signal. Then the column address is sent and latched with the ",[106,76065,76066],{},"CAS (Column Address Strobe)"," signal. This cuts the pin count roughly in half — a huge deal for chip packaging.",[25631,76069,76071,76073],{"className":76070},[13441,13442],[25635,76072,75839],{},[13438,76074,76076,76079,76084,76087,76092,76099,76106,76113,76116,76118],{"className":76075},[25641],[7026,76077],{"code":76078},"\u003Cstyle>\n  \u002F* Added fallbacks for Claude CSS Variables *\u002F\n  .step-container { \n    max-width: 640px; \n    margin: 0 auto; \n    padding: 1rem 0; \n    font-family: var(--font-sans); \n  }\n  .step-nav { display: flex; gap: 6px; margin-bottom: 16px; flex-wrap: wrap; }\n  .step-btn { \n    padding: 6px 14px; \n    border-radius: var(--border-radius-md); \n    border: 1px solid var(--color-border-tertiary); \n    background: var(--color-background-secondary); \n    color: var(--color-text-secondary); \n    font-size: 13px; \n    cursor: pointer; \n    transition: all .2s; \n  }\n  .step-btn:hover { background: var(--color-border-tertiary); }\n  .step-btn.active { background: #534AB7; color: #fff; border-color: #534AB7; }\n  .step-desc { \n    font-size: 15px; \n    line-height: 1.65; \n    color: var(--color-text-primary); \n    margin-bottom: 16px; \n    min-height: 80px; \n  }\n  .signal-row { \n    display: flex; \n    align-items: center; \n    gap: 8px; \n    margin: 6px 0; \n    font-family: var(--font-mono); \n    font-size: 13px; \n  }\n  .signal-label { \n    width: 110px; \n    text-align: right; \n    color: var(--color-text-secondary); \n    font-family: var(--font-sans); \n    font-size: 13px; \n  }\n  .pin-group { display: flex; gap: 3px; }\n  .pin { \n    width: 22px; \n    height: 26px; \n    display: flex; \n    align-items: center; \n    justify-content: center; \n    border-radius: 4px; \n    font-size: 12px; \n    font-weight: 500; \n    transition: all .3s; \n  }\n  .pin-off { \n    background: var(--color-background-tertiary); \n    color: var(--color-text-tertiary); \n  }\n  .pin-row { background: #7F77DD; color: #fff; }\n  .pin-col { background: #1D9E75; color: #fff; }\n  .pin-data { background: #D85A30; color: #fff; }\n  .signal-line { display: flex; align-items: center; gap: 4px; }\n  .sig-block { \n    height: 24px; \n    border-radius: 3px; \n    display: flex; \n    align-items: center; \n    justify-content: center; \n    font-size: 11px; \n    font-weight: 500; \n    min-width: 60px; \n    padding: 0 8px; \n    transition: all .3s; \n  }\n  .sig-high { \n    background: var(--color-background-tertiary); \n    color: var(--color-text-tertiary); \n  }\n  .sig-low { background: #534AB7; color: #fff; }\n  .sig-label { \n    font-size: 11px; \n    color: var(--color-text-tertiary); \n  }\n  .chip-state { \n    margin-top: 16px; \n    padding: 12px 16px; \n    border-radius: var(--border-radius-lg); \n    background: var(--color-background-secondary); \n    border: 1px solid var(--color-border-tertiary); \n  }\n  .chip-title { \n    font-size: 13px; \n    font-weight: 500; \n    color: var(--color-text-secondary); \n    margin-bottom: 8px; \n  }\n  .chip-detail { \n    font-size: 14px; \n    color: var(--color-text-primary); \n    line-height: 1.6; \n  }\n  .highlight { font-weight: 500; }\n  .h-purple { color: #7F77DD; }\n  .h-teal { color: #1D9E75; }\n  .h-coral { color: #D85A30; }\n\u003C\u002Fstyle>\n\n\u003Cdiv class=\"step-container\">\n  \u003Cdiv class=\"step-nav\" id=\"nav\">\u003C\u002Fdiv>\n  \u003Cdiv class=\"step-desc\" id=\"desc\">\u003C\u002Fdiv>\n  \u003Cdiv id=\"signals\">\u003C\u002Fdiv>\n  \u003Cdiv class=\"chip-state\">\n    \u003Cdiv class=\"chip-title\">Inside the chip\u003C\u002Fdiv>\n    \u003Cdiv class=\"chip-detail\" id=\"chip\">\u003C\u002Fdiv>\n  \u003C\u002Fdiv>\n\u003C\u002Fdiv>\n\u003Cscript>\nconst steps = [\n  {\n    title: \"1. Idle\",\n    desc: \"Nothing is happening. The 11 address pins are idle. RAS and CAS are both HIGH (inactive — they're active-low signals, meaning they trigger when pulled LOW). The data pins are floating.\",\n    pins: [0,0,0,0,0,0,0,0,0,0,0],\n    pinStyle: \"off\",\n    ras: \"high\", cas: \"high\",\n    chip: \"All row and column decoders are inactive. The memory array is dormant, waiting for a command.\"\n  },\n  {\n    title: \"2. Row address placed\",\n    desc: \"The memory controller puts the \u003Cspan class='h-purple highlight'>row address\u003C\u002Fspan> (e.g. row 1042 = 10000010010 in binary) onto the 11 address pins. The pins now carry the binary digits of the row number. But the chip hasn't read them yet — it's waiting for the signal.\",\n    pins: [1,0,0,0,0,0,1,0,0,1,0],\n    pinStyle: \"row\",\n    ras: \"high\", cas: \"high\",\n    chip: \"The address pins have voltages on them representing row 1042, but the chip's row latch hasn't captured them yet. Think of it like someone holding up a sign — the chip hasn't looked yet.\"\n  },\n  {\n    title: \"3. RAS goes LOW\",\n    desc: \"The memory controller pulls \u003Cspan class='h-purple highlight'>RAS low\u003C\u002Fspan>. This is the \\\"look now!\\\" signal. The chip reads the 11 address pins and \u003Cstrong>latches\u003C\u002Fstrong> (saves internally) the value as the row address. The row decoder activates row 1042, connecting all 2048 cells in that row to the sense amplifiers.\",\n    pins: [1,0,0,0,0,0,1,0,0,1,0],\n    pinStyle: \"row\",\n    ras: \"low\", cas: \"high\",\n    chip: \"Row latch captured: \u003Cspan class='h-purple'>row 1042\u003C\u002Fspan>. The entire row of 2048 capacitors is now dumping their tiny charges into the sense amplifiers, which boost them to readable voltage levels. This takes time — it's the biggest delay in a DRAM read.\"\n  },\n  {\n    title: \"4. Column address placed\",\n    desc: \"Now the memory controller \u003Cstrong>changes\u003C\u002Fstrong> the values on the \u003Cem>same 11 pins\u003C\u002Fem> to the \u003Cspan class='h-teal highlight'>column address\u003C\u002Fspan> (e.g. column 817 = 01100110001). The row address is safe — it was already latched inside the chip in step 3.\",\n    pins: [0,1,1,0,0,1,1,0,0,0,1],\n    pinStyle: \"col\",\n    ras: \"low\", cas: \"high\",\n    chip: \"Row 1042 is still active (held by the latch). All 2048 cells in that row have been amplified. The address pins now show a new number, but the chip hasn't read it yet.\"\n  },\n  {\n    title: \"5. CAS goes LOW\",\n    desc: \"The memory controller pulls \u003Cspan class='h-teal highlight'>CAS low\u003C\u002Fspan> — \\\"read the column now!\\\" The chip latches the column address, and the column decoder selects column 817 from the already-active row. The 4 bits at [row 1042, col 817] are routed to the output buffers.\",\n    pins: [0,1,1,0,0,1,1,0,0,0,1],\n    pinStyle: \"col\",\n    ras: \"low\", cas: \"low\",\n    chip: \"Column latch captured: \u003Cspan class='h-teal'>column 817\u003C\u002Fspan>. The column decoder picks 4 bits from the 2048 amplified cells. Those 4 bits are driven onto the data output pins.\"\n  },\n  {\n    title: \"6. Data appears\",\n    desc: \"The 4 data bits (e.g. \u003Cspan class='h-coral highlight'>1011\u003C\u002Fspan>) appear on the data output pins. The memory controller reads them. The full read is complete — the CPU gets its 4 bits of data.\",\n    pins: [0,1,1,0,0,1,1,0,0,0,1],\n    pinStyle: \"col\",\n    ras: \"low\", cas: \"low\",\n    data: [1,0,1,1],\n    chip: \"Data output buffers are driving \u003Cspan class='h-coral'>1011\u003C\u002Fspan> onto the 4 data pins. The memory controller captures these bits and forwards them to the CPU. Total time: ~60–70 nanoseconds for older DRAM.\"\n  },\n  {\n    title: \"7. Precharge\",\n    desc: \"RAS and CAS return HIGH. The chip deactivates the row, recharges the sense amplifiers, and gets ready for the next access. Reading a capacitor is destructive — the charge leaked out during sensing — so the sense amplifiers also \u003Cstrong>write the data back\u003C\u002Fstrong> into the row.\",\n    pins: [0,0,0,0,0,0,0,0,0,0,0],\n    pinStyle: \"off\",\n    ras: \"high\", cas: \"high\",\n    chip: \"The sense amplifiers restore the charge in all 2048 cells of row 1042 (read is destructive, so this is essential). The chip returns to idle, ready for the next row+column cycle.\"\n  }\n];\n\nconst nav = document.getElementById('nav');\nconst desc = document.getElementById('desc');\nconst signals = document.getElementById('signals');\nconst chip = document.getElementById('chip');\nlet current = 0;\n\nsteps.forEach((s, i) => {\n  const b = document.createElement('button');\n  b.className = 'step-btn';\n  b.textContent = s.title;\n  b.onclick = () => show(i);\n  nav.appendChild(b);\n});\n\nfunction show(i) {\n  current = i;\n  const s = steps[i];\n  nav.querySelectorAll('.step-btn').forEach((b,j) => b.classList.toggle('active', j===i));\n  desc.innerHTML = s.desc;\n  chip.innerHTML = s.chip;\n\n  let html = '';\n  html += '\u003Cdiv class=\"signal-row\">\u003Cspan class=\"signal-label\">Address pins\u003C\u002Fspan>\u003Cdiv class=\"pin-group\">';\n  s.pins.forEach((v,j) => {\n    const cls = s.pinStyle === 'off' ? 'pin-off' : s.pinStyle === 'row' ? 'pin-row' : 'pin-col';\n    html += `\u003Cdiv class=\"pin ${cls}\">${s.pinStyle==='off'?'—':v}\u003C\u002Fdiv>`;\n  });\n  html += '\u003C\u002Fdiv>\u003C\u002Fdiv>';\n\n  html += '\u003Cdiv class=\"signal-row\">\u003Cspan class=\"signal-label\">RAS signal\u003C\u002Fspan>';\n  html += `\u003Cdiv class=\"sig-block ${s.ras==='high'?'sig-high':'sig-low'}\">${s.ras==='high'?'HIGH (idle)':'LOW (active!)'}\u003C\u002Fdiv>`;\n  html += '\u003C\u002Fdiv>';\n\n  html += '\u003Cdiv class=\"signal-row\">\u003Cspan class=\"signal-label\">CAS signal\u003C\u002Fspan>';\n  html += `\u003Cdiv class=\"sig-block ${s.cas==='high'?'sig-high':'sig-low'}\">${s.cas==='high'?'HIGH (idle)':'LOW (active!)'}\u003C\u002Fdiv>`;\n  html += '\u003C\u002Fdiv>';\n\n  if (s.data) {\n    html += '\u003Cdiv class=\"signal-row\">\u003Cspan class=\"signal-label\">Data out\u003C\u002Fspan>\u003Cdiv class=\"pin-group\">';\n    s.data.forEach(v => { html += `\u003Cdiv class=\"pin pin-data\">${v}\u003C\u002Fdiv>`; });\n    html += '\u003C\u002Fdiv>\u003C\u002Fdiv>';\n  }\n\n  signals.innerHTML = html;\n}\n\nshow(0);\n\u003C\u002Fscript>",[12,76080,76081],{},[106,76082,76083],{},"What multiplexing actually does",[12,76085,76086],{},"Instead of 22 pins carrying row and column simultaneously, you use just 11 pins and send the address in two steps over those same wires. It's like having a single mailbox slot: first you slide in a card that says \"row 1,042\", then you slide in a second card that says \"column 817.\" The chip remembers the first card while reading the second.",[12,76088,76089],{},[106,76090,76091],{},"What RAS and CAS actually are",[12,76093,76094,76095,76098],{},"The chip needs to know ",[7289,76096,76097],{},"when"," each card arrives. That's what RAS and CAS signals do — they're timing signals (active-low pulses on dedicated pins) that say \"read the address pins NOW.\"",[12,76100,76101,76102,76105],{},"RAS (Row Address Strobe): \"The number on the address pins right now is the ",[106,76103,76104],{},"row"," address — latch it.\" The chip captures those 11 bits and stores them in an internal register.",[12,76107,76108,76109,76112],{},"CAS (Column Address Strobe): \"The number on the address pins right now is the ",[106,76110,76111],{},"column"," address — latch it.\" The chip captures those 11 bits into a second register.",[12,76114,76115],{},"So RAS and CAS are just \"now!\" signals that tell the chip which half of the address it's looking at. They're two additional pins, but you saved 11 address pins, so it's a huge net win (13 pins instead of 22).",[16,76117],{},[12,76119,76120,76121,76124],{},"The grid cells themselves don't \"remember\" any signal from the CPU. Each cell is just a tiny capacitor holding a charge (1) or not (0). All the intelligence — decoding addresses, timing the read, amplifying the faint charges — lives in the circuitry ",[7289,76122,76123],{},"surrounding"," the grid. The capacitors are passive storage; everything else is active logic that interprets the CPU's requests and routes data to the right place.",[12,76126,76127,76128,76131,76132,76135,76136,7153,76139,6959],{},"The chip also includes a ",[106,76129,76130],{},"refresh counter"," (to cycle through rows during refresh), a ",[106,76133,76134],{},"MUX"," (to select between external addresses and refresh addresses), ",[106,76137,76138],{},"row and column decoders",[106,76140,76141],{},"data I\u002FO buffers",[25631,76143,76145,76148],{"className":76144},[13441,13442],[25635,76146,76147],{},"Additional notes",[13438,76149,76151,76160,76169,76175],{"className":76150},[25641],[12,76152,76153,76156,76157,76159],{},[106,76154,76155],{},"Row and column decoders",": these are what actually ",[7289,76158,25033],{}," the latched addresses. The row decoder takes the 11-bit row number and activates one physical wire out of 2048. The column decoder does the same for columns. They're the bridge between \"the chip received address bits\" and \"the correct cells are now connected.\"",[12,76161,76162,76165,76166,76168],{},[106,76163,76164],{},"Refresh counter",": this is just a simple counter that automatically cycles through row numbers (0, 1, 2, ... 2047, 0, 1, ...) so every row gets refreshed periodically. We already covered ",[7289,76167,73259],{}," refresh is needed in the DRAM section. The counter is just the mechanism that automates it.",[12,76170,76171,76174],{},[106,76172,76173],{},"MUX (multiplexer)",": during normal operation, the row decoder receives the external address from the CPU. During refresh, it needs the address from the refresh counter instead. The MUX is a simple switch that picks between these two sources.",[12,76176,76177,76180],{},[106,76178,76179],{},"Data I\u002FO buffers",": just the circuitry that drives the data pins during a read and receives data during a write. Straightforward.",[7166,76182,76184],{"id":76183},"packaging-and-chip-pinouts","Packaging and Chip Pinouts",[12,76186,76187],{},"Memory chips come in standard packages with defined pinouts. For example, an 8 Mbit EPROM might use a 32-pin DIP (Dual In-line Package), while a 16 Mbit DRAM uses a 24-pin package. The DRAM needs fewer pins partly because of address multiplexing, a neat design win.",[25631,76189,76191,76193],{"className":76190},[13441,13442],[25635,76192,75839],{},[13438,76194,76196,76199,76202,76213],{"className":76195},[25641],[12,76197,76198],{},"A DIP (Dual In-line Package) is the physical chip you'd see on a circuit board — a rectangular plastic or ceramic body with metal pins sticking out in two parallel rows along the sides. Every pin has a specific job. For a memory chip, the pins fall into a few categories: address pins (to specify which location), data pins (to carry the bits in\u002Fout), control pins (like chip enable, read\u002Fwrite select, and for DRAM, RAS\u002FCAS), and power pins (VCC and ground).",[7026,76200],{"code":76201},"\u003Ch2 class=\"sr-only\">Side-by-side comparison of 8 Mbit EPROM (32 pins) versus 16 Mbit DRAM (24 pins) showing how address multiplexing reduces pin count.\u003C\u002Fh2>\n\u003Cstyle>\n  \u002F* Added fallbacks for Claude CSS Variables *\u002F\n  .chip { display: inline-flex; flex-direction: column; align-items: center; }\n  .chip-body { position: relative; border: 1.5px solid var(--color-text-primary); border-radius: 4px; display: flex; }\n  .pin-col { display: flex; flex-direction: column; gap: 2px; padding: 4px 0; }\n  .pin { height: 14px; width: 38px; display: flex; align-items: center; font-size: 11px; font-family: var(--font-mono); }\n  .pin-l { justify-content: flex-end; padding-right: 4px; border-right: 1.5px solid var(--color-text-primary); }\n  .pin-r { justify-content: flex-start; padding-left: 4px; border-left: 1.5px solid var(--color-text-primary); }\n  .pin-addr { color: #534AB7; }\n  .pin-data { color: #D85A30; }\n  .pin-ctrl { color: #1D9E75; }\n  .pin-pwr { color: var(--color-text-tertiary); }\n  .notch { width: 20px; height: 10px; border-bottom-left-radius: 12px; border-bottom-right-radius: 12px; border-bottom: 1.5px solid var(--color-text-primary); border-left: 1.5px solid var(--color-text-primary); border-right: 1.5px solid var(--color-text-primary); margin: 0 auto; position: relative; top: -1px; }\n  .chip-label { font-size: 11px; color: var(--color-text-secondary); writing-mode: vertical-rl; text-orientation: mixed; padding: 0 6px; display: flex; align-items: center; justify-content: center; letter-spacing: 1px; }\n  .compare { display: flex; gap: 40px; justify-content: center; flex-wrap: wrap; padding: 0.5rem 0; }\n  .section { display: flex; flex-direction: column; align-items: center; gap: 8px; }\n  .section-title { font-size: 14px; font-weight: 500; color: var(--color-text-primary); }\n  .section-sub { font-size: 12px; color: var(--color-text-secondary); text-align: center; }\n  .legend { display: flex; gap: 16px; justify-content: center; margin-top: 16px; flex-wrap: wrap; }\n  .legend-item { display: flex; align-items: center; gap: 5px; font-size: 12px; color: var(--color-text-secondary); }\n  .legend-dot { width: 8px; height: 8px; border-radius: 50%; }\n  .tally { margin-top: 12px; padding: 10px 14px; border-radius: var(--border-radius-md); background: var(--color-background-secondary); border: 1px solid var(--color-border-tertiary); font-size: 13px; color: var(--color-text-secondary); line-height: 1.6; width: 100%; max-width: 220px; }\n  .tally strong { color: var(--color-text-primary); font-weight: 500; }\n\u003C\u002Fstyle>\n\u003Cdiv class=\"compare\">\n  \u003Cdiv class=\"section\">\n    \u003Cdiv class=\"section-title\">8 Mbit EPROM\u003C\u002Fdiv>\n    \u003Cdiv class=\"section-sub\">1M locations x 8 bits\u003C\u002Fdiv>\n    \u003Cdiv class=\"chip\">\n      \u003Cdiv class=\"notch\">\u003C\u002Fdiv>\n      \u003Cdiv class=\"chip-body\">\n        \u003Cdiv class=\"pin-col\">\n          \u003Cdiv class=\"pin pin-l pin-addr\">A19\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A16\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A15\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A12\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A7\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A6\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A5\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A4\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A3\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A2\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A1\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A0\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-data\">D0\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-data\">D1\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-data\">D2\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-pwr\">GND\u003C\u002Fdiv>\n        \u003C\u002Fdiv>\n        \u003Cdiv class=\"chip-label\">EPROM\u003C\u002Fdiv>\n        \u003Cdiv class=\"pin-col\">\n          \u003Cdiv class=\"pin pin-r pin-pwr\">VCC\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-addr\">A18\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-addr\">A17\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-addr\">A14\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-addr\">A13\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-addr\">A8\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-addr\">A9\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-addr\">A11\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-ctrl\">OE\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-addr\">A10\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-ctrl\">CE\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-data\">D7\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-data\">D6\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-data\">D5\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-data\">D4\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-data\">D3\u003C\u002Fdiv>\n        \u003C\u002Fdiv>\n      \u003C\u002Fdiv>\n    \u003C\u002Fdiv>\n    \u003Cdiv class=\"tally\">\n      \u003Cspan class=\"pin-addr\">20 address pins\u003C\u002Fspan>\u003Cbr>\n      \u003Cspan class=\"pin-data\">8 data pins\u003C\u002Fspan>\u003Cbr>\n      \u003Cspan class=\"pin-ctrl\">2 control pins\u003C\u002Fspan>\u003Cbr>\n      \u003Cspan class=\"pin-pwr\">2 power pins\u003C\u002Fspan>\u003Cbr>\n      \u003Cstrong>= 32 pins total\u003C\u002Fstrong>\n    \u003C\u002Fdiv>\n  \u003C\u002Fdiv>\n\n  \u003Cdiv class=\"section\">\n    \u003Cdiv class=\"section-title\">16 Mbit DRAM\u003C\u002Fdiv>\n    \u003Cdiv class=\"section-sub\">4M locations x 4 bits\u003C\u002Fdiv>\n    \u003Cdiv class=\"chip\">\n      \u003Cdiv class=\"notch\">\u003C\u002Fdiv>\n      \u003Cdiv class=\"chip-body\">\n        \u003Cdiv class=\"pin-col\">\n          \u003Cdiv class=\"pin pin-l pin-addr\">A0\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A1\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A2\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A3\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-data\">DQ1\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A4\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A5\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A6\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A7\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-data\">DQ2\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-addr\">A8\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-l pin-pwr\">VSS\u003C\u002Fdiv>\n        \u003C\u002Fdiv>\n        \u003Cdiv class=\"chip-label\">DRAM\u003C\u002Fdiv>\n        \u003Cdiv class=\"pin-col\">\n          \u003Cdiv class=\"pin pin-r pin-pwr\">VCC\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-addr\">A10\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-addr\">A9\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-ctrl\">RAS\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-data\">DQ4\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-ctrl\">CAS\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-ctrl\">WE\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-ctrl\">OE\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-data\">DQ3\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-pwr\">NC\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-pwr\">NC\u003C\u002Fdiv>\n          \u003Cdiv class=\"pin pin-r pin-pwr\">VSS\u003C\u002Fdiv>\n        \u003C\u002Fdiv>\n      \u003C\u002Fdiv>\n    \u003C\u002Fdiv>\n    \u003Cdiv class=\"tally\">\n      \u003Cspan class=\"pin-addr\">11 address pins\u003C\u002Fspan> (multiplexed!)\u003Cbr>\n      \u003Cspan class=\"pin-data\">4 data pins\u003C\u002Fspan>\u003Cbr>\n      \u003Cspan class=\"pin-ctrl\">4 control pins\u003C\u002Fspan> (RAS, CAS, WE, OE)\u003Cbr>\n      \u003Cspan class=\"pin-pwr\">3 power + 2 NC\u003C\u002Fspan>\u003Cbr>\n      \u003Cstrong>= 24 pins total\u003C\u002Fstrong>\n    \u003C\u002Fdiv>\n  \u003C\u002Fdiv>\n\u003C\u002Fdiv>\n\u003Cdiv class=\"legend\">\n  \u003Cdiv class=\"legend-item\">\u003Cdiv class=\"legend-dot\" style=\"background:#534AB7\">\u003C\u002Fdiv>Address\u003C\u002Fdiv>\n  \u003Cdiv class=\"legend-item\">\u003Cdiv class=\"legend-dot\" style=\"background:#D85A30\">\u003C\u002Fdiv>Data\u003C\u002Fdiv>\n  \u003Cdiv class=\"legend-item\">\u003Cdiv class=\"legend-dot\" style=\"background:#1D9E75\">\u003C\u002Fdiv>Control\u003C\u002Fdiv>\n  \u003Cdiv class=\"legend-item\">\u003Cdiv class=\"legend-dot\" style=\"background:var(--color-text-tertiary)\">\u003C\u002Fdiv>Power \u002F unused\u003C\u002Fdiv>\n\u003C\u002Fdiv>",[12,76203,76204,76205,76208,76209,76212],{},"The DRAM stores ",[106,76206,76207],{},"twice as much data"," (16 Mbit vs 8 Mbit) yet uses ",[106,76210,76211],{},"fewer pins"," (24 vs 32). That's almost entirely because of address multiplexing.",[12,76214,76215,76216,76248],{},"The EPROM needs 20 address pins because it has 1M locations (",[24923,76217,76219],{"className":76218,"jax":24927},[24926],[24929,76220,76221,76226],{"style":32867,"xmlns":24932,"width":76013,"height":32869,"role":1930,"focusable":8058,"viewBox":76014,"xmlnsXLink":24936},[24938,76222,76223],{},[24941,76224],{"id":76225,"d":25928},"MJX-2-TEX-N-32",[25022,76227,76228],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,76229,76230,76235,76241],{"dataMmlNode":25028},[25022,76231,76232],{"dataMmlNode":26031},[25033,76233],{"dataC":26058,"xLinkHref":76234},"#MJX-2-TEX-N-32",[25022,76236,76237],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":26394},[25022,76238,76239],{"dataMmlNode":25031},[6977,76240,37204],{"dataVariant":32892,"transform":25025,"font-size":32893,"font-family":32894},[25022,76242,76243],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":76038},[25022,76244,76245],{"dataMmlNode":25031},[6977,76246,76247],{"dataVariant":32892,"transform":25025,"font-size":32893,"font-family":32894},"⁰"," = 1,048,576) and sends the full address all at once — no multiplexing. The DRAM needs to address 2048 rows and 2048 columns (22 address bits total), but because it sends them at different times over the same wires, it only needs 11 address pins. It trades two extra control pins (RAS and CAS) for saving 11 address pins",[7166,76250,76252],{"id":76251},"building-bigger-memory-modules","Building Bigger: Memory Modules",[12,76254,76255,76256,76259,76260,6959],{},"A single chip usually stores one bit (or a few bits) per address. To build a full ",[106,76257,76258],{},"byte-wide"," memory system, you combine multiple chips into a ",[106,76261,76262],{},"module",[12,76264,76265,76266,76269,76270,76273],{},"For example, to build a ",[106,76267,76268],{},"256 kByte module",", you can combine ",[106,76271,76272],{},"eight"," 1-bit RAM chips, each with 256k locations. All eight chips share the same address bus, and each chip contributes one bit to form an 8-bit (1-byte) data word. The Memory Address Register (MAR) feeds the address to all chips simultaneously, and the Memory Buffer Register (MBR) collects the 8 bits.",[7026,76275],{"code":76276},"\u003Ch2 class=\"sr-only\">Diagram showing 8 single-bit RAM chips wired together to form one byte-wide memory module, sharing an address bus.\u003C\u002Fh2>\n\u003Cstyle>\n  \u002F* --- CSS to style the colored chips (MAR, Chips 0-7, MBR) --- *\u002F\n  svg text {\n    font-family: var(--font-sans);\n  }\n  \n  .th { font-weight: 500; font-size: 14px; }\n  .ts { font-size: 12px; fill: var(--color-text-secondary); }\n\n  \u002F* Purple Box (MAR) *\u002F\n  .c-purple rect { \n    fill: var(--color-purple-bg); \u002F* Dark purple *\u002F\n    stroke: #534AB7; \n    stroke-width: 1.5; \n  }\n  .c-purple .th { fill: #ffffff; }\n  .c-purple .ts { fill: #c8c4f0; } \u002F* Light purple text *\u002F\n\n  \u002F* Teal Boxes (Chips 0-7) *\u002F\n  .c-teal rect { \n    fill: var(--color-teal-bg); \u002F* Dark teal *\u002F\n    stroke: #1D9E75; \n    stroke-width: 1.5; \n  }\n  .c-teal .th { fill: #ffffff; }\n  .c-teal .ts { fill: #a5d8c7; } \u002F* Light teal text *\u002F\n\n  \u002F* Coral Box (MBR) *\u002F\n  .c-coral rect { \n    fill: var(--color-coral-bg); \u002F* Dark coral *\u002F\n    stroke: #D85A30; \n    stroke-width: 1.5; \n  }\n  .c-coral .th { fill: #ffffff; }\n  .c-coral .ts { fill: #f3bda8; } \u002F* Light coral text *\u002F\n\u003C\u002Fstyle>\n\u003Csvg width=\"100%\" viewBox=\"0 0 680 520\" role=\"img\">\n  \u003Ctitle>256 kByte memory module from eight 1-bit chips\u003C\u002Ftitle>\n  \u003Cdesc>Eight chips share one address bus from the MAR, each contributing one bit to form an 8-bit word collected by the MBR.\u003C\u002Fdesc>\n  \u003Cdefs>\n    \u003Cmarker id=\"arrow\" viewBox=\"0 0 10 10\" refX=\"8\" refY=\"5\" markerWidth=\"6\" markerHeight=\"6\" orient=\"auto-start-reverse\">\n      \u003Cpath d=\"M2 1L8 5L2 9\" fill=\"none\" stroke=\"context-stroke\" stroke-width=\"1.5\" stroke-linecap=\"round\" stroke-linejoin=\"round\"\u002F>\n    \u003C\u002Fmarker>\n  \u003C\u002Fdefs>\n\n  \u003Cg class=\"c-purple\">\n    \u003Crect x=\"240\" y=\"30\" width=\"200\" height=\"44\" rx=\"8\" stroke-width=\"0.5\"\u002F>\n    \u003Ctext class=\"th\" x=\"340\" y=\"46\" text-anchor=\"middle\" dominant-baseline=\"central\">MAR\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"340\" y=\"62\" text-anchor=\"middle\" dominant-baseline=\"central\">Address: row 7, col 42\u003C\u002Ftext>\n  \u003C\u002Fg>\n\n  \u003Cline x1=\"340\" y1=\"74\" x2=\"340\" y2=\"110\" stroke=\"#534AB7\" stroke-width=\"1.5\" marker-end=\"url(#arrow)\"\u002F>\n\n  \u003Cline x1=\"80\" y1=\"120\" x2=\"600\" y2=\"120\" stroke=\"#534AB7\" stroke-width=\"2.5\" stroke-linecap=\"round\"\u002F>\n  \u003Ctext class=\"ts\" x=\"614\" y=\"124\" text-anchor=\"start\" style=\"fill:#534AB7\">Address bus\u003C\u002Ftext>\n  \u003Ctext class=\"ts\" x=\"614\" y=\"138\" text-anchor=\"start\" style=\"fill:#534AB7\">(shared)\u003C\u002Ftext>\n\n  \u003Cline x1=\"109\" y1=\"120\" x2=\"109\" y2=\"160\" stroke=\"#534AB7\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"179\" y1=\"120\" x2=\"179\" y2=\"160\" stroke=\"#534AB7\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"249\" y1=\"120\" x2=\"249\" y2=\"160\" stroke=\"#534AB7\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"319\" y1=\"120\" x2=\"319\" y2=\"160\" stroke=\"#534AB7\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"389\" y1=\"120\" x2=\"389\" y2=\"160\" stroke=\"#534AB7\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"459\" y1=\"120\" x2=\"459\" y2=\"160\" stroke=\"#534AB7\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"529\" y1=\"120\" x2=\"529\" y2=\"160\" stroke=\"#534AB7\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"599\" y1=\"120\" x2=\"599\" y2=\"160\" stroke=\"#534AB7\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n\n  \u003Cg class=\"c-teal\">\u003Crect x=\"80\"  y=\"164\" width=\"58\" height=\"140\" rx=\"6\" stroke-width=\"0.5\"\u002F>\n    \u003Ctext class=\"th\" x=\"109\" y=\"200\" text-anchor=\"middle\" dominant-baseline=\"central\" style=\"font-size:12px\">Chip 0\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"109\" y=\"220\" text-anchor=\"middle\" dominant-baseline=\"central\">256K\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"109\" y=\"236\" text-anchor=\"middle\" dominant-baseline=\"central\">x 1\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"109\" y=\"280\" text-anchor=\"middle\" dominant-baseline=\"central\">bit 0\u003C\u002Ftext>\u003C\u002Fg>\n\n  \u003Cg class=\"c-teal\">\u003Crect x=\"150\" y=\"164\" width=\"58\" height=\"140\" rx=\"6\" stroke-width=\"0.5\"\u002F>\n    \u003Ctext class=\"th\" x=\"179\" y=\"200\" text-anchor=\"middle\" dominant-baseline=\"central\" style=\"font-size:12px\">Chip 1\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"179\" y=\"220\" text-anchor=\"middle\" dominant-baseline=\"central\">256K\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"179\" y=\"236\" text-anchor=\"middle\" dominant-baseline=\"central\">x 1\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"179\" y=\"280\" text-anchor=\"middle\" dominant-baseline=\"central\">bit 1\u003C\u002Ftext>\u003C\u002Fg>\n\n  \u003Cg class=\"c-teal\">\u003Crect x=\"220\" y=\"164\" width=\"58\" height=\"140\" rx=\"6\" stroke-width=\"0.5\"\u002F>\n    \u003Ctext class=\"th\" x=\"249\" y=\"200\" text-anchor=\"middle\" dominant-baseline=\"central\" style=\"font-size:12px\">Chip 2\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"249\" y=\"220\" text-anchor=\"middle\" dominant-baseline=\"central\">256K\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"249\" y=\"236\" text-anchor=\"middle\" dominant-baseline=\"central\">x 1\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"249\" y=\"280\" text-anchor=\"middle\" dominant-baseline=\"central\">bit 2\u003C\u002Ftext>\u003C\u002Fg>\n\n  \u003Cg class=\"c-teal\">\u003Crect x=\"290\" y=\"164\" width=\"58\" height=\"140\" rx=\"6\" stroke-width=\"0.5\"\u002F>\n    \u003Ctext class=\"th\" x=\"319\" y=\"200\" text-anchor=\"middle\" dominant-baseline=\"central\" style=\"font-size:12px\">Chip 3\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"319\" y=\"220\" text-anchor=\"middle\" dominant-baseline=\"central\">256K\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"319\" y=\"236\" text-anchor=\"middle\" dominant-baseline=\"central\">x 1\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"319\" y=\"280\" text-anchor=\"middle\" dominant-baseline=\"central\">bit 3\u003C\u002Ftext>\u003C\u002Fg>\n\n  \u003Cg class=\"c-teal\">\u003Crect x=\"360\" y=\"164\" width=\"58\" height=\"140\" rx=\"6\" stroke-width=\"0.5\"\u002F>\n    \u003Ctext class=\"th\" x=\"389\" y=\"200\" text-anchor=\"middle\" dominant-baseline=\"central\" style=\"font-size:12px\">Chip 4\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"389\" y=\"220\" text-anchor=\"middle\" dominant-baseline=\"central\">256K\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"389\" y=\"236\" text-anchor=\"middle\" dominant-baseline=\"central\">x 1\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"389\" y=\"280\" text-anchor=\"middle\" dominant-baseline=\"central\">bit 4\u003C\u002Ftext>\u003C\u002Fg>\n\n  \u003Cg class=\"c-teal\">\u003Crect x=\"430\" y=\"164\" width=\"58\" height=\"140\" rx=\"6\" stroke-width=\"0.5\"\u002F>\n    \u003Ctext class=\"th\" x=\"459\" y=\"200\" text-anchor=\"middle\" dominant-baseline=\"central\" style=\"font-size:12px\">Chip 5\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"459\" y=\"220\" text-anchor=\"middle\" dominant-baseline=\"central\">256K\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"459\" y=\"236\" text-anchor=\"middle\" dominant-baseline=\"central\">x 1\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"459\" y=\"280\" text-anchor=\"middle\" dominant-baseline=\"central\">bit 5\u003C\u002Ftext>\u003C\u002Fg>\n\n  \u003Cg class=\"c-teal\">\u003Crect x=\"500\" y=\"164\" width=\"58\" height=\"140\" rx=\"6\" stroke-width=\"0.5\"\u002F>\n    \u003Ctext class=\"th\" x=\"529\" y=\"200\" text-anchor=\"middle\" dominant-baseline=\"central\" style=\"font-size:12px\">Chip 6\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"529\" y=\"220\" text-anchor=\"middle\" dominant-baseline=\"central\">256K\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"529\" y=\"236\" text-anchor=\"middle\" dominant-baseline=\"central\">x 1\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"529\" y=\"280\" text-anchor=\"middle\" dominant-baseline=\"central\">bit 6\u003C\u002Ftext>\u003C\u002Fg>\n\n  \u003Cg class=\"c-teal\">\u003Crect x=\"570\" y=\"164\" width=\"58\" height=\"140\" rx=\"6\" stroke-width=\"0.5\"\u002F>\n    \u003Ctext class=\"th\" x=\"599\" y=\"200\" text-anchor=\"middle\" dominant-baseline=\"central\" style=\"font-size:12px\">Chip 7\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"599\" y=\"220\" text-anchor=\"middle\" dominant-baseline=\"central\">256K\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"599\" y=\"236\" text-anchor=\"middle\" dominant-baseline=\"central\">x 1\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"599\" y=\"280\" text-anchor=\"middle\" dominant-baseline=\"central\">bit 7\u003C\u002Ftext>\u003C\u002Fg>\n\n  \u003Cline x1=\"109\" y1=\"304\" x2=\"109\" y2=\"340\" stroke=\"#D85A30\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"179\" y1=\"304\" x2=\"179\" y2=\"340\" stroke=\"#D85A30\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"249\" y1=\"304\" x2=\"249\" y2=\"340\" stroke=\"#D85A30\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"319\" y1=\"304\" x2=\"319\" y2=\"340\" stroke=\"#D85A30\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"389\" y1=\"304\" x2=\"389\" y2=\"340\" stroke=\"#D85A30\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"459\" y1=\"304\" x2=\"459\" y2=\"340\" stroke=\"#D85A30\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"529\" y1=\"304\" x2=\"529\" y2=\"340\" stroke=\"#D85A30\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Cline x1=\"599\" y1=\"304\" x2=\"599\" y2=\"340\" stroke=\"#D85A30\" stroke-width=\"1\" marker-end=\"url(#arrow)\"\u002F>\n\n  \u003Ctext class=\"ts\" x=\"48\" y=\"352\" text-anchor=\"end\" style=\"fill:#D85A30\">1 bit each\u003C\u002Ftext>\n\n  \u003Cline x1=\"80\" y1=\"350\" x2=\"628\" y2=\"350\" stroke=\"#D85A30\" stroke-width=\"2.5\" stroke-linecap=\"round\"\u002F>\n\n  \u003Cline x1=\"340\" y1=\"350\" x2=\"340\" y2=\"390\" stroke=\"#D85A30\" stroke-width=\"1.5\" marker-end=\"url(#arrow)\"\u002F>\n\n  \u003Cg class=\"c-coral\">\n    \u003Crect x=\"220\" y=\"396\" width=\"240\" height=\"44\" rx=\"8\" stroke-width=\"0.5\"\u002F>\n    \u003Ctext class=\"th\" x=\"340\" y=\"412\" text-anchor=\"middle\" dominant-baseline=\"central\">MBR\u003C\u002Ftext>\n    \u003Ctext class=\"ts\" x=\"340\" y=\"428\" text-anchor=\"middle\" dominant-baseline=\"central\">Collected: 10110011\u003C\u002Ftext>\n  \u003C\u002Fg>\n\n  \u003Cline x1=\"340\" y1=\"440\" x2=\"340\" y2=\"480\" stroke=\"#D85A30\" stroke-width=\"1.5\" marker-end=\"url(#arrow)\"\u002F>\n  \u003Ctext class=\"ts\" x=\"340\" y=\"500\" text-anchor=\"middle\">1 full byte to CPU\u003C\u002Ftext>\n\u003C\u002Fsvg>",[12,76278,76279],{},"Now suppose you want more capacity. You could use bigger chips, but that's expensive. Instead, you duplicate the whole 8-chip module multiple times and add a selection mechanism. A 1 MByte module groups chips into four groups (A, B, C, D), each being a 256 kByte module like the one above. A \"chip-group-enable\" signal works like a master switch, it tells one group \"you're active, respond to this address\" while the other three stay silent.",[12,76281,76282,76283,76286,76287,76290],{},"Think of it like four filing cabinets side by side, each with identical drawer numbering (addresses 0 through 256K). The address tells you ",[7289,76284,76285],{},"which drawer",", and the group-enable tells you ",[7289,76288,76289],{},"which cabinet",". Together they select one unique byte out of the full 1 MByte. This way you can keep adding cabinets (groups) to grow capacity without needing to redesign the individual chips or the addressing scheme within each group.",[6964,76292,76294],{"id":76293},"error-correction-because-bits-flip","Error Correction: Because Bits Flip",[12,76296,76297],{},"Memory isn't perfect. Errors can and do occur:",[23,76299,76300,76306],{},[26,76301,76302,76305],{},[106,76303,76304],{},"Hard failures"," are permanent physical defects — a cell that's stuck at 0 or 1.",[26,76307,76308,76311],{},[106,76309,76310],{},"Soft errors"," are random, non-destructive events. A cosmic ray or electrical noise might flip a bit, but the cell itself is fine. No permanent damage to the memory.",[12,76313,76314,76315,76318,76319,6959],{},"To detect and fix these, memory systems use ",[106,76316,76317],{},"error-correcting codes (ECC)",", most commonly ",[106,76320,76321],{},"Hamming codes",[12,76323,76324,76325,76328,76329,76331,76332,76334,76335,76338,76339,76342,76343,76346],{},"Here's the basic idea: when data (M bits) is written to memory, a function ",[7289,76326,76327],{},"f"," generates K check bits from the data. Both the M data bits and K check bits are stored together. When the data is read back, the same function ",[7289,76330,76327],{}," is applied to the M data bits to generate a ",[7289,76333,8823],{}," set of K check bits. These are ",[106,76336,76337],{},"compared"," with the stored check bits. If they match, no error. If they differ, the comparison result (called a ",[106,76340,76341],{},"syndrome",") identifies which bit flipped, and a ",[106,76344,76345],{},"corrector"," circuit fixes it. An error signal is also raised so the system knows a correction occurred.",[6964,76348,76350],{"id":76349},"advanced-dram-sdram-and-ddr","Advanced DRAM: SDRAM and DDR",[12,76352,76353],{},"Basic DRAM is asynchronous — the processor sends a request and waits an unpredictable amount of time for the data to arrive. As processors got faster, this \"waiting around\" became a serious bottleneck. Enter the advanced DRAM technologies.",[7166,76355,76357],{"id":76356},"sdram-synchronous-dram","SDRAM (Synchronous DRAM)",[12,76359,76360,76361,76364,76365,76368],{},"SDRAM changed the game by ",[106,76362,76363],{},"synchronizing memory access to an external clock",". In conventional DRAM, the CPU sends a request and then just... waits. With SDRAM, because data transfers happen in lockstep with the system clock, the CPU ",[106,76366,76367],{},"knows exactly when the data will be ready",". This means the CPU can go do something else in the meantime instead of sitting idle.",[12,76370,76371,76372,76375],{},"SDRAM also supports ",[106,76373,76374],{},"burst mode",": after providing a starting address, the chip can fire out a stream of consecutive data words on successive clock cycles without needing a new address for each one. For example, with a burst length of 4 and a CAS latency of 2, you issue one READ command and get four consecutive data outputs starting two clock cycles later.",[7166,76377,76379],{"id":76378},"ddr-sdram-double-data-rate","DDR SDRAM (Double Data Rate)",[12,76381,76382,76383,76386,76387,76389],{},"DDR took SDRAM further by transferring data on ",[106,76384,76385],{},"both edges of the clock signal"," — the rising edge ",[7289,76388,26525],{}," the falling edge. This effectively doubles the data rate without increasing the clock frequency.",[12,76391,76392,76393,76396],{},"But DDR's improvements go beyond that. Each successive DDR generation added a wider ",[106,76394,76395],{},"prefetch buffer"," — an internal buffer that reads more bits from the memory array in each access:",[7085,76398,76399,76415],{},[7088,76400,76401],{},[7091,76402,76403,76406,76409,76412],{},[7094,76404,76405],{},"Generation",[7094,76407,76408],{},"Prefetch Buffer",[7094,76410,76411],{},"Voltage",[7094,76413,76414],{},"Data Rate",[7104,76416,76417,76431,76444,76458],{},[7091,76418,76419,76422,76425,76428],{},[7109,76420,76421],{},"DDR1",[7109,76423,76424],{},"2 bits",[7109,76426,76427],{},"2.5V",[7109,76429,76430],{},"200–400 Mbps",[7091,76432,76433,76436,76438,76441],{},[7109,76434,76435],{},"DDR2",[7109,76437,75999],{},[7109,76439,76440],{},"1.8V",[7109,76442,76443],{},"400–1,066 Mbps",[7091,76445,76446,76449,76452,76455],{},[7109,76447,76448],{},"DDR3",[7109,76450,76451],{},"8 bits",[7109,76453,76454],{},"1.5V",[7109,76456,76457],{},"800–2,133 Mbps",[7091,76459,76460,76463,76465,76468],{},[7109,76461,76462],{},"DDR4",[7109,76464,76451],{},[7109,76466,76467],{},"1.2V",[7109,76469,76470],{},"2,133–4,266 Mbps",[12,76472,76473,76474,76477,76478,76481],{},"Notice the pattern: each generation ",[106,76475,76476],{},"lowers the voltage"," (less power, less heat) while ",[106,76479,76480],{},"increasing the data rate",". The internal memory array runs at roughly the same speed across generations — the speed gains come from the wider prefetch and faster I\u002FO bus. DDR4 also introduced a two-multiplexer design with bank groups, further boosting throughput.",[6964,76483,76485],{"id":76484},"flash-memory-the-best-of-both-worlds","Flash Memory: The Best of Both Worlds",[12,76487,76488],{},"Flash memory deserves its own section because it has become incredibly important in modern computing — from USB drives to SSDs to the storage in your phone.",[12,76490,76491,76492,76495],{},"First introduced in the mid-1980s, flash sits between EPROM and EEPROM in both cost and functionality. Like EEPROM, it erases electrically (no UV light). Like EPROM, it uses only one transistor per bit, achieving high density. The trade-off is that it ",[106,76493,76494],{},"cannot erase individual bytes"," — you must erase entire blocks at once.",[7166,76497,76499],{"id":76498},"how-flash-works","How Flash Works",[12,76501,76502,76503,76506],{},"A flash memory cell is a modified transistor with an extra layer: a ",[106,76504,76505],{},"floating gate"," sandwiched between the control gate and the transistor channel, surrounded by insulating oxide.",[23,76508,76509,76515],{},[26,76510,76511,76514],{},[106,76512,76513],{},"To store a 0:"," Electrons are injected onto the floating gate (by applying a high voltage to the control gate). These trapped electrons raise the transistor's threshold voltage, making it harder to turn on. When read, the cell doesn't conduct — that's interpreted as 0.",[26,76516,76517,76520],{},[106,76518,76519],{},"To store a 1:"," The floating gate has no trapped electrons (or they've been removed by erasure). The transistor turns on normally when read — that's interpreted as 1.",[7166,76522,76524],{"id":76523},"nor-vs-nand-flash","NOR vs. NAND Flash",[12,76526,76527],{},"Flash chips come in two main architectures:",[12,76529,76530,76533,76534,76537],{},[106,76531,76532],{},"NOR flash"," connects each cell individually to the bit line (like NOR gates in parallel). This allows ",[106,76535,76536],{},"random access"," to any byte — great for executing code directly from flash (called XIP, \"execute in place\"). NOR flash is typically used for firmware storage and embedded applications.",[12,76539,76540,76543,76544,76547,76548,76551],{},[106,76541,76542],{},"NAND flash"," connects cells in ",[106,76545,76546],{},"series"," (like a chain). You can't access individual bytes randomly — you read and write in pages (typically 4 kB or larger). But NAND is ",[106,76549,76550],{},"denser and cheaper"," per bit because the series connection uses less chip area. NAND flash is what's inside your SSD, USB drive, and smartphone storage.",[6964,76553,76555],{"id":76554},"looking-ahead-nonvolatile-ram-nvram","Looking Ahead: Nonvolatile RAM (NVRAM)",[12,76557,76558,76559,76562],{},"The memory world is actively pursuing technologies that combine the speed of RAM with the persistence of flash. These ",[106,76560,76561],{},"nonvolatile RAM"," (NVRAM) technologies sit in the memory hierarchy between DRAM and flash\u002FSSD, potentially offering both fast access and data retention without power.",[12,76564,76565],{},"Some promising NVRAM technologies include STT-RAM (Spin-Transfer Torque RAM), PCRAM (Phase-Change RAM), and ReRAM (Resistive RAM). Each uses a different physical mechanism to store data persistently while aiming for DRAM-like speeds. While these are still maturing, they could eventually blur the line between memory and storage.",[6964,76567,76569],{"id":76568},"summary-putting-it-all-together","Summary: Putting It All Together",[12,76571,76572],{},"Internal memory is all about trade-offs. Here's how the pieces fit together in a typical system:",[12,76574,76575,76578],{},[106,76576,76577],{},"Closest to the CPU → Fastest but smallest and most expensive:","\nRegisters → SRAM Cache (L1, L2, L3) → DRAM Main Memory → Flash\u002FSSD Storage",[12,76580,76581,76582,76585],{},"Each layer compensates for the weaknesses of the next. SRAM cache hides the relative slowness of DRAM. DRAM provides the capacity that SRAM can't afford. Flash provides persistence that DRAM can't. And the entire hierarchy works together to give you the ",[7289,76583,76584],{},"illusion"," of memory that's fast, big, cheap, and permanent — even though no single technology delivers all four.",[12,76587,76588],{},"Understanding this hierarchy — and the physics behind each layer — is one of the most fundamental insights in computer architecture. Every optimisation in modern computing, from CPU cache policies to SSD wear levelling, traces back to these core trade-offs.",[9332,76590,31966],{},{"title":180,"searchDepth":181,"depth":181,"links":76592},[76593,76594,76595,76596,76601,76604,76608,76609,76613,76617,76618],{"id":75332,"depth":181,"text":75333},{"id":75374,"depth":181,"text":75375},{"id":75441,"depth":181,"text":75442},{"id":75645,"depth":181,"text":75646,"children":76597},[76598,76599,76600],{"id":75685,"depth":7430,"text":75686},{"id":75800,"depth":7430,"text":75801},{"id":75854,"depth":7430,"text":75855},{"id":75885,"depth":181,"text":75886,"children":76602},[76603],{"id":75921,"depth":7430,"text":75922},{"id":75975,"depth":181,"text":75976,"children":76605},[76606,76607],{"id":76183,"depth":7430,"text":76184},{"id":76251,"depth":7430,"text":76252},{"id":76293,"depth":181,"text":76294},{"id":76349,"depth":181,"text":76350,"children":76610},[76611,76612],{"id":76356,"depth":7430,"text":76357},{"id":76378,"depth":7430,"text":76379},{"id":76484,"depth":181,"text":76485,"children":76614},[76615,76616],{"id":76498,"depth":7430,"text":76499},{"id":76523,"depth":7430,"text":76524},{"id":76554,"depth":181,"text":76555},{"id":76568,"depth":181,"text":76569},"2026-04-23",{},"\u002Fblog\u002FInternal-Memory-How-Your-Computer-Remembers-Things",{"title":75327,"description":180},{"loc":76621},"blog\u002FInternal-Memory-How-Your-Computer-Remembers-Things","jtwuakJ9BAZ_8sJ3nh6wLQsC6rSlzS8pX0lBUYiEz2Y",{"id":76627,"title":76628,"author":6946,"body":76629,"date":74911,"description":180,"draft":185,"edited_at":73699,"extension":186,"featured_image":194,"meta":77832,"navigation":189,"path":77833,"pinned":185,"seo":77834,"sitemap":77835,"stem":77836,"tags":194,"__hash__":77837},"blog\u002Fblog\u002FThe-Memory-Hierarchy-Understanding-Cache-Memory.md","The Memory Hierarchy: Understanding Cache Memory",{"type":9,"value":76630,"toc":77813},[76631,76633,76636,76649,76653,76666,76673,76682,76692,76695,76699,76711,76714,76728,76735,76833,76846,76850,76853,76856,76870,76876,76880,76883,76913,76977,76981,76984,76988,76991,77007,77010,77014,77024,77028,77035,77042,77072,77085,77091,77101,77227,77231,77237,77240,77264,77275,77280,77285,77379,77383,77407,77436,77439,77444,77449,77452,77564,77568,77574,77600,77604,77611,77620,77630,77634,77637,77651,77655,77658,77666,77670,77677,77704,77711,77715,77725,77728,77748,77755,77759,77762,77768,77777,77783,77787,77790,77793,77807],[6964,76632,75333],{"id":75332},[12,76634,76635],{},"Imagine you're studying at the library. You have your desk, a small shelf next to you, and the massive library stacks behind you. When you're working on an essay, do you walk to the stacks every time you need to check a single sentence? Of course not — you keep the books you're actively using right on your desk.",[12,76637,76638,76639,76642,76643,76645,76646,6959],{},"Computers face the exact same problem. The processor (CPU) needs data to work with, and where that data lives — and how quickly it can be fetched — has a ",[7289,76640,76641],{},"huge"," impact on performance. This is the core idea behind the ",[106,76644,75366],{}," and, more specifically, ",[106,76647,76648],{},"cache memory",[6964,76650,76652],{"id":76651},"the-principle-of-locality","The Principle of Locality",[12,76654,76655,76656,76658,76659,76662,76663,16419],{},"Before we dive into cache, we need to understand ",[7289,76657,73259],{}," cache works so well. The answer lies in something called the ",[106,76660,76661],{},"principle of locality"," (also known as ",[106,76664,76665],{},"locality of reference",[12,76667,76668,76669,76672],{},"When a program runs, it doesn't access memory randomly. Instead, memory accesses tend to ",[106,76670,76671],{},"cluster"," around certain locations. This clustering comes in two flavours:",[12,76674,76675,76678,76679,76681],{},[106,76676,76677],{},"Temporal locality"," — if a program accessed a piece of data recently, it's very likely to access it again soon. Think about a loop counter: the variable ",[6979,76680,10588],{}," gets read and written on every iteration, over and over. Constants, temporary variables, and working stacks all exhibit this pattern.",[12,76683,76684,76687,76688,76691],{},[106,76685,76686],{},"Spatial locality"," — if a program accessed a memory address, it's likely to access ",[7289,76689,76690],{},"nearby"," addresses soon. When you iterate through an array, you move from element 0 to element 1 to element 2 — each one sitting right next to the previous in memory.",[12,76693,76694],{},"These two tendencies are what make caching effective. If we can predict what data will be needed next (because it was used recently or lives nearby), we can keep it in a small, fast storage area and avoid the expensive trip to main memory.",[6964,76696,76698],{"id":76697},"the-memory-hierarchy","The Memory Hierarchy",[12,76700,76701,76702,7206,76705,7042,76708],{},"Designing a computer's memory system boils down to three competing questions: ",[106,76703,76704],{},"how much?",[106,76706,76707],{},"how fast?",[106,76709,76710],{},"how expensive?",[12,76712,76713],{},"Unfortunately, you can't have it all:",[23,76715,76716,76719,76722],{},[26,76717,76718],{},"Faster memory costs more per bit.",[26,76720,76721],{},"Larger memory is cheaper per bit, but slower.",[26,76723,76724,76725,76727],{},"You always want more capacity ",[7289,76726,26525],{}," more speed, but your budget disagrees.",[12,76729,76730,76731,76734],{},"The solution is a ",[106,76732,76733],{},"hierarchy"," — multiple levels of memory, each with different speed, size, and cost characteristics:",[7085,76736,76737,76753],{},[7088,76738,76739],{},[7091,76740,76741,76744,76747,76750],{},[7094,76742,76743],{},"Level",[7094,76745,76746],{},"Technology",[7094,76748,76749],{},"Typical Size",[7094,76751,76752],{},"Managed By",[7104,76754,76755,76771,76787,76802,76818],{},[7091,76756,76757,76762,76765,76768],{},[7109,76758,76759],{},[106,76760,76761],{},"Registers",[7109,76763,76764],{},"CMOS (on-chip flip-flops)",[7109,76766,76767],{},"A few hundred bytes",[7109,76769,76770],{},"Compiler",[7091,76772,76773,76778,76781,76784],{},[7109,76774,76775],{},[106,76776,76777],{},"Cache (L1\u002FL2\u002FL3)",[7109,76779,76780],{},"SRAM \u002F eDRAM",[7109,76782,76783],{},"KB to tens of MB",[7109,76785,76786],{},"Processor hardware",[7091,76788,76789,76794,76796,76799],{},[7109,76790,76791],{},[106,76792,76793],{},"Main Memory",[7109,76795,75679],{},[7109,76797,76798],{},"GB",[7109,76800,76801],{},"Operating System",[7091,76803,76804,76809,76812,76815],{},[7109,76805,76806],{},[106,76807,76808],{},"Secondary Storage",[7109,76810,76811],{},"SSD \u002F HDD",[7109,76813,76814],{},"TB",[7109,76816,76817],{},"OS \u002F User",[7091,76819,76820,76825,76828,76831],{},[7109,76821,76822],{},[106,76823,76824],{},"Offline\u002FArchival",[7109,76826,76827],{},"Tape \u002F Cloud",[7109,76829,76830],{},"Virtually unlimited",[7109,76832,76817],{},[12,76834,76835,76836,76839,76840,76842,76843,76845],{},"As you move ",[7289,76837,76838],{},"up"," the pyramid (toward registers), memory gets smaller, faster, and more expensive. As you move ",[7289,76841,71506],{}," (toward archival storage), memory gets larger, slower, and cheaper. The trick is that the principle of locality ensures you spend most of your time accessing the ",[7289,76844,13468],{}," levels.",[6964,76847,76849],{"id":76848},"what-is-cache-memory","What Is Cache Memory?",[12,76851,76852],{},"Cache is a small, fast memory that sits between the CPU and main memory. Its job is simple: keep copies of the data the processor is most likely to need next, so the processor doesn't have to wait for the (relatively) slow main memory.",[12,76854,76855],{},"When the processor needs a piece of data, it checks the cache first:",[23,76857,76858,76864],{},[26,76859,76860,76863],{},[106,76861,76862],{},"Cache hit"," — the data is in the cache. Great! The processor gets it almost instantly.",[26,76865,76866,76869],{},[106,76867,76868],{},"Cache miss"," — the data isn't in the cache. The system has to fetch it from main memory (or a lower cache level), which takes much longer.",[12,76871,9455,76872,76875],{},[106,76873,76874],{},"hit rate"," (percentage of accesses that are hits) is the single most important measure of cache performance. A well-designed cache can achieve hit rates above 90%, meaning the processor rarely has to wait.",[7166,76877,76879],{"id":76878},"key-terminology","Key Terminology",[12,76881,76882],{},"Before we go further, let's nail down a few terms:",[23,76884,76885,76891,76897,76907],{},[26,76886,76887,76890],{},[106,76888,76889],{},"Block"," — the minimum chunk of data transferred between cache and main memory. You don't fetch a single byte; you fetch an entire block at a time (taking advantage of spatial locality).",[26,76892,76893,76896],{},[106,76894,76895],{},"Line"," — a slot in the cache that can hold one block. Think of it as a shelf space.",[26,76898,76899,76902,76903,76906],{},[106,76900,76901],{},"Tag"," — a label attached to each cache line that identifies ",[7289,76904,76905],{},"which"," block from main memory is currently stored there.",[26,76908,76909,76912],{},[106,76910,76911],{},"Line size"," — the number of data bytes in a block\u002Fline (commonly 32 or 64 bytes).",[25631,76914,76916,76919],{"className":76915},[13441,13442],[25635,76917,76918],{},"Analogy",[13438,76920,76922,76925,76935],{"className":76921},[25641],[12,76923,76924],{},"If you still can't understand what the above term means, here's a more refined version:",[12,76926,76927,76928,76930,76931,76934],{},"Imagine you are writing a research paper. The ",[106,76929,76793],{}," is a massive, multi-story library down the street. The ",[106,76932,76933],{},"Cache"," is a small, fast bookshelf right on your desk.",[3603,76936,76937,76950,76959,76965],{},[26,76938,76939,76942,76943,76946,76947,76949],{},[106,76940,76941],{},"Block:"," The library has a rule: you can't check out just one book. You have to check out a whole ",[7289,76944,76945],{},"crate"," of books at once. That crate of books is the ",[106,76948,76889],{},". It's the actual data you are moving around.",[26,76951,76952,76955,76956,76958],{},[106,76953,76954],{},"Line:"," On your desk bookshelf, you have exactly 10 empty cubbies to hold crates. Each empty cubby is a ",[106,76957,76895],{},". It is the physical space reserved for the data.",[26,76960,76961,76964],{},[106,76962,76963],{},"Line Size:"," This is simply the physical dimensions of the cubby. If the \"line size\" is 64 bytes, it means that cubby is built to hold a crate that contains exactly 64 bytes of data. Every cubby (Line) and every crate (Block) is the exact same size.",[26,76966,76967,76970,76971,76974,76975,6959],{},[106,76968,76969],{},"Tag:"," Because you only have 10 cubbies, you are constantly swapping crates in and out depending on what you are researching. If you reach for cubby #3, how do you know what crate is currently in there? You stick a sticky note on the front of the cubby that says, ",[7289,76972,76973],{},"\"This cubby currently holds the crate from the Biology section.\""," That note is the ",[106,76976,76901],{},[6964,76978,76980],{"id":76979},"elements-of-cache-design","Elements of Cache Design",[12,76982,76983],{},"Designing a cache involves several interrelated decisions. Let's walk through each one.",[7166,76985,76987],{"id":76986},"_1-cache-size","1. Cache Size",[12,76989,76990],{},"How big should the cache be? There's a sweet spot:",[23,76992,76993,76998],{},[26,76994,76995,76997],{},[106,76996,57230],{}," → not enough room, too many misses.",[26,76999,77000,77002,77003,77006],{},[106,77001,57236],{}," → more expensive, physically bigger, and paradoxically ",[7289,77004,77005],{},"slightly slower"," because addressing a larger cache requires more gate logic.",[12,77008,77009],{},"In practice, L1 caches are typically 32–64 KB, L2 caches are 256 KB–1 MB, and L3 caches can be several megabytes to tens of megabytes. The exact optimal size depends heavily on the workload, so there's no single \"best\" answer.",[7166,77011,77013],{"id":77012},"_2-mapping-function","2. Mapping Function",[12,77015,77016,77017,77019,77020,77023],{},"When a block is fetched from main memory, ",[7289,77018,74635],{}," does it go in the cache? This is determined by the ",[106,77021,77022],{},"mapping function",", and there are three main approaches.",[26236,77025,77027],{"id":77026},"direct-mapping","Direct Mapping",[12,77029,77030,77031,77034],{},"Each block of main memory maps to ",[106,77032,77033],{},"exactly one"," specific cache line.",[12,77036,77037,77038,77041],{},"The mapping is simple — typically ",[6979,77039,77040],{},"cache line = block number mod number of lines",". The memory address is split into three fields:",[7085,77043,77044,77056],{},[7088,77045,77046],{},[7091,77047,77048,77050,77053],{},[7094,77049,76901],{},[7094,77051,77052],{},"Line (index)",[7094,77054,77055],{},"Word (offset)",[7104,77057,77058],{},[7091,77059,77060,77064,77068],{},[7109,77061,77062],{},[16,77063],{},[7109,77065,77066],{},[16,77067],{},[7109,77069,77070],{},[16,77071],{},[12,77073,77074,77077,77078,77080,77081,77084],{},[106,77075,77076],{},"How it works:"," The ",[7289,77079,7383],{}," field tells the cache which slot to check. The ",[7289,77082,77083],{},"tag"," field is compared against the tag stored in that slot. If they match, it's a hit. If not, it's a miss, and the existing block in that line gets evicted.",[12,77086,77087,77090],{},[106,77088,77089],{},"Pros:"," Simple, fast hardware. No searching required — go straight to the indexed line.",[12,77092,77093,77096,77097,77100],{},[106,77094,77095],{},"Cons:"," If two frequently-used blocks happen to map to the same line, they'll keep evicting each other. This is called ",[106,77098,77099],{},"thrashing",", and it can destroy performance even when the cache has plenty of empty lines elsewhere.",[25631,77102,77104,77106],{"className":77103},[13441,13442],[25635,77105,75584],{},[13438,77107,77109,77116,77127,77133,77158,77164,77167,77194,77197,77200,77221],{"className":77108},[25641],[12,77110,77111,77112,77115],{},"In a Direct Mapped cache, every single crate of books (Block) in the massive library has ",[106,77113,77114],{},"exactly one specific cubby (Line)"," it is allowed to go into on your desk. It is strict assigned seating.",[12,77117,77118,77119,77122,77123,77126],{},"Imagine your desk cache has only ",[106,77120,77121],{},"4 lines"," (numbered 0, 1, 2, and 3). The library has ",[106,77124,77125],{},"16 blocks"," (numbered 0 to 15).",[12,77128,77129,77130,24917],{},"How do we assign seats? We use modulo math (",[6979,77131,77132],{},"Block Number MOD Total Cache Lines",[23,77134,77135,77138,77141,77144,77147,77152],{},[26,77136,77137],{},"Block 0 goes to Line 0",[26,77139,77140],{},"Block 1 goes to Line 1",[26,77142,77143],{},"Block 2 goes to Line 2",[26,77145,77146],{},"Block 3 goes to Line 3",[26,77148,77149],{},[7289,77150,77151],{},"Block 4 wraps around and goes to Line 0",[26,77153,77154,77157],{},[7289,77155,77156],{},"Block 5 goes to Line 1..."," and so on.",[12,77159,77160,77161,77163],{},"Because of this rule, Cache Line 0 is the ",[7289,77162,23086],{}," place Blocks 0, 4, 8, and 12 can ever be stored.",[12,77165,77166],{},"When the CPU asks for a specific piece of data, it hands the cache an address. The cache slices this address into three parts to find the data instantly:",[3603,77168,77169,77175,77188],{},[26,77170,77171,77174],{},[106,77172,77173],{},"Line (Index):"," \"Which cubby do I check?\" The cache uses this middle chunk of the address to immediately jump to the correct physical slot. There is zero searching involved.",[26,77176,77177,77179,77180,77183,77184,77187],{},[106,77178,76969],{}," \"Is this the right crate?\" Because Line 0 could hold Block 0, 4, 8, or 12, the cache looks at the Tag (the sticky note on the cubby) to see which one is currently sitting there. If the Tag matches the address, it's a ",[106,77181,77182],{},"Cache Hit",". If it doesn't, it's a ",[106,77185,77186],{},"Cache Miss",", and the current block gets thrown out (evicted) to make room for the new one.",[26,77189,77190,77193],{},[106,77191,77192],{},"Word (Offset):"," \"Which specific book inside this crate do I read?\" Once the correct crate is confirmed, this tells the CPU exactly which byte of data to pull from the block.",[12,77195,77196],{},"Direct mapping is incredibly fast and cheap to build because the hardware never has to search—it just checks one specific location.",[12,77198,77199],{},"But imagine your code is trying to read data from Block 0 and Block 4 back-to-back in a loop.",[23,77201,77202,77208,77218],{},[26,77203,77204,77205,6959],{},"CPU asks for Block 0. It goes into ",[106,77206,77207],{},"Line 0",[26,77209,77210,77211,77214,77215,77217],{},"CPU asks for Block 4. It also ",[7289,77212,77213],{},"must"," go into ",[106,77216,77207],{},". It kicks out Block 0.",[26,77219,77220],{},"CPU asks for Block 0 again. It kicks out Block 4.",[12,77222,77223,77224,77226],{},"They are fighting over the exact same seat. Even if Cache Lines 1, 2, and 3 are completely empty, the cache will keep kicking out perfectly good data. This endless cycle of misses and evictions is called ",[106,77225,77099],{},", and it forces the CPU to constantly wait on the slow main memory.",[26236,77228,77230],{"id":77229},"associative-fully-associative-mapping","Associative (Fully Associative) Mapping",[12,77232,77233,77234,77236],{},"Any block can go in ",[106,77235,19592],{}," cache line.",[12,77238,77239],{},"The address is split into just two fields:",[7085,77241,77242,77252],{},[7088,77243,77244],{},[7091,77245,77246,77249],{},[7094,77247,77248],{},"Tag (22 bits)",[7094,77250,77251],{},"Word (2 bits)",[7104,77253,77254],{},[7091,77255,77256,77260],{},[7109,77257,77258],{},[16,77259],{},[7109,77261,77262],{},[16,77263],{},[12,77265,77266,77268,77269,77271,77272,6959],{},[106,77267,77076],{}," When looking for a block, the cache must compare the tag against ",[7289,77270,7291],{}," line simultaneously. This requires special hardware called a ",[106,77273,77274],{},"content-addressable memory (CAM)",[12,77276,77277,77279],{},[106,77278,77089],{}," Maximum flexibility — no thrashing from mapping conflicts.",[12,77281,77282,77284],{},[106,77283,77095],{}," Expensive and slow for large caches because every tag must be checked in parallel.",[25631,77286,77288,77290],{"className":77287},[13441,13442],[25635,77289,75584],{},[13438,77291,77293,77300,77303,77306,77326,77329,77332,77336,77342,77348,77358,77369,77374],{"className":77292},[25641],[12,77294,77295,77296,77299],{},"In a Fully Associative cache, there are no assigned seats. Any crate of books from the library (Block) can be placed into ",[106,77297,77298],{},"any empty cubby (Line)"," on your desk.",[12,77301,77302],{},"If your desk cache has 4 lines, and you fetch Block 0, you can put it in Line 0, 1, 2, or 3. If you fetch Block 4, it can go in any of the remaining empty lines.",[12,77304,77305],{},"Because of this open seating arrangement, the CPU address is sliced differently:",[3603,77307,77308,77316,77321],{},[26,77309,77310,7206,77312,77315],{},[106,77311,77173],{},[7289,77313,77314],{},"This no longer exists."," Because a block could be sitting anywhere, there is no specific line to jump to.",[26,77317,77318,77320],{},[106,77319,76969],{}," \"Who are you?\" Because any line can hold any block, the sticky note (Tag) on the front of the cubby must be longer and much more specific to identify exactly which crate from the entire library is sitting there.",[26,77322,77323,77325],{},[106,77324,77192],{}," Still tells the CPU exactly which byte inside the block to read.",[12,77327,77328],{},"Remember in Direct Mapping how Block 0 and Block 4 kept kicking each other out of Line 0, even when the rest of the cache was empty?",[12,77330,77331],{},"Fully Associative mapping completely solves this. If you ask for Block 0, it goes in Line 0. If you ask for Block 4, it just goes into the next empty spot (Line 1). No fighting, no thrashing.",[7166,77333,77335],{"id":77334},"the-catch-finding-the-data-the-cam","The Catch: Finding the Data (The CAM)",[12,77337,77338,77339,6959],{},"The flexibility of open seating introduces a massive new problem: ",[106,77340,77341],{},"Searching",[12,77343,77344,77345,77299],{},"When the CPU asks for Block 4, it doesn't know which cubby to check. It has to look at the sticky notes (Tags) of ",[7289,77346,77347],{},"every single cubby",[12,77349,77350,77351,77354,77355,6959],{},"In software, checking every item in a list takes time (a ",[6979,77352,77353],{},"for"," loop). But a CPU cache has to be lightning fast. To solve this, engineers use special hardware called ",[106,77356,77357],{},"Content-Addressable Memory (CAM)",[12,77359,77360,77361,77364,77365,77368],{},"Imagine you have a magical assistant for your bookshelf. Instead of reading the sticky notes one by one, you shout, ",[7289,77362,77363],{},"\"Does anyone have Block 4?!\""," and the specific cubby holding Block 4 instantly lights up. The CAM hardware allows the cache to compare the requested Tag against ",[7289,77366,77367],{},"every single line in the cache simultaneously"," in one clock cycle.",[12,77370,77371,77373],{},[106,77372,77089],{}," Maximum flexibility. You will almost never get a cache miss unless the entire cache is 100% full.",[12,77375,77376,77378],{},[106,77377,77095],{}," CAM hardware is incredibly expensive, takes up a lot of physical space on the silicon chip, and consumes a lot of power. Because of this, Fully Associative caches are usually only used for very small, highly critical caches.",[26236,77380,77382],{"id":77381},"set-associative-mapping","Set Associative Mapping",[12,77384,77385,77386,77389,77390,77393,77394,77397,77398,77400,77401,77403,77404,77406],{},"The compromise. The cache is divided into ",[106,77387,77388],{},"sets",", each containing ",[7289,77391,77392],{},"k"," lines (this is called ",[7289,77395,77396],{},"k-way"," set associative). A block maps to a specific ",[106,77399,23920],{}," (like direct mapping), but within that set it can go in ",[106,77402,19592],{}," of the ",[7289,77405,77392],{}," lines (like associative mapping).",[7085,77408,77409,77420],{},[7088,77410,77411],{},[7091,77412,77413,77415,77418],{},[7094,77414,76901],{},[7094,77416,77417],{},"Set (index)",[7094,77419,77055],{},[7104,77421,77422],{},[7091,77423,77424,77428,77432],{},[7109,77425,77426],{},[16,77427],{},[7109,77429,77430],{},[16,77431],{},[7109,77433,77434],{},[16,77435],{},[12,77437,77438],{},"For example, in a 2-way set associative cache, each set has 2 lines. A block hashes to one set, and the cache checks both lines in that set.",[12,77440,77441,77443],{},[106,77442,77089],{}," Much less thrashing than direct mapping, while requiring far less comparison hardware than full associativity.",[12,77445,77446,77448],{},[106,77447,77095],{}," Slightly more complex than direct mapping.",[12,77450,77451],{},"Most modern processors use set associative caches (commonly 4-way, 8-way, or even 16-way).",[25631,77453,77455,77457],{"className":77454},[13441,13442],[25635,77456,75584],{},[13438,77458,77460,77467,77473,77487,77494,77503,77530,77537,77540,77543],{"className":77459},[25641],[12,77461,77462,77463,77466],{},"If Direct Mapping is \"Assigned Seating\" (you must sit in chair #3) and Fully Associative is \"Open Seating\" (sit anywhere), Set Associative Mapping is the ",[106,77464,77465],{},"\"Assigned Table\""," rule.",[12,77468,77469,77470,6959],{},"Imagine we take your 4 bookshelf cubbies and divide them into ",[106,77471,77472],{},"2 distinct zones (Sets)",[23,77474,77475,77481],{},[26,77476,77477,77480],{},[106,77478,77479],{},"Set 0"," contains Cubby 0 and Cubby 1.",[26,77482,77483,77486],{},[106,77484,77485],{},"Set 1"," contains Cubby 2 and Cubby 3.",[12,77488,77489,77490,77493],{},"Because each Set has exactly 2 cubbies, we call this a ",[106,77491,77492],{},"2-Way Set Associative"," cache.",[12,77495,77496,77497,77499,77500,6959],{},"Now, when you bring a crate of books (Block) from the library, it is assigned to a specific ",[7289,77498,7045],{},", but it can go into ",[7289,77501,77502],{},"any empty cubby within that Set",[23,77504,77505,77514,77523],{},[26,77506,77507,77508,77510,77511,77513],{},"Block 0 is assigned to ",[106,77509,77479],{}," -> It can go into Cubby 0 ",[7289,77512,11286],{}," Cubby 1.",[26,77515,77516,77517,77519,77520,77522],{},"Block 1 is assigned to ",[106,77518,77485],{}," -> It can go into Cubby 2 ",[7289,77521,11286],{}," Cubby 3.",[26,77524,77525,77526,77510,77528,77513],{},"Block 2 is assigned to ",[106,77527,77479],{},[7289,77529,11286],{},[12,77531,77532,77533,77536],{},"Because a block can only go to one specific Set, the cache doesn't have to search the entire bookshelf. The CPU address gives it the ",[106,77534,77535],{},"Set Index",", so it instantly jumps to the correct zone. This eliminates the massive, expensive, slow hardware search of Fully Associative caches.",[12,77538,77539],{},"But, because there are multiple \"ways\" (slots) within that Set, it drastically reduces the thrashing problem of Direct Mapping. If your code asks for Block 0, it goes to Set 0, Cubby 0. If your code immediately asks for Block 2, it also goes to Set 0, but instead of kicking out Block 0, it just slides into Cubby 1. No thrashing!",[12,77541,77542],{},"The CPU slices the address like this:",[3603,77544,77545,77551,77559],{},[26,77546,77547,77550],{},[106,77548,77549],{},"Set (Index):"," \"Which zone do I check?\" The cache jumps immediately to this Set.",[26,77552,77553,77555,77556,6959],{},[106,77554,76969],{}," \"Are you here?\" The cache uses a small, cheap piece of CAM hardware to simultaneously check the sticky notes of ",[7289,77557,77558],{},"only the cubbies in this specific Set",[26,77560,77561,77563],{},[106,77562,77192],{}," The specific byte of data you want to read.",[7166,77565,77567],{"id":77566},"_3-replacement-algorithms","3. Replacement Algorithms",[12,77569,77570,77571,7024],{},"When a cache set (or the entire cache, for fully associative) is full and a new block needs to come in, which existing block gets evicted? For direct mapping there's no choice — there's only one candidate. But for associative and set-associative caches, we need a ",[106,77572,77573],{},"replacement algorithm",[23,77575,77576,77582,77588,77594],{},[26,77577,77578,77581],{},[106,77579,77580],{},"Least Recently Used (LRU)"," — evict the block that hasn't been accessed for the longest time. This aligns perfectly with temporal locality and is the most effective strategy. It's also the most popular because it's relatively simple to implement in hardware.",[26,77583,77584,77587],{},[106,77585,77586],{},"First-In-First-Out (FIFO)"," — evict the block that has been in the cache the longest, regardless of how recently it was accessed. Easy to implement with a circular buffer.",[26,77589,77590,77593],{},[106,77591,77592],{},"Least Frequently Used (LFU)"," — evict the block that has been accessed the fewest times. Requires a counter per line.",[26,77595,77596,77599],{},[106,77597,77598],{},"Random"," — evict a random block. Surprisingly, random replacement performs only slightly worse than LRU in many workloads and is dead simple to implement.",[7166,77601,77603],{"id":77602},"_4-write-policy","4. Write Policy",[12,77605,77606,77607,77610],{},"When the processor ",[7289,77608,77609],{},"writes"," data, what happens to the cache and main memory?",[12,77612,77613,77616,77617,77619],{},[106,77614,77615],{},"Write Through"," — every write updates ",[7289,77618,11209],{}," the cache and main memory immediately. This keeps memory always consistent and is simple to understand. The downside is that it generates a lot of memory traffic, which can become a bottleneck.",[12,77621,77622,77625,77626,77629],{},[106,77623,77624],{},"Write Back"," — writes update only the cache. The modified block is written back to main memory later, only when it gets evicted. This drastically reduces memory traffic but means main memory can be temporarily out of date. A ",[106,77627,77628],{},"dirty bit"," on each line tracks whether it's been modified.",[26236,77631,77633],{"id":77632},"what-about-write-misses","What About Write Misses?",[12,77635,77636],{},"If the processor writes to an address that isn't in the cache, there are two options:",[23,77638,77639,77645],{},[26,77640,77641,77644],{},[106,77642,77643],{},"Write allocate"," — fetch the block into the cache first, then perform the write there. This is typically paired with write-back.",[26,77646,77647,77650],{},[106,77648,77649],{},"No write allocate"," — write directly to main memory without loading the block into the cache. This is typically paired with write-through.",[7166,77652,77654],{"id":77653},"_5-line-size","5. Line Size",[12,77656,77657],{},"Bigger blocks take better advantage of spatial locality (you prefetch more neighbouring data), but there's a trade-off:",[23,77659,77660,77663],{},[26,77661,77662],{},"Larger blocks mean fewer lines in a fixed-size cache, increasing the chance of evictions.",[26,77664,77665],{},"Each miss takes longer to service because more data must be transferred.\nTypical line sizes are 32 or 64 bytes.",[7166,77667,77669],{"id":77668},"_6-number-of-caches","6. Number of Caches",[12,77671,77672,77673,77676],{},"Modern systems don't have just one cache — they use a ",[106,77674,77675],{},"multilevel"," hierarchy:",[23,77678,77679,77688,77698],{},[26,77680,77681,77684,77685,77687],{},[106,77682,77683],{},"L1 cache"," — on the same chip as the processor, tiny but extremely fast. Often ",[106,77686,14899],{}," into separate instruction and data caches (called an I-cache and D-cache) to avoid contention between the fetch and execution units.",[26,77689,77690,77693,77694,77697],{},[106,77691,77692],{},"L2 cache"," — larger and slightly slower. Usually ",[106,77695,77696],{},"unified"," (holds both instructions and data).",[26,77699,77700,77703],{},[106,77701,77702],{},"L3 cache"," — even larger, often shared across multiple processor cores.",[12,77705,77706,77707,77710],{},"Splitting L1 into instruction and data caches is important for ",[106,77708,77709],{},"pipelining",", where the processor simultaneously fetches a new instruction while executing a previous one. If both operations need the same cache, they'd conflict. Splitting eliminates this contention.",[6964,77712,77714],{"id":77713},"cache-coherency","Cache Coherency",[12,77716,77717,77718,77721,77722,32499],{},"Things get interesting, and complicated, when multiple processors (or cores) each have their own cache but share the same main memory. If Core A writes a new value to address X in its cache, Core B's cache might still hold the ",[7289,77719,77720],{},"old"," value of X. This is the ",[106,77723,77724],{},"cache coherency",[12,77726,77727],{},"Several approaches exist:",[23,77729,77730,77736,77742],{},[26,77731,77732,77735],{},[106,77733,77734],{},"Bus watching (snooping) with write-through"," — each cache controller monitors the memory bus. If it sees another core writing to an address that exists in its own cache, it invalidates that entry. Simple but depends on all caches using write-through.",[26,77737,77738,77741],{},[106,77739,77740],{},"Hardware transparency (snooping protocols)"," — dedicated hardware ensures that any cache update is propagated to all other caches. More complex, but works with write-back policies too.",[26,77743,77744,77747],{},[106,77745,77746],{},"Noncacheable memory"," — shared memory regions are simply marked as noncacheable. Every access goes directly to main memory. Simple but sacrifices performance for shared data.",[12,77749,77750,77751,77754],{},"In modern multi-core processors, sophisticated protocols like ",[106,77752,77753],{},"MESI"," (Modified, Exclusive, Shared, Invalid) handle coherency, but the fundamental ideas are the same.",[6964,77756,77758],{"id":77757},"inclusion-policy","Inclusion Policy",[12,77760,77761],{},"When you have multiple cache levels, should a block in L1 also be kept in L2? There are three schools of thought:",[12,77763,77764,77767],{},[106,77765,77766],{},"Inclusive"," — if data is in L1, it's guaranteed to also be in L2 (and L3). This simplifies coherency checks in multi-core systems because you only need to search the last-level cache to know whether any core might have a copy.",[12,77769,77770,77773,77774,77776],{},[106,77771,77772],{},"Exclusive"," — if data is in L1, it is ",[7289,77775,22210],{}," in L2. No wasted space from duplicate copies, so you effectively get more total cache capacity. The trade-off is that coherency checks become harder since you may need to search multiple levels.",[12,77778,77779,77782],{},[106,77780,77781],{},"Non-inclusive"," — data in L1 may or may not be in L2. A flexible middle ground, but with similar coherency challenges as the exclusive policy.",[6964,77784,77786],{"id":77785},"wrapping-up","Wrapping Up",[12,77788,77789],{},"Cache memory is one of those topics that sounds simple on the surface — \"just keep frequently used data nearby\" — but the design decisions run deep. From choosing a mapping function to picking a write policy and managing coherency across cores, every choice involves trade-offs between speed, complexity, cost, and correctness.",[12,77791,77792],{},"The key takeaways:",[12,77794,9455,77795,7653,77797,77799,77800,77802,77803,77806],{},[106,77796,76661],{},[7289,77798,73259],{}," caches work. Temporal and spatial locality mean that a small, fast memory can satisfy the vast majority of a processor's requests. The ",[106,77801,75366],{}," gives us the best of all worlds: the speed of small memories and the capacity of large ones. ",[106,77804,77805],{},"Cache design"," is a web of interconnected decisions — mapping, replacement, write policy, line size, levels, and inclusion — where no single choice is optimal in isolation; each must be tuned in context.",[12,77808,77809,77810,77812],{},"If you remember nothing else, remember this: the goal of the entire memory system is to create the ",[106,77811,76584],{}," that the processor has access to a very large, very fast memory — even though no such memory physically exists.",{"title":180,"searchDepth":181,"depth":181,"links":77814},[77815,77816,77817,77818,77821,77829,77830,77831],{"id":75332,"depth":181,"text":75333},{"id":76651,"depth":181,"text":76652},{"id":76697,"depth":181,"text":76698},{"id":76848,"depth":181,"text":76849,"children":77819},[77820],{"id":76878,"depth":7430,"text":76879},{"id":76979,"depth":181,"text":76980,"children":77822},[77823,77824,77825,77826,77827,77828],{"id":76986,"depth":7430,"text":76987},{"id":77012,"depth":7430,"text":77013},{"id":77566,"depth":7430,"text":77567},{"id":77602,"depth":7430,"text":77603},{"id":77653,"depth":7430,"text":77654},{"id":77668,"depth":7430,"text":77669},{"id":77713,"depth":181,"text":77714},{"id":77757,"depth":181,"text":77758},{"id":77785,"depth":181,"text":77786},{},"\u002Fblog\u002FThe-Memory-Hierarchy-Understanding-Cache-Memory",{"title":76628,"description":180},{"loc":77833},"blog\u002FThe-Memory-Hierarchy-Understanding-Cache-Memory","KodXT8URXZpZYHhv6TxzreaODZA4xeIqVxw7HDR0vyo",{"id":77839,"title":77840,"author":7,"body":77841,"date":74911,"description":78052,"draft":185,"edited_at":73699,"extension":186,"featured_image":78053,"meta":78054,"navigation":189,"path":78055,"pinned":185,"seo":78056,"sitemap":78057,"stem":78058,"tags":194,"__hash__":78059},"blog\u002Fblog\u002F小腦（一）.md","小腦（一）",{"type":9,"value":77842,"toc":78050},[77843,77846,77849,77852,77855,77866,77869,77872,77895,77897,77903,77906,77917,77920,77923,77925,77928,77960,77962,77965,77990,77992,77995,78006,78008,78011,78022,78024,78027,78035,78037,78040],[12,77844,77845],{},"看一下平衡跳掉的東西有沒有機會補上 先寫完再說 （畫餅）",[12,77847,77848],{},"平衡的定義：多個關節與肌肉，以適當的時間與正確的力度活動——具備平滑，準確，受控",[12,77850,77851],{},"包含了兩組能力表現：靈巧度（精細動作中的手指技巧）；敏捷度（姿勢控制下能迅速且平順）",[12,77853,77854],{},"形式上分成",[23,77856,77857,77860,77863],{},[26,77858,77859],{},"肢體內協調能力：單一肢段內動作",[26,77861,77862],{},"肢體間協調能力：整合兩個或以上的肢體動作",[26,77864,77865],{},"視覺與動作間協調：環境背景下，整合視覺與動作達成一目標的能力",[12,77867,77868],{},"功能障礙主要為感覺，運動，神經處理過程的整合出問題",[12,77870,77871],{},"協調檢查的目的",[23,77873,77874,77877,77880,77883,77886,77889,77892],{},[26,77875,77876],{},"隨意動作——主要測肌肉協調動作",[26,77878,77879],{},"完全功能活動的能力——以安全性及跌倒風險考量單一或一群肌肉共同完成任務能力",[26,77881,77882],{},"動作的效率與技巧程度——以省力與熟練度角度觀察",[26,77884,77885],{},"動作的啟動，控制與停止——看正常啟動，控制正常，準確停止能力",[26,77887,77888],{},"潛在病理因素的影響——以病理角度分析",[26,77890,77891],{},"動作模式時序與準確度——看時間啟動正確，動作順序正確，準確度程度表現",[26,77893,77894],{},"治療與藥物成效——看介入對動作表現有無影響",[36,77896],{},[12,77898,77899,77900,77902],{},"協調缺陷與動作系統——那動作系統一樣分成中樞跟周邊",[16,77901],{},"\n周邊一樣是指肌肉，感覺跟運動神經支配",[12,77904,77905],{},"那中樞的話Bear學者提出階層分級",[23,77907,77908,77911,77914],{},[26,77909,77910],{},"高階（聯合皮質區，前腦基底核）——跟策略有關",[26,77912,77913],{},"中階（運動皮質，小腦）——跟手段有關",[26,77915,77916],{},"低階（腦幹，脊髓）——跟執行有關",[12,77918,77919],{},"而基底核跟小腦傳出的信息不是直接到低階，而是傳到運動皮質",[12,77921,77922],{},"那感覺系統的角色是引導選擇或者調整動作反應，調整動作策略",[36,77924],{},[12,77926,77927],{},"運動皮質——跟動作有關的包括brodmann 4跟6區，在中央前腦迴那裡",[23,77929,77930,77949],{},[26,77931,77932,77933],{},"area 4——原始大腦運動皮質區(M1)，控制對側隨意動作（單關節非複雜）",[23,77934,77935],{},[26,77936,77937,77938],{},"M1——接收三個區域：本體覺皮質，小腦，基底核；共同產生協調能力",[23,77939,77940,77943,77946],{},[26,77941,77942],{},"本體覺皮質：感覺>感覺皮質>M1",[26,77944,77945],{},"小腦：小腦>視丘>運動前區",[26,77947,77948],{},"基底核：基底核>視丘>運動區",[26,77950,77951,77952],{},"area 6——在area 4 前方，上方為輔運動皮質區SMA，下方為運動前區PMA（多關節複雜協調動作）",[23,77953,77954,77957],{},[26,77955,77956],{},"SMA——啟動動作，次序動作，同雙側抓握動作",[26,77958,77959],{},"PMA ——控制Trunk, 近端肢體動作",[36,77961],{},[12,77963,77964],{},"下行運動路徑",[23,77966,77967,77975,77978,77981,77984,77987],{},[26,77968,77969,77970],{},"皮質脊髓徑（錐體徑）——最大最長最重要的路徑；負責技巧，精細控制，特別是遠端的",[23,77971,77972],{},[26,77973,77974],{},"源於4~6Area，經內囊和腦幹傳到延髓；在延腦交叉",[26,77976,77977],{},"皮質髓質徑——連到腦神經如三叉，顏面，舌下；其他連到網狀結構",[26,77979,77980],{},"四疊體脊髓徑——連到頸椎和副神經（11）：控制頸部肌肉，視覺動作任務引導頭動",[26,77982,77983],{},"網體脊髓徑——連到脊髓前角；控制肌梭，影響張力及反射",[26,77985,77986],{},"前庭脊髓徑——外側到所有level的脊髓，影響姿勢控制跟頭部動作；外側到C level，控制頭眼協調",[26,77988,77989],{},"紅核脊髓徑——人類動作控制不明顯",[36,77991],{},[12,77993,77994],{},"小腦",[23,77996,77997,78000,78003],{},[26,77998,77999],{},"主要功能：協調動作，姿勢控制，肌肉張力",[26,78001,78002],{},"機制：比較器及誤差修正機制",[26,78004,78005],{},"病理：..一堆，主要是不協調，次序出問題，亂動，還有辨距不良，回彈現象，震顫，動作學習受損",[36,78007],{},[12,78009,78010],{},"基底核",[23,78012,78013,78016,78019],{},[26,78014,78015],{},"解剖——尾狀核，皮殼，蒼白球；其他的還有黑質，視丘下核",[26,78017,78018],{},"訊息回路——輸入：動作相關區域接到皮殼；輸出：到SMA或者運動前區",[26,78020,78021],{},"機制——不明；負責控制調節，計劃選擇，自動化動作，姿勢控制",[36,78023],{},[12,78025,78026],{},"DCML",[23,78028,78029,78032],{},[26,78030,78031],{},"傳遞——區辨感覺：本體覺，觸壓覺等",[26,78033,78034],{},"病理——步態改變；感覺性步態失調；動作受損；Romberg sign陽性",[36,78036],{},[12,78038,78039],{},"感覺誘發自動化動作",[23,78041,78042],{},[26,78043,78044,78045],{},"周邊——CPG，選擇策略",[23,78046,78047],{},[26,78048,78049],{},"CPG——被啟動後，會先活化預定好的動作程式，不需要協調就能動",{"title":180,"searchDepth":181,"depth":181,"links":78051},[],"定義，組成","\u002Fimages\u002Fuploads\u002F1776676020947-78546185_p0_master1200.webp",{},"\u002Fblog\u002F小腦（一）",{"title":77840,"description":78052},{"loc":78055},"blog\u002F小腦（一）","A00sm7OK2c8_77Fy9811h8hMNw6NCtUXXjN3am3OpqM",{"id":78061,"title":78062,"author":7,"body":78063,"date":78185,"description":78186,"draft":185,"edited_at":78185,"extension":186,"featured_image":78187,"meta":78188,"navigation":189,"path":78189,"pinned":185,"seo":78190,"sitemap":78191,"stem":78192,"tags":194,"__hash__":78193},"blog\u002Fblog\u002FBalance（七）前庭障礙.md","Balance（七）前庭障礙",{"type":9,"value":78064,"toc":78183},[78065,78068,78085,78087,78090,78107,78109,78112,78120,78122,78125,78148,78150,78153,78164,78166,78169],[12,78066,78067],{},"那先說前庭的路徑",[3603,78069,78070,78073,78076,78079,78082],{},[26,78071,78072],{},"末端——內耳石迷路感覺受器：機械反應>訊號",[26,78074,78075],{},"前庭神經——第八對腦神經",[26,78077,78078],{},"前庭核與小腦——前庭核：整合視覺本體覺；小腦：較正前庭反射，調節VOR",[26,78080,78081],{},"中樞感覺路徑",[26,78083,78084],{},"大腦半球",[36,78086],{},[12,78088,78089],{},"現在來講機械性的問題",[23,78091,78092,78095],{},[26,78093,78094],{},"會引發暈眩或者BPPV",[26,78096,78097,78098,78100,78101,78103,78104,78106],{},"良性偶發性姿勢性暈眩(BPPV)——改變姿勢會暈（延遲性，15秒內），60秒內的暈眩，好發在後半規管",[16,78099],{},"\n可能的機制",[16,78102],{},"\n耳石碎片脫落，黏在了壺腹上的頂帽（耳石黏附頂帽）",[16,78105],{},"\n耳石碎片隨著內淋巴液漂流（耳石漂流半規管）",[36,78108],{},[12,78110,78111],{},"接著是周邊前庭系統傷害",[23,78113,78114,78117],{},[26,78115,78116],{},"單側前庭系統傷害——突然暈，眼球震顫（假如為自發性，可能是中樞病變），平衡失調；症狀在3~7天內緩解（大腦蔽屏）",[26,78118,78119],{},"雙側前庭系統傷害——平衡失調，振動幻視，步態失調；不會暈和噁心（因為兩邊都壞掉不會因不對稱而起衝突）",[36,78121],{},[12,78123,78124],{},"然後是中樞前庭系統傷害",[23,78126,78127,78130,78133,78142,78145],{},[26,78128,78129],{},"會引發暈的腦傷——CVA(PICA\u002FAICA\u002F椎動脈)，TBI(78%暈)，椎底循環不足，多發性硬化症",[26,78131,78132],{},"紅期——複視，鉛直眼球震顫，自發性向上跳動眼震，歪斜偏向（持續兩周以上）",[26,78134,78135,78136,78138,78139,78141],{},"眼球震顫——不會緩解",[16,78137],{},"\n鉛直眼球震顫：小腦問題（在排除前半規管問題後）",[16,78140],{},"\n鐘擺眼球震顫：先天視覺缺陷或大腦皮質問題（擺動速度相同，不會快慢）",[26,78143,78144],{},"側衝——前庭核以上的中樞傷害：側傾，頭傾，視覺感知困難",[26,78146,78147],{},"動眼傾斜反應——結合了頭傾，視覺旋轉，歪斜偏向",[36,78149],{},[12,78151,78152],{},"其他前庭系統症狀（一樣只寫老師有註解的）",[23,78154,78155,78158,78161],{},[26,78156,78157],{},"梅尼耳氏症——內淋巴液增加；症狀有低頻聲音喪失，暈，耳嗚；治療方式為使用利尿劑，吃低鹽食物",[26,78159,78160],{},"動暈症——正常的問題，原因是因為三個感覺系統正常的輸入，但不符合大腦預期，一般是因為經驗不符",[26,78162,78163],{},"多發性硬化症——侵犯到第八對腦神經導致的單側前庭問題",[36,78165],{},[12,78167,78168],{},"禁忌症",[23,78170,78171,78174,78177,78180],{},[26,78172,78173],{},"不穩定或結構未修復的病灶——梅尼耳氏症、未控制偏頭痛、外淋巴廔管、上半規管裂開症",[26,78175,78176],{},"其他警訊——突然聽力喪失，耳朵感到腫脹或壓力，嚴重耳嗚",[26,78178,78179],{},"腦脊液流出",[26,78181,78182],{},"急性頸部損傷",{"title":180,"searchDepth":181,"depth":181,"links":78184},[],"2026-04-18","好....多...","\u002Fimages\u002Fuploads\u002F1776505845269-Em3d_jeUUAA3X0N.webp",{},"\u002Fblog\u002FBalance（七）前庭障礙",{"title":78062,"description":78186},{"loc":78189},"blog\u002FBalance（七）前庭障礙","3LB2JBQlz6Bp48oWqchSfT67AWr5tyOuzTAW4rUJp04",{"id":78195,"title":78196,"author":7,"body":78197,"date":78185,"description":78333,"draft":185,"edited_at":74911,"extension":186,"featured_image":78334,"meta":78335,"navigation":189,"path":78336,"pinned":185,"seo":78337,"sitemap":78338,"stem":78339,"tags":194,"__hash__":78340},"blog\u002Fblog\u002FBalance（八）檢查.md","Balance（八）檢查",{"type":9,"value":78198,"toc":78331},[78199,78202,78205,78211,78214,78231,78233,78236,78326,78328],[12,78200,78201],{},"好了 看看這次能考多少吧 寫完這篇先去寫GMFCS（lol",[12,78203,78204],{},"首先是病史，看他以前的病有沒有像是暈眩，不平衡之類的（看有沒有跟前庭有相關的）",[12,78206,78207,78208,78210],{},"那跟平衡有相關的徵候有",[16,78209],{},"\n頭昏眼花、暈眩、頭輕飄飄、平衡失調、振動幻視",[12,78212,78213],{},"接著是持續的時間或情況",[23,78215,78216,78228],{},[26,78217,78218,78219,78221,78222,78224,78225,78227],{},"平均發作時",[16,78220],{},"\n數秒到數分鐘：BPPV",[16,78223],{},"\n數分鐘到數小時：梅內耳氏症",[16,78226],{},"\n數天：前庭神經炎，偏頭痛的頭暈",[26,78229,78230],{},"環境因素——姿勢或動作，環境（暈車）或者休息時",[36,78232],{},[12,78234,78235],{},"接著是評估部份",[23,78237,78238,78241,78273,78320,78323],{},[26,78239,78240],{},"評估表——VAS（評估頭暈，眩暈，平衡失調，眼球振顫），DHI，FDS，MSQ（評估頭暈程度）",[26,78242,78243,78244],{},"眼球動作病理（觀察眼球震顫）",[23,78245,78246,78249,78255,78270],{},[26,78247,78248],{},"眼球振顫——不自主律動擺動，而眼球轉向一邊是慢的就是壞邊，另一邊是快的就是好邊",[26,78250,78251,78252,78254],{},"自發性眼球震顫——眼會慢慢的轉到一邊，但會被快速且相反方向打斷（通常是單側急性傷害的休息時）",[16,78253],{},"\n；假如是周邊傷害能被抑制",[26,78256,78257,78258,78260,78261,78263,78264,78266,78267,78269],{},"位置性眼球震顫——透過頭部位置改變引發的，包括了",[16,78259],{},"\n偶發性：刺激半規管導致",[16,78262],{},"\n中央：因為中樞系統受傷導致",[16,78265],{},"\n靜態的：內耳石導致的（因為會連到前庭核跟小腦），主要是重力改變誘發",[16,78268],{},"\n（而以上都是可以由視覺固定抑制）",[26,78271,78272],{},"凝視引發眼球震顫——眼睛會不自主的回到中間",[26,78274,78275,78276],{},"觸發眼球震顫檢查（跑台看後面再寫，這裡只寫判讀）",[23,78277,78278,78281,78292,78303,78309],{},[26,78279,78280],{},"頭部搖動引發眼球震顫——單側前庭：（轉好邊快，轉壞邊慢）；雙側前庭：（不會眼震）；中樞傷害：（垂直眼震）",[26,78282,78283,78284],{},"視前庭交互作用檢查",[23,78285,78286,78289],{},[26,78287,78288],{},"VOR測試——症狀問病人有沒有模糊或者頭暈；他們可能會不敢快速轉頭，因為看到東西會模糊",[26,78290,78291],{},"VOR取消——效果就是頭眼同向",[26,78293,78294,78295],{},"頭部衝動測試",[23,78296,78297,78300],{},[26,78298,78299],{},"測水平半規管——單側前庭：（轉患側眼跟不上）雙側前庭：（轉兩邊都跟不上）",[26,78301,78302],{},"測鉛直半規管",[26,78304,78305,78306,78308],{},"姿勢測試",[16,78307],{},"\nHallpike-Dix Yest——測前後半規管>確定耳石在不在半規管裡（BPPV的話：在5~15秒內暈，在60秒內停止）",[26,78310,78311,78312],{},"視銳度測試",[23,78313,78314,78317],{},[26,78315,78316],{},"掃視——眼睛能迅速移動定位",[26,78318,78319],{},"追視——平滑緩慢追視移動物體",[26,78321,78322],{},"步態——失能就是flat foot，步寬變大；if 雙側前庭：速度下降，雙腳承重期上升",[26,78324,78325],{},"主觀視覺垂直（SVV）水平（SVH）測試——可評估整體耳石問題，但分不出是球狀囊還是橢圓囊問題",[36,78327],{},[12,78329,78330],{},"P145",{"title":180,"searchDepth":181,"depth":181,"links":78332},[],"怎麼辦 我還有GMFCS沒寫 小腦也還沒讀 寫完這個要先烙了","\u002Fimages\u002Fuploads\u002F1776568180512-81401517_p0_master1200.webp",{},"\u002Fblog\u002FBalance（八）檢查",{"title":78196,"description":78333},{"loc":78336},"blog\u002FBalance（八）檢查","xBjrKg1US3gjqEGK7YW8oJKoK8hpAN5wHDerIykxPWE",{"id":78342,"title":78343,"author":7,"body":78344,"date":78185,"description":78504,"draft":185,"edited_at":74911,"extension":186,"featured_image":78505,"meta":78506,"navigation":189,"path":78507,"pinned":185,"seo":78508,"sitemap":78509,"stem":78510,"tags":194,"__hash__":78511},"blog\u002Fblog\u002FBalance（六）功能恢復.md","Balance（六）功能恢復",{"type":9,"value":78345,"toc":78502},[78346,78349,78395,78397,78400,78456,78458,78461],[12,78347,78348],{},"接著講功能恢復的部份",[23,78350,78351,78374],{},[26,78352,78353,78354,78358,78359,78361,78362,78366,78367,78371,78373],{},"假如前庭壞掉怎麼辦——因為前庭壞掉，相當於就是做",[75,78355,78357],{"className":78356},[79],"裁決","的那個系統壞掉",[16,78360],{},"\n那我們能做的就是讓患者在一個",[75,78363,78365],{"className":78364},[79,78],"靜止的狀態","下，讓",[75,78368,78370],{"className":78369},[173],"視覺與本體覺輸入都處於正常",[16,78372],{},"\n大腦就會意識到是前庭出問題，減少對前庭的輸入，轉而去依賴視覺與本體覺（訊息校正）",[26,78375,78376,78377,78381,78382,78386,78388,78389,78394],{},"那要怎麼知道大腦有沒有利用大腦去做訊息的校正——",[75,78378,78380],{"className":78379},[173],"閉眼的坐在穩定的平面","上，",[75,78383,78385],{"className":78384},[79,78],"if暈就代表還沒校正好",[16,78387],{},"\n那我們就要遮住pt的眼，並在head或者shoulder處",[75,78390,78393],{"className":78391},[78392,78],"yellow-3","加重物","，放大本體覺輸入",[36,78396],{},[12,78398,78399],{},"接著是姿勢控制的參考",[23,78401,78402,78415,78425,78432,78435,78447,78453],{},[26,78403,78404,78408,78409,78411,78412,78414],{},[75,78405,78407],{"className":78406},[173],"至上向下參考（重力）","——前庭覺，提供空間相對的重力方向",[16,78410],{},"\n假如前庭失能者，當傾斜面>8度，仍然以本體覺做準",[16,78413],{},"\nankle沒有跟著地面改變而變，讓頭身腳一直維持著90度，導致不正常weight shift",[26,78416,78417,78421,78422,78424],{},[75,78418,78420],{"className":78419},[173],"至下而上參考（支撐面）","——本體覺，經由本體覺輸入，給頭的相對地面方向",[16,78423],{},"\n（足底壓力覺，當地面穩定或\u003C4度，當作平衡參考）",[26,78426,78427,78428],{},"過度依賴體感覺參考——測試：頭部扶正反應，pt遮眼坐在傾斜板上>if正常",[75,78429,78431],{"className":78430},[79,78],"頭部會與傾斜面垂直",[26,78433,78434],{},"觸碰參考——借由上肢最輕微的碰觸穩定，以定位trunk方位",[26,78436,78437,78441,78442,78446],{},[75,78438,78440],{"className":78439},[173],"本體覺","無法參考——頸椎節段差，",[75,78443,78445],{"className":78444},[173],"過度使用頸部肌肉","，頸部受傷也會干擾大腦依賴本體覺校正",[26,78448,78449,78450,78452],{},"視覺作為參考——可以觀察環境鉛直或者水平",[16,78451],{},"\nif過度依賴——pt手指前方畫8字會晃",[26,78454,78455],{},"本體覺和前庭覺整合錯誤",[36,78457],{},[12,78459,78460],{},"然後是頭部運動和視覺穩定103~106",[23,78462,78463,78466,78474,78487,78499],{},[26,78464,78465],{},"VOR故障——視覺模糊",[26,78467,78468,78469,78473],{},"視覺與前庭覺衝突——變成視覺動作敏感：轉頭的話",[75,78470,78472],{"className":78471},[79,78],"眼會反方向轉","，會暈",[26,78475,78476,78477,78481,78482,78486],{},"本體覺與前庭覺衝突——假如本體覺受損>痛或者防衛性肌縮縮>降低",[75,78478,78480],{"className":78479},[79,78],"前庭核整合",">本體覺訊息會變成",[75,78483,78485],{"className":78484},[79,78],"用皮質較正",">暈",[26,78488,78489,78490,78494,78495],{},"中樞適應不良——是指",[75,78491,78493],{"className":78492},[79,78],"大腦出問題","，即使三個系統都正常，也",[75,78496,78498],{"className":78497},[5387,78],"沒辦法穩定利用三者",[26,78500,78501],{},"視覺動作敏感——會因為動態的環境而感覺暈（即使閉上眼休息還是會有像加速的感覺）（心理狀況同樣要關注）",{"title":180,"searchDepth":181,"depth":181,"links":78503},[],"好多.....後面回到家再拿滑鼠補上顏色....","\u002Fimages\u002Fuploads\u002F1776497183577-EfVSQBDUcAANFKP.webp",{},"\u002Fblog\u002FBalance（六）功能恢復",{"title":78343,"description":78504},{"loc":78507},"blog\u002FBalance（六）功能恢復","JdryUp75e4AIbhFxznhOcbiocM5Vy3BUSoRFSCnMSJE",{"id":78513,"title":78514,"author":6946,"body":78515,"date":78185,"description":180,"draft":185,"edited_at":78185,"extension":186,"featured_image":194,"meta":79322,"navigation":189,"path":79323,"pinned":185,"seo":79324,"sitemap":79325,"stem":79326,"tags":194,"__hash__":79327},"blog\u002Fblog\u002FComputer-Systems-Architecture-Understanding-Performance.md","Computer Systems Architecture: Understanding Performance",{"type":9,"value":78516,"toc":79280},[78517,78521,78524,78531,78537,78541,78544,78547,78550,78554,78565,78569,78576,78580,78583,78587,78594,78598,78605,78608,78615,78629,78633,78636,78642,78648,78654,78658,78661,78665,78668,78672,78679,78683,78686,78690,78696,78699,78702,78706,78712,78716,78719,78723,78733,78736,78753,78756,78760,78763,78767,78770,78776,78779,78785,78788,78794,78804,78807,78813,78815,78829,78835,78839,78848,78854,78861,78865,78868,78874,78888,78891,79000,79007,79011,79014,79083,79092,79096,79100,79106,79109,79113,79119,79122,79126,79129,79132,79142,79146,79153,79167,79171,79177,79181,79198,79201,79205,79243,79246,79248,79251,79258,79263],[6964,78518,78520],{"id":78519},"why-should-we-care-about-performance","Why Should We Care About Performance?",[12,78522,78523],{},"Here's something wild to think about: the laptop you're reading this on has more computing power than an IBM mainframe from just 10–15 years ago. Processors have become so cheap that we literally throw some of them away (think disposable RFID chips). The cost of computing keeps plummeting while performance keeps skyrocketing — and that's not by accident.",[12,78525,78526,78527,78530],{},"Modern desktop applications are ",[7289,78528,78529],{},"hungry",". Image processing, 3D rendering, speech recognition, video conferencing, multimedia authoring — all of these demand serious computational muscle. On the server side, businesses rely on powerful machines for transaction processing and massive client\u002Fserver networks, and cloud providers run enormous banks of servers to handle high-volume workloads for countless clients.",[12,78532,78533,78534],{},"So when we talk about \"performance\" in computer architecture, we're really asking: ",[106,78535,78536],{},"how do we design systems that keep up with these ever-growing demands?",[6964,78538,78540],{"id":78539},"speeding-up-the-processor","Speeding Up the Processor",[12,78542,78543],{},"Modern processors don't just run instructions one after another in a straight line. They use a collection of clever techniques to squeeze out as much speed as possible. Let's walk through the big ones.",[7166,78545,78546],{"id":77709},"Pipelining",[12,78548,78549],{},"Think of pipelining like an assembly line in a factory. Instead of waiting for one instruction to finish completely before starting the next, the processor breaks execution into stages. While one instruction is being executed, the next one is already being decoded, and the one after that is being fetched. This overlap means instructions effectively complete much faster on average.",[7166,78551,78553],{"id":78552},"branch-prediction","Branch Prediction",[12,78555,78556,78557,78560,78561,78564],{},"When your code has an ",[6979,78558,78559],{},"if-else"," statement, the processor encounters a \"branch\" — it doesn't yet know which path the program will take. Rather than stalling and waiting, the processor ",[7289,78562,78563],{},"guesses"," which branch is more likely (using historical patterns) and starts executing speculatively down that path. If it guesses right, great — no time wasted. If it guesses wrong, it rolls back and takes the correct path. Modern processors guess correctly the vast majority of the time.",[7166,78566,78568],{"id":78567},"superscalar-execution","Superscalar Execution",[12,78570,78571,78572,78575],{},"A superscalar processor can issue ",[7289,78573,78574],{},"more than one instruction"," in a single clock cycle. Think of it as having multiple assembly lines running simultaneously. Instead of processing instructions one at a time, the processor identifies independent instructions and fires them off in parallel.",[7166,78577,78579],{"id":78578},"data-flow-analysis","Data Flow Analysis",[12,78581,78582],{},"The processor analyzes which instructions depend on the results of other instructions. If instruction B needs the result of instruction A, they can't run in parallel. But if instructions C and D are completely independent, the processor can reorder and execute them whenever their inputs are ready — even out of the original program order.",[7166,78584,78586],{"id":78585},"speculative-execution","Speculative Execution",[12,78588,78589,78590,78593],{},"Building on branch prediction, the processor can execute instructions ",[7289,78591,78592],{},"ahead of time"," before it's certain they'll actually be needed. If it turns out they were needed, the results are already available. If not, the results are simply discarded. This keeps the pipeline full and minimizes wasted cycles.",[6964,78595,78597],{"id":78596},"the-performance-balance-problem","The Performance Balance Problem",[12,78599,78600,78601,78604],{},"Here's a core challenge in computer design: ",[106,78602,78603],{},"not all components run at the same speed."," The processor might be blazing fast, but if it's constantly waiting for data from memory or an I\u002FO device, that speed is wasted.",[12,78606,78607],{},"Think about typical data rates across different I\u002FO devices — an Ethernet modem, a graphics display, a hard disk, an optical drive, a keyboard — they all operate at vastly different speeds. A keyboard sends data at maybe a few bytes per second, while a graphics display might need gigabytes per second.",[12,78609,78610,78611,78614],{},"The architect's job is to ",[106,78612,78613],{},"balance the system"," — adjusting the organization and architecture so the mismatch between components doesn't create bottlenecks. Some strategies include:",[23,78616,78617,78620,78623,78626],{},[26,78618,78619],{},"Making the bus (the data highway between components) wider or faster",[26,78621,78622],{},"Adding caches between the processor and main memory",[26,78624,78625],{},"Using buffering schemes so faster components don't have to wait for slower ones",[26,78627,78628],{},"Building a hierarchy of memory (registers → cache → RAM → disk) so the most frequently used data is always close to the processor",[6964,78630,78632],{"id":78631},"improving-chip-organization-and-architecture","Improving Chip Organization and Architecture",[12,78634,78635],{},"Over the decades, chip designers have pushed performance forward in three main ways:",[12,78637,78638,78641],{},[106,78639,78640],{},"1. Increasing hardware speed."," Shrinking transistors means more gates packed into less space, which raises the clock rate and reduces signal propagation time. Smaller = faster.",[12,78643,78644,78647],{},[106,78645,78646],{},"2. Bigger and faster caches."," By dedicating part of the processor chip itself to cache memory, access times drop dramatically compared to going off-chip to main memory.",[12,78649,78650,78653],{},[106,78651,78652],{},"3. Smarter organization."," Even without faster hardware, clever architectural changes — like deeper pipelines, more parallelism, and better instruction scheduling — can increase the effective speed of instruction execution.",[6964,78655,78657],{"id":78656},"the-wall-problems-with-clock-speed-and-logic-density","The Wall: Problems with Clock Speed and Logic Density",[12,78659,78660],{},"If shrinking transistors makes everything faster, why not just keep shrinking forever? Well, we've hit some very real physical limits.",[7166,78662,78664],{"id":78663},"power-and-heat","Power and Heat",[12,78666,78667],{},"As you pack more transistors together and run them faster, power consumption goes up. More power means more heat. At some point, you simply can't dissipate heat fast enough — the chip would melt or become unreliable. This is often called the \"power wall.\"",[7166,78669,78671],{"id":78670},"rc-delay","RC Delay",[12,78673,78674,78675,78678],{},"Electrons flowing through wires face resistance (R) and capacitance (C). As components shrink, the wires connecting them get thinner (higher resistance) and closer together (higher capacitance). The product R × C determines signal delay, and it actually ",[7289,78676,78677],{},"increases"," as things get smaller. So while transistors get faster, the wires between them can get slower.",[7166,78680,78682],{"id":78681},"memory-latency-and-throughput","Memory Latency and Throughput",[12,78684,78685],{},"Even if the processor can crunch numbers at incredible speed, it still has to wait for data from memory. Memory access speed (latency) and transfer speed (throughput) have historically lagged far behind processor speeds. This gap — sometimes called the \"memory wall\" — is one of the biggest challenges in modern architecture.",[6964,78687,78689],{"id":78688},"the-multicore-era","The Multicore Era",[12,78691,78692,78693],{},"Since we can't just keep cranking up the clock speed, the industry took a different approach: ",[106,78694,78695],{},"put multiple processors (cores) on a single chip.",[12,78697,78698],{},"The idea is straightforward — instead of one very fast core, use two, four, eight, or more simpler cores working in parallel. With two processors, larger caches become justified. As caches grew, it made sense to create two and then three levels of cache hierarchy on a single chip.",[12,78700,78701],{},"The catch? Software has to be written to take advantage of parallelism. A single-threaded program won't magically run faster on eight cores. This shift has had profound implications for how we write software.",[7166,78703,78705],{"id":78704},"many-integrated-core-mic","Many Integrated Core (MIC)",[12,78707,78708,78709,78711],{},"MIC takes the multicore concept further — a ",[7289,78710,37933],{}," number of general-purpose cores on a single chip. The leap in raw performance is impressive, but the challenge of writing software that effectively uses dozens or hundreds of cores is significant.",[7166,78713,78715],{"id":78714},"graphics-processing-units-gpus","Graphics Processing Units (GPUs)",[12,78717,78718],{},"GPUs were originally designed to perform parallel operations on graphics data — encoding and rendering 2D\u002F3D graphics and processing video. But their massively parallel architecture turns out to be great for any task involving repetitive computations: scientific simulations, machine learning, cryptography, and more. This is sometimes called GPGPU — General-Purpose computing on Graphics Processing Units.",[6964,78720,78722],{"id":78721},"computer-clocks-the-heartbeat-of-the-system","Computer Clocks: The Heartbeat of the System",[12,78724,78725,78726,78729,78730,78732],{},"Every digital system is driven by a ",[106,78727,78728],{},"clock"," — a quartz crystal and a converter that produce a constant, regular electrical signal. This signal is the heartbeat of the computer, and it determines ",[7289,78731,76097],{}," events take place inside the hardware.",[12,78734,78735],{},"A few key definitions:",[23,78737,78738,78744],{},[26,78739,78740,78743],{},[106,78741,78742],{},"Clock period (or clock cycle time):"," The time it takes for one complete cycle. For example, 5 nanoseconds.",[26,78745,78746,78749,78750,6959],{},[106,78747,78748],{},"Clock rate:"," The inverse of the clock period — how many cycles happen per second. If the clock period is 5 ns, then the clock rate is 1 \u002F (5 × 10⁻⁹) = ",[106,78751,78752],{},"200 MHz",[12,78754,78755],{},"So when you hear that a processor runs at \"3.5 GHz,\" that means its clock ticks 3.5 billion times per second. Each tick represents one opportunity for the processor to do work.",[6964,78757,78759],{"id":78758},"measuring-performance-cpu-time","Measuring Performance: CPU Time",[12,78761,78762],{},"Now let's get quantitative. How do we actually measure how fast a processor runs a program?",[7166,78764,78766],{"id":78765},"the-cpu-time-formula","The CPU Time Formula",[12,78768,78769],{},"The time to execute a program can be expressed as:",[6972,78771,78774],{"className":78772,"code":78773,"language":6977},[6975],"CPU time = CPU clock cycles × clock cycle time\n",[6979,78775,78773],{"__ignoreMap":180},[12,78777,78778],{},"Since clock cycle time = 1 \u002F clock rate, this is equivalent to:",[6972,78780,78783],{"className":78781,"code":78782,"language":6977},[6975],"CPU time = CPU clock cycles \u002F clock rate\n",[6979,78784,78782],{"__ignoreMap":180},[12,78786,78787],{},"How do we figure out the number of clock cycles? That depends on two things — how many instructions are in the program and how many cycles each instruction takes on average:",[6972,78789,78792],{"className":78790,"code":78791,"language":6977},[6975],"CPU clock cycles = instruction count × CPI\n",[6979,78793,78791],{"__ignoreMap":180},[12,78795,78796,78797,18372,78800,78803],{},"Where ",[106,78798,78799],{},"CPI",[106,78801,78802],{},"Cycles Per Instruction"," — the average number of clock cycles needed to execute one instruction.",[12,78805,78806],{},"Putting it all together:",[6972,78808,78811],{"className":78809,"code":78810,"language":6977},[6975],"CPU time = instruction count × CPI × clock cycle time\nCPU time = instruction count × CPI \u002F clock rate\n",[6979,78812,78810],{"__ignoreMap":180},[7166,78814,13610],{"id":13609},[12,78816,78817,78818,78821,78822,78825,78826,6959],{},"Suppose a computer has a clock rate of ",[106,78819,78820],{},"50 MHz"," and we want to run a program with ",[106,78823,78824],{},"1,000 instructions",", where the average CPI is ",[106,78827,78828],{},"3.5",[6972,78830,78833],{"className":78831,"code":78832,"language":6977},[6975],"CPU time = instruction count × CPI \u002F clock rate\n         = 1000 × 3.5 \u002F (50 × 10⁶)\n         = 3500 \u002F 50,000,000\n         = 70 microseconds\n",[6979,78834,78832],{"__ignoreMap":180},[7166,78836,78838],{"id":78837},"what-happens-when-clock-rate-changes","What Happens When Clock Rate Changes?",[12,78840,78841,78842,10720,78844,78847],{},"Say the clock rate goes from ",[106,78843,78752],{},[106,78845,78846],{},"250 MHz"," and everything else stays the same. The speedup is:",[6972,78849,78852],{"className":78850,"code":78851,"language":6977},[6975],"Speedup = (old CPU time) \u002F (new CPU time)\n        = clock rate new \u002F clock rate old\n        = 250 \u002F 200\n        = 1.25×\n",[6979,78853,78851],{"__ignoreMap":180},[12,78855,78856,78857,78860],{},"The computer is ",[106,78858,78859],{},"25% faster"," — but only if we assume the CPI and instruction count don't change, which in the real world may not always hold true.",[6964,78862,78864],{"id":78863},"computing-cpi-in-practice","Computing CPI in Practice",[12,78866,78867],{},"Different types of instructions take different numbers of cycles. A simple ALU operation might take 1 cycle, while a memory load might take 5. If we know the instruction mix (what fraction of instructions are of each type), we can calculate the overall CPI:",[6972,78869,78872],{"className":78870,"code":78871,"language":6977},[6975],"CPI = Σ (CPIᵢ × Fᵢ)\n",[6979,78873,78871],{"__ignoreMap":180},[12,78875,78796,78876,78879,78880,7153,78882,78885,78886,6959],{},[106,78877,78878],{},"CPIᵢ"," is the cycles for instruction type ",[7289,78881,10588],{},[106,78883,78884],{},"Fᵢ"," is the fraction of instructions that are type ",[7289,78887,10588],{},[7166,78889,15922],{"id":78890},"example",[7085,78892,78893,78911],{},[7088,78894,78895],{},[7091,78896,78897,78899,78903,78905,78908],{},[7094,78898,7096],{},[7094,78900,78902],{"align":78901},"center","Fraction (Fᵢ)",[7094,78904,78878],{"align":78901},[7094,78906,78907],{"align":78901},"CPIᵢ × Fᵢ",[7094,78909,78910],{"align":78901},"% of Time",[7104,78912,78913,78929,78944,78960,78975],{},[7091,78914,78915,78918,78921,78923,78926],{},[7109,78916,78917],{},"ALU",[7109,78919,78920],{"align":78901},"50%",[7109,78922,8855],{"align":78901},[7109,78924,78925],{"align":78901},"0.5",[7109,78927,78928],{"align":78901},"23%",[7091,78930,78931,78934,78937,78939,78941],{},[7109,78932,78933],{},"Load",[7109,78935,78936],{"align":78901},"20%",[7109,78938,15531],{"align":78901},[7109,78940,27020],{"align":78901},[7109,78942,78943],{"align":78901},"45%",[7091,78945,78946,78949,78952,78954,78957],{},[7109,78947,78948],{},"Store",[7109,78950,78951],{"align":78901},"10%",[7109,78953,8830],{"align":78901},[7109,78955,78956],{"align":78901},"0.3",[7109,78958,78959],{"align":78901},"14%",[7091,78961,78962,78965,78967,78969,78972],{},[7109,78963,78964],{},"Branch",[7109,78966,78936],{"align":78901},[7109,78968,8879],{"align":78901},[7109,78970,78971],{"align":78901},"0.4",[7109,78973,78974],{"align":78901},"18%",[7091,78976,78977,78982,78987,78991,78996],{},[7109,78978,78979],{},[106,78980,78981],{},"Total",[7109,78983,78984],{"align":78901},[106,78985,78986],{},"100%",[7109,78988,78989],{"align":78901},[16,78990],{},[7109,78992,78993],{"align":78901},[106,78994,78995],{},"2.2",[7109,78997,78998],{"align":78901},[106,78999,78986],{},[12,79001,79002,79003,79006],{},"So the weighted average CPI is ",[106,79004,79005],{},"2.2 cycles per instruction",". Notice that even though loads are only 20% of instructions, they account for 45% of the execution time because they're so expensive. This tells the architect where to focus optimization efforts!",[6964,79008,79010],{"id":79009},"performance-factors-what-affects-what","Performance Factors: What Affects What?",[12,79012,79013],{},"Not every design decision affects every aspect of performance. Here's a simplified view:",[7085,79015,79016,79031],{},[7088,79017,79018],{},[7091,79019,79020,79023,79026,79028],{},[7094,79021,79022],{},"Factor",[7094,79024,79025],{"align":78901},"Instruction Count",[7094,79027,78799],{"align":78901},[7094,79029,79030],{"align":78901},"Clock Rate",[7104,79032,79033,79046,79057,79070],{},[7091,79034,79035,79038,79040,79042],{},[7109,79036,79037],{},"Instruction Set Architecture (ISA)",[7109,79039,23012],{"align":78901},[7109,79041,23012],{"align":78901},[7109,79043,79044],{"align":78901},[16,79045],{},[7091,79047,79048,79051,79053,79055],{},[7109,79049,79050],{},"Compiler Technology",[7109,79052,23012],{"align":78901},[7109,79054,23012],{"align":78901},[7109,79056,23012],{"align":78901},[7091,79058,79059,79062,79066,79068],{},[7109,79060,79061],{},"Processor Implementation",[7109,79063,79064],{"align":78901},[16,79065],{},[7109,79067,23012],{"align":78901},[7109,79069,23012],{"align":78901},[7091,79071,79072,79075,79079,79081],{},[7109,79073,79074],{},"Cache and Memory Hierarchy",[7109,79076,79077],{"align":78901},[16,79078],{},[7109,79080,23012],{"align":78901},[7109,79082,23012],{"align":78901},[12,79084,79085,79086,79088,79089,79091],{},"The ISA defines ",[7289,79087,9441],{}," instructions exist (affecting instruction count and CPI). The compiler decides ",[7289,79090,76905],{}," instructions to use. The hardware implementation determines how fast each instruction actually executes.",[6964,79093,79095],{"id":79094},"beware-of-misleading-metrics","Beware of Misleading Metrics",[7166,79097,79099],{"id":79098},"mips-millions-of-instructions-per-second","MIPS (Millions of Instructions Per Second)",[6972,79101,79104],{"className":79102,"code":79103,"language":6977},[6975],"MIPS = instruction count \u002F (execution time × 10⁶)\n",[6979,79105,79103],{"__ignoreMap":180},[12,79107,79108],{},"MIPS is easy to understand and measure, but it can be deeply misleading. A processor that executes many simple instructions per second might have a higher MIPS rating than one executing fewer but more powerful instructions — even though the second processor finishes the actual task faster. MIPS doesn't account for the fact that not all instructions do the same amount of work.",[7166,79110,79112],{"id":79111},"mflops-millions-of-floating-point-operations-per-second","MFLOPS (Millions of Floating-Point Operations Per Second)",[6972,79114,79117],{"className":79115,"code":79116,"language":6977},[6975],"MFLOPS = floating-point operations \u002F (execution time × 10⁶)\n",[6979,79118,79116],{"__ignoreMap":180},[12,79120,79121],{},"MFLOPS has the same advantages and drawbacks as MIPS, with the additional limitation that it only measures floating-point work — it tells you nothing about integer performance, I\u002FO, or anything else.",[7166,79123,79125],{"id":79124},"the-mhzghz-trap","The MHz\u002FGHz Trap",[12,79127,79128],{},"1 Hertz = 1 cycle per second. So 1 GHz = 1 billion cycles per second.",[12,79130,79131],{},"It's tempting to compare processors purely by clock speed, but this ignores CPI differences. A classic example: an 800 MHz Pentium III could outperform a 1 GHz Pentium 4 on certain tasks because the Pentium III had a lower CPI — it did more useful work per cycle.",[12,79133,79134,79137,79138,79141],{},[106,79135,79136],{},"Bottom line:"," Clock speed, MIPS, and MFLOPS are all ",[7289,79139,79140],{},"partial"," metrics. Relying on any one of them alone can lead you to the wrong conclusion about which system is actually faster for your workload.",[6964,79143,79145],{"id":79144},"benchmarks-measuring-performance-properly","Benchmarks: Measuring Performance Properly",[12,79147,79148,79149,79152],{},"Since individual metrics can be misleading, the industry uses ",[106,79150,79151],{},"benchmarks"," — standardized programs designed to test real performance. A good benchmark should be:",[3603,79154,79155,79158,79161,79164],{},[26,79156,79157],{},"Written in a high-level language (so it's portable across machines)",[26,79159,79160],{},"Representative of a real programming domain (systems, numerical, commercial)",[26,79162,79163],{},"Easy to measure",[26,79165,79166],{},"Widely distributed",[7166,79168,79170],{"id":79169},"spec-the-industry-standard","SPEC: The Industry Standard",[12,79172,9455,79173,79176],{},[106,79174,79175],{},"System Performance Evaluation Corporation (SPEC)"," is an industry consortium that defines and maintains the most widely recognized benchmark suites. SPEC benchmarks are used everywhere — by researchers, hardware vendors, and buyers — to compare systems on a level playing field.",[7166,79178,79180],{"id":79179},"spec-cpu2017","SPEC CPU2017",[12,79182,79183,79184,79186,79187,7042,79190,79193,79194,79197],{},"The flagship suite for processor-intensive workloads is ",[106,79185,79180],{},". It's designed for applications that spend most of their time doing computation rather than I\u002FO. The suite consists of ",[106,79188,79189],{},"20 integer benchmarks",[106,79191,79192],{},"23 floating-point benchmarks"," written in C, C++, and Fortran, containing over ",[106,79195,79196],{},"11 million lines of code"," in total.",[12,79199,79200],{},"The benchmarks cover a fascinating range of real-world tasks: Perl interpreting, GCC compilation, route planning, video compression (x264), chess AI (alpha-beta search), Go AI (Monte Carlo tree search), Sudoku solving, weather forecasting, molecular dynamics, 3D rendering, ocean modeling, and more.",[7166,79202,79204],{"id":79203},"key-spec-terminology","Key SPEC Terminology",[23,79206,79207,79213,79219,79225,79231,79237],{},[26,79208,79209,79212],{},[106,79210,79211],{},"System under test:"," The system you're evaluating.",[26,79214,79215,79218],{},[106,79216,79217],{},"Reference machine:"," A baseline machine SPEC uses to establish reference times for each benchmark.",[26,79220,79221,79224],{},[106,79222,79223],{},"Base metric:"," Results compiled with strict, conservative compiler settings (required for all reported results).",[26,79226,79227,79230],{},[106,79228,79229],{},"Peak metric:"," Results where users can aggressively optimize compiler settings to squeeze out maximum performance.",[26,79232,79233,79236],{},[106,79234,79235],{},"Speed metric:"," How long a single task takes to complete — useful for comparing single-threaded performance.",[26,79238,79239,79242],{},[106,79240,79241],{},"Rate metric:"," How many tasks a system can complete in a given time — a throughput measure that leverages multiple processors.",[12,79244,79245],{},"The SPEC evaluation process follows a structured workflow: get the benchmark program, run it multiple times, select the median result, compute the ratio against the reference machine, and finally compute the geometric mean across all benchmarks to get a single aggregate score.",[6964,79247,77786],{"id":77785},[12,79249,79250],{},"Performance in computer architecture isn't just about having the fastest clock. It's a delicate balancing act between processor speed, memory access, I\u002FO throughput, and the software that ties it all together. As we've hit physical limits on clock speed, the industry has pivoted to parallelism — multicore, MIC, and GPUs — putting the burden on software to exploit these architectures.",[12,79252,79253,79254,79257],{},"When evaluating performance, always look beyond a single number. Understand CPI, clock rate, and instruction count together. Use standardized benchmarks like SPEC rather than relying on MIPS or raw clock speed. And remember: the fastest system is the one that finishes ",[7289,79255,79256],{},"your workload"," in the least time.",[12,79259,79260],{},[106,79261,79262],{},"Key takeaways:",[23,79264,79265,79268,79271,79274,79277],{},[26,79266,79267],{},"CPU time = instruction count × CPI \u002F clock rate — this is the fundamental equation",[26,79269,79270],{},"Clock speed alone is misleading; CPI matters just as much",[26,79272,79273],{},"Physical limits (power, RC delay, memory wall) ended the era of simple clock speed scaling",[26,79275,79276],{},"Multicore and GPU parallelism are the present and future of performance",[26,79278,79279],{},"SPEC benchmarks provide the most reliable, standardized performance comparisons",{"title":180,"searchDepth":181,"depth":181,"links":79281},[79282,79283,79290,79291,79292,79297,79301,79302,79307,79310,79311,79316,79321],{"id":78519,"depth":181,"text":78520},{"id":78539,"depth":181,"text":78540,"children":79284},[79285,79286,79287,79288,79289],{"id":77709,"depth":7430,"text":78546},{"id":78552,"depth":7430,"text":78553},{"id":78567,"depth":7430,"text":78568},{"id":78578,"depth":7430,"text":78579},{"id":78585,"depth":7430,"text":78586},{"id":78596,"depth":181,"text":78597},{"id":78631,"depth":181,"text":78632},{"id":78656,"depth":181,"text":78657,"children":79293},[79294,79295,79296],{"id":78663,"depth":7430,"text":78664},{"id":78670,"depth":7430,"text":78671},{"id":78681,"depth":7430,"text":78682},{"id":78688,"depth":181,"text":78689,"children":79298},[79299,79300],{"id":78704,"depth":7430,"text":78705},{"id":78714,"depth":7430,"text":78715},{"id":78721,"depth":181,"text":78722},{"id":78758,"depth":181,"text":78759,"children":79303},[79304,79305,79306],{"id":78765,"depth":7430,"text":78766},{"id":13609,"depth":7430,"text":13610},{"id":78837,"depth":7430,"text":78838},{"id":78863,"depth":181,"text":78864,"children":79308},[79309],{"id":78890,"depth":7430,"text":15922},{"id":79009,"depth":181,"text":79010},{"id":79094,"depth":181,"text":79095,"children":79312},[79313,79314,79315],{"id":79098,"depth":7430,"text":79099},{"id":79111,"depth":7430,"text":79112},{"id":79124,"depth":7430,"text":79125},{"id":79144,"depth":181,"text":79145,"children":79317},[79318,79319,79320],{"id":79169,"depth":7430,"text":79170},{"id":79179,"depth":7430,"text":79180},{"id":79203,"depth":7430,"text":79204},{"id":77785,"depth":181,"text":77786},{},"\u002Fblog\u002FComputer-Systems-Architecture-Understanding-Performance",{"title":78514,"description":180},{"loc":79323},"blog\u002FComputer-Systems-Architecture-Understanding-Performance","uJBODumJuxhoGq9uD6CknSUusw9f-z8fK9rvsZq0Vmw",{"id":79329,"title":79330,"author":6946,"body":79331,"date":78185,"description":180,"draft":185,"edited_at":74911,"extension":186,"featured_image":194,"meta":80774,"navigation":189,"path":80775,"pinned":185,"seo":80776,"sitemap":80777,"stem":80778,"tags":194,"__hash__":80779},"blog\u002Fblog\u002FWeek-1-—-Introduction-to-Computer-Systems.md","Introduction to Computer Systems",{"type":9,"value":79332,"toc":80745},[79333,79337,79340,79355,79358,79369,79373,79376,79380,79387,79425,79429,79436,79455,79459,79462,79483,79490,79493,79497,79508,79511,79529,79539,79543,79549,79574,79577,79580,79583,79590,79604,79607,79610,79624,79626,79633,79637,79656,79710,79713,79727,79731,79740,79744,79751,79765,79773,79777,79780,79847,79850,79854,79857,79911,79918,79921,79925,79928,79932,79936,79939,79980,79983,79987,79993,79996,79999,80025,80029,80036,80107,80114,80118,80121,80135,80138,80156,80363,80366,80370,80377,80380,80390,80397,80400,80404,80414,80418,80421,80442,80456,80460,80467,80474,80484,80487,80525,80529,80532,80552,80558,80562,80573,80592,80607,80610,80623,80627,80643,80646,80649,80669,80675,80679,80682,80689,80720,80724,80731,80734],[6964,79334,79336],{"id":79335},"why-this-series-exists","Why this series exists",[12,79338,79339],{},"Computer Systems Architecture (CSA) is one of those subjects that sounds intimidating but is actually about a very simple question:",[7351,79341,79342],{},[12,79343,79344,7206,79347,7206,79352],{},[106,79345,79346],{},"What is",[7289,79348,79349],{},[106,79350,79351],{},"inside",[106,79353,79354],{},"a computer, and how do those parts work together to run the software we write?",[12,79356,79357],{},"Every time you open a browser, play a game, or send a message, billions of tiny switches are doing an enormous amount of coordinated work under the hood. CSA is the map of that hidden world. If you are a programmer, understanding the machine below your code will make you write better, faster, and more memory-friendly programs. If you are just curious, it is genuinely one of the most elegant engineering stories of the 20th century.",[12,79359,79360,79361,79364,79365,79368],{},"In this first post we will cover the ",[106,79362,79363],{},"big picture",": the vocabulary, the mental model, and a short history that explains ",[7289,79366,79367],{},"how we got here",". Later posts will zoom in on the pieces one by one.",[6964,79370,79372],{"id":79371},"_1-computer-architecture-vs-computer-organization","1. Computer Architecture vs. Computer Organization",[12,79374,79375],{},"The very first distinction in CSA is between two words that sound like synonyms but mean different things.",[7166,79377,79379],{"id":79378},"computer-architecture","Computer Architecture",[12,79381,79382,79383,79386],{},"Architecture refers to the parts of the system that are ",[106,79384,79385],{},"visible to the programmer"," — the things that affect how a program behaves logically. Examples include:",[23,79388,79389,79404,79410,79416],{},[26,79390,9455,79391,79394,79395,7298,79398,7298,79401,7963],{},[106,79392,79393],{},"instruction set"," (what commands the CPU understands, like ",[6979,79396,79397],{},"ADD",[6979,79399,79400],{},"LOAD",[6979,79402,79403],{},"JUMP",[26,79405,9455,79406,79409],{},[106,79407,79408],{},"number of bits"," used for data (8-bit, 32-bit, 64-bit)",[26,79411,9455,79412,79415],{},[106,79413,79414],{},"I\u002FO mechanism"," (how the CPU talks to devices)",[26,79417,9455,79418,79421,79422,79424],{},[106,79419,79420],{},"addressing technique"," (how memory locations are named) If you change the architecture, programs written for the old one may no longer work. It is the ",[7289,79423,21335],{}," between hardware and software.",[7166,79426,79428],{"id":79427},"computer-organization","Computer Organization",[12,79430,79431,79432,79435],{},"Organization refers to the ",[106,79433,79434],{},"operational units and how they are interconnected"," — the actual hardware implementation that realises the architecture. Examples:",[23,79437,79438,79444,79449],{},[26,79439,79440,79443],{},[106,79441,79442],{},"Control signals"," between components",[26,79445,79446,79448],{},[106,79447,23355],{}," between the computer and peripherals",[26,79450,79451,79454],{},[106,79452,79453],{},"Memory technology"," (DRAM, SRAM, cache levels, etc.)",[7166,79456,79458],{"id":79457},"an-analogy","An analogy",[12,79460,79461],{},"Think of a car.",[23,79463,79464,79470],{},[26,79465,9455,79466,79469],{},[106,79467,79468],{},"architecture"," is the driver's interface: steering wheel, pedals, gear stick, dashboard. Every Toyota Camry driver knows how to drive any other Camry because the architecture is the same.",[26,79471,9455,79472,79475,79476,23938,79479,79482],{},[106,79473,79474],{},"organization"," is what is under the hood: the specific engine size, turbocharger, transmission design. Two Camrys can have the ",[7289,79477,79478],{},"same architecture",[7289,79480,79481],{},"different organizations"," — one is the base model, the other is the sport version. They drive the same, but one is faster and more expensive.\nThis leads to a key observation from the slides:",[7351,79484,79485],{},[12,79486,79487],{},[7289,79488,79489],{},"Same architecture but different organization → different price and performance. Architecture tends to last a long time with only minor changes, while organization changes as technology improves. By changing the organization, the user can decide the performance they want.",[12,79491,79492],{},"This is literally how Intel and AMD sell you ten different CPUs that all run Windows — same architecture (x86-64), different organizations.",[6964,79494,79496],{"id":79495},"_2-structure-and-function-the-mental-model","2. Structure and Function: the mental model",[12,79498,79499,79500,79503,79504,79507],{},"A computer is a ",[106,79501,79502],{},"complex system",". To understand complex systems, engineers use a universal trick: they break them into ",[106,79505,79506],{},"hierarchical levels",", from highest to lowest, and study each level separately. The levels then combine through the interrelationships between them.",[12,79509,79510],{},"At each level, we ask two questions:",[23,79512,79513,79519],{},[26,79514,79515,79518],{},[106,79516,79517],{},"Structure"," — How are the components inter-related? (the wiring, the shape)",[26,79520,79521,79524,79525,79528],{},[106,79522,79523],{},"Function"," — What does each component ",[7289,79526,79527],{},"do"," as part of the whole? (the behaviour)",[12,79530,79531,79532,79534,79535,79538],{},"There is a classic question that comes up here: when analysing or designing a computer, do you go ",[106,79533,31064],{}," (start from the whole system and decompose) or ",[106,79536,79537],{},"bottom-up"," (start from transistors and build up)? Both approaches have value, but in this course we will mostly go top-down: start from what a computer does, then zoom in to how it does it.",[6964,79540,79542],{"id":79541},"_3-the-four-functions-of-a-computer","3. The Four Functions of a Computer",[12,79544,79545,79546,7024],{},"No matter how fancy a computer gets — your phone, a laptop, a supercomputer — it only ever performs ",[106,79547,79548],{},"four basic functions",[3603,79550,79551,79557,79563,79569],{},[26,79552,79553,79556],{},[106,79554,79555],{},"Data Processing"," — crunching numbers, transforming data.",[26,79558,79559,79562],{},[106,79560,79561],{},"Data Storage"," — keeping data, either briefly (RAM) or for a long time (SSD, hard drive).",[26,79564,79565,79568],{},[106,79566,79567],{},"Data Movement"," — shuffling data between the computer and the outside world.",[26,79570,79571,79573],{},[106,79572,74265],{}," — orchestrating the other three, deciding what happens when.\nLet's look at each one.",[7166,79575,79555],{"id":79576},"data-processing",[12,79578,79579],{},"The computer takes data in, transforms it somehow, and produces new data out. Adding two numbers, resizing an image, decoding a video — all data processing.",[7166,79581,79561],{"id":79582},"data-storage",[12,79584,79585,79586,79589],{},"Data has to live ",[7289,79587,79588],{},"somewhere",". Storage is split into two flavours:",[23,79591,79592,79598],{},[26,79593,79594,79597],{},[106,79595,79596],{},"Short-term storage"," — fast but volatile (it disappears when the power goes off). This is RAM.",[26,79599,79600,79603],{},[106,79601,79602],{},"Long-term storage"," — slower but persistent. This is your SSD, hard drive, or USB stick.",[7166,79605,79567],{"id":79606},"data-movement",[12,79608,79609],{},"There are two sub-categories worth knowing:",[23,79611,79612,79618],{},[26,79613,79614,79617],{},[106,79615,79616],{},"Data communications"," — moving data over long distances or between remote devices (think: Wi-Fi, Ethernet, the internet).",[26,79619,79620,79623],{},[106,79621,79622],{},"Input\u002FOutput (I\u002FO)"," — moving data to and from peripherals directly connected to the computer (keyboard, mouse, screen, printer).",[7166,79625,74265],{"id":75422},[12,79627,79628,79629,79632],{},"A user-defined ",[106,79630,79631],{},"control algorithm"," decides when each of the other three functions happens, in what order, and in response to what. Without control, the other parts are just a pile of capabilities with no coordinator.",[7166,79634,79636],{"id":79635},"the-four-functions-visualised","The four functions, visualised",[12,79638,79639,79640,7298,79643,7298,79645,7298,79648,79651,79652,79655],{},"The slides use a neat diagram with four circles — ",[7289,79641,79642],{},"Movement",[7289,79644,74265],{},[7289,79646,79647],{},"Storage",[7289,79649,79650],{},"Processing"," — connected in a way that shows how control sits in the middle, orchestrating everything. Different patterns of the arrows highlight different ",[7289,79653,79654],{},"modes"," of operation:",[7085,79657,79658,79668],{},[7088,79659,79660],{},[7091,79661,79662,79665],{},[7094,79663,79664],{},"Mode",[7094,79666,79667],{},"What is happening",[7104,79669,79670,79680,79690,79700],{},[7091,79671,79672,79677],{},[7109,79673,79674],{},[106,79675,79676],{},"Data storage device",[7109,79678,79679],{},"Data moves between the external environment and storage (read\u002Fwrite).",[7091,79681,79682,79687],{},[7109,79683,79684],{},[106,79685,79686],{},"Data movement device",[7109,79688,79689],{},"Data is transferred from one peripheral or communication line to another (no storage, no processing).",[7091,79691,79692,79697],{},[7109,79693,79694],{},[106,79695,79696],{},"Data processing, storage-only",[7109,79698,79699],{},"Data already in storage is processed and written back — nothing leaves the machine.",[7091,79701,79702,79707],{},[7109,79703,79704],{},[106,79705,79706],{},"Data processing, involving external environment",[7109,79708,79709],{},"Data comes in from outside, is processed, and results go back out. This is the full pipeline.",[12,79711,79712],{},"Two important notes from the slides:",[3603,79714,79715,79721],{},[26,79716,79717,79720],{},[106,79718,79719],{},"All four operations involve the control function."," Control is always in the loop.",[26,79722,79723,79726],{},[106,79724,79725],{},"The functions can run simultaneously with multi-core processors."," In older single-core machines, things happened one at a time; today, your laptop is doing storage, movement, and processing all at once.",[6964,79728,79730],{"id":79729},"_4-structure-how-the-pieces-fit-together","4. Structure: how the pieces fit together",[12,79732,79733,79734,79736,79737,79739],{},"Now that we know ",[7289,79735,9441],{}," a computer does, let's look at ",[7289,79738,9518],{}," it is built. We will zoom in level by level.",[7166,79741,79743],{"id":79742},"level-1-the-computer-and-the-world-outside-it","Level 1: The computer and the world outside it",[12,79745,79746,79747,79750],{},"At the highest level, the computer is a single blob that interacts with the ",[106,79748,79749],{},"external environment"," through:",[23,79752,79753,79759],{},[26,79754,79755,79758],{},[106,79756,79757],{},"Peripherals"," (keyboard, mouse, monitor, printer, etc.)",[26,79760,79761,79764],{},[106,79762,79763],{},"Communication lines"," (network cables, Wi-Fi, Bluetooth)",[12,79766,79767,79768,7042,79770,79772],{},"Inside that blob, the essential jobs are ",[106,79769,79647],{},[106,79771,79650],{},". Everything else at this level is just \"the outside world\".",[7166,79774,79776],{"id":79775},"level-2-the-internal-structure-of-a-computer","Level 2: The internal structure of a computer",[12,79778,79779],{},"If we crack open the computer blob, we find four major internal components:",[7085,79781,79782,79791],{},[7088,79783,79784],{},[7091,79785,79786,79789],{},[7094,79787,79788],{},"Component",[7094,79790,7099],{},[7104,79792,79793,79806,79819,79831],{},[7091,79794,79795,79800],{},[7109,79796,79797],{},[106,79798,79799],{},"Central Processing Unit (CPU)",[7109,79801,79802,79803],{},"Controls the operation of the computer and performs data processing. ",[7289,79804,79805],{},"The brain.",[7091,79807,79808,79813],{},[7109,79809,79810],{},[106,79811,79812],{},"Main memory",[7109,79814,79815,79816],{},"Stores data. ",[7289,79817,79818],{},"The short-term workspace.",[7091,79820,79821,79825],{},[7109,79822,79823],{},[106,79824,74396],{},[7109,79826,79827,79828],{},"Moves data between the computer and its external environment. ",[7289,79829,79830],{},"The hands, eyes, and mouth.",[7091,79832,79833,79838],{},[7109,79834,79835],{},[106,79836,79837],{},"System Interconnection",[7109,79839,79840,79841,49470,79844],{},"Mechanism that provides communication among CPU, main memory, and I\u002FO. This is what we usually call the ",[106,79842,79843],{},"System Bus",[7289,79845,79846],{},"The nervous system.",[12,79848,79849],{},"Everything else you have heard of — hard drive, GPU, network card — is either part of these or hangs off them via I\u002FO.",[7166,79851,79853],{"id":79852},"level-3-inside-the-cpu","Level 3: Inside the CPU",[12,79855,79856],{},"Crack the CPU open and we find another four components:",[7085,79858,79859,79867],{},[7088,79860,79861],{},[7091,79862,79863,79865],{},[7094,79864,79788],{},[7094,79866,7099],{},[7104,79868,79869,79879,79889,79901],{},[7091,79870,79871,79876],{},[7109,79872,79873],{},[106,79874,79875],{},"Control Unit (CU)",[7109,79877,79878],{},"Controls the operation of the CPU (and, by extension, the whole computer).",[7091,79880,79881,79886],{},[7109,79882,79883],{},[106,79884,79885],{},"Arithmetic and Logic Unit (ALU)",[7109,79887,79888],{},"Performs the computer's data processing functions — the actual adding, comparing, ANDing, ORing.",[7091,79890,79891,79895],{},[7109,79892,79893],{},[106,79894,76761],{},[7109,79896,79897,79898,79900],{},"Provide tiny, super-fast storage ",[7289,79899,79351],{}," the CPU.",[7091,79902,79903,79908],{},[7109,79904,79905],{},[106,79906,79907],{},"CPU Interconnection",[7109,79909,79910],{},"Provides communication between the control unit, ALU, and registers.",[12,79912,79913,79914,79917],{},"Notice how the structure is ",[106,79915,79916],{},"recursive",": at each level, we see \"something that processes, something that stores, something that controls, something that connects them.\" This is not a coincidence — it is a reflection of the four functions from Section 3, applied at every scale.",[12,79919,79920],{},"This recursive pattern is one of the most beautiful ideas in computer architecture. We will see it again and again.",[6964,79922,79924],{"id":79923},"_5-a-brief-history-of-computers","5. A Brief History of Computers",[12,79926,79927],{},"Now the fun part. Let us walk through the four generations of computer hardware and see how each jump unlocked what came next.",[7166,79929,79931],{"id":79930},"generation-1-vacuum-tubes-1940s1950s","Generation 1: Vacuum Tubes (1940s–1950s)",[26236,79933,79935],{"id":79934},"eniac-the-first-general-purpose-electronic-computer","ENIAC — the first general-purpose electronic computer",[12,79937,79938],{},"ENIAC (Electronic Numerical Integrator and Computer) was the first general-purpose computer built with vacuum tubes. Some eye-watering stats:",[23,79940,79941,79947,79954,79960,79967,79974],{},[26,79942,79943,79944],{},"Weighed ",[106,79945,79946],{},"30 tons",[26,79948,79949,79950,79953],{},"Occupied ",[106,79951,79952],{},"1,500 square feet"," of floor space",[26,79955,79956,79957],{},"Used ",[106,79958,79959],{},"18,000 vacuum tubes",[26,79961,79962,79963,79966],{},"Consumed ",[106,79964,79965],{},"140 kW"," of power (enough to power ~100 modern homes)",[26,79968,79969,79970,79973],{},"Was a ",[106,79971,79972],{},"decimal machine",": 10 vacuum tubes were used to represent a single decimal digit",[26,79975,79976,79977],{},"Was programmed by ",[106,79978,79979],{},"manually flipping switches",[12,79981,79982],{},"Imagine programming a machine the size of a tennis court by walking around flipping thousands of switches. That was software engineering in 1946.",[26236,79984,79986],{"id":79985},"edvac-and-the-von-neumann-breakthrough","EDVAC and the Von Neumann breakthrough",[12,79988,79989,79990,6959],{},"EDVAC (Electronic Discrete Variable Computer) was proposed by John Von Neumann and introduced the idea that still defines virtually every computer today: the ",[106,79991,79992],{},"stored-program concept",[12,79994,79995],{},"Instead of rewiring the machine for every new program, you store the program in memory alongside the data, just as numbers. The computer then reads its own instructions one by one.",[12,79997,79998],{},"EDVAC also:",[23,80000,80001,80008,80014],{},[26,80002,80003,80004,80007],{},"Had the ",[106,80005,80006],{},"basic internal structure of a modern CPU"," — control unit, memory, ALU, I\u002FO (sound familiar? Section 4!)",[26,80009,79969,80010,80013],{},[106,80011,80012],{},"binary system"," — 1s and 0s instead of decimal digits",[26,80015,80016,80017,80020,80021,80024],{},"Had ",[106,80018,80019],{},"1,000 storage locations",", each ",[106,80022,80023],{},"40 bits"," wide, for both data and instructions",[26236,80026,80028],{"id":80027},"the-ias-computer-and-its-registers","The IAS computer and its registers",[12,80030,80031,80032,80035],{},"The IAS machine (built at the Institute for Advanced Study, a relative of EDVAC) is often used as the textbook example of early stored-program architecture. It introduced several specialised ",[106,80033,80034],{},"registers"," that still exist, in one form or another, in modern CPUs:",[7085,80037,80038,80047],{},[7088,80039,80040],{},[7091,80041,80042,80045],{},[7094,80043,80044],{},"Register",[7094,80046,74707],{},[7104,80048,80049,80059,80069,80078,80088,80097],{},[7091,80050,80051,80056],{},[7109,80052,80053],{},[106,80054,80055],{},"Memory Buffer Register (MBR)",[7109,80057,80058],{},"Holds a word being stored to memory or sent to I\u002FO, and vice versa.",[7091,80060,80061,80066],{},[7109,80062,80063],{},[106,80064,80065],{},"Memory Address Register (MAR)",[7109,80067,80068],{},"Specifies the memory address that will be written to or read into the MBR.",[7091,80070,80071,80075],{},[7109,80072,80073],{},[106,80074,74230],{},[7109,80076,80077],{},"Holds the 8-bit opcode of the current instruction.",[7091,80079,80080,80085],{},[7109,80081,80082],{},[106,80083,80084],{},"Instruction Buffer Register (IBR)",[7109,80086,80087],{},"Temporarily holds an instruction.",[7091,80089,80090,80094],{},[7109,80091,80092],{},[106,80093,74214],{},[7109,80095,80096],{},"Holds the address of the next instruction pair to be fetched from memory.",[7091,80098,80099,80104],{},[7109,80100,80101],{},[106,80102,80103],{},"Accumulator (AC) and Multiplier Quotient (MQ)",[7109,80105,80106],{},"Temporarily hold operands and results of ALU operations.",[12,80108,80109,80110,80113],{},"You do not need to memorise these today, but notice the pattern: ",[7289,80111,80112],{},"every register has one specific job",". CPU design is all about having the right little box in the right place.",[7166,80115,80117],{"id":80116},"generation-2-transistors-late-1950s1960s","Generation 2: Transistors (late 1950s–1960s)",[12,80119,80120],{},"In 1947, Bell Labs invented the transistor. It was a revolution:",[23,80122,80123,80129],{},[26,80124,80125,80128],{},[106,80126,80127],{},"Smaller, cheaper, and dissipated less heat"," than a vacuum tube",[26,80130,80131,80134],{},[106,80132,80133],{},"Solid-state"," device made from silicon — no fragile glass, no vacuum",[12,80136,80137],{},"To appreciate the jump, picture the difference:",[23,80139,80140,80146],{},[26,80141,9407,80142,80145],{},[106,80143,80144],{},"vacuum tube"," requires wires, metal plates, a glass capsule, and a vacuum inside. Picture a light bulb with extra plumbing.",[26,80147,9407,80148,80151,80152,80155],{},[106,80149,80150],{},"transistor"," is a tiny chunk of silicon with three leads. That's it.\nThe flagship second-generation machine was the ",[106,80153,80154],{},"IBM 7094",". Over the course of the IBM 700\u002F7000 series (see the table below), you can watch hardware improve dramatically in just 12 years:",[7085,80157,80158,80192],{},[7088,80159,80160],{},[7091,80161,80162,80165,80168,80171,80174,80177,80180,80183,80186,80189],{},[7094,80163,80164],{},"Model",[7094,80166,80167],{},"Year",[7094,80169,80170],{},"CPU Tech",[7094,80172,80173],{},"Memory Tech",[7094,80175,80176],{},"Cycle Time (µs)",[7094,80178,80179],{},"Memory (K)",[7094,80181,80182],{},"Opcodes",[7094,80184,80185],{},"Index Registers",[7094,80187,80188],{},"Floating Point",[7094,80190,80191],{},"Speed (relative to 701)",[7104,80193,80194,80223,80253,80279,80309,80337],{},[7091,80195,80196,80199,80202,80205,80208,80210,80213,80216,80218,80220],{},[7109,80197,80198],{},"701",[7109,80200,80201],{},"1952",[7109,80203,80204],{},"Vacuum tubes",[7109,80206,80207],{},"Electrostatic tubes",[7109,80209,26103],{},[7109,80211,80212],{},"2–4",[7109,80214,80215],{},"24",[7109,80217,9180],{},[7109,80219,23905],{},[7109,80221,80222],{},"1×",[7091,80224,80225,80228,80231,80233,80236,80239,80242,80245,80247,80250],{},[7109,80226,80227],{},"704",[7109,80229,80230],{},"1955",[7109,80232,80204],{},[7109,80234,80235],{},"Core",[7109,80237,80238],{},"12",[7109,80240,80241],{},"4–32",[7109,80243,80244],{},"80",[7109,80246,8830],{},[7109,80248,80249],{},"yes",[7109,80251,80252],{},"2.5×",[7091,80254,80255,80258,80261,80263,80265,80267,80269,80272,80274,80276],{},[7109,80256,80257],{},"709",[7109,80259,80260],{},"1958",[7109,80262,80204],{},[7109,80264,80235],{},[7109,80266,80238],{},[7109,80268,26058],{},[7109,80270,80271],{},"140",[7109,80273,8830],{},[7109,80275,80249],{},[7109,80277,80278],{},"4×",[7091,80280,80281,80284,80287,80292,80294,80297,80299,80302,80304,80306],{},[7109,80282,80283],{},"7090",[7109,80285,80286],{},"1960",[7109,80288,80289],{},[106,80290,80291],{},"Transistor",[7109,80293,80235],{},[7109,80295,80296],{},"2.18",[7109,80298,26058],{},[7109,80300,80301],{},"169",[7109,80303,8830],{},[7109,80305,80249],{},[7109,80307,80308],{},"25×",[7091,80310,80311,80314,80317,80319,80321,80323,80325,80328,80331,80334],{},[7109,80312,80313],{},"7094 I",[7109,80315,80316],{},"1962",[7109,80318,80291],{},[7109,80320,80235],{},[7109,80322,8879],{},[7109,80324,26058],{},[7109,80326,80327],{},"185",[7109,80329,80330],{},"7",[7109,80332,80333],{},"yes (double precision)",[7109,80335,80336],{},"30×",[7091,80338,80339,80342,80345,80347,80349,80352,80354,80356,80358,80360],{},[7109,80340,80341],{},"7094 II",[7109,80343,80344],{},"1964",[7109,80346,80291],{},[7109,80348,80235],{},[7109,80350,80351],{},"1.4",[7109,80353,26058],{},[7109,80355,80327],{},[7109,80357,80330],{},[7109,80359,80333],{},[7109,80361,80362],{},"50×",[12,80364,80365],{},"From 1× to 50× speed in twelve years, purely from hardware progress. And notice the cycle time dropping from 30 µs to 1.4 µs — that is the clock speed getting faster.",[26236,80367,80369],{"id":80368},"data-channels-giving-the-cpu-a-break","Data channels: giving the CPU a break",[12,80371,80372,80373,80376],{},"A major architectural shift with the IBM 7094 was the introduction of ",[106,80374,80375],{},"data channels",". Here is the idea:",[12,80378,80379],{},"In the old IAS design, the CPU personally supervised every byte moving to or from an I\u002FO device. That is like a CEO answering every email in the company — a terrible use of expensive brainpower.",[12,80381,9407,80382,80385,80386,80389],{},[106,80383,80384],{},"data channel"," is a small, specialised processor with its own instruction set, dedicated to I\u002FO. The CPU just signals the data channel (\"please read this file\"), and the channel does the work on its own and reports back when done. ",[106,80387,80388],{},"The burden of the CPU is reduced",", and the CPU can get on with real computing.",[12,80391,80392,80393,80396],{},"Alongside this was the ",[106,80394,80395],{},"multiplexor",", which manages how data flows to and from the CPU or memory when there are multiple channels competing for attention.",[12,80398,80399],{},"This pattern — offloading specialised work to specialised hardware so the CPU can focus — is everywhere in modern systems. GPUs, DMA controllers, network cards… they are all descendants of that 1962 data channel.",[7166,80401,80403],{"id":80402},"generation-3-integrated-circuits-1958-onwards","Generation 3: Integrated Circuits (1958 onwards)",[12,80405,80406,80407,80410,80411,6959],{},"In 1958, Jack Kilby and Robert Noyce independently invented the ",[106,80408,80409],{},"integrated circuit (IC)"," — multiple transistors, resistors, and capacitors fabricated together on a single silicon wafer. This started the era of ",[106,80412,80413],{},"microelectronics",[26236,80415,80417],{"id":80416},"the-two-building-blocks","The two building blocks",[12,80419,80420],{},"At the heart of every integrated circuit are just two primitive elements:",[23,80422,80423,80432],{},[26,80424,80425,80428,80429,6959],{},[106,80426,80427],{},"Gates"," — perform Boolean logic (AND, OR, NOT). They take inputs and produce an output when the activate signal tells them to. Gates are used for ",[106,80430,80431],{},"data processing",[26,80433,80434,80437,80438,80441],{},[106,80435,80436],{},"Memory cells"," — binary storage cells with a Read\u002FWrite signal. They are used for ",[106,80439,80440],{},"data storage",".\nAnd that is it. With enough gates and enough memory cells, wired up in the right pattern, you can build anything — including the computer you are reading this on.",[12,80443,80444,80445,7153,80448,80451,80452,80455],{},"The paths among components handle ",[106,80446,80447],{},"data movement",[106,80449,80450],{},"control signals"," control when gates and memory cells fire. Look at that: the ",[106,80453,80454],{},"four functions again",", now implemented as physical hardware elements on silicon.",[26236,80457,80459],{"id":80458},"how-ics-are-made","How ICs are made",[12,80461,80462,80463,80466],{},"The integrated circuit (transistors, resistors, and capacitors) is fabricated on a ",[106,80464,80465],{},"silicon wafer",". A single wafer can hold many chips with the same configuration of gates, memory cells, and I\u002FO connections, which are cut out and packaged individually. This is why chips keep getting cheaper: you make them by the wafer, not one at a time.",[26236,80468,80470,80471],{"id":80469},"ibm-system360-the-first-computer-family","IBM System\u002F360 — the first computer ",[7289,80472,80473],{},"family",[12,80475,80476,80477,80480,80481,6959],{},"When IBM built the System\u002F360 in 1964, they did something strategically brilliant. It used integrated circuit technology and, unfortunately, was ",[7289,80478,80479],{},"not compatible"," with the previous IBM machines — so existing customers had to rewrite their software. But in exchange, IBM offered something new: the industry's ",[106,80482,80483],{},"first planned family of computers",[12,80485,80486],{},"A \"family\" meant several models with shared characteristics, so customers could upgrade without relearning everything:",[3603,80488,80489,80495,80501,80507,80513,80519],{},[26,80490,80491,80494],{},[106,80492,80493],{},"Similar or identical instruction set"," — code written for a small model still runs on a bigger one.",[26,80496,80497,80500],{},[106,80498,80499],{},"Similar or identical operating system"," — sysadmins do not need retraining.",[26,80502,80503,80506],{},[106,80504,80505],{},"Increasing speed"," — bigger model = faster clock.",[26,80508,80509,80512],{},[106,80510,80511],{},"Increasing number of I\u002FO ports"," — bigger model = more devices can connect.",[26,80514,80515,80518],{},[106,80516,80517],{},"Increasing memory size"," — bigger model = more data can fit.",[26,80520,80521,80524],{},[106,80522,80523],{},"Increasing cost"," — you pay for what you get.\nThis is literally how every laptop lineup works today. MacBook Air → Pro → Pro Max. Same macOS, same apps, just more horsepower. IBM invented that pricing model.",[7166,80526,80528],{"id":80527},"generation-4-lsi-vlsi-ulsi-1970s-and-beyond","Generation 4: LSI, VLSI, ULSI (1970s and beyond)",[12,80530,80531],{},"As manufacturing improved, engineers kept cramming more components onto a single chip. The industry labelled successive milestones:",[23,80533,80534,80540,80546],{},[26,80535,80536,80539],{},[106,80537,80538],{},"LSI"," (Large Scale Integration)",[26,80541,80542,80545],{},[106,80543,80544],{},"VLSI"," (Very Large Scale Integration)",[26,80547,80548,80551],{},[106,80549,80550],{},"ULSI"," (Ultra Large Scale Integration)",[12,80553,9455,80554,80557],{},[106,80555,80556],{},"number of components per chip increased yearly"," — the phenomenon famously captured by Moore's Law. Two major leaps happened in this era.",[26236,80559,80561],{"id":80560},"semiconductor-memory-replaces-core-memory","Semiconductor memory replaces core memory",[12,80563,80564,80565,80568,80569,80572],{},"In the 1950s and 1960s, memory was built from ",[106,80566,80567],{},"rings of ferromagnetic material"," called ",[106,80570,80571],{},"cores",". Core memory was fast for its time, but had serious downsides:",[23,80574,80575,80581,80586],{},[26,80576,80577,80580],{},[106,80578,80579],{},"Bulky"," — imagine tiny magnetic donuts threaded on wires, by hand",[26,80582,80583],{},[106,80584,80585],{},"Expensive",[26,80587,80588,80591],{},[106,80589,80590],{},"Destructive readout"," — reading a bit erased it, so the value had to be re-written immediately afterwards",[12,80593,80594,80595,80598,80599,80602,80603,80606],{},"In ",[106,80596,80597],{},"1970, Fairchild introduced semiconductor memory",". It was ",[106,80600,80601],{},"non-destructive"," (reading did not erase the bit) and ",[106,80604,80605],{},"faster than core",". This is the ancestor of every DRAM and SRAM chip in use today.",[12,80608,80609],{},"To appreciate the insane scale of this transition, the slides end with one of my favourite images in the whole course:",[7351,80611,80612,80617],{},[12,80613,80614],{},[106,80615,80616],{},"8 Bytes of core memory vs. 8 Gigabytes on a microSD card.",[12,80618,80619,80620,6959],{},"The core memory looks like a woven sculpture the size of a brick. The microSD card is smaller than a fingernail. And the microSD holds ",[106,80621,80622],{},"one billion times more data",[26236,80624,80626],{"id":80625},"the-microprocessor-a-whole-cpu-on-one-chip","The microprocessor: a whole CPU on one chip",[12,80628,80594,80629,80632,80633,80635,80636,80639,80640,6959],{},[106,80630,80631],{},"1971, Intel released the 4004"," — the first chip to contain ",[7289,80634,23215],{}," the components of a CPU on a ",[106,80637,80638],{},"single chip",". This was the birth of the ",[106,80641,80642],{},"microprocessor",[12,80644,80645],{},"Before 1971, a CPU was a large circuit board (or several boards) full of separate chips. After 1971, \"the CPU\" became one tiny square you could hold in tweezers. This is the innovation that made personal computers, mobile phones, and embedded systems possible.",[12,80647,80648],{},"From there, microprocessor improvements have followed a few steady trends:",[23,80650,80651,80657,80663],{},[26,80652,80653,80656],{},[106,80654,80655],{},"Increase in the number of bits in the register"," (4-bit → 8-bit → 16-bit → 32-bit → 64-bit)",[26,80658,80659,80662],{},[106,80660,80661],{},"Decrease in clock switching time"," (faster clock = more instructions per second)",[26,80664,80665,80668],{},[106,80666,80667],{},"Other hardware improvements"," (pipelining, caches, branch prediction, multiple cores…)",[12,80670,80671,80672,6959],{},"The chip in your phone right now is a direct descendant of the Intel 4004 — only about ",[106,80673,80674],{},"a billion times more capable",[6964,80676,80678],{"id":80677},"recap","Recap",[12,80680,80681],{},"Let's pull all of this together.",[12,80683,80684,80685,80688],{},"We started with the question: ",[106,80686,80687],{},"what is inside a computer?"," We answered it in three layers of zoom:",[3603,80690,80691,80697,80714],{},[26,80692,80693,80696],{},[106,80694,80695],{},"Architecture vs. Organization"," — the contract with the programmer vs. the hardware implementation.",[26,80698,80699,80702,80703,7298,80706,7298,80708,7153,80711,80713],{},[106,80700,80701],{},"Structure and Function"," — at every level of the computer, there are things that ",[7289,80704,80705],{},"process",[7289,80707,19130],{},[7289,80709,80710],{},"move data",[7289,80712,75422],{},". Structure is how they are connected; function is what they do.",[26,80715,80716,80719],{},[106,80717,80718],{},"History"," — vacuum tubes gave us programmable electronic computers, transistors made them practical, integrated circuits made them affordable, and LSI\u002FVLSI put a whole computer in your pocket.\nThe same four functions (processing, storage, movement, control) reappear at every level of abstraction — from a warehouse-sized ENIAC, down to the inside of a modern CPU, down to individual gates and memory cells on silicon. Once you see this pattern, the rest of CSA is a lot less intimidating.",[6964,80721,80723],{"id":80722},"where-we-go-next","Where we go next",[12,80725,80726,80727,80730],{},"In the next post we will zoom in on ",[106,80728,80729],{},"the CPU"," — how the control unit and ALU cooperate to fetch, decode, and execute a single instruction. That is where the magic of stored-program computing actually happens.",[12,80732,80733],{},"Until then, a little homework to cement the ideas:",[23,80735,80736,80739,80742],{},[26,80737,80738],{},"Look at your own laptop or phone. Can you name its architecture (e.g. ARM, x86-64)?",[26,80740,80741],{},"For each of the four functions, point at something on your desk that performs it.",[26,80743,80744],{},"Ask yourself: in an ENIAC with 18,000 vacuum tubes, how would you debug a failure?\nSee you in Week 2.",{"title":180,"searchDepth":181,"depth":181,"links":80746},[80747,80748,80753,80754,80761,80766,80772,80773],{"id":79335,"depth":181,"text":79336},{"id":79371,"depth":181,"text":79372,"children":80749},[80750,80751,80752],{"id":79378,"depth":7430,"text":79379},{"id":79427,"depth":7430,"text":79428},{"id":79457,"depth":7430,"text":79458},{"id":79495,"depth":181,"text":79496},{"id":79541,"depth":181,"text":79542,"children":80755},[80756,80757,80758,80759,80760],{"id":79576,"depth":7430,"text":79555},{"id":79582,"depth":7430,"text":79561},{"id":79606,"depth":7430,"text":79567},{"id":75422,"depth":7430,"text":74265},{"id":79635,"depth":7430,"text":79636},{"id":79729,"depth":181,"text":79730,"children":80762},[80763,80764,80765],{"id":79742,"depth":7430,"text":79743},{"id":79775,"depth":7430,"text":79776},{"id":79852,"depth":7430,"text":79853},{"id":79923,"depth":181,"text":79924,"children":80767},[80768,80769,80770,80771],{"id":79930,"depth":7430,"text":79931},{"id":80116,"depth":7430,"text":80117},{"id":80402,"depth":7430,"text":80403},{"id":80527,"depth":7430,"text":80528},{"id":80677,"depth":181,"text":80678},{"id":80722,"depth":181,"text":80723},{},"\u002Fblog\u002FWeek-1-—-Introduction-to-Computer-Systems",{"title":79330,"description":180},{"loc":80775},"blog\u002FWeek-1-—-Introduction-to-Computer-Systems","94-DJj9_cAojlCQAlJd4kiflgnC9KQOhiMrZBfIwCTE",{"id":80781,"title":80782,"author":7,"body":80783,"date":81109,"description":81110,"draft":185,"edited_at":74911,"extension":186,"featured_image":81111,"meta":81112,"navigation":189,"path":81113,"pinned":185,"seo":81114,"sitemap":81115,"stem":81116,"tags":194,"__hash__":81117},"blog\u002Fblog\u002FBalance（五）復健.md","Balance（五）復健",{"type":9,"value":80784,"toc":81107},[80785,80788,80794,80797,80836,80838,80860,80868,80870,80898,80907,80913,80919,80928,80930,80944,80950,80956,80962,80968,80970,80973,81019,81028,81030,81064,81082,81100],[12,80786,80787],{},"對 一樣中間沒有註解的頁碼先跳過 看有沒有機會趕緊寫完再讀一次中間空的補上",[12,80789,80790,80791,80793],{},"那平衡有問題的話，要去留意有無損傷的系統包括了這四項",[16,80792],{},"\n周邊感覺能力、中樞感覺處理、中樞運動處理、周邊運動能力",[12,80795,80796],{},"那受損的處理方式在評估後分成三種做法",[23,80798,80799,80802,80818],{},[26,80800,80801],{},"暫時性——做法有給予刺激、誘發或強迫使用三種處置",[26,80803,80804,80805,80809,80810,80814,80815,80817],{},"永久性——主要考量看",[75,80806,80808],{"className":80807},[78,79],"內在替代","（能不能找類似身體功能替代）和",[75,80811,80813],{"className":80812},[78,79],"外在補償","（輔助）",[16,80816],{},"\n那做法有就替代（訓練其他類似的身體功能）、補償（輔具）和衛教",[26,80819,80820,80821,80825,80826,80830,80832,80833,80835],{},"未知——他是指說",[75,80822,80824],{"className":80823},[78,79],"症狀尚不明朗或者未定性病癥","時的處置，考量為",[75,80827,80829],{"className":80828},[78,78392],"有無禁忌症、風險收益比較、功能改變優劣",[16,80831],{},"\n假如以上考量都是OK的，那就用比較積極的方式來處理，那就刺激或強迫使用",[16,80834],{},"\n假如不OK的，那就用補償和衛教方式",[36,80837],{},[12,80839,80840,80841,80843,80844,80846,80847,80849,80850,80854,80855,80859],{},"那動作學習包含了三個階段（我怎麼感覺我在這個期中前打過了不只五次動作學習了）",[16,80842],{},"\n獲得技巧——經常失敗，品質不佳",[16,80845],{},"\n精進技巧——失敗率下降，品質持續提升",[16,80848],{},"\n保留技巧——技巧",[75,80851,80853],{"className":80852},[79,78],"保留","（不會忘掉），能",[75,80856,80858],{"className":80857},[79,78],"轉移","（類似情景適用）",[12,80861,80862,80863,80867],{},"可以透過",[75,80864,80866],{"className":80865},[79,78],"練習跟回饋","改善技巧（又 又練習 又回饋 啊啊啊啊啊）",[36,80869],{},[12,80871,80872,80873,80877,80879,80880,80884,80885,80887,80888,80890,80892,80893,80897],{},"接下來講到平衡的訓練部份，那平衡有問題的病人來說，只有兩類要區分，",[75,80874,80876],{"className":80875},[78,173],"暫時出問題和完全失能",[16,80878],{},"\n這不難理解，假如是暫時出問題，那就",[75,80881,80883],{"className":80882},[78,79],"關閉好的剩下壞的的去專心訓練","（例如前庭出問題就關閉視覺與本體覺）",[16,80886],{},"\n訓練就是難度遞增方式，先全開，關閉一個，最後只開出問題感覺輸入去訓練",[16,80889],{},[16,80891],{},"\n但假如是完全失能，變成只能",[75,80894,80896],{"className":80895},[79,78],"依賴剩下僅有","的（例如只有視覺還在，那就只能只用視覺了）",[12,80899,80900,80901,80903,80904,80906],{},"刺激本體覺輸入",[16,80902],{},"\n給一個穩定平整的平面給他踩好／干擾或關閉視覺輸入（關燈，蒙眼，頭快速轉）",[16,80905],{},"\n加強訓練的話，可以以在兩側肩膀上加重量，抑或者躺在硬的地上面（本體覺輸入up）>促使前庭覺適當較正",[12,80908,80909,80910,80912],{},"刺激視覺輸入",[16,80911],{},"\n給一個穩定且明確記的環境／雖然本體覺不能關閉但可以干擾例如站在不穩的平面上或者站在泡棉上",[12,80914,80915,80916,80918],{},"刺激前庭覺輸入",[16,80917],{},"\n關閉或者干擾視覺和本體覺（閉眼+踩泡棉）",[12,80920,80921,80922,80924,80925,80927],{},"內耳石刺激（內耳石管垂直）",[16,80923],{},"\n訓練方式有坐到站，坐在彈力球上彈跳，彈簧床上彈跳",[16,80926],{},"\n再困難就閉眼做上面的",[36,80929],{},[12,80931,80932,80933,80935,80936,80940,80941,80943],{},"那這三種感覺輸入，會隨著年齡不斷的變弱，最後在老人皆段變成失能",[16,80934],{},"\n這時我們會教導他們用",[75,80937,80939],{"className":80938},[173],"輔具","，尤其是帶輪子的助行器",[16,80942],{},"\n（雖然帶輪子的助行器比較穩，也不用花力提起來，但也會限制上肢軀幹動作，環境也有限制）",[12,80945,80946,80947,80949],{},"接著是重心控制，強調了COG的部份，主要就是看這三個",[16,80948],{},"\n中線位置（三個維度：左右、前後、旋轉）；中軸旋轉（維持中線下旋轉）；動態偏移（遠離中線再回到中線不失去平衡）",[12,80951,80952,80953,80955],{},"訓練坐姿平衡（包括身體前後左右傾斜）",[16,80954],{},"\n漸進訓練：手部（雙手>單手>放開雙手）；坐的平面（坐穩>搖版\u002F彈力球）；腳部（雙腳>翹腳>雙腳離地）",[12,80957,80958,80959,80961],{},"訓練站姿平衡（初期是硬地小幅度作緩慢的重心轉移）",[16,80960],{},"\n漸進訓練：參數改變（幅度速度變快）；加入上肢動作；BOS減少；站泡棉、閉眼、轉頭",[12,80963,80964,80965,80967],{},"訓練站到坐（傾向讓壞測承重，所以好腳向前伸）（要先學會側向重量轉移：畢竟坐姿BOS左右比前後要大）",[16,80966],{},"\n漸進訓練：椅子調低；移除扶手；換成沙發；閉眼",[36,80969],{},[12,80971,80972],{},"接下來提到踝、髖、跨步策略的訓練方式，那目的是為了讓意志變成自動動作（不用在思考）",[23,80974,80975,80985,80998],{},[26,80976,80977,80978,80982,80984],{},"踝策略——主要記得 踝策略 ",[75,80979,80981],{"className":80980},[79,78],"pelvis head同方向",[16,80983],{},"\n訓練方式：背靠牆，前方有桌子支撐，練習前後不去碰到牆或桌子的重心轉移",[26,80986,80987,80988,80992,80994,80995,80997],{},"髖策略——主要記得 髖策略 ",[75,80989,80991],{"className":80990},[79,78],"pelvis head反方向",[16,80993],{},"\n訓練方式：分成兩個場景，窄平面：平衡桿、馬路邊緣；不穩平面：泡棉、搖板",[16,80996],{},"\n一樣是跟踝策略一樣在前後左右的搖晃",[26,80999,81000,81001,81005,81007,81008,81010,81011,81013,81014,81018],{},"跨步策略——前後左右，而且",[75,81002,81004],{"className":81003},[78,79],"好腳壞腳都要練習跨步",[16,81006],{},"\n訓練方式：場地可以在樓梯或者不衡桿上練",[16,81009],{},"\n難一點就是加障礙物例如地上放雜物，或者跨過繩子",[16,81012],{},"\n那在練的時候要給的",[75,81015,81017],{"className":81016},[79],"干擾要大過髖策略","，那告不告知則比較隨意",[12,81020,81021,81022,81024,81025,81027],{},"而後就是步態訓練，那主要也是向前向後跟側走",[16,81023],{},"\n那漸進的方式也就只是BOS的變小",[16,81026],{},"\n那向後走的話，畢竟眼睛沒辦法看到後面，那就在患者背跟將中間放一顆彈力球，讓他貼著支撐安心，同時訓練向後邁步",[36,81029],{},[12,81031,81032,81033,81035,81036,81039,81040,81042,81043,81045,81046,81050,81052,81053,81056,81057,81059,81060],{},"接著是其他方面的措施，那首先是教導患者怎麼去解決跌倒之前之後的事",[16,81034],{},"\n平衡有永久性問題的話，那先從",[75,81037,4402],{"className":81038},[173],"說起",[16,81041],{},"\n例如，危險高難不穩的任務（換燈泡），環境（昏暗狹窄有雜物的路）就都不要去做了",[16,81044],{},"\n然後",[75,81047,81049],{"className":81048},[79],"教導利用輔具",[16,81051],{},"\n家中的",[75,81054,4846],{"className":81055},[79],"也是必須的",[16,81058],{},"\n最後就是",[75,81061,81063],{"className":81062},[79,78],"教導患者跌倒之後，怎麼解決",[12,81065,81066,81067,81071,81072,81076,81077,81081],{},"居家復健也是必須的，而且要記住",[75,81068,81070],{"className":81069},[79],"安全第一","，在做",[75,81073,81075],{"className":81074},[79],"不穩定的動作","也要有人",[75,81078,81080],{"className":81079},[79,78],"監管","下才去做",[12,81083,81084,81085,81087,81088,81092,81094,81095,81099],{},"然後就是雙重任務訓練，畢竟作為一個人，不可能在拿東西或者想其他東西時停下腳",[16,81086],{},"\n但平衡有問題的人確實假如分開注意力就會",[75,81089,81091],{"className":81090},[79],"增加跌倒風險",[16,81093],{},"\n那我們要訓練讓他",[75,81096,81098],{"className":81097},[79],"邊處理其他事邊走路","，簡單的做法就是在走路的過程中做100-7的計算這的讓他分心",[12,81101,81102,81103],{},"關於介入的指南，其實也就是練平衡練步態練肌肉力量，而且要",[75,81104,81106],{"className":81105},[79,78],"高強度高頻率且長時間訓練",{"title":180,"searchDepth":181,"depth":181,"links":81108},[],"2026-04-17","復健 只寫老師有註解的 不夠時間了啦","\u002Fimages\u002Fuploads\u002F1776484659450-EfoK0qsU0AEHyqr.webp",{},"\u002Fblog\u002FBalance（五）復健",{"title":80782,"description":81110},{"loc":81113},"blog\u002FBalance（五）復健","UBQ8f_CPfuFn7xarp-qKnxtxq0L7iYUSaPCQxjkBtgg",{"id":81119,"title":81120,"author":7,"body":81121,"date":81320,"description":81321,"draft":185,"edited_at":81109,"extension":186,"featured_image":81322,"meta":81323,"navigation":189,"path":81324,"pinned":185,"seo":81325,"sitemap":81326,"stem":81327,"tags":194,"__hash__":81328},"blog\u002Fblog\u002FBalance（四）評估.md","Balance（四）評估",{"type":9,"value":81122,"toc":81318},[81123,81149,81151,81154,81204,81206,81209,81278,81280,81302,81304,81307],[12,81124,81125,81126,81130,81131,4995,81135,81137,81138,81140,81141,81143,81144,81148],{},"平衡評估呢，",[75,81127,81129],{"className":81128},[5387],"沒有任何一個測試","是可以概括",[75,81132,81134],{"className":81133},[78,79],"全部平衡組成部份",[16,81136],{},"\n而目前只有BESTest比較完整的從損傷角度切入去看病人的平衡會有哪些問題，耗時較長",[16,81139],{},"\n假如平衡測試測出來有問題，不一定是因為感覺系統(例如前庭)有問題，肌肉無力也會導致誤判",[16,81142],{},"\n所以額外的也會去測",[75,81145,81147],{"className":81146},[78],"ROM、肌力、感覺統合、中樞系統","等功能",[36,81150],{},[12,81152,81153],{},"現在來講靜態站立的評估",[23,81155,81156,81159,81168,81171,81174,81183,81186],{},[26,81157,81158],{},"平靜站立——患者站好，可以對他做或者不做干援",[26,81160,81161,81162,81164,81165,81167],{},"Romberg test——雙手胸前交叉，雙腳站好，開眼先測20~30秒，再閉眼測20~30秒（關閉視覺）",[16,81163],{},"\n出現跨步反應、過度搖晃、失去平衡即屬異常",[16,81166],{},"\n假如開眼正常閉眼出現不穩，就代表脊髓後柱受損或者運動失調",[26,81169,81170],{},"Sharpened Romberg test——用腳跟碰腳尖方式，先張眼再閉眼，每次60秒，測四次",[26,81172,81173],{},"One-legged stance test——雙手抱胸，一腳knee flex 90度，一腳30秒，每隻腳都要五次",[26,81175,81176,81177,81179,81180,81182],{},"Push-and-Release test(測自動姿勢反應)——那他分成測非跨步與測跨步兩種",[16,81178],{},"\n非跨步——PT手放肩處做推放，觀察到pt腳跟或腳趾離地，代表LOS已到極限，然後突然放手，去看他平衡狀況如何",[16,81181],{},"\n跨步——讓pt重量靠在我們手上，到達LOS極限，突然放手，去看他平衡狀況如何",[26,81184,81185],{},"Nudge or push test(輕推或推)——可以提醒，也可以不提醒（預期跟非預期），去看pt的平衡狀態跟平衡策略",[26,81187,81188,81189,81191,81192,81194,81195,81197,81198,81200,81201,81203],{},"Functional reach test(對 又是他)——前提就是，能站好30秒以上，shoulder flex90度，握拳，前向傾且不能有平衡反應",[16,81190],{},"\n以下為標準數據",[16,81193],{},"\n20~40y\u002Fo 14~17inch",[16,81196],{},"\n41~69y\u002Fo 13~16inch",[16,81199],{},"\n70~87y\u002Fo 10~13inch",[16,81202],{},"\n誤差在2inch，且進步2inch以內不算進步",[36,81205],{},[12,81207,81208],{},"接下來是測感覺輸入方式",[23,81210,81211,81266],{},[26,81212,81213,81214,81216,81217,81219,81220,81222,81223,81225,81226,81228,81229,81231,81232,81234,81235,81237,81239,81240,81242,81243,81245,81247,81248,81250,81252,81253,81255,81257,81258,81260,81262,81263,81265],{},"Sensory Organization Test——這個就是逐漸關閉和擾亂感覺輸入，去評估前庭覺的方法",[16,81215],{},"\n那首先去看這六種的感覺輸入不同",[16,81218],{},"\nCondition 1 —— 視覺 本體覺 前庭覺正常給予",[16,81221],{},"\nCondition 2 —— 本體覺 前庭覺正常給予，視覺關閉",[16,81224],{},"\nCondition 3 —— 本體覺 前庭覺正常給予，視覺亂給予",[16,81227],{},"\nCondition 4 —— 視覺 前庭覺正常給予，本體覺亂給予",[16,81230],{},"\nCondition 5 —— 前庭覺正常給予，視覺關閉，本體覺亂給予",[16,81233],{},"\nCondition 6 —— 前庭覺正常給予，視覺 本體覺亂給予",[16,81236],{},[16,81238],{},"\n那講完這六種，接下來分別說明這六種如何分別配對比較",[16,81241],{},"\n首先是C1——因為三個感覺輸入都正常給，是作為標準",[16,81244],{},[16,81246],{},"\nC1vsC2——變成依賴本體覺，假如不穩，代表本體覺出問題",[16,81249],{},[16,81251],{},"\nC1vsC4——依視覺，假如不穩，代表視覺出問題",[16,81254],{},[16,81256],{},"\nC1vsC5——依賴前庭覺，假如不穩，前庭覺出問題",[16,81259],{},[16,81261],{},"\nC3+C6vsC2+C5——測試中樞判讀能力，因為都給予錯誤的東西，這時更依賴中樞來分辦該不該抑制或選擇",[16,81264],{},"\n假如CNS損傷病人在這測試就會錯亂",[26,81267,81268,81269,81271,81272,81274,81275,81277],{},"Clinical test for sensory interaction on balance(剛剛的便宜替代版)",[16,81270],{},"\npt雙腳站立手抱胸，在剛剛的六種Condition都做一次測試，各三十秒，有五次機會",[16,81273],{},"\n評分：1最小搖晃；2輕度搖晃；3中度搖晃；4跌倒",[16,81276],{},"\n正常人，六個測試都能維持30秒不跌倒",[36,81279],{},[12,81281,81282,81283,81285,81286,81290,81291,81293,81294,81296,81297,81299,81301],{},"測走路的有Time get up and go test和沒有計時版本（因為太多了 老師沒做註解我就先不寫了 讀不完啦～）",[16,81284],{},"\n測試方法——先坐在",[75,81287,81289],{"className":81288},[173,78],"堅固的椅子","上>先站起來並維持站姿>正常步速走3m>轉身>走回椅子>轉身坐下",[16,81292],{},"\n不計時版本評分——1正常；2極輕度異常；3輕度異常；4中度異常；5重度異常或跌倒",[16,81295],{},"\n計時版本評分——10正常成人；11~20正常衰弱老人或失能的人；>30功能活動有限制，有跌倒風險的人",[16,81298],{},[16,81300],{},"\nUmphred 版本: 年輕人5~7秒,低度跌倒風險老人:7~9秒,中度跌倒風險老人:10~12秒,高中度跌倒風險老人:>13秒",[36,81303],{},[12,81305,81306],{},"跳跳跳跳跳 從54頁跳到74頁的——雙重任務測試",[12,81308,81309,81310,81314,81315,81317],{},"主要就是看注意力被分散之後，對於pt來說",[75,81311,81313],{"className":81312},[78,79],"分散注意力可能會導致不穩","（評估認知與姿勢控制能力）",[16,81316],{},"\n那最簡單的做法就是在pt走路時叫他同時算數，通常都會叫他100-7再減7減下去",{"title":180,"searchDepth":181,"depth":181,"links":81319},[],"2026-04-16","評估（只有空寫老師註解）（悲 看以後有沒有空補上吧 東西好多哦","\u002Fimages\u002Fuploads\u002F1776398601020-71588436_p0_master1200.webp",{},"\u002Fblog\u002FBalance（四）評估",{"title":81120,"description":81321},{"loc":81324},"blog\u002FBalance（四）評估","eisGZc7sSRYWqYPdr0kV-Mzvk0dDSdfDNlHLskaM3Mo",{"id":81330,"title":81331,"author":7,"body":81332,"date":81622,"description":81623,"draft":185,"edited_at":81320,"extension":186,"featured_image":81624,"meta":81625,"navigation":189,"path":81626,"pinned":185,"seo":81627,"sitemap":81628,"stem":81629,"tags":194,"__hash__":81630},"blog\u002Fblog\u002FBalance（三）.md","Balance（三）",{"type":9,"value":81333,"toc":81620},[81334,81355,81379,81382,81384,81419,81432,81435,81438,81468,81470,81491,81494,81541,81544,81555,81558,81560,81563,81574,81576,81593],[12,81335,81336,81337,81341,81342,81346,81347,81351,81352,81354],{},"那中樞系統他會比較",[75,81338,81340],{"className":81339},[78,173],"左右兩邊的三套不同感覺訊息","輸入",[75,81343,81345],{"className":81344},[79,78],"是否一致","，假如有衝突，將會根據",[75,81348,81350],{"className":81349},[78392,78],"動作需求","去選擇正確的感覺輸入",[16,81353],{},"\n那單單只有一個感觀去給予訊息是不夠的",[23,81356,81357,81365,81372],{},[26,81358,81359,81360],{},"單純的本體感沒辦法區分",[75,81361,81364],{"className":81362},[78,81363],"blue-3","身體傾斜還是地面在傾斜",[26,81366,81367,81368],{},"單純的視覺也沒辦法區分是",[75,81369,81371],{"className":81370},[78,79],"人在動還是環境在動",[26,81373,81374,81375],{},"單純的前庭覺只能知道頭在動，但不知道是因為",[75,81376,81378],{"className":81377},[78,78392],"脖子轉還是軀幹轉",[12,81380,81381],{},"所以就需要大腦做多重感覺比重調整，而前庭覺也會被作為另外兩個輸入的參考指標",[36,81383],{},[12,81385,81386,81387,81389,81390,81394,81395,81399,81400,81402,81403,81407,81408,81410,81411,81415,81416,81418],{},"那動作計劃會因應個人與任務做調整",[16,81388],{},"\n意志的動作由",[75,81391,81393],{"className":81392},[78,79],"意圖，注意力，動機","而定；而",[75,81396,81398],{"className":81397},[173,78],"反射","也可以與意志做結合，例如VOR",[16,81401],{},"\n那讓任務最佳化還是得靠",[75,81404,81406],{"className":81405},[79,78],"經驗知識","，包括了動作本身，任務內容，環境因素",[16,81409],{},"\n而在計劃制定好後，除了會發到周邊動作系統上，還會",[75,81412,81414],{"className":81413},[79,78],"傳給小腦","，讓小腦根據意圖與動作去做微調修正",[16,81417],{},"\n那中樞系統出問題的病人，那在執行與調控上面能力較差，例如stroke, Parkinson disease, ataxia等",[12,81420,81421,81422,81426,81427,81431],{},"剛講完中樞 現在講周邊運動執行部份，最基本需要",[75,81423,81425],{"className":81424},[78,173],"肢體驅幹眼睛","這三樣的",[75,81428,81430],{"className":81429},[78,173],"肌肉耐力ROM","都要完整",[12,81433,81434],{},"其他系統的影響，包括了注意力，認知，記憶，情緒等",[12,81436,81437],{},"身體會透過感覺回饋與練習，不斷的調整，來促進平衡能力",[12,81439,81440,81441,81445,81446,81450,81451,81453,81454,81458,81459,81463,81464],{},"前庭的兩大反射VOR以及VSR，這兩個支撐起",[75,81442,81444],{"className":81443},[78,173],"眼頭身體","對於",[75,81447,81449],{"className":81448},[78,173],"自身以及環境的方位","感知",[16,81452],{},"\n尤其是VOR，頭在轉，眼會為了",[75,81455,81457],{"className":81456},[78,173],"不影響注視","而",[75,81460,81462],{"className":81461},[78,79],"反向方的等速轉","，達成",[75,81465,81467],{"className":81466},[78392,78],"凝視穩定",[36,81469],{},[12,81471,81472,81473,81475,81476,81480,81481,81485,81486,81490],{},"那假如遇到非預期的干擾後，身體會做出非意志動作，干擾越強，反應就越強（平衡策略等下會細說）",[16,81474],{},"\n這些反應並",[75,81477,81479],{"className":81478},[78,79],"不單獨出現","，方向並",[75,81482,81484],{"className":81483},[78,79],"不單一","，會根據當下",[75,81487,81489],{"className":81488},[78,79],"環境任務個人","等因素採取適合的反應",[12,81492,81493],{},"那先來介紹各種策略",[23,81495,81496,81511,81525,81533],{},[26,81497,81498,81499,81501,81502,81506,81507],{},"踝策略——干擾較少、慢、範圍接近身體線、地面平穩時比較適用",[16,81500],{},"\n肌肉由",[75,81503,81505],{"className":81504},[78,173],"遠端到近端","收縮，相對於ankle作擺盪，",[75,81508,81510],{"className":81509},[78,173],"head和hip方向相同",[26,81512,81513,81514,81501,81516,81520,81521],{},"髖策略——擺盪較大、快、範圍接近LOS、地面窄而不穩時使用",[16,81515],{},[75,81517,81519],{"className":81518},[78,173],"近端到遠端","收縮，相較pelvis trunk作擺盪，",[75,81522,81524],{"className":81523},[79,78],"head和hip方向相反",[26,81526,81527,81528,81532],{},"懸吊策略——借由",[75,81529,81531],{"className":81530},[78,173],"降低重心","方式如下肢彎曲，微蹲等",[26,81534,81535,81536,81540],{},"跨步策略——比原本的",[75,81537,81539],{"className":81538},[78,173],"BOS還要大","的方式來平衡（其中一腳跨出一步），抑或者伸出手抓固定物，以達到建立新的BOS",[12,81542,81543],{},"假如前庭有問題的病人，會有以下特徵",[23,81545,81546,81549,81552],{},[26,81547,81548],{},"較常使用踝策略——因為head 與腳同向，比較能降低感覺輸入衝突",[26,81550,81551],{},"使用髖策略——會有調整或限制",[26,81553,81554],{},"使用跨步策略——較常出現在自體旋轉或在踩在柔軟的地面上",[12,81556,81557],{},"那假如本體覺有問題的病人，則會使用髖策略",[36,81559],{},[12,81561,81562],{},"剛講完站姿的平衡反應，接下來講坐姿的",[23,81564,81565,81568,81571],{},[26,81566,81567],{},"向後倒——hip flexor最先收縮，接著是腹直肌跟頸屈肌",[26,81569,81570],{},"向前倒——hip extensor最先收縮，接著是trunk和neck extensor",[26,81572,81573],{},"左右倒——主要由hip abd\u002Fadd 與腰方肌收縮",[36,81575],{},[12,81577,81578,81579,81581,81582,4995,81586,81588,81589],{},"接下來講預期姿勢反應",[16,81580],{},"\n最主要就是干擾前並結合經驗，判斷當下的情況，採取最適合的反應，是",[75,81583,81585],{"className":81584},[79],"自主",[16,81587],{},"\n那假如失去了預期反應，那就只能靠",[75,81590,81592],{"className":81591},[78,79],"反應來防止跌倒",[12,81594,81595,81596,81598,81599,4995,81603,81605,81606,81610,81611,81615,81616],{},"接著講意志的姿勢動作",[16,81597],{},"\n這時重量的轉移會因為想做的動作產生改變，是",[75,81600,81602],{"className":81601},[173],"自發",[16,81604],{},"\n內容包括從",[75,81607,81609],{"className":81608},[78,173],"簡單的重量轉移","，再到",[75,81612,81614],{"className":81613},[78,173],"複雜的平衡技巧","；一樣可以",[75,81617,81619],{"className":81618},[173,78],"靠經驗去優化",{"title":180,"searchDepth":181,"depth":181,"links":81621},[],"2026-04-15","中樞神經感覺~反應姿勢","\u002Fimages\u002Fuploads\u002F1776329120376-Ef-Wny_UcAAHv_H.webp",{},"\u002Fblog\u002FBalance（三）",{"title":81331,"description":81623},{"loc":81626},"blog\u002FBalance（三）","f_KzOAqZpVIgVEBffuYwNs7BRwNuOYO19OWt55GnDT8",{"id":81632,"title":81633,"author":7,"body":81634,"date":81987,"description":81988,"draft":185,"edited_at":81987,"extension":186,"featured_image":81989,"meta":81990,"navigation":189,"path":81991,"pinned":185,"seo":81992,"sitemap":81993,"stem":81994,"tags":194,"__hash__":81995},"blog\u002Fblog\u002FCP—Rood-&-Bobath.md","CP—Rood & Bobath（一）",{"type":9,"value":81635,"toc":81985},[81636,81639,81642,81658,81675,81707,81710,81756,81758,81761,81836,81844,81925,81927,81930],[12,81637,81638],{},"(就先只寫余老師有在上課做註記的版書，讀不完啦)",[12,81640,81641],{},"1940前肌能教再育>1950年後神經成熟理論>1990年動態系統理論",[12,81643,81644,81645,81647,81648,81652,81653,81657],{},"接下來講Bobath認為的治療核心，先將高張抑制，再教導動作（1reflex,2reaction）",[16,81646],{},"\n手法有RIP(",[75,81649,81651],{"className":81650},[78,173],"異常reflex抑制",")，接著誘發動作(",[75,81654,81656],{"className":81655},[79,78],"翻正反應,平衡反應)","，再加一些本體覺和觸覺輸入",[12,81659,81660,81661,81663,81664,81666,81667,24802,81671],{},"那假如抑制了張力但還沒有正常動作出來，我們會使用key point control來誘發動作",[16,81662],{},"\n借由近端的點(中軸附近例如頭肩髖)，來引導做翻正反應，同時抑制異常反",[16,81665],{},"\n主要是",[75,81668,81670],{"className":81669},[78,79],"打破錯的感覺運動經驗",[75,81672,81674],{"className":81673},[78,78392],"重新教寶寶正確的動作形態",[12,81676,81677,81678,81680,81681,81683,81684,81686,81687,81691,81692,81694,81695,81697,81698,81702,81703],{},"現在稍微說一下RIP，像剛說的為了抑制異常張力或反射來誘發出動作出來",[16,81679],{},"\n例如寶寶想向前reach，但flexor tone 過大做不出來，我們就做RIP方式抑制張力，讓寶寶去完成reach",[16,81682],{},"\n那像是這裡提到的flexor tone過大，我們就幫他做extension",[16,81685],{},"\n在這過程中會利用到GTO的這一個機制，原理是擺位在tone過大的",[75,81688,81690],{"className":81689},[78392],"反方向","，誘發GTO關閉tone",[16,81693],{},"\n(著重在擺位)",[16,81696],{},"\n現在的RIP是改為",[75,81699,81701],{"className":81700},[173],"partial RIP","，只利用",[75,81704,81706],{"className":81705},[78,79],"Key point來達到抑制與誘發",[12,81708,81709],{},"那RIP的注意事項有如下",[23,81711,81712,81724,81736,81744,81747,81750],{},[26,81713,81714,81715,81719,81720],{},"RIP只做在",[75,81716,81718],{"className":81717},[78,79],"有痙攣","的地方，",[75,81721,81723],{"className":81722},[173],"hypotone則不需要做",[26,81725,81726,81727,81731,81732],{},"力度需",[75,81728,81730],{"className":81729},[78392,78],"緩慢施加","，因為痙攣的部位容易",[75,81733,81735],{"className":81734},[79],"contracture",[26,81737,81738,81739,81743],{},"從近端開始做，並且做",[75,81740,81742],{"className":81741},[5387,78],"旋轉以及Protraction(shoulder)是最容易打破張力","的手段",[26,81745,81746],{},"在拉開後，慢慢的放開手，讓他自己用力維持這個姿勢，誘發他做出正常的功能性動作",[26,81748,81749],{},"新的Bobath目的是做ADL動作",[26,81751,81752,81753,81755],{},"也不會在最強的痙攣部位開始做去，就像是Extensor tone過大配合有TLR 發生",[16,81754],{},"\n我們可以先將寶寶從Supine>side-lying>prone，將TLR關閉，讓張力消掉，再讓寶寶回到supine",[36,81757],{},[12,81759,81760],{},"接下來講抑制Extensor的技巧，先來講第一個",[3603,81762,81763,81766,81772,81796,81804,81825,81833],{},[26,81764,81765],{},"首先PT要坐在有靠背的椅子上保護好PT的背",[26,81767,81768,81769,81771],{},"將小孩的身體放在PT大腿上，腿面向PT，呈現total flexion姿勢",[16,81770],{},"\n一來可以借這個姿勢拉到hamstring，二來寶寶為了想看PT也會嘗試抬頭，三來可以將寶寶上肢放到他胸前",[26,81773,81774,81775,81777,81778,81782,81783,81787,81791,81792,2589],{},"那痙攣型的寶寶co-contraction(共同收縮)都比較強，所以旋轉能力較差",[16,81776],{},"\n可以在剛提到的姿勢下順便訓練旋轉，PT一手將",[75,81779,81781],{"className":81780},[78,79],"小孩的腳轉","到一側，另一手轉動寶寶的",[75,81784,81786],{"className":81785},[78,5387],"身體",[75,81788,81790],{"className":81789},[78],"到對側","（",[75,81793,81795],{"className":81794},[5387],"是身體，不是頭",[26,81797,81798,81799,81803],{},"而因為extensor張力太大，肩胛骨會retraction，所以我們可以兩手放在小孩的",[75,81800,81802],{"className":81801},[79,78],"肩胛骨內緣","後主推做protraction",[26,81805,81806,81807,81809,81810,81814,81815,81817,81818,81822,81824],{},"那小孩也因為extensor張力太大，可能沒有意識到有手的存在",[16,81808],{},"\n我們就得將他的手帶到胸前",[75,81811,81813],{"className":81812},[79,78],"進入視野範圍","，讓他開始吃或者自己的手",[16,81816],{},"\n而有些小孩會緊張會保持抓握姿勢，我們可以從",[75,81819,81821],{"className":81820},[78,79],"大拇指開始扳開慢慢張手打開",[16,81823],{},"\n讓他去做手碰手、手碰嘴、手碰腳、嘴碰嘴",[26,81826,81827,81828,81832],{},"而在抑制的過程中也可以做一些",[75,81829,81831],{"className":81830},[79,78],"誘發flexion動作","的手法，訓練屈肌，這樣也能降低extensor tone",[26,81834,81835],{},"習慣了以上姿勢後，就可以放寶寶在地上自己玩了",[12,81837,81838,81839,81843],{},"接著來講第二個，這個是",[75,81840,81842],{"className":81841},[5387,78],"利用滾筒","的方式，結合了誘發動作的方式",[3603,81845,81846,81872,81901],{},[26,81847,81848,81849,81853,81855,81856,81860,81862,81863,81867,81868,7963],{},"首先讓小孩在滾筒上side-lying，",[75,81850,81852],{"className":81851},[78,173],"手抬高將滾筒卡在腋下",[16,81854],{},"\n這時候PT要對小孩上方的髖做",[75,81857,81859],{"className":81858},[78,79],"Abd，extension和external rotation",[16,81861],{},"\n由於有一些痙攣的下肢會出現hemiflexion，所以在擺位仍然會將hip放在",[75,81864,81866],{"className":81865},[79,78],"extension","姿勢(",[75,81869,81871],{"className":81870},[78],"正常站立角度",[26,81873,81874,81875,81877,81878,81880,81881,81883,81884,81888,81892,81896,81897,2589],{},"接著按照這個上面這個姿勢教導旋轉，首先一手卡在滾筒側腋下，一手放在hip做固定",[16,81876],{},"\n然後將卡在腋下的手慢慢帶動shoulder，最後處在prone姿，而這時hip仍然被固定，所以呈軀幹扭轉姿勢",[16,81879],{},"\n這時小孩會想要翻過來，我們會慢慢的引導他的hip翻到prone姿",[16,81882],{},"\n而這時候會想要去訓練他的抬頭（雖然",[75,81885,81887],{"className":81886},[78,79],"抬頭是extension姿勢",[75,81889,81891],{"className":81890},[78],"但這個跟",[75,81893,81895],{"className":81894},[78,78392],"extensor muscle tone不一樣","，還是得教導",[75,81898,81900],{"className":81899},[79],"正確方式",[26,81902,81903,81904,81906,81907,81909,81910,81912,81913,81917,81919,81920,81924],{},"剛剛講的是從side-lying到prone，現在講反過來的從prone到side-lying",[16,81905],{},"\n一手放在pelvis，一手固定shoulder，",[16,81908],{},"\n這次先將pelvis，翻回side-lying，然後慢慢協助將shoulder帶到side-lying",[16,81911],{},"\n（而假如我們先將shoulder拉到side-lying，正常人的話也會因為被從往後拉動，肩胛骨也會被拉到",[75,81914,81916],{"className":81915},[78,79],"retraction",[16,81918],{},"\n對於CP小兒而言",[75,81921,81923],{"className":81922},[79,78],"誘發出extensor tone","，所以要先將pelvis帶走）",[36,81926],{},[12,81928,81929],{},"然後來介紹到抑制Flexor的技巧，核心是利用重力去抑制",[3603,81931,81932,81952,81967,81982],{},[26,81933,81934,81935,81939,81940,81944,81946,81947,81951],{},"讓小孩supine在球上，這過程要",[75,81936,81938],{"className":81937},[79,78],"緩慢以及穩","，不要讓小孩感到害怕，假如",[75,81941,81943],{"className":81942},[5387,78],"小孩恐懼就不要做這個",[16,81945],{},"\n因為flexor tone 高的小孩比較少動，所以",[75,81948,81950],{"className":81949},[78,79],"前庭覺十分敏感","，所以初期要多給支持，適應之後再將手慢慢放到遠端支持",[26,81953,81954,81955,81959,81960,81964,81966],{},"而通常都會在這支勢下扶穩小孩的shoulder，前後左右的輕微的搖擺，讓小孩",[75,81956,81958],{"className":81957},[173,78],"逐漸放鬆","，同時也能",[75,81961,81963],{"className":81962},[78,79],"靠重力牽拉flexor肌肉",[16,81965],{},"\n再進階就遠端例如hip去做同樣的事",[26,81968,81969,81970,81972,81973,81977,81978],{},"進一步的做法是，抓住小孩的小肢，將球往小孩的頭側推",[16,81971],{},"\n這樣因為滾動的原因變成頭反而靠近地面，呈",[75,81974,81976],{"className":81975},[173,78],"頭靠近地面","，這樣也能讓",[75,81979,81981],{"className":81980},[79,78],"extensor tone上升",[26,81983,81984],{},"還能結合旋轉去訓練，一樣也是抓住腳帶動軀幹做輕微旋轉，以降低張力",{"title":180,"searchDepth":181,"depth":181,"links":81986},[],"2026-04-14","主要是寫有特別註解的 到抑制flexor張力的部份","\u002Fimages\u002Fuploads\u002F1776176663348-EbXNw8qWkAMAVcN.webp",{},"\u002Fblog\u002FCP—Rood-&-Bobath",{"title":81633,"description":81988},{"loc":81991},"blog\u002FCP—Rood-&-Bobath","hbxSp87ZZts-QV-3eWuNwBK9q66v8uar73myHqp0-sQ",{"id":81997,"title":81998,"author":7,"body":81999,"date":81987,"description":82109,"draft":185,"edited_at":81622,"extension":186,"featured_image":82110,"meta":82111,"navigation":189,"path":82112,"pinned":185,"seo":82113,"sitemap":82114,"stem":82115,"tags":194,"__hash__":82116},"blog\u002Fblog\u002FCP—Rood-&-Bobath（二）.md","CP—Rood & Bobath（二）",{"type":9,"value":82000,"toc":82107},[82001,82004,82010,82015,82018],[12,82002,82003],{},"那剛剛一篇講的都Bobath的手法，那現在講Rood的",[12,82005,82006,82007,82009],{},"那Rood 提到說，人類有發展次序，mobility>stability>mobility on stability>skill movement",[16,82008],{},"\n也強調了可以利用感覺輸入誘發動作",[12,82011,82012],{},[1930,82013],{"alt":1932,"src":82014},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776217678571-Screenshot_2026-04-15_094736.png",[12,82016,82017],{},"那感覺刺激的部份根據強度及頻率而有這四種變化",[23,82019,82020,82026,82065,82071],{},[26,82021,82022,82023,82025],{},"快速且短暫的刺激",[16,82024],{},"\n可以用心智狀態提高，全身muscle tone 增高，反應力上升（喚醒）",[26,82027,82028,82029,82031,82032,82034,82035,82037,82039,82040,82044,82045,82047,82048,82050,82051,82055,82057,82061,82062,82064],{},"快速且重複的刺激",[16,82030],{},"\n中度，可維持心智，身體，注意力等反應，可利用前庭與本體覺持續被刺激使身體保持在適當的姿勢",[16,82033],{},"\n利用Vibration(按摩棒)，引起肌肉或肌腱反射（誘發），也可以有減敏效果",[16,82036],{},[16,82038],{},"\n這邊提到一個叫TLI的東西，用處是維持全身伸肌肌群反應，從",[75,82041,82043],{"className":82042},[79,78],"頸部開始訓練","(inversion在GA26~29就出現)",[16,82046],{},"\n操作方式為，將小孩腹部放到球上(prone姿)，並固定他的骨盤，進行前往左右搖晃，呈頭下腳上姿勢",[16,82049],{},"\n治療師採坐姿，而且不是用手在帶動，是",[75,82052,82054],{"className":82053},[78,79],"利用治療師身體帶動",[16,82056],{},[75,82058,82060],{"className":82059},[5387],"注意","：要給小孩安全感，inversion不能過久，保持頭部的正確位置",[16,82063],{},"\n避免在頭的背部(後方)施加壓力，小孩手要自由活動但不能碰到地板，持續觀察小孩反應",[26,82066,82067,82068,82070],{},"緩慢且規律重覆的刺激",[16,82069],{},"\n通常用來減敏",[26,82072,82073,82074,82076,82077,82079,82081,82082,82086,82087,82091,82092,82094,82096,82097,82101,82102,82106],{},"持續性的刺激",[16,82075],{},"\n強調翻正反應，例如抗重力時，重力能夠對姿勢穩定的肌肉去做持續刺激",[16,82078],{},[16,82080],{},"\n對於",[75,82083,82085],{"className":82084},[78,79],"痙攣","的小孩，先從",[75,82088,82090],{"className":82089},[79,78],"不敏感的地方開始","，再往較敏感的位慢慢刺激，嘴巴也一樣",[16,82093],{},[16,82095],{},"\n但對於",[75,82098,82100],{"className":82099},[78,78392],"低張","的小孩，則相反，從",[75,82103,82105],{"className":82104},[78,78392],"最敏感的地方開始","刺激",{"title":180,"searchDepth":181,"depth":181,"links":82108},[],"Rood 到結尾","\u002Fimages\u002Fuploads\u002F1776222460274-Ee4ldTMUcAEiPzF.webp",{},"\u002Fblog\u002FCP—Rood-&-Bobath（二）",{"title":81998,"description":82109},{"loc":82112},"blog\u002FCP—Rood-&-Bobath（二）","56_IFu2CIimvGKbXPweMAs8MdVF91nlzrjRw9LcNlb0",{"id":82118,"title":82119,"author":7,"body":82120,"date":81987,"description":82282,"draft":185,"edited_at":81987,"extension":186,"featured_image":82283,"meta":82284,"navigation":189,"path":82285,"pinned":185,"seo":82286,"sitemap":82287,"stem":82288,"tags":194,"__hash__":82289},"blog\u002Fblog\u002FCP補充（一）.md","CP補充（一）",{"type":9,"value":82121,"toc":82280},[82122,82125,82128,82155,82157,82163,82186,82191,82234,82237,82240,82275],[12,82123,82124],{},"好多，先把老師特別叫我們看的先讀完好了，看後面還有沒有時間讀",[12,82126,82127],{},"第七頁的這一份作業，他把腦損傷的時間點分成三個並說明了原因",[3603,82129,82130,82136,82152],{},[26,82131,82132,82133,82135],{},"產前——先天性腦部畸型，血管問題（中腦動脈阻塞），母體攝入毒素",[16,82134],{},"\n母體感染（德國痲疹，巨細胞病毒，弓形蟲病），代謝障礙，基因綜合徵",[26,82137,82138,82139,82143,82145,82146,82148,82149,82151],{},"產中——主要為",[75,82140,82142],{"className":82141},[78,173],"胎兒中毒以及產時窒息",[16,82144],{},"\n血液樣本顯示為代謝性酸中毒，pH\u003C7以及鹼剩餘>12mmol\u002FL",[16,82147],{},"\n妊娠周數>34周，早期會出現中度或重度腦病變",[16,82150],{},"\n低血糖，未經治療的黃疽，嚴重的新生兒感染亦會導致CP",[26,82153,82154],{},"產後——感染，外傷，腦膜炎，敗血症",[36,82156],{},[12,82158,82159,82160,82162],{},"腦型麻痺類別——1（雖然總結有提到，但還是再寫一次好了）",[16,82161],{},"\n分成了",[3603,82164,82165],{},[26,82166,82167,82168,82170,82171,82173,82174,82176,82177,82179,82180,82182,82183,82185],{},"肢體受影響的分佈模式——",[16,82169],{},"\n雙側痲痺(腳比手影響嚴重)",[16,82172],{},"\n單側痲痺(單側受影響)",[16,82175],{},"\n四肢痲痺（手比腳嚴重）",[16,82178],{},"\n單肢痲痺",[16,82181],{},"\n三肢痲痺",[16,82184],{},"\n下肢痲痺",[12,82187,82188],{},[1930,82189],{"alt":1932,"src":82190},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776134449314-Screenshot_2026-04-14_104024.png",[3603,82192,82193],{},[26,82194,82195,82196,82198,82199,82201,82202,82204,82205,82209,82211,82212,82216,82217,82221,82222,82224,82225,82227,82228,82230,82231,82233],{},"根據張力與動作異常形態區分——",[16,82197],{},"\n痙攣型——佔比七成，傷到錐狀束，上肢Flexor tone強+pronation，下肢Extenor tone強",[16,82200],{},"\n徐動型——佔比一成五，傷到基底核，皮質無影響，所以智力正",[16,82203],{},"\n僵直型——傷到皮質，比",[75,82206,82208],{"className":82207},[78,79],"痙攣型張力還大",[16,82210],{},"\n運動失調型——佔比一成三，傷到小腦，",[75,82213,82215],{"className":82214},[78,79],"協調能力差","且尤其是",[75,82218,82220],{"className":82219},[78392],"下肢","特別明顯，步寬大，走路搖晃",[16,82223],{},"\n低張力型——為暫時性(CP無永久低張)，之後會發展成其他類型，CP，因為low tone，所以ROM大且無法抗重力",[16,82226],{},"\n顫抖型——較徐動型有節奏，在遠端顫抖",[16,82229],{},"\n混合型",[16,82232],{},"\n無法分類型",[12,82235,82236],{},"類別——2（P8）",[12,82238,82239],{},"他提到了按照ICF思路去想各個層級寶寶的問題",[3603,82241,82242,82245,82252,82259,82267],{},[26,82243,82244],{},"首先是健康問題，受了甚麼傷",[26,82246,82247,82248],{},"其次是受限，這個傷會讓寶寶",[75,82249,82251],{"className":82250},[78,81363],"做不了哪些動作",[26,82253,82254,82255],{},"再來是侷限，這些做不了的動作會讓寶寶",[75,82256,82258],{"className":82257},[78,79],"失去甚麼功能性",[26,82260,82261,82262,82266],{},"然後是環境因素，例如失去了這些功能性的話，與",[75,82263,82265],{"className":82264},[78,173],"環境的互動","有甚麼障礙",[26,82268,82269,82270,82274],{},"與環境因素平級的有個人因素，例如寶寶的年紀，性別，家庭背景；這些都會影響寶寶的例如",[75,82271,82273],{"className":82272},[78,78392],"動機","等因素",[12,82276,82277],{},[1930,82278],{"alt":1932,"src":82279},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776135419925-Screenshot_2026-04-14_105040.png",{"title":180,"searchDepth":181,"depth":181,"links":82281},[],"前面CP的大段文字英翻中","\u002Fimages\u002Fuploads\u002F1776137123982-e925b25f25c941afa92d440eefa3b271.webp",{},"\u002Fblog\u002FCP補充（一）",{"title":82119,"description":82282},{"loc":82285},"blog\u002FCP補充（一）","8cDECsO9Ab18dGecDdRUOUB4ETPPEAjd1AS3dqL2ii4",{"id":82291,"title":82292,"author":7,"body":82293,"date":81987,"description":82432,"draft":185,"edited_at":81987,"extension":186,"featured_image":82433,"meta":82434,"navigation":189,"path":82435,"pinned":185,"seo":82436,"sitemap":82437,"stem":82438,"tags":194,"__hash__":82439},"blog\u002Fblog\u002F小兒發展（五）.md","小兒發展（五）",{"type":9,"value":82294,"toc":82430},[82295,82301,82315,82318,82338,82340,82343,82372,82374,82377,82397,82399,82402],[12,82296,82297,82298,82300],{},"接下來講到一個叫皮亞傑的人，他提出了發展理論",[16,82299],{},"\n先來講他說的四大認知階段",[3603,82302,82303,82306,82309,82312],{},[26,82304,82305],{},"感覺運動期(0-24m)——假如有嚴重殘疾的小孩可能會永遠停留在這期",[26,82307,82308],{},"前運思期(2-7y\u002Fo)——他們容易被表像所騙，沒有抽象能力；所以借由具體的東西來讓小孩理解（過家家）",[26,82310,82311],{},"具體運思期(7-12y\u002Fo)——開始有邏輯能力，但只限定自己碰過見過的事物(經驗談)，也開始有抽象能力",[26,82313,82314],{},"形式運思期(12y\u002Fo-more)——具備抽象能力，不需要真的有經驗也能有一定的發想",[12,82316,82317],{},"接著是他提到的感覺運動期",[3603,82319,82320,82323,82326,82329,82332,82335],{},[26,82321,82322],{},"反射期(0-2m)——對，就一堆反射",[26,82324,82325],{},"初級循環反應期(3-5m)——摸索自己身體",[26,82327,82328],{},"次級循環反應期(6-9m)——會開始摸索物體（例如將玩具送嘴裡咬之類的）",[26,82330,82331],{},"兩個基模協調(9-12m)——透過兩種動作模式達成目的；理解因果",[26,82333,82334],{},"三級循環反應期(1-1.5y\u002Fo)——利用試錯將任務達成；例如用不同的聲音或動作引起照顧者反應",[26,82336,82337],{},"內化方案(1.5-2y\u002Fo)——不再試錯，而是結合經驗或觀察，再完成任務",[36,82339],{},[12,82341,82342],{},"然後是個人與社會發展",[3603,82344,82345,82348,82351,82354,82357,82360,82363,82366,82369],{},[26,82346,82347],{},"兩個月——會對別人笑",[26,82349,82350],{},"六個月——自己拿玩具玩",[26,82352,82353],{},"六到八個月——開始理解陌生，不讓陌生人抱",[26,82355,82356],{},"十個月——搖手bye bye（maybe是已經將動作與語言結合?）",[26,82358,82359],{},"一歲——拿杯子喝水",[26,82361,82362],{},"十五個月——拿湯匙、脫下外套、學刷牙",[26,82364,82365],{},"兩歲——會洗手",[26,82367,82368],{},"三歲——會穿脫鞋",[26,82370,82371],{},"四歲——穿衣服，扣鈕扣，刷牙",[36,82373],{},[12,82375,82376],{},"接下來講ASD的臨床徵象（即使動作發展正常，也會有語言delay）",[3603,82378,82379,82382,82385,82388,82391,82394],{},[26,82380,82381],{},"人際關係障礙——不能互動，缺乏眼神交流注視，不尋求撫慰，無法分享，無法參與遊戲",[26,82383,82384],{},"語言溝通障礙——語句單調，不太會說話，仿說，詞性錯亂，不正確肢體語言",[26,82386,82387],{},"特殊行為問題——對刺激反應異常，動作反覆，拒絕改變(習慣或者常規)，同一僵化固執行為",[26,82389,82390],{},"注意力不集中——無法持久學習",[26,82392,82393],{},"衝動——人際關係緊張",[26,82395,82396],{},"好動——無法遵守規矩，破壞力強",[36,82398],{},[12,82400,82401],{},"吞嚥異常的臨床徵象",[3603,82403,82404,82407,82410,82413,82415,82418,82421,82424,82427],{},[26,82405,82406],{},"反覆性肺炎",[26,82408,82409],{},"營養不良",[26,82411,82412],{},"抗拒食物",[26,82414,5119],{},[26,82416,82417],{},"口中有異物感",[26,82419,82420],{},"嗆咳",[26,82422,82423],{},"口中有異味",[26,82425,82426],{},"經營嘔吐",[26,82428,82429],{},"流口水",{"title":180,"searchDepth":181,"depth":181,"links":82431},[],"皮亞傑~結束","\u002Fimages\u002Fuploads\u002F1776132158232-tHG70Gg.webp",{},"\u002Fblog\u002F小兒發展（五）",{"title":82292,"description":82432},{"loc":82435},"blog\u002F小兒發展（五）","hzbfl2qpxvvTUq5Hn8aXvAKEpMa7oP19s8AD5xnz-1Q",{"id":82441,"title":82442,"author":6946,"body":82443,"date":83251,"description":180,"draft":185,"edited_at":74911,"extension":186,"featured_image":194,"meta":83252,"navigation":189,"path":83253,"pinned":185,"seo":83254,"sitemap":83255,"stem":83256,"tags":194,"__hash__":83257},"blog\u002Fblog\u002FNetwork-Communication-Chapter-4.md","Layers 3 & 4: Routing, TCP, and UDP",{"type":9,"value":82444,"toc":83216},[82445,82449,82459,82463,82466,82497,82501,82511,82531,82535,82538,82558,82562,82565,82591,82594,82597,82601,82604,82632,82636,82642,82656,82663,82667,82674,82694,82698,82701,82707,82713,82717,82723,82727,82755,82759,82765,82779,82788,82798,82802,82809,82812,82833,82842,82846,82852,82870,82873,82877,82880,82884,82914,82921,82925,82931,82951,82954,82958,82969,82973,82976,82980,82983,82989,82993,82996,83022,83026,83029,83033,83058,83062,83073,83076,83102,83106,83195,83197,83210,83213],[6964,82446,82448],{"id":82447},"introduction","Introduction",[12,82450,82451,82452,55606,82455,82458],{},"Have you ever wondered what actually happens when you open a website or send a message online? Behind the scenes, your data goes on quite a journey — hopping through routers, getting wrapped in headers, and relying on clever protocols to arrive at the right place. In this post, we'll walk through two critical layers of networking: the ",[106,82453,82454],{},"Network Layer",[106,82456,82457],{},"Transport Layer",". By the end, you'll understand how devices find each other, how data gets routed across the globe, and why some applications need guaranteed delivery while others are happy to \"wing it.\"",[6964,82460,82462],{"id":82461},"the-network-layer","The Network Layer",[12,82464,82465],{},"The network layer is what allows devices (your phone, your laptop, a server halfway around the world) to exchange data across interconnected networks. It handles four fundamental processes:",[3603,82467,82468,82474,82483,82491],{},[26,82469,82470,82473],{},[106,82471,82472],{},"Addressing end devices"," — Every device on a network needs a unique IP address so it can be identified. Think of it as a mailing address for your device.",[26,82475,82476,82478,82479,82482],{},[106,82477,22036],{}," — When data arrives from the transport layer above, the network layer wraps it with an IP header, creating what we call a ",[7289,82480,82481],{},"packet",". This header contains all the routing information needed to get the data where it's going.",[26,82484,82485,82487,82488,82490],{},[106,82486,74748],{}," — The network layer figures out ",[7289,82489,9518],{}," to get the packet from the source to the destination, potentially through many intermediate devices.",[26,82492,82493,82496],{},[106,82494,82495],{},"De-encapsulation"," — When the packet finally arrives at the destination, the IP header is stripped off and the data is passed back up to the transport layer.",[7166,82498,82500],{"id":82499},"the-nature-of-ip","The Nature of IP",[12,82502,82503,82504,7042,82507,82510],{},"The two most common network layer protocols are ",[106,82505,82506],{},"IPv4",[106,82508,82509],{},"IPv6",". Before we dive into the differences, let's understand the basic characteristics that both share:",[23,82512,82513,82519,82525],{},[26,82514,82515,82518],{},[106,82516,82517],{},"Connectionless"," — IP doesn't establish a connection before sending data. It's like dropping a letter in a mailbox — you don't call the recipient first to let them know it's coming.",[26,82520,82521,82524],{},[106,82522,82523],{},"Best Effort (Unreliable)"," — There's no guarantee your packet will arrive. The sender doesn't even know if the destination device is turned on. (Don't worry — reliability is handled at a higher layer!)",[26,82526,82527,82530],{},[106,82528,82529],{},"Media Independent"," — IP doesn't care whether your data travels over fiber optic cable, Wi-Fi, or even a satellite link. It works the same regardless of the physical medium.",[7166,82532,82534],{"id":82533},"ipv4-and-its-limitations","IPv4 and Its Limitations",[12,82536,82537],{},"IPv4 has served us well for decades, but it comes with some significant problems:",[23,82539,82540,82546,82552],{},[26,82541,82542,82545],{},[106,82543,82544],{},"Address depletion"," — IPv4 addresses are 32-bit, giving us roughly 4.3 billion unique addresses. That sounds like a lot, but with billions of devices connected today, we've essentially run out.",[26,82547,82548,82551],{},[106,82549,82550],{},"Lack of end-to-end connectivity"," — To cope with address shortages, we use Network Address Translation (NAT), which lets many devices share a single public IP. The downside? Devices behind NAT are hidden from the outside world, which breaks certain applications that need direct connections (like peer-to-peer services).",[26,82553,82554,82557],{},[106,82555,82556],{},"Increased network complexity"," — NAT adds extra processing, introduces latency, and makes troubleshooting harder.",[7166,82559,82561],{"id":82560},"ipv6-the-solution","IPv6: The Solution",[12,82563,82564],{},"IPv6 was designed to solve these problems:",[23,82566,82567,82573,82579,82585],{},[26,82568,82569,82572],{},[106,82570,82571],{},"Massive address space"," — IPv6 uses 128-bit addresses. To put that in perspective, that's roughly 340 undecillion addresses (3.4 × 10³⁸). Every grain of sand on Earth could have its own IP address — several times over.",[26,82574,82575,82578],{},[106,82576,82577],{},"Simplified header"," — The IPv6 header has fewer fields than IPv4, which means routers can process packets more efficiently.",[26,82580,82581,82584],{},[106,82582,82583],{},"No need for NAT"," — With so many addresses available, every device can have its own globally unique address. This restores true end-to-end connectivity.",[26,82586,82587,82590],{},[106,82588,82589],{},"Built-in security"," — IPv6 natively supports IPsec for authentication and encryption, whereas in IPv4, security features were added on as an afterthought.",[6964,82592,74748],{"id":82593},"routing",[12,82595,82596],{},"Now that we understand how devices are addressed, let's look at how packets actually find their way through the network.",[7166,82598,82600],{"id":82599},"where-can-a-host-send-packets","Where Can a Host Send Packets?",[12,82602,82603],{},"A device can send packets to three types of destinations:",[23,82605,82606,82620,82626],{},[26,82607,82608,82611,82612,82615,82616,82619],{},[106,82609,82610],{},"Itself"," — By pinging the loopback address (",[6979,82613,82614],{},"127.0.0.1"," for IPv4, or ",[6979,82617,82618],{},"::1"," for IPv6). This is a quick way to test whether the device's own networking stack is working properly.",[26,82621,82622,82625],{},[106,82623,82624],{},"A local host"," — A device on the same local network. For example, another computer connected to the same Wi-Fi router.",[26,82627,82628,82631],{},[106,82629,82630],{},"A remote host"," — A device on a completely different network, like a web server on the other side of the world.",[7166,82633,82635],{"id":82634},"how-does-a-host-know-where-to-send","How Does a Host Know Where to Send?",[12,82637,82638,82639],{},"When your device wants to send a packet, it first needs to determine: ",[7289,82640,82641],{},"is the destination on my local network, or is it somewhere else?",[23,82643,82644,82650],{},[26,82645,82646,82649],{},[106,82647,82648],{},"In IPv4",", the device checks its own subnet mask along with its IP address and the destination IP address. If they're on the same subnet, the packet stays local.",[26,82651,82652,82655],{},[106,82653,82654],{},"In IPv6",", the local router advertises the network prefix to all devices, so each device knows what \"local\" looks like.",[12,82657,82658,82659,82662],{},"If the destination is on a different network, the device forwards the packet to its ",[106,82660,82661],{},"default gateway"," — usually a router that knows how to reach other networks.",[7166,82664,82666],{"id":82665},"the-routers-routing-table","The Router's Routing Table",[12,82668,82669,82670,82673],{},"Routers are the workhorses of inter-network communication. Each router maintains a ",[106,82671,82672],{},"routing table"," that contains three types of entries:",[23,82675,82676,82682,82688],{},[26,82677,82678,82681],{},[106,82679,82680],{},"Directly-connected networks"," — Networks that are physically attached to the router's own interfaces.",[26,82683,82684,82687],{},[106,82685,82686],{},"Remote networks"," — Networks that the router has learned about, either through manual configuration or dynamic routing protocols.",[26,82689,82690,82693],{},[106,82691,82692],{},"Default route"," — A fallback route used when no specific match exists in the table. Think of it as \"if you don't know where to go, send it this way.\"",[7166,82695,82697],{"id":82696},"static-vs-dynamic-routing","Static vs. Dynamic Routing",[12,82699,82700],{},"There are two approaches to populating a routing table:",[12,82702,82703,82706],{},[106,82704,82705],{},"Static routing"," is where a network administrator manually enters routes. It's simple and predictable, but if the network topology changes (say, a link goes down), someone has to manually update the routes. This doesn't scale well for large or frequently changing networks.",[12,82708,82709,82712],{},[106,82710,82711],{},"Dynamic routing"," uses protocols that allow routers to automatically discover and share route information with each other. If a link fails, routers can detect the change and find alternative paths on their own. This is far more resilient and is what powers the internet's backbone.",[6964,82714,82716],{"id":82715},"icmpv4-the-networks-diagnostic-tool","ICMPv4: The Network's Diagnostic Tool",[12,82718,9455,82719,82722],{},[106,82720,82721],{},"Internet Control Message Protocol version 4 (ICMPv4)"," isn't about delivering user data — it's a helper protocol used for diagnostics and error reporting.",[7166,82724,82726],{"id":82725},"key-icmp-messages","Key ICMP Messages",[23,82728,82729,82739,82745],{},[26,82730,82731,82734,82735,82738],{},[106,82732,82733],{},"Echo Request \u002F Echo Reply"," — This is the mechanism behind the ",[6979,82736,82737],{},"ping"," command. Your device sends an Echo Request to a target; if the target is reachable, it sends back an Echo Reply. Simple, but incredibly useful for testing connectivity.",[26,82740,82741,82744],{},[106,82742,82743],{},"Destination Unreachable"," — When a router or host can't deliver a packet (maybe the destination doesn't exist, or a service isn't running), it sends this message back to the source.",[26,82746,82747,82750,82751,82754],{},[106,82748,82749],{},"Time Exceeded"," — Every IP packet has a ",[106,82752,82753],{},"Time to Live (TTL)"," field that decreases by 1 at each router. If TTL hits 0, the packet is discarded and a Time Exceeded message is sent back. This prevents packets from looping endlessly through the network.",[7166,82756,82758],{"id":82757},"practical-tools-ping-and-traceroute","Practical Tools: Ping and Traceroute",[12,82760,82761,82764],{},[106,82762,82763],{},"Ping"," is the most common connectivity test. You can use it in a layered approach to isolate problems:",[3603,82766,82767,82773,82776],{},[26,82768,82769,82770,82772],{},"Ping the loopback (",[6979,82771,82614],{},") — Tests your own TCP\u002FIP stack.",[26,82774,82775],{},"Ping the default gateway — Tests connectivity to your local router.",[26,82777,82778],{},"Ping a remote host — Tests end-to-end connectivity.",[12,82780,82781,7959,82784,82787],{},[106,82782,82783],{},"Traceroute",[6979,82785,82786],{},"tracert"," on Windows) goes a step further. It maps out the entire path a packet takes to reach a destination, listing every router (hop) along the way. It works by sending packets with incrementally increasing TTL values, so each router along the path reveals itself via Time Exceeded messages.",[12,82789,82790,82791,82794,82795,82797],{},"Traceroute sends three packets per hop and measures the ",[106,82792,82793],{},"round-trip time (RTT)"," for each. If a packet is lost or goes unreplied, you'll see an asterisk (",[6979,82796,8128],{},") in the output — a clue that something might be wrong at that hop.",[6964,82799,82801],{"id":82800},"the-transport-layer","The Transport Layer",[12,82803,82804,82805,82808],{},"We've seen how the network layer gets packets from one device to another. But what about making sure the right ",[7289,82806,82807],{},"application"," gets the data? That's where the transport layer comes in.",[12,82810,82811],{},"The transport layer is responsible for:",[23,82813,82814,82817,82827],{},[26,82815,82816],{},"Establishing temporary communication sessions between two applications.",[26,82818,82819,82822,82823,82826],{},[106,82820,82821],{},"Segmenting"," data into smaller pieces for transmission and ",[106,82824,82825],{},"reassembling"," them on the receiving end.",[26,82828,82829,82830,6959],{},"Identifying the correct application for each data stream using ",[106,82831,82832],{},"port numbers",[12,82834,82835,82836,7042,82839,6959],{},"There are two main transport layer protocols, each with very different philosophies: ",[106,82837,82838],{},"TCP",[106,82840,82841],{},"UDP",[7166,82843,82845],{"id":82844},"port-numbers-directing-traffic-to-the-right-app","Port Numbers: Directing Traffic to the Right App",[12,82847,82848,82849,82851],{},"Both TCP and UDP use ",[106,82850,82832],{}," to identify which application should handle incoming data. When you send a request, two port numbers are involved:",[23,82853,82854,82860],{},[26,82855,82856,82859],{},[106,82857,82858],{},"Destination port"," — Tells the receiving device which service you're requesting. For example, port 80 means HTTP (web), port 443 means HTTPS, port 25 means SMTP (email).",[26,82861,82862,82865,82866,82869],{},[106,82863,82864],{},"Source port"," — A randomly generated number on the sending device that identifies ",[7289,82867,82868],{},"this specific conversation",". This is how your device can have multiple tabs open to different websites simultaneously — each tab uses a different source port.",[12,82871,82872],{},"The source and destination ports are placed in the segment header, and the segment is then encapsulated inside an IP packet for delivery.",[6964,82874,82876],{"id":82875},"transmission-control-protocol-tcp","Transmission Control Protocol (TCP)",[12,82878,82879],{},"TCP is the reliable workhorse of the internet. When your data absolutely, positively has to get there — and in the right order — TCP is the protocol for the job.",[7166,82881,82883],{"id":82882},"core-features","Core Features",[23,82885,82886,82896,82902,82908],{},[26,82887,82888,82891,82892,82895],{},[106,82889,82890],{},"Session Establishment"," — TCP is ",[7289,82893,82894],{},"connection-oriented",". Before any data flows, the two devices negotiate and establish a session. This ensures both sides are ready.",[26,82897,82898,82901],{},[106,82899,82900],{},"Reliable Delivery"," — TCP tracks every segment sent. If something gets lost or corrupted, the source retransmits it.",[26,82903,82904,82907],{},[106,82905,82906],{},"Same-Order Delivery"," — Segments are numbered and sequenced, so even if they arrive out of order, the receiving device can reassemble them correctly.",[26,82909,82910,82913],{},[106,82911,82912],{},"Flow Control"," — If the receiving device is overwhelmed (low on memory, limited bandwidth), it can tell the sender to slow down. This prevents data loss due to congestion.",[12,82915,82916,82917,82920],{},"The trade-off? All this reliability comes with overhead. Each TCP segment carries a ",[106,82918,82919],{},"20-byte header"," — considerably more than UDP's 8 bytes.",[7166,82922,82924],{"id":82923},"the-three-way-handshake","The Three-Way Handshake",[12,82926,82927,82928,7024],{},"Before TCP can transmit data, it establishes a connection through a process called the ",[106,82929,82930],{},"three-way handshake",[3603,82932,82933,82939,82945],{},[26,82934,82935,82938],{},[106,82936,82937],{},"SYN"," — The client sends a segment with the SYN (Synchronize) flag set, essentially saying \"I'd like to start a conversation.\"",[26,82940,82941,82944],{},[106,82942,82943],{},"SYN-ACK"," — The server responds with both SYN and ACK (Acknowledgement) flags, saying \"Got it, and I'd like to talk too.\"",[26,82946,82947,82950],{},[106,82948,82949],{},"ACK"," — The client sends back a final ACK, confirming the connection is established.",[12,82952,82953],{},"Once this handshake is complete, data can flow freely in both directions.",[7166,82955,82957],{"id":82956},"closing-a-connection","Closing a Connection",[12,82959,82960,82961,82964,82965,82968],{},"To close a TCP session, a similar (but simpler) process occurs using the ",[106,82962,82963],{},"FIN (Finish)"," flag. Since TCP connections are bidirectional, each direction is closed independently with a FIN segment followed by an ACK — making it a ",[106,82966,82967],{},"two-way handshake"," for each direction (four segments total to fully close).",[7166,82970,82972],{"id":82971},"handling-out-of-order-data","Handling Out-of-Order Data",[12,82974,82975],{},"Network packets don't always arrive in the order they were sent — they might take different paths or experience different delays. TCP handles this gracefully: the receiving side places incoming segments into a buffer, sorts them by sequence number, and only passes the fully reassembled data up to the application.",[7166,82977,82979],{"id":82978},"acknowledgement-and-retransmission","Acknowledgement and Retransmission",[12,82981,82982],{},"TCP ensures reliability through acknowledgements. The destination acknowledges received data, and if the source doesn't receive an ACK within a set time, it retransmits the data from the last acknowledged point.",[12,82984,9455,82985,82988],{},[106,82986,82987],{},"window size"," field in the TCP header controls how much data can be sent before an acknowledgement is required. A larger window allows more data to flow before pausing for confirmation, improving throughput on high-latency connections. This mechanism also supports flow control — the receiver can shrink the window to slow the sender down.",[7166,82990,82992],{"id":82991},"common-applications-using-tcp","Common Applications Using TCP",[12,82994,82995],{},"TCP is the go-to protocol for applications that need reliable, ordered delivery:",[23,82997,82998,83004,83010,83016],{},[26,82999,83000,83003],{},[106,83001,83002],{},"HTTP\u002FHTTPS"," — Web browsing",[26,83005,83006,83009],{},[106,83007,83008],{},"FTP"," — File transfers",[26,83011,83012,83015],{},[106,83013,83014],{},"SMTP"," — Sending email",[26,83017,83018,83021],{},[106,83019,83020],{},"Telnet"," — Remote terminal access",[6964,83023,83025],{"id":83024},"user-datagram-protocol-udp","User Datagram Protocol (UDP)",[12,83027,83028],{},"If TCP is the careful, methodical courier, UDP is the \"just throw it and hope for the best\" approach. And that's not a bad thing — for the right use case, it's exactly what you want.",[7166,83030,83032],{"id":83031},"core-characteristics","Core Characteristics",[23,83034,83035,83040,83046,83052],{},[26,83036,83037,83039],{},[106,83038,82517],{}," — No handshake, no session establishment. Data is sent immediately.",[26,83041,83042,83045],{},[106,83043,83044],{},"Unreliable delivery"," — No acknowledgements, no retransmission. If a packet is lost, it's gone.",[26,83047,83048,83051],{},[106,83049,83050],{},"No ordered reconstruction"," — Segments might arrive out of order, and UDP won't rearrange them.",[26,83053,83054,83057],{},[106,83055,83056],{},"No flow control"," — The sender transmits at whatever rate it wants, with no mechanism to slow down.",[7166,83059,83061],{"id":83060},"why-would-anyone-use-udp","Why Would Anyone Use UDP?",[12,83063,83064,83065,83068,83069,83072],{},"Because sometimes, ",[106,83066,83067],{},"speed matters more than perfection",". UDP's minimal overhead (just an 8-byte header) makes it incredibly fast and efficient. The data units in UDP are called ",[106,83070,83071],{},"datagrams",", and they're delivered on a best-effort basis.",[12,83074,83075],{},"Consider these use cases:",[23,83077,83078,83084,83090,83096],{},[26,83079,83080,83083],{},[106,83081,83082],{},"DNS"," — When your browser looks up a domain name, it needs an answer fast. If the query is lost, it can simply ask again.",[26,83085,83086,83089],{},[106,83087,83088],{},"Video streaming"," — A dropped frame here and there is barely noticeable, but buffering due to retransmission delays would ruin the experience.",[26,83091,83092,83095],{},[106,83093,83094],{},"VoIP (Voice over IP)"," — Real-time voice communication can tolerate small gaps, but it can't tolerate the delays that TCP's reliability mechanisms would introduce.",[26,83097,83098,83101],{},[106,83099,83100],{},"TFTP \u002F SNMP"," — Lightweight protocols where simplicity and speed are prioritized.",[6964,83103,83105],{"id":83104},"tcp-vs-udp-a-quick-comparison","TCP vs. UDP: A Quick Comparison",[7085,83107,83108,83118],{},[7088,83109,83110],{},[7091,83111,83112,83114,83116],{},[7094,83113,47884],{},[7094,83115,82838],{},[7094,83117,82841],{},[7104,83119,83120,83130,83141,83152,83162,83173,83184],{},[7091,83121,83122,83125,83128],{},[7109,83123,83124],{},"Connection type",[7109,83126,83127],{},"Connection-oriented (handshake)",[7109,83129,82517],{},[7091,83131,83132,83135,83138],{},[7109,83133,83134],{},"Reliability",[7109,83136,83137],{},"Guaranteed delivery with ACKs",[7109,83139,83140],{},"Best effort, no guarantees",[7091,83142,83143,83146,83149],{},[7109,83144,83145],{},"Ordering",[7109,83147,83148],{},"Segments reassembled in order",[7109,83150,83151],{},"No ordering",[7091,83153,83154,83157,83160],{},[7109,83155,83156],{},"Flow control",[7109,83158,83159],{},"Yes (window size)",[7109,83161,16113],{},[7091,83163,83164,83167,83170],{},[7109,83165,83166],{},"Header overhead",[7109,83168,83169],{},"20 bytes",[7109,83171,83172],{},"8 bytes",[7091,83174,83175,83178,83181],{},[7109,83176,83177],{},"Speed",[7109,83179,83180],{},"Slower (more overhead)",[7109,83182,83183],{},"Faster (less overhead)",[7091,83185,83186,83189,83192],{},[7109,83187,83188],{},"Use cases",[7109,83190,83191],{},"Web, email, file transfer",[7109,83193,83194],{},"Streaming, DNS, VoIP",[6964,83196,77786],{"id":77785},[12,83198,83199,83200,83202,83203,83205,83206,83209],{},"The network and transport layers work hand-in-hand to deliver data across the internet. The network layer handles addressing and routing — figuring out ",[7289,83201,74635],{}," to send your data and ",[7289,83204,9518],{}," to get it there. The transport layer ensures the data reaches the right application and decides ",[7289,83207,83208],{},"how carefully"," it needs to be delivered.",[12,83211,83212],{},"Understanding these layers gives you a solid foundation for grasping how the internet really works — from the moment you type a URL in your browser to the instant the page appears on your screen. Every packet goes through addressing, routing, encapsulation, and transport protocol handling, all happening invisibly in milliseconds.",[12,83214,83215],{},"Whether it's TCP's meticulous reliability or UDP's carefree speed, each protocol has its place. The beauty of networking is in how all these pieces fit together to create the seamless experience we take for granted every day.",{"title":180,"searchDepth":181,"depth":181,"links":83217},[83218,83219,83224,83230,83234,83237,83245,83249,83250],{"id":82447,"depth":181,"text":82448},{"id":82461,"depth":181,"text":82462,"children":83220},[83221,83222,83223],{"id":82499,"depth":7430,"text":82500},{"id":82533,"depth":7430,"text":82534},{"id":82560,"depth":7430,"text":82561},{"id":82593,"depth":181,"text":74748,"children":83225},[83226,83227,83228,83229],{"id":82599,"depth":7430,"text":82600},{"id":82634,"depth":7430,"text":82635},{"id":82665,"depth":7430,"text":82666},{"id":82696,"depth":7430,"text":82697},{"id":82715,"depth":181,"text":82716,"children":83231},[83232,83233],{"id":82725,"depth":7430,"text":82726},{"id":82757,"depth":7430,"text":82758},{"id":82800,"depth":181,"text":82801,"children":83235},[83236],{"id":82844,"depth":7430,"text":82845},{"id":82875,"depth":181,"text":82876,"children":83238},[83239,83240,83241,83242,83243,83244],{"id":82882,"depth":7430,"text":82883},{"id":82923,"depth":7430,"text":82924},{"id":82956,"depth":7430,"text":82957},{"id":82971,"depth":7430,"text":82972},{"id":82978,"depth":7430,"text":82979},{"id":82991,"depth":7430,"text":82992},{"id":83024,"depth":181,"text":83025,"children":83246},[83247,83248],{"id":83031,"depth":7430,"text":83032},{"id":83060,"depth":7430,"text":83061},{"id":83104,"depth":181,"text":83105},{"id":77785,"depth":181,"text":77786},"2026-04-13",{},"\u002Fblog\u002FNetwork-Communication-Chapter-4",{"title":82442,"description":180},{"loc":83253},"blog\u002FNetwork-Communication-Chapter-4","lwG_cLOw9vKne-6EFCJKPZOIf1Vq7iVEcFka4OSa4bU",{"id":83259,"title":83260,"author":6946,"body":83261,"date":83251,"description":180,"draft":185,"edited_at":74911,"extension":186,"featured_image":194,"meta":84139,"navigation":189,"path":84140,"pinned":185,"seo":84141,"sitemap":84142,"stem":84143,"tags":194,"__hash__":84144},"blog\u002Fblog\u002FNetwork-Communication,-Chapter-3.md","Layers 1 & 2: Physical Media, Ethernet, and Switching",{"type":9,"value":83262,"toc":84103},[83263,83267,83278,83285,83305,83309,83312,83318,83328,83354,83366,83370,83373,83406,83412,83419,83423,83426,83430,83433,83439,83445,83451,83455,83458,83490,83494,83497,83523,83526,83608,83611,83631,83635,83638,83658,83662,83668,83682,83686,83689,83695,83701,83705,83708,83728,83732,83735,83749,83753,83759,83765,83769,83776,83782,83792,83796,83799,83809,83813,83823,83826,83849,83858,83862,83869,83888,83895,83899,83902,83906,83917,83921,83924,83941,83945,83948,83952,83967,83971,84006,84010,84020,84027,84031,84042,84046,84049,84055,84069,84071,84074,84100],[6964,83264,83266],{"id":83265},"the-physical-layer-where-bits-meet-the-real-world","The Physical Layer — Where Bits Meet the Real World",[12,83268,83269,83270,83273,83274,83277],{},"Everything on a network ultimately boils down to ",[106,83271,83272],{},"bits"," — 1s and 0s. But bits are abstract; they need to be physically represented somehow in order to travel from one machine to another. That's the job of the ",[106,83275,83276],{},"physical layer"," (Layer 1 in the OSI model).",[12,83279,83280,83281,83284],{},"The physical layer takes a complete frame from the data link layer above it and converts it into ",[106,83282,83283],{},"signals"," that can be sent over the actual transmission medium. What those signals look like depends entirely on the medium:",[23,83286,83287,83293,83299],{},[26,83288,83289,83292],{},[106,83290,83291],{},"Copper cable"," → patterns of electrical pulses",[26,83294,83295,83298],{},[106,83296,83297],{},"Fiber-optic cable"," → patterns of light",[26,83300,83301,83304],{},[106,83302,83303],{},"Wireless"," → patterns of microwave\u002Fradio transmissions",[7166,83306,83308],{"id":83307},"three-functional-areas","Three Functional Areas",[12,83310,83311],{},"The physical layer standards cover three key areas:",[12,83313,83314,83317],{},[106,83315,83316],{},"1. Physical Components"," — This is all the tangible hardware: NICs (network interface cards), connectors, cables, and their specifications. Think of the RJ-45 jack you plug an Ethernet cable into — that's a physical layer component.",[12,83319,83320,83323,83324,83327],{},[106,83321,83322],{},"2. Encoding"," — Before bits are sent as signals, they're first ",[7289,83325,83326],{},"encoded",". Encoding converts raw data bits into a predefined code — a pattern that both the sender and receiver agree on. This helps with things like identifying where a frame begins and ends. Two common encoding schemes are:",[23,83329,83330,83342],{},[26,83331,83332,83335,83336,83338,83339,83341],{},[106,83333,83334],{},"Manchester Encoding",": A ",[6979,83337,9180],{}," is represented by a high-to-low voltage transition, and a ",[6979,83340,8855],{}," by a low-to-high transition. The beauty of this scheme is that there's always a transition in the middle of each bit period, which helps the receiver stay synchronized.",[26,83343,83344,83347,83348,83350,83351,83353],{},[106,83345,83346],{},"Non-Return to Zero (NRZ)",": Simpler — one voltage level means ",[6979,83349,9180],{},", another means ",[6979,83352,8855],{},". No guaranteed transition in every bit period, which can cause synchronization issues over long runs.",[12,83355,83356,83359,83360,83362,83363,83365],{},[106,83357,83358],{},"3. Signaling"," — This defines how the encoded bits are physically represented. For example, a long pulse might represent a ",[6979,83361,8855],{}," and a short pulse a ",[6979,83364,9180],{},". The standard must be agreed upon so both ends interpret the signals the same way.",[7166,83367,83369],{"id":83368},"bandwidth-throughput-and-goodput","Bandwidth, Throughput, and Goodput",[12,83371,83372],{},"These three terms are related but distinct, and confusing them is a classic beginner mistake:",[23,83374,83375,83388,83397],{},[26,83376,83377,18365,83380,83383,83384,83387],{},[106,83378,83379],{},"Bandwidth",[7289,83381,83382],{},"theoretical maximum"," capacity of a medium — how much data it ",[7289,83385,83386],{},"could"," carry. Measured in kbps, Mbps, or Gbps.",[26,83389,83390,18365,83393,83396],{},[106,83391,83392],{},"Throughput",[7289,83394,83395],{},"actual"," rate of data transfer over a period of time. It's always affected by real-world factors: traffic volume, type of traffic, network congestion, and the number of devices the data passes through.",[26,83398,83399,18365,83402,83405],{},[106,83400,83401],{},"Goodput",[7289,83403,83404],{},"usable"," data transferred — throughput minus all the overhead (session setup, acknowledgments, retransmissions, encapsulation headers). This is what you actually care about as an end user.",[12,83407,83408,83409,6959],{},"The relationship is always: ",[106,83410,83411],{},"Goodput ≤ Throughput ≤ Bandwidth",[12,83413,83414,83415,83418],{},"And then there's ",[106,83416,83417],{},"latency"," — the total time (including delays) for data to travel from point A to point B. Low latency matters a lot for real-time applications like video calls or online gaming.",[6964,83420,83422],{"id":83421},"network-media-copper-fiber-and-wireless","Network Media — Copper, Fiber, and Wireless",[12,83424,83425],{},"Now that we know the physical layer's job, let's look at the actual media that carry our signals.",[7166,83427,83429],{"id":83428},"copper-cabling","Copper Cabling",[12,83431,83432],{},"Copper is the workhorse of networking — cheap, easy to install, and widely available. The trade-off is that it's limited in distance and susceptible to electromagnetic interference (EMI). There are three main types:",[12,83434,83435,83438],{},[106,83436,83437],{},"Unshielded Twisted-Pair (UTP)"," is by far the most common. It uses pairs of wires twisted together (the twisting helps cancel out interference) and terminates with the familiar RJ-45 connector. You've almost certainly used a UTP cable before.",[12,83440,83441,83444],{},[106,83442,83443],{},"Shielded Twisted-Pair (STP)"," adds metallic shielding around the wire pairs for better noise protection. The downside? It's more expensive and harder to install. It also uses RJ-45 connectors.",[12,83446,83447,83450],{},[106,83448,83449],{},"Coaxial cable"," has a single copper conductor at its center, surrounded by insulation and a braided shield. It's used to carry radio frequency (RF) energy for things like cable internet and connecting antennas to wireless devices.",[7166,83452,83454],{"id":83453},"cable-wiring-conventions","Cable Wiring Conventions",[12,83456,83457],{},"Not all Ethernet cables are wired the same way:",[23,83459,83460,83470,83484],{},[26,83461,83462,83465,83466,83469],{},[106,83463,83464],{},"Straight-through cable",": The most common. Used to connect ",[7289,83467,83468],{},"different"," types of devices — e.g., a host to a switch, or a switch to a router.",[26,83471,83472,83475,83476,83479,83480,83483],{},[106,83473,83474],{},"Crossover cable",": Used to connect ",[7289,83477,83478],{},"similar"," devices — switch to switch, host to host, router to router. Less common today because most modern devices support ",[106,83481,83482],{},"Auto-MDIX",", which automatically detects and adjusts for the cable type.",[26,83485,83486,83489],{},[106,83487,83488],{},"Rollover cable",": A Cisco proprietary cable used specifically for connecting to a router or switch's console port for management.",[7166,83491,83493],{"id":83492},"fiber-optic-cabling","Fiber-Optic Cabling",[12,83495,83496],{},"Fiber uses pulses of light instead of electrical signals, which gives it some major advantages: much longer distances, higher bandwidth, and immunity to electromagnetic interference. It's used across several domains:",[23,83498,83499,83505,83511,83517],{},[26,83500,83501,83504],{},[106,83502,83503],{},"Enterprise networks"," for high-speed backbones",[26,83506,83507,83510],{},[106,83508,83509],{},"FTTH (Fiber-to-the-Home)"," for residential broadband",[26,83512,83513,83516],{},[106,83514,83515],{},"Long-haul networks"," spanning hundreds or thousands of kilometers",[26,83518,83519,83522],{},[106,83520,83521],{},"Submarine networks"," for transoceanic links — specially engineered cables that survive harsh undersea conditions",[12,83524,83525],{},"There are two main types of fiber:",[7085,83527,83528,83542],{},[7088,83529,83530],{},[7091,83531,83532,83536,83539],{},[7094,83533,83534],{"align":47883},[16,83535],{},[7094,83537,83538],{"align":47883},"Single-Mode Fiber (SMF)",[7094,83540,83541],{"align":47883},"Multimode Fiber (MMF)",[7104,83543,83544,83557,83570,83583,83596],{},[7091,83545,83546,83551,83554],{},[7109,83547,83548],{"align":47883},[106,83549,83550],{},"Core size",[7109,83552,83553],{"align":47883},"Very small",[7109,83555,83556],{"align":47883},"Larger",[7091,83558,83559,83564,83567],{},[7109,83560,83561],{"align":47883},[106,83562,83563],{},"Light source",[7109,83565,83566],{"align":47883},"Laser (expensive)",[7109,83568,83569],{"align":47883},"LED (cheaper)",[7091,83571,83572,83577,83580],{},[7109,83573,83574],{"align":47883},[106,83575,83576],{},"Distance",[7109,83578,83579],{"align":47883},"Hundreds of km",[7109,83581,83582],{"align":47883},"Up to ~550 m",[7091,83584,83585,83590,83593],{},[7109,83586,83587],{"align":47883},[106,83588,83589],{},"Use case",[7109,83591,83592],{"align":47883},"Long-haul, telephony, cable TV",[7109,83594,83595],{"align":47883},"LANs, short-distance links",[7091,83597,83598,83602,83605],{},[7109,83599,83600],{"align":47883},[106,83601,83379],{},[7109,83603,83604],{"align":47883},"Very high",[7109,83606,83607],{"align":47883},"Up to 10 Gbps",[12,83609,83610],{},"Common fiber connectors include:",[23,83612,83613,83619,83625],{},[26,83614,83615,83618],{},[106,83616,83617],{},"ST (Straight-Tip)",": Older bayonet-style, commonly used with multimode fiber.",[26,83620,83621,83624],{},[106,83622,83623],{},"SC (Subscriber Connector)",": Push-pull mechanism, used with both single-mode and multimode.",[26,83626,83627,83630],{},[106,83628,83629],{},"LC (Lucent Connector)",": Smaller form factor, growing in popularity, supports both fiber types.",[7166,83632,83634],{"id":83633},"wireless-media","Wireless Media",[12,83636,83637],{},"Wireless transmits data using radio or microwave frequencies. It offers mobility and convenience, but comes with its own set of challenges:",[23,83639,83640,83646,83652],{},[26,83641,83642,83645],{},[106,83643,83644],{},"Coverage",": Building materials and terrain can limit signal range significantly.",[26,83647,83648,83651],{},[106,83649,83650],{},"Interference",": Everyday devices — cordless phones, microwaves, fluorescent lights — can disrupt wireless signals.",[26,83653,83654,83657],{},[106,83655,83656],{},"Security",": Because the signal travels through the air, wireless networks require careful security management to prevent unauthorized access.",[6964,83659,83661],{"id":83660},"the-data-link-layer-framing-the-conversation","The Data Link Layer — Framing the Conversation",[12,83663,9455,83664,83667],{},[106,83665,83666],{},"data link layer"," (Layer 2) sits right above the physical layer. Its main jobs are:",[3603,83669,83670,83676,83679],{},[26,83671,83672,83673],{},"Taking Layer 3 (network layer) packets and packaging them into ",[106,83674,83675],{},"frames",[26,83677,83678],{},"Controlling how devices access the physical media",[26,83680,83681],{},"Performing error detection",[7166,83683,83685],{"id":83684},"two-sublayers","Two Sublayers",[12,83687,83688],{},"The data link layer is divided into two sublayers:",[12,83690,83691,83694],{},[106,83692,83693],{},"Logical Link Control (LLC)"," — the upper sublayer. It provides services to the network layer protocols above, essentially acting as the interface between Layer 2 and Layer 3.",[12,83696,83697,83700],{},[106,83698,83699],{},"Media Access Control (MAC)"," — the lower sublayer. This is where the rubber meets the road: it handles addressing (MAC addresses), defines how data is delimited, and manages the actual access to the physical medium. The MAC sublayer's behavior varies depending on the type of network — and a single communication might cross multiple network types as it travels from source to destination.",[7166,83702,83704],{"id":83703},"anatomy-of-a-frame","Anatomy of a Frame",[12,83706,83707],{},"A data link layer frame has three parts:",[23,83709,83710,83716,83722],{},[26,83711,83712,83715],{},[106,83713,83714],{},"Header",": Control information like source and destination addresses. Located at the beginning.",[26,83717,83718,83721],{},[106,83719,83720],{},"Data (Payload)",": The encapsulated content — includes the IP header, transport layer header, and application data.",[26,83723,83724,83727],{},[106,83725,83726],{},"Trailer",": Error detection information (like a CRC checksum), appended at the end.",[6964,83729,83731],{"id":83730},"media-access-control-who-gets-to-talk","Media Access Control — Who Gets to Talk?",[12,83733,83734],{},"When multiple devices share the same medium, there needs to be a system that determines who gets to transmit and when. The method used depends on two factors:",[23,83736,83737,83743],{},[26,83738,83739,83742],{},[106,83740,83741],{},"Topology",": How the connections between nodes appear to the data link layer.",[26,83744,83745,83748],{},[106,83746,83747],{},"Media sharing",": How nodes share the medium.",[7166,83750,83752],{"id":83751},"two-approaches","Two Approaches",[12,83754,83755,83758],{},[106,83756,83757],{},"Contention-based access"," — All nodes compete for access to the medium. Think of it like a group conversation where anyone can start talking, but everyone has an agreed-upon plan for when two people accidentally speak at the same time. The downside is that it doesn't scale well — as more nodes are added, collisions become more frequent.",[12,83760,83761,83764],{},[106,83762,83763],{},"Controlled access"," — Each node is given a designated time slot to use the medium. More orderly, but adds overhead.",[7166,83766,83768],{"id":83767},"csma-variants","CSMA Variants",[12,83770,83771,83772,83775],{},"Contention-based networks typically use ",[106,83773,83774],{},"CSMA (Carrier Sense Multiple Access)"," — devices \"listen\" to the medium before transmitting. There are two important variants:",[12,83777,83778,83781],{},[106,83779,83780],{},"CSMA\u002FCD (Collision Detection)"," — Used in wired Ethernet. A device checks if the medium is free, then transmits. If it detects that another device transmitted at the same time (a collision), all transmitting devices stop, wait a random amount of time, and try again.",[12,83783,83784,83787,83788,83791],{},[106,83785,83786],{},"CSMA\u002FCA (Collision Avoidance)"," — Used in wireless networks. Since wireless devices can't easily detect collisions while transmitting, they take a preventive approach: before sending data, a device first sends a notification of its ",[7289,83789,83790],{},"intent"," to transmit. Only after receiving clearance does it actually send the data.",[6964,83793,83795],{"id":83794},"ethernet-the-king-of-lans","Ethernet — The King of LANs",[12,83797,83798],{},"Ethernet is the most widely used LAN technology in the world. Defined by the IEEE 802.2 and 802.3 standards, it operates across both the data link layer and the physical layer.",[12,83800,83801,83802,83805,83806,6959],{},"Ethernet supports an impressive range of speeds: 10 Mbps, 100 Mbps, 1 Gbps, 10 Gbps, 40 Gbps, and 100 Gbps. The minimum frame size is ",[106,83803,83804],{},"64 bytes"," and the maximum is ",[106,83807,83808],{},"1518 bytes",[7166,83810,83812],{"id":83811},"mac-addresses","MAC Addresses",[12,83814,83815,83816,83819,83820,16419],{},"Every device on an Ethernet network needs a unique identifier — this is the ",[106,83817,83818],{},"MAC address",". It's a 48-bit value, typically written as 12 hexadecimal digits (e.g., ",[6979,83821,83822],{},"00:1A:2B:3C:4D:5E",[12,83824,83825],{},"MAC addresses are used for three types of communication:",[23,83827,83828,83834,83843],{},[26,83829,83830,83833],{},[106,83831,83832],{},"Unicast",": One-to-one. A frame sent to a single specific device.",[26,83835,83836,83839,83840,16419],{},[106,83837,83838],{},"Broadcast",": One-to-all. A frame sent to every device on the local network (destination MAC: ",[6979,83841,83842],{},"FF:FF:FF:FF:FF:FF",[26,83844,83845,83848],{},[106,83846,83847],{},"Multicast",": One-to-many. A frame sent to a group of devices that have subscribed to receive it.",[7351,83850,83851],{},[12,83852,83853,83854,83857],{},"On Windows, you can check your MAC address by running ",[6979,83855,83856],{},"ipconfig \u002Fall"," in the command prompt.",[7166,83859,83861],{"id":83860},"how-a-switch-uses-mac-addresses","How a Switch Uses MAC Addresses",[12,83863,83864,83865,83868],{},"An Ethernet switch maintains a ",[106,83866,83867],{},"MAC address table"," that maps MAC addresses to its physical ports. When a frame arrives:",[3603,83870,83871,83881],{},[26,83872,83873,83874,83877,83878,6959],{},"The switch reads the ",[106,83875,83876],{},"source MAC address"," and records which port it came from — this is how the table is ",[7289,83879,83880],{},"dynamically built",[26,83882,83883,83884,83887],{},"The switch then looks up the ",[106,83885,83886],{},"destination MAC address"," in its table to decide which port to forward the frame to.",[12,83889,83890,83891,83894],{},"If the destination MAC isn't in the table yet, the switch ",[7289,83892,83893],{},"floods"," the frame out all ports (except the one it arrived on) — essentially asking, \"Who has this address?\"",[6964,83896,83898],{"id":83897},"switching-making-smart-forwarding-decisions","Switching — Making Smart Forwarding Decisions",[12,83900,83901],{},"Switches can use different methods to forward frames, each with its own trade-off between speed and reliability:",[7166,83903,83905],{"id":83904},"store-and-forward-switching","Store-and-Forward Switching",[12,83907,83908,83909,83912,83913,83916],{},"The switch waits until it has received the ",[106,83910,83911],{},"entire frame",", stores it in a buffer, and performs an error check using a ",[106,83914,83915],{},"CRC (Cyclic Redundancy Check)",". Only if the frame passes the check is it forwarded. This is the most reliable method but adds a small delay.",[7166,83918,83920],{"id":83919},"cut-through-switching","Cut-Through Switching",[12,83922,83923],{},"The switch starts forwarding the frame as soon as it reads the destination address — before the entire frame has arrived. This is faster, but it forwards corrupted frames too since there's no error check. There are two variants:",[23,83925,83926,83932],{},[26,83927,83928,83931],{},[106,83929,83930],{},"Fast-forward switching",": Forwards immediately after reading the destination address. Lowest latency, but no error checking at all.",[26,83933,83934,83937,83938,83940],{},[106,83935,83936],{},"Fragment-free switching",": Waits for the first ",[106,83939,83804],{}," before forwarding. Why 64 bytes? Because most collision-related errors occur within the first 64 bytes of a frame, so this catches the majority of bad frames while still being faster than store-and-forward.",[7166,83942,83944],{"id":83943},"frame-buffering","Frame Buffering",[12,83946,83947],{},"Switches also use buffering to handle congestion. If the destination port is busy, the switch stores the frame temporarily until the port is available. This prevents frame loss during traffic spikes.",[6964,83949,83951],{"id":83950},"arp-translating-addresses","ARP — Translating Addresses",[12,83953,83954,83955,83958,83959,83962,83963,83966],{},"Here's a fundamental problem in networking: the network layer uses ",[106,83956,83957],{},"IP addresses"," to identify devices, but the data link layer uses ",[106,83960,83961],{},"MAC addresses",". When a device wants to send data to another device on the same local network, it knows the destination's IP address but needs the MAC address to build a frame. This is where ",[106,83964,83965],{},"ARP (Address Resolution Protocol)"," comes in.",[7166,83968,83970],{"id":83969},"how-arp-works","How ARP Works",[3603,83972,83973,83980,83983,83996,84003],{},[26,83974,83975,83976,83979],{},"The sender checks its ",[106,83977,83978],{},"ARP table"," (a local cache of IP-to-MAC mappings).",[26,83981,83982],{},"If a mapping exists → great, use it to build the frame.",[26,83984,83985,83986,83989,83990,83992,83993],{},"If no mapping exists → the sender broadcasts an ",[106,83987,83988],{},"ARP request"," to the entire local network (destination MAC: ",[6979,83991,83842],{},"), essentially asking: ",[7289,83994,83995],{},"\"Who has this IP address? Tell me your MAC address.\"",[26,83997,83998,83999,84002],{},"The device with the matching IP address responds with an ",[106,84000,84001],{},"ARP reply"," containing its MAC address.",[26,84004,84005],{},"The sender stores this mapping in its ARP table for future use.",[7166,84007,84009],{"id":84008},"what-about-remote-networks","What About Remote Networks?",[12,84011,84012,84013,84016,84017,84019],{},"If the destination IP address is on a ",[106,84014,84015],{},"different network",", the device doesn't ARP for the destination directly. Instead, it looks up the MAC address of its ",[106,84018,82661],{}," (the router) in the ARP table. The frame is sent to the router, which then handles forwarding it to the correct network.",[12,84021,84022,84023,84026],{},"If no device responds to an ARP request, the packet is simply ",[106,84024,84025],{},"dropped"," — because without a MAC address, a frame simply cannot be constructed.",[7166,84028,84030],{"id":84029},"arp-table-management","ARP Table Management",[12,84032,84033,84034,84037,84038,84041],{},"ARP entries don't last forever. Each entry has a ",[106,84035,84036],{},"cache timer"," — on Windows, entries typically expire after ",[106,84039,84040],{},"15 to 45 seconds"," if unused. This keeps the table fresh and prevents stale mappings.",[7166,84043,84045],{"id":84044},"arp-security-concerns","ARP Security Concerns",[12,84047,84048],{},"ARP has two notable issues:",[12,84050,84051,84054],{},[106,84052,84053],{},"1. Overhead",": Since ARP requests are broadcasts, every device on the local network must process them. If many devices start ARP-ing at the same time (e.g., during a network boot storm), it can cause a temporary performance dip.",[12,84056,84057,84060,84061,84064,84065,84068],{},[106,84058,84059],{},"2. ARP Spoofing \u002F Poisoning",": Because ARP has no built-in authentication, a malicious actor can send fake ARP replies, claiming that ",[7289,84062,84063],{},"their"," MAC address maps to another device's IP (like the default gateway). Victims then unknowingly send traffic to the attacker instead — a classic ",[106,84066,84067],{},"man-in-the-middle attack",". This is why techniques like Dynamic ARP Inspection (DAI) and static ARP entries exist as countermeasures.",[6964,84070,77786],{"id":77785},[12,84072,84073],{},"In this chapter, we traced the journey of data from the abstract world of bits down to the physical signals on a wire (or through the air), and back up through the framing and addressing mechanisms that make local network communication possible. Here's a quick mental model:",[23,84075,84076,84082,84088,84094],{},[26,84077,84078,84081],{},[106,84079,84080],{},"Physical Layer",": Converts bits into signals (electrical, light, or radio) and defines the hardware.",[26,84083,84084,84087],{},[106,84085,84086],{},"Data Link Layer",": Packages data into frames, handles MAC addressing, and manages media access.",[26,84089,84090,84093],{},[106,84091,84092],{},"Ethernet",": The dominant LAN technology, using MAC addresses and switches to deliver frames.",[26,84095,84096,84099],{},[106,84097,84098],{},"ARP",": The bridge between IP addresses (Layer 3) and MAC addresses (Layer 2).",[12,84101,84102],{},"Understanding these layers is essential because every higher-level protocol — HTTP, DNS, SSH, you name it — ultimately relies on these mechanisms to move data across the wire. Getting comfortable with these fundamentals will make everything else in networking click.",{"title":180,"searchDepth":181,"depth":181,"links":84104},[84105,84109,84115,84119,84123,84127,84132,84138],{"id":83265,"depth":181,"text":83266,"children":84106},[84107,84108],{"id":83307,"depth":7430,"text":83308},{"id":83368,"depth":7430,"text":83369},{"id":83421,"depth":181,"text":83422,"children":84110},[84111,84112,84113,84114],{"id":83428,"depth":7430,"text":83429},{"id":83453,"depth":7430,"text":83454},{"id":83492,"depth":7430,"text":83493},{"id":83633,"depth":7430,"text":83634},{"id":83660,"depth":181,"text":83661,"children":84116},[84117,84118],{"id":83684,"depth":7430,"text":83685},{"id":83703,"depth":7430,"text":83704},{"id":83730,"depth":181,"text":83731,"children":84120},[84121,84122],{"id":83751,"depth":7430,"text":83752},{"id":83767,"depth":7430,"text":83768},{"id":83794,"depth":181,"text":83795,"children":84124},[84125,84126],{"id":83811,"depth":7430,"text":83812},{"id":83860,"depth":7430,"text":83861},{"id":83897,"depth":181,"text":83898,"children":84128},[84129,84130,84131],{"id":83904,"depth":7430,"text":83905},{"id":83919,"depth":7430,"text":83920},{"id":83943,"depth":7430,"text":83944},{"id":83950,"depth":181,"text":83951,"children":84133},[84134,84135,84136,84137],{"id":83969,"depth":7430,"text":83970},{"id":84008,"depth":7430,"text":84009},{"id":84029,"depth":7430,"text":84030},{"id":84044,"depth":7430,"text":84045},{"id":77785,"depth":181,"text":77786},{},"\u002Fblog\u002FNetwork-Communication,-Chapter-3",{"title":83260,"description":180},{"loc":84140},"blog\u002FNetwork-Communication,-Chapter-3","FNiLv6XaOOv8LlffkUJGdoupIGuASkQFpUx2tfCHnZg",{"id":84146,"title":84147,"author":6946,"body":84148,"date":83251,"description":180,"draft":185,"edited_at":74911,"extension":186,"featured_image":194,"meta":85270,"navigation":189,"path":85271,"pinned":185,"seo":85272,"sitemap":85273,"stem":85274,"tags":194,"__hash__":85275},"blog\u002Fblog\u002FNetwork-Communication,-Chapter-6.md","IPv6: Addressing, SLAAC, and NDP",{"type":9,"value":84149,"toc":85223},[84150,84154,84157,84172,84179,84186,84199,84203,84207,84222,84228,84231,84235,84242,84248,84255,84271,84277,84281,84288,84294,84304,84308,84314,84318,84325,84329,84336,84340,84351,84363,84367,84374,84378,84385,84402,84405,84460,84466,84477,84481,84492,84515,84518,84522,84528,84538,84542,84548,84555,84559,84576,84588,84592,84595,84599,84602,84606,84616,84619,84647,84650,84653,84659,84663,84670,84674,84681,84685,84695,84698,84721,84727,84730,84734,84741,84745,84748,84789,84793,84796,84807,84810,84814,84824,84827,84897,84901,84906,84909,84927,84931,84943,84946,84950,84953,84957,84963,84967,84978,84982,84988,84992,85002,85006,85016,85022,85025,85031,85035,85045,85061,85067,85086,85090],[6964,84151,84153],{"id":84152},"why-do-we-need-ipv6","Why Do We Need IPv6?",[12,84155,84156],{},"The Internet was never meant to be this big.",[12,84158,84159,84160,84163,84164,84167,84168,84171],{},"When IPv4 was designed, its 32-bit address space gave us roughly ",[106,84161,84162],{},"4.3 billion addresses"," — a number that seemed inexhaustible at the time. But the digital world had other plans. On ",[106,84165,84166],{},"January 31, 2011",", the Internet Assigned Numbers Authority (IANA) allocated the last two ",[6979,84169,84170],{},"\u002F8"," IPv4 address blocks to the Regional Internet Registries (RIRs). In other words, we officially ran out of fresh IPv4 addresses to hand out.",[12,84173,84174,84175,84178],{},"And the demand is only growing. The Internet is no longer just for computers, tablets, and smartphones. We're entering the era of the ",[106,84176,84177],{},"Internet of Things (IoT)"," — where everything from cars and medical sensors to household appliances and environmental monitors are connecting online. Each of these devices needs an address.",[12,84180,84181,84182,84185],{},"IPv4 tried to cope with workarounds like ",[106,84183,84184],{},"NAT (Network Address Translation)",", which lets multiple devices share a single public IP. But NAT introduces complexity, breaks certain applications, and is ultimately a band-aid, not a cure.",[12,84187,84188,84190,84191,84194,84195,84198],{},[106,84189,82509],{}," is the real solution. With a ",[106,84192,84193],{},"128-bit address space",", it provides approximately ",[106,84196,84197],{},"3.4 × 10³⁸ addresses"," — enough for every grain of sand on Earth to have its own IP address, many times over.",[6964,84200,84202],{"id":84201},"understanding-ipv6-addresses","Understanding IPv6 Addresses",[7166,84204,84206],{"id":84205},"the-format","The Format",[12,84208,84209,84210,84213,84214,84217,84218,84221],{},"An IPv6 address is ",[106,84211,84212],{},"128 bits"," long, written as ",[106,84215,84216],{},"eight groups of four hexadecimal digits",", separated by colons. Each group (called a ",[106,84219,84220],{},"hextet",") represents 16 bits.",[6972,84223,84226],{"className":84224,"code":84225,"language":6977},[6975],"Full format: X:X:X:X:X:X:X:X   (where each X = four hex digits)\n \nExample:    2001:0DB8:ACAD:0001:0000:0000:0000:0100\n",[6979,84227,84225],{"__ignoreMap":180},[12,84229,84230],{},"That's a lot of characters. Fortunately, there are two shorthand rules to make things more manageable.",[7166,84232,84234],{"id":84233},"rule-1-drop-leading-zeros","Rule 1: Drop Leading Zeros",[12,84236,84237,84238,84241],{},"Within any hextet, you can omit ",[106,84239,84240],{},"leading zeros"," (zeros at the front). For example:",[6972,84243,84246],{"className":84244,"code":84245,"language":6977},[6975],"Full:        2001:0DB8:0000:1111:0000:0000:0000:0200\nShortened:   2001: DB8:   0:1111:   0:   0:   0: 200\n",[6979,84247,84245],{"__ignoreMap":180},[7166,84249,84251,84252],{"id":84250},"rule-2-replace-consecutive-all-zero-groups-with","Rule 2: Replace Consecutive All-Zero Groups with ",[6979,84253,84254],{},"::",[12,84256,9407,84257,10476,84260,84262,84263,7206,84265,84267,84268,84270],{},[106,84258,84259],{},"double colon",[6979,84261,84254],{},") can replace one or more consecutive hextets that are all zeros. But you can only use ",[6979,84264,84254],{},[106,84266,18152],{}," per address — otherwise, it would be ambiguous how many zero groups each ",[6979,84269,84254],{}," represents.",[6972,84272,84275],{"className":84273,"code":84274,"language":6977},[6975],"Full:        2001:0DB8:0000:0000:0000:0000:0000:0100\nCompressed:  2001:DB8::100\n",[6979,84276,84274],{"__ignoreMap":180},[7166,84278,84280],{"id":84279},"prefix-length-notation","Prefix Length Notation",[12,84282,84283,84284,84287],{},"Just like IPv4 uses a subnet mask, IPv6 uses a ",[106,84285,84286],{},"prefix length"," to indicate the network portion of an address. It's written with a slash followed by a number:",[6972,84289,84292],{"className":84290,"code":84291,"language":6977},[6975],"2001:0DB8:000A::\u002F64\n      ↑                  ↑\n  Network portion    Prefix length (first 64 bits = network)\n",[6979,84293,84291],{"__ignoreMap":180},[12,84295,84296,84297,84300,84301,16419],{},"An IPv6 address is typically split into two halves: the first 64 bits identify the network (the ",[106,84298,84299],{},"prefix","), and the last 64 bits identify the device on that network (the ",[106,84302,84303],{},"Interface ID",[6964,84305,84307],{"id":84306},"types-of-ipv6-addresses","Types of IPv6 Addresses",[12,84309,84310,84311,84313],{},"IPv6 defines ",[106,84312,71274],{}," main categories of addresses:",[7166,84315,84317],{"id":84316},"_1-unicast-one-to-one","1. Unicast — \"One-to-One\"",[12,84319,84320,84321,84324],{},"A unicast address uniquely identifies ",[106,84322,84323],{},"a single interface"," on a device. When you send a packet to a unicast address, it goes to exactly one destination. Think of it as a direct letter to a specific mailbox.",[7166,84326,84328],{"id":84327},"_2-multicast-one-to-many","2. Multicast — \"One-to-Many\"",[12,84330,84331,84332,84335],{},"A multicast address delivers a packet to ",[106,84333,84334],{},"multiple destinations"," simultaneously. Instead of sending separate copies, the network efficiently distributes one packet to an entire group. Think of it as a group chat message.",[7166,84337,84339],{"id":84338},"_3-anycast-one-to-nearest","3. Anycast — \"One-to-Nearest\"",[12,84341,84342,84343,84346,84347,84350],{},"An anycast address can be assigned to ",[106,84344,84345],{},"multiple devices",", but a packet sent to it is delivered only to the ",[106,84348,84349],{},"nearest one"," (determined by routing distance). Think of it as calling the closest branch of a restaurant chain — the system routes you to the nearest location.",[7351,84352,84353],{},[12,84354,84355,84358,84359,84362],{},[106,84356,84357],{},"Note:"," Unlike IPv4, IPv6 has ",[106,84360,84361],{},"no broadcast"," address. The function of broadcast is handled by multicast in IPv6.",[6964,84364,84366],{"id":84365},"ipv6-unicast-addresses-in-detail","IPv6 Unicast Addresses in Detail",[12,84368,84369,84370,84373],{},"There are ",[106,84371,84372],{},"six"," types of unicast addresses. Let's walk through each one.",[7166,84375,84377],{"id":84376},"global-unicast-address-gua","Global Unicast Address (GUA)",[12,84379,84380,84381,84384],{},"This is the IPv6 equivalent of a ",[106,84382,84383],{},"public IPv4 address"," — globally unique and routable across the Internet.",[12,84386,84387,84388,84391,84392,84395,84396,10720,84399,6959],{},"Currently, only addresses starting with the binary bits ",[6979,84389,84390],{},"001"," (the range ",[6979,84393,84394],{},"2000::\u002F3",") are being assigned as global unicast. That covers all addresses from ",[6979,84397,84398],{},"2000::",[6979,84400,84401],{},"3FFF::",[12,84403,84404],{},"A global unicast address has three parts:",[7085,84406,84407,84420],{},[7088,84408,84409],{},[7091,84410,84411,84414,84417],{},[7094,84412,84413],{},"Part",[7094,84415,84416],{},"Bits",[7094,84418,84419],{},"Description",[7104,84421,84422,84435,84448],{},[7091,84423,84424,84429,84432],{},[7109,84425,84426],{},[106,84427,84428],{},"Global Routing Prefix",[7109,84430,84431],{},"Typically 48 bits",[7109,84433,84434],{},"Assigned by the ISP or RIR; identifies the organization's network on the global Internet",[7091,84436,84437,84442,84445],{},[7109,84438,84439],{},[106,84440,84441],{},"Subnet ID",[7109,84443,84444],{},"Typically 16 bits",[7109,84446,84447],{},"Used by the organization to create internal subnets",[7091,84449,84450,84454,84457],{},[7109,84451,84452],{},[106,84453,84303],{},[7109,84455,84456],{},"64 bits",[7109,84458,84459],{},"Identifies the specific device interface; similar to the host portion in IPv4",[6972,84461,84464],{"className":84462,"code":84463,"language":6977},[6975],"|\u003C--- 48 bits --->|\u003C- 16 bits ->|\u003C-------- 64 bits -------->|\n  Global Routing       Subnet          Interface ID\n     Prefix              ID\n",[6979,84465,84463],{"__ignoreMap":180},[12,84467,84468,84469,84472,84473,84476],{},"So a ",[6979,84470,84471],{},"\u002F48"," routing prefix + 16-bit Subnet ID gives you a ",[6979,84474,84475],{},"\u002F64"," prefix for each subnet.",[7166,84478,84480],{"id":84479},"link-local-address","Link-Local Address",[12,84482,84483,84484,84487,84488,84491],{},"Link-local addresses are used for communication ",[106,84485,84486],{},"within a single network segment"," (link). They are ",[106,84489,84490],{},"not routable"," — routers will not forward packets with a link-local source or destination beyond the local link.",[23,84493,84494,84502,84509,84512],{},[26,84495,84496,7206,84499],{},[106,84497,84498],{},"Range:",[6979,84500,84501],{},"FE80::\u002F10",[26,84503,84504,84505,84508],{},"Every IPv6-enabled interface ",[106,84506,84507],{},"automatically"," gets a link-local address",[26,84510,84511],{},"Used for neighbor discovery, router advertisements, and as the default gateway address",[26,84513,84514],{},"Can also be configured manually as a static address",[12,84516,84517],{},"Think of link-local as a \"neighborhood-only\" address — devices on the same link can talk to each other, but the conversation stays local.",[7166,84519,84521],{"id":84520},"loopback-address","Loopback Address",[6972,84523,84526],{"className":84524,"code":84525,"language":6977},[6975],"::1\n",[6979,84527,84525],{"__ignoreMap":180},[12,84529,84530,84531,84533,84534,84537],{},"The loopback address is the IPv6 version of ",[6979,84532,82614],{}," in IPv4. A device uses it to ",[106,84535,84536],{},"send a packet to itself"," — useful for testing whether the IPv6 stack is working. It cannot be assigned to a physical interface.",[7166,84539,84541],{"id":84540},"unspecified-address","Unspecified Address",[6972,84543,84546],{"className":84544,"code":84545,"language":6977},[6975],"::\n",[6979,84547,84545],{"__ignoreMap":180},[12,84549,84550,84551,84554],{},"The all-zeros address is used as a ",[106,84552,84553],{},"placeholder source address"," when a device doesn't yet have an address assigned (for example, during the initial boot-up process). It cannot be assigned to an interface or used as a destination.",[7166,84556,84558],{"id":84557},"unique-local-address","Unique Local Address",[12,84560,84561,84562,84565,84566,7067,84569,84572,84573,84575],{},"These are the IPv6 equivalent of ",[106,84563,84564],{},"private IPv4 addresses"," (like ",[6979,84567,84568],{},"192.168.x.x",[6979,84570,84571],{},"10.x.x.x","). They are meant for communication within a site or between a few sites, and are ",[106,84574,84490],{}," on the global Internet.",[23,84577,84578],{},[26,84579,84580,7206,84582,10720,84585],{},[106,84581,84498],{},[6979,84583,84584],{},"FC00::\u002F7",[6979,84586,84587],{},"FDFF::\u002F7",[7166,84589,84591],{"id":84590},"embedded-ipv4-address","Embedded IPv4 Address",[12,84593,84594],{},"A special unicast address format that embeds an IPv4 address within an IPv6 address, used during transition from IPv4 to IPv6.",[6964,84596,84598],{"id":84597},"how-devices-get-their-ipv6-addresses","How Devices Get Their IPv6 Addresses",[12,84600,84601],{},"One of the powerful features of IPv6 is its flexible addressing model. There are three main ways a device can obtain its IPv6 address.",[7166,84603,84605],{"id":84604},"option-1-slaac-stateless-address-auto-configuration","Option 1: SLAAC (Stateless Address Auto-Configuration)",[12,84607,84608,84609,84612,84613,6959],{},"SLAAC lets a device configure its own address ",[106,84610,84611],{},"without a DHCPv6 server",". The process relies on ",[106,84614,84615],{},"ICMPv6 Router Advertisement (RA) messages",[12,84617,84618],{},"Here's how it works:",[3603,84620,84621,84634,84641],{},[26,84622,84623,84624,84627,84628,83992,84631],{},"A device boots up and sends a ",[106,84625,84626],{},"Router Solicitation (RS)"," message to the all-routers multicast address (",[6979,84629,84630],{},"FF02::2",[7289,84632,84633],{},"\"Are there any routers out there? What network am I on?\"",[26,84635,84636,84637,84640],{},"A router responds with a ",[106,84638,84639],{},"Router Advertisement (RA)"," containing the network prefix, prefix length, and default gateway information.",[26,84642,84643,84644,84646],{},"The device combines the prefix with its own ",[106,84645,84303],{}," (often generated via EUI-64, more on this below) to form a complete global unicast address.",[12,84648,84649],{},"Routers also send RA messages periodically (roughly every 200 seconds), so even devices that missed the initial exchange will eventually get their information.",[12,84651,84652],{},"To enable a Cisco router for IPv6 routing, you need the command:",[6972,84654,84657],{"className":84655,"code":84656,"language":6977},[6975],"Router(config)# ipv6 unicast-routing\n",[6979,84658,84656],{"__ignoreMap":180},[7166,84660,84662],{"id":84661},"option-2-stateless-dhcpv6-slaac-dhcpv6","Option 2: Stateless DHCPv6 (SLAAC + DHCPv6)",[12,84664,84665,84666,84669],{},"In this hybrid approach, SLAAC handles the address and prefix, while a ",[106,84667,84668],{},"DHCPv6 server"," provides additional information such as DNS server addresses and domain names. The DHCPv6 server doesn't track which addresses are assigned — hence \"stateless.\"",[7166,84671,84673],{"id":84672},"option-3-stateful-dhcpv6","Option 3: Stateful DHCPv6",[12,84675,84676,84677,84680],{},"This is closest to traditional DHCP in IPv4. A ",[106,84678,84679],{},"stateful DHCPv6 server"," assigns the full addressing information, including the global unicast address, prefix length, and DNS server addresses. The server maintains a record of which addresses are assigned to which devices.",[6964,84682,84684],{"id":84683},"the-eui-64-process","The EUI-64 Process",[12,84686,84687,84688,84691,84692,6959],{},"When using SLAAC, a device needs to generate a 64-bit Interface ID. One common method is ",[106,84689,84690],{},"EUI-64"," (Extended Unique Identifier, 64-bit), which derives the Interface ID from the device's ",[106,84693,84694],{},"48-bit MAC address",[12,84696,84697],{},"Here's the process:",[3603,84699,84700,84706,84715],{},[26,84701,84702,84705],{},[106,84703,84704],{},"Split"," the 48-bit MAC address into two halves (24 bits each)",[26,84707,84708,84710,84711,84714],{},[106,84709,13398],{}," the hex value ",[6979,84712,84713],{},"FFFE"," (16 bits) in the middle",[26,84716,84717,84720],{},[106,84718,84719],{},"Flip"," the 7th bit (the Universal\u002FLocal bit) of the first byte",[6972,84722,84725],{"className":84723,"code":84724,"language":6977},[6975],"MAC Address:       FC:99:47:75:CE:E0\n \nStep 1 — Split:    FC:99:47  |  75:CE:E0\nStep 2 — Insert:   FC:99:47:FF:FE:75:CE:E0\nStep 3 — Flip 7th: FE:99:47:FF:FE:75:CE:E0\n \nInterface ID:      FE99:47FF:FE75:CEE0\n",[6979,84726,84724],{"__ignoreMap":180},[12,84728,84729],{},"This gives each device a unique Interface ID derived from its hardware address.",[6964,84731,84733],{"id":84732},"ipv6-multicast-addresses","IPv6 Multicast Addresses",[12,84735,84736,84737,84740],{},"Since IPv6 eliminates broadcast, ",[106,84738,84739],{},"multicast"," takes on a bigger role. There are two key types:",[7166,84742,84744],{"id":84743},"assigned-multicast","Assigned Multicast",[12,84746,84747],{},"These are reserved addresses for well-known groups:",[7085,84749,84750,84762],{},[7088,84751,84752],{},[7091,84753,84754,84757,84760],{},[7094,84755,84756],{},"Address",[7094,84758,84759],{},"Group",[7094,84761,16271],{},[7104,84763,84764,84777],{},[7091,84765,84766,84771,84774],{},[7109,84767,84768],{},[6979,84769,84770],{},"FF02::1",[7109,84772,84773],{},"All-nodes",[7109,84775,84776],{},"Every IPv6-enabled device on the link receives packets sent here — this is the closest thing to a broadcast",[7091,84778,84779,84783,84786],{},[7109,84780,84781],{},[6979,84782,84630],{},[7109,84784,84785],{},"All-routers",[7109,84787,84788],{},"Only IPv6 routers join this group; hosts use it to send Router Solicitations",[7166,84790,84792],{"id":84791},"solicited-node-multicast","Solicited-Node Multicast",[12,84794,84795],{},"This is a clever mechanism for efficient neighbor discovery. Instead of broadcasting to every device, IPv6 creates a special multicast address that targets only the device(s) with a matching address suffix.",[12,84797,84798,84799,84802,84803,84806],{},"A solicited-node address is formed by combining the fixed prefix ",[6979,84800,84801],{},"FF02:0:0:0:0:1:FF00::\u002F104"," with the ",[106,84804,84805],{},"last 24 bits"," of a device's unicast address. Only devices sharing those same 24 bits will process the packet — everyone else ignores it.",[12,84808,84809],{},"This means address resolution in IPv6 is far more efficient than ARP broadcasts in IPv4.",[6964,84811,84813],{"id":84812},"icmpv6-and-neighbor-discovery-protocol-ndp","ICMPv6 and Neighbor Discovery Protocol (NDP)",[12,84815,84816,84819,84820,84823],{},[106,84817,84818],{},"ICMPv6"," is the upgraded messaging protocol for IPv6, and it plays a much larger role than its IPv4 counterpart. A key part of ICMPv6 is the ",[106,84821,84822],{},"Neighbor Discovery Protocol (NDP)",", which handles several functions that ARP and other protocols managed in IPv4.",[12,84825,84826],{},"NDP uses five message types:",[7085,84828,84829,84840],{},[7088,84830,84831],{},[7091,84832,84833,84835,84838],{},[7094,84834,74847],{},[7094,84836,84837],{},"Abbreviation",[7094,84839,16271],{},[7104,84841,84842,84853,84864,84875,84886],{},[7091,84843,84844,84847,84850],{},[7109,84845,84846],{},"Router Solicitation",[7109,84848,84849],{},"RS",[7109,84851,84852],{},"Hosts ask routers for network configuration info",[7091,84854,84855,84858,84861],{},[7109,84856,84857],{},"Router Advertisement",[7109,84859,84860],{},"RA",[7109,84862,84863],{},"Routers announce prefix, gateway, and configuration options",[7091,84865,84866,84869,84872],{},[7109,84867,84868],{},"Neighbor Solicitation",[7109,84870,84871],{},"NS",[7109,84873,84874],{},"Resolve IPv6 address to a MAC address (replaces ARP); also used for Duplicate Address Detection",[7091,84876,84877,84880,84883],{},[7109,84878,84879],{},"Neighbor Advertisement",[7109,84881,84882],{},"NA",[7109,84884,84885],{},"Response to NS, providing the requested MAC address",[7091,84887,84888,84891,84894],{},[7109,84889,84890],{},"Redirect",[7109,84892,84893],{},"—",[7109,84895,84896],{},"Informs a host of a better next-hop router for a destination",[7166,84898,84900],{"id":84899},"duplicate-address-detection-dad","Duplicate Address Detection (DAD)",[12,84902,84903,84904,6959],{},"Before a device starts using a new IPv6 address (whether global unicast or link-local), it should verify that no other device on the link already has the same address. This is called ",[106,84905,84900],{},[12,84907,84908],{},"The process:",[3603,84910,84911,84918,84924],{},[26,84912,84913,84914,84917],{},"The device sends a ",[106,84915,84916],{},"Neighbor Solicitation (NS)"," message with its own tentative IPv6 address as the target",[26,84919,84920,84921],{},"If another device already uses this address, it replies with a ",[106,84922,84923],{},"Neighbor Advertisement (NA)",[26,84925,84926],{},"If no NA is received within a timeout period, the address is considered unique and safe to use",[7166,84928,84930],{"id":84929},"address-resolution-replacing-arp","Address Resolution (Replacing ARP)",[12,84932,84933,84934,84936,84937,84940,84941,84002],{},"When a device knows the IPv6 address of a destination on the same link but not its MAC address, it sends a ",[106,84935,84868],{}," to the destination's ",[106,84938,84939],{},"solicited-node multicast address",". The destination responds with a ",[106,84942,84879],{},[12,84944,84945],{},"This is more efficient than IPv4's ARP, which broadcasts to every device on the network.",[6964,84947,84949],{"id":84948},"transitioning-from-ipv4-to-ipv6","Transitioning from IPv4 to IPv6",[12,84951,84952],{},"The Internet can't switch to IPv6 overnight. The migration relies on three main strategies:",[7166,84954,84956],{"id":84955},"dual-stack","Dual Stack",[12,84958,84959,84960,84962],{},"Devices run ",[106,84961,11209],{}," IPv4 and IPv6 simultaneously. They have both an IPv4 and IPv6 address and can communicate using either protocol depending on what the destination supports. This is the most common transition approach today.",[7166,84964,84966],{"id":84965},"tunneling","Tunneling",[12,84968,84969,84970,84973,84974,84977],{},"When an IPv6 packet needs to cross an ",[106,84971,84972],{},"IPv4-only"," network, it gets ",[106,84975,84976],{},"encapsulated"," inside an IPv4 packet for transit, then unwrapped at the other end. The IPv4 network acts as a tunnel, carrying the IPv6 traffic without needing to understand it.",[7166,84979,84981],{"id":84980},"translation-nat64","Translation (NAT64)",[12,84983,84984,84987],{},[106,84985,84986],{},"NAT64"," enables IPv6-only devices to communicate with IPv4-only devices by translating between the two protocols — similar in concept to how NAT works in IPv4. This is useful when dual-stack isn't an option and the two endpoints speak different protocol versions.",[6964,84989,84991],{"id":84990},"ipv6-subnetting","IPv6 Subnetting",[12,84993,84994,84995,84998,84999,6959],{},"Subnetting in IPv6 has a fundamentally different philosophy from IPv4. In IPv4, subnetting is primarily about ",[106,84996,84997],{},"conserving addresses",". In IPv6, with its enormous address space, subnetting is about ",[106,85000,85001],{},"organizing your network logically and hierarchically",[7166,85003,85005],{"id":85004},"the-standard-approach","The Standard Approach",[12,85007,85008,85009,85011,85012,85015],{},"With a typical ",[6979,85010,84471],{}," global routing prefix, you have a ",[106,85013,85014],{},"16-bit Subnet ID"," to work with. That gives you:",[6972,85017,85020],{"className":85018,"code":85019,"language":6977},[6975],"2¹⁶ = 65,536 possible \u002F64 subnets\n",[6979,85021,85019],{"__ignoreMap":180},[12,85023,85024],{},"And you haven't borrowed a single bit from the Interface ID. Each subnet still has a full 64-bit host portion, supporting an astronomical number of devices.",[6972,85026,85029],{"className":85027,"code":85028,"language":6977},[6975],"Address Block: 2001:0DB8:ACAD::\u002F48\n \nSubnets:\n  2001:0DB8:ACAD:0000::\u002F64\n  2001:0DB8:ACAD:0001::\u002F64\n  2001:0DB8:ACAD:0002::\u002F64\n  ... (up to)\n  2001:0DB8:ACAD:FFFF::\u002F64\n",[6979,85030,85028],{"__ignoreMap":180},[7166,85032,85034],{"id":85033},"subnetting-on-a-nibble-boundary","Subnetting on a Nibble Boundary",[12,85036,85037,85038,85040,85041,85044],{},"In some cases (often for security, to limit the number of hosts per subnet), you can extend the subnet prefix beyond ",[6979,85039,84475],{}," by borrowing bits from the Interface ID. When doing this, the best practice is to subnet on a ",[106,85042,85043],{},"nibble boundary"," — meaning you extend in increments of 4 bits (one hex digit).",[12,85046,85047,85048,10720,85050,85053,85054,85057,85058,85060],{},"For example, extending from ",[6979,85049,84475],{},[6979,85051,85052],{},"\u002F68"," borrows 4 bits from the Interface ID, reducing it from 64 to 60 bits. This creates ",[106,85055,85056],{},"16 sub-subnets"," within each ",[6979,85059,84475],{}," subnet, each supporting fewer hosts.",[6972,85062,85065],{"className":85063,"code":85064,"language":6977},[6975],"\u002F64 prefix → extend by 4 bits → \u002F68 prefix\n  Interface ID shrinks: 64 bits → 60 bits\n  Sub-subnets per \u002F64:  2⁴ = 16\n",[6979,85066,85064],{"__ignoreMap":180},[12,85068,85069,85070,7298,85072,7298,85075,7298,85078,7298,85081,7298,85083,85085],{},"Nibble boundaries (",[6979,85071,84471],{},[6979,85073,85074],{},"\u002F52",[6979,85076,85077],{},"\u002F56",[6979,85079,85080],{},"\u002F60",[6979,85082,84475],{},[6979,85084,85052],{},", ...) keep the math clean because each nibble corresponds to exactly one hexadecimal digit.",[6964,85087,85089],{"id":85088},"quick-reference-summary","Quick Reference Summary",[7085,85091,85092,85102],{},[7088,85093,85094],{},[7091,85095,85096,85099],{},[7094,85097,85098],{},"Concept",[7094,85100,85101],{},"Key Detail",[7104,85103,85104,85112,85120,85131,85141,85151,85165,85174,85183,85192,85200,85207,85215],{},[7091,85105,85106,85109],{},[7109,85107,85108],{},"IPv6 address size",[7109,85110,85111],{},"128 bits (vs. 32 bits for IPv4)",[7091,85113,85114,85117],{},[7109,85115,85116],{},"Address notation",[7109,85118,85119],{},"8 hextets of 4 hex digits, separated by colons",[7091,85121,85122,85125],{},[7109,85123,85124],{},"Shorthand rules",[7109,85126,85127,85128,85130],{},"Drop leading zeros; replace consecutive all-zero groups with ",[6979,85129,84254],{}," (once only)",[7091,85132,85133,85136],{},[7109,85134,85135],{},"Global Unicast",[7109,85137,85138,85140],{},[6979,85139,84394],{}," — publicly routable, similar to public IPv4 addresses",[7091,85142,85143,85146],{},[7109,85144,85145],{},"Link-Local",[7109,85147,85148,85150],{},[6979,85149,84501],{}," — local link only, auto-generated, not routable",[7091,85152,85153,85156],{},[7109,85154,85155],{},"Unique Local",[7109,85157,85158,85160,85161,8046,85163],{},[6979,85159,84584],{}," — site-private, similar to ",[6979,85162,84571],{},[6979,85164,84568],{},[7091,85166,85167,85170],{},[7109,85168,85169],{},"Loopback",[7109,85171,85172],{},[6979,85173,82618],{},[7091,85175,85176,85179],{},[7109,85177,85178],{},"Multicast (all-nodes)",[7109,85180,85181],{},[6979,85182,84770],{},[7091,85184,85185,85188],{},[7109,85186,85187],{},"Multicast (all-routers)",[7109,85189,85190],{},[6979,85191,84630],{},[7091,85193,85194,85197],{},[7109,85195,85196],{},"SLAAC",[7109,85198,85199],{},"Stateless auto-configuration using RA messages",[7091,85201,85202,85204],{},[7109,85203,84690],{},[7109,85205,85206],{},"Generates Interface ID from MAC address",[7091,85208,85209,85212],{},[7109,85210,85211],{},"DAD",[7109,85213,85214],{},"Duplicate Address Detection using NS\u002FNA messages",[7091,85216,85217,85220],{},[7109,85218,85219],{},"Transition methods",[7109,85221,85222],{},"Dual Stack, Tunneling, NAT64",{"title":180,"searchDepth":181,"depth":181,"links":85224},[85225,85226,85233,85238,85246,85251,85252,85256,85260,85265,85269],{"id":84152,"depth":181,"text":84153},{"id":84201,"depth":181,"text":84202,"children":85227},[85228,85229,85230,85232],{"id":84205,"depth":7430,"text":84206},{"id":84233,"depth":7430,"text":84234},{"id":84250,"depth":7430,"text":85231},"Rule 2: Replace Consecutive All-Zero Groups with ::",{"id":84279,"depth":7430,"text":84280},{"id":84306,"depth":181,"text":84307,"children":85234},[85235,85236,85237],{"id":84316,"depth":7430,"text":84317},{"id":84327,"depth":7430,"text":84328},{"id":84338,"depth":7430,"text":84339},{"id":84365,"depth":181,"text":84366,"children":85239},[85240,85241,85242,85243,85244,85245],{"id":84376,"depth":7430,"text":84377},{"id":84479,"depth":7430,"text":84480},{"id":84520,"depth":7430,"text":84521},{"id":84540,"depth":7430,"text":84541},{"id":84557,"depth":7430,"text":84558},{"id":84590,"depth":7430,"text":84591},{"id":84597,"depth":181,"text":84598,"children":85247},[85248,85249,85250],{"id":84604,"depth":7430,"text":84605},{"id":84661,"depth":7430,"text":84662},{"id":84672,"depth":7430,"text":84673},{"id":84683,"depth":181,"text":84684},{"id":84732,"depth":181,"text":84733,"children":85253},[85254,85255],{"id":84743,"depth":7430,"text":84744},{"id":84791,"depth":7430,"text":84792},{"id":84812,"depth":181,"text":84813,"children":85257},[85258,85259],{"id":84899,"depth":7430,"text":84900},{"id":84929,"depth":7430,"text":84930},{"id":84948,"depth":181,"text":84949,"children":85261},[85262,85263,85264],{"id":84955,"depth":7430,"text":84956},{"id":84965,"depth":7430,"text":84966},{"id":84980,"depth":7430,"text":84981},{"id":84990,"depth":181,"text":84991,"children":85266},[85267,85268],{"id":85004,"depth":7430,"text":85005},{"id":85033,"depth":7430,"text":85034},{"id":85088,"depth":181,"text":85089},{},"\u002Fblog\u002FNetwork-Communication,-Chapter-6",{"title":84147,"description":180},{"loc":85271},"blog\u002FNetwork-Communication,-Chapter-6","amEF0Us40CgRO0UbSe1K9AABqIQ3ei12pDYcd_0jUlA",{"id":85277,"title":85278,"author":6946,"body":85279,"date":83251,"description":180,"draft":185,"edited_at":74911,"extension":186,"featured_image":194,"meta":86499,"navigation":189,"path":86500,"pinned":185,"seo":86501,"sitemap":86502,"stem":86503,"tags":194,"__hash__":86504},"blog\u002Fblog\u002FNetwork-Communications,-Chapter-5.md","IPv4 Addressing, Subnetting, and VLSM",{"type":9,"value":85280,"toc":86473},[85281,85285,85295,85309,85315,85325,85329,85340,85349,85441,85460,85464,85474,85477,85481,85484,85497,85512,85524,85528,85532,85538,85562,85566,85573,85623,85629,85636,85639,85643,85649,85655,85659,85666,85747,85759,85763,85770,85774,85778,85785,85795,85799,85812,85815,85857,85860,85864,85871,85885,85888,85892,85901,85908,85918,85938,85944,86002,86015,86019,86022,86027,86122,86127,86248,86252,86266,86290,86295,86334,86337,86341,86344,86354,86360,86377,86381,86384,86390,86404,86410,86416,86419],[6964,85282,85284],{"id":85283},"what-is-ipv4","What is IPv4?",[12,85286,85287,85288,85291,85292,85294],{},"Every device on a network needs an address, just like every house on a street needs a number. In networking, that address is called an ",[106,85289,85290],{},"IP address",", and the most widely used version is ",[106,85293,82506],{}," (Internet Protocol version 4).",[12,85296,85297,85298,85301,85302,85305,85306,7024],{},"An IPv4 address is a ",[106,85299,85300],{},"32-bit number",", but instead of writing out all 32 ones and zeros, we break it into four groups of 8 bits (called ",[106,85303,85304],{},"octets",") and write each group as a decimal number separated by dots. This is known as ",[106,85307,85308],{},"dotted decimal notation",[6972,85310,85313],{"className":85311,"code":85312,"language":6977},[6975],"A.B.C.D\n",[6979,85314,85312],{"__ignoreMap":180},[12,85316,85317,85318,85321,85322,6959],{},"Each octet ranges from ",[106,85319,85320],{},"0 to 255",", so a typical IPv4 address looks like ",[6979,85323,85324],{},"192.168.10.10",[7166,85326,85328],{"id":85327},"the-two-parts-of-an-ip-address","The Two Parts of an IP Address",[12,85330,85331,85332,85335,85336,85339],{},"Here's the key idea: within those 32 bits, some of them identify ",[106,85333,85334],{},"which network"," the device belongs to, and the rest identify ",[106,85337,85338],{},"which host"," (device) it is on that network.",[12,85341,85342,85343,85345,85346,7024],{},"Take the address ",[6979,85344,85324],{}," with a subnet mask of ",[6979,85347,85348],{},"255.255.255.0",[7085,85350,85351,85371],{},[7088,85352,85353],{},[7091,85354,85355,85359,85362,85365,85368],{},[7094,85356,85357],{},[16,85358],{},[7094,85360,85361],{},"Octet 1",[7094,85363,85364],{},"Octet 2",[7094,85366,85367],{},"Octet 3",[7094,85369,85370],{},"Octet 4",[7104,85372,85373,85390,85408,85424],{},[7091,85374,85375,85380,85383,85386,85388],{},[7109,85376,85377],{},[106,85378,85379],{},"IPv4 Address",[7109,85381,85382],{},"192",[7109,85384,85385],{},"168",[7109,85387,17383],{},[7109,85389,17383],{},[7091,85391,85392,85397,85400,85403,85406],{},[7109,85393,85394],{},[106,85395,85396],{},"Binary",[7109,85398,85399],{},"11000000",[7109,85401,85402],{},"10101000",[7109,85404,85405],{},"00001010",[7109,85407,85405],{},[7091,85409,85410,85415,85418,85420,85422],{},[7109,85411,85412],{},[106,85413,85414],{},"Subnet Mask",[7109,85416,85417],{},"255",[7109,85419,85417],{},[7109,85421,85417],{},[7109,85423,9180],{},[7091,85425,85426,85431,85434,85436,85438],{},[7109,85427,85428],{},[106,85429,85430],{},"Mask in Binary",[7109,85432,85433],{},"11111111",[7109,85435,85433],{},[7109,85437,85433],{},[7109,85439,85440],{},"00000000",[12,85442,85443,85444,85447,85448,85451,85452,85455,85456,85459],{},"The 1s in the subnet mask mark the ",[106,85445,85446],{},"network portion",", and the 0s mark the ",[106,85449,85450],{},"host portion",". So in this case, ",[6979,85453,85454],{},"192.168.10"," is the network, and ",[6979,85457,85458],{},".10"," identifies the specific device.",[7166,85461,85463],{"id":85462},"subnet-masks-and-the-anding-process","Subnet Masks and the ANDing Process",[12,85465,9407,85466,85469,85470,85473],{},[106,85467,85468],{},"subnet mask"," tells us exactly where the boundary lies between the network bits and the host bits. When you assign an IP address to a device, the subnet mask is used to determine the network address through a process called ",[106,85471,85472],{},"ANDing"," — a bitwise AND operation between the IP address and the subnet mask.",[12,85475,85476],{},"Think of it this way: ANDing is like laying the mask on top of the address and only letting through the network part.",[6964,85478,85480],{"id":85479},"three-types-of-addresses-in-every-network","Three Types of Addresses in Every Network",[12,85482,85483],{},"Within any network, there are three special types of addresses you need to know:",[12,85485,85486,85489,85490,9019,85493,85496],{},[106,85487,85488],{},"Network Address"," — This is the \"name\" of the network itself. All host bits are set to 0. You can't assign this to a device; it's used to refer to the network as a whole. For example, in ",[6979,85491,85492],{},"192.168.1.0\u002F24",[6979,85494,85495],{},".0"," means all host bits are zero.",[12,85498,85499,85502,85503,85505,85506,85508,85509,6959],{},[106,85500,85501],{},"Broadcast Address"," — This is the address used to send a message to ",[7289,85504,7291],{}," device on the network. All host bits are set to 1. For the network ",[6979,85507,85492],{},", the broadcast address is ",[6979,85510,85511],{},"192.168.1.255",[12,85513,85514,85517,85518,43657,85521,6959],{},[106,85515,85516],{},"Host Addresses"," — These are all the addresses in between the network address and the broadcast address. These are the ones you actually assign to devices. In our example, the usable host range is ",[6979,85519,85520],{},"192.168.1.1",[6979,85522,85523],{},"192.168.1.254",[6964,85525,85527],{"id":85526},"types-of-ipv4-addresses","Types of IPv4 Addresses",[7166,85529,85531],{"id":85530},"unicast-multicast-and-beyond","Unicast, Multicast, and Beyond",[12,85533,85534,85537],{},[106,85535,85536],{},"Unicast addresses"," (0.0.0.0 to 223.255.255.255) are used for one-to-one communication — one sender, one receiver. This is the most common type.",[12,85539,85540,85543,85544,85547,85548,10720,85551,85554,85555,10720,85558,85561],{},[106,85541,85542],{},"Multicast addresses"," (224.0.0.0 to 239.255.255.255) are used for one-to-many communication. A device sends a single packet, and it gets delivered to a ",[7289,85545,85546],{},"group"," of interested receivers. Within this range, ",[6979,85549,85550],{},"224.0.0.0",[6979,85552,85553],{},"224.0.0.255"," are reserved for link-local multicast, while ",[6979,85556,85557],{},"224.0.1.0",[6979,85559,85560],{},"238.255.255.255"," are globally scoped.",[7166,85563,85565],{"id":85564},"public-vs-private-addresses","Public vs. Private Addresses",[12,85567,85568,85569,85572],{},"Not every IP address is meant to be seen on the public internet. ",[106,85570,85571],{},"Private addresses"," (defined in RFC 1918) are reserved for use within internal networks. If your device doesn't need direct internet access, it can use a private address. The three private ranges are:",[7085,85574,85575,85588],{},[7088,85576,85577],{},[7091,85578,85579,85582,85585],{},[7094,85580,85581],{},"Range",[7094,85583,85584],{},"CIDR Notation",[7094,85586,85587],{},"Typical Use",[7104,85589,85590,85601,85612],{},[7091,85591,85592,85595,85598],{},[7109,85593,85594],{},"10.0.0.0 – 10.255.255.255",[7109,85596,85597],{},"10.0.0.0\u002F8",[7109,85599,85600],{},"Large enterprises",[7091,85602,85603,85606,85609],{},[7109,85604,85605],{},"172.16.0.0 – 172.31.255.255",[7109,85607,85608],{},"172.16.0.0\u002F12",[7109,85610,85611],{},"Medium networks",[7091,85613,85614,85617,85620],{},[7109,85615,85616],{},"192.168.0.0 – 192.168.255.255",[7109,85618,85619],{},"192.168.0.0\u002F16",[7109,85621,85622],{},"Home & small office",[12,85624,85625,85628],{},[106,85626,85627],{},"Public addresses"," are everything else in the unicast range. These are globally unique and routable on the internet — they're what web servers, cloud services, and any publicly accessible host use.",[12,85630,85631,85632,85635],{},"One critical rule: ",[106,85633,85634],{},"private addresses cannot be routed over the internet."," If a device with a private address needs to reach the internet, a router must perform Network Address Translation (NAT) to swap the private address for a public one.",[12,85637,85638],{},"In a typical network setup, devices on the intranet use private addresses, servers in the DMZ (demilitarized zone) use public addresses, and a router connects everything to the internet.",[7166,85640,85642],{"id":85641},"special-addresses","Special Addresses",[12,85644,85645,85648],{},[106,85646,85647],{},"Loopback (127.0.0.1)"," — This is the \"talk to yourself\" address. When a device sends traffic to 127.0.0.1, it's sending it right back to itself. It's commonly used for testing whether the TCP\u002FIP stack on a machine is working.",[12,85650,85651,85654],{},[106,85652,85653],{},"Link-Local (169.254.0.0\u002F16)"," — If a device can't get an IP address from a DHCP server, the operating system may automatically assign it an address in this range. You've probably seen this happen when your Wi-Fi connects but \"has no internet\" — the device falls back to a link-local address.",[6964,85656,85658],{"id":85657},"classful-addressing-historical-context","Classful Addressing (Historical Context)",[12,85660,85661,85662,85665],{},"In the early days of the internet, IP addresses were divided into classes. While ",[106,85663,85664],{},"classful addressing is largely obsolete today"," (replaced by CIDR), understanding it helps you make sense of older documentation and networking exams.",[7085,85667,85668,85683],{},[7088,85669,85670],{},[7091,85671,85672,85674,85677,85680],{},[7094,85673,23887],{},[7094,85675,85676],{},"1st Octet Range",[7094,85678,85679],{},"Network\u002FHost Split",[7094,85681,85682],{},"Default Subnet Mask",[7104,85684,85685,85698,85711,85723,85735],{},[7091,85686,85687,85689,85692,85695],{},[7109,85688,13631],{},[7109,85690,85691],{},"1–127",[7109,85693,85694],{},"N.H.H.H",[7109,85696,85697],{},"255.0.0.0",[7091,85699,85700,85702,85705,85708],{},[7109,85701,13639],{},[7109,85703,85704],{},"128–191",[7109,85706,85707],{},"N.N.H.H",[7109,85709,85710],{},"255.255.0.0",[7091,85712,85713,85715,85718,85721],{},[7109,85714,13642],{},[7109,85716,85717],{},"192–223",[7109,85719,85720],{},"N.N.N.H",[7109,85722,85348],{},[7091,85724,85725,85727,85730,85733],{},[7109,85726,13635],{},[7109,85728,85729],{},"224–239",[7109,85731,85732],{},"Multicast (no mask)",[7109,85734,84893],{},[7091,85736,85737,85739,85742,85745],{},[7109,85738,7402],{},[7109,85740,85741],{},"240–255",[7109,85743,85744],{},"Experimental",[7109,85746,84893],{},[12,85748,85749,85750,85752,85753,85755,85756,13335],{},"The first few bits of the first octet determine the class: Class A starts with ",[6979,85751,9180],{},", Class B with ",[6979,85754,17383],{},", Class C with ",[6979,85757,85758],{},"110",[7166,85760,85762],{"id":85761},"who-manages-ip-address-assignments","Who Manages IP Address Assignments?",[12,85764,85765,85766,85769],{},"Public IP addresses are managed by five ",[106,85767,85768],{},"Regional Internet Registries (RIRs)",", each responsible for a different part of the world: ARIN (North America), RIPE NCC (Europe & Middle East), APNIC (Asia-Pacific — this covers Taiwan!), AfriNIC (Africa), and LACNIC (Latin America & Caribbean).",[6964,85771,85773],{"id":85772},"subnetting-dividing-networks-into-smaller-pieces","Subnetting: Dividing Networks into Smaller Pieces",[7166,85775,85777],{"id":85776},"why-subnet","Why Subnet?",[12,85779,85780,85781,85784],{},"Imagine a single network with thousands of devices. Every time any device sends a broadcast (and they do this often), ",[7289,85782,85783],{},"every other device"," on the network has to process that broadcast. This creates a huge amount of unnecessary traffic, slowing everything down.",[12,85786,85787,85790,85791,85794],{},[106,85788,85789],{},"Subnetting"," is the solution. It divides one large network into smaller ",[106,85792,85793],{},"subnets",", each with its own broadcast domain. Broadcasts in one subnet don't affect devices in another. The result: less congestion, better performance, and easier management.",[7166,85796,85798],{"id":85797},"how-subnets-communicate","How Subnets Communicate",[12,85800,85801,85802,85805,85806,85809,85810,6959],{},"Devices on the ",[106,85803,85804],{},"same subnet"," can talk to each other directly. But to communicate across subnets, traffic must pass through a ",[106,85807,85808],{},"router",". Each device uses the router interface on its local subnet as its ",[106,85811,82661],{},[12,85813,85814],{},"For example, consider a university network with a single router (R1) connected to the internet. The network is divided into four subnets:",[23,85816,85817,85827,85837,85847],{},[26,85818,85819,85822,85823,85826],{},[106,85820,85821],{},"Administration"," — LAN 1: ",[6979,85824,85825],{},"10.0.1.0\u002F24"," (via interface G0\u002F0)",[26,85828,85829,85832,85833,85836],{},[106,85830,85831],{},"Students"," — LAN 2: ",[6979,85834,85835],{},"10.0.2.0\u002F24"," (via interface G0\u002F1)",[26,85838,85839,85842,85843,85846],{},[106,85840,85841],{},"Human Resources"," — LAN 3: ",[6979,85844,85845],{},"10.0.3.0\u002F24"," (via interface G0\u002F3)",[26,85848,85849,85852,85853,85856],{},[106,85850,85851],{},"Accounting"," — LAN 4: ",[6979,85854,85855],{},"10.0.4.0\u002F24"," (via interface G0\u002F2)",[12,85858,85859],{},"Each subnet is isolated. A student's computer can't directly broadcast to an accounting workstation — traffic between them goes through R1.",[7166,85861,85863],{"id":85862},"the-math-behind-subnetting","The Math Behind Subnetting",[12,85865,85866,85867,85870],{},"Subnetting works by ",[106,85868,85869],{},"borrowing bits"," from the host portion of an address and using them as additional network bits. The key formulas are:",[23,85872,85873,85879],{},[26,85874,85875,85878],{},[106,85876,85877],{},"Number of subnets"," = 2^(borrowed bits)",[26,85880,85881,85884],{},[106,85882,85883],{},"Number of usable hosts per subnet"," = 2^(remaining host bits) − 2",[12,85886,85887],{},"We subtract 2 because every subnet reserves one address for the network address and one for the broadcast address.",[7166,85889,85891],{"id":85890},"worked-example-subnetting-a-24-network","Worked Example: Subnetting a \u002F24 Network",[12,85893,85894,85895,85897,85898,85900],{},"Let's subnet ",[6979,85896,85492],{}," by borrowing ",[106,85899,76424],{}," from the host portion.",[12,85902,85903,85904,85907],{},"The original last octet in binary: ",[6979,85905,85906],{},"00 | 000000"," (after borrowing 2 bits, the first 2 become subnet bits, and 6 remain as host bits).",[12,85909,85910,85911,85914,85915,85917],{},"The new subnet mask becomes ",[6979,85912,85913],{},"255.255.255.192"," (or \u002F26), since those 2 extra bits in the last octet give us ",[6979,85916,85399],{}," = 192.",[23,85919,85920,85929],{},[26,85921,85922,85925,85926],{},[106,85923,85924],{},"Number of subnets:"," 2² = ",[106,85927,85928],{},"4 subnets",[26,85930,85931,85934,85935],{},[106,85932,85933],{},"Hosts per subnet:"," 2⁶ = 64 total, minus 2 = ",[106,85936,85937],{},"62 usable hosts",[12,85939,85940,85941,24917],{},"For the first subnet (",[6979,85942,85943],{},"192.168.1.0\u002F26",[7085,85945,85946,85959],{},[7088,85947,85948],{},[7091,85949,85950,85953,85956],{},[7094,85951,85952],{},"Type",[7094,85954,85955],{},"Binary (last octet)",[7094,85957,85958],{},"Decimal",[7104,85960,85961,85971,85981,85992],{},[7091,85962,85963,85965,85968],{},[7109,85964,85488],{},[7109,85966,85967],{},"00 000000",[7109,85969,85970],{},"192.168.1.0",[7091,85972,85973,85976,85979],{},[7109,85974,85975],{},"First Host",[7109,85977,85978],{},"00 000001",[7109,85980,85520],{},[7091,85982,85983,85986,85989],{},[7109,85984,85985],{},"Last Host",[7109,85987,85988],{},"00 111110",[7109,85990,85991],{},"192.168.1.62",[7091,85993,85994,85996,85999],{},[7109,85995,83838],{},[7109,85997,85998],{},"00 111111",[7109,86000,86001],{},"192.168.1.63",[12,86003,86004,86005,19180,86008,86011,86012,6959],{},"The next subnet starts at ",[6979,86006,86007],{},"192.168.1.64",[6979,86009,86010],{},"192.168.1.128",", and finally ",[6979,86013,86014],{},"192.168.1.192",[7166,86016,86018],{"id":86017},"subnetting-reference-tables","Subnetting Reference Tables",[12,86020,86021],{},"Here are quick reference tables for common subnetting scenarios.",[12,86023,86024],{},[106,86025,86026],{},"Subnetting a \u002F24 Network:",[7085,86028,86029,86044],{},[7088,86030,86031],{},[7091,86032,86033,86036,86038,86041],{},[7094,86034,86035],{},"Prefix",[7094,86037,85414],{},[7094,86039,86040],{},"# of Subnets",[7094,86042,86043],{},"# of Usable Hosts",[7104,86045,86046,86059,86071,86083,86097,86110],{},[7091,86047,86048,86051,86054,86056],{},[7109,86049,86050],{},"\u002F25",[7109,86052,86053],{},"255.255.255.128",[7109,86055,8879],{},[7109,86057,86058],{},"126",[7091,86060,86061,86064,86066,86068],{},[7109,86062,86063],{},"\u002F26",[7109,86065,85913],{},[7109,86067,8348],{},[7109,86069,86070],{},"62",[7091,86072,86073,86076,86079,86081],{},[7109,86074,86075],{},"\u002F27",[7109,86077,86078],{},"255.255.255.224",[7109,86080,31184],{},[7109,86082,26103],{},[7091,86084,86085,86088,86091,86094],{},[7109,86086,86087],{},"\u002F28",[7109,86089,86090],{},"255.255.255.240",[7109,86092,86093],{},"16",[7109,86095,86096],{},"14",[7091,86098,86099,86102,86105,86107],{},[7109,86100,86101],{},"\u002F29",[7109,86103,86104],{},"255.255.255.248",[7109,86106,26058],{},[7109,86108,86109],{},"6",[7091,86111,86112,86115,86118,86120],{},[7109,86113,86114],{},"\u002F30",[7109,86116,86117],{},"255.255.255.252",[7109,86119,25144],{},[7109,86121,8879],{},[12,86123,86124],{},[106,86125,86126],{},"Subnetting a \u002F16 Network:",[7085,86128,86129,86141],{},[7088,86130,86131],{},[7091,86132,86133,86135,86137,86139],{},[7094,86134,86035],{},[7094,86136,85414],{},[7094,86138,86040],{},[7094,86140,86043],{},[7104,86142,86143,86156,86169,86182,86195,86208,86221,86235],{},[7091,86144,86145,86148,86151,86153],{},[7109,86146,86147],{},"\u002F17",[7109,86149,86150],{},"255.255.128.0",[7109,86152,8879],{},[7109,86154,86155],{},"32,766",[7091,86157,86158,86161,86164,86166],{},[7109,86159,86160],{},"\u002F18",[7109,86162,86163],{},"255.255.192.0",[7109,86165,8348],{},[7109,86167,86168],{},"16,382",[7091,86170,86171,86174,86177,86179],{},[7109,86172,86173],{},"\u002F19",[7109,86175,86176],{},"255.255.224.0",[7109,86178,31184],{},[7109,86180,86181],{},"8,190",[7091,86183,86184,86187,86190,86192],{},[7109,86185,86186],{},"\u002F20",[7109,86188,86189],{},"255.255.240.0",[7109,86191,86093],{},[7109,86193,86194],{},"4,094",[7091,86196,86197,86200,86203,86205],{},[7109,86198,86199],{},"\u002F21",[7109,86201,86202],{},"255.255.248.0",[7109,86204,26058],{},[7109,86206,86207],{},"2,046",[7091,86209,86210,86213,86216,86218],{},[7109,86211,86212],{},"\u002F22",[7109,86214,86215],{},"255.255.252.0",[7109,86217,25144],{},[7109,86219,86220],{},"1,022",[7091,86222,86223,86226,86229,86232],{},[7109,86224,86225],{},"\u002F23",[7109,86227,86228],{},"255.255.254.0",[7109,86230,86231],{},"128",[7109,86233,86234],{},"510",[7091,86236,86237,86240,86242,86245],{},[7109,86238,86239],{},"\u002F24",[7109,86241,85348],{},[7109,86243,86244],{},"256",[7109,86246,86247],{},"254",[7166,86249,86251],{"id":86250},"worked-example-subnetting-a-16-network","Worked Example: Subnetting a \u002F16 Network",[12,86253,86254,86255,86258,86259,86262,86263,86265],{},"Consider the Class B network ",[6979,86256,86257],{},"172.16.0.0\u002F16",". If we borrow ",[106,86260,86261],{},"7 bits"," from the host portion, the new mask becomes ",[6979,86264,86228],{}," (or \u002F23).",[23,86267,86268,86276,86282],{},[26,86269,86270,86272,86273],{},[106,86271,85924],{}," 2⁷ = ",[106,86274,86275],{},"128 subnets",[26,86277,86278,86281],{},[106,86279,86280],{},"Remaining host bits:"," 9",[26,86283,86284,86286,86287],{},[106,86285,85933],{}," 2⁹ − 2 = ",[106,86288,86289],{},"510 usable hosts",[12,86291,85940,86292,24917],{},[6979,86293,86294],{},"172.16.0.0\u002F23",[7085,86296,86297,86305],{},[7088,86298,86299],{},[7091,86300,86301,86303],{},[7094,86302,85952],{},[7094,86304,84756],{},[7104,86306,86307,86313,86320,86327],{},[7091,86308,86309,86311],{},[7109,86310,85488],{},[7109,86312,86294],{},[7091,86314,86315,86317],{},[7109,86316,85975],{},[7109,86318,86319],{},"172.16.0.1\u002F23",[7091,86321,86322,86324],{},[7109,86323,85985],{},[7109,86325,86326],{},"172.16.0.254\u002F23",[7091,86328,86329,86331],{},[7109,86330,83838],{},[7109,86332,86333],{},"172.16.1.255",[12,86335,86336],{},"Notice how the subnet spans two values in the third octet (0 and 1) because we have 9 host bits that cross the octet boundary.",[6964,86338,86340],{"id":86339},"variable-length-subnet-mask-vlsm","Variable Length Subnet Mask (VLSM)",[12,86342,86343],{},"Standard subnetting gives every subnet the same size, but real-world networks rarely have equal needs. A headquarters might need 40 hosts while a small branch only needs 10. Giving both subnets the same \u002F26 mask wastes addresses in the smaller branch.",[12,86345,86346,86349,86350,86353],{},[106,86347,86348],{},"VLSM"," (Variable Length Subnet Mask) solves this by allowing you to ",[106,86351,86352],{},"subnet a subnet"," — using different mask lengths for different parts of the network.",[12,86355,86356,86357,86359],{},"For example, with ",[6979,86358,86257],{},", you might create seven \u002F27 subnets (30 hosts each) for your larger offices, then take one of the remaining blocks and further divide it into eight smaller \u002F30 subnets (2 hosts each) for point-to-point router links.",[12,86361,86362,86363,7042,86366,86369,86370,7042,86373,86376],{},"In a real enterprise topology, subnet assignments might look like: the corporate headquarters gets ",[6979,86364,86365],{},"172.16.0.0\u002F26",[6979,86367,86368],{},"172.16.0.64\u002F26",", Branch 1 gets ",[6979,86371,86372],{},"172.16.0.128\u002F26",[6979,86374,86375],{},"172.16.0.192\u002F26",", and so on — each sized to match the actual number of hosts needed at that site.",[6964,86378,86380],{"id":86379},"planning-your-address-scheme","Planning Your Address Scheme",[12,86382,86383],{},"When designing a network's addressing plan, there are several key decisions to make:",[12,86385,86386,86389],{},[106,86387,86388],{},"Determine your needs"," — How many subnets do you need? How many hosts per subnet? These two numbers will guide which private address block to use and how many bits to borrow.",[12,86391,86392,86395,86396,86399,86400,86403],{},[106,86393,86394],{},"Decide on static vs. dynamic addressing"," — Servers, printers, and network infrastructure should have ",[106,86397,86398],{},"static (fixed) IP addresses"," so they're always reachable at the same address. Regular workstations and mobile devices can use ",[106,86401,86402],{},"DHCP"," to receive addresses automatically.",[12,86405,86406,86409],{},[106,86407,86408],{},"Prevent duplicate addresses"," — Every host in the entire internetwork must have a unique address. Overlapping subnets or duplicate assignments will cause connectivity issues that are painful to debug.",[12,86411,86412,86415],{},[106,86413,86414],{},"Plan for access control and monitoring"," — Grouping devices by function into subnets makes it easier to apply security policies (like firewall rules) and monitor performance. For instance, putting all servers in one subnet lets you apply stricter access controls to that subnet.",[6964,86417,86418],{"id":17905},"Quick Recap",[23,86420,86421,86428,86438,86451,86460,86465,86470],{},[26,86422,86423,86425,86426,16419],{},[106,86424,82506],{}," addresses are 32-bit numbers written in dotted decimal (e.g., ",[6979,86427,85520],{},[26,86429,86430,86431,23384,86433,86435,86436,6959],{},"Every address has a ",[106,86432,85446],{},[106,86434,85450],{},", determined by the ",[106,86437,85468],{},[26,86439,86440,86441,14022,86444,7153,86447,86450],{},"Each network has a ",[106,86442,86443],{},"network address",[106,86445,86446],{},"broadcast address",[106,86448,86449],{},"usable host addresses"," in between.",[26,86452,86453,86455,86456,86459],{},[106,86454,85571],{}," (10.x.x.x, 172.16–31.x.x, 192.168.x.x) are for internal use; ",[106,86457,86458],{},"public addresses"," are for the internet.",[26,86461,86462,86464],{},[106,86463,85789],{}," divides large networks into smaller ones by borrowing host bits, reducing broadcast traffic and improving manageability.",[26,86466,86467,86469],{},[106,86468,86348],{}," lets you create subnets of different sizes to match real-world requirements.",[26,86471,86472],{},"Good address planning considers subnet sizing, static vs. dynamic allocation, uniqueness, and security.",{"title":180,"searchDepth":181,"depth":181,"links":86474},[86475,86479,86480,86485,86488,86496,86497,86498],{"id":85283,"depth":181,"text":85284,"children":86476},[86477,86478],{"id":85327,"depth":7430,"text":85328},{"id":85462,"depth":7430,"text":85463},{"id":85479,"depth":181,"text":85480},{"id":85526,"depth":181,"text":85527,"children":86481},[86482,86483,86484],{"id":85530,"depth":7430,"text":85531},{"id":85564,"depth":7430,"text":85565},{"id":85641,"depth":7430,"text":85642},{"id":85657,"depth":181,"text":85658,"children":86486},[86487],{"id":85761,"depth":7430,"text":85762},{"id":85772,"depth":181,"text":85773,"children":86489},[86490,86491,86492,86493,86494,86495],{"id":85776,"depth":7430,"text":85777},{"id":85797,"depth":7430,"text":85798},{"id":85862,"depth":7430,"text":85863},{"id":85890,"depth":7430,"text":85891},{"id":86017,"depth":7430,"text":86018},{"id":86250,"depth":7430,"text":86251},{"id":86339,"depth":181,"text":86340},{"id":86379,"depth":181,"text":86380},{"id":17905,"depth":181,"text":86418},{},"\u002Fblog\u002FNetwork-Communications,-Chapter-5",{"title":85278,"description":180},{"loc":86500},"blog\u002FNetwork-Communications,-Chapter-5","G7K7S74XGoCd5HrJFs78LckJ0B9YuyOuPnERUaxxuHY",{"id":86506,"title":86507,"author":7,"body":86508,"date":83251,"description":86651,"draft":185,"edited_at":83251,"extension":186,"featured_image":86652,"meta":86653,"navigation":189,"path":86654,"pinned":185,"seo":86655,"sitemap":86656,"stem":86657,"tags":194,"__hash__":86658},"blog\u002Fblog\u002F小兒發展（四）.md","小兒發展（四）",{"type":9,"value":86509,"toc":86649},[86510,86513,86536,86538,86541,86567,86569,86572,86592,86594,86597,86611,86613,86616,86630,86632,86635],[12,86511,86512],{},"接下來講到括約肌控制發展（大小便控制）",[3603,86514,86515,86518,86527,86533],{},[26,86516,86517],{},"十五個月——能告訴爸媽想上廁所或者褲子濕了",[26,86519,86520,86521,86523,86524,86526],{},"十八個月——白天褲子不太會濕，當然也有意外；",[16,86522],{},"\n這邊有一位學者研究說假如太早訓練寶寶大小便，會有焦慮症問題；",[16,86525],{},"\n所以歐美都偏向包尿布到兩三歲，但東方人則偏好早點訓練",[26,86528,86529,86530,86532],{},"兩到三歲——半夜讓小孩上廁所就整晚不會尿床",[16,86531],{},"\n有研究表明，晚上會叫尿床的都比較神經質(講義原句XD)",[26,86534,86535],{},"三歲——能自己上廁所但擦不乾淨，要到六歲才能擦乾淨",[36,86537],{},[12,86539,86540],{},"接著講ADL的發展",[3603,86542,86543,86546,86549,86555,86558,86561,86564],{},[26,86544,86545],{},"剛出生到二十四周——可以用吸吮吞嚥反射以及杯子喝水，但後者較難，需要嘴唇及吞嚥協調才不會嗆到",[26,86547,86548],{},"六個月——開始長牙，能夠吃一些固體食物例如餅乾，這時也已經可以訓練舌頭口腔的協調",[26,86550,86551,86552,86554],{},"十五個月——用湯匙進食(但foreman supination還沒成熟，所以靠轉動湯匙進食)，可以吃半固體例如果醬；",[16,86553],{},"\n也開始可以脫襪子和鞋子",[26,86556,86557],{},"十八個月——會脫襪子鞋子，將拉鏈拉開，不需要轉動湯匙進食",[26,86559,86560],{},"兩歲——可以穿脫襪子鞋子褲子",[26,86562,86563],{},"三歲——除了背後有鈕扣以外，都會穿脫衣服了；分不清左右，要到六歲才能分清",[26,86565,86566],{},"五歲——會綁鞋子",[36,86568],{},[12,86570,86571],{},"社會性情緒發展部份",[3603,86573,86574,86577,86583,86586,86589],{},[26,86575,86576],{},"剛出生到六個月——認得懂媽媽，找不到媽媽，不知道媽媽意思是甚麼，逗他會笑",[26,86578,86579,86580,86582],{},"六到十二個月——十分依賴，認得且會得媽媽",[16,86581],{},"\n這個階段很重要，形成安全連結，以媽媽作為探索世界的據點，後面才能發展社會性",[26,86584,86585],{},"十二到十八個月——有物體恆久的概念，例如媽媽只是暫時離開，會回來",[26,86587,86588],{},"十八到二十四個月——與媽媽分開會有焦慮；開始可以接觸陌生環境",[26,86590,86591],{},"二十四到三十六個月——越來越適應陌生環境，上托兒所",[36,86593],{},[12,86595,86596],{},"自我概念發展部份",[3603,86598,86599,86602,86605,86608],{},[26,86600,86601],{},"剛出生到六個月——動作控制不佳；稍微分得出自我跟非自我",[26,86603,86604],{},"六到十二個月——可以自我翻身；自我跟環境分得開，有自我概念",[26,86606,86607],{},"十八到二十四個月——對自己名字敏感",[26,86609,86610],{},"二十四到三十六個月——看到男女上廁所的差異，對性別有概念",[36,86612],{},[12,86614,86615],{},"情緒發展部份",[3603,86617,86618,86621,86624,86627],{},[26,86619,86620],{},"剛出生到六個月——早期只有哭跟高興表情",[26,86622,86623],{},"六到十二個月——開始有其他情緒例如厭惡，害怕，憤怒",[26,86625,86626],{},"十二到十八個月——從高興分得出喜愛，有喜歡的人事物，也喜歡做讓自己有成就感的事",[26,86628,86629],{},"十八到二十四個月——能夠用語言表達情緒",[36,86631],{},[12,86633,86634],{},"依賴及獨立發展",[3603,86636,86637,86640,86643],{},[26,86638,86639],{},"剛出生到十二個月——完全依賴照顧",[26,86641,86642],{},"十二到十八個月——知道自己是個體後，想自我尋求",[26,86644,86645,86646,86648],{},"二十四到三十六個月——覺得自己了不起，但遇到無法達成的事還是有挫折感；",[16,86647],{},"\n在適當協助下明白甚麼事能做跟不能做，建立初步信心",{"title":180,"searchDepth":181,"depth":181,"links":86650},[],"括約肌控制~依賴獨立發展","\u002Fimages\u002Fuploads\u002F1776052282656-EXjYORHUcAALULe.webp",{},"\u002Fblog\u002F小兒發展（四）",{"title":86507,"description":86651},{"loc":86654},"blog\u002F小兒發展（四）","iCO6Xr2Vz0v9l6ulRBKlWBt93Dzu8k8I4HDGsX-Dbpc",{"id":86660,"title":86661,"author":6946,"body":86662,"date":83251,"description":180,"draft":185,"edited_at":83251,"extension":186,"featured_image":194,"meta":91787,"navigation":189,"path":91788,"pinned":185,"seo":91789,"sitemap":91790,"stem":91791,"tags":194,"__hash__":91792},"blog\u002Fblog\u002F決策樹.md","決策樹",{"type":9,"value":86663,"toc":91762},[86664,86668,86671,86674,86679,86685,86688,86695,86712,86715,86719,86722,86759,86763,86766,86772,86875,86881,86885,86888,86892,86898,87111,87170,88358,88361,88375,88396,88587,88731,88734,88748,88752,88758,89085,89177,89264,89891,89898,89901,89904,89981,90095,90100,90330,90335,90611,90616,90880,90883,90887,90893,90922,90931,90935,90938,91069,91215,91218,91237,91240,91247,91250,91256,91259,91278,91285,91288,91291,91387,91432,91435,91438,91441,91445,91448,91454,91460,91464,91467,91484,91527,91530,91544,91548,91647,91688,91691,91694,91711,91714,91731,91735,91738,91757,91760],[6964,86665,86667],{"id":86666},"決策樹是什麼","決策樹是什麼？",[12,86669,86670],{},"決策樹就跟它的名字一樣，是一棵用來做決策的樹狀模型。每個內部節點會針對某個特徵提出一個問題，每條分支代表一個可能的答案，而每片葉子就是最終的分類結果。",[12,86672,86673],{},"來看一個經典例子：今天該不該去打網球？",[12,86675,86676],{},[1930,86677],{"alt":1932,"src":86678},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776069090146-Screenshot_2026-04-13_at_4.30.55_PM.png",[6972,86680,86683],{"className":86681,"code":86682,"language":6977},[6975],"                   天氣狀況\n                 \u002F    |     \\\n              晴天   陰天    雨天\n               \u002F      |       \\\n            濕度      Yes     風力\n           \u002F   \\             \u002F    \\\n         高    正常        強     弱\n         \u002F       \\         \u002F       \\\n        No       Yes      No      Yes\n",[6979,86684,86682],{"__ignoreMap":180},[12,86686,86687],{},"這棵樹會先檢查天氣狀況。如果是陰天，什麼都不用管，去打球就對了（Yes）；如果是晴天，就接著檢查濕度；如果是雨天，就看風速大不大。邏輯非常簡單又好懂對吧？",[12,86689,86690,86691,86694],{},"我們也可以把這棵樹寫成一組邏輯規則（叫做",[106,86692,86693],{},"析取範式，disjunctive normal form","）：",[23,86696,86697,86702,86707],{},[26,86698,86699,86700],{},"（天氣 = 晴天）∧（濕度 = 正常）→ ",[106,86701,16122],{},[26,86703,86704,86705],{},"（天氣 = 陰天）→ ",[106,86706,16122],{},[26,86708,86709,86710],{},"（天氣 = 雨天）∧（風力 = 弱）→ ",[106,86711,16122],{},[12,86713,86714],{},"這也是決策樹最棒的地方之一：你永遠可以把它轉換成人類看得懂的 if-then 規則。",[6964,86716,86718],{"id":86717},"為什麼要用決策樹","為什麼要用決策樹？",[12,86720,86721],{},"決策樹有幾個很不錯的特性，讓它很適合作為分類問題的入門方式：",[23,86723,86724,86730,86736,86742,86748],{},[26,86725,86726,86729],{},[106,86727,86728],{},"離散和連續特徵都能處理。"," 像顏色（紅、藍、綠）這種離散特徵，決策樹處理起來很自然。至於像溫度這種連續數值，決策樹也能透過設定門檻來切分（例如：溫度 \u003C 30°C vs. ≥ 30°C）。",[26,86731,86732,86735],{},[106,86733,86734],{},"天生就能表達「或(OR)」的邏輯。"," 有時候分類的界線是由多個條件組合而成的，決策樹處理這種情況得心應手。",[26,86737,86738,86741],{},[106,86739,86740],{},"對雜訊(Noise)有一定的容忍度。"," 就算資料裡有一些標記錯誤的樣本，或是特徵值有點誤差，它還是能運作得不錯。",[26,86743,86744,86747],{},[106,86745,86746],{},"能處理缺失值。"," 就算某筆資料少了某個特徵的值，演算法依然有辦法處理。",[26,86749,86750,86751,86754,86755,86758],{},"分為兩種：",[106,86752,86753],{},"分類樹 (Classification trees)"," 會在葉節點輸出離散的類別標籤；而",[106,86756,86757],{},"迴歸樹 (Regression trees)"," 則是預測出連續的實數值。",[6964,86760,86762],{"id":86761},"怎麼建構一棵決策樹","怎麼建構一棵決策樹？",[12,86764,86765],{},"建構決策樹的核心是一個「由上而下、遞迴、分而治之 (divide-and-conquer)」的過程。虛擬碼大概長這樣：",[6972,86767,86770],{"className":86768,"code":86769,"language":6977},[6975],"DTree(examples, features):\n    if 所有 examples 都屬於同一個類別：\n        return 一個標有該類別的葉節點\n\n    if 已經沒有 features 可以用了：\n        return 一個標有多數類別的葉節點\n\n    挑一個「最好的」特徵 F\n    為 F 建立一個節點\n    對 F 的每個可能值 v：\n        取出 F = v 的資料子集 (examples_v)\n        如果 examples_v 是空的：\n            接上一個標有 examples 多數類別的葉子\n        否則：\n            接上 DTree(examples_v, features - {F}) 的結果\n\n    回傳以這個節點為根的子樹\n",[6979,86771,86769],{"__ignoreMap":180},[25631,86773,86775,86778],{"className":86774},[13441,13442],[25635,86776,86777],{},"虛擬碼詳解",[13438,86779,86781,86784,86787,86791,86801,86804,86808,86817,86824,86828,86837,86844],{"className":86780},[25641],[12,86782,86783],{},"因為我看了好幾次都看不太懂，所以寫一個白話文版本，希望能幫助理解：",[12,86785,86786],{},"把建構決策樹想像成「在一個大箱子裡，把混在一起的水果分門別類」的過程。",[7166,86788,86790],{"id":86789},"守則一已經分乾淨了就直接貼標籤結束","守則一：已經分乾淨了，就直接貼標籤（結束）",[7351,86792,86793],{},[12,86794,86795,7206,86798],{},[106,86796,86797],{},"對應虛擬碼：",[6979,86799,86800],{},"if 所有 examples 都屬於同一個類別：return 葉節點",[12,86802,86803],{},"如果你現在看著眼前的這堆水果，發現「哇！裡面百分之百都是紅蘋果」，那你就不用再去想還要問什麼問題了。直接拿一張貼紙寫上「蘋果」貼上去，這堆水果的分類工作就完美結束。",[7166,86805,86807],{"id":86806},"守則二問題問完了但還是混在一起少數服從多數結束","守則二：問題問完了但還是混在一起，少數服從多數（結束）",[7351,86809,86810],{},[12,86811,86812,7206,86814],{},[106,86813,86797],{},[6979,86815,86816],{},"if 已經沒有 features 可以用了：return 多數類別的葉節點",[12,86818,86819,86820,86823],{},"假設你手上只有三個問題能問：「是不是圓的？」、「是不是紅的？」、「是不是甜的？」。 如果你把這三個問題都問完了，把水果分到最後一小堆時，發現裡面還有「3 顆紅蘋果」跟「1 顆長得很像蘋果的紅芭樂」混在一起，但你已經",[106,86821,86822],{},"沒有問題可以問了","，怎麼辦？ 沒辦法，只能妥協！既然蘋果佔多數，你就閉著眼睛給這堆水果貼上「蘋果」的標籤，結束這回合。",[7166,86825,86827],{"id":86826},"守則三還沒分乾淨就找個好問題切兩半然後重複守則繼續","守則三：還沒分乾淨，就找個好問題切兩半，然後「重複守則」（繼續）",[7351,86829,86830],{},[12,86831,86832,7206,86834],{},[106,86833,86797],{},[6979,86835,86836],{},"挑一個「最好的」特徵 F... 分子集... 接著呼叫 DTree(子集)",[12,86838,86839,86840,86843],{},"如果眼前的這堆水果還是蘋果、香蕉、芭樂大混雜，你就必須",[106,86841,86842],{},"找一個最能把牠們區分開來的好問題","。",[3603,86845,86846,86849,86859],{},[26,86847,86848],{},"你問：「是不是黃色的？」",[26,86850,86851,86852,86855,86856,86843],{},"你把水果分成兩堆：",[106,86853,86854],{},"「是黃色的放左邊」","、",[106,86857,86858],{},"「不是黃色的放右邊」",[26,86860,86861,86862],{},"遞迴過程：\n",[23,86863,86864,86870],{},[26,86865,86866,86867,86843],{},"針對左邊那一堆，你假裝自己是剛上班，",[106,86868,86869],{},"重新從「守則一」開始檢查這堆水果",[26,86871,86872,86873,86843],{},"針對右邊那一堆，你也",[106,86874,86869],{},[12,86876,86877,86878],{},"整個流程的關鍵問題就是：",[106,86879,86880],{},"要怎麼挑「最好的」特徵？",[6964,86882,86884],{"id":86883},"熵-entropy-與資訊增益-information-gain","熵 (Entropy) 與資訊增益 (Information Gain)",[12,86886,86887],{},"這裡就要請出資訊理論了。我們的目標是選一個能把資料分得最「乾淨」的特徵，也就是讓每個分支盡可能只剩下單一類別。",[7166,86889,86891],{"id":86890},"熵entropy","熵（Entropy）",[12,86893,86894,86897],{},[106,86895,86896],{},"熵","衡量的是一個集合的混亂程度或不純度。以二元分類來說：",[12,86899,86900],{},[24923,86901,86903],{"className":86902,"jax":24927},[24926],[24929,86904,86905,86954],{"style":24931,"xmlns":24932,"width":25862,"height":24934,"role":1930,"focusable":8058,"viewBox":25863,"xmlnsXLink":24936},[24938,86906,86907,86910,86912,86915,86918,86921,86924,86927,86929,86932,86934,86936,86938,86940,86942,86944,86947,86949,86952],{},[24941,86908],{"id":86909,"d":25869},"MJX-1-TEX-I-1D438",[24941,86911],{"id":32160,"d":25873},[24941,86913],{"id":86914,"d":25877},"MJX-1-TEX-I-1D461",[24941,86916],{"id":86917,"d":25881},"MJX-1-TEX-I-1D45F",[24941,86919],{"id":86920,"d":25885},"MJX-1-TEX-I-1D45C",[24941,86922],{"id":86923,"d":25889},"MJX-1-TEX-I-1D45D",[24941,86925],{"id":86926,"d":25893},"MJX-1-TEX-I-1D466",[24941,86928],{"id":24943,"d":24944},[24941,86930],{"id":86931,"d":25900},"MJX-1-TEX-I-1D446",[24941,86933],{"id":24987,"d":24988},[24941,86935],{"id":24971,"d":24972},[24941,86937],{"id":32151,"d":25910},[24941,86939],{"id":47918,"d":25914},[24941,86941],{"id":24959,"d":24960},[24941,86943],{"id":24963,"d":24964},[24941,86945],{"id":86946,"d":25924},"MJX-1-TEX-N-67",[24941,86948],{"id":76019,"d":25928},[24941,86950],{"id":86951,"d":180},"MJX-1-TEX-N-2061",[24941,86953],{"id":47921,"d":25935},[25022,86955,86956],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,86957,86958,86963,86967,86972,86977,86982,86987,86992,86996,87001,87005,87009,87013,87023,87038,87043,87047,87057,87061,87065,87075,87089,87093,87097,87107],{"dataMmlNode":25028},[25022,86959,86960],{"dataMmlNode":25942},[25033,86961],{"dataC":25945,"xLinkHref":86962},"#MJX-1-TEX-I-1D438",[25022,86964,86965],{"dataMmlNode":25942,"transform":25949},[25033,86966],{"dataC":25952,"xLinkHref":32206},[25022,86968,86969],{"dataMmlNode":25942,"transform":25956},[25033,86970],{"dataC":25959,"xLinkHref":86971},"#MJX-1-TEX-I-1D461",[25022,86973,86974],{"dataMmlNode":25942,"transform":25963},[25033,86975],{"dataC":25966,"xLinkHref":86976},"#MJX-1-TEX-I-1D45F",[25022,86978,86979],{"dataMmlNode":25942,"transform":25970},[25033,86980],{"dataC":25973,"xLinkHref":86981},"#MJX-1-TEX-I-1D45C",[25022,86983,86984],{"dataMmlNode":25942,"transform":25977},[25033,86985],{"dataC":25980,"xLinkHref":86986},"#MJX-1-TEX-I-1D45D",[25022,86988,86989],{"dataMmlNode":25942,"transform":25984},[25033,86990],{"dataC":25987,"xLinkHref":86991},"#MJX-1-TEX-I-1D466",[25022,86993,86994],{"dataMmlNode":25031,"transform":25991},[25033,86995],{"dataC":25035,"xLinkHref":25036},[25022,86997,86998],{"dataMmlNode":25942,"transform":25997},[25033,86999],{"dataC":26000,"xLinkHref":87000},"#MJX-1-TEX-I-1D446",[25022,87002,87003],{"dataMmlNode":25031,"transform":26004},[25033,87004],{"dataC":25108,"xLinkHref":25109},[25022,87006,87007],{"dataMmlNode":25031,"transform":26010},[25033,87008],{"dataC":25078,"xLinkHref":25079},[25022,87010,87011],{"dataMmlNode":25031,"transform":26016},[25033,87012],{"dataC":26019,"xLinkHref":32188},[25022,87014,87015,87019],{"dataMmlNode":26023,"transform":26024},[25022,87016,87017],{"dataMmlNode":25942},[25033,87018],{"dataC":25980,"xLinkHref":86986},[25022,87020,87021],{"dataMmlNode":26031,"transform":26032},[25033,87022],{"dataC":26035,"xLinkHref":47932},[25022,87024,87025,87034],{"dataMmlNode":26023,"transform":26039},[25022,87026,87027,87029,87031],{"dataMmlNode":25942},[25033,87028],{"dataC":25057,"xLinkHref":25058},[25033,87030],{"dataC":25062,"xLinkHref":25063,"transform":26048},[25033,87032],{"dataC":26051,"xLinkHref":87033,"transform":25049},"#MJX-1-TEX-N-67",[25022,87035,87036],{"dataMmlNode":26031,"transform":26055},[25033,87037],{"dataC":26058,"xLinkHref":76028},[25022,87039,87040],{"dataMmlNode":25031,"transform":26062},[25033,87041],{"dataC":26065,"xLinkHref":87042},"#MJX-1-TEX-N-2061",[25022,87044,87045],{"dataMmlNode":25031,"transform":26062},[25033,87046],{"dataC":25035,"xLinkHref":25036},[25022,87048,87049,87053],{"dataMmlNode":26023,"transform":26073},[25022,87050,87051],{"dataMmlNode":25942},[25033,87052],{"dataC":25980,"xLinkHref":86986},[25022,87054,87055],{"dataMmlNode":26031,"transform":26032},[25033,87056],{"dataC":26035,"xLinkHref":47932},[25022,87058,87059],{"dataMmlNode":25031,"transform":26084},[25033,87060],{"dataC":25108,"xLinkHref":25109},[25022,87062,87063],{"dataMmlNode":25031,"transform":26089},[25033,87064],{"dataC":26019,"xLinkHref":32188},[25022,87066,87067,87071],{"dataMmlNode":26023,"transform":26094},[25022,87068,87069],{"dataMmlNode":25942},[25033,87070],{"dataC":25980,"xLinkHref":86986},[25022,87072,87073],{"dataMmlNode":26031,"transform":26032},[25033,87074],{"dataC":26103,"xLinkHref":47935},[25022,87076,87077,87085],{"dataMmlNode":26023,"transform":26107},[25022,87078,87079,87081,87083],{"dataMmlNode":25942},[25033,87080],{"dataC":25057,"xLinkHref":25058},[25033,87082],{"dataC":25062,"xLinkHref":25063,"transform":26048},[25033,87084],{"dataC":26051,"xLinkHref":87033,"transform":25049},[25022,87086,87087],{"dataMmlNode":26031,"transform":26055},[25033,87088],{"dataC":26058,"xLinkHref":76028},[25022,87090,87091],{"dataMmlNode":25031,"transform":26122},[25033,87092],{"dataC":26065,"xLinkHref":87042},[25022,87094,87095],{"dataMmlNode":25031,"transform":26122},[25033,87096],{"dataC":25035,"xLinkHref":25036},[25022,87098,87099,87103],{"dataMmlNode":26023,"transform":26131},[25022,87100,87101],{"dataMmlNode":25942},[25033,87102],{"dataC":25980,"xLinkHref":86986},[25022,87104,87105],{"dataMmlNode":26031,"transform":26032},[25033,87106],{"dataC":26103,"xLinkHref":47935},[25022,87108,87109],{"dataMmlNode":25031,"transform":26142},[25033,87110],{"dataC":25108,"xLinkHref":25109},[12,87112,87113,87114,87141,87142,87169],{},"其中 ",[24923,87115,87117],{"className":87116,"jax":24927},[24926],[24929,87118,87119,87126],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,87120,87121,87124],{},[24941,87122],{"id":87123,"d":25889},"MJX-2-TEX-I-1D45D",[24941,87125],{"id":42606,"d":25914},[25022,87127,87128],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,87129,87130],{"dataMmlNode":25028},[25022,87131,87132,87137],{"dataMmlNode":26023},[25022,87133,87134],{"dataMmlNode":25942},[25033,87135],{"dataC":25980,"xLinkHref":87136},"#MJX-2-TEX-I-1D45D",[25022,87138,87139],{"dataMmlNode":26031,"transform":26032},[25033,87140],{"dataC":26035,"xLinkHref":42643}," 是正例的比例，",[24923,87143,87145],{"className":87144,"jax":24927},[24926],[24929,87146,87147,87154],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,87148,87149,87152],{},[24941,87150],{"id":87151,"d":25889},"MJX-3-TEX-I-1D45D",[24941,87153],{"id":48105,"d":25935},[25022,87155,87156],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,87157,87158],{"dataMmlNode":25028},[25022,87159,87160,87165],{"dataMmlNode":26023},[25022,87161,87162],{"dataMmlNode":25942},[25033,87163],{"dataC":25980,"xLinkHref":87164},"#MJX-3-TEX-I-1D45D",[25022,87166,87167],{"dataMmlNode":26031,"transform":26032},[25033,87168],{"dataC":26103,"xLinkHref":48119}," 是反例的比例。",[25631,87171,87173,87176],{"className":87172},[13441,13442],[25635,87174,87175],{},"公式詳解",[13438,87177,87179,87182,87242,87245,87248,87405,87491,87544,87572,87575,88013,88084,88088,88091,88294,88301,88304],{"className":87178},[25641],[12,87180,87181],{},"我們把這個公式一步一步才開，看看到底是什麼意思？",[7166,87183,87185,87186,87213,87214,87241],{"id":87184},"第一步-和-是什麼","第一步：",[24923,87187,87189],{"className":87188,"jax":24927},[24926],[24929,87190,87191,87198],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,87192,87193,87196],{},[24941,87194],{"id":87195,"d":25889},"MJX-4-TEX-I-1D45D",[24941,87197],{"id":42701,"d":25914},[25022,87199,87200],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,87201,87202],{"dataMmlNode":25028},[25022,87203,87204,87209],{"dataMmlNode":26023},[25022,87205,87206],{"dataMmlNode":25942},[25033,87207],{"dataC":25980,"xLinkHref":87208},"#MJX-4-TEX-I-1D45D",[25022,87210,87211],{"dataMmlNode":26031,"transform":26032},[25033,87212],{"dataC":26035,"xLinkHref":42742}," 和 ",[24923,87215,87217],{"className":87216,"jax":24927},[24926],[24929,87218,87219,87226],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,87220,87221,87224],{},[24941,87222],{"id":87223,"d":25889},"MJX-5-TEX-I-1D45D",[24941,87225],{"id":48182,"d":25935},[25022,87227,87228],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,87229,87230],{"dataMmlNode":25028},[25022,87231,87232,87237],{"dataMmlNode":26023},[25022,87233,87234],{"dataMmlNode":25942},[25033,87235],{"dataC":25980,"xLinkHref":87236},"#MJX-5-TEX-I-1D45D",[25022,87238,87239],{"dataMmlNode":26031,"transform":26032},[25033,87240],{"dataC":26103,"xLinkHref":48199}," 是什麼？",[12,87243,87244],{},"這最簡單。它們就只是「比例」而已（所有的比例加起來要是 1）。",[12,87246,87247],{},"假設箱子裡只有「蘋果 (1)」和「香蕉 (0)」：",[23,87249,87250,87278,87308],{},[26,87251,87252,87277],{},[24923,87253,87255],{"className":87254,"jax":24927},[24926],[24929,87256,87257,87263],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,87258,87259,87261],{},[24941,87260],{"id":25888,"d":25889},[24941,87262],{"id":25913,"d":25914},[25022,87264,87265],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,87266,87267],{"dataMmlNode":25028},[25022,87268,87269,87273],{"dataMmlNode":26023},[25022,87270,87271],{"dataMmlNode":25942},[25033,87272],{"dataC":25980,"xLinkHref":25981},[25022,87274,87275],{"dataMmlNode":26031,"transform":26032},[25033,87276],{"dataC":26035,"xLinkHref":26036}," = 蘋果佔的比例。",[26,87279,87280,87307],{},[24923,87281,87283],{"className":87282,"jax":24927},[24926],[24929,87284,87285,87292],{"style":26153,"xmlns":24932,"width":26154,"height":26155,"role":1930,"focusable":8058,"viewBox":26156,"xmlnsXLink":24936},[24938,87286,87287,87289],{},[24941,87288],{"id":26161,"d":25889},[24941,87290],{"id":87291,"d":25935},"MJX-7-TEX-N-30",[25022,87293,87294],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,87295,87296],{"dataMmlNode":25028},[25022,87297,87298,87302],{"dataMmlNode":26023},[25022,87299,87300],{"dataMmlNode":25942},[25033,87301],{"dataC":25980,"xLinkHref":26175},[25022,87303,87304],{"dataMmlNode":26031,"transform":26032},[25033,87305],{"dataC":26103,"xLinkHref":87306},"#MJX-7-TEX-N-30"," = 香蕉佔的比例。",[26,87309,87310,87311,24802,87360,86843],{},"如果一半一半，那 ",[24923,87312,87314],{"className":87313,"jax":24927},[24926],[24929,87315,87316,87332],{"style":26153,"xmlns":24932,"width":26337,"height":26436,"role":1930,"focusable":8058,"viewBox":26437,"xmlnsXLink":24936},[24938,87317,87318,87320,87322,87325,87327,87329],{},[24941,87319],{"id":26191,"d":25889},[24941,87321],{"id":48460,"d":25914},[24941,87323],{"id":87324,"d":24972},"MJX-8-TEX-N-3D",[24941,87326],{"id":26194,"d":25935},[24941,87328],{"id":48469,"d":26357},[24941,87330],{"id":87331,"d":26660},"MJX-8-TEX-N-35",[25022,87333,87334],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,87335,87336,87346,87351],{"dataMmlNode":25028},[25022,87337,87338,87342],{"dataMmlNode":26023},[25022,87339,87340],{"dataMmlNode":25942},[25033,87341],{"dataC":25980,"xLinkHref":26205},[25022,87343,87344],{"dataMmlNode":26031,"transform":26032},[25033,87345],{"dataC":26035,"xLinkHref":48490},[25022,87347,87348],{"dataMmlNode":25031,"transform":26380},[25033,87349],{"dataC":25078,"xLinkHref":87350},"#MJX-8-TEX-N-3D",[25022,87352,87353,87355,87357],{"dataMmlNode":26031,"transform":26386},[25033,87354],{"dataC":26103,"xLinkHref":26210},[25033,87356],{"dataC":26392,"xLinkHref":48514,"transform":26394},[25033,87358],{"dataC":26717,"xLinkHref":87359,"transform":25049},"#MJX-8-TEX-N-35",[24923,87361,87363],{"className":87362,"jax":24927},[24926],[24929,87364,87365,87378],{"style":26153,"xmlns":24932,"width":26337,"height":26436,"role":1930,"focusable":8058,"viewBox":26437,"xmlnsXLink":24936},[24938,87366,87367,87369,87371,87373,87376],{},[24941,87368],{"id":26249,"d":25889},[24941,87370],{"id":42928,"d":25935},[24941,87372],{"id":42922,"d":24972},[24941,87374],{"id":87375,"d":26357},"MJX-9-TEX-N-2E",[24941,87377],{"id":42957,"d":26660},[25022,87379,87380],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,87381,87382,87392,87396],{"dataMmlNode":25028},[25022,87383,87384,87388],{"dataMmlNode":26023},[25022,87385,87386],{"dataMmlNode":25942},[25033,87387],{"dataC":25980,"xLinkHref":26263},[25022,87389,87390],{"dataMmlNode":26031,"transform":26032},[25033,87391],{"dataC":26103,"xLinkHref":43006},[25022,87393,87394],{"dataMmlNode":25031,"transform":26380},[25033,87395],{"dataC":25078,"xLinkHref":42993},[25022,87397,87398,87400,87403],{"dataMmlNode":26031,"transform":26386},[25033,87399],{"dataC":26103,"xLinkHref":43006},[25033,87401],{"dataC":26392,"xLinkHref":87402,"transform":26394},"#MJX-9-TEX-N-2E",[25033,87404],{"dataC":26717,"xLinkHref":43215,"transform":25049},[7166,87406,87408,87409,87490],{"id":87407},"第二步-是什麼意思","第二步：",[24923,87410,87412],{"className":87411,"jax":24927},[24926],[24929,87413,87416,87442],{"style":24931,"xmlns":24932,"width":87414,"height":24934,"role":1930,"focusable":8058,"viewBox":87415,"xmlnsXLink":24936},"8.915ex","0 -750 3940.2 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2 為底的對數）。",[12,87545,87546,87547,7206,87550,7206,87569],{},"請直接把它翻譯成白話文：",[106,87548,87549],{},"「為了解決一個發生機率是",[106,87551,87552],{},[24923,87553,87555],{"className":87554,"jax":24927},[24926],[24929,87556,87557,87561],{"style":26153,"xmlns":24932,"width":49454,"height":26155,"role":1930,"focusable":8058,"viewBox":49455,"xmlnsXLink":24936},[24938,87558,87559],{},[24941,87560],{"id":26344,"d":25889},[25022,87562,87563],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,87564,87565],{"dataMmlNode":25028},[25022,87566,87567],{"dataMmlNode":25942},[25033,87568],{"dataC":25980,"xLinkHref":26372},[106,87570,87571],{},"的事件，你需要問幾個『是非題』？」",[12,87573,87574],{},"數學家發現了一個規律：",[23,87576,87577,87729,87892],{},[26,87578,87579,87580,87627,87628,87630,87631,86843],{},"如果抽中蘋果的機率是 50% 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25% (",[24923,87733,87735],{"className":87734,"jax":24927},[24926],[24929,87736,87739,87758],{"style":26153,"xmlns":24932,"width":87737,"height":26436,"role":1930,"focusable":8058,"viewBox":87738,"xmlnsXLink":24936},"8.178ex","0 -666 3614.6 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100% (",[24923,87896,87898],{"className":87897,"jax":24927},[24926],[24929,87899,87902,87913],{"style":26153,"xmlns":24932,"width":87900,"height":26436,"role":1930,"focusable":8058,"viewBox":87901,"xmlnsXLink":24936},"5.286ex","0 -666 2336.6 860",[24938,87903,87904,87907,87910],{},[24941,87905],{"id":87906,"d":25889},"MJX-17-TEX-I-1D45D",[24941,87908],{"id":87909,"d":24972},"MJX-17-TEX-N-3D",[24941,87911],{"id":87912,"d":25914},"MJX-17-TEX-N-31",[25022,87914,87915],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,87916,87917,87922,87927],{"dataMmlNode":25028},[25022,87918,87919],{"dataMmlNode":25942},[25033,87920],{"dataC":25980,"xLinkHref":87921},"#MJX-17-TEX-I-1D45D",[25022,87923,87924],{"dataMmlNode":25031,"transform":87612},[25033,87925],{"dataC":25078,"xLinkHref":87926},"#MJX-17-TEX-N-3D",[25022,87928,87929],{"dataMmlNode":26031,"transform":87618},[25033,87930],{"dataC":26035,"xLinkHref":87931},"#MJX-17-TEX-N-31",")，你根本不用問。在數學上：",[24923,87934,87936],{"className":87935,"jax":24927},[24926],[24929,87937,87940,87964],{"style":24931,"xmlns":24932,"width":87938,"height":24934,"role":1930,"focusable":8058,"viewBox":87939,"xmlnsXLink":24936},"13.056ex","0 -750 5770.8 1000",[24938,87941,87942,87944,87946,87948,87950,87952,87954,87956,87958,87960,87962],{},[24941,87943],{"id":26669,"d":25910},[24941,87945],{"id":26635,"d":24960},[24941,87947],{"id":26638,"d":24964},[24941,87949],{"id":26641,"d":25924},[24941,87951],{"id":26644,"d":25928},[24941,87953],{"id":26647,"d":180},[24941,87955],{"id":26650,"d":24944},[24941,87957],{"id":26672,"d":25914},[24941,87959],{"id":26663,"d":24988},[24941,87961],{"id":26666,"d":24972},[24941,87963],{"id":26653,"d":25935},[25022,87965,87966],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,87967,87968,87972,87986,87990,87994,87998,88003,88008],{"dataMmlNode":25028},[25022,87969,87970],{"dataMmlNode":25031},[25033,87971],{"dataC":26019,"xLinkHref":26736},[25022,87973,87974,87982],{"dataMmlNode":26023,"transform":87453},[25022,87975,87976,87978,87980],{"dataMmlNode":25942},[25033,87977],{"dataC":25057,"xLinkHref":26683},[25033,87979],{"dataC":25062,"xLinkHref":26686,"transform":26048},[25033,87981],{"dataC":26051,"xLinkHref":26689,"transform":25049},[25022,87983,87984],{"dataMmlNode":26031,"transform":26055},[25033,87985],{"dataC":26058,"xLinkHref":26694},[25022,87987,87988],{"dataMmlNode":25031,"transform":87472},[25033,87989],{"dataC":26065,"xLinkHref":26700},[25022,87991,87992],{"dataMmlNode":25031,"transform":87472},[25033,87993],{"dataC":25035,"xLinkHref":26705},[25022,87995,87996],{"dataMmlNode":26031,"transform":87482},[25033,87997],{"dataC":26035,"xLinkHref":26742},[25022,87999,88001],{"dataMmlNode":25031,"transform":88000},"translate(3548.2,0)",[25033,88002],{"dataC":25108,"xLinkHref":26724},[25022,88004,88006],{"dataMmlNode":25031,"transform":88005},"translate(4215,0)",[25033,88007],{"dataC":25078,"xLinkHref":26730},[25022,88009,88011],{"dataMmlNode":26031,"transform":88010},"translate(5270.8,0)",[25033,88012],{"dataC":26103,"xLinkHref":26711},[12,88014,88015,88016,7206,88081],{},"所以，",[106,88017,88018],{},[24923,88019,88021],{"className":88020,"jax":24927},[24926],[24929,88022,88023,88043],{"style":24931,"xmlns":24932,"width":87414,"height":24934,"role":1930,"focusable":8058,"viewBox":87415,"xmlnsXLink":24936},[24938,88024,88025,88027,88029,88031,88033,88035,88037,88039,88041],{},[24941,88026],{"id":26789,"d":25910},[24941,88028],{"id":26756,"d":24960},[24941,88030],{"id":26759,"d":24964},[24941,88032],{"id":26762,"d":25924},[24941,88034],{"id":26765,"d":25928},[24941,88036],{"id":26768,"d":180},[24941,88038],{"id":26771,"d":24944},[24941,88040],{"id":49246,"d":25889},[24941,88042],{"id":26783,"d":24988},[25022,88044,88045],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88046,88047,88051,88065,88069,88073,88077],{"dataMmlNode":25028},[25022,88048,88049],{"dataMmlNode":25031},[25033,88050],{"dataC":26019,"xLinkHref":26853},[25022,88052,88053,88061],{"dataMmlNode":26023,"transform":87453},[25022,88054,88055,88057,88059],{"dataMmlNode":25942},[25033,88056],{"dataC":25057,"xLinkHref":26800},[25033,88058],{"dataC":25062,"xLinkHref":26803,"transform":26048},[25033,88060],{"dataC":26051,"xLinkHref":26806,"transform":25049},[25022,88062,88063],{"dataMmlNode":26031,"transform":26055},[25033,88064],{"dataC":26058,"xLinkHref":26811},[25022,88066,88067],{"dataMmlNode":25031,"transform":87472},[25033,88068],{"dataC":26065,"xLinkHref":26816},[25022,88070,88071],{"dataMmlNode":25031,"transform":87472},[25033,88072],{"dataC":25035,"xLinkHref":26821},[25022,88074,88075],{"dataMmlNode":25942,"transform":87482},[25033,88076],{"dataC":25980,"xLinkHref":49262},[25022,88078,88079],{"dataMmlNode":25031,"transform":87487},[25033,88080],{"dataC":25108,"xLinkHref":26841},[106,88082,88083],{},"代表的就是「找這種東西需要花費的問題數量」。",[7166,88085,88087],{"id":88086},"第三步為什麼要相乘又相加","第三步：為什麼要「相乘」又「相加」？",[12,88089,88090],{},"公式是長這樣的：",[12,88092,88093],{},[24923,88094,88096],{"className":88095,"jax":24927},[24926],[24929,88097,88100,88137],{"style":24931,"xmlns":24932,"width":88098,"height":24934,"role":1930,"focusable":8058,"viewBox":88099,"xmlnsXLink":24936},"35.119ex","0 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,88295,88296,88297,88300],{},"這在數學上叫做「加權平均數」。因為你每次伸手進去抽，有時候抽到蘋果，有時候抽到香蕉，所以我們要把所有的情況",[106,88298,88299],{},"平均","起來看。",[12,88302,88303],{},"整句公式的白話文翻譯就是：",[12,88305,88306,7206,88309,7206,88331,7206,88334,7206,88355],{},[106,88307,88308],{},"（抽中蘋果的機率",[106,88310,88311],{},[24923,88312,88314],{"className":88313,"jax":24927},[24926],[24929,88315,88319,88323],{"style":88316,"xmlns":24932,"width":26869,"height":88317,"role":1930,"focusable":8058,"viewBox":88318,"xmlnsXLink":24936},"vertical-align: 0.02ex;","1.09ex","0 -491 778 482",[24938,88320,88321],{},[24941,88322],{"id":26906,"d":26907},[25022,88324,88325],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88326,88327],{"dataMmlNode":25028},[25022,88328,88329],{"dataMmlNode":25031},[25033,88330],{"dataC":26950,"xLinkHref":26951},[106,88332,88333],{},"找蘋果要問的問題數） 加上 （抽中香蕉的機率",[106,88335,88336],{},[24923,88337,88339],{"className":88338,"jax":24927},[24926],[24929,88340,88341,88346],{"style":88316,"xmlns":24932,"width":26869,"height":88317,"role":1930,"focusable":8058,"viewBox":88318,"xmlnsXLink":24936},[24938,88342,88343],{},[24941,88344],{"id":88345,"d":26907},"MJX-22-TEX-N-D7",[25022,88347,88348],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88349,88350],{"dataMmlNode":25028},[25022,88351,88352],{"dataMmlNode":25031},[25033,88353],{"dataC":26950,"xLinkHref":88354},"#MJX-22-TEX-N-D7",[106,88356,88357],{},"找香蕉要問的問題數） ＝ 平均每次要問幾個問題（也就是 Entropy）",[12,88359,88360],{},"直覺上怎麼理解：",[23,88362,88363,88369],{},[26,88364,88365,88366,88368],{},"如果所有樣本都是同一類 → 熵 = ",[106,88367,9180],{},"（完全純淨，沒有不確定性）",[26,88370,88371,88372,88374],{},"如果樣本剛好正反各半（50\u002F50）→ 熵 = ",[106,88373,8855],{},"（最大的不確定性）",[12,88376,88377,88378,88395],{},"如果是包含 ",[24923,88379,88381],{"className":88380,"jax":24927},[24926],[24929,88382,88383,88387],{"style":25196,"xmlns":24932,"width":27037,"height":27038,"role":1930,"focusable":8058,"viewBox":27039,"xmlnsXLink":24936},[24938,88384,88385],{},[24941,88386],{"id":27114,"d":27045},[25022,88388,88389],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88390,88391],{"dataMmlNode":25028},[25022,88392,88393],{"dataMmlNode":25942},[25033,88394],{"dataC":27054,"xLinkHref":27230}," 個類別的多分類問題，公式就會推廣成：",[12,88397,88398],{},[24923,88399,88401],{"className":88400,"jax":24927},[24926],[24929,88402,88403,88455],{"style":27869,"xmlns":24932,"width":27870,"height":27871,"role":1930,"focusable":8058,"viewBox":27872,"xmlnsXLink":24936},[24938,88404,88405,88407,88409,88411,88413,88415,88417,88419,88421,88423,88425,88427,88430,88433,88435,88438,88440,88443,88446,88449,88452],{},[24941,88406],{"id":27352,"d":25869},[24941,88408],{"id":27329,"d":25873},[24941,88410],{"id":27355,"d":25877},[24941,88412],{"id":27358,"d":25881},[24941,88414],{"id":27361,"d":25885},[24941,88416],{"id":27364,"d":25889},[24941,88418],{"id":27367,"d":25893},[24941,88420],{"id":27332,"d":24944},[24941,88422],{"id":27335,"d":25900},[24941,88424],{"id":27346,"d":24988},[24941,88426],{"id":27349,"d":24972},[24941,88428],{"id":88429,"d":27911},"MJX-24-TEX-SO-2211",[24941,88431],{"id":88432,"d":27045},"MJX-24-TEX-I-1D450",[24941,88434],{"id":27326,"d":27108},[24941,88436],{"id":88437,"d":25914},"MJX-24-TEX-N-31",[24941,88439],{"id":27370,"d":25910},[24941,88441],{"id":88442,"d":24960},"MJX-24-TEX-N-6C",[24941,88444],{"id":88445,"d":24964},"MJX-24-TEX-N-6F",[24941,88447],{"id":88448,"d":25924},"MJX-24-TEX-N-67",[24941,88450],{"id":88451,"d":25928},"MJX-24-TEX-N-32",[24941,88453],{"id":88454,"d":180},"MJX-24-TEX-N-2061",[25022,88456,88457],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88458,88459,88463,88467,88471,88475,88479,88483,88487,88491,88495,88499,88503,88532,88536,88546,88564,88569,88573,88583],{"dataMmlNode":25028},[25022,88460,88461],{"dataMmlNode":25942},[25033,88462],{"dataC":25945,"xLinkHref":27482},[25022,88464,88465],{"dataMmlNode":25942,"transform":25949},[25033,88466],{"dataC":25952,"xLinkHref":27438},[25022,88468,88469],{"dataMmlNode":25942,"transform":25956},[25033,88470],{"dataC":25959,"xLinkHref":27493},[25022,88472,88473],{"dataMmlNode":25942,"transform":25963},[25033,88474],{"dataC":25966,"xLinkHref":27499},[25022,88476,88477],{"dataMmlNode":25942,"transform":25970},[25033,88478],{"dataC":25973,"xLinkHref":27505},[25022,88480,88481],{"dataMmlNode":25942,"transform":25977},[25033,88482],{"dataC":25980,"xLinkHref":27511},[25022,88484,88485],{"dataMmlNode":25942,"transform":25984},[25033,88486],{"dataC":25987,"xLinkHref":27517},[25022,88488,88489],{"dataMmlNode":25031,"transform":25991},[25033,88490],{"dataC":25035,"xLinkHref":27444},[25022,88492,88493],{"dataMmlNode":25942,"transform":25997},[25033,88494],{"dataC":26000,"xLinkHref":27450},[25022,88496,88497],{"dataMmlNode":25031,"transform":26004},[25033,88498],{"dataC":25108,"xLinkHref":27470},[25022,88500,88501],{"dataMmlNode":25031,"transform":26010},[25033,88502],{"dataC":25078,"xLinkHref":27476},[25022,88504,88505,88510,88517],{"dataMmlNode":27194,"transform":26016},[25022,88506,88507],{"dataMmlNode":25031},[25033,88508],{"dataC":27199,"xLinkHref":88509},"#MJX-24-TEX-SO-2211",[25022,88511,88512],{"dataMmlNode":27203,"transform":28007,"dataMjxTexclass":27205},[25022,88513,88514],{"dataMmlNode":25942},[25033,88515],{"dataC":27054,"xLinkHref":88516},"#MJX-24-TEX-I-1D450",[25022,88518,88519,88523,88527],{"dataMmlNode":27203,"transform":28015,"dataMjxTexclass":27205},[25022,88520,88521],{"dataMmlNode":25942},[25033,88522],{"dataC":27210,"xLinkHref":27432},[25022,88524,88525],{"dataMmlNode":25031,"transform":27214},[25033,88526],{"dataC":25078,"xLinkHref":27476},[25022,88528,88529],{"dataMmlNode":26031,"transform":27219},[25033,88530],{"dataC":26035,"xLinkHref":88531},"#MJX-24-TEX-N-31",[25022,88533,88534],{"dataMmlNode":25031,"transform":28032},[25033,88535],{"dataC":26019,"xLinkHref":27538},[25022,88537,88538,88542],{"dataMmlNode":26023,"transform":28038},[25022,88539,88540],{"dataMmlNode":25942},[25033,88541],{"dataC":25980,"xLinkHref":27511},[25022,88543,88544],{"dataMmlNode":25942,"transform":26032},[25033,88545],{"dataC":27210,"xLinkHref":27432},[25022,88547,88548,88559],{"dataMmlNode":26023,"transform":28049},[25022,88549,88550,88553,88556],{"dataMmlNode":25942},[25033,88551],{"dataC":25057,"xLinkHref":88552},"#MJX-24-TEX-N-6C",[25033,88554],{"dataC":25062,"xLinkHref":88555,"transform":26048},"#MJX-24-TEX-N-6F",[25033,88557],{"dataC":26051,"xLinkHref":88558,"transform":25049},"#MJX-24-TEX-N-67",[25022,88560,88561],{"dataMmlNode":26031,"transform":26055},[25033,88562],{"dataC":26058,"xLinkHref":88563},"#MJX-24-TEX-N-32",[25022,88565,88566],{"dataMmlNode":25031,"transform":28068},[25033,88567],{"dataC":26065,"xLinkHref":88568},"#MJX-24-TEX-N-2061",[25022,88570,88571],{"dataMmlNode":25031,"transform":28068},[25033,88572],{"dataC":25035,"xLinkHref":27444},[25022,88574,88575,88579],{"dataMmlNode":26023,"transform":28078},[25022,88576,88577],{"dataMmlNode":25942},[25033,88578],{"dataC":25980,"xLinkHref":27511},[25022,88580,88581],{"dataMmlNode":25942,"transform":26032},[25033,88582],{"dataC":27210,"xLinkHref":27432},[25022,88584,88585],{"dataMmlNode":25031,"transform":28089},[25033,88586],{"dataC":25108,"xLinkHref":27470},[13438,88588,88590],{"className":88589},[13441,13442],[12,88591,88592,7206,88616,7206,88637,7206,88660,7206,88681,7206,88683,7206,88704,7206,88707,7206,88728],{},[106,88593,88594,88595,88615],{},"就只是第一個公式加了一個Sigma(",[24923,88596,88598],{"className":88597,"jax":24927},[24926],[24929,88599,88601,88606],{"style":24931,"xmlns":24932,"width":33756,"height":24934,"role":1930,"focusable":8058,"viewBox":88600,"xmlnsXLink":24936},"0 -750 1056 1000",[24938,88602,88603],{},[24941,88604],{"id":88605,"d":27911},"MJX-25-TEX-SO-2211",[25022,88607,88608],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88609,88610],{"dataMmlNode":25028},[25022,88611,88612],{"dataMmlNode":25031},[25033,88613],{"dataC":27199,"xLinkHref":88614},"#MJX-25-TEX-SO-2211","). 翻譯過來（第 1 種水果的機率",[106,88617,88618],{},[24923,88619,88621],{"className":88620,"jax":24927},[24926],[24929,88622,88623,88628],{"style":88316,"xmlns":24932,"width":26869,"height":88317,"role":1930,"focusable":8058,"viewBox":88318,"xmlnsXLink":24936},[24938,88624,88625],{},[24941,88626],{"id":88627,"d":26907},"MJX-26-TEX-N-D7",[25022,88629,88630],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88631,88632],{"dataMmlNode":25028},[25022,88633,88634],{"dataMmlNode":25031},[25033,88635],{"dataC":26950,"xLinkHref":88636},"#MJX-26-TEX-N-D7",[106,88638,88639,88640,88659],{},"找它要問的問題數）",[24923,88641,88643],{"className":88642,"jax":24927},[24926],[24929,88644,88645,88650],{"style":26868,"xmlns":24932,"width":26869,"height":26870,"role":1930,"focusable":8058,"viewBox":26871,"xmlnsXLink":24936},[24938,88646,88647],{},[24941,88648],{"id":88649,"d":30313},"MJX-27-TEX-N-2B",[25022,88651,88652],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88653,88654],{"dataMmlNode":25028},[25022,88655,88656],{"dataMmlNode":25031},[25033,88657],{"dataC":30393,"xLinkHref":88658},"#MJX-27-TEX-N-2B","（第 2 種水果的機率",[106,88661,88662],{},[24923,88663,88665],{"className":88664,"jax":24927},[24926],[24929,88666,88667,88672],{"style":88316,"xmlns":24932,"width":26869,"height":88317,"role":1930,"focusable":8058,"viewBox":88318,"xmlnsXLink":24936},[24938,88668,88669],{},[24941,88670],{"id":88671,"d":26907},"MJX-28-TEX-N-D7",[25022,88673,88674],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88675,88676],{"dataMmlNode":25028},[25022,88677,88678],{"dataMmlNode":25031},[25033,88679],{"dataC":26950,"xLinkHref":88680},"#MJX-28-TEX-N-D7",[106,88682,88639],{},[106,88684,88685],{},[24923,88686,88688],{"className":88687,"jax":24927},[24926],[24929,88689,88690,88695],{"style":26868,"xmlns":24932,"width":26869,"height":26870,"role":1930,"focusable":8058,"viewBox":26871,"xmlnsXLink":24936},[24938,88691,88692],{},[24941,88693],{"id":88694,"d":30313},"MJX-29-TEX-N-2B",[25022,88696,88697],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88698,88699],{"dataMmlNode":25028},[25022,88700,88701],{"dataMmlNode":25031},[25033,88702],{"dataC":30393,"xLinkHref":88703},"#MJX-29-TEX-N-2B",[106,88705,88706],{},"... 一路加到第",[106,88708,88709],{},[24923,88710,88712],{"className":88711,"jax":24927},[24926],[24929,88713,88714,88719],{"style":25196,"xmlns":24932,"width":27037,"height":27038,"role":1930,"focusable":8058,"viewBox":27039,"xmlnsXLink":24936},[24938,88715,88716],{},[24941,88717],{"id":88718,"d":27045},"MJX-30-TEX-I-1D450",[25022,88720,88721],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88722,88723],{"dataMmlNode":25028},[25022,88724,88725],{"dataMmlNode":25942},[25033,88726],{"dataC":27054,"xLinkHref":88727},"#MJX-30-TEX-I-1D450",[106,88729,88730],{},"種水果 ＝ 整個箱子的平均問題數 (Entropy)",[12,88732,88733],{},"你可以把熵想成：如果隨機抽一筆資料，平均需要幾個 bit 才能表達它的類別。越不確定，需要的 bit 就越多。",[13438,88735,88737,88740],{"className":88736},[13441,13442],[12,88738,88739],{},"或者這麼說：如果你閉著眼睛從箱子裡隨機抽一顆水果，你平均需要問幾個『是非題』，才能100%確定你抽到的是什麼水果。",[23,88741,88742,88745],{},[26,88743,88744],{},"如果你什麼都不用問就知道答案 -> 不確定性極低 -> 熵為 0。",[26,88746,88747],{},"如果你要問很多個問題才能找出答案 -> 不確定性很高 -> 熵很大。",[7166,88749,88751],{"id":88750},"資訊增益information-gain","資訊增益（Information Gain）",[12,88753,88754,88757],{},[106,88755,88756],{},"資訊增益","衡量的是：用某個特徵做分割之後，熵減少了多少（也就是不確定性降低了多少）。",[24923,88759,88761],{"className":88760,"jax":24927,"display":13594},[24926],[24929,88762,88763,88838],{"style":27310,"xmlns":24932,"width":27311,"height":27312,"role":1930,"focusable":8058,"viewBox":27313,"xmlnsXLink":24936},[24938,88764,88765,88768,88771,88774,88777,88780,88783,88785,88788,88791,88793,88796,88799,88802,88805,88807,88809,88811,88814,88817,88820,88823,88825,88827,88830,88833,88836],{},[24941,88766],{"id":88767,"d":27319},"MJX-31-TEX-I-1D43A",[24941,88769],{"id":88770,"d":27323},"MJX-31-TEX-I-1D44E",[24941,88772],{"id":88773,"d":27108},"MJX-31-TEX-I-1D456",[24941,88775],{"id":88776,"d":25873},"MJX-31-TEX-I-1D45B",[24941,88778],{"id":88779,"d":24944},"MJX-31-TEX-N-28",[24941,88781],{"id":88782,"d":25900},"MJX-31-TEX-I-1D446",[24941,88784],{"id":49909,"d":27339},[24941,88786],{"id":88787,"d":27343},"MJX-31-TEX-I-1D439",[24941,88789],{"id":88790,"d":24988},"MJX-31-TEX-N-29",[24941,88792],{"id":49915,"d":24972},[24941,88794],{"id":88795,"d":25869},"MJX-31-TEX-I-1D438",[24941,88797],{"id":88798,"d":25877},"MJX-31-TEX-I-1D461",[24941,88800],{"id":88801,"d":25881},"MJX-31-TEX-I-1D45F",[24941,88803],{"id":88804,"d":25885},"MJX-31-TEX-I-1D45C",[24941,88806],{"id":49906,"d":25889},[24941,88808],{"id":49924,"d":25893},[24941,88810],{"id":49921,"d":25910},[24941,88812],{"id":88813,"d":27104},"MJX-31-TEX-LO-2211",[24941,88815],{"id":88816,"d":27377},"MJX-31-TEX-I-1D463",[24941,88818],{"id":88819,"d":27381},"MJX-31-TEX-N-2208",[24941,88821],{"id":88822,"d":27385},"MJX-31-TEX-I-1D449",[24941,88824],{"id":49903,"d":27389},[24941,88826],{"id":35091,"d":27393},[24941,88828],{"id":88829,"d":27397},"MJX-31-TEX-I-1D452",[24941,88831],{"id":88832,"d":27401},"MJX-31-TEX-I-1D460",[24941,88834],{"id":88835,"d":27405},"MJX-31-TEX-N-7C",[24941,88837],{"id":35094,"d":27409},[25022,88839,88840],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,88841,88842,88847,88852,88857,88862,88867,88872,88876,88881,88886,88890,88895,88899,88904,88909,88914,88918,88922,88926,88930,88934,88938,88996,89035,89039,89043,89047,89051,89055,89059,89063,89067,89071,89081],{"dataMmlNode":25028},[25022,88843,88844],{"dataMmlNode":25942},[25033,88845],{"dataC":27418,"xLinkHref":88846},"#MJX-31-TEX-I-1D43A",[25022,88848,88849],{"dataMmlNode":25942,"transform":27422},[25033,88850],{"dataC":27425,"xLinkHref":88851},"#MJX-31-TEX-I-1D44E",[25022,88853,88854],{"dataMmlNode":25942,"transform":27429},[25033,88855],{"dataC":27210,"xLinkHref":88856},"#MJX-31-TEX-I-1D456",[25022,88858,88859],{"dataMmlNode":25942,"transform":27435},[25033,88860],{"dataC":25952,"xLinkHref":88861},"#MJX-31-TEX-I-1D45B",[25022,88863,88864],{"dataMmlNode":25031,"transform":27441},[25033,88865],{"dataC":25035,"xLinkHref":88866},"#MJX-31-TEX-N-28",[25022,88868,88869],{"dataMmlNode":25942,"transform":27447},[25033,88870],{"dataC":26000,"xLinkHref":88871},"#MJX-31-TEX-I-1D446",[25022,88873,88874],{"dataMmlNode":25031,"transform":27453},[25033,88875],{"dataC":27456,"xLinkHref":49947},[25022,88877,88878],{"dataMmlNode":25942,"transform":27460},[25033,88879],{"dataC":27463,"xLinkHref":88880},"#MJX-31-TEX-I-1D439",[25022,88882,88883],{"dataMmlNode":25031,"transform":27467},[25033,88884],{"dataC":25108,"xLinkHref":88885},"#MJX-31-TEX-N-29",[25022,88887,88888],{"dataMmlNode":25031,"transform":27473},[25033,88889],{"dataC":25078,"xLinkHref":49957},[25022,88891,88892],{"dataMmlNode":25942,"transform":27479},[25033,88893],{"dataC":25945,"xLinkHref":88894},"#MJX-31-TEX-I-1D438",[25022,88896,88897],{"dataMmlNode":25942,"transform":27485},[25033,88898],{"dataC":25952,"xLinkHref":88861},[25022,88900,88901],{"dataMmlNode":25942,"transform":27490},[25033,88902],{"dataC":25959,"xLinkHref":88903},"#MJX-31-TEX-I-1D461",[25022,88905,88906],{"dataMmlNode":25942,"transform":27496},[25033,88907],{"dataC":25966,"xLinkHref":88908},"#MJX-31-TEX-I-1D45F",[25022,88910,88911],{"dataMmlNode":25942,"transform":27502},[25033,88912],{"dataC":25973,"xLinkHref":88913},"#MJX-31-TEX-I-1D45C",[25022,88915,88916],{"dataMmlNode":25942,"transform":27508},[25033,88917],{"dataC":25980,"xLinkHref":49942},[25022,88919,88920],{"dataMmlNode":25942,"transform":27514},[25033,88921],{"dataC":25987,"xLinkHref":49990},[25022,88923,88924],{"dataMmlNode":25031,"transform":27520},[25033,88925],{"dataC":25035,"xLinkHref":88866},[25022,88927,88928],{"dataMmlNode":25942,"transform":27525},[25033,88929],{"dataC":26000,"xLinkHref":88871},[25022,88931,88932],{"dataMmlNode":25031,"transform":27530},[25033,88933],{"dataC":25108,"xLinkHref":88885},[25022,88935,88936],{"dataMmlNode":25031,"transform":27535},[25033,88937],{"dataC":26019,"xLinkHref":49983},[25022,88939,88940,88945],{"dataMmlNode":27541,"transform":27542},[25022,88941,88942],{"dataMmlNode":25031,"transform":27545},[25033,88943],{"dataC":27199,"xLinkHref":88944},"#MJX-31-TEX-LO-2211",[25022,88946,88947,88952,88957,88962,88966,88970,88974,88979,88984,88988,88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是 ",[24923,89118,89120],{"className":89119,"jax":24927},[24926],[24929,89121,89122,89127],{"style":27778,"xmlns":24932,"width":27779,"height":27780,"role":1930,"focusable":8058,"viewBox":27781,"xmlnsXLink":24936},[24938,89123,89124],{},[24941,89125],{"id":89126,"d":25900},"MJX-33-TEX-I-1D446",[25022,89128,89129],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,89130,89131],{"dataMmlNode":25028},[25022,89132,89133],{"dataMmlNode":25942},[25033,89134],{"dataC":26000,"xLinkHref":89135},"#MJX-33-TEX-I-1D446"," 裡面特徵 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減號前面的 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",[24923,89370,89372],{"className":89371,"jax":24927},[24926],[24929,89373,89374,89379],{"style":27778,"xmlns":24932,"width":27779,"height":27780,"role":1930,"focusable":8058,"viewBox":27781,"xmlnsXLink":24936},[24938,89375,89376],{},[24941,89377],{"id":89378,"d":25900},"MJX-41-TEX-I-1D446",[25022,89380,89381],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,89382,89383],{"dataMmlNode":25028},[25022,89384,89385],{"dataMmlNode":25942},[25033,89386],{"dataC":26000,"xLinkHref":89387},"#MJX-41-TEX-I-1D446"," 的混亂程度。",[23,89390,89391],{},[26,89392,89393,89396,89397,89399],{},[106,89394,89395],{},"白話文："," 在你問任何問題之前，如果你閉著眼睛抽一顆水果，平均需要問幾個是非題？（例如原本很亂，需要問 ",[106,89398,8879],{}," 個問題）。",[7166,89401,89403],{"id":89402},"_2-減號後面的那一長串分割後的加權平均混亂度","2. 減號後面的那一長串：分割後的「加權平均」混亂度",[12,89405,89406,89407,89654],{},"這一串 ",[24923,89408,89410],{"className":89409,"jax":24927},[24926],[24929,89411,89414,89479],{"style":28580,"xmlns":24932,"width":89412,"height":28582,"role":1930,"focusable":8058,"viewBox":89413,"xmlnsXLink":24936},"29.763ex","0 -1046.4 13155.3 1592.8",[24938,89415,89416,89419,89422,89425,89428,89431,89434,89437,89440,89443,89445,89447,89449,89452,89455,89458,89461,89464,89467,89470,89473,89476],{},[24941,89417],{"id":89418,"d":27911},"MJX-42-TEX-SO-2211",[24941,89420],{"id":89421,"d":27377},"MJX-42-TEX-I-1D463",[24941,89423],{"id":89424,"d":27381},"MJX-42-TEX-N-2208",[24941,89426],{"id":89427,"d":27385},"MJX-42-TEX-I-1D449",[24941,89429],{"id":89430,"d":27323},"MJX-42-TEX-I-1D44E",[24941,89432],{"id":89433,"d":27389},"MJX-42-TEX-I-1D459",[24941,89435],{"id":89436,"d":27393},"MJX-42-TEX-I-1D462",[24941,89438],{"id":89439,"d":27397},"MJX-42-TEX-I-1D452",[24941,89441],{"id":89442,"d":27401},"MJX-42-TEX-I-1D460",[24941,89444],{"id":35863,"d":24944},[24941,89446],{"id":29245,"d":27343},[24941,89448],{"id":35872,"d":24988},[24941,89450],{"id":89451,"d":27405},"MJX-42-TEX-N-7C",[24941,89453],{"id":89454,"d":25900},"MJX-42-TEX-I-1D446",[24941,89456],{"id":89457,"d":27409},"MJX-42-TEX-N-22C5",[24941,89459],{"id":89460,"d":25869},"MJX-42-TEX-I-1D438",[24941,89462],{"id":89463,"d":25873},"MJX-42-TEX-I-1D45B",[24941,89465],{"id":89466,"d":25877},"MJX-42-TEX-I-1D461",[24941,89468],{"id":89469,"d":25881},"MJX-42-TEX-I-1D45F",[24941,89471],{"id":89472,"d":25885},"MJX-42-TEX-I-1D45C",[24941,89474],{"id":89475,"d":25889},"MJX-42-TEX-I-1D45D",[24941,89477],{"id":89478,"d":25893},"MJX-42-TEX-I-1D466",[25022,89480,89481],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,89482,89483,89544,89585,89591,89597,89603,89609,89615,89621,89627,89633,89638,89649],{"dataMmlNode":25028},[25022,89484,89485,89490],{"dataMmlNode":27541},[25022,89486,89487],{"dataMmlNode":25031},[25033,89488],{"dataC":27199,"xLinkHref":89489},"#MJX-42-TEX-SO-2211",[25022,89491,89492,89497,89502,89507,89512,89517,89522,89527,89532,89536,89540],{"dataMmlNode":27203,"transform":28015,"dataMjxTexclass":27205},[25022,89493,89494],{"dataMmlNode":25942},[25033,89495],{"dataC":27556,"xLinkHref":89496},"#MJX-42-TEX-I-1D463",[25022,89498,89499],{"dataMmlNode":25031,"transform":27560},[25033,89500],{"dataC":27563,"xLinkHref":89501},"#MJX-42-TEX-N-2208",[25022,89503,89504],{"dataMmlNode":25942,"transform":27567},[25033,89505],{"dataC":27570,"xLinkHref":89506},"#MJX-42-TEX-I-1D449",[25022,89508,89509],{"dataMmlNode":25942,"transform":27574},[25033,89510],{"dataC":27425,"xLinkHref":89511},"#MJX-42-TEX-I-1D44E",[25022,89513,89514],{"dataMmlNode":25942,"transform":27579},[25033,89515],{"dataC":27582,"xLinkHref":89516},"#MJX-42-TEX-I-1D459",[25022,89518,89519],{"dataMmlNode":25942,"transform":27586},[25033,89520],{"dataC":27589,"xLinkHref":89521},"#MJX-42-TEX-I-1D462",[25022,89523,89524],{"dataMmlNode":25942,"transform":27593},[25033,89525],{"dataC":27596,"xLinkHref":89526},"#MJX-42-TEX-I-1D452",[25022,89528,89529],{"dataMmlNode":25942,"transform":27600},[25033,89530],{"dataC":27603,"xLinkHref":89531},"#MJX-42-TEX-I-1D460",[25022,89533,89534],{"dataMmlNode":25031,"transform":27607},[25033,89535],{"dataC":25035,"xLinkHref":35904},[25022,89537,89538],{"dataMmlNode":25942,"transform":27612},[25033,89539],{"dataC":27463,"xLinkHref":29254},[25022,89541,89542],{"dataMmlNode":25031,"transform":27617},[25033,89543],{"dataC":25108,"xLinkHref":35927},[25022,89545,89547,89569,89583],{"dataMmlNode":27622,"transform":89546},"translate(5394.2,0)",[25022,89548,89549,89554,89565],{"dataMmlNode":27626,"transform":28837},[25022,89550,89551],{"dataMmlNode":25031,"transform":27630},[25033,89552],{"dataC":27633,"xLinkHref":89553},"#MJX-42-TEX-N-7C",[25022,89555,89556,89561],{"dataMmlNode":26023,"transform":26048},[25022,89557,89558],{"dataMmlNode":25942},[25033,89559],{"dataC":26000,"xLinkHref":89560},"#MJX-42-TEX-I-1D446",[25022,89562,89563],{"dataMmlNode":25942,"transform":27643},[25033,89564],{"dataC":27556,"xLinkHref":89496},[25022,89566,89567],{"dataMmlNode":25031,"transform":27648},[25033,89568],{"dataC":27633,"xLinkHref":89553},[25022,89570,89571,89575,89579],{"dataMmlNode":27626,"transform":28859},[25022,89572,89573],{"dataMmlNode":25031,"transform":27630},[25033,89574],{"dataC":27633,"xLinkHref":89553},[25022,89576,89577],{"dataMmlNode":25942,"transform":26048},[25033,89578],{"dataC":26000,"xLinkHref":89560},[25022,89580,89581],{"dataMmlNode":25031,"transform":27664},[25033,89582],{"dataC":27633,"xLinkHref":89553},[27668,89584],{"width":28874,"height":27671,"x":27672,"y":27673},[25022,89586,89588],{"dataMmlNode":25031,"transform":89587},"translate(7184.2,0)",[25033,89589],{"dataC":27679,"xLinkHref":89590},"#MJX-42-TEX-N-22C5",[25022,89592,89594],{"dataMmlNode":25942,"transform":89593},"translate(7684.4,0)",[25033,89595],{"dataC":25945,"xLinkHref":89596},"#MJX-42-TEX-I-1D438",[25022,89598,89600],{"dataMmlNode":25942,"transform":89599},"translate(8448.4,0)",[25033,89601],{"dataC":25952,"xLinkHref":89602},"#MJX-42-TEX-I-1D45B",[25022,89604,89606],{"dataMmlNode":25942,"transform":89605},"translate(9048.4,0)",[25033,89607],{"dataC":25959,"xLinkHref":89608},"#MJX-42-TEX-I-1D461",[25022,89610,89612],{"dataMmlNode":25942,"transform":89611},"translate(9409.4,0)",[25033,89613],{"dataC":25966,"xLinkHref":89614},"#MJX-42-TEX-I-1D45F",[25022,89616,89618],{"dataMmlNode":25942,"transform":89617},"translate(9860.4,0)",[25033,89619],{"dataC":25973,"xLinkHref":89620},"#MJX-42-TEX-I-1D45C",[25022,89622,89624],{"dataMmlNode":25942,"transform":89623},"translate(10345.4,0)",[25033,89625],{"dataC":25980,"xLinkHref":89626},"#MJX-42-TEX-I-1D45D",[25022,89628,89630],{"dataMmlNode":25942,"transform":89629},"translate(10848.4,0)",[25033,89631],{"dataC":25987,"xLinkHref":89632},"#MJX-42-TEX-I-1D466",[25022,89634,89636],{"dataMmlNode":25031,"transform":89635},"translate(11338.4,0)",[25033,89637],{"dataC":25035,"xLinkHref":35904},[25022,89639,89641,89645],{"dataMmlNode":26023,"transform":89640},"translate(11727.4,0)",[25022,89642,89643],{"dataMmlNode":25942},[25033,89644],{"dataC":26000,"xLinkHref":89560},[25022,89646,89647],{"dataMmlNode":25942,"transform":27643},[25033,89648],{"dataC":27556,"xLinkHref":89496},[25022,89650,89652],{"dataMmlNode":25031,"transform":89651},"translate(12766.3,0)",[25033,89653],{"dataC":25108,"xLinkHref":35927}," 看起來最嚇人，但其實它只是在算「切成好幾小堆後，平均還要問幾個問題」。",[12,89656,89657,89658,89717],{},"為什麼要乘上 ",[24923,89659,89661],{"className":89660,"jax":24927},[24926],[24929,89662,89663,89673],{"style":28580,"xmlns":24932,"width":29394,"height":28582,"role":1930,"focusable":8058,"viewBox":29395,"xmlnsXLink":24936},[24938,89664,89665,89668,89671],{},[24941,89666],{"id":89667,"d":27405},"MJX-43-TEX-N-7C",[24941,89669],{"id":89670,"d":25900},"MJX-43-TEX-I-1D446",[24941,89672],{"id":29265,"d":27377},[25022,89674,89675],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,89676,89677],{"dataMmlNode":25028},[25022,89678,89679,89701,89715],{"dataMmlNode":27622},[25022,89680,89681,89686,89697],{"dataMmlNode":27626,"transform":28837},[25022,89682,89683],{"dataMmlNode":25031,"transform":27630},[25033,89684],{"dataC":27633,"xLinkHref":89685},"#MJX-43-TEX-N-7C",[25022,89687,89688,89693],{"dataMmlNode":26023,"transform":26048},[25022,89689,89690],{"dataMmlNode":25942},[25033,89691],{"dataC":26000,"xLinkHref":89692},"#MJX-43-TEX-I-1D446",[25022,89694,89695],{"dataMmlNode":25942,"transform":27643},[25033,89696],{"dataC":27556,"xLinkHref":29274},[25022,89698,89699],{"dataMmlNode":25031,"transform":27648},[25033,89700],{"dataC":27633,"xLinkHref":89685},[25022,89702,89703,89707,89711],{"dataMmlNode":27626,"transform":28859},[25022,89704,89705],{"dataMmlNode":25031,"transform":27630},[25033,89706],{"dataC":27633,"xLinkHref":89685},[25022,89708,89709],{"dataMmlNode":25942,"transform":26048},[25033,89710],{"dataC":26000,"xLinkHref":89692},[25022,89712,89713],{"dataMmlNode":25031,"transform":27664},[25033,89714],{"dataC":27633,"xLinkHref":89685},[27668,89716],{"width":28874,"height":27671,"x":27672,"y":27673},"（子集的比例）呢？",[12,89719,89720,89721,89724],{},"這叫做",[106,89722,89723],{},"加權平均","。想像你把 100 顆水果用「是不是紅色？」切成兩堆：",[23,89726,89727,89733],{},[26,89728,89729,89732],{},[106,89730,89731],{},"左邊堆（紅色）："," 有 99 顆，裡面有蘋果也有小番茄，很亂（Entropy 很高）。",[26,89734,89735,89738],{},[106,89736,89737],{},"右邊堆（不是紅色）："," 只有 1 顆，剛好是黃色香蕉，很純（Entropy = 0）。",[12,89740,89741,89742],{},"因為左邊那堆佔了絕大多數（99%），所以你下一次伸手抽水果，有 99% 的機率會抽到左邊那堆，你還是得面對很高的混亂度。因此，",[106,89743,89744],{},"哪一堆的水果數量越多，它對整體混亂度的影響力（權重）就應該越大。",[23,89746,89747,89807],{},[26,89748,89749,89806],{},[24923,89750,89752],{"className":89751,"jax":24927},[24926],[24929,89753,89754,89763],{"style":28580,"xmlns":24932,"width":29394,"height":28582,"role":1930,"focusable":8058,"viewBox":29395,"xmlnsXLink":24936},[24938,89755,89756,89759,89761],{},[24941,89757],{"id":89758,"d":27405},"MJX-44-TEX-N-7C",[24941,89760],{"id":29315,"d":25900},[24941,89762],{"id":29318,"d":27377},[25022,89764,89765],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,89766,89767],{"dataMmlNode":25028},[25022,89768,89769,89790,89804],{"dataMmlNode":27622},[25022,89770,89771,89776,89786],{"dataMmlNode":27626,"transform":28837},[25022,89772,89773],{"dataMmlNode":25031,"transform":27630},[25033,89774],{"dataC":27633,"xLinkHref":89775},"#MJX-44-TEX-N-7C",[25022,89777,89778,89782],{"dataMmlNode":26023,"transform":26048},[25022,89779,89780],{"dataMmlNode":25942},[25033,89781],{"dataC":26000,"xLinkHref":29372},[25022,89783,89784],{"dataMmlNode":25942,"transform":27643},[25033,89785],{"dataC":27556,"xLinkHref":29377},[25022,89787,89788],{"dataMmlNode":25031,"transform":27648},[25033,89789],{"dataC":27633,"xLinkHref":89775},[25022,89791,89792,89796,89800],{"dataMmlNode":27626,"transform":28859},[25022,89793,89794],{"dataMmlNode":25031,"transform":27630},[25033,89795],{"dataC":27633,"xLinkHref":89775},[25022,89797,89798],{"dataMmlNode":25942,"transform":26048},[25033,89799],{"dataC":26000,"xLinkHref":29372},[25022,89801,89802],{"dataMmlNode":25031,"transform":27664},[25033,89803],{"dataC":27633,"xLinkHref":89775},[27668,89805],{"width":28874,"height":27671,"x":27672,"y":27673}," ＝ 這小堆水果佔整個箱子的比例。",[26,89808,89809,89890],{},[24923,89810,89812],{"className":89811,"jax":24927},[24926],[24929,89813,89814,89839],{"style":24931,"xmlns":24932,"width":29285,"height":24934,"role":1930,"focusable":8058,"viewBox":29286,"xmlnsXLink":24936},[24938,89815,89816,89819,89821,89823,89825,89827,89829,89831,89833,89835,89837],{},[24941,89817],{"id":89818,"d":25869},"MJX-45-TEX-I-1D438",[24941,89820],{"id":45784,"d":25873},[24941,89822],{"id":45787,"d":25877},[24941,89824],{"id":45769,"d":25881},[24941,89826],{"id":45795,"d":25885},[24941,89828],{"id":50726,"d":25889},[24941,89830],{"id":36512,"d":25893},[24941,89832],{"id":36497,"d":24944},[24941,89834],{"id":29403,"d":25900},[24941,89836],{"id":29406,"d":27377},[24941,89838],{"id":36506,"d":24988},[25022,89840,89841],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,89842,89843,89848,89852,89856,89860,89864,89868,89872,89876,89886],{"dataMmlNode":25028},[25022,89844,89845],{"dataMmlNode":25942},[25033,89846],{"dataC":25945,"xLinkHref":89847},"#MJX-45-TEX-I-1D438",[25022,89849,89850],{"dataMmlNode":25942,"transform":25949},[25033,89851],{"dataC":25952,"xLinkHref":45847},[25022,89853,89854],{"dataMmlNode":25942,"transform":25956},[25033,89855],{"dataC":25959,"xLinkHref":45852},[25022,89857,89858],{"dataMmlNode":25942,"transform":25963},[25033,89859],{"dataC":25966,"xLinkHref":45822},[25022,89861,89862],{"dataMmlNode":25942,"transform":25970},[25033,89863],{"dataC":25973,"xLinkHref":45881},[25022,89865,89866],{"dataMmlNode":25942,"transform":25977},[25033,89867],{"dataC":25980,"xLinkHref":50756},[25022,89869,89870],{"dataMmlNode":25942,"transform":25984},[25033,89871],{"dataC":25987,"xLinkHref":36554},[25022,89873,89874],{"dataMmlNode":25031,"transform":25991},[25033,89875],{"dataC":25035,"xLinkHref":36521},[25022,89877,89878,89882],{"dataMmlNode":26023,"transform":25997},[25022,89879,89880],{"dataMmlNode":25942},[25033,89881],{"dataC":26000,"xLinkHref":29426},[25022,89883,89884],{"dataMmlNode":25942,"transform":27643},[25033,89885],{"dataC":27556,"xLinkHref":29431},[25022,89887,89888],{"dataMmlNode":25031,"transform":29380},[25033,89889],{"dataC":25108,"xLinkHref":36542}," ＝ 這小堆水果自己的混亂度（還要問幾個問題）。",[12,89892,89893,89894,89897],{},"白話來說：資訊增益 ＝ 分割前的熵 − 分割後各子集熵的加權平均。",[106,89895,89896],{},"資訊增益最高","的那個特徵就是我們的首選。",[7166,89899,89900],{"id":89900},"實際算一次",[12,89902,89903],{},"來看四筆資料，特徵是（大小、顏色、形狀），分類是正（+）或負（−）：",[7085,89905,89906,89925],{},[7088,89907,89908],{},[7091,89909,89910,89913,89916,89919,89922],{},[7094,89911,89912],{"align":47883},"編號",[7094,89914,89915],{"align":47883},"大小",[7094,89917,89918],{"align":47883},"顏色",[7094,89920,89921],{"align":47883},"形狀",[7094,89923,89924],{"align":47883},"類別",[7104,89926,89927,89942,89955,89968],{},[7091,89928,89929,89931,89934,89937,89940],{},[7109,89930,8855],{"align":47883},[7109,89932,89933],{"align":47883},"大",[7109,89935,89936],{"align":47883},"紅",[7109,89938,89939],{"align":47883},"圓形",[7109,89941,8302],{"align":47883},[7091,89943,89944,89946,89949,89951,89953],{},[7109,89945,8879],{"align":47883},[7109,89947,89948],{"align":47883},"小",[7109,89950,89936],{"align":47883},[7109,89952,89939],{"align":47883},[7109,89954,8302],{"align":47883},[7091,89956,89957,89959,89961,89963,89966],{},[7109,89958,8830],{"align":47883},[7109,89960,89948],{"align":47883},[7109,89962,89936],{"align":47883},[7109,89964,89965],{"align":47883},"方形",[7109,89967,30013],{"align":47883},[7091,89969,89970,89972,89974,89977,89979],{},[7109,89971,8348],{"align":47883},[7109,89973,89933],{"align":47883},[7109,89975,89976],{"align":47883},"藍",[7109,89978,89939],{"align":47883},[7109,89980,30013],{"align":47883},[12,89982,89983,89984,86843],{},"整組資料有 2 正 2 負，所以 ",[24923,89985,89987],{"className":89986,"jax":24927},[24926],[24929,89988,89989,90029],{"style":24931,"xmlns":24932,"width":30035,"height":24934,"role":1930,"focusable":8058,"viewBox":30036,"xmlnsXLink":24936},[24938,89990,89991,89994,89997,90000,90003,90006,90008,90010,90012,90015,90017,90020,90023,90026],{},[24941,89992],{"id":89993,"d":25869},"MJX-46-TEX-I-1D438",[24941,89995],{"id":89996,"d":25873},"MJX-46-TEX-I-1D45B",[24941,89998],{"id":89999,"d":25877},"MJX-46-TEX-I-1D461",[24941,90001],{"id":90002,"d":25881},"MJX-46-TEX-I-1D45F",[24941,90004],{"id":90005,"d":25885},"MJX-46-TEX-I-1D45C",[24941,90007],{"id":50776,"d":25889},[24941,90009],{"id":36572,"d":25893},[24941,90011],{"id":36569,"d":24944},[24941,90013],{"id":90014,"d":25900},"MJX-46-TEX-I-1D446",[24941,90016],{"id":36584,"d":24988},[24941,90018],{"id":90019,"d":24972},"MJX-46-TEX-N-3D",[24941,90021],{"id":90022,"d":25914},"MJX-46-TEX-N-31",[24941,90024],{"id":90025,"d":26357},"MJX-46-TEX-N-2E",[24941,90027],{"id":90028,"d":25935},"MJX-46-TEX-N-30",[25022,90030,90031],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,90032,90033,90038,90043,90048,90053,90058,90062,90066,90070,90075,90079,90084],{"dataMmlNode":25028},[25022,90034,90035],{"dataMmlNode":25942},[25033,90036],{"dataC":25945,"xLinkHref":90037},"#MJX-46-TEX-I-1D438",[25022,90039,90040],{"dataMmlNode":25942,"transform":25949},[25033,90041],{"dataC":25952,"xLinkHref":90042},"#MJX-46-TEX-I-1D45B",[25022,90044,90045],{"dataMmlNode":25942,"transform":25956},[25033,90046],{"dataC":25959,"xLinkHref":90047},"#MJX-46-TEX-I-1D461",[25022,90049,90050],{"dataMmlNode":25942,"transform":25963},[25033,90051],{"dataC":25966,"xLinkHref":90052},"#MJX-46-TEX-I-1D45F",[25022,90054,90055],{"dataMmlNode":25942,"transform":25970},[25033,90056],{"dataC":25973,"xLinkHref":90057},"#MJX-46-TEX-I-1D45C",[25022,90059,90060],{"dataMmlNode":25942,"transform":25977},[25033,90061],{"dataC":25980,"xLinkHref":50785},[25022,90063,90064],{"dataMmlNode":25942,"transform":25984},[25033,90065],{"dataC":25987,"xLinkHref":36598},[25022,90067,90068],{"dataMmlNode":25031,"transform":25991},[25033,90069],{"dataC":25035,"xLinkHref":36593},[25022,90071,90072],{"dataMmlNode":25942,"transform":25997},[25033,90073],{"dataC":26000,"xLinkHref":90074},"#MJX-46-TEX-I-1D446",[25022,90076,90077],{"dataMmlNode":25031,"transform":26004},[25033,90078],{"dataC":25108,"xLinkHref":36627},[25022,90080,90081],{"dataMmlNode":25031,"transform":26010},[25033,90082],{"dataC":25078,"xLinkHref":90083},"#MJX-46-TEX-N-3D",[25022,90085,90086,90089,90092],{"dataMmlNode":26031,"transform":26016},[25033,90087],{"dataC":26035,"xLinkHref":90088},"#MJX-46-TEX-N-31",[25033,90090],{"dataC":26392,"xLinkHref":90091,"transform":26394},"#MJX-46-TEX-N-2E",[25033,90093],{"dataC":26103,"xLinkHref":90094,"transform":25049},"#MJX-46-TEX-N-30",[12,90096,90097],{},[106,90098,90099],{},"用大小來分：",[23,90101,90102,90148,90194],{},[26,90103,90104,90105],{},"大 → 1+, 1− → ",[24923,90106,90108],{"className":90107,"jax":24927},[24926],[24929,90109,90110,90125],{"style":26868,"xmlns":24932,"width":30166,"height":30167,"role":1930,"focusable":8058,"viewBox":30168,"xmlnsXLink":24936},[24938,90111,90112,90114,90116,90119,90122],{},[24941,90113],{"id":50801,"d":25869},[24941,90115],{"id":36655,"d":24972},[24941,90117],{"id":90118,"d":25914},"MJX-47-TEX-N-31",[24941,90120],{"id":90121,"d":26357},"MJX-47-TEX-N-2E",[24941,90123],{"id":90124,"d":25935},"MJX-47-TEX-N-30",[25022,90126,90127],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,90128,90129,90133,90137],{"dataMmlNode":25028},[25022,90130,90131],{"dataMmlNode":25942},[25033,90132],{"dataC":25945,"xLinkHref":50810},[25022,90134,90135],{"dataMmlNode":25031,"transform":30197},[25033,90136],{"dataC":25078,"xLinkHref":36728},[25022,90138,90139,90142,90145],{"dataMmlNode":26031,"transform":30203},[25033,90140],{"dataC":26035,"xLinkHref":90141},"#MJX-47-TEX-N-31",[25033,90143],{"dataC":26392,"xLinkHref":90144,"transform":26394},"#MJX-47-TEX-N-2E",[25033,90146],{"dataC":26103,"xLinkHref":90147,"transform":25049},"#MJX-47-TEX-N-30",[26,90149,90150,90151],{},"小 → 1+, 1− → ",[24923,90152,90154],{"className":90153,"jax":24927},[24926],[24929,90155,90156,90171],{"style":26868,"xmlns":24932,"width":30166,"height":30167,"role":1930,"focusable":8058,"viewBox":30168,"xmlnsXLink":24936},[24938,90157,90158,90160,90162,90165,90168],{},[24941,90159],{"id":50821,"d":25869},[24941,90161],{"id":36909,"d":24972},[24941,90163],{"id":90164,"d":25914},"MJX-48-TEX-N-31",[24941,90166],{"id":90167,"d":26357},"MJX-48-TEX-N-2E",[24941,90169],{"id":90170,"d":25935},"MJX-48-TEX-N-30",[25022,90172,90173],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,90174,90175,90179,90183],{"dataMmlNode":25028},[25022,90176,90177],{"dataMmlNode":25942},[25033,90178],{"dataC":25945,"xLinkHref":50830},[25022,90180,90181],{"dataMmlNode":25031,"transform":30197},[25033,90182],{"dataC":25078,"xLinkHref":36973},[25022,90184,90185,90188,90191],{"dataMmlNode":26031,"transform":30203},[25033,90186],{"dataC":26035,"xLinkHref":90187},"#MJX-48-TEX-N-31",[25033,90189],{"dataC":26392,"xLinkHref":90190,"transform":26394},"#MJX-48-TEX-N-2E",[25033,90192],{"dataC":26103,"xLinkHref":90193,"transform":25049},"#MJX-48-TEX-N-30",[26,90195,90196,90329],{},[24923,90197,90199],{"className":90198,"jax":24927},[24926],[24929,90200,90201,90238],{"style":24931,"xmlns":24932,"width":30270,"height":24934,"role":1930,"focusable":8058,"viewBox":30271,"xmlnsXLink":24936},[24938,90202,90203,90206,90209,90211,90214,90216,90219,90222,90224,90226,90228,90231,90234,90236],{},[24941,90204],{"id":90205,"d":27319},"MJX-49-TEX-I-1D43A",[24941,90207],{"id":90208,"d":27323},"MJX-49-TEX-I-1D44E",[24941,90210],{"id":46258,"d":27108},[24941,90212],{"id":90213,"d":25873},"MJX-49-TEX-I-1D45B",[24941,90215],{"id":46261,"d":24972},[24941,90217],{"id":90218,"d":25914},"MJX-49-TEX-N-31",[24941,90220],{"id":90221,"d":25910},"MJX-49-TEX-N-2212",[24941,90223],{"id":46285,"d":24944},[24941,90225],{"id":29575,"d":25935},[24941,90227],{"id":29578,"d":26357},[24941,90229],{"id":90230,"d":26660},"MJX-49-TEX-N-35",[24941,90232],{"id":90233,"d":26907},"MJX-49-TEX-N-D7",[24941,90235],{"id":46264,"d":30313},[24941,90237],{"id":46291,"d":24988},[25022,90239,90240],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,90241,90242,90247,90252,90256,90261,90265,90270,90275,90279,90288,90293,90297,90301,90309,90313,90317,90321,90325],{"dataMmlNode":25028},[25022,90243,90244],{"dataMmlNode":25942},[25033,90245],{"dataC":27418,"xLinkHref":90246},"#MJX-49-TEX-I-1D43A",[25022,90248,90249],{"dataMmlNode":25942,"transform":27422},[25033,90250],{"dataC":27425,"xLinkHref":90251},"#MJX-49-TEX-I-1D44E",[25022,90253,90254],{"dataMmlNode":25942,"transform":27429},[25033,90255],{"dataC":27210,"xLinkHref":46307},[25022,90257,90258],{"dataMmlNode":25942,"transform":27435},[25033,90259],{"dataC":25952,"xLinkHref":90260},"#MJX-49-TEX-I-1D45B",[25022,90262,90263],{"dataMmlNode":25031,"transform":30343},[25033,90264],{"dataC":25078,"xLinkHref":46312},[25022,90266,90267],{"dataMmlNode":26031,"transform":30349},[25033,90268],{"dataC":26035,"xLinkHref":90269},"#MJX-49-TEX-N-31",[25022,90271,90272],{"dataMmlNode":25031,"transform":30355},[25033,90273],{"dataC":26019,"xLinkHref":90274},"#MJX-49-TEX-N-2212",[25022,90276,90277],{"dataMmlNode":25031,"transform":30361},[25033,90278],{"dataC":25035,"xLinkHref":46389},[25022,90280,90281,90283,90285],{"dataMmlNode":26031,"transform":30367},[25033,90282],{"dataC":26103,"xLinkHref":29591},[25033,90284],{"dataC":26392,"xLinkHref":29594,"transform":26394},[25033,90286],{"dataC":26717,"xLinkHref":90287,"transform":25049},"#MJX-49-TEX-N-35",[25022,90289,90290],{"dataMmlNode":25031,"transform":30379},[25033,90291],{"dataC":26950,"xLinkHref":90292},"#MJX-49-TEX-N-D7",[25022,90294,90295],{"dataMmlNode":26031,"transform":30385},[25033,90296],{"dataC":26035,"xLinkHref":90269},[25022,90298,90299],{"dataMmlNode":25031,"transform":30390},[25033,90300],{"dataC":30393,"xLinkHref":46327},[25022,90302,90303,90305,90307],{"dataMmlNode":26031,"transform":30397},[25033,90304],{"dataC":26103,"xLinkHref":29591},[25033,90306],{"dataC":26392,"xLinkHref":29594,"transform":26394},[25033,90308],{"dataC":26717,"xLinkHref":90287,"transform":25049},[25022,90310,90311],{"dataMmlNode":25031,"transform":30406},[25033,90312],{"dataC":26950,"xLinkHref":90292},[25022,90314,90315],{"dataMmlNode":26031,"transform":30411},[25033,90316],{"dataC":26035,"xLinkHref":90269},[25022,90318,90319],{"dataMmlNode":25031,"transform":30416},[25033,90320],{"dataC":25108,"xLinkHref":46401},[25022,90322,90323],{"dataMmlNode":25031,"transform":30422},[25033,90324],{"dataC":25078,"xLinkHref":46312},[25022,90326,90327],{"dataMmlNode":26031,"transform":30427},[25033,90328],{"dataC":26103,"xLinkHref":29591}," — 完全沒用！",[12,90331,90332],{},[106,90333,90334],{},"用顏色來分：",[23,90336,90337,90399,90433],{},[26,90338,90339,90340],{},"紅 → 2+, 1− → ",[24923,90341,90343],{"className":90342,"jax":24927},[24926],[24929,90344,90345,90368],{"style":26868,"xmlns":24932,"width":30446,"height":30167,"role":1930,"focusable":8058,"viewBox":30447,"xmlnsXLink":24936},[24938,90346,90347,90350,90353,90356,90359,90362,90365],{},[24941,90348],{"id":90349,"d":25869},"MJX-50-TEX-I-1D438",[24941,90351],{"id":90352,"d":24972},"MJX-50-TEX-N-3D",[24941,90354],{"id":90355,"d":25935},"MJX-50-TEX-N-30",[24941,90357],{"id":90358,"d":26357},"MJX-50-TEX-N-2E",[24941,90360],{"id":90361,"d":29754},"MJX-50-TEX-N-39",[24941,90363],{"id":90364,"d":25914},"MJX-50-TEX-N-31",[24941,90366],{"id":90367,"d":30471},"MJX-50-TEX-N-38",[25022,90369,90370],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,90371,90372,90377,90382],{"dataMmlNode":25028},[25022,90373,90374],{"dataMmlNode":25942},[25033,90375],{"dataC":25945,"xLinkHref":90376},"#MJX-50-TEX-I-1D438",[25022,90378,90379],{"dataMmlNode":25031,"transform":30197},[25033,90380],{"dataC":25078,"xLinkHref":90381},"#MJX-50-TEX-N-3D",[25022,90383,90384,90387,90390,90393,90396],{"dataMmlNode":26031,"transform":30203},[25033,90385],{"dataC":26103,"xLinkHref":90386},"#MJX-50-TEX-N-30",[25033,90388],{"dataC":26392,"xLinkHref":90389,"transform":26394},"#MJX-50-TEX-N-2E",[25033,90391],{"dataC":29772,"xLinkHref":90392,"transform":25049},"#MJX-50-TEX-N-39",[25033,90394],{"dataC":26035,"xLinkHref":90395,"transform":26835},"#MJX-50-TEX-N-31",[25033,90397],{"dataC":30502,"xLinkHref":90398,"transform":30504},"#MJX-50-TEX-N-38",[26,90400,90401,90402],{},"藍 → 0+, 1− → ",[24923,90403,90405],{"className":90404,"jax":24927},[24926],[24929,90406,90407,90416],{"style":26868,"xmlns":24932,"width":30513,"height":30167,"role":1930,"focusable":8058,"viewBox":30514,"xmlnsXLink":24936},[24938,90408,90409,90412,90414],{},[24941,90410],{"id":90411,"d":25869},"MJX-51-TEX-I-1D438",[24941,90413],{"id":37266,"d":24972},[24941,90415],{"id":29710,"d":25935},[25022,90417,90418],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,90419,90420,90425,90429],{"dataMmlNode":25028},[25022,90421,90422],{"dataMmlNode":25942},[25033,90423],{"dataC":25945,"xLinkHref":90424},"#MJX-51-TEX-I-1D438",[25022,90426,90427],{"dataMmlNode":25031,"transform":30197},[25033,90428],{"dataC":25078,"xLinkHref":37335},[25022,90430,90431],{"dataMmlNode":26031,"transform":30203},[25033,90432],{"dataC":26103,"xLinkHref":29735},[26,90434,90435,90610],{},[24923,90436,90438],{"className":90437,"jax":24927},[24926],[24929,90439,90440,90493],{"style":24931,"xmlns":24932,"width":30552,"height":24934,"role":1930,"focusable":8058,"viewBox":30553,"xmlnsXLink":24936},[24938,90441,90442,90445,90447,90449,90451,90454,90456,90459,90462,90464,90467,90470,90473,90476,90478,90481,90484,90487,90490],{},[24941,90443],{"id":90444,"d":27319},"MJX-52-TEX-I-1D43A",[24941,90446],{"id":46615,"d":27323},[24941,90448],{"id":46618,"d":27108},[24941,90450],{"id":46621,"d":25873},[24941,90452],{"id":90453,"d":24972},"MJX-52-TEX-N-3D",[24941,90455],{"id":29760,"d":25914},[24941,90457],{"id":90458,"d":25910},"MJX-52-TEX-N-2212",[24941,90460],{"id":90461,"d":24944},"MJX-52-TEX-N-28",[24941,90463],{"id":29763,"d":25935},[24941,90465],{"id":90466,"d":26357},"MJX-52-TEX-N-2E",[24941,90468],{"id":90469,"d":26361},"MJX-52-TEX-N-37",[24941,90471],{"id":90472,"d":26660},"MJX-52-TEX-N-35",[24941,90474],{"id":90475,"d":26907},"MJX-52-TEX-N-D7",[24941,90477],{"id":29753,"d":29754},[24941,90479],{"id":90480,"d":30471},"MJX-52-TEX-N-38",[24941,90482],{"id":90483,"d":30313},"MJX-52-TEX-N-2B",[24941,90485],{"id":90486,"d":25928},"MJX-52-TEX-N-32",[24941,90488],{"id":90489,"d":24988},"MJX-52-TEX-N-29",[24941,90491],{"id":90492,"d":26455},"MJX-52-TEX-N-33",[25022,90494,90495],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,90496,90497,90502,90506,90510,90514,90519,90523,90528,90533,90546,90551,90564,90569,90580,90584,90588,90593,90597],{"dataMmlNode":25028},[25022,90498,90499],{"dataMmlNode":25942},[25033,90500],{"dataC":27418,"xLinkHref":90501},"#MJX-52-TEX-I-1D43A",[25022,90503,90504],{"dataMmlNode":25942,"transform":27422},[25033,90505],{"dataC":27425,"xLinkHref":46652},[25022,90507,90508],{"dataMmlNode":25942,"transform":27429},[25033,90509],{"dataC":27210,"xLinkHref":46657},[25022,90511,90512],{"dataMmlNode":25942,"transform":27435},[25033,90513],{"dataC":25952,"xLinkHref":46662},[25022,90515,90516],{"dataMmlNode":25031,"transform":30343},[25033,90517],{"dataC":25078,"xLinkHref":90518},"#MJX-52-TEX-N-3D",[25022,90520,90521],{"dataMmlNode":26031,"transform":30349},[25033,90522],{"dataC":26035,"xLinkHref":29788},[25022,90524,90525],{"dataMmlNode":25031,"transform":30355},[25033,90526],{"dataC":26019,"xLinkHref":90527},"#MJX-52-TEX-N-2212",[25022,90529,90530],{"dataMmlNode":25031,"transform":30361},[25033,90531],{"dataC":25035,"xLinkHref":90532},"#MJX-52-TEX-N-28",[25022,90534,90535,90537,90540,90543],{"dataMmlNode":26031,"transform":30367},[25033,90536],{"dataC":26103,"xLinkHref":29791},[25033,90538],{"dataC":26392,"xLinkHref":90539,"transform":26394},"#MJX-52-TEX-N-2E",[25033,90541],{"dataC":26397,"xLinkHref":90542,"transform":25049},"#MJX-52-TEX-N-37",[25033,90544],{"dataC":26717,"xLinkHref":90545,"transform":26835},"#MJX-52-TEX-N-35",[25022,90547,90548],{"dataMmlNode":25031,"transform":30673},[25033,90549],{"dataC":26950,"xLinkHref":90550},"#MJX-52-TEX-N-D7",[25022,90552,90553,90555,90557,90559,90561],{"dataMmlNode":26031,"transform":30679},[25033,90554],{"dataC":26103,"xLinkHref":29791},[25033,90556],{"dataC":26392,"xLinkHref":90539,"transform":26394},[25033,90558],{"dataC":29772,"xLinkHref":29773,"transform":25049},[25033,90560],{"dataC":26035,"xLinkHref":29788,"transform":26835},[25033,90562],{"dataC":30502,"xLinkHref":90563,"transform":30504},"#MJX-52-TEX-N-38",[25022,90565,90566],{"dataMmlNode":25031,"transform":30694},[25033,90567],{"dataC":30393,"xLinkHref":90568},"#MJX-52-TEX-N-2B",[25022,90570,90571,90573,90575,90578],{"dataMmlNode":26031,"transform":30700},[25033,90572],{"dataC":26103,"xLinkHref":29791},[25033,90574],{"dataC":26392,"xLinkHref":90539,"transform":26394},[25033,90576],{"dataC":26058,"xLinkHref":90577,"transform":25049},"#MJX-52-TEX-N-32",[25033,90579],{"dataC":26717,"xLinkHref":90545,"transform":26835},[25022,90581,90582],{"dataMmlNode":25031,"transform":30712},[25033,90583],{"dataC":26950,"xLinkHref":90550},[25022,90585,90586],{"dataMmlNode":26031,"transform":30717},[25033,90587],{"dataC":26103,"xLinkHref":29791},[25022,90589,90590],{"dataMmlNode":25031,"transform":30722},[25033,90591],{"dataC":25108,"xLinkHref":90592},"#MJX-52-TEX-N-29",[25022,90594,90595],{"dataMmlNode":25031,"transform":30728},[25033,90596],{"dataC":25078,"xLinkHref":90518},[25022,90598,90599,90601,90603,90606,90608],{"dataMmlNode":26031,"transform":30733},[25033,90600],{"dataC":26103,"xLinkHref":29791},[25033,90602],{"dataC":26392,"xLinkHref":90539,"transform":26394},[25033,90604],{"dataC":26486,"xLinkHref":90605,"transform":25049},"#MJX-52-TEX-N-33",[25033,90607],{"dataC":26035,"xLinkHref":29788,"transform":26835},[25033,90609],{"dataC":26035,"xLinkHref":29788,"transform":30504}," — 好多了。",[12,90612,90613],{},[106,90614,90615],{},"用形狀來分：",[23,90617,90618,90680,90712],{},[26,90619,90620,90621],{},"圓形 → 2+, 1− → ",[24923,90622,90624],{"className":90623,"jax":24927},[24926],[24929,90625,90626,90649],{"style":26868,"xmlns":24932,"width":30446,"height":30167,"role":1930,"focusable":8058,"viewBox":30447,"xmlnsXLink":24936},[24938,90627,90628,90631,90634,90637,90640,90643,90646],{},[24941,90629],{"id":90630,"d":25869},"MJX-53-TEX-I-1D438",[24941,90632],{"id":90633,"d":24972},"MJX-53-TEX-N-3D",[24941,90635],{"id":90636,"d":25935},"MJX-53-TEX-N-30",[24941,90638],{"id":90639,"d":26357},"MJX-53-TEX-N-2E",[24941,90641],{"id":90642,"d":29754},"MJX-53-TEX-N-39",[24941,90644],{"id":90645,"d":25914},"MJX-53-TEX-N-31",[24941,90647],{"id":90648,"d":30471},"MJX-53-TEX-N-38",[25022,90650,90651],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,90652,90653,90658,90663],{"dataMmlNode":25028},[25022,90654,90655],{"dataMmlNode":25942},[25033,90656],{"dataC":25945,"xLinkHref":90657},"#MJX-53-TEX-I-1D438",[25022,90659,90660],{"dataMmlNode":25031,"transform":30197},[25033,90661],{"dataC":25078,"xLinkHref":90662},"#MJX-53-TEX-N-3D",[25022,90664,90665,90668,90671,90674,90677],{"dataMmlNode":26031,"transform":30203},[25033,90666],{"dataC":26103,"xLinkHref":90667},"#MJX-53-TEX-N-30",[25033,90669],{"dataC":26392,"xLinkHref":90670,"transform":26394},"#MJX-53-TEX-N-2E",[25033,90672],{"dataC":29772,"xLinkHref":90673,"transform":25049},"#MJX-53-TEX-N-39",[25033,90675],{"dataC":26035,"xLinkHref":90676,"transform":26835},"#MJX-53-TEX-N-31",[25033,90678],{"dataC":30502,"xLinkHref":90679,"transform":30504},"#MJX-53-TEX-N-38",[26,90681,90682,90683],{},"方形 → 0+, 1− → ",[24923,90684,90686],{"className":90685,"jax":24927},[24926],[24929,90687,90688,90696],{"style":26868,"xmlns":24932,"width":30513,"height":30167,"role":1930,"focusable":8058,"viewBox":30514,"xmlnsXLink":24936},[24938,90689,90690,90692,90694],{},[24941,90691],{"id":30041,"d":25869},[24941,90693],{"id":30071,"d":24972},[24941,90695],{"id":30080,"d":25935},[25022,90697,90698],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,90699,90700,90704,90708],{"dataMmlNode":25028},[25022,90701,90702],{"dataMmlNode":25942},[25033,90703],{"dataC":25945,"xLinkHref":30089},[25022,90705,90706],{"dataMmlNode":25031,"transform":30197},[25033,90707],{"dataC":25078,"xLinkHref":30139},[25022,90709,90710],{"dataMmlNode":26031,"transform":30203},[25033,90711],{"dataC":26103,"xLinkHref":30150},[26,90713,90714,90879],{},[24923,90715,90717],{"className":90716,"jax":24927},[24926],[24929,90718,90719,90767],{"style":24931,"xmlns":24932,"width":30552,"height":24934,"role":1930,"focusable":8058,"viewBox":30553,"xmlnsXLink":24936},[24938,90720,90721,90724,90726,90728,90730,90732,90734,90736,90738,90740,90742,90745,90748,90751,90754,90757,90760,90762,90764],{},[24941,90722],{"id":90723,"d":27319},"MJX-55-TEX-I-1D43A",[24941,90725],{"id":46965,"d":27323},[24941,90727],{"id":37600,"d":27108},[24941,90729],{"id":46970,"d":25873},[24941,90731],{"id":30176,"d":24972},[24941,90733],{"id":30179,"d":25914},[24941,90735],{"id":37623,"d":25910},[24941,90737],{"id":37597,"d":24944},[24941,90739],{"id":30185,"d":25935},[24941,90741],{"id":30182,"d":26357},[24941,90743],{"id":90744,"d":26361},"MJX-55-TEX-N-37",[24941,90746],{"id":90747,"d":26660},"MJX-55-TEX-N-35",[24941,90749],{"id":90750,"d":26907},"MJX-55-TEX-N-D7",[24941,90752],{"id":90753,"d":29754},"MJX-55-TEX-N-39",[24941,90755],{"id":90756,"d":30471},"MJX-55-TEX-N-38",[24941,90758],{"id":90759,"d":30313},"MJX-55-TEX-N-2B",[24941,90761],{"id":37629,"d":25928},[24941,90763],{"id":37603,"d":24988},[24941,90765],{"id":90766,"d":26455},"MJX-55-TEX-N-33",[25022,90768,90769],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,90770,90771,90776,90780,90784,90788,90792,90796,90800,90804,90816,90821,90835,90840,90850,90854,90858,90862,90866],{"dataMmlNode":25028},[25022,90772,90773],{"dataMmlNode":25942},[25033,90774],{"dataC":27418,"xLinkHref":90775},"#MJX-55-TEX-I-1D43A",[25022,90777,90778],{"dataMmlNode":25942,"transform":27422},[25033,90779],{"dataC":27425,"xLinkHref":47001},[25022,90781,90782],{"dataMmlNode":25942,"transform":27429},[25033,90783],{"dataC":27210,"xLinkHref":37658},[25022,90785,90786],{"dataMmlNode":25942,"transform":27435},[25033,90787],{"dataC":25952,"xLinkHref":47010},[25022,90789,90790],{"dataMmlNode":25031,"transform":30343},[25033,90791],{"dataC":25078,"xLinkHref":30200},[25022,90793,90794],{"dataMmlNode":26031,"transform":30349},[25033,90795],{"dataC":26035,"xLinkHref":30206},[25022,90797,90798],{"dataMmlNode":25031,"transform":30355},[25033,90799],{"dataC":26019,"xLinkHref":37695},[25022,90801,90802],{"dataMmlNode":25031,"transform":30361},[25033,90803],{"dataC":25035,"xLinkHref":37653},[25022,90805,90806,90808,90810,90813],{"dataMmlNode":26031,"transform":30367},[25033,90807],{"dataC":26103,"xLinkHref":30212},[25033,90809],{"dataC":26392,"xLinkHref":30209,"transform":26394},[25033,90811],{"dataC":26397,"xLinkHref":90812,"transform":25049},"#MJX-55-TEX-N-37",[25033,90814],{"dataC":26717,"xLinkHref":90815,"transform":26835},"#MJX-55-TEX-N-35",[25022,90817,90818],{"dataMmlNode":25031,"transform":30673},[25033,90819],{"dataC":26950,"xLinkHref":90820},"#MJX-55-TEX-N-D7",[25022,90822,90823,90825,90827,90830,90832],{"dataMmlNode":26031,"transform":30679},[25033,90824],{"dataC":26103,"xLinkHref":30212},[25033,90826],{"dataC":26392,"xLinkHref":30209,"transform":26394},[25033,90828],{"dataC":29772,"xLinkHref":90829,"transform":25049},"#MJX-55-TEX-N-39",[25033,90831],{"dataC":26035,"xLinkHref":30206,"transform":26835},[25033,90833],{"dataC":30502,"xLinkHref":90834,"transform":30504},"#MJX-55-TEX-N-38",[25022,90836,90837],{"dataMmlNode":25031,"transform":30694},[25033,90838],{"dataC":30393,"xLinkHref":90839},"#MJX-55-TEX-N-2B",[25022,90841,90842,90844,90846,90848],{"dataMmlNode":26031,"transform":30700},[25033,90843],{"dataC":26103,"xLinkHref":30212},[25033,90845],{"dataC":26392,"xLinkHref":30209,"transform":26394},[25033,90847],{"dataC":26058,"xLinkHref":37743,"transform":25049},[25033,90849],{"dataC":26717,"xLinkHref":90815,"transform":26835},[25022,90851,90852],{"dataMmlNode":25031,"transform":30712},[25033,90853],{"dataC":26950,"xLinkHref":90820},[25022,90855,90856],{"dataMmlNode":26031,"transform":30717},[25033,90857],{"dataC":26103,"xLinkHref":30212},[25022,90859,90860],{"dataMmlNode":25031,"transform":30722},[25033,90861],{"dataC":25108,"xLinkHref":37663},[25022,90863,90864],{"dataMmlNode":25031,"transform":30728},[25033,90865],{"dataC":25078,"xLinkHref":30200},[25022,90867,90868,90870,90872,90875,90877],{"dataMmlNode":26031,"transform":30733},[25033,90869],{"dataC":26103,"xLinkHref":30212},[25033,90871],{"dataC":26392,"xLinkHref":30209,"transform":26394},[25033,90873],{"dataC":26486,"xLinkHref":90874,"transform":25049},"#MJX-55-TEX-N-33",[25033,90876],{"dataC":26035,"xLinkHref":30206,"transform":26835},[25033,90878],{"dataC":26035,"xLinkHref":30206,"transform":30504}," — 跟顏色一樣好。",[12,90881,90882],{},"所以第一刀我們會選顏色或形狀（兩者 gain 都是 0.311），絕對不會選大小（gain = 0）。",[6964,90884,90886],{"id":90885},"id3-演算法","ID3 演算法",[12,90888,90889,90890,90892],{},"我們一直在講的這個方法，正式名稱叫做 ",[106,90891,31055],{},"，是 J. Ross Quinlan 在 1979 年發明的。它巧妙地運用了 Shannon 的資訊理論 (1948) 來挑選特徵。它的幾個重要特色包含：",[23,90894,90895,90902,90908,90915],{},[26,90896,90897,90898,90901],{},"採用",[106,90899,90900],{},"由上而下 (top-down)"," 建構，不走回頭路 (no backtracking)。",[26,90903,90904,90905,90907],{},"每一步都貪心地選擇",[106,90906,89896],{},"的特徵。",[26,90909,90910,90911,90914],{},"在非葉節點的分支上",[106,90912,90913],{},"遞迴","執行，直到所有資料都被分類完畢。",[26,90916,90917,90918,90921],{},"會搜尋",[106,90919,90920],{},"整個資料集","來建構這棵樹。",[12,90923,90924,90925,90927,90928,90930],{},"後來這個演算法被改良成了 ",[106,90926,31088],{}," (Quinlan, 1993)，可以處理連續數值、缺失值，還加入了剪枝機制。同一時期，Breiman 和 Friedman 也開發了另一個很有名的平行發展演算法，叫做 ",[106,90929,31092],{}," (Classification and Regression Trees)。",[6964,90932,90934],{"id":90933},"有趣的例子辛普森家庭角色分類","有趣的例子：辛普森家庭角色分類",[12,90936,90937],{},"假設我們想用三個特徵（頭髮長度、體重、年齡）來分類辛普森家庭的角色是男是女。",[7085,90939,90940,90959],{},[7088,90941,90942],{},[7091,90943,90944,90947,90950,90953,90956],{},[7094,90945,90946],{"align":47883},"角色",[7094,90948,90949],{"align":47883},"頭髮長度 (吋)",[7094,90951,90952],{"align":47883},"體重 (磅)",[7094,90954,90955],{"align":47883},"年齡",[7094,90957,90958],{"align":47883},"性別",[7104,90960,90961,90973,90985,90997,91009,91021,91033,91045,91057],{},[7091,90962,90963,90965,90967,90969,90971],{},[7109,90964,31127],{"align":47883},[7109,90966,31130],{"align":47883},[7109,90968,31133],{"align":47883},[7109,90970,31136],{"align":47883},[7109,90972,31139],{"align":47883},[7091,90974,90975,90977,90979,90981,90983],{},[7109,90976,31144],{"align":47883},[7109,90978,31147],{"align":47883},[7109,90980,31150],{"align":47883},[7109,90982,29597],{"align":47883},[7109,90984,31155],{"align":47883},[7091,90986,90987,90989,90991,90993,90995],{},[7109,90988,31160],{"align":47883},[7109,90990,31163],{"align":47883},[7109,90992,31166],{"align":47883},[7109,90994,17383],{"align":47883},[7109,90996,31139],{"align":47883},[7091,90998,90999,91001,91003,91005,91007],{},[7109,91000,31175],{"align":47883},[7109,91002,31178],{"align":47883},[7109,91004,31181],{"align":47883},[7109,91006,31184],{"align":47883},[7109,91008,31155],{"align":47883},[7091,91010,91011,91013,91015,91017,91019],{},[7109,91012,31191],{"align":47883},[7109,91014,31194],{"align":47883},[7109,91016,31197],{"align":47883},[7109,91018,8855],{"align":47883},[7109,91020,31155],{"align":47883},[7091,91022,91023,91025,91027,91029,91031],{},[7109,91024,31206],{"align":47883},[7109,91026,31209],{"align":47883},[7109,91028,31212],{"align":47883},[7109,91030,31215],{"align":47883},[7109,91032,31139],{"align":47883},[7091,91034,91035,91037,91039,91041,91043],{},[7109,91036,31222],{"align":47883},[7109,91038,31225],{"align":47883},[7109,91040,31228],{"align":47883},[7109,91042,31231],{"align":47883},[7109,91044,31155],{"align":47883},[7091,91046,91047,91049,91051,91053,91055],{},[7109,91048,31238],{"align":47883},[7109,91050,31147],{"align":47883},[7109,91052,31243],{"align":47883},[7109,91054,30502],{"align":47883},[7109,91056,31139],{"align":47883},[7091,91058,91059,91061,91063,91065,91067],{},[7109,91060,31252],{"align":47883},[7109,91062,31178],{"align":47883},[7109,91064,31257],{"align":47883},[7109,91066,31260],{"align":47883},[7109,91068,31139],{"align":47883},[12,91070,91071,91072,86843],{},"資料裡有 4 女 5 男，初始的",[24923,91073,91075],{"className":91074,"jax":24927},[24926],[24929,91076,91077,91127],{"style":24931,"xmlns":24932,"width":31271,"height":24934,"role":1930,"focusable":8058,"viewBox":31272,"xmlnsXLink":24936},[24938,91078,91079,91081,91083,91086,91089,91092,91095,91098,91100,91103,91106,91109,91112,91114,91116,91118,91120,91122,91125],{},[24941,91080],{"id":30225,"d":25869},[24941,91082],{"id":47112,"d":25873},[24941,91084],{"id":91085,"d":25877},"MJX-56-TEX-I-1D461",[24941,91087],{"id":91088,"d":25881},"MJX-56-TEX-I-1D45F",[24941,91090],{"id":91091,"d":25885},"MJX-56-TEX-I-1D45C",[24941,91093],{"id":91094,"d":25889},"MJX-56-TEX-I-1D45D",[24941,91096],{"id":91097,"d":25893},"MJX-56-TEX-I-1D466",[24941,91099],{"id":37789,"d":24944},[24941,91101],{"id":91102,"d":29582},"MJX-56-TEX-N-34",[24941,91104],{"id":91105,"d":27343},"MJX-56-TEX-I-1D439",[24941,91107],{"id":91108,"d":27339},"MJX-56-TEX-N-2C",[24941,91110],{"id":91111,"d":26660},"MJX-56-TEX-N-35",[24941,91113],{"id":47100,"d":31314},[24941,91115],{"id":37798,"d":24988},[24941,91117],{"id":30228,"d":24972},[24941,91119],{"id":30237,"d":25935},[24941,91121],{"id":30234,"d":26357},[24941,91123],{"id":91124,"d":29754},"MJX-56-TEX-N-39",[24941,91126],{"id":30231,"d":25914},[25022,91128,91129],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91130,91131,91135,91139,91144,91149,91154,91159,91164,91168,91173,91178,91183,91188,91192,91196,91200],{"dataMmlNode":25028},[25022,91132,91133],{"dataMmlNode":25942},[25033,91134],{"dataC":25945,"xLinkHref":30246},[25022,91136,91137],{"dataMmlNode":25942,"transform":25949},[25033,91138],{"dataC":25952,"xLinkHref":47171},[25022,91140,91141],{"dataMmlNode":25942,"transform":25956},[25033,91142],{"dataC":25959,"xLinkHref":91143},"#MJX-56-TEX-I-1D461",[25022,91145,91146],{"dataMmlNode":25942,"transform":25963},[25033,91147],{"dataC":25966,"xLinkHref":91148},"#MJX-56-TEX-I-1D45F",[25022,91150,91151],{"dataMmlNode":25942,"transform":25970},[25033,91152],{"dataC":25973,"xLinkHref":91153},"#MJX-56-TEX-I-1D45C",[25022,91155,91156],{"dataMmlNode":25942,"transform":25977},[25033,91157],{"dataC":25980,"xLinkHref":91158},"#MJX-56-TEX-I-1D45D",[25022,91160,91161],{"dataMmlNode":25942,"transform":25984},[25033,91162],{"dataC":25987,"xLinkHref":91163},"#MJX-56-TEX-I-1D466",[25022,91165,91166],{"dataMmlNode":25031,"transform":25991},[25033,91167],{"dataC":25035,"xLinkHref":37813},[25022,91169,91170],{"dataMmlNode":26031,"transform":25997},[25033,91171],{"dataC":29597,"xLinkHref":91172},"#MJX-56-TEX-N-34",[25022,91174,91175],{"dataMmlNode":25942,"transform":31384},[25033,91176],{"dataC":27463,"xLinkHref":91177},"#MJX-56-TEX-I-1D439",[25022,91179,91180],{"dataMmlNode":25031,"transform":31390},[25033,91181],{"dataC":27456,"xLinkHref":91182},"#MJX-56-TEX-N-2C",[25022,91184,91185],{"dataMmlNode":26031,"transform":31396},[25033,91186],{"dataC":26717,"xLinkHref":91187},"#MJX-56-TEX-N-35",[25022,91189,91190],{"dataMmlNode":25942,"transform":31402},[25033,91191],{"dataC":31405,"xLinkHref":47142},[25022,91193,91194],{"dataMmlNode":25031,"transform":31409},[25033,91195],{"dataC":25108,"xLinkHref":37836},[25022,91197,91198],{"dataMmlNode":25031,"transform":31415},[25033,91199],{"dataC":25078,"xLinkHref":30251},[25022,91201,91202,91204,91206,91209,91211,91213],{"dataMmlNode":26031,"transform":31421},[25033,91203],{"dataC":26103,"xLinkHref":30262},[25033,91205],{"dataC":26392,"xLinkHref":30259,"transform":26394},[25033,91207],{"dataC":29772,"xLinkHref":91208,"transform":25049},"#MJX-56-TEX-N-39",[25033,91210],{"dataC":29772,"xLinkHref":91208,"transform":26835},[25033,91212],{"dataC":26035,"xLinkHref":30256,"transform":30504},[25033,91214],{"dataC":26035,"xLinkHref":30256,"transform":31438},[12,91216,91217],{},"我們計算每個特徵的資訊增益來找最佳切分點：",[23,91219,91220,91225,91231],{},[26,91221,91222,31449],{},[106,91223,91224],{},"頭髮長度 ≤ 5?",[26,91226,91227,91230],{},[106,91228,91229],{},"體重 ≤ 160?"," → Gain = 0.5900 ← 贏了！",[26,91232,91233,91236],{},[106,91234,91235],{},"年齡 ≤ 40?"," → Gain = 0.0183",[12,91238,91239],{},"「體重」給了我們最棒的切分。所有體重超過 160 磅的角色都是男生（這條分支的熵 = 0，搞定！）。但體重 ≤ 160 的群體裡還是混雜著 (4女, 1男)，所以我們要繼續往下遞迴。",[12,91241,91242,91243,91246],{},"在第二次切分時，我們發現 ",[106,91244,91245],{},"頭髮長度 ≤ 2"," 完美地把剩下的人分開了：Bart（短頭髮，男生）對上其他女生（長頭髮）。",[12,91248,91249],{},"最終的決策樹：",[6972,91251,91254],{"className":91252,"code":91253,"language":6977},[6975],"            體重 ≤ 160?\n            \u002F           \\\n          是              否\n          \u002F               \\\n   頭髮長度 ≤ 2?         男性\n      \u002F        \\\n    是          否\n    \u002F             \\\n  男性           女性\n",[6979,91255,91253],{"__ignoreMap":180},[12,91257,91258],{},"寫成規則就是：",[23,91260,91261,91267,91272],{},[26,91262,91263,91264],{},"如果體重 > 160 → ",[106,91265,91266],{},"男性",[26,91268,91269,91270],{},"否則如果頭髮長度 ≤ 2 → ",[106,91271,91266],{},[26,91273,91274,91275],{},"否則 → ",[106,91276,91277],{},"女性",[12,91279,91280,91281,91284],{},"現在如果「漫畫男 (Comic Book Guy)」出現了（頭髮: 8\", 體重: 290, 年齡: 38），決策樹會判斷：體重 290 > 160 → ",[106,91282,91283],{},"男生","。完全正確！",[6964,91286,91287],{"id":91287},"過擬合問題",[12,91289,91290],{},"不過有個陷阱：一棵能完美分對所有訓練資料的樹，不見得就是一棵好樹。",[12,91292,91293,91296,91297,91326,91327,91346,91347,91366,91367,91386],{},[106,91294,91295],{},"過擬合（Overfitting）"," 就是模型把訓練資料學得「太好了」——連雜訊和偶然的巧合都一起學進去了——結果在新資料上反而表現更差。正式地說，如果存在另一個假設 ",[24923,91298,91300],{"className":91299,"jax":24927},[24926],[24929,91301,91302,91310],{"style":25196,"xmlns":24932,"width":31547,"height":31548,"role":1930,"focusable":8058,"viewBox":31549,"xmlnsXLink":24936},[24938,91303,91304,91307],{},[24941,91305],{"id":91306,"d":31530},"MJX-57-TEX-I-210E",[24941,91308],{"id":91309,"d":31558},"MJX-57-TEX-V-2032",[25022,91311,91312],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91313,91314],{"dataMmlNode":25028},[25022,91315,91316,91321],{"dataMmlNode":31565},[25022,91317,91318],{"dataMmlNode":25942},[25033,91319],{"dataC":31539,"xLinkHref":91320},"#MJX-57-TEX-I-210E",[25022,91322,91323],{"dataMmlNode":25031,"transform":31573},[25033,91324],{"dataC":31576,"xLinkHref":91325},"#MJX-57-TEX-V-2032","，它的訓練誤差比 ",[24923,91328,91330],{"className":91329,"jax":24927},[24926],[24929,91331,91332,91337],{"style":25196,"xmlns":24932,"width":31522,"height":31523,"role":1930,"focusable":8058,"viewBox":31524,"xmlnsXLink":24936},[24938,91333,91334],{},[24941,91335],{"id":91336,"d":31530},"MJX-58-TEX-I-210E",[25022,91338,91339],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91340,91341],{"dataMmlNode":25028},[25022,91342,91343],{"dataMmlNode":25942},[25033,91344],{"dataC":31539,"xLinkHref":91345},"#MJX-58-TEX-I-210E"," 高，但測試誤差卻比 ",[24923,91348,91350],{"className":91349,"jax":24927},[24926],[24929,91351,91352,91357],{"style":25196,"xmlns":24932,"width":31522,"height":31523,"role":1930,"focusable":8058,"viewBox":31524,"xmlnsXLink":24936},[24938,91353,91354],{},[24941,91355],{"id":91356,"d":31530},"MJX-59-TEX-I-210E",[25022,91358,91359],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91360,91361],{"dataMmlNode":25028},[25022,91362,91363],{"dataMmlNode":25942},[25033,91364],{"dataC":31539,"xLinkHref":91365},"#MJX-59-TEX-I-210E"," 低，那 ",[24923,91368,91370],{"className":91369,"jax":24927},[24926],[24929,91371,91372,91377],{"style":25196,"xmlns":24932,"width":31522,"height":31523,"role":1930,"focusable":8058,"viewBox":31524,"xmlnsXLink":24936},[24938,91373,91374],{},[24941,91375],{"id":91376,"d":31530},"MJX-60-TEX-I-210E",[25022,91378,91379],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91380,91381],{"dataMmlNode":25028},[25022,91382,91383],{"dataMmlNode":25942},[25033,91384],{"dataC":31539,"xLinkHref":91385},"#MJX-60-TEX-I-210E"," 就是過擬合了。",[12,91388,91389,91390,91431],{},"打個比方：假設你要驗證歐姆定律 ",[24923,91391,91393],{"className":91392,"jax":24927},[24926],[24929,91394,91395,91408],{"style":26868,"xmlns":24932,"width":31595,"height":31596,"role":1930,"focusable":8058,"viewBox":31597,"xmlnsXLink":24936},[24938,91396,91397,91400,91402,91405],{},[24941,91398],{"id":91399,"d":27385},"MJX-61-TEX-I-1D449",[24941,91401],{"id":30768,"d":24972},[24941,91403],{"id":91404,"d":29814},"MJX-61-TEX-I-1D43C",[24941,91406],{"id":91407,"d":31612},"MJX-61-TEX-I-1D445",[25022,91409,91410],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91411,91412,91417,91421,91426],{"dataMmlNode":25028},[25022,91413,91414],{"dataMmlNode":25942},[25033,91415],{"dataC":27570,"xLinkHref":91416},"#MJX-61-TEX-I-1D449",[25022,91418,91419],{"dataMmlNode":25031,"transform":31624},[25033,91420],{"dataC":25078,"xLinkHref":30797},[25022,91422,91423],{"dataMmlNode":25942,"transform":31630},[25033,91424],{"dataC":29851,"xLinkHref":91425},"#MJX-61-TEX-I-1D43C",[25022,91427,91428],{"dataMmlNode":25942,"transform":31636},[25033,91429],{"dataC":31639,"xLinkHref":91430},"#MJX-61-TEX-I-1D445","，量了 10 個數據點，然後硬是用一個 9 次方多項式的複雜曲線去畫它。，這條線確實會完美穿過每一個點。但這條彎彎曲曲的線拿來預測新的測量值，效果絕對比一條簡單的直線還要差得多。",[12,91433,91434],{},"決策樹也是一樣的道理。資料裡的雜訊會讓演算法多長出一些不必要的分支，這些分支只是捕捉到了隨機性，不是真正的規律。",[7166,91436,91437],{"id":91437},"雜訊怎麼造成過擬合",[12,91439,91440],{},"回到前面顏色\u002F形狀的例子，假設有一筆訓練資料被標錯了。為了把這一筆也分對，決策樹可能需要額外用一些不相關的特徵（像大小）來多分幾層。結果就是一棵更大、更複雜，但泛化能力更差的樹。",[6964,91442,91444],{"id":91443},"剪枝-pruning過擬合的解藥","剪枝 (Pruning)：過擬合的解藥",[12,91446,91447],{},"要解決過度擬合，主要有兩種策略：",[12,91449,91450,91453],{},[106,91451,91452],{},"預剪枝（Prepruning）"," — 在樹還沒完全長滿之前，就提早喊停。例如，當某個節點的資料量太少，覺得切分下去不可靠時，就不切了。",[12,91455,91456,91459],{},[106,91457,91458],{},"後剪枝（Postpruning）"," — 先讓整棵樹長好長滿，然後再把那些對「泛化能力（面對新資料的能力）」沒有幫助的樹枝剪掉。通常這種方法比較有效，因為實務上我們很難預測什麼時候該提早喊停。",[7166,91461,91463],{"id":91462},"簡化誤差剪枝reduced-error-pruning","簡化誤差剪枝（Reduced Error Pruning）",[12,91465,91466],{},"這是一種很常見的事後剪枝法：",[3603,91468,91469,91472,91475,91478,91481],{},[26,91470,91471],{},"把你的資料分成「生長集 (grow set)」和「驗證集 (validation set)」。",[26,91473,91474],{},"用生長集建構一棵完整的決策樹。",[26,91476,91477],{},"針對每一個內部節點，試著把它的子樹整個拔掉，變成一個葉節點（類別就用該節點的多數決）。",[26,91479,91480],{},"如果拔掉之後，模型在驗證集上的準確率提升了（或沒變差），就永遠剪掉它。",[26,91482,91483],{},"重複這個動作，直到再剪下去準確率會掉為止。",[12,91485,91486,91487,91506,91507,91526],{},"這方法的缺點是你得「犧牲」一部分訓練資料拿去當驗證集。另一個替代方案是：跑好幾輪不同的隨機分割，記錄每次剪枝後的樹的複雜度，取平均值 ",[24923,91488,91490],{"className":91489,"jax":24927},[24926],[24929,91491,91492,91497],{"style":27778,"xmlns":24932,"width":56558,"height":27780,"role":1930,"focusable":8058,"viewBox":56559,"xmlnsXLink":24936},[24938,91493,91494],{},[24941,91495],{"id":91496,"d":56463},"MJX-62-TEX-I-1D436",[25022,91498,91499],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91500,91501],{"dataMmlNode":25028},[25022,91502,91503],{"dataMmlNode":25942},[25033,91504],{"dataC":56500,"xLinkHref":91505},"#MJX-62-TEX-I-1D436","，然後用全部資料從頭長一棵樹，長到複雜度 ",[24923,91508,91510],{"className":91509,"jax":24927},[24926],[24929,91511,91512,91517],{"style":27778,"xmlns":24932,"width":56558,"height":27780,"role":1930,"focusable":8058,"viewBox":56559,"xmlnsXLink":24936},[24938,91513,91514],{},[24941,91515],{"id":91516,"d":56463},"MJX-63-TEX-I-1D436",[25022,91518,91519],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91520,91521],{"dataMmlNode":25028},[25022,91522,91523],{"dataMmlNode":25942},[25033,91524],{"dataC":56500,"xLinkHref":91525},"#MJX-63-TEX-I-1D436"," 就停。",[7166,91528,91529],{"id":91529},"其他剪枝方法",[23,91531,91532,91538],{},[26,91533,91534,91537],{},[106,91535,91536],{},"統計檢定"," ： 檢查節點上的資料規律是真的有統計顯著性，還是只是隨機出現的。",[26,91539,91540,91543],{},[106,91541,91542],{},"最小描述長度（MDL）"," ： 權衡一下：「多長出這些樹枝的複雜度」跟「直接把例外情況死記下來」，哪一個比較划算？",[6964,91545,91547],{"id":91546},"運算複雜度-computational-complexity","運算複雜度 (Computational Complexity)",[12,91549,91550,91551,91606,91607,91626,91627,91646],{},"在最壞的情況下（長出一棵完整的樹，而且每條路徑都測試了所有特徵），建構決策樹需要花費 ",[24923,91552,91554],{"className":91553,"jax":24927},[24926],[24929,91555,91556,91573],{"style":24931,"xmlns":24932,"width":31732,"height":31733,"role":1930,"focusable":8058,"viewBox":31734,"xmlnsXLink":24936},[24938,91557,91558,91561,91563,91565,91568,91571],{},[24941,91559],{"id":91560,"d":31740},"MJX-64-TEX-I-1D442",[24941,91562],{"id":31298,"d":24944},[24941,91564],{"id":31280,"d":25873},[24941,91566],{"id":91567,"d":28310},"MJX-64-TEX-I-1D45A",[24941,91569],{"id":91570,"d":25928},"MJX-64-TEX-N-32",[24941,91572],{"id":31317,"d":24988},[25022,91574,91575],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91576,91577,91582,91586,91590,91602],{"dataMmlNode":25028},[25022,91578,91579],{"dataMmlNode":25942},[25033,91580],{"dataC":31764,"xLinkHref":91581},"#MJX-64-TEX-I-1D442",[25022,91583,91584],{"dataMmlNode":25031,"transform":31768},[25033,91585],{"dataC":25035,"xLinkHref":31376},[25022,91587,91588],{"dataMmlNode":25942,"transform":27567},[25033,91589],{"dataC":25952,"xLinkHref":31346},[25022,91591,91592,91597],{"dataMmlNode":31565,"transform":31779},[25022,91593,91594],{"dataMmlNode":25942},[25033,91595],{"dataC":28479,"xLinkHref":91596},"#MJX-64-TEX-I-1D45A",[25022,91598,91599],{"dataMmlNode":26031,"transform":31787},[25033,91600],{"dataC":26058,"xLinkHref":91601},"#MJX-64-TEX-N-32",[25022,91603,91604],{"dataMmlNode":25031,"transform":31793},[25033,91605],{"dataC":25108,"xLinkHref":31412}," 的時間，其中 ",[24923,91608,91610],{"className":91609,"jax":24927},[24926],[24929,91611,91612,91617],{"style":25196,"xmlns":24932,"width":31803,"height":27038,"role":1930,"focusable":8058,"viewBox":31804,"xmlnsXLink":24936},[24938,91613,91614],{},[24941,91615],{"id":91616,"d":25873},"MJX-65-TEX-I-1D45B",[25022,91618,91619],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91620,91621],{"dataMmlNode":25028},[25022,91622,91623],{"dataMmlNode":25942},[25033,91624],{"dataC":25952,"xLinkHref":91625},"#MJX-65-TEX-I-1D45B"," 是資料筆數，",[24923,91628,91630],{"className":91629,"jax":24927},[24926],[24929,91631,91632,91637],{"style":25196,"xmlns":24932,"width":31825,"height":27038,"role":1930,"focusable":8058,"viewBox":31826,"xmlnsXLink":24936},[24938,91633,91634],{},[24941,91635],{"id":91636,"d":28310},"MJX-66-TEX-I-1D45A",[25022,91638,91639],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91640,91641],{"dataMmlNode":25028},[25022,91642,91643],{"dataMmlNode":25942},[25033,91644],{"dataC":28479,"xLinkHref":91645},"#MJX-66-TEX-I-1D45A"," 是特徵數量。",[12,91648,91649,91650,91669,91670,91687],{},"但實際上，樹很少會長到那麼完整，通常時間複雜度跟 ",[24923,91651,91653],{"className":91652,"jax":24927},[24926],[24929,91654,91655,91660],{"style":25196,"xmlns":24932,"width":31803,"height":27038,"role":1930,"focusable":8058,"viewBox":31804,"xmlnsXLink":24936},[24938,91656,91657],{},[24941,91658],{"id":91659,"d":25873},"MJX-67-TEX-I-1D45B",[25022,91661,91662],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91663,91664],{"dataMmlNode":25028},[25022,91665,91666],{"dataMmlNode":25942},[25033,91667],{"dataC":25952,"xLinkHref":91668},"#MJX-67-TEX-I-1D45B"," 還有 ",[24923,91671,91673],{"className":91672,"jax":24927},[24926],[24929,91674,91675,91679],{"style":25196,"xmlns":24932,"width":31825,"height":27038,"role":1930,"focusable":8058,"viewBox":31826,"xmlnsXLink":24936},[24938,91676,91677],{},[24941,91678],{"id":31749,"d":28310},[25022,91680,91681],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,91682,91683],{"dataMmlNode":25028},[25022,91684,91685],{"dataMmlNode":25942},[25033,91686],{"dataC":28479,"xLinkHref":31784}," 都差不多是呈現線性關係。這也是為什麼決策樹可以很好地擴充應用到大型資料集上。",[6964,91689,91690],{"id":91690},"優點與限制",[7166,91692,91693],{"id":91693},"優點",[23,91695,91696,91699,91702,91705,91708],{},[26,91697,91698],{},"能產出人類可以直接看懂的解釋規則。",[26,91700,91701],{},"訓練速度快，預測新資料的速度也很快。",[26,91703,91704],{},"數值型跟類別型的特徵都能無痛處理。",[26,91706,91707],{},"對雜訊跟缺失值有不錯的容忍度。",[26,91709,91710],{},"不需要先做特徵縮放 (Feature scaling) 或正規化。",[7166,91712,91713],{"id":91713},"限制",[23,91715,91716,91719,91722,91725,91728],{},[26,91717,91718],{},"非常容易過度擬合 (Overfitting)，特別是在資料量很小的時候。",[26,91720,91721],{},"一次只能測試一個特徵（只能切出平行於坐標軸的邊界），很難捕捉到對角線或傾斜的決策邊界。",[26,91723,91724],{},"資訊增益演算法天生會偏袒那些「擁有很多種不同數值」的特徵。",[26,91726,91727],{},"貪心演算法沒辦法保證你能找到「全域最佳」的樹（事實上，要找到最小且一致的樹是一個 NP-hard 問題）。",[26,91729,91730],{},"只能給出絕對的分類結果，沒辦法告訴你它有「多大機率」或「多大信心」覺得是這個類別。",[6964,91732,91734],{"id":91733},"接下來呢","接下來呢？",[12,91736,91737],{},"學會決策樹後，其實你已經為學習更強大的演算法打下了基礎：",[23,91739,91740,91746,91752],{},[26,91741,91742,91745],{},[106,91743,91744],{},"隨機森林 (Random Forests)"," — 隨機挑選資料和特徵，建構出一大堆決策樹，最後大家一起投票表決。",[26,91747,91748,91751],{},[106,91749,91750],{},"梯度提升樹 (Gradient Boosted Trees，像是 XGBoost, LightGBM)"," — 也是種一堆樹，但它們是循序漸進的，每一棵新樹都在努力修正前一棵樹犯的錯誤。",[26,91753,91754,91756],{},[106,91755,31955],{}," — Quinlan 對 ID3 的升級版，更聰明地處理連續特徵、缺失值和剪枝。",[12,91758,91759],{},"這些「集成學習 (Ensemble)」的方法，是目前業界在處理結構化表格資料時最成功也最常勝的演算法——而它們全都是建立在我們今天學到的決策樹概念之上！",[9332,91761,31966],{},{"title":180,"searchDepth":181,"depth":181,"links":91763},[91764,91765,91766,91767,91772,91773,91774,91777,91781,91782,91786],{"id":86666,"depth":181,"text":86667},{"id":86717,"depth":181,"text":86718},{"id":86761,"depth":181,"text":86762},{"id":86883,"depth":181,"text":86884,"children":91768},[91769,91770,91771],{"id":86890,"depth":7430,"text":86891},{"id":88750,"depth":7430,"text":88751},{"id":89900,"depth":7430,"text":89900},{"id":90885,"depth":181,"text":90886},{"id":90933,"depth":181,"text":90934},{"id":91287,"depth":181,"text":91287,"children":91775},[91776],{"id":91437,"depth":7430,"text":91437},{"id":91443,"depth":181,"text":91444,"children":91778},[91779,91780],{"id":91462,"depth":7430,"text":91463},{"id":91529,"depth":7430,"text":91529},{"id":91546,"depth":181,"text":91547},{"id":91690,"depth":181,"text":91690,"children":91783},[91784,91785],{"id":91693,"depth":7430,"text":91693},{"id":91713,"depth":7430,"text":91713},{"id":91733,"depth":181,"text":91734},{},"\u002Fblog\u002F決策樹",{"title":86661,"description":180},{"loc":91788},"blog\u002F決策樹","fPO17x7dW9X0wlzz1wmHkPIrbWN_nOTrhkwaFZfRdM4",{"id":91794,"title":91795,"author":7,"body":91796,"date":92020,"description":92021,"draft":185,"edited_at":83251,"extension":186,"featured_image":92022,"meta":92023,"navigation":189,"path":92024,"pinned":185,"seo":92025,"sitemap":92026,"stem":92027,"tags":194,"__hash__":92028},"blog\u002Fblog\u002F小兒發展（三）.md","小兒發展（三）",{"type":9,"value":91797,"toc":92018},[91798,91801,91804,91833,91835,91838,91855,91857,91860,91886,91888,91891,91924,91930,91949,91955,91981,91984,91987,92010,92013],[12,91799,91800],{},"寫著寫著發現這一課東西好多（悲",[12,91802,91803],{},"接著講精細動作的發展過程",[3603,91805,91806,91809,91812,91815,91818,91821,91824,91827,91830],{},[26,91807,91808],{},"四個月——可以合掌",[26,91810,91811],{},"五個月——手可以向前伸並抓東西",[26,91813,91814],{},"七個月——握緊奶瓶",[26,91816,91817],{},"十個月——拍手",[26,91819,91820],{},"一歲——亂塗鴉",[26,91822,91823],{},"一歲半——疊兩到四個積木",[26,91825,91826],{},"兩歲——疊四到八個積木；畫直線",[26,91828,91829],{},"三歲——畫圓圈",[26,91831,91832],{},"五歲——畫出人形",[36,91834],{},[12,91836,91837],{},"假如有發展遲緩的問題，會有以下徵候",[23,91839,91840,91843,91846,91849,91852],{},[26,91841,91842],{},"以翻身作為移動方式",[26,91844,91845],{},"手腳不協調的爬行方式",[26,91847,91848],{},"W坐姿",[26,91850,91851],{},"兔子跳",[26,91853,91854],{},"一歲前出現明顯慣用手(因為一歲前大多為雙手並用)",[36,91856],{},[12,91858,91859],{},"而寶寶出現以下徵候，可能他有CP問題",[23,91861,91862,91865,91868,91871,91874,91877,91880,91883],{},[26,91863,91864],{},"躁動",[26,91866,91867],{},"缺乏精力",[26,91869,91870],{},"吸吮無力伴隨舌頭往前突",[26,91872,91873],{},"六個月大頭部控制不佳",[26,91875,91876],{},"口腔過度敏感",[26,91878,91879],{},"強直性咬合反射",[26,91881,91882],{},"不對稱動作形態",[26,91884,91885],{},"異常姿勢",[36,91887],{},[12,91889,91890],{},"接著講正常的語言發展，分成了語言準備期以及語言發展期",[12,91892,91893,91894,91896,91897,91901,91902,91906,91908,91909,91913,91914,91918,91919,91923],{},"首先是語言準備期，分成了表達跟理解兩個面向",[16,91895],{},"\n這時候寶寶還小，表達還只能靠",[75,91898,91900],{"className":91899},[78],"反射發聲","，有哭聲和verbal play(發一聲奇怪的聲音探索自己的聲音)，或者",[75,91903,91905],{"className":91904},[78],"學身邊的人咿咿呀呀",[16,91907],{},"\n而這個時候",[75,91910,91912],{"className":91911},[78,173],"理解能力比表達能力還要發展得快","，知覺部份開始分得出",[75,91915,91917],{"className":91916},[173],"語氣快慢聲源","等，也能",[75,91920,91922],{"className":91921},[173],"理解一些簡單的短語","例如yes or no等",[12,91925,91926,91927,91929],{},"然後是語言發展期，這時年齡為十二個月以上，發音能力基本清淅正確",[16,91928],{},"\n而語法構造則分為以下部份",[23,91931,91932,91940],{},[26,91933,91934,91935,91939],{},"句子表達能力——分成剛開始學的不完整句子，",[75,91936,91938],{"className":91937},[78,173],"只講關鍵的詞","；和發展到後面的會講完整句子",[26,91941,91942,91943,91945,91946,91948],{},"理解能力的話——十二個月已經聽懂簡單的句子指令；到兩三歲已經能理解和記住故事內容",[16,91944],{},"\n六歲理解被動句；七歲理解雙重否定句；之後能理解語句結構例如名 動 賓 謂 定 狀 補之類",[16,91947],{},"\n最後也能理解對哪個人說甚麼樣的話",[12,91950,91951,91952,91954],{},"然後講一下正常語言發展能力，雖然跟聽覺有部份重合，",[16,91953],{},"\n但我個人解讀是因為wernicke跟broca區是連在一起的會有重複的發展形態應該說得過去(?)",[3603,91956,91957,91960,91963,91966,91969,91972,91975,91978],{},[26,91958,91959],{},"四個月——頭尋找聲源",[26,91961,91962],{},"六到七個月——仿說",[26,91964,91965],{},"九到十二個月——叫爸爸媽媽",[26,91967,91968],{},"一歲到一歲半——只會講一兩個字且沒有意義",[26,91970,91971],{},"兩歲到兩歲半——講片語（例如像鸚鵡般學大人說啥）",[26,91973,91974],{},"三歲——開始能講句子，講顏色，倒數十，說出身體部位",[26,91976,91977],{},"四歲——能說出數種顏色",[26,91979,91980],{},"五歲——計算五個積木",[12,91982,91983],{},"(所以看起來三歲已經具備認知能力?)",[12,91985,91986],{},"那假如語言發展異常會有以下徵象",[23,91988,91989,91992,91995,91998,92001,92004,92007],{},[26,91990,91991],{},"一歲前——太安靜",[26,91993,91994],{},"兩歲——還沒有出現有意思的字詞",[26,91996,91997],{},"三歲——沒有出現任何句子",[26,91999,92000],{},"四歲——有構音困難",[26,92002,92003],{},"五歲——說話結巴",[26,92005,92006],{},"年紀越大話越少",[26,92008,92009],{},"鼻音重，說話小聲，不當的臉部表情，說話會喘",[12,92011,92012],{},"然後，對，還有個訪說測驗，但我完全看不懂注音，原樣上圖好了(悲",[12,92014,92015],{},[1930,92016],{"alt":1932,"src":92017},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1776047406679-Screenshot_2026-04-13_102943.png",{"title":180,"searchDepth":181,"depth":181,"links":92019},[],"2026-04-12","精細動作~語言發展","\u002Fimages\u002Fuploads\u002F1776047258825-ElqW51yVoAI09Jg.webp",{},"\u002Fblog\u002F小兒發展（三）",{"title":91795,"description":92021},{"loc":92024},"blog\u002F小兒發展（三）","cGrD26dwyUjVzawlHZHH_OnTqXRpbSPGvcZEZluSbRk",{"id":92030,"title":92031,"author":7,"body":92032,"date":92020,"description":92184,"draft":185,"edited_at":92020,"extension":186,"featured_image":92185,"meta":92186,"navigation":189,"path":92187,"pinned":185,"seo":92188,"sitemap":92189,"stem":92190,"tags":194,"__hash__":92191},"blog\u002Fblog\u002F小兒發展（二）.md","小兒發展（二）",{"type":9,"value":92033,"toc":92182},[92034,92037,92054,92056,92059,92087,92089,92092,92118,92120,92123,92137,92139,92142,92156,92158,92161,92169,92171,92174],[12,92035,92036],{},"那正常的寶寶下肢的生理變化也會按照一個叫鐘擺現象的過程發展",[3603,92038,92039,92042,92045,92048,92051],{},[26,92040,92041],{},"新生兒——中度的膝內翻（O型腿）",[26,92043,92044],{},"六個月——輕度的O型腿",[26,92046,92047],{},"一歲半——直的",[26,92049,92050],{},"三歲半——輕度的膝外翻（X型腿）",[26,92052,92053],{},"五到七歲——直的",[36,92055],{},[12,92057,92058],{},"而步態的發展會按照以下時期發展",[3603,92060,92061,92064,92072,92075,92078,92081,92084],{},[26,92062,92063],{},"反射踏步（靠反射）",[26,92065,92066,92067,2589],{},"抑制或靜止期（不過這裡有實驗提到，是因為脂肪比肌肉重，導致反射出不來，並不是因為高階抑制低制—— ",[2330,92068,92071],{"href":92069,"rel":92070},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002Fmotor-learning?series=%E5%B0%8F%E5%85%92%E7%89%A9%E6%B2%BB",[2334],"倒數第二段",[26,92073,92074],{},"過渡期",[26,92076,92077],{},"刻意跨步",[26,92079,92080],{},"獨立行走",[26,92082,92083],{},"Heell-toe步態",[26,92085,92086],{},"整合或直立行走成熟期",[36,92088],{},[12,92090,92091],{},"接著來看不同系統的發展進程，從出生至九個月",[23,92093,92094,92100,92106,92112],{},[26,92095,92096,92097,92099],{},"神經系統——在九個月的時候感覺系統已經成熟（視覺，本體覺，前庭覺）",[16,92098],{},"\n尤其是視覺發展遠大於粗大動作",[26,92101,92102,92103,92105],{},"身體機能——寶寶的脂肪站比高，在六個月時體重就已經是剛出生兩倍，一歲時還來到了三倍",[16,92104],{},"\n那關節部份——剛出生的寶寶：Hip會呈flexion contracture, ER>IR；膝內翻；假如承重時，會呈現旋前足",[26,92107,92108,92109,92111],{},"抗重力肌力——Hip flexor(訓練方法是supine kicking)；Hip extensor(訓練方法是prone, 爬行, 跪姿)",[16,92110],{},"\nHip abduction(訓練方法是扶物側行)",[26,92113,92114,92115,92117],{},"功能性步態——那在扶著的情況下寶寶的腳會呈現弓形腿(bowed legs)，拆開來看是",[16,92116],{},"\n步寬外展；外轉(O型腿)；髖屈曲；足跟外翻",[36,92119],{},[12,92121,92122],{},"時間來到了九個月到第十五個月之間的發展進程",[3603,92124,92125,92128,92131],{},[26,92126,92127],{},"神經系統——開始能在視覺刺激或引導情況下行走，且開始具備有CPG的能力(反射抑或身體機能成熟以達成)",[26,92129,92130],{},"力學因素——重心較高(在下胸椎層級)，BOS左右積大前後面積小，直立姿勢下肌力尚有不足",[26,92132,92133,92134,92136],{},"功能性步態——雖然能獨自行走，但還是會呈現弓形腿(bowed legs)，拆開來看是",[16,92135],{},"\n步頻增加；步寬大；髖和膝關節屈曲角度增加；全足著地；擺盪期有垂足",[36,92138],{},[12,92140,92141],{},"接下來時間來到了十八個月到二十四個月的發展進程",[3603,92143,92144,92150],{},[26,92145,92146,92147,92149],{},"力學因素——重心下降(由於腿快速生長)，BOS減少",[16,92148],{},"\nROM的話O型腿消失，但有旋前足出現",[26,92151,92152,92153,92155],{},"功能性步態——擺盪期學會用慣性甩出腳，支撐腳也開始會伸直承重；",[16,92154],{},"\n足跟著地還不穩定；Initical contact有knee屈曲(在足跟著地發展後出現)",[36,92157],{},[12,92159,92160],{},"時間繼續來到了三歲到三歲半的發展進程",[3603,92162,92163,92166],{},[26,92164,92165],{},"力學因素——ROM的話膝外翻(X型腿)，且有旋前足出現",[26,92167,92168],{},"功能性步態——足跟著地穩定，且出現Knee flexion",[36,92170],{},[12,92172,92173],{},"最後時間來到了六歲到七歲的發展進程",[3603,92175,92176,92179],{},[26,92177,92178],{},"力學因素——已經能直立行走，並且不在有旋前足",[26,92180,92181],{},"功能性步態——與成人一樣",{"title":180,"searchDepth":181,"depth":181,"links":92183},[],"步態~步態發展","\u002Fimages\u002Fuploads\u002F1775979234822-139196433_p0_master1200.webp",{},"\u002Fblog\u002F小兒發展（二）",{"title":92031,"description":92184},{"loc":92187},"blog\u002F小兒發展（二）","tgAyAngzwVGZGqD2GsTT7J1O1RchC3cKRE-f381-bqo",{"id":92193,"title":92194,"author":7,"body":92195,"date":92394,"description":92395,"draft":185,"edited_at":92394,"extension":186,"featured_image":92396,"meta":92397,"navigation":189,"path":92398,"pinned":185,"seo":92399,"sitemap":92400,"stem":92401,"tags":194,"__hash__":92402},"blog\u002Fblog\u002FGait-analysis小兒（一）.md","Gait analysis小兒（一）",{"type":9,"value":92196,"toc":92392},[92197,92200,92203,92220,92223,92234,92237,92254,92280,92286,92289,92306,92315,92326,92341,92347,92375,92378],[12,92198,92199],{},"我也不知道這一課要怎麼說開篇語，還是乖乖按著老師的講對寫好了",[12,92201,92202],{},"那下肢雙關節肌肉有以下",[3603,92204,92205,92208,92211,92214,92217],{},[26,92206,92207],{},"腰大肌——Hip Flexion, Trunk Flexion",[26,92209,92210],{},"腿後肌——Hip Extension, Knee Flexion",[26,92212,92213],{},"股直肌——Hip Flexion, Knee Extension",[26,92215,92216],{},"腓腸肌——Knee Flexion, PF",[26,92218,92219],{},"縫匠肌——Hip Flexion, Hip ER",[12,92221,92222],{},"那單關節的肌肉有",[3603,92224,92225,92228,92231],{},[26,92226,92227],{},"臀大肌——Hip Extension, Hip ER",[26,92229,92230],{},"脛前肌——DF, inversion",[26,92232,92233],{},"腓骨長肌——PF, eversion",[12,92235,92236],{},"那以下動作的抗重力與無抗重力的擺位姿勢為",[3603,92238,92239,92242,92245,92248,92251],{},[26,92240,92241],{},"Hip Extension——Prone 抗重力, side-lying 無抗重力",[26,92243,92244],{},"Hip Flexion——Sitting 抗重力, side-lying 無抗重力",[26,92246,92247],{},"Hip Abduction——Side-lying 抗重力, supine 無抗重力",[26,92249,92250],{},"Knee Extension——Sitting 抗重力, side-lying 無抗重力",[26,92252,92253],{},"PF——Standing 抗重力, prone 無抗重力",[12,92255,92256,92257,92259,92260,92264,92265,92269,92271,92272,92274,92275,92279],{},"接下來講一個異常步態——Crouch gait",[16,92258],{},"\n為甚麼他會呈現屈曲狀態，主要是因為",[75,92261,92263],{"className":92262},[78,79],"雙關節肌肉張力過強","，而",[75,92266,92268],{"className":92267},[78,173],"單關節肌肉力量不足",[16,92270],{},"\n不是單純的肌肉太緊，也是一種肌力失衡的表現",[16,92273],{},"\n其中，在",[75,92276,92278],{"className":92277},[78],"腿固定時(stand)，屈曲大腿的肌肉作用，會讓骨盤前傾","(像是髂腰肌)",[12,92281,92282,92283,92285],{},"接著來講步態的參數",[16,92284],{},"\nStep length(步距,左右之間距離),Stride length(步幅,左右左的左左之間距離),Speed(速度),Cadence(步頻)",[12,92287,92288],{},"這邊提到一個中風後會出現的代償步態",[23,92290,92291,92294,92297,92300,92303],{},[26,92292,92293],{},"步速下降",[26,92295,92296],{},"步距縮短或不對稱",[26,92298,92299],{},"步寬變大(BOS增加)",[26,92301,92302],{},"雙腳站立期變長",[26,92304,92305],{},"依賴手部支撐",[12,92307,92308,92309,92311,92312,92314],{},"接著講Gait cycle，分別為站立期以及擺盪期",[16,92310],{},"\n而站立期要去做承重以及單腳支撐的任務，擺盪期則要做肢體向前的任務",[16,92313],{},"\n完成這三個任務又分成了這八個動作",[3603,92316,92317,92320,92323],{},[26,92318,92319],{},"承重——Initial contact和Loading Response",[26,92321,92322],{},"單腳支撐——Mid-stance, Terminal Stance和Pre-swing",[26,92324,92325],{},"肢體向前——Pre-swing, Initial swing, Mid-swing和Terminal Swing",[12,92327,92328,92329,92333,92334,92336,92337],{},"而Pre-swing雖然腳尖還沒離地，但",[75,92330,92332],{"className":92331},[78,173],"重心其實已經轉移去另一隻腳","了",[16,92335],{},"\n其中 ",[75,92338,92340],{"className":92339},[78],"Initial contact和Pre-swing為雙腳承重期",[12,92342,92343,92344,92346],{},"我們現在攤開來看站立期和擺盪期下肢各關節變化",[16,92345],{},"\n首先是站立期",[3603,92348,92349,92357,92360,92366,92372],{},[26,92350,92351,92352,92356],{},"在雙腳承重期，",[75,92353,92355],{"className":92354},[78,173],"足部會放平","，準備Push Off",[26,92358,92359],{},"Hip的話，在Initical Contact的Flexion到Terminal Stance的Hyperextension",[26,92361,92362,92363,92365],{},"Knee的話，會從Flexion姿勢，到Mid-stance的Extension，再變回Pre-swing的Flexion",[16,92364],{},"\n(Flexion>Extension>Flexion)",[26,92367,92368,92369,92371],{},"Ankle的話，在Initical Contact的Neutral姿勢，Loading Response的PF，",[16,92370],{},"\nMid-stance的DF，最後是Terminal Stance\u002FPre-swing的PF（Neutral>PF>DF>PF）",[26,92373,92374],{},"Pelvis的話，基本保持水平，但在Terminal stance會略微前傾",[12,92376,92377],{},"然後是擺盪期",[3603,92379,92380,92383,92386,92389],{},[26,92381,92382],{},"Knee的話，在Initical swing時會呈現Flexion，在Terminal swing會Full-extension",[26,92384,92385],{},"Hip的話，從Initical swing的Hyperextension，在Terminal swing會變成Flexion",[26,92387,92388],{},"Ankle的話，保持DF狀態，使腳尖不會碰到地板",[26,92390,92391],{},"Pelvis的話，會Rotation",{"title":180,"searchDepth":181,"depth":181,"links":92393},[],"2026-04-11","肌肉~步態各分期關節姿勢","\u002Fimages\u002Fuploads\u002F1775885400167-37100769_p0.png",{},"\u002Fblog\u002FGait-analysis小兒（一）",{"title":92194,"description":92395},{"loc":92398},"blog\u002FGait-analysis小兒（一）","twQeon3uGQlDYDjG3t_oGlDC93CraJNGOOTRTTxdazk",{"id":92404,"title":92405,"author":7,"body":92406,"date":92394,"description":92595,"draft":185,"edited_at":92394,"extension":186,"featured_image":92596,"meta":92597,"navigation":189,"path":92598,"pinned":185,"seo":92599,"sitemap":92600,"stem":92601,"tags":194,"__hash__":92602},"blog\u002Fblog\u002FGait-analysis小兒（二）.md","Gait analysis小兒（二）",{"type":9,"value":92407,"toc":92593},[92408,92411,92525,92528,92564,92576,92582],[12,92409,92410],{},"現在我們將八個步態分期拆開了細看",[3603,92412,92413,92437,92451,92469,92479,92500,92519,92522],{},[26,92414,92415,92416,92420,92424,92425,92429,92433,92434,92436],{},"Initical Contact——用",[75,92417,92419],{"className":92418},[78],"股四頭肌和脛前肌做",[75,92421,92423],{"className":92422},[78,79],"離心收縮","，這時",[75,92426,92428],{"className":92427},[78],"腿後肌做",[75,92430,92432],{"className":92431},[78,173],"向心收縮","協同加速穩定",[16,92435],{},"\n（減速與準備承重）",[26,92438,92439,92440,92444,92447,92448,92450],{},"Loading Response——",[75,92441,92443],{"className":92442},[78],"肌內側肌與脛前肌做",[75,92445,92423],{"className":92446},[78,79],"，而臀大肌以及腿後肌輔助Hip加速，將重心向前移動",[16,92449],{},"\n（避震）",[26,92452,92453,92454,92264,92458,92462,92465,92466,92468],{},"Mid-stance——膝蓋後方有關節囊結構的緣故，肌四頭肌幾乎",[75,92455,92457],{"className":92456},[173],"不需用力",[75,92459,92461],{"className":92460},[78],"比目魚肌此時做",[75,92463,92423],{"className":92464},[78,79],"控制脛骨速度",[16,92467],{},"\n（省力）",[26,92470,92471,92472,92476,92478],{},"Terminal stance——",[75,92473,92475],{"className":92474},[78,173],"腓腸肌與比目魚肌做向心收縮",[16,92477],{},"\n（產生推進力）",[26,92480,92481,92482,92486,24802,92489,92491,92492,92496],{},"Pre-swing——假如是慢速走路，會利用",[75,92483,92485],{"className":92484},[78],"縫匠肌(Sartorius)和肌薄肌(Graillis)做",[75,92487,92432],{"className":92488},[78,173],[16,92490],{},"\n快速行走則靠",[75,92493,92495],{"className":92494},[78],"肌直肌做",[75,92497,92499],{"className":92498},[78,78392],"等長收縮",[26,92501,92502,92503,92507,92510,92512,92513,92516],{},"Initial swing——在慢速走路中，",[75,92504,92506],{"className":92505},[78],"股薄肌與縫匠肌會進行",[75,92508,92432],{"className":92509},[78,173],[16,92511],{},"\n而快速走路中，",[75,92514,92495],{"className":92515},[78],[75,92517,92499],{"className":92518},[78,78392],[26,92520,92521],{},"Mid swing——靠慣性推進，基本沒有肌肉參與",[26,92523,92524],{},"Terminal swing——腿後肌做等長收縮，限制Knee extension速度",[12,92526,92527],{},"接著來講小腿後肌群無力的話會有甚麼問題",[23,92529,92530,92540],{},[26,92531,92532,92533,24802,92537,92539],{},"Mid-stannce——比目魚肌無力的話，會讓",[75,92534,92536],{"className":92535},[79,78],"Ankle做DF",[16,92538],{},"\n那由於代償原因，Knee會做flexion來讓腳踩在地面，結果變成股四頭肌必須要加入收縮才能穩定站立",[26,92541,92542,92543,92547,92549,92550,92554,92555,92559,92560],{},"Terminal stance和Pre-swing——腓腸肌力量不足的話，",[75,92544,92546],{"className":92545},[78,79],"沒辦法產生足夠的推進力",[16,92548],{},"\n只好讓",[75,92551,92553],{"className":92552},[79,78],"髖屈肌收縮","提起整段下肢，確保",[75,92556,92558],{"className":92557},[173],"腳尖離地","，導致",[75,92561,92563],{"className":92562},[5387],"步距變短",[12,92565,92566,92567,92569,92570,92572,92573,92575],{},"最後講小孩的步態發展過程",[16,92568],{},"\n在18個月就會出現交互雙手擺動以及腳跟著地（Heel strike）的形態",[16,92571],{},"\n2歲的小孩懂得矢狀面(Sagittal-plane)關節旋轉",[16,92574],{},"\n7歲的小孩步態已經能達到成人標準了",[12,92577,92578,92579,92581],{},"CP的小孩在五歲時——有54%不需要輔具獨立行走，16%需要輔具行走，30%無法行走",[16,92580],{},"\n那CP小孩預測能力方面也分成了三個時段",[3603,92583,92584,92587,92590],{},[26,92585,92586],{},"9~18個月——頭部控制能力",[26,92588,92589],{},"24個月——不需要支撐能獨自坐穩",[26,92591,92592],{},"30個月——做爬行",{"title":180,"searchDepth":181,"depth":181,"links":92594},[],"八分期肌肉拆解，腿後肌力不足的問題以及小兒步態發展過程","\u002Fimages\u002Fuploads\u002F1775898442489-136249570_p0_master1200.webp",{},"\u002Fblog\u002FGait-analysis小兒（二）",{"title":92405,"description":92595},{"loc":92598},"blog\u002FGait-analysis小兒（二）","x7y-Smp4poLa_EJUkNUpEi-ow03rff_xJb4B-xlHN1k",{"id":92604,"title":92605,"author":7,"body":92606,"date":92394,"description":92750,"draft":185,"edited_at":92020,"extension":186,"featured_image":92751,"meta":92752,"navigation":189,"path":92753,"pinned":185,"seo":92754,"sitemap":92755,"stem":92756,"tags":194,"__hash__":92757},"blog\u002Fblog\u002F小兒發展（一）.md","小兒發展（一）",{"type":9,"value":92607,"toc":92748},[92608,92611,92626,92632,92635,92661,92670,92672,92675,92692,92694,92697],[12,92609,92610],{},"（PS：別問我為甚麼寫完步態動作寫習跟CP才回頭來寫這個，我真忘了你信嗎）",[12,92612,92613,92614,92616,92617,92619,92620,92622,92623,92625],{},"那先來講發展遲緩的定義，它不單指運動層面上，還包括了",[16,92615],{},"\n知覺，語言，心理，社會，情緒等層面",[16,92618],{},"\n而這個動作遲緩有標準的",[16,92621],{},"\n發展過程成未達到正常小孩的90%，或者在評估測驗分數上落後兩個標準差，即屬發展遲緩",[16,92624],{},"\n根據WHO的數據，發生率為6~8%",[12,92627,92628,92629,92631],{},"那高危族群有以下",[16,92630],{},"\n視障兒童，聽障兒童，心理行為障礙兒童，生理狀況障礙兒童，家庭問題兒童",[12,92633,92634],{},"接下來講寶寶的正常發育過程，先從視角說起，如下",[3603,92636,92637,92640,92643,92646,92649,92652,92655,92658],{},[26,92638,92639],{},"剛出生數天——眨眼反射",[26,92641,92642],{},"六周——注射物體",[26,92644,92645],{},"二到三個月——對明亮的東西感興趣",[26,92647,92648],{},"四個月——協調眼球轉動",[26,92650,92651],{},"一歲——影像認知發展完成，能看清輪廓",[26,92653,92654],{},"三歲——能分辨紅黃藍綠",[26,92656,92657],{},"五到六歲——可以分辨大多數顏色",[26,92659,92660],{},"八歲以前——有遠視現象（我問了Gemini也看不太懂就是了）",[12,92662,92663,92664,92666,92667,92669],{},"那關於視覺發展異常的問題包括了",[16,92665],{},"\n視覺注視不良，視線不隨物體移動，對熟悉的臉孔或物體缺乏反應 手眼協調異常",[16,92668],{},"\n眼睛外觀異常，因視覺問題的代償性頭部姿勢異常，只能看近物，怕光",[36,92671],{},[12,92673,92674],{},"接著是聽覺發展，如下",[3603,92676,92677,92680,92683,92686,92689],{},[26,92678,92679],{},"四個月——追尋聲音或說話來源",[26,92681,92682],{},"七到八個月——被叫名字有反應",[26,92684,92685],{},"十個月——仿說（無意義內容）",[26,92687,92688],{},"一到一歲半——初步聽懂簡單的話",[26,92690,92691],{},"兩歲以後——鸚鵡式學說話",[36,92693],{},[12,92695,92696],{},"接下來是粗大動作的發展，如下",[3603,92698,92699,92702,92705,92708,92711,92714,92717,92720,92723,92731,92739,92742,92745],{},[26,92700,92701],{},"兩個月——不穩定的抬頭控制",[26,92703,92704],{},"四個月——抬頭控制適當",[26,92706,92707],{},"五個月——翻身（先學會Prone to supine, 接著是supine to prone）",[26,92709,92710],{},"七個月——坐起來",[26,92712,92713],{},"八個月——爬行",[26,92715,92716],{},"九個月——需要支撐的站起來",[26,92718,92719],{},"十個月——僅需要極小輔助的從坐到站",[26,92721,92722],{},"一歲——獨立行走",[26,92724,92725,92726,92730],{},"兩歲——跑，",[75,92727,92729],{"className":92728},[173],"上樓梯","，踢球，投球，蹲下來",[26,92732,92733,92734,92738],{},"三歲——",[75,92735,92737],{"className":92736},[79],"下樓梯","，雙腳跳，騎三輪車",[26,92740,92741],{},"四歲——單腳跳",[26,92743,92744],{},"五歲——走直線",[26,92746,92747],{},"六歲——騎腳踏車",{"title":180,"searchDepth":181,"depth":181,"links":92749},[],"定義~粗大動作發展","\u002Fimages\u002Fuploads\u002F1775957410970-EebbBUPUcAYSTE9.webp",{},"\u002Fblog\u002F小兒發展（一）",{"title":92605,"description":92750},{"loc":92753},"blog\u002F小兒發展（一）","PRBPkkWsOaM11PriYRM9bPFgTo_bGZSIOZZuLMzA1sk",{"id":92759,"title":92760,"author":6946,"body":92761,"date":92394,"description":180,"draft":185,"edited_at":92020,"extension":186,"featured_image":194,"meta":98609,"navigation":189,"path":98610,"pinned":185,"seo":98611,"sitemap":98612,"stem":98613,"tags":194,"__hash__":98614},"blog\u002Fblog\u002F資料前處理與迴歸分析.md","資料前處理與迴歸分析",{"type":9,"value":92762,"toc":98589},[92763,92767,92774,92777,92781,92787,92802,92808,92814,93047,93053,93059,93073,93076,93079,93099,93103,93106,93109,93115,93126,93161,93164,93167,93433,93514,93518,93521,93638,93644,93691,93695,93701,93862,94254,94337,94341,94344,94453,94456,94573,94944,94971,94974,94979,94982,95159,95722,95729,95733,95743,95748,95776,95781,95785,95790,95793,95820,95827,95831,95838,95841,96026,96383,96410,96414,96459,96466,96628,96631,96775,96778,96785,97030,97310,97313,97631,98239,98250,98255,98444,98491,98494,98587],[6964,92764,92766],{"id":92765},"資料前處理為什麼清理資料這麼重要","資料前處理：為什麼清理資料這麼重要？",[12,92768,92769,92770,92773],{},"在我們訓練任何機器學習模型之前，必須先面對現實中亂七八糟的原始數據。",[106,92771,92772],{},"資料前處理 (Data preprocessing)"," 就是把這些數據變成好看的報告的過程。想像一下，你在煮飯前總得先洗菜、切菜吧？如果食材又髒又沒切過，是做不出好菜的。",[12,92775,92776],{},"原始數據通常會有一堆問題，像是異常值 (outliers)、缺失值 (missing values)、無意義的組合，還有亂七八糟的單位。前處理就是要把這些混亂，變成模型真正能看得懂、學得會的東西。",[7166,92778,92780],{"id":92779},"資料前處理的-7-個步驟","資料前處理的 7 個步驟",[12,92782,92783,92786],{},[106,92784,92785],{},"1. 取得資料集 (Acquire the dataset)。"," 從各個來源收集資料，然後把它們整合成適當的格式。資料集的結構會根據你的領域而有所不同（比如商業資料跟醫療資料就長得不一樣）。",[12,92788,92789,92792,92793,92795,92796,92798,92799,92801],{},[106,92790,92791],{},"2. 匯入關鍵函式庫 (Import crucial libraries)。"," 在 Python 裡，有三個函式庫是前處理的鐵三角：",[106,92794,32037],{},"（科學運算）、",[106,92797,32041],{},"（資料操作與分析）和 ",[106,92800,32045],{},"（2D 繪圖跟視覺化）。",[12,92803,92804,92807],{},[106,92805,92806],{},"3. 匯入資料集 (Import the dataset)。"," 把資料載入你的工作環境。記得確認你的工作目錄 (working directory) 設定正確，程式才找得到檔案喔。",[12,92809,92810,92813],{},[106,92811,92812],{},"4. 處理缺失值 (Handle missing values)。"," 這點超級重要！如果你無視缺失資料，模型就會得出錯誤的結論。常見的做法包含用該欄位的平均數或中位數來填補，或是乾脆把缺漏的那一整列刪掉。",[25631,92815,92817,92820],{"className":92816},[13441,13442],[25635,92818,92819],{},"欄位？列？都是什麼？",[13438,92821,92823,92826,92852,92855,92858,92922,92929,93036],{"className":92822},[25641],[12,92824,92825],{},"為了更容易理解，我們可以把資料想像成一個 Excel 表格。在資料科學中，表格的兩個維度分別代表：",[23,92827,92828,92842],{},[26,92829,92830,92833,92834,92837,92838,92841],{},[106,92831,92832],{},"欄位 (Column \u002F 垂直方向)："," 代表一個特定的 ",[106,92835,92836],{},"「特徵 (Feature)」"," 或 ",[106,92839,92840],{},"「變數」","。例如：每個人的「年齡」、「身高」或「收入」。",[26,92843,92844,92847,92848,92851],{},[106,92845,92846],{},"列 (Row \u002F 水平方向)："," 代表 ",[106,92849,92850],{},"「一筆完整的資料 (Observation \u002F Record)」","。例如：某位特定使用者的所有基本資料。",[7166,92853,92854],{"id":92854},"實際範例解析",[12,92856,92857],{},"假設我們有一個如下的客戶資料表：",[7085,92859,92860,92879],{},[7088,92861,92862],{},[7091,92863,92864,92869,92874],{},[7094,92865,92866],{"align":47883},[106,92867,92868],{},"姓名",[7094,92870,92871],{"align":47883},[106,92872,92873],{},"年齡 (欄位)",[7094,92875,92876],{"align":47883},[106,92877,92878],{},"收入 (欄位)",[7104,92880,92881,92894,92909],{},[7091,92882,92883,92889,92891],{},[7109,92884,92885,92886,7963],{"align":47883},"小明 (",[106,92887,92888],{},"這是一列 Row",[7109,92890,17409],{"align":47883},[7109,92892,92893],{"align":47883},"50,000",[7091,92895,92896,92901,92906],{},[7109,92897,92898,92899,7963],{"align":47883},"小華 (",[106,92900,92888],{},[7109,92902,92903],{"align":47883},[106,92904,92905],{},"(缺失值)",[7109,92907,92908],{"align":47883},"60,000",[7091,92910,92911,92916,92919],{},[7109,92912,92913,92914,7963],{"align":47883},"小美 (",[106,92915,92888],{},[7109,92917,92918],{"align":47883},"27",[7109,92920,92921],{"align":47883},"55,000",[12,92923,92924,92925,92928],{},"當我們提到",[106,92926,92927],{},"用該欄位的平均數或中位數填補","，它的執行動作如下：",[3603,92930,92931,92934,92941,93033],{},[26,92932,92933],{},"系統發現「小華」的「年齡」空缺了。",[26,92935,92936,92937,92940],{},"系統會去尋找「年齡」",[106,92938,92939],{},"這個欄位 (Column)"," 裡其他已知的數值（也就是小明的 25 歲和小美的 27 歲）。",[26,92942,92943,92944,86843],{},"計算這個欄位的平均數：",[24923,92945,92947],{"className":92946,"jax":24927},[24926],[24929,92948,92951,92975],{"style":24931,"xmlns":24932,"width":92949,"height":24934,"role":1930,"focusable":8058,"viewBox":92950,"xmlnsXLink":24936},"16.593ex","0 -750 7334 1000",[24938,92952,92953,92955,92957,92960,92963,92966,92968,92970,92972],{},[24941,92954],{"id":24943,"d":24944},[24941,92956],{"id":76019,"d":25928},[24941,92958],{"id":92959,"d":26660},"MJX-1-TEX-N-35",[24941,92961],{"id":92962,"d":30313},"MJX-1-TEX-N-2B",[24941,92964],{"id":92965,"d":26361},"MJX-1-TEX-N-37",[24941,92967],{"id":24987,"d":24988},[24941,92969],{"id":32165,"d":29707},[24941,92971],{"id":24971,"d":24972},[24941,92973],{"id":92974,"d":42964},"MJX-1-TEX-N-36",[25022,92976,92977],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,92978,92979,92983,92990,92996,93003,93008,93015,93020,93025],{"dataMmlNode":25028},[25022,92980,92981],{"dataMmlNode":25031},[25033,92982],{"dataC":25035,"xLinkHref":25036},[25022,92984,92985,92987],{"dataMmlNode":26031,"transform":25040},[25033,92986],{"dataC":26058,"xLinkHref":76028},[25033,92988],{"dataC":26717,"xLinkHref":92989,"transform":26394},"#MJX-1-TEX-N-35",[25022,92991,92993],{"dataMmlNode":25031,"transform":92992},"translate(1611.2,0)",[25033,92994],{"dataC":30393,"xLinkHref":92995},"#MJX-1-TEX-N-2B",[25022,92997,92998,93000],{"dataMmlNode":26031,"transform":41596},[25033,92999],{"dataC":26058,"xLinkHref":76028},[25033,93001],{"dataC":26397,"xLinkHref":93002,"transform":26394},"#MJX-1-TEX-N-37",[25022,93004,93006],{"dataMmlNode":25031,"transform":93005},"translate(3611.4,0)",[25033,93007],{"dataC":25108,"xLinkHref":25109},[25022,93009,93011],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":93010},"translate(4000.4,0)",[25022,93012,93013],{"dataMmlNode":25031},[25033,93014],{"dataC":29726,"xLinkHref":32219},[25022,93016,93018],{"dataMmlNode":26031,"transform":93017},"translate(4500.4,0)",[25033,93019],{"dataC":26058,"xLinkHref":76028},[25022,93021,93023],{"dataMmlNode":25031,"transform":93022},"translate(5278.2,0)",[25033,93024],{"dataC":25078,"xLinkHref":25079},[25022,93026,93028,93030],{"dataMmlNode":26031,"transform":93027},"translate(6334,0)",[25033,93029],{"dataC":26058,"xLinkHref":76028},[25033,93031],{"dataC":31136,"xLinkHref":93032,"transform":26394},"#MJX-1-TEX-N-36",[26,93034,93035],{},"最後把 26 填入小華年齡的缺失值中。",[12,93037,93038,93039,93042,93043,93046],{},"而後半段提到的",[106,93040,93041],{},"把缺漏的那一整列刪掉","，則是另一種暴力的做法：只要發現小華有一項資料沒填（年齡缺失），就把「小華」",[106,93044,93045],{},"這一整筆資料（Row）"," 直接從表格中剔除，連帶他完好的收入資料也不採納了。",[12,93048,93049,93052],{},[106,93050,93051],{},"5. 類別資料編碼 (Encode categorical data)。"," 機器學習模型是靠數學運作的——它們只懂數字，不懂「紅色」或「男生」這種標籤文字。編碼的方法有虛擬變數 (dummy variables)、標籤編碼 (label encoding，把類別變成整數) 以及獨熱編碼 (one-hot encoding，為每個類別建立二元欄位)。",[12,93054,93055,93058],{},[106,93056,93057],{},"6. 特徵縮放 (Feature scaling)。"," 當你的特徵範圍差超多的時候（例如年齡 0–100 歲 vs 薪水 0–1,000,000 元），有些演算法會被搞混。特徵縮放就是把變數標準化到同一個範圍裡。最常用的兩個方法是 Min-Max 縮放和標準化 (Standardization \u002F z-score)。",[12,93060,93061,93064,93065,93068,93069,93072],{},[106,93062,93063],{},"7. 切割資料集 (Split the dataset)。"," 把資料分成",[106,93066,93067],{},"訓練集 (training set)","（給模型學習用）和",[106,93070,93071],{},"測試集 (test set)","（用來驗證模型好壞）。最常見的切法是跟著八二法則走：80\u002F20，或者 70\u002F30、60\u002F40 也很常見。",[7166,93074,93075],{"id":93075},"其他的前處理技巧",[12,93077,93078],{},"除了這 7 個核心步驟，你可能還會遇到：",[23,93080,93081,93087,93093],{},[26,93082,93083,93086],{},[106,93084,93085],{},"資料整合 (Data Integration)"," — 把來自不同來源系統的資料，合併成單一的資料集。",[26,93088,93089,93092],{},[106,93090,93091],{},"資料轉換 (Data Transformation)"," — 把資料從一種格式轉成另一種格式（e.g.: source system format → destination format）。",[26,93094,93095,93098],{},[106,93096,93097],{},"資料離散化 (Data Discretization)"," — 把連續數值變成離散的區間（比如把確切的年齡，變成「18–25 歲」、「26–35 歲」這種年齡層）。",[6964,93100,93102],{"id":93101},"第二部分迴歸分析-預測連續數值","第二部分：迴歸分析 — 預測連續數值",[12,93104,93105],{},"現在資料乾淨了，我們來看看怎麼做預測。先從一個經典的例子開始。",[7166,93107,93108],{"id":93108},"房價問題",[12,93110,93111,93112],{},"想像一下，我們有一份包含 47 間房子的資料集，裡面有居住面積（平方英尺）跟價格（千元馬幣）。問題很簡單：",[106,93113,93114],{},"給定一間房子的面積，我們能預測它的價格嗎？",[12,93116,93117,93118,93121,93122,93125],{},"這是一個",[106,93119,93120],{},"監督式學習 (supervised learning)"," 的問題。我們有「輸入-輸出」的配對資料，並想找出它們之間的關聯規則。因為輸出（價格）是一個連續的數值，所以這具體來說叫作",[106,93123,93124],{},"迴歸 (regression)"," 問題。",[25631,93127,93129,93132],{"className":93128},[13441,13442],[25635,93130,93131],{},"補充說明：連續的數值？迴歸？都是什麼？",[13438,93133,93135,93141,93147,93153,93156],{"className":93134},[25641],[12,93136,93137,93140],{},[106,93138,93139],{},"連續的數值是可以無限細分、有大小之分的數字","。它們通常是「測量」出來的，而不是一個一個「數」出來的。它包含了所有的小數和分數。比如，房價可以是 50 萬馬幣，也可以是 50.5 萬馬幣，甚至是 50.512 萬馬幣。只要儀器或計算夠精確，數字可以無限切分下去。生活中的面積、身高、體重、溫度、時間，都是連續數值。",[12,93142,93143,93146],{},[106,93144,93145],{},"相對的概念是「離散數值」(Discrete Value)："," 這是只能跳躍式出現、通常是整數的數字。例如：「這間房子有幾間臥室？」答案只能是 1 間、2 間或 3 間，不可能有 2.38 間臥室。",[12,93148,93149,93150,86843],{},"在機器學習的領域裡，",[106,93151,93152],{},"只要你的目標是預測一個「連續的數值」，我們就把這個任務稱為「迴歸」",[12,93154,93155],{},"迴歸的核心概念，就是在茫茫的數據資料中，找出一條**「最符合趨勢的線」**（Line of Best Fit），藉此理解輸入（面積）與輸出（價格）之間的數學關係。",[12,93157,93158],{},[1930,93159],{"alt":1932,"src":93160},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775983903268-1_N1-K-A43_98pYZ27fnupDA.webp",[7166,93162,93163],{"id":93163},"數學符號約定",[12,93165,93166],{},"在深入之前，我們先來約定一下符號：",[23,93168,93169,93231,93291,93375,93395],{},[26,93170,93171,93212,93213,93230],{},[24923,93172,93174],{"className":93173,"jax":24927},[24926],[24929,93175,93176,93187],{"style":25196,"xmlns":24932,"width":32537,"height":32538,"role":1930,"focusable":8058,"viewBox":32539,"xmlnsXLink":24936},[24938,93177,93178,93180,93182,93185],{},[24941,93179],{"id":32285,"d":32148},[24941,93181],{"id":32282,"d":24944},[24941,93183],{"id":93184,"d":27108},"MJX-2-TEX-I-1D456",[24941,93186],{"id":32303,"d":24988},[25022,93188,93189],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93190,93191],{"dataMmlNode":25028},[25022,93192,93193,93197],{"dataMmlNode":31565},[25022,93194,93195],{"dataMmlNode":25942},[25033,93196],{"dataC":32181,"xLinkHref":32337},[25022,93198,93199,93203,93208],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},[25022,93200,93201],{"dataMmlNode":25031},[25033,93202],{"dataC":25035,"xLinkHref":32332},[25022,93204,93205],{"dataMmlNode":25942,"transform":25040},[25033,93206],{"dataC":27210,"xLinkHref":93207},"#MJX-2-TEX-I-1D456",[25022,93209,93210],{"dataMmlNode":25031,"transform":32577},[25033,93211],{"dataC":25108,"xLinkHref":32370}," = 輸入特徵 (例如第 ",[24923,93214,93216],{"className":93215,"jax":24927},[24926],[24929,93217,93218,93222],{"style":25196,"xmlns":24932,"width":28237,"height":28238,"role":1930,"focusable":8058,"viewBox":28239,"xmlnsXLink":24936},[24938,93219,93220],{},[24941,93221],{"id":32549,"d":27108},[25022,93223,93224],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93225,93226],{"dataMmlNode":25028},[25022,93227,93228],{"dataMmlNode":25942},[25033,93229],{"dataC":27210,"xLinkHref":32574}," 間房子的面積)",[26,93232,93233,93272,93273,93290],{},[24923,93234,93236],{"className":93235,"jax":24927},[24926],[24929,93237,93238,93248],{"style":29805,"xmlns":24932,"width":32590,"height":32591,"role":1930,"focusable":8058,"viewBox":32592,"xmlnsXLink":24936},[24938,93239,93240,93242,93244,93246],{},[24941,93241],{"id":32597,"d":25893},[24941,93243],{"id":32600,"d":24944},[24941,93245],{"id":32603,"d":27108},[24941,93247],{"id":32606,"d":24988},[25022,93249,93250],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93251,93252],{"dataMmlNode":25028},[25022,93253,93254,93258],{"dataMmlNode":31565},[25022,93255,93256],{"dataMmlNode":25942},[25033,93257],{"dataC":25987,"xLinkHref":32617},[25022,93259,93260,93264,93268],{"dataMmlNode":27203,"transform":32620,"dataMjxTexclass":27205},[25022,93261,93262],{"dataMmlNode":25031},[25033,93263],{"dataC":25035,"xLinkHref":32625},[25022,93265,93266],{"dataMmlNode":25942,"transform":25040},[25033,93267],{"dataC":27210,"xLinkHref":32630},[25022,93269,93270],{"dataMmlNode":25031,"transform":32577},[25033,93271],{"dataC":25108,"xLinkHref":32635}," = 輸出\u002F目標變數 (例如第 ",[24923,93274,93276],{"className":93275,"jax":24927},[24926],[24929,93277,93278,93282],{"style":25196,"xmlns":24932,"width":28237,"height":28238,"role":1930,"focusable":8058,"viewBox":28239,"xmlnsXLink":24936},[24938,93279,93280],{},[24941,93281],{"id":32658,"d":27108},[25022,93283,93284],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93285,93286],{"dataMmlNode":25028},[25022,93287,93288],{"dataMmlNode":25942},[25033,93289],{"dataC":27210,"xLinkHref":32692}," 間房子的價格)",[26,93292,93293,93374],{},[24923,93294,93296],{"className":93295,"jax":24927},[24926],[24929,93297,93298,93315],{"style":24931,"xmlns":24932,"width":32646,"height":32647,"role":1930,"focusable":8058,"viewBox":32648,"xmlnsXLink":24936},[24938,93299,93300,93302,93305,93308,93310,93313],{},[24941,93301],{"id":25896,"d":24944},[24941,93303],{"id":93304,"d":32148},"MJX-6-TEX-I-1D465",[24941,93306],{"id":93307,"d":27108},"MJX-6-TEX-I-1D456",[24941,93309],{"id":25903,"d":24988},[24941,93311],{"id":93312,"d":27339},"MJX-6-TEX-N-2C",[24941,93314],{"id":25892,"d":25893},[25022,93316,93317],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93318,93319,93323,93345,93350,93370],{"dataMmlNode":25028},[25022,93320,93321],{"dataMmlNode":25031},[25033,93322],{"dataC":25035,"xLinkHref":25994},[25022,93324,93325,93330],{"dataMmlNode":31565,"transform":25040},[25022,93326,93327],{"dataMmlNode":25942},[25033,93328],{"dataC":32181,"xLinkHref":93329},"#MJX-6-TEX-I-1D465",[25022,93331,93332,93336,93341],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},[25022,93333,93334],{"dataMmlNode":25031},[25033,93335],{"dataC":25035,"xLinkHref":25994},[25022,93337,93338],{"dataMmlNode":25942,"transform":25040},[25033,93339],{"dataC":27210,"xLinkHref":93340},"#MJX-6-TEX-I-1D456",[25022,93342,93343],{"dataMmlNode":25031,"transform":32577},[25033,93344],{"dataC":25108,"xLinkHref":26007},[25022,93346,93347],{"dataMmlNode":25031,"transform":32699},[25033,93348],{"dataC":27456,"xLinkHref":93349},"#MJX-6-TEX-N-2C",[25022,93351,93352,93356],{"dataMmlNode":31565,"transform":32705},[25022,93353,93354],{"dataMmlNode":25942},[25033,93355],{"dataC":25987,"xLinkHref":25988},[25022,93357,93358,93362,93366],{"dataMmlNode":27203,"transform":32620,"dataMjxTexclass":27205},[25022,93359,93360],{"dataMmlNode":25031},[25033,93361],{"dataC":25035,"xLinkHref":25994},[25022,93363,93364],{"dataMmlNode":25942,"transform":25040},[25033,93365],{"dataC":27210,"xLinkHref":93340},[25022,93367,93368],{"dataMmlNode":25031,"transform":32577},[25033,93369],{"dataC":25108,"xLinkHref":26007},[25022,93371,93372],{"dataMmlNode":25031,"transform":32727},[25033,93373],{"dataC":25108,"xLinkHref":26007}," = 一筆訓練樣本",[26,93376,93377,93394],{},[24923,93378,93380],{"className":93379,"jax":24927},[24926],[24929,93381,93382,93386],{"style":25196,"xmlns":24932,"width":31825,"height":27038,"role":1930,"focusable":8058,"viewBox":31826,"xmlnsXLink":24936},[24938,93383,93384],{},[24941,93385],{"id":48375,"d":28310},[25022,93387,93388],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93389,93390],{"dataMmlNode":25028},[25022,93391,93392],{"dataMmlNode":25942},[25033,93393],{"dataC":28479,"xLinkHref":48438}," = 訓練樣本的總數",[26,93396,93397,93432],{},[24923,93398,93400],{"className":93399,"jax":24927},[24926],[24929,93401,93402,93412],{"style":24931,"xmlns":24932,"width":32764,"height":24934,"role":1930,"focusable":8058,"viewBox":32765,"xmlnsXLink":24936},[24938,93403,93404,93406,93408,93410],{},[24941,93405],{"id":32820,"d":31530},[24941,93407],{"id":32823,"d":24944},[24941,93409],{"id":32826,"d":32148},[24941,93411],{"id":32829,"d":24988},[25022,93413,93414],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93415,93416,93420,93424,93428],{"dataMmlNode":25028},[25022,93417,93418],{"dataMmlNode":25942},[25033,93419],{"dataC":31539,"xLinkHref":32838},[25022,93421,93422],{"dataMmlNode":25031,"transform":32791},[25033,93423],{"dataC":25035,"xLinkHref":32843},[25022,93425,93426],{"dataMmlNode":25942,"transform":32797},[25033,93427],{"dataC":32181,"xLinkHref":32848},[25022,93429,93430],{"dataMmlNode":25031,"transform":32803},[25033,93431],{"dataC":25108,"xLinkHref":32853}," = 我們的假設函數 (hypothesis function), 也就是我們想學出來的預測器",[12,93434,93435,93436,93455,93456,93493,93494,93513],{},"我們的目標是：找到一個函數 ",[24923,93437,93439],{"className":93438,"jax":24927},[24926],[24929,93440,93441,93446],{"style":25196,"xmlns":24932,"width":31522,"height":31523,"role":1930,"focusable":8058,"viewBox":31524,"xmlnsXLink":24936},[24938,93442,93443],{},[24941,93444],{"id":93445,"d":31530},"MJX-9-TEX-I-210E",[25022,93447,93448],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93449,93450],{"dataMmlNode":25028},[25022,93451,93452],{"dataMmlNode":25942},[25033,93453],{"dataC":31539,"xLinkHref":93454},"#MJX-9-TEX-I-210E","，讓 ",[24923,93457,93459],{"className":93458,"jax":24927},[24926],[24929,93460,93461,93472],{"style":24931,"xmlns":24932,"width":32764,"height":24934,"role":1930,"focusable":8058,"viewBox":32765,"xmlnsXLink":24936},[24938,93462,93463,93466,93468,93470],{},[24941,93464],{"id":93465,"d":31530},"MJX-10-TEX-I-210E",[24941,93467],{"id":43317,"d":24944},[24941,93469],{"id":32906,"d":32148},[24941,93471],{"id":43320,"d":24988},[25022,93473,93474],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93475,93476,93481,93485,93489],{"dataMmlNode":25028},[25022,93477,93478],{"dataMmlNode":25942},[25033,93479],{"dataC":31539,"xLinkHref":93480},"#MJX-10-TEX-I-210E",[25022,93482,93483],{"dataMmlNode":25031,"transform":32791},[25033,93484],{"dataC":25035,"xLinkHref":43340},[25022,93486,93487],{"dataMmlNode":25942,"transform":32797},[25033,93488],{"dataC":32181,"xLinkHref":32915},[25022,93490,93491],{"dataMmlNode":25031,"transform":32803},[25033,93492],{"dataC":25108,"xLinkHref":43350}," 能準確預測 ",[24923,93495,93497],{"className":93496,"jax":24927},[24926],[24929,93498,93499,93504],{"style":29805,"xmlns":24932,"width":54359,"height":54360,"role":1930,"focusable":8058,"viewBox":54361,"xmlnsXLink":24936},[24938,93500,93501],{},[24941,93502],{"id":93503,"d":25893},"MJX-11-TEX-I-1D466",[25022,93505,93506],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93507,93508],{"dataMmlNode":25028},[25022,93509,93510],{"dataMmlNode":25942},[25033,93511],{"dataC":25987,"xLinkHref":93512},"#MJX-11-TEX-I-1D466"," 的值。",[7166,93515,93517],{"id":93516},"線性迴歸-linear-regression","線性迴歸 (Linear Regression)",[12,93519,93520],{},"最簡單的假設就是線性函數。",[24923,93522,93524],{"className":93523,"jax":24927,"display":13594},[24926],[24929,93525,93526,93552],{"style":24931,"xmlns":24932,"width":32929,"height":24934,"role":1930,"focusable":8058,"viewBox":32930,"xmlnsXLink":24936},[24938,93527,93528,93531,93533,93536,93538,93540,93542,93544,93547,93549],{},[24941,93529],{"id":93530,"d":31530},"MJX-12-TEX-I-210E",[24941,93532],{"id":43425,"d":24944},[24941,93534],{"id":93535,"d":32148},"MJX-12-TEX-I-1D465",[24941,93537],{"id":43428,"d":24988},[24941,93539],{"id":26350,"d":24972},[24941,93541],{"id":33080,"d":32951},[24941,93543],{"id":26353,"d":25935},[24941,93545],{"id":93546,"d":30313},"MJX-12-TEX-N-2B",[24941,93548],{"id":26347,"d":25914},[24941,93550],{"id":93551,"d":25928},"MJX-12-TEX-N-32",[25022,93553,93554],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93555,93556,93561,93565,93570,93574,93578,93588,93593,93603,93613,93617,93628],{"dataMmlNode":25028},[25022,93557,93558],{"dataMmlNode":25942},[25033,93559],{"dataC":31539,"xLinkHref":93560},"#MJX-12-TEX-I-210E",[25022,93562,93563],{"dataMmlNode":25031,"transform":32791},[25033,93564],{"dataC":25035,"xLinkHref":43448},[25022,93566,93567],{"dataMmlNode":25942,"transform":32797},[25033,93568],{"dataC":32181,"xLinkHref":93569},"#MJX-12-TEX-I-1D465",[25022,93571,93572],{"dataMmlNode":25031,"transform":32803},[25033,93573],{"dataC":25108,"xLinkHref":43459},[25022,93575,93576],{"dataMmlNode":25031,"transform":32989},[25033,93577],{"dataC":25078,"xLinkHref":26383},[25022,93579,93580,93584],{"dataMmlNode":26023,"transform":32995},[25022,93581,93582],{"dataMmlNode":25942},[25033,93583],{"dataC":33000,"xLinkHref":33089},[25022,93585,93586],{"dataMmlNode":26031,"transform":33004},[25033,93587],{"dataC":26103,"xLinkHref":26389},[25022,93589,93590],{"dataMmlNode":25031,"transform":33010},[25033,93591],{"dataC":30393,"xLinkHref":93592},"#MJX-12-TEX-N-2B",[25022,93594,93595,93599],{"dataMmlNode":26023,"transform":33016},[25022,93596,93597],{"dataMmlNode":25942},[25033,93598],{"dataC":33000,"xLinkHref":33089},[25022,93600,93601],{"dataMmlNode":26031,"transform":33004},[25033,93602],{"dataC":26035,"xLinkHref":26377},[25022,93604,93605,93609],{"dataMmlNode":26023,"transform":33027},[25022,93606,93607],{"dataMmlNode":25942},[25033,93608],{"dataC":32181,"xLinkHref":93569},[25022,93610,93611],{"dataMmlNode":26031,"transform":33034},[25033,93612],{"dataC":26035,"xLinkHref":26377},[25022,93614,93615],{"dataMmlNode":25031,"transform":33039},[25033,93616],{"dataC":30393,"xLinkHref":93592},[25022,93618,93619,93623],{"dataMmlNode":26023,"transform":33044},[25022,93620,93621],{"dataMmlNode":25942},[25033,93622],{"dataC":33000,"xLinkHref":33089},[25022,93624,93625],{"dataMmlNode":26031,"transform":33004},[25033,93626],{"dataC":26058,"xLinkHref":93627},"#MJX-12-TEX-N-32",[25022,93629,93630,93634],{"dataMmlNode":26023,"transform":33056},[25022,93631,93632],{"dataMmlNode":25942},[25033,93633],{"dataC":32181,"xLinkHref":93569},[25022,93635,93636],{"dataMmlNode":26031,"transform":33034},[25033,93637],{"dataC":26058,"xLinkHref":93627},[13438,93639,93641],{"className":93640},[13441,13442],[12,93642,93643],{},"預測的房價 = (基本底價) + (面積的影響力 × 面積) + (臥室的影響力 × 臥室數)",[12,93645,93646,93647,93664,93665,93668,93669,93672,93673,93690],{},"這些 ",[24923,93648,93650],{"className":93649,"jax":24927},[24926],[24929,93651,93652,93656],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,93653,93654],{},[24941,93655],{"id":33107,"d":32951},[25022,93657,93658],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93659,93660],{"dataMmlNode":25028},[25022,93661,93662],{"dataMmlNode":25942},[25033,93663],{"dataC":33000,"xLinkHref":33116}," 值被稱為",[106,93666,93667],{},"參數 (parameters)","（或",[106,93670,93671],{},"權重 weights","）。我們的工作就是要找出最準確的 ",[24923,93674,93676],{"className":93675,"jax":24927},[24926],[24929,93677,93678,93682],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,93679,93680],{},[24941,93681],{"id":33149,"d":32951},[25022,93683,93684],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,93685,93686],{"dataMmlNode":25028},[25022,93687,93688],{"dataMmlNode":25942},[25033,93689],{"dataC":33000,"xLinkHref":33200}," 值。但該怎麼找呢？",[7166,93692,93694],{"id":93693},"成本函數-the-cost-function","成本函數 (The Cost 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841},[25022,94077,94078],{"dataMmlNode":25031,"transform":32577},[25033,94079],{"dataC":25108,"xLinkHref":26724}," 是這間房子在現實中的",[106,94082,94083],{},"真實成交價",[12,94085,94086],{},"把它們相減，就是在算**「我們猜錯了多少錢（誤差）」**。例如，我們預測賣 50 萬，實際賣了 60 萬，誤差就是 -10 萬。",[12,94088,94089,94090],{},"2. 平方",[24923,94091,94093],{"className":94092,"jax":24927},[24926],[24929,94094,94097,94107],{"style":24931,"xmlns":24932,"width":94095,"height":31733,"role":1930,"focusable":8058,"viewBox":94096,"xmlnsXLink":24936},"5.766ex","0 -833.9 2548.6 1083.9",[24938,94098,94099,94101,94103,94105],{},[24941,94100],{"id":26771,"d":24944},[24941,94102],{"id":26777,"d":26357},[24941,94104],{"id":26783,"d":24988},[24941,94106],{"id":26765,"d":25928},[25022,94108,94109],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94110,94111,94115,94119,94124,94129],{"dataMmlNode":25028},[25022,94112,94113],{"dataMmlNode":25031},[25033,94114],{"dataC":25035,"xLinkHref":26821},[25022,94116,94117],{"dataMmlNode":25031,"transform":25040},[25033,94118],{"dataC":26392,"xLinkHref":26829},[25022,94120,94122],{"dataMmlNode":25031,"transform":94121},"translate(833.7,0)",[25033,94123],{"dataC":26392,"xLinkHref":26829},[25022,94125,94127],{"dataMmlNode":25031,"transform":94126},"translate(1278.3,0)",[25033,94128],{"dataC":26392,"xLinkHref":26829},[25022,94130,94131,94135],{"dataMmlNode":31565,"transform":25059},[25022,94132,94133],{"dataMmlNode":25031},[25033,94134],{"dataC":25108,"xLinkHref":26841},[25022,94136,94137],{"dataMmlNode":26031,"transform":37418},[25033,94138],{"dataC":26058,"xLinkHref":26811},[12,94140,94141],{},"這裡把誤差拿來「平方」，有兩個很重要的原因：",[23,94143,94144,94150],{},[26,94145,94146,94149],{},[106,94147,94148],{},"把負數變成正數："," 如果我們高估了 10 萬（誤差 +10）或低估了 10 萬（誤差 -10），對預測來說都是一樣糟糕的。平方之後，負號就不見了，我們只看「誤差的幅度」。",[26,94151,94152,94155],{},[106,94153,94154],{},"嚴厲懲罰「大錯誤」："," 平方有一個特性，會讓越大的數字膨脹得越快（例如 2 的平方是 4，但 10 的平方是 100）。這等於是在告訴機器：「小誤差還可以接受，但如果預測錯得太離譜，你的分數會扣得非常非常重！」",[12,94157,94158,94159],{},"3. 加總",[24923,94160,94162],{"className":94161,"jax":24927},[24926],[24929,94163,94164,94169],{"style":24931,"xmlns":24932,"width":33756,"height":24934,"role":1930,"focusable":8058,"viewBox":88600,"xmlnsXLink":24936},[24938,94165,94166],{},[24941,94167],{"id":94168,"d":27911},"MJX-20-TEX-SO-2211",[25022,94170,94171],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94172,94173],{"dataMmlNode":25028},[25022,94174,94175],{"dataMmlNode":25031},[25033,94176],{"dataC":27199,"xLinkHref":94177},"#MJX-20-TEX-SO-2211",[12,94179,94180,94181,86843],{},"我們不能只看一間房子的誤差，而是要把資料集裡全部 47 間房子的「誤差平方」全部加起來，算出一個總分。這就是為什麼這個方法被稱為",[106,94182,94183],{},"最小平方法 (Least Squares)",[3603,94185,94186],{},[26,94187,94188],{},[24923,94189,94191],{"className":94190,"jax":24927},[24926],[24929,94192,94193,94199],{"style":44803,"xmlns":24932,"width":44804,"height":44805,"role":1930,"focusable":8058,"viewBox":44806,"xmlnsXLink":24936},[24938,94194,94195,94197],{},[24941,94196],{"id":26903,"d":25914},[24941,94198],{"id":26919,"d":25928},[25022,94200,94201],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94202,94203],{"dataMmlNode":25028},[25022,94204,94205,94209,94213],{"dataMmlNode":27622},[25022,94206,94207],{"dataMmlNode":26031,"transform":44821},[25033,94208],{"dataC":26035,"xLinkHref":26944},[25022,94210,94211],{"dataMmlNode":26031,"transform":44826},[25033,94212],{"dataC":26058,"xLinkHref":26970},[27668,94214],{"width":44831,"height":27671,"x":27672,"y":27673},[12,94216,94217,94218,94221,94222,94253],{},"為什麼前面要乘一個二分之一？其實這只是一個",[106,94219,94220],{},"數學上的小偷吃步","。 在機器學習的下一步，我們需要用到微積分來尋找最小誤差。當平方的那個 2 掉下來相乘時，剛好可以跟 ",[24923,94223,94225],{"className":94224,"jax":24927},[24926],[24929,94226,94227,94235],{"style":44803,"xmlns":24932,"width":44804,"height":44805,"role":1930,"focusable":8058,"viewBox":44806,"xmlnsXLink":24936},[24938,94228,94229,94232],{},[24941,94230],{"id":94231,"d":25914},"MJX-22-TEX-N-31",[24941,94233],{"id":94234,"d":25928},"MJX-22-TEX-N-32",[25022,94236,94237],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94238,94239],{"dataMmlNode":25028},[25022,94240,94241,94246,94251],{"dataMmlNode":27622},[25022,94242,94243],{"dataMmlNode":26031,"transform":44821},[25033,94244],{"dataC":26035,"xLinkHref":94245},"#MJX-22-TEX-N-31",[25022,94247,94248],{"dataMmlNode":26031,"transform":44826},[25033,94249],{"dataC":26058,"xLinkHref":94250},"#MJX-22-TEX-N-32",[27668,94252],{"width":44831,"height":27671,"x":27672,"y":27673}," 抵銷掉，讓後面的計算過程變得更乾淨漂亮。它完全不會影響我們尋找最佳答案的結果。",[12,94255,94256,94295,94296,94299,94300,86843],{},[24923,94257,94259],{"className":94258,"jax":24927},[24926],[24929,94260,94261,94273],{"style":24931,"xmlns":24932,"width":33337,"height":24934,"role":1930,"focusable":8058,"viewBox":33338,"xmlnsXLink":24936},[24938,94262,94263,94266,94268,94271],{},[24941,94264],{"id":94265,"d":33143},"MJX-23-TEX-I-1D43D",[24941,94267],{"id":27091,"d":24944},[24941,94269],{"id":94270,"d":32951},"MJX-23-TEX-I-1D703",[24941,94272],{"id":27097,"d":24988},[25022,94274,94275],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94276,94277,94282,94286,94291],{"dataMmlNode":25028},[25022,94278,94279],{"dataMmlNode":25942},[25033,94280],{"dataC":33187,"xLinkHref":94281},"#MJX-23-TEX-I-1D43D",[25022,94283,94284],{"dataMmlNode":25031,"transform":33191},[25033,94285],{"dataC":25035,"xLinkHref":27176},[25022,94287,94288],{"dataMmlNode":25942,"transform":33197},[25033,94289],{"dataC":33000,"xLinkHref":94290},"#MJX-23-TEX-I-1D703",[25022,94292,94293],{"dataMmlNode":25031,"transform":33203},[25033,94294],{"dataC":25108,"xLinkHref":27186}," 越小，代表我們的模型越貼合資料。所以我們要想辦法",[106,94297,94298],{},"最小化"," ",[24923,94301,94303],{"className":94302,"jax":24927},[24926],[24929,94304,94305,94316],{"style":24931,"xmlns":24932,"width":33337,"height":24934,"role":1930,"focusable":8058,"viewBox":33338,"xmlnsXLink":24936},[24938,94306,94307,94310,94312,94314],{},[24941,94308],{"id":94309,"d":33143},"MJX-24-TEX-I-1D43D",[24941,94311],{"id":27332,"d":24944},[24941,94313],{"id":33838,"d":32951},[24941,94315],{"id":27346,"d":24988},[25022,94317,94318],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94319,94320,94325,94329,94333],{"dataMmlNode":25028},[25022,94321,94322],{"dataMmlNode":25942},[25033,94323],{"dataC":33187,"xLinkHref":94324},"#MJX-24-TEX-I-1D43D",[25022,94326,94327],{"dataMmlNode":25031,"transform":33191},[25033,94328],{"dataC":25035,"xLinkHref":27444},[25022,94330,94331],{"dataMmlNode":25942,"transform":33197},[25033,94332],{"dataC":33000,"xLinkHref":33879},[25022,94334,94335],{"dataMmlNode":25031,"transform":33203},[25033,94336],{"dataC":25108,"xLinkHref":27470},[7166,94338,94340],{"id":94339},"梯度下降-gradient-descent-尋找最佳參數","梯度下降 (Gradient Descent) — 尋找最佳參數",[12,94342,94343],{},"梯度下降是一個迭代的優化演算法。它的直觀概念是這樣的：",[3603,94345,94346,94367,94411,94450],{},[26,94347,94348,94349,94366],{},"隨便猜一個 ",[24923,94350,94352],{"className":94351,"jax":24927},[24926],[24929,94353,94354,94358],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,94355,94356],{},[24941,94357],{"id":34049,"d":32951},[25022,94359,94360],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94361,94362],{"dataMmlNode":25028},[25022,94363,94364],{"dataMmlNode":25942},[25033,94365],{"dataC":33000,"xLinkHref":34087}," 的起始值。",[26,94368,94369,94370,86843],{},"計算成本 ",[24923,94371,94373],{"className":94372,"jax":24927},[24926],[24929,94374,94375,94388],{"style":24931,"xmlns":24932,"width":33337,"height":24934,"role":1930,"focusable":8058,"viewBox":33338,"xmlnsXLink":24936},[24938,94376,94377,94379,94382,94385],{},[24941,94378],{"id":44464,"d":33143},[24941,94380],{"id":94381,"d":24944},"MJX-26-TEX-N-28",[24941,94383],{"id":94384,"d":32951},"MJX-26-TEX-I-1D703",[24941,94386],{"id":94387,"d":24988},"MJX-26-TEX-N-29",[25022,94389,94390],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94391,94392,94396,94401,94406],{"dataMmlNode":25028},[25022,94393,94394],{"dataMmlNode":25942},[25033,94395],{"dataC":33187,"xLinkHref":44473},[25022,94397,94398],{"dataMmlNode":25031,"transform":33191},[25033,94399],{"dataC":25035,"xLinkHref":94400},"#MJX-26-TEX-N-28",[25022,94402,94403],{"dataMmlNode":25942,"transform":33197},[25033,94404],{"dataC":33000,"xLinkHref":94405},"#MJX-26-TEX-I-1D703",[25022,94407,94408],{"dataMmlNode":25031,"transform":33203},[25033,94409],{"dataC":25108,"xLinkHref":94410},"#MJX-26-TEX-N-29",[26,94412,94413,94414,94449],{},"朝著能讓 ",[24923,94415,94417],{"className":94416,"jax":24927},[24926],[24929,94418,94419,94429],{"style":24931,"xmlns":24932,"width":33337,"height":24934,"role":1930,"focusable":8058,"viewBox":33338,"xmlnsXLink":24936},[24938,94420,94421,94423,94425,94427],{},[24941,94422],{"id":34206,"d":33143},[24941,94424],{"id":34209,"d":24944},[24941,94426],{"id":34212,"d":32951},[24941,94428],{"id":34215,"d":24988},[25022,94430,94431],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94432,94433,94437,94441,94445],{"dataMmlNode":25028},[25022,94434,94435],{"dataMmlNode":25942},[25033,94436],{"dataC":33187,"xLinkHref":34259},[25022,94438,94439],{"dataMmlNode":25031,"transform":33191},[25033,94440],{"dataC":25035,"xLinkHref":34264},[25022,94442,94443],{"dataMmlNode":25942,"transform":33197},[25033,94444],{"dataC":33000,"xLinkHref":34269},[25022,94446,94447],{"dataMmlNode":25031,"transform":33203},[25033,94448],{"dataC":25108,"xLinkHref":34274}," 減少最多的方向（最陡的下坡方向）稍微調整 θ。",[26,94451,94452],{},"重複這個動作，直到收斂（走到谷底）。",[12,94454,94455],{},"更新規則長這樣：",[24923,94457,94459],{"className":94458,"jax":24927,"display":13594},[24926],[24929,94460,94461,94486],{"style":33594,"xmlns":24932,"width":33595,"height":33596,"role":1930,"focusable":8058,"viewBox":33597,"xmlnsXLink":24936},[24938,94462,94463,94465,94467,94470,94472,94474,94477,94480,94482,94484],{},[24941,94464],{"id":34425,"d":32951},[24941,94466],{"id":34444,"d":33606},[24941,94468],{"id":94469,"d":33610},"MJX-28-TEX-N-3A",[24941,94471],{"id":34422,"d":24972},[24941,94473],{"id":44666,"d":25910},[24941,94475],{"id":94476,"d":33620},"MJX-28-TEX-I-1D6FC",[24941,94478],{"id":94479,"d":33624},"MJX-28-TEX-I-1D715",[24941,94481],{"id":44630,"d":33143},[24941,94483],{"id":34410,"d":24944},[24941,94485],{"id":34419,"d":24988},[25022,94487,94488],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94489,94490,94500,94511,94521,94525,94530],{"dataMmlNode":25028},[25022,94491,94492,94496],{"dataMmlNode":26023},[25022,94493,94494],{"dataMmlNode":25942},[25033,94495],{"dataC":33000,"xLinkHref":34504},[25022,94497,94498],{"dataMmlNode":25942,"transform":33004},[25033,94499],{"dataC":33647,"xLinkHref":34624},[25022,94501,94502,94507],{"dataMmlNode":25031,"transform":33651},[25022,94503,94504],{"dataMmlNode":6977},[25033,94505],{"dataC":33656,"xLinkHref":94506},"#MJX-28-TEX-N-3A",[25022,94508,94509],{"dataMmlNode":6977,"transform":26048},[25033,94510],{"dataC":25078,"xLinkHref":34496},[25022,94512,94513,94517],{"dataMmlNode":26023,"transform":33665},[25022,94514,94515],{"dataMmlNode":25942},[25033,94516],{"dataC":33000,"xLinkHref":34504},[25022,94518,94519],{"dataMmlNode":25942,"transform":33004},[25033,94520],{"dataC":33647,"xLinkHref":34624},[25022,94522,94523],{"dataMmlNode":25031,"transform":33676},[25033,94524],{"dataC":26019,"xLinkHref":44760},[25022,94526,94527],{"dataMmlNode":25942,"transform":33682},[25033,94528],{"dataC":33685,"xLinkHref":94529},"#MJX-28-TEX-I-1D6FC",[25022,94531,94532,94555,94571],{"dataMmlNode":27622,"transform":33689},[25022,94533,94534,94539,94543,94547,94551],{"dataMmlNode":27626,"transform":33692},[25022,94535,94536],{"dataMmlNode":25942},[25033,94537],{"dataC":33697,"xLinkHref":94538},"#MJX-28-TEX-I-1D715",[25022,94540,94541],{"dataMmlNode":25942,"transform":33701},[25033,94542],{"dataC":33187,"xLinkHref":44675},[25022,94544,94545],{"dataMmlNode":25031,"transform":33707},[25033,94546],{"dataC":25035,"xLinkHref":34462},[25022,94548,94549],{"dataMmlNode":25942,"transform":33712},[25033,94550],{"dataC":33000,"xLinkHref":34504},[25022,94552,94553],{"dataMmlNode":25031,"transform":33717},[25033,94554],{"dataC":25108,"xLinkHref":34485},[25022,94556,94557,94561],{"dataMmlNode":27626,"transform":33722},[25022,94558,94559],{"dataMmlNode":25942},[25033,94560],{"dataC":33697,"xLinkHref":94538},[25022,94562,94563,94567],{"dataMmlNode":26023,"transform":33701},[25022,94564,94565],{"dataMmlNode":25942},[25033,94566],{"dataC":33000,"xLinkHref":34504},[25022,94568,94569],{"dataMmlNode":25942,"transform":33004},[25033,94570],{"dataC":33647,"xLinkHref":34624},[27668,94572],{"width":33739,"height":27671,"x":27672,"y":27673},[25631,94574,94576,94578],{"className":94575},[13441,13442],[25635,94577,87175],{},[13438,94579,94581,94587,94594,94597,94655,94721,94808,94816,94840,94849,94853,94938,94941],{"className":94580},[25641],[12,94582,94583,94586],{},[5610,94584,94585],{},"其實看不懂半點"," 我盡力說說看",[12,94588,94589,94590,94593],{},"想像一個情境：",[106,94591,94592],{},"你被蒙住了雙眼，丟在一座高山上的某個地方。你的目標是「走到這座山的最低谷」。"," 你看不見，只能用腳感覺周圍地面的傾斜程度。你會怎麼做？你一定是用腳探一探，感覺哪邊是「最陡的下坡」，然後往那個方向踏出一步。接著再用腳感覺一下，再往最陡的下坡走一步。不斷重複，直到你覺得四周都平了，你就知道自己到達谷底了。",[12,94595,94596],{},"現在我們來看看我們的公式，把它拆開來看：",[26236,94598,94600,94601,94626,94627,94654],{"id":94599},"_1與準備更新位置","1. ",[24923,94602,94604],{"className":94603,"jax":24927},[24926],[24929,94605,94608,94614],{"style":48965,"xmlns":24932,"width":56007,"height":94606,"role":1930,"focusable":8058,"viewBox":94607,"xmlnsXLink":24936},"2.057ex","0 -705 881 909",[24938,94609,94610,94612],{},[24941,94611],{"id":34690,"d":32951},[24941,94613],{"id":34695,"d":33606},[25022,94615,94616],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94617,94618,94622],{"dataMmlNode":25028},[25022,94619,94620],{"dataMmlNode":25942},[25033,94621],{"dataC":33000,"xLinkHref":34746},[25022,94623,94624],{"dataMmlNode":25942,"transform":70896},[25033,94625],{"dataC":33647,"xLinkHref":34767},"​ 與 ",[24923,94628,94630],{"className":94629,"jax":24927},[24926],[24929,94631,94632,94639],{"style":26868,"xmlns":24932,"width":33756,"height":26870,"role":1930,"focusable":8058,"viewBox":33757,"xmlnsXLink":24936},[24938,94633,94634,94637],{},[24941,94635],{"id":94636,"d":33610},"MJX-30-TEX-N-3A",[24941,94638],{"id":34970,"d":24972},[25022,94640,94641],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94642,94643],{"dataMmlNode":25028},[25022,94644,94645,94650],{"dataMmlNode":25031},[25022,94646,94647],{"dataMmlNode":6977},[25033,94648],{"dataC":33656,"xLinkHref":94649},"#MJX-30-TEX-N-3A",[25022,94651,94652],{"dataMmlNode":6977,"transform":26048},[25033,94653],{"dataC":25078,"xLinkHref":35006}," （準備更新位置）",[23,94656,94657,94688],{},[26,94658,94659,94687],{},[106,94660,94661,94686],{},[24923,94662,94664],{"className":94663,"jax":24927},[24926],[24929,94665,94666,94673],{"style":48965,"xmlns":24932,"width":56007,"height":94606,"role":1930,"focusable":8058,"viewBox":94607,"xmlnsXLink":24936},[24938,94667,94668,94671],{},[24941,94669],{"id":94670,"d":32951},"MJX-31-TEX-I-1D703",[24941,94672],{"id":49912,"d":33606},[25022,94674,94675],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94676,94677,94682],{"dataMmlNode":25028},[25022,94678,94679],{"dataMmlNode":25942},[25033,94680],{"dataC":33000,"xLinkHref":94681},"#MJX-31-TEX-I-1D703",[25022,94683,94684],{"dataMmlNode":25942,"transform":70896},[25033,94685],{"dataC":33647,"xLinkHref":49952},"​："," 這是我們的「權重」（例如：面積對房價的影響力）。你可以把它想像成你現在站在山坡上的**「座標位置」**。",[26,94689,94690,94720],{},[106,94691,94692,94719],{},[24923,94693,94695],{"className":94694,"jax":24927},[24926],[24929,94696,94697,94704],{"style":26868,"xmlns":24932,"width":33756,"height":26870,"role":1930,"focusable":8058,"viewBox":33757,"xmlnsXLink":24936},[24938,94698,94699,94702],{},[24941,94700],{"id":94701,"d":33610},"MJX-32-TEX-N-3A",[24941,94703],{"id":28171,"d":24972},[25022,94705,94706],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94707,94708],{"dataMmlNode":25028},[25022,94709,94710,94715],{"dataMmlNode":25031},[25022,94711,94712],{"dataMmlNode":6977},[25033,94713],{"dataC":33656,"xLinkHref":94714},"#MJX-32-TEX-N-3A",[25022,94716,94717],{"dataMmlNode":6977,"transform":26048},[25033,94718],{"dataC":25078,"xLinkHref":28189},"："," 在程式語言裡，這個符號唸作 \"Assign\"（賦值）。它的意思是**「把右邊算出來的新結果，取代掉左邊舊的東西」**。所以整行公式的意思是：「我的新位置，將會等於右邊算出來的結果」。",[26236,94722,94724,94725,94807],{"id":94723},"_2腳底下的坡度","2. ",[24923,94726,94728],{"className":94727,"jax":24927},[24926],[24929,94729,94734,94753],{"style":94730,"xmlns":24932,"width":94731,"height":94732,"role":1930,"focusable":8058,"viewBox":94733,"xmlnsXLink":24936},"vertical-align: -1.253ex;","4.909ex","3.622ex","0 -1047.1 2169.6 1600.8",[24938,94735,94736,94739,94742,94745,94748,94751],{},[24941,94737],{"id":94738,"d":33624},"MJX-33-TEX-I-1D715",[24941,94740],{"id":94741,"d":33143},"MJX-33-TEX-I-1D43D",[24941,94743],{"id":94744,"d":24944},"MJX-33-TEX-N-28",[24941,94746],{"id":94747,"d":32951},"MJX-33-TEX-I-1D703",[24941,94749],{"id":94750,"d":24988},"MJX-33-TEX-N-29",[24941,94752],{"id":50066,"d":33606},[25022,94754,94755],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94756,94757],{"dataMmlNode":25028},[25022,94758,94759,94787,94804],{"dataMmlNode":27622},[25022,94760,94762,94767,94772,94777,94782],{"dataMmlNode":27626,"transform":94761},"translate(220,516.8) scale(0.707)",[25022,94763,94764],{"dataMmlNode":25942},[25033,94765],{"dataC":33697,"xLinkHref":94766},"#MJX-33-TEX-I-1D715",[25022,94768,94769],{"dataMmlNode":25942,"transform":33701},[25033,94770],{"dataC":33187,"xLinkHref":94771},"#MJX-33-TEX-I-1D43D",[25022,94773,94774],{"dataMmlNode":25031,"transform":33707},[25033,94775],{"dataC":25035,"xLinkHref":94776},"#MJX-33-TEX-N-28",[25022,94778,94779],{"dataMmlNode":25942,"transform":33712},[25033,94780],{"dataC":33000,"xLinkHref":94781},"#MJX-33-TEX-I-1D703",[25022,94783,94784],{"dataMmlNode":25031,"transform":33717},[25033,94785],{"dataC":25108,"xLinkHref":94786},"#MJX-33-TEX-N-29",[25022,94788,94790,94794],{"dataMmlNode":27626,"transform":94789},"translate(586.5,-345.6) scale(0.707)",[25022,94791,94792],{"dataMmlNode":25942},[25033,94793],{"dataC":33697,"xLinkHref":94766},[25022,94795,94796,94800],{"dataMmlNode":26023,"transform":33701},[25022,94797,94798],{"dataMmlNode":25942},[25033,94799],{"dataC":33000,"xLinkHref":94781},[25022,94801,94802],{"dataMmlNode":25942,"transform":33004},[25033,94803],{"dataC":33647,"xLinkHref":50075},[27668,94805],{"width":94806,"height":27671,"x":27672,"y":27673},1929.6," （腳底下的坡度）",[23,94809,94810,94813],{},[26,94811,94812],{},"這坨看起來最可怕的符號叫「偏導數」(Partial Derivative)。別管微積分了，它的白話文就叫作「坡度（slope）」。",[26,94814,94815],{},"它代表你現在腳踩的地方「有多陡」以及「哪邊是上坡」。如果山坡很陡，這個算出來的數字就很大；如果很平緩，數字就很小。",[26236,94817,94819,94820,94839],{"id":94818},"_3-減號我們要往下坡走","3. 減號 ",[24923,94821,94823],{"className":94822,"jax":24927},[24926],[24929,94824,94825,94830],{"style":26868,"xmlns":24932,"width":26869,"height":26870,"role":1930,"focusable":8058,"viewBox":26871,"xmlnsXLink":24936},[24938,94826,94827],{},[24941,94828],{"id":94829,"d":25910},"MJX-34-TEX-N-2212",[25022,94831,94832],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94833,94834],{"dataMmlNode":25028},[25022,94835,94836],{"dataMmlNode":25031},[25033,94837],{"dataC":26019,"xLinkHref":94838},"#MJX-34-TEX-N-2212"," （我們要往下坡走）",[23,94841,94842],{},[26,94843,94844,94845,94848],{},"剛剛說偏導數算出來的是「上坡」的方向。但我們的目標是走到谷底（減少誤差），所以我們必須在公式裡加一個",[106,94846,94847],{},"減號","，強迫機器**「反著坡度走，往下坡前進」**。",[26236,94850,94852],{"id":94851},"_4αalpha步伐大小","4. α (Alpha，步伐大小)",[23,94854,94855,94882],{},[26,94856,94857,94858,94877,94878,94881],{},"這個 ",[24923,94859,94861],{"className":94860,"jax":24927},[24926],[24929,94862,94863,94868],{"style":25196,"xmlns":24932,"width":33802,"height":27038,"role":1930,"focusable":8058,"viewBox":33803,"xmlnsXLink":24936},[24938,94864,94865],{},[24941,94866],{"id":94867,"d":33620},"MJX-35-TEX-I-1D6FC",[25022,94869,94870],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94871,94872],{"dataMmlNode":25028},[25022,94873,94874],{"dataMmlNode":25942},[25033,94875],{"dataC":33685,"xLinkHref":94876},"#MJX-35-TEX-I-1D6FC"," 叫作",[106,94879,94880],{},"學習率 (Learning Rate)","。在下山的故事裡，它就是你的**「步伐大小」**。",[26,94883,94884,94885],{},"機器在下山時，步伐不能亂踩。",[23,94886,94887,94913],{},[26,94888,94889,94912],{},[106,94890,94891,94892,94911],{},"如果 ",[24923,94893,94895],{"className":94894,"jax":24927},[24926],[24929,94896,94897,94902],{"style":25196,"xmlns":24932,"width":33802,"height":27038,"role":1930,"focusable":8058,"viewBox":33803,"xmlnsXLink":24936},[24938,94898,94899],{},[24941,94900],{"id":94901,"d":33620},"MJX-36-TEX-I-1D6FC",[25022,94903,94904],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94905,94906],{"dataMmlNode":25028},[25022,94907,94908],{"dataMmlNode":25942},[25033,94909],{"dataC":33685,"xLinkHref":94910},"#MJX-36-TEX-I-1D6FC","太大："," 你的步伐太大，可能會一腳從山谷左邊直接跨到山谷右邊，甚至越爬越高，永遠找不到最低點。",[26,94914,94915,94937],{},[106,94916,94891,94917,94936],{},[24923,94918,94920],{"className":94919,"jax":24927},[24926],[24929,94921,94922,94927],{"style":25196,"xmlns":24932,"width":33802,"height":27038,"role":1930,"focusable":8058,"viewBox":33803,"xmlnsXLink":24936},[24938,94923,94924],{},[24941,94925],{"id":94926,"d":33620},"MJX-37-TEX-I-1D6FC",[25022,94928,94929],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94930,94931],{"dataMmlNode":25028},[25022,94932,94933],{"dataMmlNode":25942},[25033,94934],{"dataC":33685,"xLinkHref":94935},"#MJX-37-TEX-I-1D6FC"," 太小："," 你用螞蟻的步伐下山，雖然很安全，但機器會算到天荒地老才走到谷底。",[12,94939,94940],{},"所以公式，翻譯過來大概是：",[12,94942,94943],{},"新的位置 := 舊的位置 - ( 步伐大小 × 腳底下的陡峭程度 )",[12,94945,94946,94947,94966,94967,94970],{},"這裡的 ",[24923,94948,94950],{"className":94949,"jax":24927},[24926],[24929,94951,94952,94957],{"style":25196,"xmlns":24932,"width":33802,"height":27038,"role":1930,"focusable":8058,"viewBox":33803,"xmlnsXLink":24936},[24938,94953,94954],{},[24941,94955],{"id":94956,"d":33620},"MJX-38-TEX-I-1D6FC",[25022,94958,94959],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,94960,94961],{"dataMmlNode":25028},[25022,94962,94963],{"dataMmlNode":25942},[25033,94964],{"dataC":33685,"xLinkHref":94965},"#MJX-38-TEX-I-1D6FC"," 是",[106,94968,94969],{},"學習率 (learning rate)","——它控制了你每次踏出的步伐有多大。步伐太大會衝過頭；太小又會走得很慢，收斂超級久。",[12,94972,94973],{},"把偏導數 (partial derivative) 展開，",[12,94975,94976],{},[1930,94977],{"alt":1932,"src":94978},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775986283712-Screenshot_2026-04-12_at_5.30.47_PM.png",[12,94980,94981],{},"更新規則會變成：",[24923,94983,94985],{"className":94984,"jax":24927,"display":13594},[24926],[24929,94986,94987,95022],{"style":33830,"xmlns":24932,"width":33831,"height":33832,"role":1930,"focusable":8058,"viewBox":33833,"xmlnsXLink":24936},[24938,94988,94989,94991,94993,94996,94998,95001,95004,95006,95009,95012,95014,95016,95019],{},[24941,94990],{"id":35750,"d":32951},[24941,94992],{"id":35753,"d":33606},[24941,94994],{"id":94995,"d":33610},"MJX-39-TEX-N-3A",[24941,94997],{"id":50250,"d":24972},[24941,94999],{"id":95000,"d":30313},"MJX-39-TEX-N-2B",[24941,95002],{"id":95003,"d":33620},"MJX-39-TEX-I-1D6FC",[24941,95005],{"id":45278,"d":24944},[24941,95007],{"id":95008,"d":25893},"MJX-39-TEX-I-1D466",[24941,95010],{"id":95011,"d":27108},"MJX-39-TEX-I-1D456",[24941,95013],{"id":45284,"d":24988},[24941,95015],{"id":50279,"d":25910},[24941,95017],{"id":95018,"d":31530},"MJX-39-TEX-I-210E",[24941,95020],{"id":95021,"d":32148},"MJX-39-TEX-I-1D465",[25022,95023,95024],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,95025,95026,95036,95047,95057,95062,95067,95071,95093,95097,95102,95106,95127,95131,95135],{"dataMmlNode":25028},[25022,95027,95028,95032],{"dataMmlNode":26023},[25022,95029,95030],{"dataMmlNode":25942},[25033,95031],{"dataC":33000,"xLinkHref":35764},[25022,95033,95034],{"dataMmlNode":25942,"transform":33004},[25033,95035],{"dataC":33647,"xLinkHref":35769},[25022,95037,95038,95043],{"dataMmlNode":25031,"transform":33651},[25022,95039,95040],{"dataMmlNode":6977},[25033,95041],{"dataC":33656,"xLinkHref":95042},"#MJX-39-TEX-N-3A",[25022,95044,95045],{"dataMmlNode":6977,"transform":26048},[25033,95046],{"dataC":25078,"xLinkHref":50301},[25022,95048,95049,95053],{"dataMmlNode":26023,"transform":33665},[25022,95050,95051],{"dataMmlNode":25942},[25033,95052],{"dataC":33000,"xLinkHref":35764},[25022,95054,95055],{"dataMmlNode":25942,"transform":33004},[25033,95056],{"dataC":33647,"xLinkHref":35769},[25022,95058,95059],{"dataMmlNode":25031,"transform":33676},[25033,95060],{"dataC":30393,"xLinkHref":95061},"#MJX-39-TEX-N-2B",[25022,95063,95064],{"dataMmlNode":25942,"transform":33682},[25033,95065],{"dataC":33685,"xLinkHref":95066},"#MJX-39-TEX-I-1D6FC",[25022,95068,95069],{"dataMmlNode":25031,"transform":33689},[25033,95070],{"dataC":25035,"xLinkHref":45298},[25022,95072,95073,95078],{"dataMmlNode":31565,"transform":33922},[25022,95074,95075],{"dataMmlNode":25942},[25033,95076],{"dataC":25987,"xLinkHref":95077},"#MJX-39-TEX-I-1D466",[25022,95079,95080,95084,95089],{"dataMmlNode":27203,"transform":33304,"dataMjxTexclass":27205},[25022,95081,95082],{"dataMmlNode":25031},[25033,95083],{"dataC":25035,"xLinkHref":45298},[25022,95085,95086],{"dataMmlNode":25942,"transform":25040},[25033,95087],{"dataC":27210,"xLinkHref":95088},"#MJX-39-TEX-I-1D456",[25022,95090,95091],{"dataMmlNode":25031,"transform":32577},[25033,95092],{"dataC":25108,"xLinkHref":45308},[25022,95094,95095],{"dataMmlNode":25031,"transform":33943},[25033,95096],{"dataC":26019,"xLinkHref":50390},[25022,95098,95099],{"dataMmlNode":25942,"transform":33948},[25033,95100],{"dataC":31539,"xLinkHref":95101},"#MJX-39-TEX-I-210E",[25022,95103,95104],{"dataMmlNode":25031,"transform":33954},[25033,95105],{"dataC":25035,"xLinkHref":45298},[25022,95107,95108,95113],{"dataMmlNode":31565,"transform":33959},[25022,95109,95110],{"dataMmlNode":25942},[25033,95111],{"dataC":32181,"xLinkHref":95112},"#MJX-39-TEX-I-1D465",[25022,95114,95115,95119,95123],{"dataMmlNode":27203,"transform":33270,"dataMjxTexclass":27205},[25022,95116,95117],{"dataMmlNode":25031},[25033,95118],{"dataC":25035,"xLinkHref":45298},[25022,95120,95121],{"dataMmlNode":25942,"transform":25040},[25033,95122],{"dataC":27210,"xLinkHref":95088},[25022,95124,95125],{"dataMmlNode":25031,"transform":32577},[25033,95126],{"dataC":25108,"xLinkHref":45308},[25022,95128,95129],{"dataMmlNode":25031,"transform":33981},[25033,95130],{"dataC":25108,"xLinkHref":45308},[25022,95132,95133],{"dataMmlNode":25031,"transform":33986},[25033,95134],{"dataC":25108,"xLinkHref":45308},[25022,95136,95137,95141,95155],{"dataMmlNode":33991,"transform":33992},[25022,95138,95139],{"dataMmlNode":25942},[25033,95140],{"dataC":32181,"xLinkHref":95112},[25022,95142,95143,95147,95151],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,95144,95145],{"dataMmlNode":25031},[25033,95146],{"dataC":25035,"xLinkHref":45298},[25022,95148,95149],{"dataMmlNode":25942,"transform":25040},[25033,95150],{"dataC":27210,"xLinkHref":95088},[25022,95152,95153],{"dataMmlNode":25031,"transform":32577},[25033,95154],{"dataC":25108,"xLinkHref":45308},[25022,95156,95157],{"dataMmlNode":25942,"transform":34014},[25033,95158],{"dataC":33647,"xLinkHref":35769},[25631,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1143.3",[24938,95372,95373,95375,95377,95379,95381,95384,95386],{},[24941,95374],{"id":35863,"d":24944},[24941,95376],{"id":89478,"d":25893},[24941,95378],{"id":35869,"d":27108},[24941,95380],{"id":35872,"d":24988},[24941,95382],{"id":95383,"d":25910},"MJX-42-TEX-N-2212",[24941,95385],{"id":35860,"d":31530},[24941,95387],{"id":35866,"d":32148},[25022,95389,95390],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,95391,95392,95396,95416,95422,95427,95432,95453,95458],{"dataMmlNode":25028},[25022,95393,95394],{"dataMmlNode":25031},[25033,95395],{"dataC":25035,"xLinkHref":35904},[25022,95397,95398,95402],{"dataMmlNode":31565,"transform":25040},[25022,95399,95400],{"dataMmlNode":25942},[25033,95401],{"dataC":25987,"xLinkHref":89632},[25022,95403,95404,95408,95412],{"dataMmlNode":27203,"transform":32620,"dataMjxTexclass":27205},[25022,95405,95406],{"dataMmlNode":25031},[25033,95407],{"dataC":25035,"xLinkHref":35904},[25022,95409,95410],{"dataMmlNode":25942,"transform":25040},[25033,95411],{"dataC":27210,"xLinkHref":35922},[25022,95413,95414],{"dataMmlNode":25031,"transform":32577},[25033,95415],{"dataC":25108,"xLinkHref":35927},[25022,95417,95419],{"dataMmlNode":25031,"transform":95418},"translate(1978.3,0)",[25033,95420],{"dataC":26019,"xLinkHref":95421},"#MJX-42-TEX-N-2212",[25022,95423,95425],{"dataMmlNode":25942,"transform":95424},"translate(2978.5,0)",[25033,95426],{"dataC":31539,"xLinkHref":35899},[25022,95428,95430],{"dataMmlNode":25031,"transform":95429},"translate(3554.5,0)",[25033,95431],{"dataC":25035,"xLinkHref":35904},[25022,95433,95435,95439],{"dataMmlNode":31565,"transform":95434},"translate(3943.5,0)",[25022,95436,95437],{"dataMmlNode":25942},[25033,95438],{"dataC":32181,"xLinkHref":35911},[25022,95440,95441,95445,95449],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},[25022,95442,95443],{"dataMmlNode":25031},[25033,95444],{"dataC":25035,"xLinkHref":35904},[25022,95446,95447],{"dataMmlNode":25942,"transform":25040},[25033,95448],{"dataC":27210,"xLinkHref":35922},[25022,95450,95451],{"dataMmlNode":25031,"transform":32577},[25033,95452],{"dataC":25108,"xLinkHref":35927},[25022,95454,95456],{"dataMmlNode":25031,"transform":95455},"translate(5392.6,0)",[25033,95457],{"dataC":25108,"xLinkHref":35927},[25022,95459,95461],{"dataMmlNode":25031,"transform":95460},"translate(5781.6,0)",[25033,95462],{"dataC":25108,"xLinkHref":35927},[23,95464,95465,95509],{},[26,95466,95467,94966,95506,86843],{},[24923,95468,95470],{"className":95469,"jax":24927},[24926],[24929,95471,95472,95482],{"style":29805,"xmlns":24932,"width":32590,"height":32591,"role":1930,"focusable":8058,"viewBox":32592,"xmlnsXLink":24936},[24938,95473,95474,95476,95478,95480],{},[24941,95475],{"id":36043,"d":25893},[24941,95477],{"id":36006,"d":24944},[24941,95479],{"id":36024,"d":27108},[24941,95481],{"id":36012,"d":24988},[25022,95483,95484],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,95485,95486],{"dataMmlNode":25028},[25022,95487,95488,95492],{"dataMmlNode":31565},[25022,95489,95490],{"dataMmlNode":25942},[25033,95491],{"dataC":25987,"xLinkHref":36184},[25022,95493,95494,95498,95502],{"dataMmlNode":27203,"transform":32620,"dataMjxTexclass":27205},[25022,95495,95496],{"dataMmlNode":25031},[25033,95497],{"dataC":25035,"xLinkHref":36076},[25022,95499,95500],{"dataMmlNode":25942,"transform":25040},[25033,95501],{"dataC":27210,"xLinkHref":36122},[25022,95503,95504],{"dataMmlNode":25031,"transform":32577},[25033,95505],{"dataC":25108,"xLinkHref":36086},[106,95507,95508],{},"現實成交價",[26,95510,95511,95564,95565,86843],{},[24923,95512,95514],{"className":95513,"jax":24927},[24926],[24929,95515,95516,95528],{"style":24931,"xmlns":24932,"width":35684,"height":32647,"role":1930,"focusable":8058,"viewBox":35685,"xmlnsXLink":24936},[24938,95517,95518,95520,95522,95524,95526],{},[24941,95519],{"id":36291,"d":31530},[24941,95521],{"id":29312,"d":24944},[24941,95523],{"id":36296,"d":32148},[24941,95525],{"id":36282,"d":27108},[24941,95527],{"id":29321,"d":24988},[25022,95529,95530],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,95531,95532,95536,95540,95560],{"dataMmlNode":25028},[25022,95533,95534],{"dataMmlNode":25942},[25033,95535],{"dataC":31539,"xLinkHref":36390},[25022,95537,95538],{"dataMmlNode":25031,"transform":32791},[25033,95539],{"dataC":25035,"xLinkHref":29365},[25022,95541,95542,95546],{"dataMmlNode":31565,"transform":32797},[25022,95543,95544],{"dataMmlNode":25942},[25033,95545],{"dataC":32181,"xLinkHref":36401},[25022,95547,95548,95552,95556],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},[25022,95549,95550],{"dataMmlNode":25031},[25033,95551],{"dataC":25035,"xLinkHref":29365},[25022,95553,95554],{"dataMmlNode":25942,"transform":25040},[25033,95555],{"dataC":27210,"xLinkHref":36370},[25022,95557,95558],{"dataMmlNode":25031,"transform":32577},[25033,95559],{"dataC":25108,"xLinkHref":29383},[25022,95561,95562],{"dataMmlNode":25031,"transform":34488},[25033,95563],{"dataC":25108,"xLinkHref":29383}," 是機器的",[106,95566,95567],{},"預測價",[26236,95569,95571,95572],{"id":95570},"_3-這個特徵的存在感","3. 這個特徵的「存在感」：",[24923,95573,95575],{"className":95574,"jax":24927},[24926],[24929,95576,95577,95590],{"style":33830,"xmlns":24932,"width":32537,"height":33832,"role":1930,"focusable":8058,"viewBox":35798,"xmlnsXLink":24936},[24938,95578,95579,95581,95583,95585,95587],{},[24941,95580],{"id":36503,"d":32148},[24941,95582],{"id":36497,"d":24944},[24941,95584],{"id":45778,"d":27108},[24941,95586],{"id":36506,"d":24988},[24941,95588],{"id":95589,"d":33606},"MJX-45-TEX-I-1D457",[25022,95591,95592],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,95593,95594],{"dataMmlNode":25028},[25022,95595,95596,95600,95614],{"dataMmlNode":33991},[25022,95597,95598],{"dataMmlNode":25942},[25033,95599],{"dataC":32181,"xLinkHref":36536},[25022,95601,95602,95606,95610],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,95603,95604],{"dataMmlNode":25031},[25033,95605],{"dataC":25035,"xLinkHref":36521},[25022,95607,95608],{"dataMmlNode":25942,"transform":25040},[25033,95609],{"dataC":27210,"xLinkHref":45837},[25022,95611,95612],{"dataMmlNode":25031,"transform":32577},[25033,95613],{"dataC":25108,"xLinkHref":36542},[25022,95615,95616],{"dataMmlNode":25942,"transform":34014},[25033,95617],{"dataC":33647,"xLinkHref":95618},"#MJX-45-TEX-I-1D457",[12,95620,95621,95622,95671],{},"為什麼最後還要乘上一個 ",[24923,95623,95625],{"className":95624,"jax":24927},[24926],[24929,95626,95627,95641],{"style":33830,"xmlns":24932,"width":32537,"height":33832,"role":1930,"focusable":8058,"viewBox":35798,"xmlnsXLink":24936},[24938,95628,95629,95631,95633,95636,95638],{},[24941,95630],{"id":36581,"d":32148},[24941,95632],{"id":36569,"d":24944},[24941,95634],{"id":95635,"d":27108},"MJX-46-TEX-I-1D456",[24941,95637],{"id":36584,"d":24988},[24941,95639],{"id":95640,"d":33606},"MJX-46-TEX-I-1D457",[25022,95642,95643],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,95644,95645],{"dataMmlNode":25028},[25022,95646,95647,95651,95666],{"dataMmlNode":33991},[25022,95648,95649],{"dataMmlNode":25942},[25033,95650],{"dataC":32181,"xLinkHref":36621},[25022,95652,95653,95657,95662],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,95654,95655],{"dataMmlNode":25031},[25033,95656],{"dataC":25035,"xLinkHref":36593},[25022,95658,95659],{"dataMmlNode":25942,"transform":25040},[25033,95660],{"dataC":27210,"xLinkHref":95661},"#MJX-46-TEX-I-1D456",[25022,95663,95664],{"dataMmlNode":25031,"transform":32577},[25033,95665],{"dataC":25108,"xLinkHref":36627},[25022,95667,95668],{"dataMmlNode":25942,"transform":34014},[25033,95669],{"dataC":33647,"xLinkHref":95670},"#MJX-46-TEX-I-1D457","​（例如這間房子的實際面積）呢？這是在計算「責任歸屬」。 想像一下，你今天預測房價大翻車，總得找個特徵來怪罪並調整吧？",[23,95673,95674,95698],{},[26,95675,95676,95677,95694,95695,86843],{},"如果這間房子的面積超大（x 數值很大），那它對這次錯誤預測的「責任」就很大。乘上這個大數字後，面積的權重 (",[24923,95678,95680],{"className":95679,"jax":24927},[24926],[24929,95681,95682,95686],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,95683,95684],{},[24941,95685],{"id":36646,"d":32951},[25022,95687,95688],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,95689,95690],{"dataMmlNode":25028},[25022,95691,95692],{"dataMmlNode":25942},[25033,95693],{"dataC":33000,"xLinkHref":36711},") 就會被",[106,95696,95697],{},"大幅度地調整",[26,95699,95700],{},"相反地，如果今天這間房子根本沒有游泳池（游泳池數量 x=0），那房價算錯關游泳池什麼事？乘上 0 之後，游泳池的權重就不會被調整。",[12,95702,95703,95704,95721],{},"也就是，我要更新這個條件的重要性 (",[24923,95705,95707],{"className":95706,"jax":24927},[24926],[24929,95708,95709,95713],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,95710,95711],{},[24941,95712],{"id":36900,"d":32951},[25022,95714,95715],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,95716,95717],{"dataMmlNode":25028},[25022,95718,95719],{"dataMmlNode":25942},[25033,95720],{"dataC":33000,"xLinkHref":36956},")。更新的幅度，取決於我這次『猜錯了多少錢』，再乘上這間房子『這個條件本身的數值有多大』。",[12,95723,95724,95725,95728],{},"這就是 ",[106,95726,95727],{},"LMS (Least Mean Squares) 更新規則","，也叫作 Widrow-Hoff 學習規則。注意看一個細節：更新的幅度跟誤差是成正比的。當預測值很接近實際值時，調整的幅度就會很微小；當誤差很大的時候，調整的幅度就會很大。",[7166,95730,95732],{"id":95731},"批量-batch-vs-隨機-stochastic-梯度下降","批量 (Batch) vs 隨機 (Stochastic) 梯度下降",[12,95734,95735,95738,95739,95742],{},[106,95736,95737],{},"批量梯度下降 (Batch Gradient Descent)"," 會在做單次更新前，把",[106,95740,95741],{},"所有","訓練樣本的梯度加總起來。它很精準，但遇到大資料集的時候會跑得非常慢。",[12,95744,95745],{},[1930,95746],{"alt":1932,"src":95747},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775987318720-Screenshot_2026-04-12_at_5.47.44_PM.png",[12,95749,95750,95753,95754,95757,95758,95775],{},[106,95751,95752],{},"隨機\u002F增量梯度下降 (Stochastic \u002F Incremental Gradient Descent)"," 則是每看完",[106,95755,95756],{},"一筆","訓練樣本就立刻更新一次 ",[24923,95759,95761],{"className":95760,"jax":24927},[24926],[24929,95762,95763,95767],{"style":33072,"xmlns":24932,"width":33073,"height":33074,"role":1930,"focusable":8058,"viewBox":33075,"xmlnsXLink":24936},[24938,95764,95765],{},[24941,95766],{"id":37134,"d":32951},[25022,95768,95769],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,95770,95771],{"dataMmlNode":25028},[25022,95772,95773],{"dataMmlNode":25942},[25033,95774],{"dataC":33000,"xLinkHref":37143},"。它的速度快很多，在實務上通常效果也不錯，雖然走向收斂的路徑會比較曲折、充滿雜訊。",[12,95777,95778],{},[1930,95779],{"alt":1932,"src":95780},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775987315179-Screenshot_2026-04-12_at_5.47.53_PM.png",[7166,95782,95784],{"id":95783},"欠擬合-underfitting-vs-過擬合-overfitting","欠擬合 (Underfitting) vs 過擬合 (Overfitting)",[12,95786,95787],{},[1930,95788],{"alt":1932,"src":95789},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775987112930-Screenshot_2026-04-12_at_5.44.51_PM.png",[12,95791,95792],{},"這是機器學習裡最重要的概念之一！想像一下用不同的模型來擬合我們的房價資料：",[23,95794,95795,95805,95811],{},[26,95796,95797,95798,95801,95802,86843],{},"一條",[106,95799,95800],{},"直線"," (線性擬合) 可能太簡單了——它抓不到資料的真實弧度。這叫做",[106,95803,95804],{},"欠擬合 (underfitting)",[26,95806,95807,95810],{},[106,95808,95809],{},"二次方程式"," (加上 x2) 擬合得比較好，抓到了更多結構。",[26,95812,95813,95816,95817,86843],{},[106,95814,95815],{},"五次方多項式"," 可以完美穿過每一個訓練資料點——但如果拿它來預測新房子的價格，結果一定會慘不忍睹。這叫做",[106,95818,95819],{},"過擬合 (overfitting)",[12,95821,95822,95823,95826],{},"一個只會死背訓練資料的模型，不一定是個好模型。我們要的是能對未知的新資料做出準確預測（具備",[106,95824,95825],{},"泛化能力 generalization","）的模型。",[7166,95828,95830],{"id":95829},"局部加權線性迴歸-locally-weighted-linear-regression","局部加權線性迴歸 (Locally Weighted Linear Regression)",[12,95832,95833,95834,95837],{},"一般的線性迴歸是用一條全域的線來擬合所有資料。",[106,95835,95836],{},"局部加權線性迴歸 (LWLR)"," 換了個思路：在做每次預測時，它會給附近的訓練點比較大的權重，離得遠的點權重就給得比較小。",[12,95839,95840],{},"每個訓練樣本的權重通常會用高斯函數來算：",[12,95842,95843],{},[24923,95844,95846],{"className":95845,"jax":24927},[24926],[24929,95847,95850,95892],{"style":36892,"xmlns":24932,"width":95848,"height":36894,"role":1930,"focusable":8058,"viewBox":95849,"xmlnsXLink":24936},"22.333ex","0 -1149.5 9871.4 1799",[24938,95851,95852,95855,95857,95859,95861,95863,95866,95869,95872,95875,95878,95881,95883,95886,95889],{},[24941,95853],{"id":95854,"d":37254},"MJX-50-TEX-I-1D464",[24941,95856],{"id":46435,"d":24944},[24941,95858],{"id":46429,"d":27108},[24941,95860],{"id":46441,"d":24988},[24941,95862],{"id":90352,"d":24972},[24941,95864],{"id":95865,"d":25259},"MJX-50-TEX-N-65",[24941,95867],{"id":95868,"d":37273},"MJX-50-TEX-N-78",[24941,95870],{"id":95871,"d":37277},"MJX-50-TEX-N-70",[24941,95873],{"id":95874,"d":180},"MJX-50-TEX-N-2061",[24941,95876],{"id":95877,"d":34234},"MJX-50-TEX-LO-28",[24941,95879],{"id":95880,"d":25910},"MJX-50-TEX-N-2212",[24941,95882],{"id":37188,"d":32148},[24941,95884],{"id":95885,"d":25928},"MJX-50-TEX-N-32",[24941,95887],{"id":95888,"d":37296},"MJX-50-TEX-I-1D70F",[24941,95890],{"id":95891,"d":34250},"MJX-50-TEX-LO-29",[25022,95893,95894],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,95895,95896,95917,95921,95932,95937],{"dataMmlNode":25028},[25022,95897,95898,95903],{"dataMmlNode":31565},[25022,95899,95900],{"dataMmlNode":25942},[25033,95901],{"dataC":37310,"xLinkHref":95902},"#MJX-50-TEX-I-1D464",[25022,95904,95905,95909,95913],{"dataMmlNode":27203,"transform":37564,"dataMjxTexclass":27205},[25022,95906,95907],{"dataMmlNode":25031},[25033,95908],{"dataC":25035,"xLinkHref":46484},[25022,95910,95911],{"dataMmlNode":25942,"transform":25040},[25033,95912],{"dataC":27210,"xLinkHref":46474},[25022,95914,95915],{"dataMmlNode":25031,"transform":32577},[25033,95916],{"dataC":25108,"xLinkHref":46494},[25022,95918,95919],{"dataMmlNode":25031,"transform":37332},[25033,95920],{"dataC":25078,"xLinkHref":90381},[25022,95922,95923,95926,95929],{"dataMmlNode":25942,"transform":37338},[25033,95924],{"dataC":25323,"xLinkHref":95925},"#MJX-50-TEX-N-65",[25033,95927],{"dataC":31181,"xLinkHref":95928,"transform":37345},"#MJX-50-TEX-N-78",[25033,95930],{"dataC":31215,"xLinkHref":95931,"transform":37349},"#MJX-50-TEX-N-70",[25022,95933,95934],{"dataMmlNode":25031,"transform":37352},[25033,95935],{"dataC":26065,"xLinkHref":95936},"#MJX-50-TEX-N-2061",[25022,95938,95939,95944,95949,96020],{"dataMmlNode":27626,"transform":37358},[25022,95940,95941],{"dataMmlNode":25031,"transform":27630},[25033,95942],{"dataC":25035,"xLinkHref":95943},"#MJX-50-TEX-LO-28",[25022,95945,95946],{"dataMmlNode":25031,"transform":34318},[25033,95947],{"dataC":26019,"xLinkHref":95948},"#MJX-50-TEX-N-2212",[25022,95950,95952,95999,96017],{"dataMmlNode":27622,"transform":95951},"translate(1375,0)",[25022,95953,95954,95958,95978,95982,95987],{"dataMmlNode":27626,"transform":94761},[25022,95955,95956],{"dataMmlNode":25031},[25033,95957],{"dataC":25035,"xLinkHref":46484},[25022,95959,95960,95964],{"dataMmlNode":31565,"transform":25040},[25022,95961,95962],{"dataMmlNode":25942},[25033,95963],{"dataC":32181,"xLinkHref":37197},[25022,95965,95966,95970,95974],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},[25022,95967,95968],{"dataMmlNode":25031},[25033,95969],{"dataC":25035,"xLinkHref":46484},[25022,95971,95972],{"dataMmlNode":25942,"transform":25040},[25033,95973],{"dataC":27210,"xLinkHref":46474},[25022,95975,95976],{"dataMmlNode":25031,"transform":32577},[25033,95977],{"dataC":25108,"xLinkHref":46494},[25022,95979,95980],{"dataMmlNode":25031,"transform":32699},[25033,95981],{"dataC":26019,"xLinkHref":95948},[25022,95983,95985],{"dataMmlNode":25942,"transform":95984},"translate(2616.1,0)",[25033,95986],{"dataC":32181,"xLinkHref":37197},[25022,95988,95990,95994],{"dataMmlNode":31565,"transform":95989},"translate(3188.1,0)",[25022,95991,95992],{"dataMmlNode":25031},[25033,95993],{"dataC":25108,"xLinkHref":46494},[25022,95995,95996],{"dataMmlNode":26031,"transform":37418},[25033,95997],{"dataC":26058,"xLinkHref":95998},"#MJX-50-TEX-N-32",[25022,96000,96002,96006],{"dataMmlNode":27626,"transform":96001},"translate(1106,-377.4) scale(0.707)",[25022,96003,96004],{"dataMmlNode":26031},[25033,96005],{"dataC":26058,"xLinkHref":95998},[25022,96007,96008,96013],{"dataMmlNode":31565,"transform":26394},[25022,96009,96010],{"dataMmlNode":25942},[25033,96011],{"dataC":37435,"xLinkHref":96012},"#MJX-50-TEX-I-1D70F",[25022,96014,96015],{"dataMmlNode":26031,"transform":37439},[25033,96016],{"dataC":26058,"xLinkHref":95998},[27668,96018],{"width":96019,"height":27671,"x":27672,"y":27673},3038.1,[25022,96021,96023],{"dataMmlNode":25031,"transform":96022},"translate(4653.1,0) translate(0 -0.5)",[25033,96024],{"dataC":25108,"xLinkHref":96025},"#MJX-50-TEX-LO-29",[25631,96027,96029,96031],{"className":96028},[13441,13442],[25635,96030,93867],{},[13438,96032,96034,96037,96047,96115,96173,96210,96215,96261,96267,96287,96309,96361,96365,96372],{"className":96033},[25641],[12,96035,96036],{},"這個應該不在考試範圍裡面，看個大概就行。",[12,96038,96039,96042,96043,96046],{},[106,96040,96041],{},"LWLR 就像是找了一個深耕當地的「資深房仲」","。當你要賣一間 30 坪的房子時，這位房仲不會管那些 100 坪的豪宅賣多少錢，他",[106,96044,96045],{},"只會特別去查「附近同樣也是 28 到 32 坪左右」的房子成交價","，用這些極度相似的房子來幫你估價。",[12,96048,96049,96050],{},"「差異程度」或「距離」：",[24923,96051,96053],{"className":96052,"jax":24927},[24926],[24929,96054,96055,96069],{"style":24931,"xmlns":24932,"width":37783,"height":32647,"role":1930,"focusable":8058,"viewBox":37784,"xmlnsXLink":24936},[24938,96056,96057,96059,96061,96063,96065,96067],{},[24941,96058],{"id":37257,"d":24944},[24941,96060],{"id":37289,"d":32148},[24941,96062],{"id":37260,"d":27108},[24941,96064],{"id":37263,"d":24988},[24941,96066],{"id":37286,"d":25910},[24941,96068],{"id":37292,"d":25928},[25022,96070,96071],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96072,96073,96077,96097,96101,96105],{"dataMmlNode":25028},[25022,96074,96075],{"dataMmlNode":25031},[25033,96076],{"dataC":25035,"xLinkHref":37319},[25022,96078,96079,96083],{"dataMmlNode":31565,"transform":25040},[25022,96080,96081],{"dataMmlNode":25942},[25033,96082],{"dataC":32181,"xLinkHref":37384},[25022,96084,96085,96089,96093],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},[25022,96086,96087],{"dataMmlNode":25031},[25033,96088],{"dataC":25035,"xLinkHref":37319},[25022,96090,96091],{"dataMmlNode":25942,"transform":25040},[25033,96092],{"dataC":27210,"xLinkHref":37324},[25022,96094,96095],{"dataMmlNode":25031,"transform":32577},[25033,96096],{"dataC":25108,"xLinkHref":37329},[25022,96098,96099],{"dataMmlNode":25031,"transform":37401},[25033,96100],{"dataC":26019,"xLinkHref":37368},[25022,96102,96103],{"dataMmlNode":25942,"transform":37406},[25033,96104],{"dataC":32181,"xLinkHref":37384},[25022,96106,96107,96111],{"dataMmlNode":31565,"transform":37411},[25022,96108,96109],{"dataMmlNode":25031},[25033,96110],{"dataC":25108,"xLinkHref":37329},[25022,96112,96113],{"dataMmlNode":26031,"transform":37418},[25033,96114],{"dataC":26058,"xLinkHref":37421},[23,96116,96117,96124,96170],{},[26,96118,96119,96120,96123],{},"x 是你",[106,96121,96122],{},"現在想估價","的房子（例如：30坪）。",[26,96125,96126,96165,96166,96169],{},[24923,96127,96129],{"className":96128,"jax":24927},[24926],[24929,96130,96131,96141],{"style":25196,"xmlns":24932,"width":32537,"height":32538,"role":1930,"focusable":8058,"viewBox":32539,"xmlnsXLink":24936},[24938,96132,96133,96135,96137,96139],{},[24941,96134],{"id":51142,"d":32148},[24941,96136],{"id":90461,"d":24944},[24941,96138],{"id":46618,"d":27108},[24941,96140],{"id":90489,"d":24988},[25022,96142,96143],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96144,96145],{"dataMmlNode":25028},[25022,96146,96147,96151],{"dataMmlNode":31565},[25022,96148,96149],{"dataMmlNode":25942},[25033,96150],{"dataC":32181,"xLinkHref":51155},[25022,96152,96153,96157,96161],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},[25022,96154,96155],{"dataMmlNode":25031},[25033,96156],{"dataC":25035,"xLinkHref":90532},[25022,96158,96159],{"dataMmlNode":25942,"transform":25040},[25033,96160],{"dataC":27210,"xLinkHref":46657},[25022,96162,96163],{"dataMmlNode":25031,"transform":32577},[25033,96164],{"dataC":25108,"xLinkHref":90592}," 是資料庫裡的",[106,96167,96168],{},"歷史成交房子","（例如：一間 80 坪的房子）。",[26,96171,96172],{},"把它們相減再平方，算出來的就是這兩間房子的「差距有多大」。差距越大，這個數字就越大。",[12,96174,96175,96209],{},[24923,96176,96178],{"className":96177,"jax":24927},[24926],[24929,96179,96183,96194],{"style":26153,"xmlns":24932,"width":96180,"height":96181,"role":1930,"focusable":8058,"viewBox":96182,"xmlnsXLink":24936},"3.457ex","1.452ex","0 -448 1528 642",[24938,96184,96185,96188,96191],{},[24941,96186],{"id":96187,"d":25259},"MJX-53-TEX-N-65",[24941,96189],{"id":96190,"d":37273},"MJX-53-TEX-N-78",[24941,96192],{"id":96193,"d":37277},"MJX-53-TEX-N-70",[25022,96195,96196],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96197,96198],{"dataMmlNode":25028},[25022,96199,96200,96203,96206],{"dataMmlNode":25942},[25033,96201],{"dataC":25323,"xLinkHref":96202},"#MJX-53-TEX-N-65",[25033,96204],{"dataC":31181,"xLinkHref":96205,"transform":37345},"#MJX-53-TEX-N-78",[25033,96207],{"dataC":31215,"xLinkHref":96208,"transform":37349},"#MJX-53-TEX-N-70"," 是指數函數（Exponential），配合前面的負號，它畫出來會是一個中間高、兩邊低的「鐘型曲線」或稱高斯曲線。",[12,96211,96212],{},[1930,96213],{"alt":1932,"src":96214},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775988429324-0_9POrHcDjgD6eXS-f.webp",[12,96216,96217,96220,96221,96260],{},[106,96218,96219],{},"如果兩間房子很像（距離近）："," 算出來的權重 ",[24923,96222,96224],{"className":96223,"jax":24927},[24926],[24929,96225,96226,96236],{"style":25196,"xmlns":24932,"width":28582,"height":32538,"role":1930,"focusable":8058,"viewBox":37538,"xmlnsXLink":24936},[24938,96227,96228,96230,96232,96234],{},[24941,96229],{"id":37543,"d":37254},[24941,96231],{"id":30062,"d":24944},[24941,96233],{"id":37548,"d":27108},[24941,96235],{"id":30068,"d":24988},[25022,96237,96238],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96239,96240],{"dataMmlNode":25028},[25022,96241,96242,96246],{"dataMmlNode":31565},[25022,96243,96244],{"dataMmlNode":25942},[25033,96245],{"dataC":37310,"xLinkHref":37561},[25022,96247,96248,96252,96256],{"dataMmlNode":27203,"transform":37564,"dataMjxTexclass":27205},[25022,96249,96250],{"dataMmlNode":25031},[25033,96251],{"dataC":25035,"xLinkHref":30124},[25022,96253,96254],{"dataMmlNode":25942,"transform":25040},[25033,96255],{"dataC":27210,"xLinkHref":37573},[25022,96257,96258],{"dataMmlNode":25031,"transform":32577},[25033,96259],{"dataC":25108,"xLinkHref":30134}," 會非常接近 1（代表重要，必須強烈參考）。",[12,96262,96263,96266],{},[106,96264,96265],{},"如果兩間房子差十萬八千里（距離遠）："," 這個函數會讓權重「墜崖式」地急速暴跌到趨近於 0。這等於房仲在說：「那間 80 坪的豪宅對我們這間 30 坪的估價毫無參考價值，直接忽略！」",[12,96268,96269,96270],{},"「參考範圍」的大小：",[24923,96271,96273],{"className":96272,"jax":24927},[24926],[24929,96274,96275,96279],{"style":37459,"xmlns":24932,"width":37460,"height":37461,"role":1930,"focusable":8058,"viewBox":37462,"xmlnsXLink":24936},[24938,96276,96277],{},[24941,96278],{"id":37632,"d":37296},[25022,96280,96281],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96282,96283],{"dataMmlNode":25028},[25022,96284,96285],{"dataMmlNode":25942},[25033,96286],{"dataC":37435,"xLinkHref":37756},[12,96288,94857,96289,96308],{},[24923,96290,96292],{"className":96291,"jax":24927},[24926],[24929,96293,96294,96299],{"style":37459,"xmlns":24932,"width":37460,"height":37461,"role":1930,"focusable":8058,"viewBox":37462,"xmlnsXLink":24936},[24938,96295,96296],{},[24941,96297],{"id":96298,"d":37296},"MJX-56-TEX-I-1D70F",[25022,96300,96301],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96302,96303],{"dataMmlNode":25028},[25022,96304,96305],{"dataMmlNode":25942},[25033,96306],{"dataC":37435,"xLinkHref":96307},"#MJX-56-TEX-I-1D70F","是你可以自己設定的參數，它決定了房仲的「目光有多寬」。",[23,96310,96311,96336],{},[26,96312,96313,96335],{},[106,96314,94891,96315,96334],{},[24923,96316,96318],{"className":96317,"jax":24927},[24926],[24929,96319,96320,96325],{"style":37459,"xmlns":24932,"width":37460,"height":37461,"role":1930,"focusable":8058,"viewBox":37462,"xmlnsXLink":24936},[24938,96321,96322],{},[24941,96323],{"id":96324,"d":37296},"MJX-57-TEX-I-1D70F",[25022,96326,96327],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96328,96329],{"dataMmlNode":25028},[25022,96330,96331],{"dataMmlNode":25942},[25033,96332],{"dataC":37435,"xLinkHref":96333},"#MJX-57-TEX-I-1D70F"," 很大："," 鐘型曲線會很平緩。這代表房仲比較隨和，連 40 坪、50 坪的房子也會加減參考一下。",[26,96337,96338,96360],{},[106,96339,94891,96340,96359],{},[24923,96341,96343],{"className":96342,"jax":24927},[24926],[24929,96344,96345,96350],{"style":37459,"xmlns":24932,"width":37460,"height":37461,"role":1930,"focusable":8058,"viewBox":37462,"xmlnsXLink":24936},[24938,96346,96347],{},[24941,96348],{"id":96349,"d":37296},"MJX-58-TEX-I-1D70F",[25022,96351,96352],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96353,96354],{"dataMmlNode":25028},[25022,96355,96356],{"dataMmlNode":25942},[25033,96357],{"dataC":37435,"xLinkHref":96358},"#MJX-58-TEX-I-1D70F"," 很小："," 鐘型曲線會非常尖銳、極度狹窄。這代表房仲非常嚴格，只肯參考 29.5 坪到 30.5 坪的房子，只要差一點點，權重就直接給 0。",[7166,96362,96364],{"id":96363},"什麼叫無母數-non-parametric","什麼叫「無母數 (non-parametric)」？",[12,96366,96367,96368,96371],{},"一般的線性迴歸，機器只要把 θ0​、θ1​、θ2​ 這三個參數學好，它就可以把那 47 間房子的歷史資料",[106,96369,96370],{},"全部刪掉","了，因為規律已經濃縮在那三個參數裡。",[12,96373,96374,96375,96378,96379,96382],{},"但 LWLR 是「無母數」的，這意味著它",[106,96376,96377],{},"沒有固定的公式參數","。它就像一個不記筆記、全靠臨時抱佛腳的學生。每次有人來問一間新房子的價格，它就必須",[106,96380,96381],{},"把那 47 間歷史資料全部拿出來重新比對一次、重新算一次附近的權重","。雖然計算量變大很多，但它的預測會比一條死板的直線靈活且準確非常多！",[12,96384,96385,96386,96405,96406,96409],{},"參數 ",[24923,96387,96389],{"className":96388,"jax":24927},[24926],[24929,96390,96391,96396],{"style":37459,"xmlns":24932,"width":37460,"height":37461,"role":1930,"focusable":8058,"viewBox":37462,"xmlnsXLink":24936},[24938,96392,96393],{},[24941,96394],{"id":96395,"d":37296},"MJX-59-TEX-I-1D70F",[25022,96397,96398],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96399,96400],{"dataMmlNode":25028},[25022,96401,96402],{"dataMmlNode":25942},[25033,96403],{"dataC":37435,"xLinkHref":96404},"#MJX-59-TEX-I-1D70F"," (tau) 決定了權重隨距離衰減的速度有多快。LWLR 是一種",[106,96407,96408],{},"無母數 (non-parametric)"," 的方法：它沒有一組固定的參數，而是每次有新的查詢點時，都要重新針對附近資料擬合一次。",[6964,96411,96413],{"id":96412},"第三部分邏輯迴歸-logistic-regression-當輸出是離散值","第三部分：邏輯迴歸 (Logistic Regression) — 當輸出是離散值",[12,96415,96416,96417,96434,96435,96454,96455,96458],{},"前面講的都是建立在 ",[24923,96418,96420],{"className":96419,"jax":24927},[24926],[24929,96421,96422,96426],{"style":29805,"xmlns":24932,"width":54359,"height":54360,"role":1930,"focusable":8058,"viewBox":54361,"xmlnsXLink":24936},[24938,96423,96424],{},[24941,96425],{"id":51459,"d":25893},[25022,96427,96428],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96429,96430],{"dataMmlNode":25028},[25022,96431,96432],{"dataMmlNode":25942},[25033,96433],{"dataC":25987,"xLinkHref":51523},"是連續數值的狀況。但如果 ",[24923,96436,96438],{"className":96437,"jax":24927},[24926],[24929,96439,96440,96445],{"style":29805,"xmlns":24932,"width":54359,"height":54360,"role":1930,"focusable":8058,"viewBox":54361,"xmlnsXLink":24936},[24938,96441,96442],{},[24941,96443],{"id":96444,"d":25893},"MJX-61-TEX-I-1D466",[25022,96446,96447],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96448,96449],{"dataMmlNode":25028},[25022,96450,96451],{"dataMmlNode":25942},[25033,96452],{"dataC":25987,"xLinkHref":96453},"#MJX-61-TEX-I-1D466"," 只能是 ",[106,96456,96457],{},"0 或 1"," 呢？（比如判斷垃圾郵件 vs 正常郵件，或是腫瘤是惡性 vs 良性）。",[12,96460,96461,96462,96465],{},"我們沒辦法用單純的線性函數，因為它有可能輸出小於 0 或大於 1 的值。所以，我們要把它包裝進 ",[106,96463,96464],{},"Sigmoid (邏輯) 函數","裡面：",[12,96467,96468],{},[24923,96469,96471],{"className":96470,"jax":24927},[24926],[24929,96472,96475,96510],{"style":39252,"xmlns":24932,"width":96473,"height":39254,"role":1930,"focusable":8058,"viewBox":96474,"xmlnsXLink":24936},"23.767ex","0 -864.9 10504.8 1450.3",[24938,96476,96477,96480,96482,96484,96486,96488,96491,96494,96497,96499,96502,96504,96507],{},[24941,96478],{"id":96479,"d":31530},"MJX-62-TEX-I-210E",[24941,96481],{"id":51767,"d":24944},[24941,96483],{"id":51764,"d":32148},[24941,96485],{"id":51772,"d":24988},[24941,96487],{"id":30830,"d":24972},[24941,96489],{"id":96490,"d":71087},"MJX-62-TEX-I-1D454",[24941,96492],{"id":96493,"d":32951},"MJX-62-TEX-I-1D703",[24941,96495],{"id":96496,"d":38734},"MJX-62-TEX-I-1D447",[24941,96498],{"id":38083,"d":25914},[24941,96500],{"id":96501,"d":30313},"MJX-62-TEX-N-2B",[24941,96503],{"id":38076,"d":27397},[24941,96505],{"id":96506,"d":25910},"MJX-62-TEX-N-2212",[24941,96508],{"id":96509,"d":180},"MJX-62-TEX-N-200B",[25022,96511,96512],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96513,96514,96519,96523,96527,96531,96535,96540,96545,96558,96563,96568,96573,96620],{"dataMmlNode":25028},[25022,96515,96516],{"dataMmlNode":25942},[25033,96517],{"dataC":31539,"xLinkHref":96518},"#MJX-62-TEX-I-210E",[25022,96520,96521],{"dataMmlNode":25031,"transform":32791},[25033,96522],{"dataC":25035,"xLinkHref":51793},[25022,96524,96525],{"dataMmlNode":25942,"transform":32797},[25033,96526],{"dataC":32181,"xLinkHref":51786},[25022,96528,96529],{"dataMmlNode":25031,"transform":32803},[25033,96530],{"dataC":25108,"xLinkHref":51802},[25022,96532,96533],{"dataMmlNode":25031,"transform":32989},[25033,96534],{"dataC":25078,"xLinkHref":30847},[25022,96536,96537],{"dataMmlNode":25942,"transform":32995},[25033,96538],{"dataC":71139,"xLinkHref":96539},"#MJX-62-TEX-I-1D454",[25022,96541,96543],{"dataMmlNode":25031,"transform":96542},"translate(3736.6,0)",[25033,96544],{"dataC":25035,"xLinkHref":51793},[25022,96546,96548,96553],{"dataMmlNode":31565,"transform":96547},"translate(4125.6,0)",[25022,96549,96550],{"dataMmlNode":25942},[25033,96551],{"dataC":33000,"xLinkHref":96552},"#MJX-62-TEX-I-1D703",[25022,96554,96555],{"dataMmlNode":25942,"transform":39416},[25033,96556],{"dataC":38825,"xLinkHref":96557},"#MJX-62-TEX-I-1D447",[25022,96559,96561],{"dataMmlNode":25942,"transform":96560},"translate(5175.4,0)",[25033,96562],{"dataC":32181,"xLinkHref":51786},[25022,96564,96566],{"dataMmlNode":25031,"transform":96565},"translate(5747.4,0)",[25033,96567],{"dataC":25108,"xLinkHref":51802},[25022,96569,96571],{"dataMmlNode":25031,"transform":96570},"translate(6414.1,0)",[25033,96572],{"dataC":25078,"xLinkHref":30847},[25022,96574,96576,96580,96618],{"dataMmlNode":27622,"transform":96575},"translate(7469.9,0)",[25022,96577,96578],{"dataMmlNode":26031,"transform":39331},[25033,96579],{"dataC":26035,"xLinkHref":38110},[25022,96581,96582,96586,96591],{"dataMmlNode":27626,"transform":39337},[25022,96583,96584],{"dataMmlNode":26031},[25033,96585],{"dataC":26035,"xLinkHref":38110},[25022,96587,96588],{"dataMmlNode":25031,"transform":26394},[25033,96589],{"dataC":30393,"xLinkHref":96590},"#MJX-62-TEX-N-2B",[25022,96592,96593,96597],{"dataMmlNode":31565,"transform":26835},[25022,96594,96595],{"dataMmlNode":25942},[25033,96596],{"dataC":27596,"xLinkHref":38094},[25022,96598,96599,96604,96614],{"dataMmlNode":27203,"transform":38808,"dataMjxTexclass":27205},[25022,96600,96601],{"dataMmlNode":25031},[25033,96602],{"dataC":26019,"xLinkHref":96603},"#MJX-62-TEX-N-2212",[25022,96605,96606,96610],{"dataMmlNode":31565,"transform":25049},[25022,96607,96608],{"dataMmlNode":25942},[25033,96609],{"dataC":33000,"xLinkHref":96552},[25022,96611,96612],{"dataMmlNode":25942,"transform":39369},[25033,96613],{"dataC":38825,"xLinkHref":96557},[25022,96615,96616],{"dataMmlNode":25942,"transform":39375},[25033,96617],{"dataC":32181,"xLinkHref":51786},[27668,96619],{"width":39380,"height":27671,"x":27672,"y":27673},[25022,96621,96623],{"dataMmlNode":27203,"dataMjxTexclass":27205,"transform":96622},"translate(10504.8,0)",[25022,96624,96625],{"dataMmlNode":25031},[25033,96626],{"dataC":44737,"xLinkHref":96627},"#MJX-62-TEX-N-200B",[12,96629,96630],{},"Sigmoid 函數可以把任何實數壓縮到 (0,1) 的範圍內，這樣我們就可以把它解釋成「機率」。如果 h(x)≥0.5，我們就預測 y=1；否則就預測 y=0。",[25631,96632,96634,96636],{"className":96633},[13441,13442],[25635,96635,87175],{},[13438,96637,96639,96766,96769],{"className":96638},[25641],[12,96640,96641,96642,96679,96680,96765],{},"**",[24923,96643,96645],{"className":96644,"jax":24927},[24926],[24929,96646,96647,96658],{"style":25196,"xmlns":24932,"width":39389,"height":39390,"role":1930,"focusable":8058,"viewBox":39391,"xmlnsXLink":24936},[24938,96648,96649,96652,96655],{},[24941,96650],{"id":96651,"d":32951},"MJX-63-TEX-I-1D703",[24941,96653],{"id":96654,"d":38734},"MJX-63-TEX-I-1D447",[24941,96656],{"id":96657,"d":32148},"MJX-63-TEX-I-1D465",[25022,96659,96660],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96661,96662,96674],{"dataMmlNode":25028},[25022,96663,96664,96669],{"dataMmlNode":31565},[25022,96665,96666],{"dataMmlNode":25942},[25033,96667],{"dataC":33000,"xLinkHref":96668},"#MJX-63-TEX-I-1D703",[25022,96670,96671],{"dataMmlNode":25942,"transform":39416},[25033,96672],{"dataC":38825,"xLinkHref":96673},"#MJX-63-TEX-I-1D447",[25022,96675,96676],{"dataMmlNode":25942,"transform":39422},[25033,96677],{"dataC":32181,"xLinkHref":96678},"#MJX-63-TEX-I-1D465","是什麼？**其實就是我們上面學到的計價公式的縮寫版（也就是 ",[24923,96681,96683],{"className":96682,"jax":24927},[24926],[24929,96684,96688,96703],{"style":61403,"xmlns":24932,"width":96685,"height":96686,"role":1930,"focusable":8058,"viewBox":96687,"xmlnsXLink":24936},"13.546ex","1.97ex","0 -705 5987.4 870.6",[24938,96689,96690,96693,96695,96697,96699,96701],{},[24941,96691],{"id":96692,"d":32951},"MJX-64-TEX-I-1D703",[24941,96694],{"id":31323,"d":25935},[24941,96696],{"id":51902,"d":30313},[24941,96698],{"id":31332,"d":25914},[24941,96700],{"id":51893,"d":32148},[24941,96702],{"id":31326,"d":26357},[25022,96704,96705],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96706,96707,96718,96723,96734,96745,96750,96755,96760],{"dataMmlNode":25028},[25022,96708,96709,96714],{"dataMmlNode":26023},[25022,96710,96711],{"dataMmlNode":25942},[25033,96712],{"dataC":33000,"xLinkHref":96713},"#MJX-64-TEX-I-1D703",[25022,96715,96716],{"dataMmlNode":26031,"transform":33004},[25033,96717],{"dataC":26103,"xLinkHref":31424},[25022,96719,96721],{"dataMmlNode":25031,"transform":96720},"translate(1127.8,0)",[25033,96722],{"dataC":30393,"xLinkHref":51955},[25022,96724,96726,96730],{"dataMmlNode":26023,"transform":96725},"translate(2128,0)",[25022,96727,96728],{"dataMmlNode":25942},[25033,96729],{"dataC":33000,"xLinkHref":96713},[25022,96731,96732],{"dataMmlNode":26031,"transform":33004},[25033,96733],{"dataC":26035,"xLinkHref":31435},[25022,96735,96737,96741],{"dataMmlNode":26023,"transform":96736},"translate(3033.6,0)",[25022,96738,96739],{"dataMmlNode":25942},[25033,96740],{"dataC":32181,"xLinkHref":51926},[25022,96742,96743],{"dataMmlNode":26031,"transform":33034},[25033,96744],{"dataC":26035,"xLinkHref":31435},[25022,96746,96748],{"dataMmlNode":25031,"transform":96747},"translate(4042.1,0)",[25033,96749],{"dataC":30393,"xLinkHref":51955},[25022,96751,96753],{"dataMmlNode":25031,"transform":96752},"translate(4820.1,0)",[25033,96754],{"dataC":26392,"xLinkHref":31427},[25022,96756,96758],{"dataMmlNode":25031,"transform":96757},"translate(5264.8,0)",[25033,96759],{"dataC":26392,"xLinkHref":31427},[25022,96761,96763],{"dataMmlNode":25031,"transform":96762},"translate(5709.4,0)",[25033,96764],{"dataC":26392,"xLinkHref":31427}," 算出來的那個直線數值）。",[12,96767,96768],{},"它就像一台「數值壓縮機」。無論你把多大或多小的數字（哪怕是 +10000 或 −10000）丟進這個公式裡，它都會把它「擠壓」成一個介於 0 到 1 之間的小數。",[12,96770,96771,96774],{},[106,96772,96773],{},"為什麼要壓在 0 到 1 之間？"," 因為這正好就是「機率」！ 如果公式算出來是 0.85，機器就是在告訴你：「我有 85% 的把握這是一封垃圾郵件（y=1）。」既然大於 0.5（50%），我們就大膽預測它是垃圾郵件。",[7166,96776,96777],{"id":96777},"擬合邏輯迴歸",[12,96779,96780,96781,96784],{},"這次我們不用最小化平方誤差了，而是改用",[106,96782,96783],{},"最大概似估計 (Maximum Likelihood Estimation, MLE)","。我們的假設是：",[23,96786,96787,96904],{},[26,96788,96789],{},[24923,96790,96792],{"className":96791,"jax":24927},[24926],[24929,96793,96796,96825],{"style":24931,"xmlns":24932,"width":96794,"height":24934,"role":1930,"focusable":8058,"viewBox":96795,"xmlnsXLink":24936},"21.338ex","0 -750 9431.3 1000",[24938,96797,96798,96801,96803,96806,96808,96810,96813,96815,96818,96821,96823],{},[24941,96799],{"id":96800,"d":39730},"MJX-65-TEX-I-1D443",[24941,96802],{"id":51996,"d":24944},[24941,96804],{"id":96805,"d":25893},"MJX-65-TEX-I-1D466",[24941,96807],{"id":52005,"d":24972},[24941,96809],{"id":52008,"d":25914},[24941,96811],{"id":96812,"d":27405},"MJX-65-TEX-N-2223",[24941,96814],{"id":51999,"d":32148},[24941,96816],{"id":96817,"d":39752},"MJX-65-TEX-N-3B",[24941,96819],{"id":96820,"d":32951},"MJX-65-TEX-I-1D703",[24941,96822],{"id":52002,"d":24988},[24941,96824],{"id":31529,"d":31530},[25022,96826,96827],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,96828,96829,96834,96838,96843,96847,96851,96857,96862,96868,96874,96879,96884,96889,96894,96899],{"dataMmlNode":25028},[25022,96830,96831],{"dataMmlNode":25942},[25033,96832],{"dataC":39770,"xLinkHref":96833},"#MJX-65-TEX-I-1D443",[25022,96835,96836],{"dataMmlNode":25031,"transform":39774},[25033,96837],{"dataC":25035,"xLinkHref":52034},[25022,96839,96840],{"dataMmlNode":25942,"transform":39780},[25033,96841],{"dataC":25987,"xLinkHref":96842},"#MJX-65-TEX-I-1D466",[25022,96844,96845],{"dataMmlNode":25031,"transform":39786},[25033,96846],{"dataC":25078,"xLinkHref":52052},[25022,96848,96849],{"dataMmlNode":26031,"transform":39792},[25033,96850],{"dataC":26035,"xLinkHref":52083},[25022,96852,96854],{"dataMmlNode":25031,"transform":96853},"translate(3741.3,0)",[25033,96855],{"dataC":49171,"xLinkHref":96856},"#MJX-65-TEX-N-2223",[25022,96858,96860],{"dataMmlNode":25942,"transform":96859},"translate(4297.1,0)",[25033,96861],{"dataC":32181,"xLinkHref":52040},[25022,96863,96865],{"dataMmlNode":25031,"transform":96864},"translate(4869.1,0)",[25033,96866],{"dataC":39813,"xLinkHref":96867},"#MJX-65-TEX-N-3B",[25022,96869,96871],{"dataMmlNode":25942,"transform":96870},"translate(5313.8,0)",[25033,96872],{"dataC":33000,"xLinkHref":96873},"#MJX-65-TEX-I-1D703",[25022,96875,96877],{"dataMmlNode":25031,"transform":96876},"translate(5782.8,0)",[25033,96878],{"dataC":25108,"xLinkHref":52046},[25022,96880,96882],{"dataMmlNode":25031,"transform":96881},"translate(6449.6,0)",[25033,96883],{"dataC":25078,"xLinkHref":52052},[25022,96885,96887],{"dataMmlNode":25942,"transform":96886},"translate(7505.3,0)",[25033,96888],{"dataC":31539,"xLinkHref":31540},[25022,96890,96892],{"dataMmlNode":25031,"transform":96891},"translate(8081.3,0)",[25033,96893],{"dataC":25035,"xLinkHref":52034},[25022,96895,96897],{"dataMmlNode":25942,"transform":96896},"translate(8470.3,0)",[25033,96898],{"dataC":32181,"xLinkHref":52040},[25022,96900,96902],{"dataMmlNode":25031,"transform":96901},"translate(9042.3,0)",[25033,96903],{"dataC":25108,"xLinkHref":52046},[26,96905,96906],{},[24923,96907,96909],{"className":96908,"jax":24927},[24926],[24929,96910,96913,96947],{"style":24931,"xmlns":24932,"width":96911,"height":24934,"role":1930,"focusable":8058,"viewBox":96912,"xmlnsXLink":24936},"25.235ex","0 -750 11153.8 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第一個公式：預測「會發生」的機率",[13438,97037,97039,97146,97151,97156,97162,97166,97289,97293,97298,97307],{"className":97038},[25641],[12,97040,97041],{},[24923,97042,97044],{"className":97043,"jax":24927},[24926],[24929,97045,97046,97076],{"style":24931,"xmlns":24932,"width":96794,"height":24934,"role":1930,"focusable":8058,"viewBox":96795,"xmlnsXLink":24936},[24938,97047,97048,97051,97053,97056,97058,97060,97063,97065,97068,97071,97073],{},[24941,97049],{"id":97050,"d":39730},"MJX-67-TEX-I-1D443",[24941,97052],{"id":52264,"d":24944},[24941,97054],{"id":97055,"d":25893},"MJX-67-TEX-I-1D466",[24941,97057],{"id":31605,"d":24972},[24941,97059],{"id":52278,"d":25914},[24941,97061],{"id":97062,"d":27405},"MJX-67-TEX-N-2223",[24941,97064],{"id":52267,"d":32148},[24941,97066],{"id":97067,"d":39752},"MJX-67-TEX-N-3B",[24941,97069],{"id":97070,"d":32951},"MJX-67-TEX-I-1D703",[24941,97072],{"id":52270,"d":24988},[24941,97074],{"id":97075,"d":31530},"MJX-67-TEX-I-210E",[25022,97077,97078],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,97079,97080,97085,97089,97094,97098,97102,97107,97111,97116,97121,97125,97129,97134,97138,97142],{"dataMmlNode":25028},[25022,97081,97082],{"dataMmlNode":25942},[25033,97083],{"dataC":39770,"xLinkHref":97084},"#MJX-67-TEX-I-1D443",[25022,97086,97087],{"dataMmlNode":25031,"transform":39774},[25033,97088],{"dataC":25035,"xLinkHref":52314},[25022,97090,97091],{"dataMmlNode":25942,"transform":39780},[25033,97092],{"dataC":25987,"xLinkHref":97093},"#MJX-67-TEX-I-1D466",[25022,97095,97096],{"dataMmlNode":25031,"transform":39786},[25033,97097],{"dataC":25078,"xLinkHref":31627},[25022,97099,97100],{"dataMmlNode":26031,"transform":39792},[25033,97101],{"dataC":26035,"xLinkHref":52339},[25022,97103,97104],{"dataMmlNode":25031,"transform":96853},[25033,97105],{"dataC":49171,"xLinkHref":97106},"#MJX-67-TEX-N-2223",[25022,97108,97109],{"dataMmlNode":25942,"transform":96859},[25033,97110],{"dataC":32181,"xLinkHref":52319},[25022,97112,97113],{"dataMmlNode":25031,"transform":96864},[25033,97114],{"dataC":39813,"xLinkHref":97115},"#MJX-67-TEX-N-3B",[25022,97117,97118],{"dataMmlNode":25942,"transform":96870},[25033,97119],{"dataC":33000,"xLinkHref":97120},"#MJX-67-TEX-I-1D703",[25022,97122,97123],{"dataMmlNode":25031,"transform":96876},[25033,97124],{"dataC":25108,"xLinkHref":52324},[25022,97126,97127],{"dataMmlNode":25031,"transform":96881},[25033,97128],{"dataC":25078,"xLinkHref":31627},[25022,97130,97131],{"dataMmlNode":25942,"transform":96886},[25033,97132],{"dataC":31539,"xLinkHref":97133},"#MJX-67-TEX-I-210E",[25022,97135,97136],{"dataMmlNode":25031,"transform":96891},[25033,97137],{"dataC":25035,"xLinkHref":52314},[25022,97139,97140],{"dataMmlNode":25942,"transform":96896},[25033,97141],{"dataC":32181,"xLinkHref":52319},[25022,97143,97144],{"dataMmlNode":25031,"transform":96901},[25033,97145],{"dataC":25108,"xLinkHref":52324},[12,97147,97148],{},[106,97149,97150],{},"白話文翻譯：",[7351,97152,97153],{},[12,97154,97155],{},"「在我們已知這筆資料 (x) 和現在的模型標準 (θ) 的條件下，這件事**『會發生 (y=1)』**的機率是多少？」",[12,97157,97158,97161],{},[106,97159,97160],{},"答案："," 就是我們的魔法壓縮機 h(x) 算出來的數字！ 例如：模型算出來 h(x)=0.8，這行公式就是在說：「這封信是垃圾郵件的機率是 80%。」",[7166,97163,97165],{"id":97164},"_2-第二個公式預測不會發生的機率","2. 第二個公式：預測「不會發生」的機率",[12,97167,97168],{},[24923,97169,97171],{"className":97170,"jax":24927},[24926],[24929,97172,97173,97209],{"style":24931,"xmlns":24932,"width":96911,"height":24934,"role":1930,"focusable":8058,"viewBox":96912,"xmlnsXLink":24936},[24938,97174,97175,97178,97180,97183,97186,97189,97192,97194,97197,97200,97202,97204,97206],{},[24941,97176],{"id":97177,"d":39730},"MJX-68-TEX-I-1D443",[24941,97179],{"id":31743,"d":24944},[24941,97181],{"id":97182,"d":25893},"MJX-68-TEX-I-1D466",[24941,97184],{"id":97185,"d":24972},"MJX-68-TEX-N-3D",[24941,97187],{"id":97188,"d":25935},"MJX-68-TEX-N-30",[24941,97190],{"id":97191,"d":27405},"MJX-68-TEX-N-2223",[24941,97193],{"id":38288,"d":32148},[24941,97195],{"id":97196,"d":39752},"MJX-68-TEX-N-3B",[24941,97198],{"id":97199,"d":32951},"MJX-68-TEX-I-1D703",[24941,97201],{"id":31755,"d":24988},[24941,97203],{"id":52481,"d":25914},[24941,97205],{"id":52490,"d":25910},[24941,97207],{"id":97208,"d":31530},"MJX-68-TEX-I-210E",[25022,97210,97211],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,97212,97213,97218,97222,97227,97232,97237,97242,97246,97251,97256,97260,97264,97268,97272,97277,97281,97285],{"dataMmlNode":25028},[25022,97214,97215],{"dataMmlNode":25942},[25033,97216],{"dataC":39770,"xLinkHref":97217},"#MJX-68-TEX-I-1D443",[25022,97219,97220],{"dataMmlNode":25031,"transform":39774},[25033,97221],{"dataC":25035,"xLinkHref":31771},[25022,97223,97224],{"dataMmlNode":25942,"transform":39780},[25033,97225],{"dataC":25987,"xLinkHref":97226},"#MJX-68-TEX-I-1D466",[25022,97228,97229],{"dataMmlNode":25031,"transform":39786},[25033,97230],{"dataC":25078,"xLinkHref":97231},"#MJX-68-TEX-N-3D",[25022,97233,97234],{"dataMmlNode":26031,"transform":39792},[25033,97235],{"dataC":26103,"xLinkHref":97236},"#MJX-68-TEX-N-30",[25022,97238,97239],{"dataMmlNode":25031,"transform":96853},[25033,97240],{"dataC":49171,"xLinkHref":97241},"#MJX-68-TEX-N-2223",[25022,97243,97244],{"dataMmlNode":25942,"transform":96859},[25033,97245],{"dataC":32181,"xLinkHref":38297},[25022,97247,97248],{"dataMmlNode":25031,"transform":96864},[25033,97249],{"dataC":39813,"xLinkHref":97250},"#MJX-68-TEX-N-3B",[25022,97252,97253],{"dataMmlNode":25942,"transform":96870},[25033,97254],{"dataC":33000,"xLinkHref":97255},"#MJX-68-TEX-I-1D703",[25022,97257,97258],{"dataMmlNode":25031,"transform":96876},[25033,97259],{"dataC":25108,"xLinkHref":31796},[25022,97261,97262],{"dataMmlNode":25031,"transform":96881},[25033,97263],{"dataC":25078,"xLinkHref":97231},[25022,97265,97266],{"dataMmlNode":26031,"transform":96886},[25033,97267],{"dataC":26035,"xLinkHref":52507},[25022,97269,97270],{"dataMmlNode":25031,"transform":97007},[25033,97271],{"dataC":26019,"xLinkHref":52526},[25022,97273,97274],{"dataMmlNode":25942,"transform":97012},[25033,97275],{"dataC":31539,"xLinkHref":97276},"#MJX-68-TEX-I-210E",[25022,97278,97279],{"dataMmlNode":25031,"transform":97017},[25033,97280],{"dataC":25035,"xLinkHref":31771},[25022,97282,97283],{"dataMmlNode":25942,"transform":97022},[25033,97284],{"dataC":32181,"xLinkHref":38297},[25022,97286,97287],{"dataMmlNode":25031,"transform":97027},[25033,97288],{"dataC":25108,"xLinkHref":31796},[12,97290,97291],{},[106,97292,97150],{},[7351,97294,97295],{},[12,97296,97297],{},"「在同樣的條件下，這件事**『不會發生 (y=0)』**的機率是多少？」",[12,97299,97300,97302,97303,97306],{},[106,97301,97160],{}," 因為結果只有「是」跟「否」兩種可能，既然全部的機率加起來必須是 100%（也就是數字 1），那麼「不會發生」的機率，當然就是 ",[106,97304,97305],{},"1 減掉「會發生」的機率","！",[12,97308,97309],{},"延續上面的例子：既然是垃圾郵件的機率是 80% (0.8)，那它是正常郵件的機率就是 1−0.8=0.2 (也就是 20%)。",[12,97311,97312],{},"它的對數概似函數 (Log-likelihood) 長這樣（推導過程省略）（應該也沒什麼用，看看就好）：",[12,97314,97315],{},[24923,97316,97318],{"className":97317,"jax":24927},[24926],[24929,97319,97324,97383],{"style":97320,"xmlns":24932,"width":97321,"height":97322,"role":1930,"focusable":8058,"viewBox":97323,"xmlnsXLink":24936},"vertical-align: -0.791ex;","52.038ex","2.812ex","0 -893.3 23000.8 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(MLE)"," 的目標，就是要「最大化」這個函數。白話來說，就是想盡辦法讓機器「對正確答案充滿自信」。",[12,98240,98241,98242,98245,98246,98249],{},"我們要想辦法",[106,98243,98244],{},"最大化","這個函數，用的方法是——你猜對了——",[106,98247,98248],{},"梯度下降","。算出來的更新規則跟線性迴歸的簡直像得出奇：",[12,98251,98252],{},[1930,98253],{"alt":1932,"src":98254},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775990017729-Screenshot_2026-04-12_at_6.33.22_PM.png",[12,98256,98257],{},[24923,98258,98260],{"className":98259,"jax":24927},[24926],[24929,98261,98262,98302],{"style":33830,"xmlns":24932,"width":33831,"height":33832,"role":1930,"focusable":8058,"viewBox":33833,"xmlnsXLink":24936},[24938,98263,98264,98267,98270,98273,98276,98279,98282,98285,98288,98291,98294,98297,98300],{},[24941,98265],{"id":98266,"d":32951},"MJX-84-TEX-I-1D703",[24941,98268],{"id":98269,"d":33606},"MJX-84-TEX-I-1D457",[24941,98271],{"id":98272,"d":33610},"MJX-84-TEX-N-3A",[24941,98274],{"id":98275,"d":24972},"MJX-84-TEX-N-3D",[24941,98277],{"id":98278,"d":30313},"MJX-84-TEX-N-2B",[24941,98280],{"id":98281,"d":33620},"MJX-84-TEX-I-1D6FC",[24941,98283],{"id":98284,"d":24944},"MJX-84-TEX-N-28",[24941,98286],{"id":98287,"d":25893},"MJX-84-TEX-I-1D466",[24941,98289],{"id":98290,"d":27108},"MJX-84-TEX-I-1D456",[24941,98292],{"id":98293,"d":24988},"MJX-84-TEX-N-29",[24941,98295],{"id":98296,"d":25910},"MJX-84-TEX-N-2212",[24941,98298],{"id":98299,"d":31530},"MJX-84-TEX-I-210E",[24941,98301],{"id":39012,"d":32148},[25022,98303,98304],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,98305,98306,98318,98330,98340,98345,98350,98355,98378,98383,98388,98392,98412,98416,98420],{"dataMmlNode":25028},[25022,98307,98308,98313],{"dataMmlNode":26023},[25022,98309,98310],{"dataMmlNode":25942},[25033,98311],{"dataC":33000,"xLinkHref":98312},"#MJX-84-TEX-I-1D703",[25022,98314,98315],{"dataMmlNode":25942,"transform":33004},[25033,98316],{"dataC":33647,"xLinkHref":98317},"#MJX-84-TEX-I-1D457",[25022,98319,98320,98325],{"dataMmlNode":25031,"transform":33651},[25022,98321,98322],{"dataMmlNode":6977},[25033,98323],{"dataC":33656,"xLinkHref":98324},"#MJX-84-TEX-N-3A",[25022,98326,98327],{"dataMmlNode":6977,"transform":26048},[25033,98328],{"dataC":25078,"xLinkHref":98329},"#MJX-84-TEX-N-3D",[25022,98331,98332,98336],{"dataMmlNode":26023,"transform":33665},[25022,98333,98334],{"dataMmlNode":25942},[25033,98335],{"dataC":33000,"xLinkHref":98312},[25022,98337,98338],{"dataMmlNode":25942,"transform":33004},[25033,98339],{"dataC":33647,"xLinkHref":98317},[25022,98341,98342],{"dataMmlNode":25031,"transform":33676},[25033,98343],{"dataC":30393,"xLinkHref":98344},"#MJX-84-TEX-N-2B",[25022,98346,98347],{"dataMmlNode":25942,"transform":33682},[25033,98348],{"dataC":33685,"xLinkHref":98349},"#MJX-84-TEX-I-1D6FC",[25022,98351,98352],{"dataMmlNode":25031,"transform":33689},[25033,98353],{"dataC":25035,"xLinkHref":98354},"#MJX-84-TEX-N-28",[25022,98356,98357,98362],{"dataMmlNode":31565,"transform":33922},[25022,98358,98359],{"dataMmlNode":25942},[25033,98360],{"dataC":25987,"xLinkHref":98361},"#MJX-84-TEX-I-1D466",[25022,98363,98364,98368,98373],{"dataMmlNode":27203,"transform":32620,"dataMjxTexclass":27205},[25022,98365,98366],{"dataMmlNode":25031},[25033,98367],{"dataC":25035,"xLinkHref":98354},[25022,98369,98370],{"dataMmlNode":25942,"transform":25040},[25033,98371],{"dataC":27210,"xLinkHref":98372},"#MJX-84-TEX-I-1D456",[25022,98374,98375],{"dataMmlNode":25031,"transform":32577},[25033,98376],{"dataC":25108,"xLinkHref":98377},"#MJX-84-TEX-N-29",[25022,98379,98380],{"dataMmlNode":25031,"transform":33943},[25033,98381],{"dataC":26019,"xLinkHref":98382},"#MJX-84-TEX-N-2212",[25022,98384,98385],{"dataMmlNode":25942,"transform":33948},[25033,98386],{"dataC":31539,"xLinkHref":98387},"#MJX-84-TEX-I-210E",[25022,98389,98390],{"dataMmlNode":25031,"transform":33954},[25033,98391],{"dataC":25035,"xLinkHref":98354},[25022,98393,98394,98398],{"dataMmlNode":31565,"transform":33959},[25022,98395,98396],{"dataMmlNode":25942},[25033,98397],{"dataC":32181,"xLinkHref":39021},[25022,98399,98400,98404,98408],{"dataMmlNode":27203,"transform":32565,"dataMjxTexclass":27205},[25022,98401,98402],{"dataMmlNode":25031},[25033,98403],{"dataC":25035,"xLinkHref":98354},[25022,98405,98406],{"dataMmlNode":25942,"transform":25040},[25033,98407],{"dataC":27210,"xLinkHref":98372},[25022,98409,98410],{"dataMmlNode":25031,"transform":32577},[25033,98411],{"dataC":25108,"xLinkHref":98377},[25022,98413,98414],{"dataMmlNode":25031,"transform":33981},[25033,98415],{"dataC":25108,"xLinkHref":98377},[25022,98417,98418],{"dataMmlNode":25031,"transform":33986},[25033,98419],{"dataC":25108,"xLinkHref":98377},[25022,98421,98422,98426,98440],{"dataMmlNode":33991,"transform":33992},[25022,98423,98424],{"dataMmlNode":25942},[25033,98425],{"dataC":32181,"xLinkHref":39021},[25022,98427,98428,98432,98436],{"dataMmlNode":27203,"transform":33999,"dataMjxTexclass":27205},[25022,98429,98430],{"dataMmlNode":25031},[25033,98431],{"dataC":25035,"xLinkHref":98354},[25022,98433,98434],{"dataMmlNode":25942,"transform":25040},[25033,98435],{"dataC":27210,"xLinkHref":98372},[25022,98437,98438],{"dataMmlNode":25031,"transform":32577},[25033,98439],{"dataC":25108,"xLinkHref":98377},[25022,98441,98442],{"dataMmlNode":25942,"transform":34014},[25033,98443],{"dataC":33647,"xLinkHref":98317},[12,98445,98446,98447,98490],{},"公式長得一模一樣，但別忘了，這裡的 ",[24923,98448,98450],{"className":98449,"jax":24927},[24926],[24929,98451,98452,98466],{"style":24931,"xmlns":24932,"width":32764,"height":24934,"role":1930,"focusable":8058,"viewBox":32765,"xmlnsXLink":24936},[24938,98453,98454,98457,98460,98463],{},[24941,98455],{"id":98456,"d":31530},"MJX-85-TEX-I-210E",[24941,98458],{"id":98459,"d":24944},"MJX-85-TEX-N-28",[24941,98461],{"id":98462,"d":32148},"MJX-85-TEX-I-1D465",[24941,98464],{"id":98465,"d":24988},"MJX-85-TEX-N-29",[25022,98467,98468],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,98469,98470,98475,98480,98485],{"dataMmlNode":25028},[25022,98471,98472],{"dataMmlNode":25942},[25033,98473],{"dataC":31539,"xLinkHref":98474},"#MJX-85-TEX-I-210E",[25022,98476,98477],{"dataMmlNode":25031,"transform":32791},[25033,98478],{"dataC":25035,"xLinkHref":98479},"#MJX-85-TEX-N-28",[25022,98481,98482],{"dataMmlNode":25942,"transform":32797},[25033,98483],{"dataC":32181,"xLinkHref":98484},"#MJX-85-TEX-I-1D465",[25022,98486,98487],{"dataMmlNode":25031,"transform":32803},[25033,98488],{"dataC":25108,"xLinkHref":98489},"#MJX-85-TEX-N-29"," 已經是 Sigmoid 函數，不再是線性函數囉！（第三部分重點差不多就是這個了，上面推導過程應該不重要）",[6964,98492,98493],{"id":98493},"總結",[7085,98495,98496,98506],{},[7088,98497,98498],{},[7091,98499,98500,98503],{},[7094,98501,98502],{"align":47883},"主題",[7094,98504,98505],{"align":47883},"核心概念",[7104,98507,98508,98518,98528,98538,98547,98557,98567,98577],{},[7091,98509,98510,98515],{},[7109,98511,98512],{"align":47883},[106,98513,98514],{},"資料前處理",[7109,98516,98517],{"align":47883},"在訓練模型前，清理並轉換原始資料。",[7091,98519,98520,98525],{},[7109,98521,98522],{"align":47883},[106,98523,98524],{},"線性迴歸",[7109,98526,98527],{"align":47883},"用線性模型來預測連續的數值。",[7091,98529,98530,98535],{},[7109,98531,98532],{"align":47883},[106,98533,98534],{},"成本函數 (J)",[7109,98536,98537],{"align":47883},"衡量我們的預測錯得有多離譜。",[7091,98539,98540,98544],{},[7109,98541,98542],{"align":47883},[106,98543,98248],{},[7109,98545,98546],{"align":47883},"透過迭代調整參數，想辦法把成本降到最低。",[7091,98548,98549,98554],{},[7109,98550,98551],{"align":47883},[106,98552,98553],{},"批量 vs 隨機 GD",[7109,98555,98556],{"align":47883},"看完所有樣本才更新 vs 看完一筆樣本就更新。",[7091,98558,98559,98564],{},[7109,98560,98561],{"align":47883},[106,98562,98563],{},"欠擬合 \u002F 過擬合",[7109,98565,98566],{"align":47883},"模型太簡單抓不到規律 vs 模型太複雜只會死背。",[7091,98568,98569,98574],{},[7109,98570,98571],{"align":47883},[106,98572,98573],{},"局部加權 LR",[7109,98575,98576],{"align":47883},"預測時，給距離近的資料點比較大的影響力。",[7091,98578,98579,98584],{},[7109,98580,98581],{"align":47883},[106,98582,98583],{},"邏輯迴歸",[7109,98585,98586],{"align":47883},"使用 Sigmoid 函數來分類離散的結果 (如 0 或 1)。",[9332,98588,31966],{},{"title":180,"searchDepth":181,"depth":181,"links":98590},[98591,98595,98605,98608],{"id":92765,"depth":181,"text":92766,"children":98592},[98593,98594],{"id":92779,"depth":7430,"text":92780},{"id":93075,"depth":7430,"text":93075},{"id":93101,"depth":181,"text":93102,"children":98596},[98597,98598,98599,98600,98601,98602,98603,98604],{"id":93108,"depth":7430,"text":93108},{"id":93163,"depth":7430,"text":93163},{"id":93516,"depth":7430,"text":93517},{"id":93693,"depth":7430,"text":93694},{"id":94339,"depth":7430,"text":94340},{"id":95731,"depth":7430,"text":95732},{"id":95783,"depth":7430,"text":95784},{"id":95829,"depth":7430,"text":95830},{"id":96412,"depth":181,"text":96413,"children":98606},[98607],{"id":96777,"depth":7430,"text":96777},{"id":98493,"depth":181,"text":98493},{},"\u002Fblog\u002F資料前處理與迴歸分析",{"title":92760,"description":180},{"loc":98610},"blog\u002F資料前處理與迴歸分析","YU-EoYC5o7fpA80jQhf2vxD4LYAYKTpjFb7bZhY4Tm4",{"id":98616,"title":98617,"author":7,"body":98618,"date":98838,"description":98839,"draft":185,"edited_at":83251,"extension":186,"featured_image":98840,"meta":98841,"navigation":189,"path":98842,"pinned":185,"seo":98843,"sitemap":98844,"stem":98845,"tags":194,"__hash__":98846},"blog\u002Fblog\u002FMotor-Learning小兒（二）.md","Motor Learning小兒（二）",{"type":9,"value":98619,"toc":98836},[98620,98623,98629,98638,98641,98675,98678],[12,98621,98622],{},"阿對的，又要來講動作學習的定義了，雖然之前就有講過",[12,98624,98625,98626,98628],{},"那動作學習是一個過程，這個過程是借由學習或者經驗產生相當永久的動作行為(long-term effect)",[16,98627],{},"\n動作學習就是過程，而動作表現是動作做出來的結果。",[12,98630,98631,98632,98634,98635,98637],{},"而這邊講道Positive和Negative Sign",[16,98633],{},"\nPositive是指正常人沒有，但病人身上會出現的異常動作 ",[16,98636],{},"\nNegative則相反，正常人有的，但病人身上無法出現",[12,98639,98640],{},"對於動作學習有三大要素",[3603,98642,98643,98646,98655],{},[26,98644,98645],{},"偵測練習——初期PT可以直接監測教導，而後讓病人自己監測自己動作，並且利用內在回饋方式，有利於long-term effect",[26,98647,98648,98649,98651,98652,98654],{},"動機——那對於動作的學習，肯定是動機最為重要，這樣能讓學習過程變佳，那有以下幾點可以引發動機",[16,98650],{},"\n利用喜歡的東西作為目標去做、任務是有趣的、從容易的去做起讓他們有成熟感激發動機",[16,98653],{},"\n同儕競爭互相鼓勵、最後才是給予獎勵作為誘因激發動機",[26,98656,98657,98658,98662,98663,98665,98666,98670,98671],{},"對於目標的清淅程度——就是",[75,98659,98661],{"className":98660},[173],"任務具體","並且病人不會理解錯，像是抬高手拿東西而不是單純抬高手",[16,98664],{},"\n而任務",[75,98667,98669],{"className":98668},[79],"越具體","，更能",[75,98672,98674],{"className":98673},[79,78],"引導寶寶產生興趣",[12,98676,98677],{},"那動作學習有以下三個考量點",[3603,98679,98680,98729,98775],{},[26,98681,98682,98683,98687,98688,98692,98693,7963,98698,98700,98701,98703,98704,98708,98709,98711,98712,98715,98716,98720,98721,98723,98724,98728],{},"轉移——意思是",[75,98684,98686],{"className":98685},[78,173],"相同相似的技巧","在不同情景下可以變化",[75,98689,98691],{"className":98690},[79,78],"產生新的","動作型態 (這裡也有說到 ",[2330,98694,98697],{"href":98695,"rel":98696},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E5%8B%95%E4%BD%9C%E5%AD%B8%E7%BF%92%EF%BC%88%E5%85%AB%EF%BC%89%E7%B7%B4%E7%BF%92%E6%96%B9%E5%BC%8F-2?series=Motor%20learning",[2334],"在最後一段",[16,98699],{},"\n栗子：一樣是站起來，從椅子上站起來與亞洲蹲的情況下站起來，這兩者都是Hip和Knee的Extend才能達成，當然後者更難",[16,98702],{},"\n所以我們可以借由",[75,98705,98707],{"className":98706},[173],"先學簡單","且技巧相同相似的動作，慢慢的去訓練到難的或具有功能性ADL等的動作",[16,98710],{},"\n也可以透過",[75,98713,4846],{"className":98714},[78,79],"達成這一點，例如說寶寶的手還沒有能握住細筷子的能力，可以用",[75,98717,98719],{"className":98718},[78],"加粗","的方式讓寶寶更好握住",[16,98722],{},"\n但記得靠轉移來學習新的動作型態",[75,98725,98727],{"className":98726},[78,79],"不能一下子改太多太難","參數",[26,98730,98731,98732,24802,98737,2589,98741,98743,98744,98748,98749,98751,98752,98756,98758,98759,98763,98764,98766,98767,98771,98772,98774],{},"接著是練習的方式，對沒錯，之前有讀過，現在我只精簡寫一下好了，詳寫的看這兩篇（",[2330,98733,98736],{"href":98734,"rel":98735},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E5%8B%95%E4%BD%9C%E5%AD%B8%E7%BF%92%EF%BC%88%E4%B8%83%EF%BC%89%E7%B7%B4%E7%BF%92%E6%96%B9%E5%BC%8F-1?series=Motor%20learning",[2334],"一",[2330,98738,98740],{"href":98695,"rel":98739},[2334],"二",[16,98742],{},"\n集中VS分散，分散休息時間更長，對",[75,98745,98747],{"className":98746},[78],"學習較難或易疲勞病人","來說更佳",[16,98750],{},"\n段落VS隨機，隨機較難，但通常對",[75,98753,98755],{"className":98754},[173,78],"形成長期記憶以記轉移效果更有",[16,98757],{},"\n固定VS變異，變異在",[75,98760,98762],{"className":98761},[79,78],"轉移效果上","更佳",[16,98765],{},"\n全部VS分段，分段有",[75,98768,98770],{"className":98769},[78,79],"減負效果","，在學習較難動作時可以運用，將同一套動作數個技巧慢慢拆分練習",[16,98773],{},"\n心像練習，在腦中模擬動作型態",[26,98776,98777,98778,24802,98782,24802,98786,24802,98791,7963,98796,98798,98799,98801,98802,98804,98805,2589,98809,98811,98812,98816,98817,98819,98820,98824,98825,98827,98828,98830,98831,98835],{},"最後是回饋方式，對，又來了，一樣在這裡寫精寫的，詳細看..哇，我居然分開了四篇來寫(",[2330,98779,98736],{"href":98780,"rel":98781},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E5%8B%95%E4%BD%9C%E5%AD%B8%E7%BF%92%EF%BC%88%E4%B9%9D%EF%BC%89%E5%9B%9E%E9%A5%8B-1?series=Motor%20learning",[2334],[2330,98783,98740],{"href":98784,"rel":98785},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E5%8B%95%E4%BD%9C%E5%AD%B8%E7%BF%92%EF%BC%88%E5%8D%81%EF%BC%89%E5%9B%9E%E9%A5%8B-2-%E6%93%B4%E5%A2%9E%E6%80%A7(KR)?series=Motor%20learning",[2334],[2330,98787,98790],{"href":98788,"rel":98789},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E5%8B%95%E4%BD%9C%E5%AD%B8%E7%BF%92%EF%BC%88%E5%8D%81%E4%B8%80%EF%BC%89%E5%9B%9E%E9%A5%8B-3-%E6%93%B4%E5%A2%9E%E6%80%A7(KP)?series=Motor%20learning",[2334],"三",[2330,98792,98795],{"href":98793,"rel":98794},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E5%8B%95%E4%BD%9C%E5%AD%B8%E7%BF%92%EF%BC%88%E5%8D%81%E4%B8%80%EF%BC%89%E5%9B%9E%E9%A5%8B-4-(%E9%96%93%E9%9A%94+%E7%89%A9%E7%90%86%E5%BC%95%E5%B0%8E)?series=Motor%20learning",[2334],"四",[16,98797],{},"\nKR與KP，KR效度比KP更佳",[16,98800],{},"\n總結VS延時VS立即，總結是多次動作後給予回饋，延時為動作後隔一小段時間給予回饋，這兩種都比立即回饋效果更佳",[16,98803],{},"\n（",[2330,98806,98808],{"href":98793,"rel":98807},[2334],"這裡第一段有提到",[16,98810],{},"\n回饋的方法，",[2330,98813,98815],{"href":98784,"rel":98814},[2334],"這一篇的後半段","有提到，可以再複習一下",[16,98818],{},"\n先設定一個",[75,98821,98823],{"className":98822},[173],"區間","，達到的都算得分",[16,98826],{},"\n回饋也可以做遞減，逐漸減少依賴",[16,98829],{},"\n假如病人",[75,98832,98834],{"className":98833},[79,78],"主動要求回饋","，我們再給予，也會更佳",{"title":180,"searchDepth":181,"depth":181,"links":98837},[],"2026-04-10","Motor Learning定義~結束","\u002Fimages\u002Fuploads\u002F1775875520542-37128694_p0_master1200.jpg",{},"\u002Fblog\u002FMotor-Learning小兒（二）",{"title":98617,"description":98839},{"loc":98842},"blog\u002FMotor-Learning小兒（二）","vCmIeebtfGfHQk7UoNr_yn1XC2gwpWX_B8fsdXmUsJc",{"id":98848,"title":98849,"author":7,"body":98850,"date":99105,"description":99106,"draft":185,"edited_at":99105,"extension":186,"featured_image":99107,"meta":99108,"navigation":189,"path":99109,"pinned":185,"seo":99110,"sitemap":99111,"stem":99112,"tags":194,"__hash__":99113},"blog\u002Fblog\u002FCVA（九）復健—手部-(i).md","CVA（九）復健—手部 (i)",{"type":9,"value":98851,"toc":99103},[98852,98855,98873,98924,98985,98990,99025,99032,99049,99062,99074,99077,99100],[12,98853,98854],{},"有跑台的內容我先跳過",[12,98856,98857,98858,98860,98861,98863,98864,98866,98867,98869,98870,98872],{},"Brunnstrom說手部在中風後早期會有這五種反應",[16,98859],{},"\n本體覺牽引反應(Proprioceptive traction response)",[16,98862],{},"\n真正的抓握反射(True grasp reflex)",[16,98865],{},"\n本能性抓握反射(Instinctive grasp reaction)",[16,98868],{},"\n本能性規避反應(Instinctive avoiding reaction)",[16,98871],{},"\n索格氏手指現象(Sougue's finger phenomenon)",[12,98874,98875,98876,98878,98879,98883,98884,98888,98890,98891,98895,98899,98900,98904,98905,98907,98908,98912,98914,98915,98917,98918,98920],{},"先來講本體覺牽引反應",[16,98877],{},"\n他的機制是說，假如對上肢或者下肢的其中一個",[75,98880,98882],{"className":98881},[78,173],"關節","的屈肌肌群做",[75,98885,98887],{"className":98886},[5387],"快速的牽拉或者伸展",[16,98889],{},"\n會使被",[75,98892,98894],{"className":98893},[79],"同一個肢體",[75,98896,98898],{"className":98897},[5387,78],"所有關節的屈肌","一起",[75,98901,98903],{"className":98902},[173,78],"被誘發收縮","，讓整段肢體一起縮短",[16,98906],{},"\n主要會被這兩種反射",[75,98909,98911],{"className":98910},[78,6760],"促進或者抑制",[16,98913],{},"\nTNR(Tonic neck reflexes)——尤其是ATNR",[16,98916],{},"\n翻正反射(Body-righting reflexes)",[16,98919],{},[75,98921,98923],{"className":98922},[79],"（利用本體感覺進行誘發）",[12,98925,98926,98927,98929,98930,98934,98935,98938,98939,98943,98944,98948,98949,98951,98952,98954,98955,98957,98958,98962,98963,98965,98966,98970,98971,98975,98976,98980,98981,4995],{},"接著是真正的抓握反射",[16,98928],{},"\n它的誘發方式是，",[75,98931,98933],{"className":98932},[173],"必須要","從",[75,98936,81519],{"className":98937},[78,79],"並經過",[75,98940,98942],{"className":98941},[78392],"掌心橈側部","給予",[75,98945,98947],{"className":98946},[78],"深層壓力","（利用觸覺誘發）",[16,98950],{},"\n這個反射有兩個過程",[16,98953],{},"\n首先是捕捉期，手指的屈曲會快速出現，試圖抓住物體",[16,98956],{},"\n接著是持續抓握期，這個時候屈曲會持續用力收縮，假如物體",[75,98959,98961],{"className":98960},[173],"試圖掙脫","，會讓抓握力度繼續增加（反射變強）",[16,98964],{},"\n那",[75,98967,98969],{"className":98968},[78],"掌指關節","(MP)以及",[75,98972,98974],{"className":98973},[78],"指間關節","(IP)會一起收縮並且呈現",[75,98977,98979],{"className":98978},[79],"內收(Add)","，但",[75,98982,98984],{"className":98983},[173,78],"拇指的力量是最弱",[12,98986,98987],{},[1930,98988],{"alt":1932,"src":98989},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775725402541-Screenshot_2026-04-09_170119.png",[12,98991,98992,98993,98995,98996,99000,99004,99008,99009,99011,99012,99016,99018,99019,99021,99022,99024],{},"現在講的是本能性抓握反射",[16,98994],{},"\n那它的機制跟剛剛的真正抓握不同，不需要從近到遠，只需要",[75,98997,98999],{"className":98998},[78,79],"靜止讓",[75,99001,99003],{"className":99002},[78,78392],"手掌",[75,99005,99007],{"className":99006},[78,79],"碰到","就能觸",[16,99010],{},"\n手指會不自主的握成拳頭，假如物體不拿開手是張不開的",[75,99013,99015],{"className":99014},[78],"（即使像Brunnstrom4 5期的病人能自主張開手也一樣）",[16,99017],{},"\n那有兩種方法可以釋放這個抓握",[16,99020],{},"\n刺激手背（誘發拮抗肌）",[16,99023],{},"\n腕屈曲+放在大腿上休息（因為手指在flex的情況下被帶到wrist flex, 伸肌會被拉到, 借由這個原理讓被動張力來拉開手指）",[12,99026,99027,99028],{},"那Brunnstrom覺得抓握反射也有幾個發展分期",[75,99029,99031],{"className":99030},[5387],"(低階到高階)",[3603,99033,99034,99037,99040,99043,99046],{},[26,99035,99036],{},"牽張反射（最原始的）——在脊髓，並且是單節段的",[26,99038,99039],{},"本體感覺牽引反射——高於脊髓，並且為多節段的（像剛說到的牽拉一個關節可以連帶誘發同一肢段的關節一起收縮）",[26,99041,99042],{},"真正的抓握反射——皮層下反應（subcortical response）",[26,99044,99045],{},"本能性抓握反射——有部份皮層使用（這裡是恢復過程重要的指標，比起真正抓握還要有功能性--by gemini）",[26,99047,99048],{},"抑制本能性抓握反射——高階皮層",[12,99050,99051,99052,99054,99055,99059,99061],{},"然後講本能性規避反應",[16,99053],{},"\n雖然它一樣從近到遠去滑動誘發，但他的前提是將手向前抬高到一定高度，這樣誘發出來的型態會變成手指拇指",[75,99056,99058],{"className":99057},[79],"hyperextension",[16,99060],{},"\n而這種反應會體現在頂葉受損的病人上",[12,99063,99064,99065,99070,99071,99073],{},"最後講的是索格氏手指現象 (",[2330,99066,99069],{"href":99067,"rel":99068},"https:\u002F\u002Fwww.youtube.com\u002Fshorts\u002Fk9i59cNYq5E",[2334],"這是影片","，雖然看起來沒有想象中的一下子就鬆開，但看起來也已經是不像上面第二第三點說的抓死)",[16,99072],{},"\n他提到，假如將手臂整個抬高到90度以上，可以讓手指自動出現伸展（對癱瘓的手指也有用）",[12,99075,99076],{},"Brunnstrom對於手部的功能恢復分成了七個階段",[3603,99078,99079,99082,99085,99088,99091,99094,99097],{},[26,99080,99081],{},"肌腱反射回歸並且亢奮(Tendon reflexes return and become hyperactive)",[26,99083,99084],{},"痙攣與阻力增加(Spasticity and resistance develop)",[26,99086,99087],{},"借由本體覺刺激誘發手指主動屈曲(Voluntary finger flexion occurs by proprioceptive stimuli)",[26,99089,99090],{},"本體覺牽引反射出現(Proprioceptive traction response appear)",[26,99092,99093],{},"脫離本體覺刺激的手部控制(Control of hand movement without proprioceptive stimuli)",[26,99095,99096],{},"刺激掌心誘發屈曲，痙攣下降(Grasp is greatly reinforced by tactile stimuli in palm, spasticity is declining)",[26,99098,99099],{},"真正的抓握反射出現(True grasp reflex appear)",[12,99101,99102],{},"後面的我留給跑台寫好了",{"title":180,"searchDepth":181,"depth":181,"links":99104},[],"2026-04-09","講brunnstrom手指五個反應~功能恢復","\u002Fimages\u002Fuploads\u002F1775730146879-139326331_p1.png",{},"\u002Fblog\u002FCVA（九）復健—手部-(i)",{"title":98849,"description":99106},{"loc":99109},"blog\u002FCVA（九）復健—手部-(i)","bsqAJwJ58wMf5-lcMFABpNnYIvmUwQtnszJLu7m1nxc",{"id":99115,"title":99116,"author":6946,"body":99117,"date":99105,"description":180,"draft":185,"edited_at":6934,"extension":186,"featured_image":194,"meta":99772,"navigation":189,"path":99773,"pinned":185,"seo":99774,"sitemap":99775,"stem":99776,"tags":194,"__hash__":99777},"blog\u002Fblog\u002FNetwork-Communication,-Chapter-1.md","Introduction to Networking Infrastructure",{"type":9,"value":99118,"toc":99743},[99119,99123,99129,99136,99140,99150,99153,99177,99181,99184,99188,99191,99195,99198,99222,99255,99259,99262,99282,99286,99293,99307,99310,99314,99317,99374,99378,99381,99387,99393,99397,99400,99423,99427,99431,99494,99498,99501,99521,99528,99560,99564,99567,99571,99578,99582,99585,99589,99596,99599,99605,99625,99629,99632,99664,99668,99671,99709,99713,99720,99735,99737,99740],[6964,99120,99122],{"id":99121},"why-should-you-care-about-networks","Why Should You Care About Networks?",[12,99124,99125,99126,6959],{},"We live in a hyper-connected world. News travels worldwide in seconds, we store our photos and documents in the cloud, and video calls with people across the globe are just a click away. Behind all of this is one thing: ",[106,99127,99128],{},"computer networks",[12,99130,99131,99132,99135],{},"The idea of the ",[106,99133,99134],{},"Internet of Everything (IoE)"," captures this well — it's the convergence of people, processes, data, and things, all linked together through networked connections. Understanding how networks work is the first step to understanding the digital infrastructure that powers modern life.",[6964,99137,99139],{"id":99138},"hosts-clients-and-servers","Hosts, Clients, and Servers",[12,99141,99142,99143,7959,99146,99149],{},"Every device that participates directly in network communication is called a ",[106,99144,99145],{},"host",[106,99147,99148],{},"end device","). Your laptop, your phone, a web server — all hosts. They can send and receive messages on the network.",[12,99151,99152],{},"In modern networks, an end device can play one of three roles:",[23,99154,99155,99161,99167],{},[26,99156,99157,99160],{},[106,99158,99159],{},"Client"," — requests information or services (e.g. your browser asking for a web page).",[26,99162,99163,99166],{},[106,99164,99165],{},"Server"," — provides information or services (e.g. a machine serving that web page). Servers run specialised software for each service they offer.",[26,99168,99169,99172,99173,99176],{},[106,99170,99171],{},"Both"," — in a ",[106,99174,99175],{},"peer-to-peer (P2P) network",", a single host can act as client and server simultaneously. Think of file-sharing applications where your computer both downloads from and uploads to other users.",[6964,99178,99180],{"id":99179},"network-components","Network Components",[12,99182,99183],{},"A network's infrastructure breaks down into three broad categories:",[7166,99185,99187],{"id":99186},"_1-end-devices","1. End Devices",[12,99189,99190],{},"These are the devices at the \"edges\" of the network — the ones people actually interact with. Examples include laptops, smartphones, network printers, VoIP phones, security cameras, and tablets.",[7166,99192,99194],{"id":99193},"_2-intermediary-devices","2. Intermediary Devices",[12,99196,99197],{},"These are the behind-the-scenes workhorses that make sure data gets where it needs to go:",[23,99199,99200,99206,99216],{},[26,99201,99202,99205],{},[106,99203,99204],{},"Switches and wireless access points"," — provide network access by connecting end devices together.",[26,99207,99208,99211,99212,99215],{},[106,99209,99210],{},"Routers"," — connect separate networks to form an ",[7289,99213,99214],{},"internetwork",", directing traffic between them.",[26,99217,99218,99221],{},[106,99219,99220],{},"Firewalls"," — enforce security policies by filtering unauthorised traffic.",[25631,99223,99225,99228],{"className":99224},[13441,13442],[25635,99226,99227],{},"What's different between a Switch, Router and Modem?",[13438,99229,99231],{"className":99230},[25641],[3603,99232,99233,99239,99249],{},[26,99234,99235,99238],{},[106,99236,99237],{},"Modem",": Your ISP sends the internet to your house via analog signals through cables (like coax, fiber optic, or phone lines). Your computers, however, only speak in digital signals (1s and 0s). The modem acts as a translator, turning the ISP's signal into a digital signal your devices can understand, and vice versa.",[26,99240,99241,99244,99245,99248],{},[106,99242,99243],{},"Router",": The router sits just inside your modem. It assigns a unique local IP address to every device in your house. When you request a webpage on your phone, the router remembers that ",[7289,99246,99247],{},"your phone"," asked for it, sends the request out through the modem, and when the website's data comes back, the router ensures it goes to your phone and not your smart TV.",[26,99250,99251,99254],{},[106,99252,99253],{},"Switch",": While a router connects your home to the outside world, a switch just links your home devices to each other. It gives you more ports to plug things in. When a computer plugged into port 1 wants to send a file to a printer plugged into port 4, the switch directs that data straight from port 1 to port 4 without bothering the rest of the network.",[7166,99256,99258],{"id":99257},"_3-network-media","3. Network Media",[12,99260,99261],{},"Data needs a physical (or wireless) channel to travel through. The three common types of media are:",[23,99263,99264,99270,99276],{},[26,99265,99266,99269],{},[106,99267,99268],{},"Metallic wires"," (e.g. copper cables like Ethernet)",[26,99271,99272,99275],{},[106,99273,99274],{},"Glass or plastic fibres"," (fibre optic cables — fast and long-range)",[26,99277,99278,99281],{},[106,99279,99280],{},"Wireless transmission"," (radio waves — Wi-Fi, Bluetooth, cellular)",[6964,99283,99285],{"id":99284},"network-representations-and-topologies","Network Representations and Topologies",[12,99287,99288,99289,99292],{},"When networks get large, we need diagrams to make sense of them. These ",[106,99290,99291],{},"topology diagrams"," come in two flavours:",[23,99294,99295,99301],{},[26,99296,99297,99300],{},[106,99298,99299],{},"Physical topology"," — shows where devices are physically located, which ports are in use, and how cables are actually installed.",[26,99302,99303,99306],{},[106,99304,99305],{},"Logical topology"," — shows how devices are logically connected, including their IP addresses and port assignments.",[12,99308,99309],{},"Both views are essential. The physical topology helps you troubleshoot hardware, while the logical topology helps you understand how data flows.",[6964,99311,99313],{"id":99312},"types-of-networks","Types of Networks",[12,99315,99316],{},"Networks come in every size:",[7085,99318,99319,99327],{},[7088,99320,99321],{},[7091,99322,99323,99325],{},[7094,99324,48089],{"align":47883},[7094,99326,84419],{"align":47883},[7104,99328,99329,99339,99350,99360],{},[7091,99330,99331,99336],{},[7109,99332,99333],{"align":47883},[106,99334,99335],{},"Home network",[7109,99337,99338],{"align":47883},"A few devices sharing printers, files, and music.",[7091,99340,99341,99347],{},[7109,99342,99343,99346],{"align":47883},[106,99344,99345],{},"SOHO"," (Small Office \u002F Home Office)",[7109,99348,99349],{"align":47883},"Supports remote work — advertising, ordering supplies, communicating with clients.",[7091,99351,99352,99357],{},[7109,99353,99354],{"align":47883},[106,99355,99356],{},"Medium to large",[7109,99358,99359],{"align":47883},"Corporate or campus networks with hundreds to thousands of hosts across multiple locations.",[7091,99361,99362,99367],{},[7109,99363,99364],{"align":47883},[106,99365,99366],{},"The Internet",[7109,99368,99369,99370,99373],{"align":47883},"A network ",[7289,99371,99372],{},"of"," networks, connecting hundreds of millions of computers worldwide.",[7166,99375,99377],{"id":99376},"lan-vs-wan","LAN vs. WAN",[12,99379,99380],{},"The two most fundamental network types are:",[12,99382,99383,99386],{},[106,99384,99385],{},"Local Area Network (LAN)"," — covers a small geographical area like a home, school, or office building. Usually managed by a single person or organisation, and offers high-speed connections to devices within that area.",[12,99388,99389,99392],{},[106,99390,99391],{},"Wide Area Network (WAN)"," — spans large geographical areas, connecting LANs across cities, countries, or even continents. Typically managed by multiple service providers, and the links between LANs tend to be slower than what you'd get inside a LAN.",[7166,99394,99396],{"id":99395},"internet-intranet-and-extranet","Internet, Intranet, and Extranet",[12,99398,99399],{},"These three terms often confuse beginners, so here's a simple breakdown:",[23,99401,99402,99408,99417],{},[26,99403,99404,99407],{},[106,99405,99406],{},"Internet"," — the global, public collection of interconnected networks using common standards.",[26,99409,99410,99413,99414,99416],{},[106,99411,99412],{},"Intranet"," — a ",[7289,99415,7656],{}," network of LANs and WANs belonging to an organisation, accessible only to its members.",[26,99418,99419,99422],{},[106,99420,99421],{},"Extranet"," — like an intranet, but with controlled access granted to specific external users (e.g. suppliers or partner hospitals).",[6964,99424,99426],{"id":99425},"internet-connection-types","Internet Connection Types",[7166,99428,99430],{"id":99429},"for-homes-and-small-offices","For Homes and Small Offices",[7085,99432,99433,99442],{},[7088,99434,99435],{},[7091,99436,99437,99439],{},[7094,99438,85952],{"align":47883},[7094,99440,99441],{"align":47883},"How it Works",[7104,99443,99444,99454,99464,99474,99484],{},[7091,99445,99446,99451],{},[7109,99447,99448],{"align":47883},[106,99449,99450],{},"Cable",[7109,99452,99453],{"align":47883},"Shares the same cable as cable TV. High bandwidth, always-on.",[7091,99455,99456,99461],{},[7109,99457,99458],{"align":47883},[106,99459,99460],{},"DSL (Digital Subscriber Line)",[7109,99462,99463],{"align":47883},"Runs over copper telephone lines, with a much higher frequency than human voice. Home users typically use ADSL (faster download than upload).",[7091,99465,99466,99471],{},[7109,99467,99468],{"align":47883},[106,99469,99470],{},"Cellular",[7109,99472,99473],{"align":47883},"Connects via cell towers (3G \u002F 4G \u002F 5G). Great for mobile access.",[7091,99475,99476,99481],{},[7109,99477,99478],{"align":47883},[106,99479,99480],{},"Satellite",[7109,99482,99483],{"align":47883},"Useful in remote areas, but needs a clear line of sight to the satellite.",[7091,99485,99486,99491],{},[7109,99487,99488],{"align":47883},[106,99489,99490],{},"Dial-up",[7109,99492,99493],{"align":47883},"Uses a phone line and modem. Very slow — mostly a relic, but still exists in some areas.",[7166,99495,99497],{"id":99496},"for-businesses","For Businesses",[12,99499,99500],{},"Businesses often need faster, more reliable, and dedicated connections:",[23,99502,99503,99509,99515],{},[26,99504,99505,99508],{},[106,99506,99507],{},"Dedicated Leased Lines"," — reserved circuits rented monthly or yearly for private data\u002Fvoice networking. It guarantees 100% of the bandwidth all the time.",[26,99510,99511,99514],{},[106,99512,99513],{},"Metro Ethernet"," — Metro Ethernet takes that incredibly fast, highly reliable local network technology and stretches it across an entire city. Metro Ethernet allows a company to connect their downtown office to their suburban warehouse.",[26,99516,99517,99520],{},[106,99518,99519],{},"Business DSL (SDSL)"," — symmetric speeds (equal upload and download), unlike consumer ADSL.",[12,99522,99523,99524,99527],{},"A key modern trend is ",[106,99525,99526],{},"converged networks"," — carrying voice, video, and data all on a single network infrastructure.",[13438,99529,99531,99534,99541,99552,99555,99557],{"className":99530},[13441,13442],[12,99532,99533],{},"A note: In modern days, this (converged networks) is just standard behaviour. The note is talking about historical shift in how networks are built.",[12,99535,99536,99537,99540],{},"Imagine an office building in the 1990s. The IT and facilities teams literally had to run ",[106,99538,99539],{},"three completely separate sets of cables"," through the walls.",[3603,99542,99543,99546,99549],{},[26,99544,99545],{},"Copper telephone wires to every desk just for phone calls.",[26,99547,99548],{},"Coaxial cables to conference rooms and lobbies just for video\u002Ftelevision.",[26,99550,99551],{},"Ethernet cables to the computers just for data and internet access.",[12,99553,99554],{},"Eventually, engineers realised that a voice call and a video stream could just be chopped up into digital 1s and 0s, exactly like an email or a webpage.",[16,99556],{},[12,99558,99559],{},"Once voice and video became digital data, businesses could rip out the old phone lines and TV cables. Now, they just run one single, robust Ethernet\u002FWi-Fi network. That single infrastructure carries the data, the video (like Zoom), and the voice (VoIP—Voice over Internet Protocol).",[6964,99561,99563],{"id":99562},"what-makes-a-network-reliable","What Makes a Network \"Reliable\"?",[12,99565,99566],{},"Good network architecture addresses four fundamental characteristics:",[7166,99568,99570],{"id":99569},"fault-tolerance","Fault Tolerance",[12,99572,99573,99574,99577],{},"A fault-tolerant network limits the damage when something breaks. The key idea is ",[106,99575,99576],{},"redundancy"," — having multiple paths between source and destination so that if one path fails, traffic is instantly rerouted through another.",[7166,99579,99581],{"id":99580},"scalability","Scalability",[12,99583,99584],{},"A scalable network can grow — adding new users, devices, and services — without degrading performance for existing users. Following accepted standards and protocols is what makes this possible.",[7166,99586,99588],{"id":99587},"quality-of-service-qos","Quality of Service (QoS)",[12,99590,99591,99592,99595],{},"When voice, video, and data all share the same network, congestion can happen. QoS is the mechanism that manages this by ",[106,99593,99594],{},"prioritising"," traffic. For example, a router with QoS can give voice calls priority over a large file download, so your call doesn't break up.",[7166,99597,83656],{"id":99598},"security",[12,99600,99601,99602,7024],{},"Network security protects both the infrastructure itself and the data flowing through it. It rests on three pillars, sometimes called the ",[106,99603,99604],{},"CIA triad",[3603,99606,99607,99613,99619],{},[26,99608,99609,99612],{},[106,99610,99611],{},"Confidentiality"," — only authorised recipients can access the data.",[26,99614,99615,99618],{},[106,99616,99617],{},"Integrity"," — data hasn't been tampered with during transmission.",[26,99620,99621,99624],{},[106,99622,99623],{},"Availability"," — authorised users can access services when they need to.",[6964,99626,99628],{"id":99627},"network-trends-to-watch","Network Trends to Watch",[12,99630,99631],{},"A few trends shaping today's networks:",[23,99633,99634,99640,99646,99652,99658],{},[26,99635,99636,99639],{},[106,99637,99638],{},"BYOD (Bring Your Own Device)"," — people use their personal devices on business and campus networks. Convenient, but raises security questions.",[26,99641,99642,99645],{},[106,99643,99644],{},"Online Collaboration"," — tools for real-time messaging, video, and file sharing (think Webex, Teams, Slack, etc.).",[26,99647,99648,99651],{},[106,99649,99650],{},"Cloud Computing"," — storing data and running applications on remote servers. Comes in four flavours: public, private, hybrid, and community clouds.",[26,99653,99654,99657],{},[106,99655,99656],{},"Smart Home Technology"," — everyday appliances connecting to the network and becoming automated.",[26,99659,99660,99663],{},[106,99661,99662],{},"Powerline Networking"," — using existing electrical wiring to carry network data, eliminating the need for extra cables.",[6964,99665,99667],{"id":99666},"network-security-threats","Network Security Threats",[12,99669,99670],{},"Finally, let's talk about the bad stuff. Common external threats include:",[23,99672,99673,99679,99685,99691,99697,99703],{},[26,99674,99675,99678],{},[106,99676,99677],{},"Malware"," (viruses, worms, Trojans) — malicious code running on your device.",[26,99680,99681,99684],{},[106,99682,99683],{},"Spyware and adware"," — secretly installed software that collects your information.",[26,99686,99687,99690],{},[106,99688,99689],{},"Zero-day attacks"," — exploits that target vulnerabilities on the very day they're discovered, before patches exist.",[26,99692,99693,99696],{},[106,99694,99695],{},"Denial of Service (DoS)"," — attacks that overwhelm and crash network services.",[26,99698,99699,99702],{},[106,99700,99701],{},"Data interception and theft"," — capturing private information as it travels across the network.",[26,99704,99705,99708],{},[106,99706,99707],{},"Identity theft"," — stealing login credentials to access private data.",[7166,99710,99712],{"id":99711},"how-do-we-defend-against-these","How Do We Defend Against These?",[12,99714,99715,99716,99719],{},"For home and small office networks, the basics are antivirus\u002Fantispyware software and ",[106,99717,99718],{},"firewall filtering"," to block unauthorised access.",[12,99721,99722,99723,99726,99727,99730,99731,99734],{},"Larger corporate networks add more layers: dedicated firewall systems, ",[106,99724,99725],{},"Access Control Lists (ACLs)"," for fine-grained traffic filtering, ",[106,99728,99729],{},"Intrusion Prevention Systems (IPS)"," to catch fast-spreading threats, and ",[106,99732,99733],{},"VPNs"," to give remote workers secure access.",[6964,99736,77786],{"id":77785},[12,99738,99739],{},"This chapter covered the foundations: what networks are made of, how they're classified, how we connect to the internet, what makes a network reliable, and how we keep it secure. These concepts form the bedrock for everything else in networking.",[12,99741,99742],{},"In the next chapter, we'll dive deeper into how data actually moves across these networks. Stay tuned!",{"title":180,"searchDepth":181,"depth":181,"links":99744},[99745,99746,99747,99752,99753,99757,99761,99767,99768,99771],{"id":99121,"depth":181,"text":99122},{"id":99138,"depth":181,"text":99139},{"id":99179,"depth":181,"text":99180,"children":99748},[99749,99750,99751],{"id":99186,"depth":7430,"text":99187},{"id":99193,"depth":7430,"text":99194},{"id":99257,"depth":7430,"text":99258},{"id":99284,"depth":181,"text":99285},{"id":99312,"depth":181,"text":99313,"children":99754},[99755,99756],{"id":99376,"depth":7430,"text":99377},{"id":99395,"depth":7430,"text":99396},{"id":99425,"depth":181,"text":99426,"children":99758},[99759,99760],{"id":99429,"depth":7430,"text":99430},{"id":99496,"depth":7430,"text":99497},{"id":99562,"depth":181,"text":99563,"children":99762},[99763,99764,99765,99766],{"id":99569,"depth":7430,"text":99570},{"id":99580,"depth":7430,"text":99581},{"id":99587,"depth":7430,"text":99588},{"id":99598,"depth":7430,"text":83656},{"id":99627,"depth":181,"text":99628},{"id":99666,"depth":181,"text":99667,"children":99769},[99770],{"id":99711,"depth":7430,"text":99712},{"id":77785,"depth":181,"text":77786},{},"\u002Fblog\u002FNetwork-Communication,-Chapter-1",{"title":99116,"description":180},{"loc":99773},"blog\u002FNetwork-Communication,-Chapter-1","t48yv3uwq7eJ2V4zbkN5suWOSdMymph3TdmQc731kMw",{"id":99779,"title":99780,"author":6946,"body":99781,"date":99105,"description":180,"draft":185,"edited_at":100791,"extension":186,"featured_image":194,"meta":100792,"navigation":189,"path":100793,"pinned":185,"seo":100794,"sitemap":100795,"stem":100796,"tags":194,"__hash__":100797},"blog\u002Fblog\u002FNetwork-Communication,-Chapter-2.md","How Data is Packaged: The TCP\u002FIP Model",{"type":9,"value":99782,"toc":100766},[99783,99787,99790,99797,99800,99826,99830,99833,99887,99891,99901,99908,99912,99919,99923,99930,99934,99937,99941,99948,99964,99967,99971,99981,100060,100067,100127,100131,100138,100164,100275,100279,100286,100297,100303,100481,100485,100488,100498,100501,100527,100531,100534,100605,100608,100612,100615,100663,100667,100670,100674,100683,100687,100690,100723,100725],[6964,99784,99786],{"id":99785},"why-do-we-need-rules-for-communication","Why Do We Need \"Rules\" for Communication?",[12,99788,99789],{},"Think about a regular face-to-face conversation. Even without realising it, you follow a bunch of unspoken rules — you wait for your turn to speak, you use a language both sides understand, and you expect some kind of acknowledgment that the other person heard you. Networking is exactly the same.",[12,99791,99792,99793,99796],{},"In networking, these rules are called ",[106,99794,99795],{},"protocols",". Every time a message travels from a source to a destination — whether it's an email, a video call, or a simple ping — protocols govern how that happens.",[12,99798,99799],{},"For any communication to work, you need at least these things in place:",[23,99801,99802,99808,99814,99820],{},[26,99803,99804,99807],{},[106,99805,99806],{},"An identified sender and receiver"," — both ends need to be known.",[26,99809,99810,99813],{},[106,99811,99812],{},"A common language and grammar"," — otherwise the message is just noise.",[26,99815,99816,99819],{},[106,99817,99818],{},"Speed and timing of delivery"," — both sides need to agree on pacing.",[26,99821,99822,99825],{},[106,99823,99824],{},"Confirmation or acknowledgment"," — how do you know the message arrived?",[7166,99827,99829],{"id":99828},"common-computer-protocol-concepts","Common Computer Protocol Concepts",[12,99831,99832],{},"Protocols in computer networks handle several key jobs:",[23,99834,99835,99841,99847,99853,99881],{},[26,99836,99837,99840],{},[106,99838,99839],{},"Message encoding"," — Converting information into a transmittable format (and decoding it on the other end). Think of it like translating your thoughts into words before speaking.",[26,99842,99843,99846],{},[106,99844,99845],{},"Message formatting and encapsulation"," — Structuring the message so the receiver knows how to read it, much like writing a letter with a proper header, body, and signature.",[26,99848,99849,99852],{},[106,99850,99851],{},"Message size"," — Just like you wouldn't dump an entire essay on someone in one breath, networks break messages into manageable chunks.",[26,99854,99855,99858,99859],{},[106,99856,99857],{},"Message timing"," — This covers three things:",[23,99860,99861,99866,99872],{},[26,99862,99863,99865],{},[106,99864,83156],{},": How much data can be sent, and how fast.",[26,99867,99868,99871],{},[106,99869,99870],{},"Response timeout",": How long to wait for a reply before assuming something went wrong.",[26,99873,99874,99877,99878,99880],{},[106,99875,99876],{},"Access method",": Deciding ",[7289,99879,76097],{}," a device is allowed to send — you can't have everyone shouting at once.",[26,99882,99883,99886],{},[106,99884,99885],{},"Message delivery options"," — Sometimes you send to one person (unicast), sometimes to a group (multicast), and sometimes to everyone nearby (broadcast).",[6964,99888,99890],{"id":99889},"network-protocols-the-common-language-of-devices","Network Protocols: The Common Language of Devices",[12,99892,99893,99894,99897,99898,6959],{},"A single message sent over a network usually involves ",[106,99895,99896],{},"multiple protocols"," working together, each handling a different part of the job. A group of these inter-related protocols is called a ",[106,99899,99900],{},"protocol suite",[12,99902,99903,99904,99907],{},"The most important protocol suite you'll encounter is ",[106,99905,99906],{},"TCP\u002FIP"," — it's the backbone of the Internet. TCP\u002FIP is implemented in software, hardware, or both on every device that connects to a network.",[7166,99909,99911],{"id":99910},"the-layered-model","The Layered Model",[12,99913,99914,99915,99918],{},"To make sense of how all these protocols interact, we use a ",[106,99916,99917],{},"layered model",". The TCP\u002FIP model organises protocols into layers, where each layer has a specific role and communicates with the layers directly above and below it. This separation makes it easier to design, troubleshoot, and update networks — you can swap out a protocol at one layer without breaking everything else.",[7166,99920,99922],{"id":99921},"who-makes-the-rules","Who Makes the Rules?",[12,99924,99925,99926,99929],{},"Network standards are developed by ",[106,99927,99928],{},"standards organisations"," — vendor-neutral, non-profit bodies whose job is to promote open standards so that devices from different manufacturers can all work together. You'll hear names like IEEE, IETF, and ICANN come up often in networking.",[6964,99931,99933],{"id":99932},"data-encapsulation-wrapping-your-data-for-delivery","Data Encapsulation: Wrapping Your Data for Delivery",[12,99935,99936],{},"Imagine you want to send a huge video file across the Internet. Sending it as one giant uninterrupted stream would hog the entire communication channel and create massive delays for everyone else. That's clearly not practical.",[7166,99938,99940],{"id":99939},"segmentation-and-multiplexing","Segmentation and Multiplexing",[12,99942,99943,99944,99947],{},"The solution is ",[106,99945,99946],{},"segmentation"," — breaking the data into smaller, more manageable pieces before sending. This has two big benefits:",[3603,99949,99950,99955],{},[26,99951,99952,99954],{},[106,99953,83134],{}," — If one small piece gets lost, you only resend that piece, not the entire file.",[26,99956,99957,99960,99961,99963],{},[106,99958,99959],{},"Multiplexing"," — Smaller pieces from ",[7289,99962,83468],{}," conversations can be interleaved on the same network, so multiple users can share the same link simultaneously.",[12,99965,99966],{},"Each segment goes through the same process to make sure it reaches the right destination and can be reassembled into the original message.",[7166,99968,99970],{"id":99969},"the-encapsulation-process","The Encapsulation Process",[12,99972,99973,99974,99977,99978,7024],{},"As data travels down through the protocol layers, each layer adds its own header (and sometimes trailer) information. This is the ",[106,99975,99976],{},"encapsulation process",". At each layer, the data takes on a different form, called a ",[106,99979,99980],{},"Protocol Data Unit (PDU)",[7085,99982,99983,99995],{},[7088,99984,99985],{},[7091,99986,99987,99989,99992],{},[7094,99988,74704],{"align":47883},[7094,99990,99991],{"align":47883},"PDU Name",[7094,99993,99994],{"align":47883},"What It Represents",[7104,99996,99997,100010,100023,100036,100049],{},[7091,99998,99999,100002,100007],{},[7109,100000,100001],{"align":47883},"Application",[7109,100003,100004],{"align":47883},[106,100005,100006],{},"Data",[7109,100008,100009],{"align":47883},"The raw information from the app",[7091,100011,100012,100015,100020],{},[7109,100013,100014],{"align":47883},"Transport",[7109,100016,100017],{"align":47883},[106,100018,100019],{},"Segment",[7109,100021,100022],{"align":47883},"Data + port info for process-to-process delivery",[7091,100024,100025,100028,100033],{},[7109,100026,100027],{"align":47883},"Internet (Network)",[7109,100029,100030],{"align":47883},[106,100031,100032],{},"Packet",[7109,100034,100035],{"align":47883},"Segment + IP addresses for host-to-host delivery",[7091,100037,100038,100041,100046],{},[7109,100039,100040],{"align":47883},"Network Access (Data Link)",[7109,100042,100043],{"align":47883},[106,100044,100045],{},"Frame",[7109,100047,100048],{"align":47883},"Packet + MAC addresses for hop-to-hop delivery",[7091,100050,100051,100053,100057],{},[7109,100052,74716],{"align":47883},[7109,100054,100055],{"align":47883},[106,100056,84416],{},[7109,100058,100059],{"align":47883},"The actual 1s and 0s on the wire",[12,100061,100062,100063,100066],{},"When the data arrives at the destination, the reverse happens — ",[106,100064,100065],{},"de-encapsulation",". Each layer strips off its header, passing the remaining data up to the next layer until the application gets the original message.",[25631,100068,100070,100073],{"className":100069},[13441,13442],[25635,100071,100072],{},"The Encapsulation Process (Packing the Box)",[13438,100074,100076],{"className":100075},[25641],[23,100077,100078,100084,100094,100108,100121],{},[26,100079,100080,100083],{},[106,100081,100082],{},"Application Layer (Data):"," This is the actual birthday card and the gift you want to send. It is the raw information you care about.",[26,100085,100086,100089,100090,100093],{},[106,100087,100088],{},"Transport Layer (Segment):"," You take the gift and put it in a small inner envelope. You write *\"To John, from Jane\"*on it. In networking, this layer adds the ",[106,100091,100092],{},"port number",", which makes sure the data goes to the exact right application (John) and not someone else in the house. Because the data might be too big, this layer will also chop it up into manageable pieces (segments).",[26,100095,100096,100099,100100,100103,100104,100107],{},[106,100097,100098],{},"Internet Layer (Packet):"," You put that inner envelope inside a standard cardboard shipping box. On the outside, you write the final destination address: ",[7289,100101,100102],{},"\"123 Main St, New York, NY.\""," In networking, this layer adds the ",[106,100105,100106],{},"IP Address",", ensuring the box can navigate the global internet to find the correct final building (host-to-host delivery).",[26,100109,100110,100113,100114,100103,100117,100120],{},[106,100111,100112],{},"Network Access Layer (Frame):"," The post office takes your box and loads it into a specific local delivery truck. They stick a temporary barcode on the outside of the box that just says, ",[7289,100115,100116],{},"\"Take this to the airport depot.\"",[106,100118,100119],{},"MAC Address",". It doesn't care about the final destination in New York; it only cares about successfully moving the box to the very next local stop (hop-to-hop delivery).",[26,100122,100123,100126],{},[106,100124,100125],{},"Physical Layer (Bits):"," The truck physically drives down the road. In networking, this is the actual physical electricity pulsing over copper wires, or light flashing down a fiber optic cable, represented as 1s and 0s.",[6964,100128,100130],{"id":100129},"data-access-two-kinds-of-addresses","Data Access: Two Kinds of Addresses",[12,100132,100133,100134,100137],{},"Getting data from point A to point B actually involves ",[106,100135,100136],{},"two different addressing systems"," working at different layers:",[23,100139,100140,100154],{},[26,100141,100142,100145,100146,100149,100150,100153],{},[106,100143,100144],{},"Network layer (IP) addresses"," — These are responsible for delivering the packet from the ",[7289,100147,100148],{},"original source"," to the ",[7289,100151,100152],{},"final destination",", even across different networks. Think of this as the mailing address on an envelope.",[26,100155,100156,100159,100160,100163],{},[106,100157,100158],{},"Data link layer (MAC) addresses"," — These handle delivery between one network interface card (NIC) and the next, on the ",[7289,100161,100162],{},"same local network",". Think of this as the \"hand it to the next person in the chain\" step.",[25631,100165,100167,100170],{"className":100166},[13441,13442],[25635,100168,100169],{},"Deep dive: Why VPNs work but MAC bans are weird",[13438,100171,100173,100177,100183,100189,100193,100196,100200,100205,100216,100223,100226,100235,100247,100253],{"className":100172},[25641],[7166,100174,100176],{"id":100175},"mac-address-vs-ip-address","MAC address vs IP address",[12,100178,100179,100182],{},[106,100180,100181],{},"MAC Address = Who you are"," (Like your DNA). It is permanently burned into your device at the factory.",[12,100184,100185,100188],{},[106,100186,100187],{},"IP Address = Where you are"," (Like your street address). It changes depending on where you connect. If you take your laptop from your house to a coffee shop, your MAC address stays exactly the same, but your IP address changes because you are on a new network.",[7166,100190,100192],{"id":100191},"the-ip-ban-vpns-the-network-layer","The IP Ban & VPNs (The Network Layer)",[12,100194,100195],{},"When a remote website or game server bans your IP address, they are blocking your \"street address.\" You turn on a VPN. A VPN acts as a middleman. You send your data to the VPN server, and the VPN server forwards it to the website. The website only sees the VPN's \"street address\" (IP address), not yours. Because your IP is hidden, you bypass the ban.",[7166,100197,100199],{"id":100198},"the-mac-ban-the-data-link-layer","The MAC Ban (The Data Link Layer)",[12,100201,100202],{},[106,100203,100204],{},"MAC addresses never leave your local network.",[12,100206,100207,100208,100211,100212,100215],{},"When your computer sends a packet of data to a server in another country, it hands the packet to your home router. Your router literally ",[106,100209,100210],{},"strips off your computer's MAC address",", throws it in the trash, and slaps on the ",[7289,100213,100214],{},"router's own MAC address"," before sending it to the next stop. This happens at every single \"hop\" across the internet.",[12,100217,100218,100219,100222],{},"Because of this, ",[106,100220,100221],{},"a remote website cannot ban your MAC address over the internet because they literally cannot see it."," Only your local router can see your MAC address.",[12,100224,100225],{},"So, how do MAC bans actually work, and are they circumventable?",[12,100227,100228,100231,100232,100234],{},[106,100229,100230],{},"1. Local Network MAC Bans (The Wi-Fi Ban)"," If you misbehave at school or work, the IT admin can tell the local Wi-Fi router, \"Do not let the MAC address belonging to John's phone connect.\" Since the router is on the ",[7289,100233,100162],{}," as your phone, it sees your MAC address and blocks you.",[23,100236,100237],{},[26,100238,100239,100242,100243,100246],{},[106,100240,100241],{},"Is it circumventable? YES."," Even though the MAC address is burned into your hardware, you can use a software trick called ",[106,100244,100245],{},"MAC Spoofing",". You simply tell your phone or computer's operating system to wear a \"mask\" and broadcast a fake, randomly generated MAC address. The router sees the new fake MAC, doesn't recognize it as banned, and lets you right back on. (In fact, modern iOS and Android phones do this automatically for privacy now!).",[12,100248,100249,100252],{},[106,100250,100251],{},"2. Hardware Bans (The Anti-Cheat Ban)"," If websites can't see your MAC address over the internet, how do game developers \"hardware ban\" cheaters?",[23,100254,100255,100269],{},[26,100256,100257,100258,7067,100261,100264,100265,100268],{},"They don't do it over the internet; they do it locally. Games like ",[7289,100259,100260],{},"Valorant",[7289,100262,100263],{},"Call of Duty"," make you install aggressive Anti-Cheat software. That software runs ",[7289,100266,100267],{},"locally"," on your actual physical machine, reads your motherboard's MAC address (and serial numbers of your CPU and hard drive), and sends a list of those numbers to the game server.",[26,100270,100271,100274],{},[106,100272,100273],{},"Is it circumventable? YES, but it's much harder."," Because the anti-cheat is looking deeply at multiple pieces of hardware, simple MAC spoofing usually isn't enough. People use complex \"hardware spoofers\" to trick the anti-cheat, but it is a constant game of cat-and-mouse.",[7166,100276,100278],{"id":100277},"same-network-vs-different-network","Same Network vs. Different Network",[12,100280,100281,100282,100285],{},"When sender and receiver are on the ",[106,100283,100284],{},"same network",", the frame goes directly to the destination device using its MAC address.",[12,100287,100288,100289,100292,100293,100296],{},"When they're on ",[106,100290,100291],{},"different networks",", things get more interesting. The IP addresses still point to the original source and final destination, but the frame is first sent to the ",[106,100294,100295],{},"router (default gateway)",". The router then re-wraps (re-encapsulates) the data in a new frame destined for the next hop, and so on, until the packet reaches the destination network.",[12,100298,100299,100300],{},"This is a fundamental concept: ",[106,100301,100302],{},"IP addresses stay the same end-to-end, but MAC addresses change at every hop.",[25631,100304,100306,100309],{"className":100305},[13441,13442],[25635,100307,100308],{},"NAT \u002F Port walkthrough",[13438,100310,100312,100323,100337,100350,100356,100373,100395,100407,100433,100436,100445,100450,100456,100465,100471],{"className":100311},[25641],[12,100313,100314,100315,100318,100319,100322],{},"Let's say, you want to connect to Google (",[6979,100316,100317],{},"8.8.8.8",") from your computer (",[6979,100320,100321],{},"192.168.0.15","). So, it builds an IP Packet (the cardboard shipping box).",[23,100324,100325,100331],{},[26,100326,100327,100330],{},[106,100328,100329],{},"Source IP:"," 192.168.0.15 (You)",[26,100332,100333,100336],{},[106,100334,100335],{},"Destination IP:"," 8.8.8.8 (Google)",[12,100338,100339,100340,100343,100344,100346,100347],{},"This box is now sealed. These IP addresses ",[106,100341,100342],{},"will not change"," for the entire journey. Your computer looks at Google's IP address (",[6979,100345,100317],{},") and realizes, ",[7289,100348,100349],{},"\"Google is not on my local Wi-Fi network. I need to hand this box to the router so it can leave the house.\"",[12,100351,100352,100353,100355],{},"To get the box to the router, your computer encapsulates the Packet inside a ",[106,100354,100045],{}," (the local delivery truck). It puts local MAC address labels on the outside of this truck:",[23,100357,100358,100364],{},[26,100359,100360,100363],{},[106,100361,100362],{},"Source MAC:"," Your Computer's MAC Address",[26,100365,100366,100369,100370],{},[106,100367,100368],{},"Destination MAC:"," Your Router's MAC Address ",[7289,100371,100372],{},"(Notice it does NOT use Google's MAC address!)",[12,100374,100375,100376,100379,100380,100382,100383,100386,100387,100390,100391,100394],{},"Your computer sends this Frame over the Wi-Fi or Ethernet cable. Because the Destination MAC belongs to your router, the router says, ",[7289,100377,100378],{},"\"Hey, this truck is for me!\""," and catches it. The router \"opens\" the Frame (it unloads the delivery truck). It looks at the inner IP Packet (the cardboard box) and sees the Destination IP is ",[6979,100381,100317],{}," (Google). The router says, ",[7289,100384,100385],{},"\"Okay, I need to send this to the internet provider next.\""," The router ",[106,100388,100389],{},"throws away your original Frame"," (the local truck is done with its job). The router builds a ",[106,100392,100393],{},"brand new Frame"," (a new truck) to drive the packet to the next stop (your ISP). The new Frame has:",[23,100396,100397,100402],{},[26,100398,100399,100401],{},[106,100400,100362],{}," Your Router's MAC Address",[26,100403,100404,100406],{},[106,100405,100368],{}," The ISP Router's MAC Address",[12,100408,100409,100410,100412,100413,100416,100417,100420,100421,100424,100425,100428,100429,100432],{},"It hides your local IP address (",[6979,100411,100321],{},"), and replaces your local IP with your house's ",[106,100414,100415],{},"Public IP Address"," (the single address your ISP assigned to your modem, say ",[6979,100418,100419],{},"203.0.113.5","). To remember that ",[7289,100422,100423],{},"your specific computer"," made this request, the router assigns a random, unique ",[106,100426,100427],{},"Port Number"," (let's say ",[6979,100430,100431],{},"#5001",") to the packet. The router then writes this in its NAT Table ledger:",[12,100434,100435],{},"    \"If any mail comes back from Google addressed to Port #5001, give it to the computer at 192.168.0.15.\"",[12,100437,100438,100439,100441,100442,100444],{},"Google receives the request. As far as Google is concerned, the request came from ",[6979,100440,100419],{}," on Port ",[6979,100443,100431],{},". Google has no idea your specific computer exists, nor does it know your MAC address.",[12,100446,100447,100448,6959],{},"Google packages up the webpage and sends it back to your house's Public IP, specifically tagging it for Port ",[6979,100449,100431],{},[12,100451,100452,100453],{},"The webpage arrives at your router. The router looks at the package and says, ",[7289,100454,100455],{},"\"Okay, this is from Google, and it's tagged for Port #5001.\"",[12,100457,100458,100459,100461,100462],{},"It opens up its NAT Table ledger, finds the entry for ",[6979,100460,100431],{},", and sees the note it left for itself: ",[7289,100463,100464],{},"\"Ah! Port #5001 means this belongs to 192.168.0.15.\"",[12,100466,100467,100468,100470],{},"Now that the router knows the internal IP address (",[6979,100469,100321],{},") that the data belongs to, it needs to hand the data across the house to your computer.",[12,100472,100473,100474,100476,100477,100480],{},"The router looks up your computer's IP address in its local directory to find your ",[106,100475,100119],{},". It packages Google's webpage into a brand new ",[106,100478,100479],{},"Local Frame",", slaps your computer's MAC address on the destination label, and pushes it out over the Wi-Fi. Your computer sees its MAC address, catches the frame, and loads the webpage!",[6964,100482,100484],{"id":100483},"network-operating-systems","Network Operating Systems",[12,100486,100487],{},"Now that we know how data travels across networks, how do we actually control the devices—like routers and switches—that are directing all this traffic? To do that, we need to look at the software running on the hardware.",[12,100489,100490,100491,100494,100495,6959],{},"Routers, switches, and other infrastructure devices don't run Windows or macOS — they run specialised ",[106,100492,100493],{},"network operating systems",". On Cisco devices, this is the ",[106,100496,100497],{},"Cisco IOS (Internetwork Operating System)",[12,100499,100500],{},"A few key points about how IOS works:",[23,100502,100503,100510,100516],{},[26,100504,100505,100506,100509],{},"The IOS file is stored in ",[106,100507,100508],{},"flash memory"," (non-volatile, survives reboots).",[26,100511,100512,100513,100515],{},"On boot, the IOS is copied from flash into ",[106,100514,75475],{},", where it runs while the device is operating.",[26,100517,100518,100519,100522,100523,100526],{},"Like any OS, it has a ",[106,100520,100521],{},"kernel"," (talks to hardware) and a ",[106,100524,100525],{},"shell"," (the interface you interact with), which can be a CLI or GUI.",[7166,100528,100530],{"id":100529},"ways-to-access-the-cli","Ways to Access the CLI",[12,100532,100533],{},"There are several ways to get into a device's command-line interface:",[7085,100535,100536,100547],{},[7088,100537,100538],{},[7091,100539,100540,100542,100544],{},[7094,100541,7922],{"align":47883},[7094,100543,85952],{"align":47883},[7094,100545,100546],{"align":47883},"Notes",[7104,100548,100549,100562,100574,100590],{},[7091,100550,100551,100556,100559],{},[7109,100552,100553],{"align":47883},[106,100554,100555],{},"Console",[7109,100557,100558],{"align":47883},"Out-of-band",[7109,100560,100561],{"align":47883},"Physical cable to the console port. Used for initial setup and maintenance. Works even if the network is down.",[7091,100563,100564,100569,100571],{},[7109,100565,100566],{"align":47883},[106,100567,100568],{},"AUX",[7109,100570,100558],{"align":47883},[7109,100572,100573],{"align":47883},"Older method using a modem and phone line. Also doesn't require network services.",[7091,100575,100576,100580,100583],{},[7109,100577,100578],{"align":47883},[106,100579,83020],{},[7109,100581,100582],{"align":47883},"In-band",[7109,100584,100585,100586,100589],{"align":47883},"Remote access over the network via a virtual interface. Requires the network to be up. ",[106,100587,100588],{},"Not secure","— data is sent in plain text.",[7091,100591,100592,100597,100599],{},[7109,100593,100594],{"align":47883},[106,100595,100596],{},"SSH",[7109,100598,100582],{"align":47883},[7109,100600,100601,100602],{"align":47883},"Like Telnet, but with encryption and stronger authentication. ",[106,100603,100604],{},"Always prefer SSH over Telnet.",[12,100606,100607],{},"Common terminal emulator programs include PuTTY, Tera Term, SecureCRT, and the macOS Terminal.",[7166,100609,100611],{"id":100610},"cli-modes","CLI Modes",[12,100613,100614],{},"The Cisco IOS CLI is organised in a hierarchy of modes, each with different levels of access:",[3603,100616,100617,100625,100637,100649],{},[26,100618,100619,10476,100622,100624],{},[106,100620,100621],{},"User EXEC mode",[6979,100623,8279],{},") — Limited monitoring commands. You land here by default.",[26,100626,100627,10476,100630,100633,100634,6959],{},[106,100628,100629],{},"Privileged EXEC mode",[6979,100631,100632],{},"#",") — Full access to monitoring and management commands. Enter with ",[6979,100635,100636],{},"enable",[26,100638,100639,10476,100642,100645,100646,6959],{},[106,100640,100641],{},"Global configuration mode",[6979,100643,100644],{},"(config)#",") — Used to make device-wide configuration changes. Enter with ",[6979,100647,100648],{},"configure terminal",[26,100650,100651,100654,100655,100658,100659,100662],{},[106,100652,100653],{},"Specific configuration modes"," — Sub-modes like interface configuration (",[6979,100656,100657],{},"(config-if)#",") or line configuration (",[6979,100660,100661],{},"(config-line)#",") for configuring specific components.",[6964,100664,100666],{"id":100665},"basic-commands-securing-your-device","Basic Commands: Securing Your Device",[12,100668,100669],{},"Once you're in the CLI, some of the first things you'll want to do are give your device a name and lock it down with passwords.",[7166,100671,100673],{"id":100672},"setting-a-hostname","Setting a Hostname",[12,100675,100676,100677,100680,100681,6959],{},"Hostnames help you identify devices on the network. A descriptive hostname (like ",[6979,100678,100679],{},"HQ-Router-1",") is much more useful than the default ",[6979,100682,99243],{},[7166,100684,100686],{"id":100685},"passwords-you-should-know","Passwords You Should Know",[12,100688,100689],{},"Passwords are the first line of defence against unauthorised access:",[23,100691,100692,100698,100711,100717],{},[26,100693,100694,100697],{},[106,100695,100696],{},"Enable password"," — Restricts access to privileged EXEC mode (stored in plain text — avoid using this alone).",[26,100699,100700,100703,100704,100707,100708,6959],{},[106,100701,100702],{},"Enable secret"," — Same purpose as above, but the password is ",[106,100705,100706],{},"encrypted",". Always use this instead of ",[6979,100709,100710],{},"enable password",[26,100712,100713,100716],{},[106,100714,100715],{},"Console password"," — Protects the physical console port.",[26,100718,100719,100722],{},[106,100720,100721],{},"VTY password"," — Protects remote access via Telnet\u002FSSH.",[6964,100724,47762],{"id":47761},[23,100726,100727,100730,100736,100744,100752,100759],{},[26,100728,100729],{},"Protocols are the rules that govern all network communication — from encoding to timing to delivery.",[26,100731,9455,100732,100735],{},[106,100733,100734],{},"TCP\u002FIP model"," organises protocols into layers, each with a distinct job.",[26,100737,100738,100740,100741,100743],{},[106,100739,22036],{}," wraps data with headers at each layer; ",[106,100742,100065],{}," unwraps them at the destination.",[26,100745,100746,100748,100749,100751],{},[106,100747,83957],{}," handle end-to-end delivery; ",[106,100750,83961],{}," handle hop-by-hop delivery.",[26,100753,100754,100755,100758],{},"Network devices run specialised operating systems like ",[106,100756,100757],{},"Cisco IOS",", accessed through console, Telnet, or SSH.",[26,100760,100761,100762,100765],{},"Always ",[106,100763,100764],{},"secure your devices"," with hostnames, encrypted passwords, and SSH access.",{"title":180,"searchDepth":181,"depth":181,"links":100767},[100768,100771,100775,100779,100782,100786,100790],{"id":99785,"depth":181,"text":99786,"children":100769},[100770],{"id":99828,"depth":7430,"text":99829},{"id":99889,"depth":181,"text":99890,"children":100772},[100773,100774],{"id":99910,"depth":7430,"text":99911},{"id":99921,"depth":7430,"text":99922},{"id":99932,"depth":181,"text":99933,"children":100776},[100777,100778],{"id":99939,"depth":7430,"text":99940},{"id":99969,"depth":7430,"text":99970},{"id":100129,"depth":181,"text":100130,"children":100780},[100781],{"id":100277,"depth":7430,"text":100278},{"id":100483,"depth":181,"text":100484,"children":100783},[100784,100785],{"id":100529,"depth":7430,"text":100530},{"id":100610,"depth":7430,"text":100611},{"id":100665,"depth":181,"text":100666,"children":100787},[100788,100789],{"id":100672,"depth":7430,"text":100673},{"id":100685,"depth":7430,"text":100686},{"id":47761,"depth":181,"text":47762},"2026-05-19",{},"\u002Fblog\u002FNetwork-Communication,-Chapter-2",{"title":99780,"description":180},{"loc":100793},"blog\u002FNetwork-Communication,-Chapter-2","bvCZzF7GxG3qrC-YRuynwdmrAxMAkTqWy9gJZUJAlc0",{"id":100799,"title":100800,"author":7,"body":100801,"date":101053,"description":101054,"draft":185,"edited_at":81622,"extension":186,"featured_image":101055,"meta":101056,"navigation":189,"path":101057,"pinned":185,"seo":101058,"sitemap":101059,"stem":101060,"tags":194,"__hash__":101061},"blog\u002Fblog\u002FBalance（二）前庭覺-(i).md","Balance（二）前庭覺",{"type":9,"value":100802,"toc":101051},[100803,100835,100838,100851,100899,100908,100924,100926,100941,100956,100971,101005,101015,101040,101042],[12,100804,100805,100806,100808,100809,100813,100815,100816,100820,100821,100823,100824,100828,100829,100831,100832,100834],{},"接著講到前庭覺",[16,100807],{},"\n它主要是提供了關於",[75,100810,100812],{"className":100811},[5387],"頭部相對重力及動作的資訊",[16,100814],{},"\n那左右兩邊前庭都會提供",[75,100817,100819],{"className":100818},[5387],"相同(重複)","的感覺訊息，假如一邊壞掉了，也會有另一邊做補充",[16,100822],{},"\n也是唯一能夠定義",[75,100825,100827],{"className":100826},[5387],"自體有沒有動","的感覺，能分清是別人歪掉還是自己歪掉",[16,100830],{},"\n假若當視覺與軀體感覺有衝突，前庭覺就會介入並給出最終答案（但假若前庭覺壞掉了，那就會引發頭昏眼花）",[16,100833],{},"\n甚至還可以活化抗重力肌肉來提供適當的維持姿勢能力",[12,100836,100837],{},"那前庭覺也分成了周邊的跟中樞的兩個組件",[12,100839,100840,100841,100845,100846,100850],{},"先來講周邊的前庭覺，主要是前庭迷路，由",[75,100842,100844],{"className":100843},[173],"半規管","以及",[75,100847,100849],{"className":100848},[173],"內耳石","組成",[12,100852,100853,100854,100856,100857,100861,100862,100864,100865,100869,100870,100872,100876,100877,100879,100883,100884,100886,100890,100891,100893,100894,100898],{},"那半規管的構造，就是一個充滿液體的環狀結構，裡面包含了纖毛細胞",[16,100855],{},"\n而這個構造能提供",[75,100858,100860],{"className":100859},[5387],"頭部旋轉及角速度","的訊息",[16,100863],{},"\n而左右耳各三個半規管（前後外半規管），且彼此",[75,100866,100868],{"className":100867},[173],"互相垂直","，而分別為這樣組合來掌管不同的方向",[16,100871],{},[75,100873,100875],{"className":100874},[79],"左右外半規管","——左右轉頭",[16,100878],{},[75,100880,100882],{"className":100881},[173],"左前+右後半規管","——側前方傾斜",[16,100885],{},[75,100887,100889],{"className":100888},[78392],"右前+左後半規管","——另一個方向的側前方傾斜",[16,100892],{},"\n而這三個平面都十分接近眼睛",[75,100895,100897],{"className":100896},[173],"外眼肌肉","，所以是容易控住外眼肌肉的路徑",[12,100900,100901,100902,100904,100905,100907],{},"而半規管的開口有一個叫壺腹，它裡面有動跟靜的纖毛，這兩個纖毛互動可以產生過極化以及去極化",[16,100903],{},"\n去極化——偵測頭部速度增加（靜纖毛靠近動纖毛）",[16,100906],{},"\n過極化——偵測頭部速度減少（靜纖毛遠離動纖毛）",[12,100909,100910,100911,100861,100915,100917,100918,100920,100921,100923],{},"那周邊前庭覺的第二個零件就是內耳石，與剛剛半規管是管旋轉的不同，它管的是",[75,100912,100914],{"className":100913},[5387],"重力和直線加速度",[16,100916],{},"\n那內耳石也有兩個構造",[16,100919],{},"\n橢圓囊（管水平方向）——側彎頭部刺激最大",[16,100922],{},"\n球狀囊（管垂直方向）——前後彎頭部刺激最大",[36,100925],{},[12,100927,100928,100929,100931,100932,100936,100937],{},"接著講中樞的前庭覺",[16,100930],{},"\n那剛剛在周邊的前庭覺接收到的訊息會先經過",[75,100933,100935],{"className":100934},[6760],"前庭耳蝸腦神經","再到小腦和延腦之間的",[75,100938,100940],{"className":100939},[5387],"前庭核",[12,100942,100943,100944,100946,100947,100949,100950,100952,100953,100955],{},"而前庭核一共有四顆，以下是它們的分工",[16,100945],{},"\n上&內核——主要是管眼睛，負責凝視穩定",[16,100948],{},"\n內核——還要協調眼睛跟頭部的動作，以及肢體&軀體動作感覺輸入",[16,100951],{},"\n外核——主要調控肌肉跟軀體協調（姿勢穩定）",[16,100954],{},"\n下核——連接其他神經核以及網狀系統",[12,100957,100958,100959,100961,100962,100964,100965,100967,100968,100970],{},"那前庭核接收了之後會往這四個地方發送信號",[16,100960],{},"\n動眼中心——調控VOR",[16,100963],{},"\n脊髓——分成內側前庭脊髓徑（控制頭頸部移動）和外側前庭脊髓徑（身體的移動）",[16,100966],{},"\n小腦——較正VOR",[16,100969],{},"\n大腦深處（例如網狀系統）——引發嘔吐噁心",[12,100972,100973,100974,100976,100977,100981,100983,100987,100989,100990,100994,100995,100999,101000],{},"接著講前庭的運作邏輯",[16,100975],{},"\n前庭覺它的放電一般會維持在70-100PPS，即使是",[75,100978,100980],{"className":100979},[5387],"睡覺或者沒在動也不會歸零",[16,100982],{},[75,100984,100986],{"className":100985},[78],"放電的多寡取決於速度的變化",[16,100988],{},"\n且它會有",[75,100991,100993],{"className":100992},[173],"對稱性","，假如頭部的角速度或者水平速度發生改變，同側的",[75,100996,100998],{"className":100997},[79],"放電會增加","，對側的將會",[75,101001,101004],{"className":101002},[101003],"teal-3","減少",[12,101006,101007,101008,101012,101014],{},"接著來講VOR的特性——用來",[75,101009,101011],{"className":101010},[5387],"維持凝視穩定",[16,101013],{},"\n它是建立在注視上面的，假如眼盯前方，頭部往左轉，那半規管就會被觸發，讓眼睛就會自動往右走",[12,101016,101017,101018,101020,101021,101025,101026,101028,101029,101032,101033,101028,101035,101039],{},"那對於轉向的速度也有三種調節能力",[16,101019],{},"\n頭部速度小於",[75,101022,101024],{"className":101023},[81363],"每秒60度","——就是自己眼睛能夠跟上的，不需要特別調節（平滑追蹤）",[16,101027],{},"\n頭部速度大於",[75,101030,101024],{"className":101031},[79],"——需要使用VOR協助",[16,101034],{},[75,101036,101038],{"className":101037},[173],"每秒350度","——連凝視都沒辦法維持了，會採用跳視(saccade)方式",[36,101041],{},[12,101043,101044,101045,101047,101048,101050],{},"那這邊提到一個叫推-拉機制的東西，那記得上面我們說的半規管兩邊，方向上互為一對",[16,101046],{},"\n而假如一對的一邊半規管被興奮，那另一邊的半規管則會被抑制",[16,101049],{},"\n（轉向哪邊 哪邊就會被興奮 對面則會被抑制 且這麼抑制只會降到0）",{"title":180,"searchDepth":181,"depth":181,"links":101052},[],"2026-04-06","周邊以及中樞前庭的構造及運作方式","\u002Fimages\u002Fuploads\u002F1775545346184-141163959_p0.png",{},"\u002Fblog\u002FBalance（二）前庭覺-(i)",{"title":100800,"description":101054},{"loc":101057},"blog\u002FBalance（二）前庭覺-(i)","6ogKFXo4XYEMBmR1kxUkjSgvTFj9cWhOfEF371RLey0",{"id":101063,"title":101064,"author":7,"body":101065,"date":101053,"description":101312,"draft":185,"edited_at":101053,"extension":186,"featured_image":101313,"meta":101314,"navigation":189,"path":101315,"pinned":185,"seo":101316,"sitemap":101317,"stem":101318,"tags":194,"__hash__":101319},"blog\u002Fblog\u002FCVA（七）評估-(i).md","CVA（七）評估 (i)",{"type":9,"value":101066,"toc":101310},[101067,101073,101076,101092,101101,101104,101123,101138,101161,101202,101207,101210,101257,101283,101298],[12,101068,101069,101072],{},[5610,101070,101071],{},"CVA好像有點多 看不到頭哦","  目前進度計量 76\u002F292頁",[12,101074,101075],{},"那CVA評估方向有分這幾種",[12,101077,101078,101079,101081,101085,101086,101088,101089,101091],{},"首先是評估認知跟意識的量表，包括了",[16,101080],{},[75,101082,101084],{"className":101083},[78,5387],"JOMAC","——Judgment(常識\u002F判斷)、Orientation(定向感)、Memory、Abstract thinking(抽象思考)、Calculation(計算)",[16,101087],{},"\nMMSE",[16,101090],{},"\nGCS",[12,101093,101094,101095,101097,101098,101100],{},"接著是功能量表，包括了",[16,101096],{},"\nBarthel index(巴氏量表)",[16,101099],{},"\nFuntional Independent measure(FIM)——這個比巴氏量表測得還要更精細",[12,101102,101103],{},"肌張力評估，有Modified Ashworth Scale(MAS)",[12,101105,101106,101107,101109,101113,101114,101116,101117,101119,101120,101122],{},"動作功能整合測試，包括了",[16,101108],{},[75,101110,101112],{"className":101111},[78,5387],"Brunnstorm Method","——分了六期 而第二期會開始出現協同動作(區分)",[16,101115],{},"\nFugl-Meyer Assessment——這個是根據布朗斯壯量表改良的",[16,101118],{},"\nMotor Assessment Scale",[16,101121],{},"\nNIH Stroke Scale",[12,101124,101125,101126,7206,101130,101134,101135,101137],{},"我們先來講布朗斯壯（跑台要跑哦）（可以再複習一下這兩份 ",[2330,101127,98736],{"href":101128,"rel":101129},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E5%91%A8%E5%9C%8D%E7%90%86%E8%AB%96%E5%AD%B8%E6%B4%BE%EF%BC%88%E4%BA%8C%EF%BC%89%E9%9A%8E%E5%B1%A4%E7%90%86%E8%AB%96?series=Motor%20Control",[2334],[2330,101131,98740],{"href":101132,"rel":101133},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E5%8B%95%E4%BD%9C%E5%AD%B8%E7%BF%92%EF%BC%88%E5%8D%81%E4%B8%89%EF%BC%89%E8%87%A8%E5%BA%8A%E6%87%89%E7%94%A8?series=Motor%20learning",[2334]," 的階層理論跟布朗斯壯的描述）",[16,101136],{},"\n首先來測的是感覺，分了四種測試方向",[23,101139,101140,101146,101152,101158],{},[26,101141,101142,101143,101145],{},"被動運動覺（本體感覺）——蒙住病人眼，分別移動病人的肩、肘、前臂、腕、手指、下肢",[16,101144],{},"\n（這樣才可以知道病人需不需要一直專注看著自己腳來移動）（患測）",[26,101147,101148,101149,101151],{},"指尖辨識——蒙住病人眼，碰病人的指尖看他分不分得出來是碰哪根",[16,101150],{},"\n（測解覺跟大腦皮層的整合能力）（患測）",[26,101153,101154,101155,101157],{},"足底感覺——拿毛巾分別放在病人的前足，後足，內足，外足測他看能不能分不分得出",[16,101156],{},"\n（這跟平衡有關）（患測）",[26,101159,101160],{},"動態運動覺——蒙住病人眼，PT手抓患測帶著移動，讓病人的好手跟著這個速度跟方向跟著動",[12,101162,101163,101164,101166,101167,101169,101170,101174,101175,101179,101180,101182,101183,101187,101188,101190,101191,101193,101194,101198,101199,101201],{},"然後就是布朗斯壯的運動測試，它分成了六期，稍微解釋一下",[16,101165],{},"\n首先是第一期，沒有張力也沒有動作的",[16,101168],{},"\n第二期，它有",[75,101171,101173],{"className":101172},[5387],"聯合動作出現","(伴隨協同動作)（是用來",[75,101176,101178],{"className":101177},[5387],"區分第一期跟第二期","的指標），而且開始有痙攣出現，但還沒有主動動作",[16,101181],{},"\n第三期，痙攣是",[75,101184,101186],{"className":101185},[1149],"最強","的，雖然開始有主動動作，但還是被協同動作主導",[16,101189],{},"\n第四期，這時痙攣開始下降，而且動作也開始偏離協同動作模式",[16,101192],{},"\n第五期，痙攣",[75,101195,101197],{"className":101196},[6760],"最弱","，基本上也沒有協同動作出現",[16,101200],{},"\n第六期，無痙攣，並且可以做到單關節動作，動作開始是協調的(雖然還有些笨拙)",[12,101203,101204],{},[1930,101205],{"alt":1932,"src":101206},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775437999044-Gemini_Generated_Image_5mnz195mnz195mnz.png",[12,101208,101209],{},"emmm，跑台的話看後面要不要再開一章做整理好了",[12,101211,101212,101213,101215,101216,101218,101219,101223,101224,101226,101227,101231,101232,101234,101235,101239,101240,101242,101243,101245,101246,101248,101249,101251,101252,2589],{},"接著講測試張力的MAS(Modified Ashworth Scale)，它分成了六個等級",[16,101214],{},"\nGrade 0  ——正常人",[16,101217],{},"\nGrade 1  ——到關節活動度",[75,101220,101222],{"className":101221},[5387],"末端","才有張力出現",[16,101225],{},"\nGrade 1+  ——",[75,101228,101230],{"className":101229},[1149],"後半段","關節活動度出現張力",[16,101233],{},"\nGrade 2  ——",[75,101236,101238],{"className":101237},[6760],"大半段","活動度都有出現張力",[16,101241],{},"\nGrade 3  ——很難拉得動，但還是能拉",[16,101244],{},"\nGrade 4  ——拉不動了",[16,101247],{},"\n（補充，因為張力會受到速度影響(速度相依)，而另一個表MTS是MAS再把速度整合進去的版本）",[16,101250],{},"\n（Clasp-knife Response(摺刀反應)意思是在測試張力過程中，阻力突然消失了，代表上運動神經元受損——",[2330,101253,101256],{"href":101254,"rel":101255},"https:\u002F\u002Fwww.youtube.com\u002Fwatch?v=JJ9Hg1hZS9Q",[2334],"補一條影片",[12,101258,101259,101260,101262,101263,101265,101266,101268,101269,101271,101272,101276,101277,101279,101280,98882],{},"然後來講FMA，它主要看五大面向加起來113個測試項目",[16,101261],{},"\n動作功能——包括了上下肢",[16,101264],{},"\n感覺功能——分為了Light touch兩面(輕觸覺)和Position Sense八個(位置覺)",[16,101267],{},"\n平衡——七個測試（坐姿三個，站姿四個）",[16,101270],{},"\n關節活動度——",[75,101273,101275],{"className":101274},[79],"九個","關節測ROM",[16,101278],{},"\n關節疼痛——一樣是",[75,101281,101275],{"className":101282},[79],[12,101284,101285,101286,101288,101289,101291,101292,101294,101295,101297],{},"單論上肢就總分126分，包括了",[16,101287],{},"\nMotor(66)",[16,101290],{},"\nSensation(12)",[16,101293],{},"\nROM(24)",[16,101296],{},"\nPain(24)",[12,101299,101300,101301,101303,101304,101306,101307,101309],{},"那FMA的評分是0~2分來看的",[16,101302],{},"\n0分是完全沒有動作",[16,101305],{},"\n1分是能做到一部份抑或者不完美",[16,101308],{},"\n2分是流暢完美的完成",{"title":180,"searchDepth":181,"depth":181,"links":101311},[],"評估 Brunnstrom~FMA","\u002Fimages\u002Fuploads\u002F1775441596804-HE-2PhpbAAAPbL6.png",{},"\u002Fblog\u002FCVA（七）評估-(i)",{"title":101064,"description":101312},{"loc":101315},"blog\u002FCVA（七）評估-(i)","Zfllyd4sSVYTMUdR_GCYlWHqBhtWA_1XVBnrhtWDF74",{"id":101321,"title":101322,"author":7,"body":101323,"date":101053,"description":101523,"draft":185,"edited_at":101053,"extension":186,"featured_image":101524,"meta":101525,"navigation":189,"path":101526,"pinned":185,"seo":101527,"sitemap":101528,"stem":101529,"tags":194,"__hash__":101530},"blog\u002Fblog\u002FCVA（八）評估-(ii).md","CVA（八）評估 (ii)",{"type":9,"value":101324,"toc":101521},[101325,101328,101361,101382,101418,101427,101432,101480,101495,101515],[12,101326,101327],{},"繼續講評估",[12,101329,101330,101331,101333,101334,101336,101337,101339,101340,101342,101343,101345,101346,101348,101349,101351,101352,101354,101355,101357,101358,101360],{},"首先是Motor Assessment Scale",[16,101332],{},"\n它這個表比起上一份說的布朗斯壯（粗大動作）以及FMA（精細動作）不同",[16,101335],{},"\n這個更著重於功能性部份(ADL)，更加臨床，它分成了八個項目",[16,101338],{},"\nSupine to side lying",[16,101341],{},"\nSupine to sitting over the edge of a bed(仰躺到床緣再坐起)",[16,101344],{},"\nBalanced sitting",[16,101347],{},"\nSitting to Standing",[16,101350],{},"\nWalking",[16,101353],{},"\nUpper-arm function",[16,101356],{},"\nHand movements",[16,101359],{},"\nAdvance hand activites",[12,101362,101363,101364,101366,101367,101369,101370,101372,101373,101375,101376,101378,101379,101381],{},"接著講NIHSS的部份，它分成了11評估項目，一共42分，越高分越嚴重",[16,101365],{},"\n0分就是正常",[16,101368],{},"\n1~4分是輕度中風",[16,101371],{},"\n5~14分是中度中風",[16,101374],{},"\n15~24分是中重度重風",[16,101377],{},"\n24分以上是十分嚴重的中風",[16,101380],{},"\n（而10分以下的預後會有60%~70%機會恢復很好，20分以上預後會變十分差）",[12,101383,101384,101385,101387,101388,101390,101391,101393,101394,101396,101397,101399,101400,101402,101403,101405,101406,101408,101409,101411,101412,101414,101415,101417],{},"那十一項評分項目包括了",[16,101386],{},"\n意識——意識的評分又分成了三類（水準，問題，指令）",[16,101389],{},"\n眼球運動",[16,101392],{},"\n視野",[16,101395],{},"\n面部肌肉",[16,101398],{},"\n上肢運動",[16,101401],{},"\n下肢運動",[16,101404],{},"\n肢體運動失調",[16,101407],{},"\n感覺功能",[16,101410],{},"\n語言功能",[16,101413],{},"\n構音困難",[16,101416],{},"\n忽略症",[12,101419,101420,101421,101426],{},"下圖是來自網上抓的台中榮民總醫院的",[2330,101422,101425],{"href":101423,"rel":101424},"https:\u002F\u002Fwww.vghtc.gov.tw\u002FUploadFiles\u002FWebFiles\u002FWebPagesFiles\u002FFiles\u002Ff3802ff1-0016-44af-939f-fb045f9d5a4a\u002F20190419%20NIHSS%E4%BB%8B%E7%B4%B9%E5%8F%8A%E6%87%89%E7%94%A8.pdf",[2334],"NIHSS 中風量表之介紹與應用"," 雖然它這邊有分了左右的上下肢體運動 但總分一樣是42",[12,101428,101429],{},[1930,101430],{"alt":1932,"src":101431},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775443710135-Screenshot_2026-04-06_103756.png",[12,101433,101434,101435,101437,101438,101440,101441,101443,101444,101446,101447,101449,101450,101452,101453,101455,101456,101458,101459,101461,101462,101464,101465,101467,101468,101470,101471,101473,101474,101476,101477,101479],{},"然後來講伯格氏平衡量表Berg Balance Scale，它原本是來預測老人的跌倒風險的，但發現用在CVA上也不錯",[16,101436],{},"\n有以下十四點",[16,101439],{},"\nSitting unsupported——不支撐坐著",[16,101442],{},"\nChange of position--sitting to standing——坐到站",[16,101445],{},"\nChange of position--standing to sitting——站到坐",[16,101448],{},"\nTransfers——轉移位",[16,101451],{},"\nStanding unsupported——不支撐站著",[16,101454],{},"\nStanding with eyes closed——閉眼站著",[16,101457],{},"\nStanding with feet together——雙腳拼攏站著",[16,101460],{},"\nTandem standing——腳尖碰腳跟站著",[16,101463],{},"\nStanding on one leg——單腳站",[16,101466],{},"\nTurning Trunk(feet fix)——轉軀幹但腳不動",[16,101469],{},"\nRetrieving objects from floor——從地上撿東西",[16,101472],{},"\nTurning 360 degrees——轉身360度",[16,101475],{},"\nStool stepping——用一個踏板輪流讓左右腳上去下來（重量轉移）",[16,101478],{},"\nReaching forward while standing——站立前往伸手",[12,101481,101482,101483,101485,101486,101488,101489,101491,101492,101494],{},"而這個表的得分會長這樣",[16,101484],{},"\n0~20分——一定會跌倒，建議坐輪椅",[16,101487],{},"\n21~40分——中度跌倒風險，建議用輔具",[16,101490],{},"\n41~56分——低跌倒風險，可以獨立行走",[16,101493],{},"\n（而老師有補充低於36分就一定會跌倒，45分以內也會高機率跌倒）",[12,101496,101497,101498,101500,101501,101505,101506,101508,101509,101511,101512,101514],{},"最後講的是Functional Reach Test，這是一個可以快速中風或者老人的跌倒風險測試",[16,101499],{},"\n那最基本測試對象",[75,101502,101504],{"className":101503},[5387],"需要這兩點能力","（獨立站立30秒，肩至少能平舉90度）",[16,101507],{},"\n然後是判讀的部分",[16,101510],{},"\n假如病人能向前伸6inch(15cm)以下，代表他跌倒風險高",[16,101513],{},"\n假如病人能向前伸6~10inch(25cm)，代表跌倒風險是中",[12,101516,101517,101518,101520],{},"接下來有一大堆跑台的東西了 我之後再整理（啊好多啊~",[16,101519],{},"\n目前進度50\u002F292頁",{"title":180,"searchDepth":181,"depth":181,"links":101522},[],"Motor Assessment Scale~Functional Reach Test","\u002Fimages\u002Fuploads\u002F1775445542448-135083397_p0.png",{},"\u002Fblog\u002FCVA（八）評估-(ii)",{"title":101322,"description":101523},{"loc":101526},"blog\u002FCVA（八）評估-(ii)","A55pnDiozacPmO2cI-2mTWShY3GQKE0CuZoTjuReemw",{"id":101532,"title":101533,"author":7,"body":101534,"date":101053,"description":101670,"draft":185,"edited_at":101053,"extension":186,"featured_image":101671,"meta":101672,"navigation":189,"path":101673,"pinned":185,"seo":101674,"sitemap":101675,"stem":101676,"tags":194,"__hash__":101677},"blog\u002Fblog\u002F平衡與前庭失調（一）.md","平衡與前庭失調（一）",{"type":9,"value":101535,"toc":101668},[101536,101539,101545,101562,101574,101586,101592,101597,101600,101603,101622,101628,101636,101645,101662],[12,101537,101538],{},"小兒的Motor Learning也稍微講到一些平衡的內容，不過這一課是直接攤開來講，努力吃吧，加油",[12,101540,101541,101542,101544],{},"首先來講定義",[16,101543],{},"\n維持平衡需要持續接收整合感覺輸入，接著去制定活動的計劃並執行，以達到維持直立平衡的目標。",[12,101546,101547,101548,101550,101551,101555,101556,101558,101559,101561],{},"那平衡要達成，其實就是在處理這三個參數",[16,101549],{},"\nCOG——重心在站立時會在",[75,101552,101554],{"className":101553},[5387],"S2","，且重心會按照肢體動作\u002F位置而改變",[16,101557],{},"\nBOS——這個區域當然就是越大越穩，越小越不穩（站著時是雙腳之間的範圍，坐著的是大腿和臀部構成的範圍）",[16,101560],{},"\nLOS——就是在原本BOS不變情況下，身體能在各個方向傾斜到的最遠距離",[12,101563,101564,101565,101567,101568,101570,101571,101573],{},"那維持平衡的模式，會分成三種類型（預防跌倒的三種反應模式）",[16,101566],{},"\n預期性——它是按照經驗來看的，主要是活動前會採取這個",[16,101569],{},"\n反應性——靠臨場反應",[16,101572],{},"\n適應性——主要是看任務或者環境來調整的",[12,101575,101576,101577,101579,101580,101582,101583,101585],{},"最後是環境的參數，那它會改變包括剛剛說到的最基本參數，以及因應這三個而改變",[16,101578],{},"\n地面性質——平坦草地VS沙漠",[16,101581],{},"\n視覺環境——昏暗環境VS光亮環境",[16,101584],{},"\n意圖跟選擇的任務——主動去做某些動作VS下意識（被動的，為了挽回某些情況）做的某些動作",[12,101587,101588,101589,101591],{},"那關於人類平衡的方式，這邊介紹一個觀點叫系統方式",[16,101590],{},"\n它是指說動態平衡的條件是按照這三點組成：個人、任務、環境（對 很熟悉對吧）",[12,101593,101594],{},[1930,101595],{"alt":1932,"src":101596},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775460234558-Screenshot_2026-04-06_151838.png",[12,101598,101599],{},"那我就跳過講解這三要素好了，直接上圖",[12,101601,101602],{},"那接下來講周邊神經的接收，那跟姿勢控制相關的有這三個：軀體感覺、視覺、前庭覺",[12,101604,101605,101606,101610,101611,101615,101616,101618,101619,101621],{},"首先是視覺，那視角也是分成了",[75,101607,101609],{"className":101608},[79],"中間","眼睛清淅看到的，還有",[75,101612,101614],{"className":101613},[173],"周圍","不是直接注視的兩個部份",[16,101617],{},"\n那先來講中間直接注視的部份，主要是用來辦別環境方位包括",[16,101620],{},"\n判斷垂直，物體的動作（其他東西有沒有在動），環境裡的其他因素",[12,101623,101624,101625,101627],{},"然後講周圍的視角，主要是本體相對環境的的動作，包括了頭的晃動跟身體的晃動",[16,101626],{},"\n而上面就相當於視覺的本體感覺",[12,101629,101630,101631,101635],{},"那視角對於個人與",[75,101632,101634],{"className":101633},[5387],"任務及環境的互動","十分重要（抓起物品，變動的環境）",[12,101637,101638,101639,101641,101642,101644],{},"那視覺評估有兩種",[16,101640],{},"\nFocal vision視銳度檢查",[16,101643],{},"\nAmbient vision視野檢查",[12,101646,101647,101648,101650,101651,100845,101655,101659,101661],{},"接著講到了軀體感覺，主要是包括了關節，肌肉，肌腱，皮膚給的訊息",[16,101649],{},"\n主要是負責",[75,101652,101654],{"className":101653},[5387],"直立姿勢控制",[75,101656,101658],{"className":101657},[5387],"自動姿勢反應",[16,101660],{},"\n相關的功能動作還包括了牽張反射，Flex withdraw&Crosses extensor reflex(踩到釘子，同側腳屈曲，對側腳伸直)",[12,101663,101664,101665,101667],{},"那常見會有這些病會影響軀體感覺",[16,101666],{},"\n包括了 糖尿病導致的神經病變，周邊血管疾病，SCI，截肢",{"title":180,"searchDepth":181,"depth":181,"links":101669},[],"定義~軀體感覺","\u002Fimages\u002Fuploads\u002F1775464559030-136205774_p1_master1200.jpg",{},"\u002Fblog\u002F平衡與前庭失調（一）",{"title":101533,"description":101670},{"loc":101673},"blog\u002F平衡與前庭失調（一）","aZvykZYcZXC_PJM32BpefYnVTh8_AJK4pm7ottmkZwE",{"id":101679,"title":101680,"author":7,"body":101681,"date":101919,"description":101920,"draft":185,"edited_at":101921,"extension":186,"featured_image":101922,"meta":101923,"navigation":189,"path":101924,"pinned":185,"seo":101925,"sitemap":101926,"stem":101927,"tags":194,"__hash__":101928},"blog\u002Fblog\u002FCVA（六）.md","CVA（六）",{"type":9,"value":101682,"toc":101917},[101683,101699,101733,101753,101755,101758,101770,101788,101797,101800,101862,101896],[12,101684,101685,101686,101688,101689,101691,101692,101694,101695],{},"那從時期來分PT能幹嘛 分成了",[16,101687],{},"\n急性期，復健期，回歸社會環境",[16,101690],{},"\n那CVA運動功能恢復時長需要數月到數年",[16,101693],{},"\n而復健黃金期只有",[75,101696,101698],{"className":101697},[5387],"三到六個月",[12,101700,101701,101702,101704,101705,101709,101710,101712,101713,101717,101718,101720,101721,101725,101726,101728,101729],{},"CVA有相關的復健指標來預測",[16,101703],{},"\n第一點，假如復健期",[75,101706,101708],{"className":101707},[6760],"第一個月","表現很差，那中風後的半年的進步空間也會有限（86%）",[16,101711],{},"\n第二點，假如",[75,101714,101716],{"className":101715},[5387],"前兩周內","病人上肢有回復，那代表那個病人以後上肢功能性一定會完全康復",[16,101719],{},"\n第三點，假如",[75,101722,101724],{"className":101723},[6760],"中風後24天","內沒有回復握力，這樣上肢的功能性恢復一定很差",[16,101727],{},"\n第四點，假如第一周內腿沒動作，第四周內手沒有動作，那病人預後",[75,101730,101732],{"className":101731},[5387],"極差",[12,101734,101735,101736,101738,101739,101741,101742,101744,101745,101747,101748,101752],{},"從預後結果來看",[16,101737],{},"\n功能性是完全有機會康復的（只不能多次中風的話，難度會增大）",[16,101740],{},"\n就算是嚴重受損者例如包括認知功能受損，依然有機會讓他們功能性變好",[16,101743],{},"\n而且年齡不是決定預後的因素，主要要看動機以及受損程度",[16,101746],{},"\n尤其是",[75,101749,101751],{"className":101750},[6760],"年輕人","，因為年輕人在CVA上偏向出血型，出血型致死率更高",[36,101754],{},[12,101756,101757],{},"接著講損傷導致的限制跟失能部份",[12,101759,101760,101761,101763,101764,101766,101767,101769],{},"主要的損傷會以以下三點呈現",[16,101762],{},"\n第一點，癱瘓（MMT0分）或者無力",[16,101765],{},"\n第二點，高張力或低張力",[16,101768],{},"\n第三點，肌肉的啟動也有所改變",[12,101771,101772,101773,101775,101776,101778,101779,101781,101782,101784,101785,101787],{},"接著來講次級傷害，包括但不限於",[16,101774],{},"\n生理曲線與活動度改變",[16,101777],{},"\n肌肉與軟組織長度改變",[16,101780],{},"\n疼痛",[16,101783],{},"\n水腫",[16,101786],{},"\n等等等等.........",[12,101789,101790,101791,101793,101794,101796],{},"然後講動作上的問題，主要能觀察到這三種問題",[16,101792],{},"\n首先是動作喪失，異常動作模式，以及代償",[16,101795],{},"\n那這三個會漸漸影響ADL的獨立性，變得要別人來照顧",[12,101798,101799],{},"那中風病人會在復原中出現協同動作，可以分成上下肢以及屈曲伸直來看",[23,101801,101802,101814,101826,101839],{},[26,101803,101804,101805,24802,101809,101813],{},"那首先是上肢的屈曲協同動作——肩胛骨後縮和上提，",[75,101806,101808],{"className":101807},[6760],"肩外轉外旋",[75,101810,101812],{"className":101811},[5387],"肘屈曲","，前臂旋後，手指手腕屈曲",[26,101815,101816,101817,24802,101821,101825],{},"接著是上肢的伸展協同動作——肩胛骨前伸，",[75,101818,101820],{"className":101819},[5387],"肩內收內旋",[75,101822,101824],{"className":101823},[6760],"肘伸直","，前臂旋前，手腕手指屈曲",[26,101827,101828,101829,101833,101834,101838],{},"然後是下肢的屈曲協同動作——骨盤後縮上提(？)，髖",[75,101830,101832],{"className":101831},[5387],"彎曲外展外旋","，膝",[75,101835,101837],{"className":101836},[6760],"彎曲90度","，踝背曲內翻，腳趾背屈",[26,101840,101841,101842,101846,101847,24802,101852,101856,101857,101861],{},"最後是下肢的伸展協同動作——髖",[75,101843,101845],{"className":101844},[6760],"伸直","內旋",[75,101848,101851],{"className":101849},[101850],"bg-yellow-2","內收",[75,101853,101855],{"className":101854},[5387],"膝伸直","，踝",[75,101858,101860],{"className":101859},[5387],"蹠曲","內收，腳趾蹠曲",[12,101863,101864,101865,101867,101868,101872,101873,101875,101876,101880,101882,101883,101885,101886,101890,101892,101893,101895],{},"在中風病人身上還能看到聯合反應，簡單來說是病人健側用力，患側也會不自主用力",[16,101866],{},"\n上肢的話這個聯合反應是",[75,101869,101871],{"className":101870},[6760],"相同方向","的（例如我健側肘用力屈曲，患側也會肘屈曲）",[16,101874],{},"\n而下肢的則",[75,101877,101879],{"className":101878},[5387],"相反（內收、外展除外",[16,101881],{},"\n也會有這三種特殊反應",[16,101884],{},"\n那打哈欠、打噴嚏、咳嗽的話，它們的",[75,101887,101889],{"className":101888},[5387],"吸氣期會產生屈曲的協同動作",[16,101891],{},"\n而假如患側上肢有動作，那麼同側的下肢也會有動作（可以反過來，下肢先有動作讓上肢也有動作）(Homolateral Synkinesis)",[16,101894],{},"\n那結合剛剛上面說的因為下肢內收外展是同向，所以健側內收或外展，患側一樣是內收或外展",[12,101897,101898,101899,101901,101902,101904,101905,101907,101908,101910,101911,101913,101914,101916],{},"最後來講功能性限制，主要有以下六點",[16,101900],{},"\nADL",[16,101903],{},"\n床上活動能力",[16,101906],{},"\n轉移位",[16,101909],{},"\n平衡與姿勢控制",[16,101912],{},"\n移動能力",[16,101915],{},"\n上肢的功能性動作",{"title":180,"searchDepth":181,"depth":181,"links":101918},[],"2026-04-03","PT能幹嘛；受損導致的限制或失能","2026-04-07","\u002Fimages\u002Fuploads\u002F1775208758679-138277104_p0_master1200.jpg",{},"\u002Fblog\u002FCVA（六）",{"title":101680,"description":101920},{"loc":101924},"blog\u002FCVA（六）","wvQ7PvvQYioEBeqkdWkPnHLp6m6hRVGhWSvr1GkFiPg",{"id":101930,"title":101931,"author":7,"body":101932,"date":101919,"description":102182,"draft":185,"edited_at":101921,"extension":186,"featured_image":102183,"meta":102184,"navigation":189,"path":102185,"pinned":185,"seo":102186,"sitemap":102187,"stem":102188,"tags":194,"__hash__":102189},"blog\u002Fblog\u002FCVA（四）.md","CVA（五）",{"type":9,"value":101933,"toc":102180},[101934,101940,101954,101963,101978,101984,102005,102024,102051,102053,102056,102074,102092,102095,102118,102124,102141,102162,102171],[12,101935,101936,101937,101939],{},"在急性期我們能做甚麼呢，首先要看類型",[16,101938],{},"\n先來講血栓型跟TIA，接著要看的是缺血部位",[12,101941,101942,101943,101945,101946,101950,101951,101953],{},"一般來說缺血有分兩種變體 ",[16,101944],{},"\n一個叫中心區（80%～100%缺血），那這個區域細胞會在",[75,101947,101949],{"className":101948},[5387],"幾分鐘內死亡","（梗塞）",[16,101952],{},"\n一個叫半影區（50%～80%缺血），這個區域的話，假如趕緊搶救，是能救得回來的（手術上也是專門搶救這個區域）",[12,101955,101956,101957,101959,101960,101962],{},"接著來看做甚麼處置 ",[16,101958],{},"\n首先是預防 預防勝於治療這是肯定的了",[16,101961],{},"\n能做的是維持血壓及血流",[12,101964,101965,101966,101968,101969,101971,101972,101974,101975,101977],{},"但假如不幸的話，有兩種手段",[16,101967],{},"\n首先是藥物 有三種類型的藥物",[16,101970],{},"\n第一類是Tissue plasminogen activator（老師給翻譯是組織性血栓溶解因子,t-PA）：在90-180分鐘內最有效",[16,101973],{},"\n第二類是抗凝血藥：有肝素，華法林",[16,101976],{},"\n第三類是抗血小板藥：有著名的阿斯匹靈",[12,101979,101980,101981,101983],{},"第二種手段就是手術，有兩種手術類型",[16,101982],{},"\n血栓內膜切開術（直接對堵住區域的斑塊\u002F血栓清理）和血管移植",[12,101985,101986,101987,101989,101990,92333,101994,101996,101997,102001,102002,102004],{},"然後來看栓塞型 雖然他跟血栓型類似",[16,101988],{},"\n但主因是因為",[75,101991,101993],{"className":101992},[5387],"別的地方的東西經血流漂到腦血管並堵住",[16,101995],{},"\n所以治療目的應主要放在",[75,101998,102000],{"className":101999},[5387],"源頭上","例如心臟或大血管",[16,102003],{},"\n也有三種是高危險問題：心房震顫，心肌梗塞，人工瓣膜",[12,102006,102007,102008,102010,102011,102015,102016,102020,102021,102023],{},"接著是高血壓性的出血型",[16,102009],{},"\n一般來說，其實",[75,102012,102014],{"className":102013},[6760],"手術跟降壓","都",[75,102017,102019],{"className":102018},[5387],"沒有很好","的對這個類型做處置",[16,102022],{},"\n所以能做得只有預防和長期控制（長期吃降血壓藥）",[12,102025,102026,102027,102029,102030,102032,102033,102041,102042,102044,102045,102047,102048,102050],{},"最後來講動脈瘤破裂的出血型 有以下幾種措施",[16,102028],{},"\n手術",[16,102031],{},"\n床上休養（",[7289,102034,102035],{},[106,102036,102037],{},[75,102038,102040],{"className":102039},[5387],"絕對的床上休息","）4~6weeks",[16,102043],{},"\n降低動脈血壓",[16,102046],{},"\n抗癲癇藥物",[16,102049],{},"\n系統性抗纖維蛋白溶解劑",[36,102052],{},[12,102054,102055],{},"接著講併發症的相關處置，總得來說有七個大類",[12,102057,102058,102059,102061,102062,102066,102067,102071,102073],{},"肯定得先來講最大問題的痙攣了，也就是高張力",[16,102060],{},"\n那關於，管不管高張這個問題，其實吵了很久，因為，高張他",[75,102063,102065],{"className":102064},[173],"沒辦法做動作","，低張又軟癱也沒法",[75,102068,102070],{"className":102069},[173],"維持姿勢",[16,102072],{},"\n所以普遍會從這兩個方向去看待，影不影響動作，以及影不影響功能性（ADL）",[12,102075,102076,102077,102079,102080,102082,102083,102085,102086,102088,102089,102091],{},"那抑制痙攣有兩種手段，吃藥以及做手術",[16,102078],{},"\n首先講吃藥，用藥也分成了對周邊抑或者對中樞去作用",[16,102081],{},"\n這部份我只介紹老師打星的藥物",[16,102084],{},"\n首先對中樞的，藥名叫Diazepam，可以抑制掉整個中樞系統，但副作用有嗜睡，焦慮或者全身無力",[16,102087],{},"\n接著是周邊的，有兩種手段",[16,102090],{},"\n一種名字叫肉毒桿菌，另一種叫做脊髓內注射貝可芬（這個量極需要極小且不易嗜，效果也極佳）",[12,102093,102094],{},"手術的部份有三種，肌腱切開術，神經切除術，以後選擇性後神經根切除術",[12,102096,102097,102098,102100,102101,102105,102106,102108,102109,102111,102112,102114,102115,102117],{},"接著講第二大類，癲癇",[16,102099],{},"\n癲癇也是最麻煩的一個問題，我們給病人教導了一些東西後，遇到癲癇，教過的東西他都會",[75,102102,102104],{"className":102103},[5387],"全部忘記","，相當於reset掉",[16,102107],{},"\n機率上來看，57%的病人會在第一周發生癲癇，而88%的病人會在第一年內發生癲癇",[16,102110],{},"\n類型上，血栓或者栓塞這兩個會比較早發生，而出血型會比較晚發生",[16,102113],{},"\n那治療癲癇當然就是用抗癲癇藥物了",[16,102116],{},"\n副作用的話會有：嗜睡，運動失調，注意力不集中，記憶力變差",[12,102119,102120,102121,102123],{},"第三大類，呼吸系統也會有影響",[16,102122],{},"\n主因有兩種，因為癱瘓導致的功能不全，以及因為痙攣導致的需氧增加（或者異常動作模式）",[12,102125,102126,102127,102129,102130,102132,102133,102135,102136,102140],{},"第四大類，創傷",[16,102128],{},"\n這很容易理解，中風了嘛，肯定有不穩定不平衡，抑或者不對稱姿勢導致跌導",[16,102131],{},"\n那常見的骨折部位有：肱骨，手腕，髖部",[16,102134],{},"\n這邊有小補充，偏癱病人的",[75,102137,102139],{"className":102138},[6760],"偏癱側肢體","會高機率有骨質疏鬆症（因為不常用\u002F廢用）",[12,102142,102143,102144,102146,102147,102149,102150,102152,102153,102155,102156,102158,102159,102161],{},"第五大類，血栓性靜脈炎跟深層靜脈栓塞",[16,102145],{},"\n這個問題其實在復健早期階段就會被發現了（而中風病人在中風前血管就已經有硬化或者狹窄問題）",[16,102148],{},"\n那深層靜脈栓塞成因有這幾點",[16,102151],{},"\n血流改變：流速變慢",[16,102154],{},"\n血管壁受損",[16,102157],{},"\n血流凝固時間改變：血液變稠",[16,102160],{},"\n患側姿體的不活動與垂放姿勢：不活動肌肉不收縮就沒法回流；垂放加上重力影響也沒法回流",[12,102163,102164,102165,102167,102168,102170],{},"第六大類，反射性交感神經失養症",[16,102166],{},"\n症狀有灼燒感",[16,102169],{},"\n處置上可以用交感神經阻斷劑，或者利用類固醇",[12,102172,102173,102174,102176,102177,102179],{},"最後一類，疼痛",[16,102175],{},"\n病人經常抱怨肩膀痛",[16,102178],{},"\n處置上是局部注射類固醇",{"title":180,"searchDepth":181,"depth":181,"links":102181},[],"急性期醫生能幹嘛以及併發症的相關處置","\u002Fimages\u002Fuploads\u002F1775193545170-142911098_p0_master1200.jpg",{},"\u002Fblog\u002FCVA（四）",{"title":101931,"description":102182},{"loc":102185},"blog\u002FCVA（四）","tUz4Y3ceEDBIka5e8fPnn7NDycAqdXGw4Qfxck8vuo4",{"id":102191,"title":102192,"author":6946,"body":102193,"date":106421,"description":106422,"draft":185,"edited_at":106423,"extension":186,"featured_image":106424,"meta":106425,"navigation":189,"path":106426,"pinned":185,"seo":106427,"sitemap":106428,"stem":106429,"tags":194,"__hash__":106430},"blog\u002Fblog\u002F機器學習導論.md","機器學習導論",{"type":9,"value":102194,"toc":106399},[102195,102198,102201,102204,102207,102211,102214,102221,102224,102238,102242,102245,102248,102279,102285,102288,102291,102317,102320,102323,102326,102358,102362,102365,102379,102383,102386,102406,102409,102470,102476,102479,102482,102508,102512,102515,102519,102561,102776,102818,102822,102825,103158,103161,103464,103523,103526,103533,103546,103550,103553,103903,103988,105747,105750,105761,105768,105928,106165,106174,106178,106181,106207,106222,106306,106310,106313,106339,106342,106345,106348,106362,106365,106368,106397],[6964,102196,102197],{"id":102197},"前言",[12,102199,102200],{},"最近大學剛上第二學期，我發現機器學習算是挺難的一個科目。剛好我覺得可能會有人對機器學習感興趣，但不知道怎麽下手，或者想要大概看一眼機器學習是個什麽東東，再決定要不要深入學習。所以我想要基於大學講義，寫一篇對新手比較友好的文章，帶你揭開機器學習的神秘面紗~",[12,102202,102203],{},"當然，我的講義是英文寫的，在翻譯的時候難免會有錯誤。在遇到專用名詞的時候，我都會附上英文名詞，也盡量用通俗易懂的方式講解。這篇文章假設你已經有一定的編程基礎，但是不需要對機器學習方面有任何經驗。",[12,102205,102206],{},"準備好的話，我們就開始囉！",[6964,102208,102210],{"id":102209},"到底什麼是學習","到底什麼是「學習」？",[12,102212,102213],{},"在開始聊「機器」學習之前，我們不妨先退一步想想：一般所謂的「學習」到底是什麼意思？",[12,102215,102216,102217,102220],{},"諾貝爾獎得主 Herbert Simon 曾給過一個很精準的定義：",[106,102218,102219],{},"「任何讓系統透過經驗，進而提升效能的過程。」"," 這句話非常直白，而且不僅適用於人類，套用在電腦上也完全講得通。",[12,102222,102223],{},"當我們希望機器去「學習」並不斷進步時，丟給它們的任務通常可以分成兩大類：",[23,102225,102226,102232],{},[26,102227,102228,102231],{},[106,102229,102230],{},"分類 (Classification)"," ： 把東西歸類到不同的標籤下（例如：「這封 email 是不是垃圾郵件？」）",[26,102233,102234,102237],{},[106,102235,102236],{},"問題解決 \u002F 規劃 \u002F 控制 (Problem solving \u002F Planning \u002F Control)"," ： 採取一連串的行動來達成某個目標（例如：「機器人該怎麼走才能離開迷宮？」）",[6964,102239,102241],{"id":102240},"機器學習ai-到底差在哪","機器學習、AI 到底差在哪？",[12,102243,102244],{},"在看新聞或滑文章時，你可能常看到人工智慧 (AI)、機器學習 (ML)、深度學習 (DL) 和資料科學 (Data Science) 這幾個詞被混著用。雖然它們彼此相關，但其實不完全一樣。",[12,102246,102247],{},"你可以把它們想像成一組同心圓：",[23,102249,102250,102256,102262,102268],{},[26,102251,102252,102255],{},[106,102253,102254],{},"人工智慧 (AI)"," 是最廣泛的概念。它指的是創造能夠模仿人類智慧行為的機器。需要注意的是，AI 只在乎結果看起來聰不聰明，不在乎背後是怎麼做到的。如果你今天讓工程師寫個幾千行 if-else，那也算AI。事實上，在電腦發展的前幾十年，大家都是這樣寫 AI 的。",[26,102257,102258,102261],{},[106,102259,102260],{},"機器學習 (ML)"," 是 AI 的一個分支。跟人工手寫 if-else 規則不同，ML 是讓系統自己從資料裡面找出規律，然後用這些規律來做預測或決策。",[26,102263,102264,102267],{},[106,102265,102266],{},"深度學習 (DL)"," 又是機器學習的再往下一個分支，主要是用多層的「神經網路」來處理像影像辨識、自然語言處理等超複雜的問題。",[26,102269,102270,102273,102274,102276],{},[106,102271,102272],{},"資料科學 (Data Science)"," 則與上述所有領域都有交集。它是一門更廣的學科，涵蓋了怎麼用統計學、科學方法和演算法，從資料裡挖出有價值的資訊。",[16,102275],{},[1930,102277],{"alt":1932,"src":102278},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1775145963610-Screenshot_2026-04-03_000510.png",[12,102280,102281,102282],{},"簡單來說：",[106,102283,102284],{},"機器學習的重點在於「用訓練資料建立模型來做預測」，而不是靠人去手寫規則。",[6964,102286,102287],{"id":102287},"機器學習的四大種類",[12,102289,102290],{},"根據電腦系統學習的方式，我們可以把機器學習分成幾個主要類別：",[23,102292,102293,102299,102305,102311],{},[26,102294,102295,102298],{},[106,102296,102297],{},"監督式學習 (Supervised Learning)"," ： 就像你給模型看題目和標準答案。你餵給模型一堆附帶答案的範例，讓它自己去搞清楚輸入和輸出之間的關係。這包含了預測連續數字的「迴歸」(Regression，像是預測房價)，以及預測類別的「分類」(Classification，像是分辨垃圾郵件)。",[26,102300,102301,102304],{},[106,102302,102303],{},"非監督式學習 (Unsupervised Learning)"," ： 模型只拿到一堆沒有標準答案（標籤）的資料，必須自己找出裡面的架構。這裡最常見的任務是「分群」(Clustering)，把長得像的資料分在同一組。",[26,102306,102307,102310],{},[106,102308,102309],{},"強化學習 (Reinforcement Learning)"," ： 模型透過與環境互動來學習。它會採取行動，然後得到獎勵或懲罰，最後慢慢試出最好的策略。你可以想像成用零食跟指令在訓練狗狗。",[26,102312,102313,102316],{},[106,102314,102315],{},"自監督式學習 (Self-Supervised Learning)"," ： 這算是比較新的玩法。模型會根據資料本身「自己生出標籤」，幫自己出考題（例如把句子裡的某個單字挖空，然後叫模型練習預測那個字是什麼）。",[6964,102318,102319],{"id":102319},"實際應用",[7166,102321,102322],{"id":102322},"分類應用範例",[12,102324,102325],{},"分類在我們的日常生活中無所不在：",[23,102327,102328,102334,102340,102346,102352],{},[26,102329,102330,102333],{},[106,102331,102332],{},"醫療診斷"," ： 這張 X 光片是否有肺炎的跡象？",[26,102335,102336,102339],{},[106,102337,102338],{},"垃圾郵件過濾"," ： 這封 email 該收進收件箱還是垃圾信箱？",[26,102341,102342,102345],{},[106,102343,102344],{},"詐欺偵測"," ： 這筆信用卡交易是否可疑？",[26,102347,102348,102351],{},[106,102349,102350],{},"推薦系統"," ： 你可能會喜歡聽哪首歌、看哪部電影？",[26,102353,102354,102357],{},[106,102355,102356],{},"語音與手寫辨識"," ： 將口語或手寫文字轉換為數位文字。",[7166,102359,102361],{"id":102360},"問題解決-規劃-控制範例","問題解決 \u002F 規劃 \u002F 控制範例",[12,102363,102364],{},"在這些任務中，代理程式 (Agent) 會根據環境自己採取行動來達成目標：",[23,102366,102367,102370,102373,102376],{},[26,102368,102369],{},"下西洋棋、圍棋等棋盤遊戲",[26,102371,102372],{},"開在路上的自動駕駛汽車",[26,102374,102375],{},"控制機器人或遊戲角色",[26,102377,102378],{},"無人機或直升機的自主飛行",[6964,102380,102382],{"id":102381},"定義學習任務的萬用公式tpe","定義學習任務的萬用公式：T、P、E",[12,102384,102385],{},"當你想構思一個機器學習的專案時，Tom Mitchell 提出的一個框架超好用。每一個學習任務都可以用三個元素來定義：",[23,102387,102388,102394,102400],{},[26,102389,102390,102393],{},[106,102391,102392],{},"T (任務, Task)"," ： 你想讓系統嘗試做什麽？",[26,102395,102396,102399],{},[106,102397,102398],{},"P (效能, Performance)"," ： 我們應該如何衡量成功與否？",[26,102401,102402,102405],{},[106,102403,102404],{},"E (經驗, Experience)"," ： 系統要從什麼資料中學習？",[12,102407,102408],{},"這裡有幾個具體的範例：",[7085,102410,102411,102424],{},[7088,102412,102413],{},[7091,102414,102415,102418,102421],{},[7094,102416,102417],{"align":47883},"任務 (T)",[7094,102419,102420],{"align":47883},"效能 (P)",[7094,102422,102423],{"align":47883},"經驗 (E)",[7104,102425,102426,102437,102448,102459],{},[7091,102427,102428,102431,102434],{},[7109,102429,102430],{"align":47883},"下西洋跳棋",[7109,102432,102433],{"align":47883},"贏得比賽的概率",[7109,102435,102436],{"align":47883},"自我對弈的練習賽",[7091,102438,102439,102442,102445],{},[7109,102440,102441],{"align":47883},"辨識手寫單字",[7109,102443,102444],{"align":47883},"正確分類單字的概率",[7109,102446,102447],{"align":47883},"一堆有標註標準答案的手寫圖檔",[7091,102449,102450,102453,102456],{},[7109,102451,102452],{"align":47883},"在高速公路上駕駛",[7109,102454,102455],{"align":47883},"發生人為判定錯誤前的平均行駛距離",[7109,102457,102458],{"align":47883},"記錄人類駕駛的影像與方向盤指令",[7091,102460,102461,102464,102467],{},[7109,102462,102463],{"align":47883},"垃圾郵件分類",[7109,102465,102466],{"align":47883},"正確分類電子郵件的概率",[7109,102468,102469],{"align":47883},"帶有人工標籤的電子郵件資料庫",[12,102471,102472,102473],{},"在開始任何機器學習專案之前，這個框架非常適合用來釐清思緒。問問自己：",[7289,102474,102475],{},"我的 T、P 和 E 分別是什麼？",[6964,102477,102478],{"id":102478},"設計學習系統",[12,102480,102481],{},"當你著手建立一個機器學習系統時，有四個關鍵的設計決策：",[3603,102483,102484,102490,102496,102502],{},[26,102485,102486,102489],{},[106,102487,102488],{},"選擇訓練經驗"," ： 系統將從哪種資料中學習？是直接的（有標籤的輸入-輸出對），還是間接的（例如只知道一盤棋最後的輸贏，但不知道中間每一步的好壞）？",[26,102491,102492,102495],{},[106,102493,102494],{},"選擇目標函數"," ： 系統究竟應該學習什麼？以跳棋程式為例，它要學的可能是一個「評估函數」，用來給當下的棋盤局勢打分數。",[26,102497,102498,102501],{},[106,102499,102500],{},"選擇表達方式"," ：這個目標函數要長什麼樣子？可以是查詢表 (lookup tables)、線性函數 (linear function)、決策樹 (decision tree)、神經網路等 (neural network)。",[26,102503,102504,102507],{},[106,102505,102506],{},"選擇學習演算法"," ： 系統要用什麼方法去找到那個「最好的函數」？這可能是梯度下降法 (gradient decent)、動態規劃 (dynamic programming)、演化演算法等 (evolutionary algorithms)。",[6964,102509,102511],{"id":102510},"具體範例教電腦下西洋跳棋","具體範例：教電腦下西洋跳棋",[12,102513,102514],{},"好，理論講了一堆，我們來把所有東西串起來。我們來看一個經典的老例子：Arthur Samuel 在 1959 年寫的跳棋程式。這可是史上最早的機器學習系統之一。",[7166,102516,102518],{"id":102517},"目標函數-target-function","目標函數 (Target Function)",[12,102520,102521,102522,7206,102525,102560],{},"我們想讓系統學到一個",[106,102523,102524],{},"評估函數",[24923,102526,102528],{"className":102527,"jax":24927},[24926],[24929,102529,102530,102540],{"style":24931,"xmlns":24932,"width":42536,"height":24934,"role":1930,"focusable":8058,"viewBox":42537,"xmlnsXLink":24936},[24938,102531,102532,102534,102536,102538],{},[24941,102533],{"id":42542,"d":27385},[24941,102535],{"id":24943,"d":24944},[24941,102537],{"id":42547,"d":42548},[24941,102539],{"id":24987,"d":24988},[25022,102541,102542],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,102543,102544,102548,102552,102556],{"dataMmlNode":25028},[25022,102545,102546],{"dataMmlNode":25942},[25033,102547],{"dataC":27570,"xLinkHref":42559},[25022,102549,102550],{"dataMmlNode":25031,"transform":42562},[25033,102551],{"dataC":25035,"xLinkHref":25036},[25022,102553,102554],{"dataMmlNode":25942,"transform":42567},[25033,102555],{"dataC":42570,"xLinkHref":42571},[25022,102557,102558],{"dataMmlNode":25031,"transform":42574},[25033,102559],{"dataC":25108,"xLinkHref":25109},"，你給它一個棋盤的狀態，它就回傳一個分數，告訴你現在的局勢對我們有多有利。在理想情況下：",[23,102562,102563,102619,102685,102735],{},[26,102564,102565,102566],{},"如果這盤棋贏定了，",[24923,102567,102569],{"className":102568,"jax":24927},[24926],[24929,102570,102571,102587],{"style":24931,"xmlns":24932,"width":42587,"height":24934,"role":1930,"focusable":8058,"viewBox":42588,"xmlnsXLink":24936},[24938,102572,102573,102575,102577,102579,102581,102583,102585],{},[24941,102574],{"id":42593,"d":27385},[24941,102576],{"id":32282,"d":24944},[24941,102578],{"id":42598,"d":42548},[24941,102580],{"id":32303,"d":24988},[24941,102582],{"id":42603,"d":24972},[24941,102584],{"id":42606,"d":25914},[24941,102586],{"id":42609,"d":25935},[25022,102588,102589],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,102590,102591,102595,102599,102603,102607,102611],{"dataMmlNode":25028},[25022,102592,102593],{"dataMmlNode":25942},[25033,102594],{"dataC":27570,"xLinkHref":42618},[25022,102596,102597],{"dataMmlNode":25031,"transform":42562},[25033,102598],{"dataC":25035,"xLinkHref":32332},[25022,102600,102601],{"dataMmlNode":25942,"transform":42567},[25033,102602],{"dataC":42570,"xLinkHref":42627},[25022,102604,102605],{"dataMmlNode":25031,"transform":42574},[25033,102606],{"dataC":25108,"xLinkHref":32370},[25022,102608,102609],{"dataMmlNode":25031,"transform":42634},[25033,102610],{"dataC":25078,"xLinkHref":42637},[25022,102612,102613,102615,102617],{"dataMmlNode":26031,"transform":42640},[25033,102614],{"dataC":26035,"xLinkHref":42643},[25033,102616],{"dataC":26103,"xLinkHref":42646,"transform":26394},[25033,102618],{"dataC":26103,"xLinkHref":42646,"transform":29730},[26,102620,102621,102622],{},"如果這盤棋輸定了，",[24923,102623,102625],{"className":102624,"jax":24927},[24926],[24929,102626,102627,102647],{"style":24931,"xmlns":24932,"width":42679,"height":24934,"role":1930,"focusable":8058,"viewBox":42680,"xmlnsXLink":24936},[24938,102628,102629,102632,102634,102636,102638,102640,102643,102645],{},[24941,102630],{"id":102631,"d":27385},"MJX-3-TEX-I-1D449",[24941,102633],{"id":25230,"d":24944},[24941,102635],{"id":42661,"d":42548},[24941,102637],{"id":25276,"d":24988},[24941,102639],{"id":25251,"d":24972},[24941,102641],{"id":102642,"d":25910},"MJX-3-TEX-N-2212",[24941,102644],{"id":48102,"d":25914},[24941,102646],{"id":48105,"d":25935},[25022,102648,102649],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,102650,102651,102656,102660,102664,102668,102672,102677],{"dataMmlNode":25028},[25022,102652,102653],{"dataMmlNode":25942},[25033,102654],{"dataC":27570,"xLinkHref":102655},"#MJX-3-TEX-I-1D449",[25022,102657,102658],{"dataMmlNode":25031,"transform":42562},[25033,102659],{"dataC":25035,"xLinkHref":25285},[25022,102661,102662],{"dataMmlNode":25942,"transform":42567},[25033,102663],{"dataC":42570,"xLinkHref":42670},[25022,102665,102666],{"dataMmlNode":25031,"transform":42574},[25033,102667],{"dataC":25108,"xLinkHref":25351},[25022,102669,102670],{"dataMmlNode":25031,"transform":42634},[25033,102671],{"dataC":25078,"xLinkHref":25312},[25022,102673,102674],{"dataMmlNode":25031,"transform":42640},[25033,102675],{"dataC":26019,"xLinkHref":102676},"#MJX-3-TEX-N-2212",[25022,102678,102679,102681,102683],{"dataMmlNode":26031,"transform":42739},[25033,102680],{"dataC":26035,"xLinkHref":48116},[25033,102682],{"dataC":26103,"xLinkHref":48119,"transform":26394},[25033,102684],{"dataC":26103,"xLinkHref":48119,"transform":29730},[26,102686,102687,102688],{},"如果是平手，",[24923,102689,102691],{"className":102690,"jax":24927},[24926],[24929,102692,102693,102707],{"style":24931,"xmlns":24932,"width":42775,"height":24934,"role":1930,"focusable":8058,"viewBox":42776,"xmlnsXLink":24936},[24938,102694,102695,102697,102699,102701,102703,102705],{},[24941,102696],{"id":42685,"d":27385},[24941,102698],{"id":32600,"d":24944},[24941,102700],{"id":42690,"d":42548},[24941,102702],{"id":32606,"d":24988},[24941,102704],{"id":42695,"d":24972},[24941,102706],{"id":42704,"d":25935},[25022,102708,102709],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,102710,102711,102715,102719,102723,102727,102731],{"dataMmlNode":25028},[25022,102712,102713],{"dataMmlNode":25942},[25033,102714],{"dataC":27570,"xLinkHref":42713},[25022,102716,102717],{"dataMmlNode":25031,"transform":42562},[25033,102718],{"dataC":25035,"xLinkHref":32625},[25022,102720,102721],{"dataMmlNode":25942,"transform":42567},[25033,102722],{"dataC":42570,"xLinkHref":42722},[25022,102724,102725],{"dataMmlNode":25031,"transform":42574},[25033,102726],{"dataC":25108,"xLinkHref":32635},[25022,102728,102729],{"dataMmlNode":25031,"transform":42634},[25033,102730],{"dataC":25078,"xLinkHref":42731},[25022,102732,102733],{"dataMmlNode":26031,"transform":42640},[25033,102734],{"dataC":26103,"xLinkHref":42745},[26,102736,102737,102738,102775],{},"在其他情況下，",[24923,102739,102741],{"className":102740,"jax":24927},[24926],[24929,102742,102743,102754],{"style":24931,"xmlns":24932,"width":42536,"height":24934,"role":1930,"focusable":8058,"viewBox":42537,"xmlnsXLink":24936},[24938,102744,102745,102748,102750,102752],{},[24941,102746],{"id":102747,"d":27385},"MJX-5-TEX-I-1D449",[24941,102749],{"id":25386,"d":24944},[24941,102751],{"id":42757,"d":42548},[24941,102753],{"id":25423,"d":24988},[25022,102755,102756],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,102757,102758,102763,102767,102771],{"dataMmlNode":25028},[25022,102759,102760],{"dataMmlNode":25942},[25033,102761],{"dataC":27570,"xLinkHref":102762},"#MJX-5-TEX-I-1D449",[25022,102764,102765],{"dataMmlNode":25031,"transform":42562},[25033,102766],{"dataC":25035,"xLinkHref":25448},[25022,102768,102769],{"dataMmlNode":25942,"transform":42567},[25033,102770],{"dataC":42570,"xLinkHref":42766},[25022,102772,102773],{"dataMmlNode":25031,"transform":42574},[25033,102774],{"dataC":25108,"xLinkHref":25499}," 就等於如果雙方都完美發揮，最後能拿到的最佳分數。",[12,102777,102778,102779,102814,102815,86843],{},"但問題來了：要算出精確的",[24923,102780,102782],{"className":102781,"jax":24927},[24926],[24929,102783,102784,102794],{"style":24931,"xmlns":24932,"width":42536,"height":24934,"role":1930,"focusable":8058,"viewBox":42537,"xmlnsXLink":24936},[24938,102785,102786,102788,102790,102792],{},[24941,102787],{"id":42781,"d":27385},[24941,102789],{"id":25896,"d":24944},[24941,102791],{"id":42786,"d":42548},[24941,102793],{"id":25903,"d":24988},[25022,102795,102796],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,102797,102798,102802,102806,102810],{"dataMmlNode":25028},[25022,102799,102800],{"dataMmlNode":25942},[25033,102801],{"dataC":27570,"xLinkHref":42801},[25022,102803,102804],{"dataMmlNode":25031,"transform":42562},[25033,102805],{"dataC":25035,"xLinkHref":25994},[25022,102807,102808],{"dataMmlNode":25942,"transform":42567},[25033,102809],{"dataC":42570,"xLinkHref":42810},[25022,102811,102812],{"dataMmlNode":25031,"transform":42574},[25033,102813],{"dataC":25108,"xLinkHref":26007},"，電腦得把整個遊戲的所有可能性都算過一遍，其運算量是天文數字。所以我們只能退而求其次，找一個",[106,102816,102817],{},"近似值",[7166,102819,102821],{"id":102820},"線性近似-linear-approximation","線性近似 (Linear Approximation)",[12,102823,102824],{},"我們可以把評估函數設計成各種「棋盤特徵」的加權總和：",[12,102826,102827],{},[24923,102828,102830],{"className":102829,"jax":24927},[24926],[24929,102831,102832,102884],{"style":24931,"xmlns":24932,"width":42904,"height":24934,"role":1930,"focusable":8058,"viewBox":42905,"xmlnsXLink":24936},[24938,102833,102834,102837,102839,102841,102843,102846,102849,102851,102854,102856,102859,102861,102863,102866,102869,102872,102875,102878,102881],{},[24941,102835],{"id":102836,"d":27385},"MJX-7-TEX-I-1D449",[24941,102838],{"id":32773,"d":24944},[24941,102840],{"id":42832,"d":42548},[24941,102842],{"id":32779,"d":24988},[24941,102844],{"id":102845,"d":24972},"MJX-7-TEX-N-3D",[24941,102847],{"id":102848,"d":37254},"MJX-7-TEX-I-1D464",[24941,102850],{"id":87291,"d":25935},[24941,102852],{"id":102853,"d":30313},"MJX-7-TEX-N-2B",[24941,102855],{"id":26164,"d":25914},[24941,102857],{"id":102858,"d":27409},"MJX-7-TEX-N-22C5",[24941,102860],{"id":26161,"d":25889},[24941,102862],{"id":48369,"d":25928},[24941,102864],{"id":102865,"d":25881},"MJX-7-TEX-I-1D45F",[24941,102867],{"id":102868,"d":26455},"MJX-7-TEX-N-33",[24941,102870],{"id":102871,"d":42951},"MJX-7-TEX-I-1D458",[24941,102873],{"id":102874,"d":29582},"MJX-7-TEX-N-34",[24941,102876],{"id":102877,"d":26660},"MJX-7-TEX-N-35",[24941,102879],{"id":102880,"d":25877},"MJX-7-TEX-I-1D461",[24941,102882],{"id":102883,"d":42964},"MJX-7-TEX-N-36",[25022,102885,102886],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,102887,102888,102893,102897,102901,102905,102910,102921,102926,102936,102941,102945,102949,102953,102957,102961,102965,102975,102979,102984,102988,102992,102996,103000,103004,103015,103019,103023,103028,103032,103036,103040,103044,103055,103059,103063,103067,103071,103075,103079,103083,103094,103098,103102,103107,103111,103115,103119,103123,103134,103138,103142,103146,103150,103154],{"dataMmlNode":25028},[25022,102889,102890],{"dataMmlNode":25942},[25033,102891],{"dataC":27570,"xLinkHref":102892},"#MJX-7-TEX-I-1D449",[25022,102894,102895],{"dataMmlNode":25031,"transform":42562},[25033,102896],{"dataC":25035,"xLinkHref":32794},[25022,102898,102899],{"dataMmlNode":25942,"transform":42567},[25033,102900],{"dataC":42570,"xLinkHref":42841},[25022,102902,102903],{"dataMmlNode":25031,"transform":42574},[25033,102904],{"dataC":25108,"xLinkHref":32806},[25022,102906,102907],{"dataMmlNode":25031,"transform":42634},[25033,102908],{"dataC":25078,"xLinkHref":102909},"#MJX-7-TEX-N-3D",[25022,102911,102912,102917],{"dataMmlNode":26023,"transform":42640},[25022,102913,102914],{"dataMmlNode":25942},[25033,102915],{"dataC":37310,"xLinkHref":102916},"#MJX-7-TEX-I-1D464",[25022,102918,102919],{"dataMmlNode":26031,"transform":43003},[25033,102920],{"dataC":26103,"xLinkHref":87306},[25022,102922,102923],{"dataMmlNode":25031,"transform":43009},[25033,102924],{"dataC":30393,"xLinkHref":102925},"#MJX-7-TEX-N-2B",[25022,102927,102928,102932],{"dataMmlNode":26023,"transform":43015},[25022,102929,102930],{"dataMmlNode":25942},[25033,102931],{"dataC":37310,"xLinkHref":102916},[25022,102933,102934],{"dataMmlNode":26031,"transform":43003},[25033,102935],{"dataC":26035,"xLinkHref":26180},[25022,102937,102938],{"dataMmlNode":25031,"transform":43026},[25033,102939],{"dataC":27679,"xLinkHref":102940},"#MJX-7-TEX-N-22C5",[25022,102942,102943],{"dataMmlNode":25942,"transform":43032},[25033,102944],{"dataC":42570,"xLinkHref":42841},[25022,102946,102947],{"dataMmlNode":25942,"transform":43037},[25033,102948],{"dataC":25980,"xLinkHref":26175},[25022,102950,102951],{"dataMmlNode":25031,"transform":43042},[25033,102952],{"dataC":25035,"xLinkHref":32794},[25022,102954,102955],{"dataMmlNode":25942,"transform":43047},[25033,102956],{"dataC":42570,"xLinkHref":42841},[25022,102958,102959],{"dataMmlNode":25031,"transform":43052},[25033,102960],{"dataC":25108,"xLinkHref":32806},[25022,102962,102963],{"dataMmlNode":25031,"transform":43057},[25033,102964],{"dataC":30393,"xLinkHref":102925},[25022,102966,102967,102971],{"dataMmlNode":26023,"transform":43062},[25022,102968,102969],{"dataMmlNode":25942},[25033,102970],{"dataC":37310,"xLinkHref":102916},[25022,102972,102973],{"dataMmlNode":26031,"transform":43003},[25033,102974],{"dataC":26058,"xLinkHref":48410},[25022,102976,102977],{"dataMmlNode":25031,"transform":43074},[25033,102978],{"dataC":27679,"xLinkHref":102940},[25022,102980,102981],{"dataMmlNode":25942,"transform":43079},[25033,102982],{"dataC":25966,"xLinkHref":102983},"#MJX-7-TEX-I-1D45F",[25022,102985,102986],{"dataMmlNode":25942,"transform":43085},[25033,102987],{"dataC":25980,"xLinkHref":26175},[25022,102989,102990],{"dataMmlNode":25031,"transform":43090},[25033,102991],{"dataC":25035,"xLinkHref":32794},[25022,102993,102994],{"dataMmlNode":25942,"transform":43095},[25033,102995],{"dataC":42570,"xLinkHref":42841},[25022,102997,102998],{"dataMmlNode":25031,"transform":43100},[25033,102999],{"dataC":25108,"xLinkHref":32806},[25022,103001,103002],{"dataMmlNode":25031,"transform":43105},[25033,103003],{"dataC":30393,"xLinkHref":102925},[25022,103005,103006,103010],{"dataMmlNode":26023,"transform":43110},[25022,103007,103008],{"dataMmlNode":25942},[25033,103009],{"dataC":37310,"xLinkHref":102916},[25022,103011,103012],{"dataMmlNode":26031,"transform":43003},[25033,103013],{"dataC":26486,"xLinkHref":103014},"#MJX-7-TEX-N-33",[25022,103016,103017],{"dataMmlNode":25031,"transform":43122},[25033,103018],{"dataC":27679,"xLinkHref":102940},[25022,103020,103021],{"dataMmlNode":25942,"transform":43127},[25033,103022],{"dataC":42570,"xLinkHref":42841},[25022,103024,103025],{"dataMmlNode":25942,"transform":43132},[25033,103026],{"dataC":43135,"xLinkHref":103027},"#MJX-7-TEX-I-1D458",[25022,103029,103030],{"dataMmlNode":25031,"transform":43139},[25033,103031],{"dataC":25035,"xLinkHref":32794},[25022,103033,103034],{"dataMmlNode":25942,"transform":43144},[25033,103035],{"dataC":42570,"xLinkHref":42841},[25022,103037,103038],{"dataMmlNode":25031,"transform":43149},[25033,103039],{"dataC":25108,"xLinkHref":32806},[25022,103041,103042],{"dataMmlNode":25031,"transform":43154},[25033,103043],{"dataC":30393,"xLinkHref":102925},[25022,103045,103046,103050],{"dataMmlNode":26023,"transform":43159},[25022,103047,103048],{"dataMmlNode":25942},[25033,103049],{"dataC":37310,"xLinkHref":102916},[25022,103051,103052],{"dataMmlNode":26031,"transform":43003},[25033,103053],{"dataC":29597,"xLinkHref":103054},"#MJX-7-TEX-N-34",[25022,103056,103057],{"dataMmlNode":25031,"transform":43171},[25033,103058],{"dataC":27679,"xLinkHref":102940},[25022,103060,103061],{"dataMmlNode":25942,"transform":43176},[25033,103062],{"dataC":25966,"xLinkHref":102983},[25022,103064,103065],{"dataMmlNode":25942,"transform":43181},[25033,103066],{"dataC":43135,"xLinkHref":103027},[25022,103068,103069],{"dataMmlNode":25031,"transform":43186},[25033,103070],{"dataC":25035,"xLinkHref":32794},[25022,103072,103073],{"dataMmlNode":25942,"transform":43191},[25033,103074],{"dataC":42570,"xLinkHref":42841},[25022,103076,103077],{"dataMmlNode":25031,"transform":43196},[25033,103078],{"dataC":25108,"xLinkHref":32806},[25022,103080,103081],{"dataMmlNode":25031,"transform":43201},[25033,103082],{"dataC":30393,"xLinkHref":102925},[25022,103084,103085,103089],{"dataMmlNode":26023,"transform":43206},[25022,103086,103087],{"dataMmlNode":25942},[25033,103088],{"dataC":37310,"xLinkHref":102916},[25022,103090,103091],{"dataMmlNode":26031,"transform":43003},[25033,103092],{"dataC":26717,"xLinkHref":103093},"#MJX-7-TEX-N-35",[25022,103095,103096],{"dataMmlNode":25031,"transform":43218},[25033,103097],{"dataC":27679,"xLinkHref":102940},[25022,103099,103100],{"dataMmlNode":25942,"transform":43223},[25033,103101],{"dataC":42570,"xLinkHref":42841},[25022,103103,103104],{"dataMmlNode":25942,"transform":43228},[25033,103105],{"dataC":25959,"xLinkHref":103106},"#MJX-7-TEX-I-1D461",[25022,103108,103109],{"dataMmlNode":25031,"transform":43234},[25033,103110],{"dataC":25035,"xLinkHref":32794},[25022,103112,103113],{"dataMmlNode":25942,"transform":43239},[25033,103114],{"dataC":42570,"xLinkHref":42841},[25022,103116,103117],{"dataMmlNode":25031,"transform":43244},[25033,103118],{"dataC":25108,"xLinkHref":32806},[25022,103120,103121],{"dataMmlNode":25031,"transform":43249},[25033,103122],{"dataC":30393,"xLinkHref":102925},[25022,103124,103125,103129],{"dataMmlNode":26023,"transform":43254},[25022,103126,103127],{"dataMmlNode":25942},[25033,103128],{"dataC":37310,"xLinkHref":102916},[25022,103130,103131],{"dataMmlNode":26031,"transform":43003},[25033,103132],{"dataC":31136,"xLinkHref":103133},"#MJX-7-TEX-N-36",[25022,103135,103136],{"dataMmlNode":25031,"transform":43266},[25033,103137],{"dataC":27679,"xLinkHref":102940},[25022,103139,103140],{"dataMmlNode":25942,"transform":43271},[25033,103141],{"dataC":25966,"xLinkHref":102983},[25022,103143,103144],{"dataMmlNode":25942,"transform":43276},[25033,103145],{"dataC":25959,"xLinkHref":103106},[25022,103147,103148],{"dataMmlNode":25031,"transform":43281},[25033,103149],{"dataC":25035,"xLinkHref":32794},[25022,103151,103152],{"dataMmlNode":25942,"transform":43286},[25033,103153],{"dataC":42570,"xLinkHref":42841},[25022,103155,103156],{"dataMmlNode":25031,"transform":43291},[25033,103157],{"dataC":25108,"xLinkHref":32806},[12,103159,103160],{},"其中：",[23,103162,103163,103196,103238,103284,103328,103374,103418],{},[26,103164,103165,103195],{},[24923,103166,103168],{"className":103167,"jax":24927},[24926],[24929,103169,103173,103180],{"style":61403,"xmlns":24932,"width":103170,"height":103171,"role":1930,"focusable":8058,"viewBox":103172,"xmlnsXLink":24936},"2.608ex","1.377ex","0 -443 1152.6 608.6",[24938,103174,103175,103178],{},[24941,103176],{"id":103177,"d":37254},"MJX-8-TEX-I-1D464",[24941,103179],{"id":26194,"d":25935},[25022,103181,103182],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,103183,103184],{"dataMmlNode":25028},[25022,103185,103186,103191],{"dataMmlNode":26023},[25022,103187,103188],{"dataMmlNode":25942},[25033,103189],{"dataC":37310,"xLinkHref":103190},"#MJX-8-TEX-I-1D464",[25022,103192,103193],{"dataMmlNode":26031,"transform":43003},[25033,103194],{"dataC":26103,"xLinkHref":26210}," = 基礎底分 (Bias)，也就是即使棋盤上什麼特徵都還沒有，系統預設的初始局勢分數。",[26,103197,103198,103237],{},[24923,103199,103201],{"className":103200,"jax":24927},[24926],[24929,103202,103203,103213],{"style":24931,"xmlns":24932,"width":43306,"height":24934,"role":1930,"focusable":8058,"viewBox":43307,"xmlnsXLink":24936},[24938,103204,103205,103207,103209,103211],{},[24941,103206],{"id":42916,"d":42548},[24941,103208],{"id":26249,"d":25889},[24941,103210],{"id":42913,"d":24944},[24941,103212],{"id":42919,"d":24988},[25022,103214,103215],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,103216,103217,103221,103225,103229,103233],{"dataMmlNode":25028},[25022,103218,103219],{"dataMmlNode":25942},[25033,103220],{"dataC":42570,"xLinkHref":42983},[25022,103222,103223],{"dataMmlNode":25942,"transform":43332},[25033,103224],{"dataC":25980,"xLinkHref":26263},[25022,103226,103227],{"dataMmlNode":25031,"transform":43337},[25033,103228],{"dataC":25035,"xLinkHref":42978},[25022,103230,103231],{"dataMmlNode":25942,"transform":43343},[25033,103232],{"dataC":42570,"xLinkHref":42983},[25022,103234,103235],{"dataMmlNode":25031,"transform":25328},[25033,103236],{"dataC":25108,"xLinkHref":42988}," = 黑棋的數量",[26,103239,103240,103283],{},[24923,103241,103243],{"className":103242,"jax":24927},[24926],[24929,103244,103245,103258],{"style":24931,"xmlns":24932,"width":43359,"height":24934,"role":1930,"focusable":8058,"viewBox":43360,"xmlnsXLink":24936},[24938,103246,103247,103250,103252,103254,103256],{},[24941,103248],{"id":103249,"d":25881},"MJX-10-TEX-I-1D45F",[24941,103251],{"id":26278,"d":25889},[24941,103253],{"id":43317,"d":24944},[24941,103255],{"id":43312,"d":42548},[24941,103257],{"id":43320,"d":24988},[25022,103259,103260],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,103261,103262,103267,103271,103275,103279],{"dataMmlNode":25028},[25022,103263,103264],{"dataMmlNode":25942},[25033,103265],{"dataC":25966,"xLinkHref":103266},"#MJX-10-TEX-I-1D45F",[25022,103268,103269],{"dataMmlNode":25942,"transform":43386},[25033,103270],{"dataC":25980,"xLinkHref":26292},[25022,103272,103273],{"dataMmlNode":25031,"transform":43391},[25033,103274],{"dataC":25035,"xLinkHref":43340},[25022,103276,103277],{"dataMmlNode":25942,"transform":43396},[25033,103278],{"dataC":42570,"xLinkHref":43329},[25022,103280,103281],{"dataMmlNode":25031,"transform":43402},[25033,103282],{"dataC":25108,"xLinkHref":43350}," = 紅棋的數量",[26,103285,103286,103327],{},[24923,103287,103289],{"className":103288,"jax":24927},[24926],[24929,103290,103291,103302],{"style":24931,"xmlns":24932,"width":43413,"height":24934,"role":1930,"focusable":8058,"viewBox":43414,"xmlnsXLink":24936},[24938,103292,103293,103295,103298,103300],{},[24941,103294],{"id":43372,"d":42548},[24941,103296],{"id":103297,"d":42951},"MJX-11-TEX-I-1D458",[24941,103299],{"id":32938,"d":24944},[24941,103301],{"id":32944,"d":24988},[25022,103303,103304],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,103305,103306,103310,103315,103319,103323],{"dataMmlNode":25028},[25022,103307,103308],{"dataMmlNode":25942},[25033,103309],{"dataC":42570,"xLinkHref":43399},[25022,103311,103312],{"dataMmlNode":25942,"transform":43332},[25033,103313],{"dataC":43135,"xLinkHref":103314},"#MJX-11-TEX-I-1D458",[25022,103316,103317],{"dataMmlNode":25031,"transform":43445},[25033,103318],{"dataC":25035,"xLinkHref":32976},[25022,103320,103321],{"dataMmlNode":25942,"transform":43451},[25033,103322],{"dataC":42570,"xLinkHref":43399},[25022,103324,103325],{"dataMmlNode":25031,"transform":43456},[25033,103326],{"dataC":25108,"xLinkHref":32986}," = 黑棋國王的數量",[26,103329,103330,103373],{},[24923,103331,103333],{"className":103332,"jax":24927},[24926],[24929,103334,103335,103348],{"style":24931,"xmlns":24932,"width":43468,"height":24934,"role":1930,"focusable":8058,"viewBox":43469,"xmlnsXLink":24936},[24938,103336,103337,103340,103342,103344,103346],{},[24941,103338],{"id":103339,"d":25881},"MJX-12-TEX-I-1D45F",[24941,103341],{"id":43422,"d":42951},[24941,103343],{"id":43425,"d":24944},[24941,103345],{"id":43419,"d":42548},[24941,103347],{"id":43428,"d":24988},[25022,103349,103350],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,103351,103352,103357,103361,103365,103369],{"dataMmlNode":25028},[25022,103353,103354],{"dataMmlNode":25942},[25033,103355],{"dataC":25966,"xLinkHref":103356},"#MJX-12-TEX-I-1D45F",[25022,103358,103359],{"dataMmlNode":25942,"transform":43386},[25033,103360],{"dataC":43135,"xLinkHref":43442},[25022,103362,103363],{"dataMmlNode":25031,"transform":37349},[25033,103364],{"dataC":25035,"xLinkHref":43448},[25022,103366,103367],{"dataMmlNode":25942,"transform":43508},[25033,103368],{"dataC":42570,"xLinkHref":43437},[25022,103370,103371],{"dataMmlNode":25031,"transform":43514},[25033,103372],{"dataC":25108,"xLinkHref":43459}," = 紅棋國王的數量",[26,103375,103376,103417],{},[24923,103377,103379],{"className":103378,"jax":24927},[24926],[24929,103380,103381,103392],{"style":24931,"xmlns":24932,"width":43526,"height":24934,"role":1930,"focusable":8058,"viewBox":43527,"xmlnsXLink":24936},[24938,103382,103383,103385,103388,103390],{},[24941,103384],{"id":43483,"d":42548},[24941,103386],{"id":103387,"d":25877},"MJX-13-TEX-I-1D461",[24941,103389],{"id":43480,"d":24944},[24941,103391],{"id":43486,"d":24988},[25022,103393,103394],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,103395,103396,103400,103405,103409,103413],{"dataMmlNode":25028},[25022,103397,103398],{"dataMmlNode":25942},[25033,103399],{"dataC":42570,"xLinkHref":43511},[25022,103401,103402],{"dataMmlNode":25942,"transform":43332},[25033,103403],{"dataC":25959,"xLinkHref":103404},"#MJX-13-TEX-I-1D461",[25022,103406,103407],{"dataMmlNode":25031,"transform":43556},[25033,103408],{"dataC":25035,"xLinkHref":43505},[25022,103410,103411],{"dataMmlNode":25942,"transform":43561},[25033,103412],{"dataC":42570,"xLinkHref":43511},[25022,103414,103415],{"dataMmlNode":25031,"transform":43566},[25033,103416],{"dataC":25108,"xLinkHref":43517}," = 受到威脅的黑棋數量",[26,103419,103420,103463],{},[24923,103421,103423],{"className":103422,"jax":24927},[24926],[24929,103424,103425,103438],{"style":24931,"xmlns":24932,"width":43577,"height":24934,"role":1930,"focusable":8058,"viewBox":43578,"xmlnsXLink":24936},[24938,103426,103427,103430,103432,103434,103436],{},[24941,103428],{"id":103429,"d":25881},"MJX-14-TEX-I-1D45F",[24941,103431],{"id":43535,"d":25877},[24941,103433],{"id":33146,"d":24944},[24941,103435],{"id":43532,"d":42548},[24941,103437],{"id":33152,"d":24988},[25022,103439,103440],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,103441,103442,103447,103451,103455,103459],{"dataMmlNode":25028},[25022,103443,103444],{"dataMmlNode":25942},[25033,103445],{"dataC":25966,"xLinkHref":103446},"#MJX-14-TEX-I-1D45F",[25022,103448,103449],{"dataMmlNode":25942,"transform":43386},[25033,103450],{"dataC":25959,"xLinkHref":43553},[25022,103452,103453],{"dataMmlNode":25031,"transform":43610},[25033,103454],{"dataC":25035,"xLinkHref":33194},[25022,103456,103457],{"dataMmlNode":25942,"transform":43615},[25033,103458],{"dataC":42570,"xLinkHref":43548},[25022,103460,103461],{"dataMmlNode":25031,"transform":43621},[25033,103462],{"dataC":25108,"xLinkHref":33206}," = 受到威脅的紅棋數量",[12,103465,103466,103467,103494,103495,103522],{},"這些權重（",[24923,103468,103470],{"className":103469,"jax":24927},[24926],[24929,103471,103472,103479],{"style":61403,"xmlns":24932,"width":103170,"height":103171,"role":1930,"focusable":8058,"viewBox":103172,"xmlnsXLink":24936},[24938,103473,103474,103477],{},[24941,103475],{"id":103476,"d":37254},"MJX-15-TEX-I-1D464",[24941,103478],{"id":87748,"d":25935},[25022,103480,103481],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,103482,103483],{"dataMmlNode":25028},[25022,103484,103485,103490],{"dataMmlNode":26023},[25022,103486,103487],{"dataMmlNode":25942},[25033,103488],{"dataC":37310,"xLinkHref":103489},"#MJX-15-TEX-I-1D464",[25022,103491,103492],{"dataMmlNode":26031,"transform":43003},[25033,103493],{"dataC":26103,"xLinkHref":87775}," 到 ",[24923,103496,103498],{"className":103497,"jax":24927},[24926],[24929,103499,103500,103507],{"style":61403,"xmlns":24932,"width":103170,"height":103171,"role":1930,"focusable":8058,"viewBox":103172,"xmlnsXLink":24936},[24938,103501,103502,103504],{},[24941,103503],{"id":43639,"d":37254},[24941,103505],{"id":103506,"d":42964},"MJX-16-TEX-N-36",[25022,103508,103509],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,103510,103511],{"dataMmlNode":25028},[25022,103512,103513,103517],{"dataMmlNode":26023},[25022,103514,103515],{"dataMmlNode":25942},[25033,103516],{"dataC":37310,"xLinkHref":43648},[25022,103518,103519],{"dataMmlNode":26031,"transform":43003},[25033,103520],{"dataC":31136,"xLinkHref":103521},"#MJX-16-TEX-N-36","）就是系統需要「學習」的東西。說白了，學習的過程就是在調整這些數字。",[7166,103524,103525],{"id":103525},"透過間接經驗進行訓練",[12,103527,103528,103529,103532],{},"因為我們是讓電腦自己跟自己下棋，沒辦法在它每走一步時都告訴它「這步得幾分」。取而代之的是，我們用",[106,103530,103531],{},"時序差分學習 (Temporal difference learning)","：電腦會根據「下一步」的局勢預測，來修正「這一步」的分數估計。下了幾千幾萬盤之後，遊戲結尾那些很準確的分數，就會慢慢「回傳」並影響遊戲早期的步數。",[13438,103534,103536,103539],{"className":103535},[13441,13442],[12,103537,103538],{},"想像你在打游戲。你不會每點一下滑鼠、每放一個技能，系統就立刻告訴你「這步 +1 分」或「這步 -1 分」對吧？你只有在最後輸掉的那一刻，才知道「啊，這局輸了」。",[12,103540,103541,103542,103545],{},"這種「只有最後結果，沒有中間每一步的評分」就叫做",[106,103543,103544],{},"間接經驗","。電腦在學下跳棋的時候也是這樣，它必須自己去覆盤思考：「到底是我開局第一步走錯，還是最後一步下錯，才導致最後輸掉的？」",[7166,103547,103549],{"id":103548},"lms-最小均方-演算法","LMS (最小均方) 演算法",[12,103551,103552],{},"為了一直調整權重，我們會用到「梯度下降法」。每看到一個訓練範例，就做兩件事：",[3603,103554,103555,103779],{},[26,103556,103557,103558,103560,103561,103563,86843,103644,103646],{},"計算誤差：",[16,103559],{},"既然我們不知道這步真正該得幾分，我們就拿「下一步的預測分數」當作這步的臨時標準答案，我們稱之為",[16,103562],{},[24923,103564,103566],{"className":103565,"jax":24927},[24926],[24929,103567,103568,103594],{"style":24931,"xmlns":24932,"width":44166,"height":24934,"role":1930,"focusable":8058,"viewBox":44167,"xmlnsXLink":24936},[24938,103569,103570,103573,103576,103579,103582,103584,103587,103589,103592],{},[24941,103571],{"id":103572,"d":27385},"MJX-17-TEX-I-1D449",[24941,103574],{"id":103575,"d":25877},"MJX-17-TEX-I-1D461",[24941,103577],{"id":103578,"d":25881},"MJX-17-TEX-I-1D45F",[24941,103580],{"id":103581,"d":27323},"MJX-17-TEX-I-1D44E",[24941,103583],{"id":93988,"d":27108},[24941,103585],{"id":103586,"d":25873},"MJX-17-TEX-I-1D45B",[24941,103588],{"id":93982,"d":24944},[24941,103590],{"id":103591,"d":42548},"MJX-17-TEX-I-1D44F",[24941,103593],{"id":93991,"d":24988},[25022,103595,103596],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,103597,103598,103631,103635,103640],{"dataMmlNode":25028},[25022,103599,103600,103605],{"dataMmlNode":26023},[25022,103601,103602],{"dataMmlNode":25942},[25033,103603],{"dataC":27570,"xLinkHref":103604},"#MJX-17-TEX-I-1D449",[25022,103606,103607,103612,103617,103622,103626],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,103608,103609],{"dataMmlNode":25942},[25033,103610],{"dataC":25959,"xLinkHref":103611},"#MJX-17-TEX-I-1D461",[25022,103613,103614],{"dataMmlNode":25942,"transform":43828},[25033,103615],{"dataC":25966,"xLinkHref":103616},"#MJX-17-TEX-I-1D45F",[25022,103618,103619],{"dataMmlNode":25942,"transform":43610},[25033,103620],{"dataC":27425,"xLinkHref":103621},"#MJX-17-TEX-I-1D44E",[25022,103623,103624],{"dataMmlNode":25942,"transform":43838},[25033,103625],{"dataC":27210,"xLinkHref":94023},[25022,103627,103628],{"dataMmlNode":25942,"transform":43844},[25033,103629],{"dataC":25952,"xLinkHref":103630},"#MJX-17-TEX-I-1D45B",[25022,103632,103633],{"dataMmlNode":25031,"transform":44227},[25033,103634],{"dataC":25035,"xLinkHref":94005},[25022,103636,103637],{"dataMmlNode":25942,"transform":44232},[25033,103638],{"dataC":42570,"xLinkHref":103639},"#MJX-17-TEX-I-1D44F",[25022,103641,103642],{"dataMmlNode":25031,"transform":44238},[25033,103643],{"dataC":25108,"xLinkHref":94028},[16,103645],{},[24923,103647,103649],{"className":103648,"jax":24927},[24926],[24929,103650,103651,103679],{"style":24931,"xmlns":24932,"width":43719,"height":24934,"role":1930,"focusable":8058,"viewBox":43720,"xmlnsXLink":24936},[24938,103652,103653,103655,103657,103659,103661,103663,103665,103667,103669,103671,103673,103675,103677],{},[24941,103654],{"id":43725,"d":27397},[24941,103656],{"id":43728,"d":25881},[24941,103658],{"id":43731,"d":25885},[24941,103660],{"id":26650,"d":24944},[24941,103662],{"id":43736,"d":42548},[24941,103664],{"id":26663,"d":24988},[24941,103666],{"id":26666,"d":24972},[24941,103668],{"id":43743,"d":27385},[24941,103670],{"id":43746,"d":25877},[24941,103672],{"id":43749,"d":27323},[24941,103674],{"id":43752,"d":27108},[24941,103676],{"id":43755,"d":25873},[24941,103678],{"id":26669,"d":25910},[25022,103680,103681],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,103682,103683,103687,103691,103695,103699,103703,103707,103711,103715,103719,103747,103751,103755,103759,103763,103767,103771,103775],{"dataMmlNode":25028},[25022,103684,103685],{"dataMmlNode":25942},[25033,103686],{"dataC":27596,"xLinkHref":43766},[25022,103688,103689],{"dataMmlNode":25942,"transform":43769},[25033,103690],{"dataC":25966,"xLinkHref":43772},[25022,103692,103693],{"dataMmlNode":25942,"transform":43775},[25033,103694],{"dataC":25966,"xLinkHref":43772},[25022,103696,103697],{"dataMmlNode":25942,"transform":43780},[25033,103698],{"dataC":25973,"xLinkHref":43783},[25022,103700,103701],{"dataMmlNode":25942,"transform":43786},[25033,103702],{"dataC":25966,"xLinkHref":43772},[25022,103704,103705],{"dataMmlNode":25031,"transform":43791},[25033,103706],{"dataC":25035,"xLinkHref":26705},[25022,103708,103709],{"dataMmlNode":25942,"transform":43796},[25033,103710],{"dataC":42570,"xLinkHref":43799},[25022,103712,103713],{"dataMmlNode":25031,"transform":43802},[25033,103714],{"dataC":25108,"xLinkHref":26724},[25022,103716,103717],{"dataMmlNode":25031,"transform":43807},[25033,103718],{"dataC":25078,"xLinkHref":26730},[25022,103720,103721,103725],{"dataMmlNode":26023,"transform":43812},[25022,103722,103723],{"dataMmlNode":25942},[25033,103724],{"dataC":27570,"xLinkHref":43817},[25022,103726,103727,103731,103735,103739,103743],{"dataMmlNode":27203,"transform":43820,"dataMjxTexclass":27205},[25022,103728,103729],{"dataMmlNode":25942},[25033,103730],{"dataC":25959,"xLinkHref":43825},[25022,103732,103733],{"dataMmlNode":25942,"transform":43828},[25033,103734],{"dataC":25966,"xLinkHref":43772},[2502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這些具體的英文縮寫，統統改叫",[24923,104443,104445],{"className":104444,"jax":24927},[24926],[24929,104446,104448,104455],{"style":29805,"xmlns":24932,"width":54622,"height":29807,"role":1930,"focusable":8058,"viewBox":104447,"xmlnsXLink":24936},"0 -705 926.6 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5033,104641],{"dataC":29864,"xLinkHref":104642},"#MJX-33-TEX-I-1D453",[25022,104644,104645],{"dataMmlNode":26031,"transform":43996},[25033,104646],{"dataC":26058,"xLinkHref":44939},[106,104648,104567],{},[106,104650,104651],{},[24923,104652,104654],{"className":104653,"jax":24927},[24926],[24929,104655,104656,104672],{"style":24931,"xmlns":24932,"width":43359,"height":24934,"role":1930,"focusable":8058,"viewBox":43360,"xmlnsXLink":24936},[24938,104657,104658,104661,104663,104666,104669],{},[24941,104659],{"id":104660,"d":25881},"MJX-34-TEX-I-1D45F",[24941,104662],{"id":50093,"d":25889},[24941,104664],{"id":104665,"d":24944},"MJX-34-TEX-N-28",[24941,104667],{"id":104668,"d":42548},"MJX-34-TEX-I-1D44F",[24941,104670],{"id":104671,"d":24988},"MJX-34-TEX-N-29",[25022,104673,104674],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,104675,104676,104681,104685,104690,104695],{"dataMmlNode":25028},[25022,104677,104678],{"dataMmlNode":25942},[25033,104679],{"dataC":25966,"xLinkHref":104680},"#MJX-34-TEX-I-1D45F",[25022,104682,104683],{"dataMmlNode":25942,"transform":43386},[25033,104684],{"dataC":25980,"xLinkHref":50117},[25022,104686,104687],{"dataMmlNode":25031,"transform":43391},[25033,104688],{"dataC":25035,"xLinkHref":104689},"#MJX-34-TEX-N-28",[25022,104691,104692],{"dataMmlNode":25942,"transform":43396},[25033,104693],{"dataC":42570,"xLinkHref":104694},"#MJX-34-TEX-I-1D44F",[25022,104696,104697],{"dataMmlNode":25031,"transform":43402},[25033,104698],{"dataC":25108,"xLinkHref":104699},"#MJX-34-TEX-N-29","：代表「紅棋的數量」。",[26,104702,104703,7206,104734,7206,104736,104781],{},[106,104704,104705],{},[24923,104706,104708],{"className":104707,"jax":24927},[24926],[24929,104709,104710,104718],{"style":29805,"xmlns":24932,"width":54622,"height":29807,"role":1930,"focusable":8058,"viewBox":104447,"xmlnsXLink":24936},[24938,104711,104712,104715],{},[24941,104713],{"id":104714,"d":29821},"MJX-35-TEX-I-1D453",[24941,104716],{"id":104717,"d":26455},"MJX-35-TEX-N-33",[25022,104719,104720],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,104721,104722],{"dataMmlNode":25028},[25022,104723,104724,104729],{"dataMmlNode":26023},[25022,104725,104726],{"dataMmlNode":25942},[25033,104727],{"dataC":29864,"xLinkHref":104728},"#MJX-35-TEX-I-1D453",[25022,104730,104731],{"dataMmlNode":26031,"transform":43996},[25033,104732],{"dataC":26486,"xLinkHref":104733},"#MJX-35-TEX-N-33",[106,104735,104567],{},[106,104737,104738],{},[24923,104739,104741],{"className":104740,"jax":24927},[24926],[24929,104742,104743,104755],{"style":24931,"xmlns":24932,"width":43413,"height":24934,"role":1930,"focusable":8058,"viewBox":43414,"xmlnsXLink":24936},[24938,104744,104745,104748,104751,104753],{},[24941,104746],{"id":104747,"d":42548},"MJX-36-TEX-I-1D44F",[24941,104749],{"id":104750,"d":42951},"MJX-36-TEX-I-1D458",[24941,104752],{"id":28600,"d":24944},[24941,104754],{"id":28612,"d":24988},[25022,104756,104757],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,104758,104759,104764,104769,104773,104777],{"dataMmlNode":25028},[25022,104760,104761],{"dataMmlNode":25942},[25033,104762],{"dataC":42570,"xLinkHref":104763},"#MJX-36-TEX-I-1D44F",[25022,104765,104766],{"dataMmlNode":25942,"transform":43332},[25033,104767],{"dataC":43135,"xLinkHref":104768},"#MJX-36-TEX-I-1D458",[25022,104770,104771],{"dataMmlNode":25031,"transform":43445},[25033,104772],{"dataC":25035,"xLinkHref":28695},[25022,104774,104775],{"dataMmlNode":25942,"transform":43451},[25033,104776],{"dataC":42570,"xLinkHref":104763},[25022,104778,104779],{"dataMmlNode":25031,"transform":43456},[25033,104780],{"dataC":25108,"xLinkHref":28715},"：代表「黑棋國王的數量」。",[12,104783,104784],{},[106,104785,104786],{},"舉個具體的例子：",[12,104788,104789,104790,104809,104810,104813,104814,86843],{},"假設你現在正下到一半（棋盤狀態 ",[24923,104791,104793],{"className":104792,"jax":24927},[24926],[24929,104794,104795,104800],{"style":25196,"xmlns":24932,"width":42655,"height":31523,"role":1930,"focusable":8058,"viewBox":42656,"xmlnsXLink":24936},[24938,104796,104797],{},[24941,104798],{"id":104799,"d":42548},"MJX-37-TEX-I-1D44F",[25022,104801,104802],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,104803,104804],{"dataMmlNode":25028},[25022,104805,104806],{"dataMmlNode":25942},[25033,104807],{"dataC":42570,"xLinkHref":104808},"#MJX-37-TEX-I-1D44F","），你定睛一看，發現畫面上目前有 ",[106,104811,104812],{},"5 顆黑棋","，而且",[106,104815,104816],{},"沒有黑棋國王",[12,104818,104819,104820,104837],{},"那麼，對這個棋盤 ",[24923,104821,104823],{"className":104822,"jax":24927},[24926],[24929,104824,104825,104829],{"style":25196,"xmlns":24932,"width":42655,"height":31523,"role":1930,"focusable":8058,"viewBox":42656,"xmlnsXLink":24936},[24938,104826,104827],{},[24941,104828],{"id":45166,"d":42548},[25022,104830,104831],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,104832,104833],{"dataMmlNode":25028},[25022,104834,104835],{"dataMmlNode":25942},[25033,104836],{"dataC":42570,"xLinkHref":45226}," 來說：",[23,104839,104840,104888],{},[26,104841,104842,104843],{},"第 1 個特徵值 ",[106,104844,104845],{},[24923,104846,104848],{"className":104847,"jax":24927},[24926],[24929,104849,104852,104864],{"style":29805,"xmlns":24932,"width":104850,"height":29807,"role":1930,"focusable":8058,"viewBox":104851,"xmlnsXLink":24936},"6.245ex","0 -705 2760.1 910",[24938,104853,104854,104857,104859,104861],{},[24941,104855],{"id":104856,"d":29821},"MJX-39-TEX-I-1D453",[24941,104858],{"id":50259,"d":25914},[24941,104860],{"id":50250,"d":24972},[24941,104862],{"id":104863,"d":26660},"MJX-39-TEX-N-35",[25022,104865,104866],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,104867,104868,104879,104883],{"dataMmlNode":25028},[25022,104869,104870,104875],{"dataMmlNode":26023},[25022,104871,104872],{"dataMmlNode":25942},[25033,104873],{"dataC":29864,"xLinkHref":104874},"#MJX-39-TEX-I-1D453",[25022,104876,104877],{"dataMmlNode":26031,"transform":43996},[25033,104878],{"dataC":26035,"xLinkHref":50324},[25022,104880,104881],{"dataMmlNode":25031,"transform":54498},[25033,104882],{"dataC":25078,"xLinkHref":50301},[25022,104884,104885],{"dataMmlNode":26031,"transform":54504},[25033,104886],{"dataC":26717,"xLinkHref":104887},"#MJX-39-TEX-N-35",[26,104889,104890,104891],{},"第 3 個特徵值 ",[106,104892,104893],{},[24923,104894,104896],{"className":104895,"jax":24927},[24926],[24929,104897,104898,104910],{"style":29805,"xmlns":24932,"width":104850,"height":29807,"role":1930,"focusable":8058,"viewBox":104851,"xmlnsXLink":24936},[24938,104899,104900,104902,104905,104908],{},[24941,104901],{"id":45335,"d":29821},[24941,104903],{"id":104904,"d":26455},"MJX-40-TEX-N-33",[24941,104906],{"id":104907,"d":24972},"MJX-40-TEX-N-3D",[24941,104909],{"id":35782,"d":25935},[25022,104911,104912],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,104913,104914,104925,104930],{"dataMmlNode":25028},[25022,104915,104916,104920],{"dataMmlNode":26023},[25022,104917,104918],{"dataMmlNode":25942},[25033,104919],{"dataC":29864,"xLinkHref":45396},[25022,104921,104922],{"dataMmlNode":26031,"transform":43996},[25033,104923],{"dataC":26486,"xLinkHref":104924},"#MJX-40-TEX-N-33",[25022,104926,104927],{"dataMmlNode":25031,"transform":54498},[25033,104928],{"dataC":25078,"xLinkHref":104929},"#MJX-40-TEX-N-3D",[25022,104931,104932],{"dataMmlNode":26031,"transform":54504},[25033,104933],{"dataC":26103,"xLinkHref":35791},[12,104935,104936,104963,104964,7206,104984,104987],{},[24923,104937,104939],{"className":104938,"jax":24927},[24926],[24929,104940,104941,104948],{"style":29805,"xmlns":24932,"width":45482,"height":29807,"role":1930,"focusable":8058,"viewBox":45483,"xmlnsXLink":24936},[24938,104942,104943,104946],{},[24941,104944],{"id":104945,"d":29821},"MJX-41-TEX-I-1D453",[24941,104947],{"id":35808,"d":27108},[25022,104949,104950],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,104951,104952],{"dataMmlNode":25028},[25022,104953,104954,104959],{"dataMmlNode":26023},[25022,104955,104956],{"dataMmlNode":25942},[25033,104957],{"dataC":29864,"xLinkHref":104958},"#MJX-41-TEX-I-1D453",[25022,104960,104961],{"dataMmlNode":25942,"transform":43996},[25033,104962],{"dataC":27210,"xLinkHref":35835}," 其實就是在問：「",[106,104965,104966,104967],{},"現在盤面上，第",[24923,104968,104970],{"className":104969,"jax":24927},[24926],[24929,104971,104972,104976],{"style":25196,"xmlns":24932,"width":28237,"height":28238,"role":1930,"focusable":8058,"viewBox":28239,"xmlnsXLink":24936},[24938,104973,104974],{},[24941,104975],{"id":35869,"d":27108},[25022,104977,104978],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,104979,104980],{"dataMmlNode":25028},[25022,104981,104982],{"dataMmlNode":25942},[25033,104983],{"dataC":27210,"xLinkHref":35922},[106,104985,104986],{},"號特徵有幾個？","」",[7166,104989,104991,104992,105017],{"id":104990},"為什麼更新公式裡需要乘上","為什麼更新公式裡需要乘上 ",[24923,104993,104995],{"className":104994,"jax":24927},[24926],[24929,104996,104997,105003],{"style":29805,"xmlns":24932,"width":45482,"height":29807,"role":1930,"focusable":8058,"viewBox":45483,"xmlnsXLink":24936},[24938,104998,104999,105001],{},[24941,105000],{"id":45523,"d":29821},[24941,105002],{"id":36024,"d":27108},[25022,105004,105005],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105006,105007],{"dataMmlNode":25028},[25022,105008,105009,105013],{"dataMmlNode":26023},[25022,105010,105011],{"dataMmlNode":25942},[25033,105012],{"dataC":29864,"xLinkHref":45540},[25022,105014,105015],{"dataMmlNode":25942,"transform":43996},[25033,105016],{"dataC":27210,"xLinkHref":36122},"？",[12,105019,105020,105021,105046],{},"了解了 ",[24923,105022,105024],{"className":105023,"jax":24927},[24926],[24929,105025,105026,105032],{"style":29805,"xmlns":24932,"width":45482,"height":29807,"role":1930,"focusable":8058,"viewBox":45483,"xmlnsXLink":24936},[24938,105027,105028,105030],{},[24941,105029],{"id":45596,"d":29821},[24941,105031],{"id":36282,"d":27108},[25022,105033,105034],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105035,105036],{"dataMmlNode":25028},[25022,105037,105038,105042],{"dataMmlNode":26023},[25022,105039,105040],{"dataMmlNode":25942},[25033,105041],{"dataC":29864,"xLinkHref":45747},[25022,105043,105044],{"dataMmlNode":25942,"transform":43996},[25033,105045],{"dataC":27210,"xLinkHref":36370}," 是什麼之後，我們回頭看 LMS 更新權重的公式，你會發現一個非常聰明的設計：",[12,105048,105049],{},[24923,105050,105052],{"className":105051,"jax":24927},[24926],[24929,105053,105054,105085],{"style":24931,"xmlns":24932,"width":43896,"height":24934,"role":1930,"focusable":8058,"viewBox":43897,"xmlnsXLink":24936},[24938,105055,105056,105059,105061,105063,105066,105069,105071,105073,105075,105077,105079,105081,105083],{},[24941,105057],{"id":105058,"d":37254},"MJX-45-TEX-I-1D464",[24941,105060],{"id":45778,"d":27108},[24941,105062],{"id":45790,"d":24972},[24941,105064],{"id":105065,"d":30313},"MJX-45-TEX-N-2B",[24941,105067],{"id":105068,"d":27045},"MJX-45-TEX-I-1D450",[24941,105070],{"id":45805,"d":27409},[24941,105072],{"id":45808,"d":29821},[24941,105074],{"id":45781,"d":27397},[24941,105076],{"id":45769,"d":25881},[24941,105078],{"id":45795,"d":25885},[24941,105080],{"id":36497,"d":24944},[24941,105082],{"id":45800,"d":42548},[24941,105084],{"id":36506,"d":24988},[25022,105086,105087],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105088,105089,105100,105104,105114,105119,105124,105128,105138,105142,105146,105150,105154,105158,105162,105166,105170],{"dataMmlNode":25028},[25022,105090,105091,105096],{"dataMmlNode":26023},[25022,105092,105093],{"dataMmlNode":25942},[25033,105094],{"dataC":37310,"xLinkHref":105095},"#MJX-45-TEX-I-1D464",[25022,105097,105098],{"dataMmlNode":25942,"transform":43003},[25033,105099],{"dataC":27210,"xLinkHref":45837},[25022,105101,105102],{"dataMmlNode":25031,"transform":43954},[25033,105103],{"dataC":25078,"xLinkHref":45857},[25022,105105,105106,105110],{"dataMmlNode":26023,"transform":43959},[25022,105107,105108],{"dataMmlNode":25942},[25033,105109],{"dataC":37310,"xLinkHref":105095},[25022,105111,105112],{"dataMmlNode":25942,"transform":43003},[25033,105113],{"dataC":27210,"xLinkHref":45837},[25022,105115,105116],{"dataMmlNode":25031,"transform":43970},[25033,105117],{"dataC":30393,"xLinkHref":105118},"#MJX-45-TEX-N-2B",[25022,105120,105121],{"dataMmlNode":25942,"transform":43976},[25033,105122],{"dataC":27054,"xLinkHref":105123},"#MJX-45-TEX-I-1D450",[25022,105125,105126],{"dataMmlNode":25031,"transform":43982},[25033,105127],{"dataC":27679,"xLinkHref":45908},[25022,105129,105130,105134],{"dataMmlNode":26023,"transform":43988},[25022,105131,105132],{"dataMmlNode":25942},[25033,105133],{"dataC":29864,"xLinkHref":45918},[25022,105135,105136],{"dataMmlNode":25942,"transform":43996},[25033,105137],{"dataC":27210,"xLinkHref":45837},[25022,105139,105140],{"dataMmlNode":25031,"transform":44001},[25033,105141],{"dataC":27679,"xLinkHref":45908},[25022,105143,105144],{"dataMmlNode":25942,"transform":44006},[25033,105145],{"dataC":27596,"xLinkHref":45842},[25022,105147,105148],{"dataMmlNode":25942,"transform":44012},[25033,105149],{"dataC":25966,"xLinkHref":45822},[25022,105151,105152],{"dataMmlNode":25942,"transform":44018},[25033,105153],{"dataC":25966,"xLinkHref":45822},[25022,105155,105156],{"dataMmlNode":25942,"transform":44023},[25033,105157],{"dataC":25973,"xLinkHref":45881},[25022,105159,105160],{"dataMmlNode":25942,"transform":44029},[25033,105161],{"dataC":25966,"xLinkHref":45822},[25022,105163,105164],{"dataMmlNode":25031,"transform":44034},[25033,105165],{"dataC":25035,"xLinkHref":36521},[25022,105167,105168],{"dataMmlNode":25942,"transform":44039},[25033,105169],{"dataC":42570,"xLinkHref":45897},[25022,105171,105172],{"dataMmlNode":25031,"transform":44045},[25033,105173],{"dataC":25108,"xLinkHref":36542},[12,105175,105176,105177,105194,105195,105220,105221,105248],{},"你看，要更新第",[24923,105178,105180],{"className":105179,"jax":24927},[24926],[24929,105181,105182,105186],{"style":25196,"xmlns":24932,"width":28237,"height":28238,"role":1930,"focusable":8058,"viewBox":28239,"xmlnsXLink":24936},[24938,105183,105184],{},[24941,105185],{"id":95635,"d":27108},[25022,105187,105188],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105189,105190],{"dataMmlNode":25028},[25022,105191,105192],{"dataMmlNode":25942},[25033,105193],{"dataC":27210,"xLinkHref":95661}," 個權重 (",[24923,105196,105198],{"className":105197,"jax":24927},[24926],[24929,105199,105200,105206],{"style":27744,"xmlns":24932,"width":44951,"height":44952,"role":1930,"focusable":8058,"viewBox":44953,"xmlnsXLink":24936},[24938,105201,105202,105204],{},[24941,105203],{"id":45963,"d":37254},[24941,105205],{"id":36667,"d":27108},[25022,105207,105208],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105209,105210],{"dataMmlNode":25028},[25022,105211,105212,105216],{"dataMmlNode":26023},[25022,105213,105214],{"dataMmlNode":25942},[25033,105215],{"dataC":37310,"xLinkHref":46006},[25022,105217,105218],{"dataMmlNode":25942,"transform":43003},[25033,105219],{"dataC":27210,"xLinkHref":36760},") 時，後面乘上了一個 ",[106,105222,105223],{},[24923,105224,105226],{"className":105225,"jax":24927},[24926],[24929,105227,105228,105234],{"style":29805,"xmlns":24932,"width":45482,"height":29807,"role":1930,"focusable":8058,"viewBox":45483,"xmlnsXLink":24936},[24938,105229,105230,105232],{},[24941,105231],{"id":46123,"d":29821},[24941,105233],{"id":36927,"d":27108},[25022,105235,105236],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105237,105238],{"dataMmlNode":25028},[25022,105239,105240,105244],{"dataMmlNode":26023},[25022,105241,105242],{"dataMmlNode":25942},[25033,105243],{"dataC":29864,"xLinkHref":46233},[25022,105245,105246],{"dataMmlNode":25942,"transform":43996},[25033,105247],{"dataC":27210,"xLinkHref":37019},"。為什麼要這樣做？",[12,105250,105251],{},[106,105252,105253],{},"這是一個「誰惹禍，誰負責」的機制。",[12,105255,105256,105257,105296],{},"延續上面的例子，假設這步棋我們預測錯誤了，產生了誤差 (",[24923,105258,105260],{"className":105259,"jax":24927},[24926],[24929,105261,105264,105272],{"style":25196,"xmlns":24932,"width":105262,"height":27038,"role":1930,"focusable":8058,"viewBox":105263,"xmlnsXLink":24936},"5.213ex","0 -442 2304 453",[24938,105265,105266,105268,105270],{},[24941,105267],{"id":46276,"d":27397},[24941,105269],{"id":46279,"d":25881},[24941,105271],{"id":46282,"d":25885},[25022,105273,105274],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105275,105276,105280,105284,105288,105292],{"dataMmlNode":25028},[25022,105277,105278],{"dataMmlNode":25942},[25033,105279],{"dataC":27596,"xLinkHref":46361},[25022,105281,105282],{"dataMmlNode":25942,"transform":43769},[25033,105283],{"dataC":25966,"xLinkHref":46367},[25022,105285,105286],{"dataMmlNode":25942,"transform":43775},[25033,105287],{"dataC":25966,"xLinkHref":46367},[25022,105289,105290],{"dataMmlNode":25942,"transform":43780},[25033,105291],{"dataC":25973,"xLinkHref":46378},[25022,105293,105294],{"dataMmlNode":25942,"transform":43786},[25033,105295],{"dataC":25966,"xLinkHref":46367},")。現在演算法要來微調那些「旋鈕（權重）」了。",[3603,105298,105299,105504],{},[26,105300,105301,105332,105334,105335,105376,105377,86843,105466,105468,7206,105471,105498,105499,105501,105503],{},[106,105302,105303,105304,105331],{},"黑棋國王的旋鈕 (",[24923,105305,105307],{"className":105306,"jax":24927},[24926],[24929,105308,105309,105316],{"style":61403,"xmlns":24932,"width":103170,"height":103171,"role":1930,"focusable":8058,"viewBox":103172,"xmlnsXLink":24936},[24938,105310,105311,105313],{},[24941,105312],{"id":95854,"d":37254},[24941,105314],{"id":105315,"d":26455},"MJX-50-TEX-N-33",[25022,105317,105318],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105319,105320],{"dataMmlNode":25028},[25022,105321,105322,105326],{"dataMmlNode":26023},[25022,105323,105324],{"dataMmlNode":25942},[25033,105325],{"dataC":37310,"xLinkHref":95902},[25022,105327,105328],{"dataMmlNode":26031,"transform":43003},[25033,105329],{"dataC":26486,"xLinkHref":105330},"#MJX-50-TEX-N-33",")：",[16,105333],{},"因為盤面上根本沒有黑棋國王 (",[24923,105336,105338],{"className":105337,"jax":24927},[24926],[24929,105339,105340,105352],{"style":29805,"xmlns":24932,"width":104850,"height":29807,"role":1930,"focusable":8058,"viewBox":104851,"xmlnsXLink":24936},[24938,105341,105342,105345,105348,105350],{},[24941,105343],{"id":105344,"d":29821},"MJX-51-TEX-I-1D453",[24941,105346],{"id":105347,"d":26455},"MJX-51-TEX-N-33",[24941,105349],{"id":37266,"d":24972},[24941,105351],{"id":29710,"d":25935},[25022,105353,105354],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105355,105356,105368,105372],{"dataMmlNode":25028},[25022,105357,105358,105363],{"dataMmlNode":26023},[25022,105359,105360],{"dataMmlNode":25942},[25033,105361],{"dataC":29864,"xLinkHref":105362},"#MJX-51-TEX-I-1D453",[25022,105364,105365],{"dataMmlNode":26031,"transform":43996},[25033,105366],{"dataC":26486,"xLinkHref":105367},"#MJX-51-TEX-N-33",[25022,105369,105370],{"dataMmlNode":25031,"transform":54498},[25033,105371],{"dataC":25078,"xLinkHref":37335},[25022,105373,105374],{"dataMmlNode":26031,"transform":54504},[25033,105375],{"dataC":26103,"xLinkHref":29735},")，所以公式後半段會變成： ",[24923,105378,105380],{"className":105379,"jax":24927},[24926],[24929,105381,105384,105404],{"style":26868,"xmlns":24932,"width":105382,"height":39628,"role":1930,"focusable":8058,"viewBox":105383,"xmlnsXLink":24936},"14.741ex","0 -666 6515.4 748",[24938,105385,105386,105389,105392,105394,105397,105399,105402],{},[24941,105387],{"id":105388,"d":27045},"MJX-52-TEX-I-1D450",[24941,105390],{"id":105391,"d":27409},"MJX-52-TEX-N-22C5",[24941,105393],{"id":29763,"d":25935},[24941,105395],{"id":105396,"d":27397},"MJX-52-TEX-I-1D452",[24941,105398],{"id":46612,"d":25881},[24941,105400],{"id":105401,"d":25885},"MJX-52-TEX-I-1D45C",[24941,105403],{"id":90453,"d":24972},[25022,105405,105406],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105407,105408,105413,105419,105424,105429,105435,105440,105445,105451,105456,105461],{"dataMmlNode":25028},[25022,105409,105410],{"dataMmlNode":25942},[25033,105411],{"dataC":27054,"xLinkHref":105412},"#MJX-52-TEX-I-1D450",[25022,105414,105416],{"dataMmlNode":25031,"transform":105415},"translate(655.2,0)",[25033,105417],{"dataC":27679,"xLinkHref":105418},"#MJX-52-TEX-N-22C5",[25022,105420,105422],{"dataMmlNode":26031,"transform":105421},"translate(1155.4,0)",[25033,105423],{"dataC":26103,"xLinkHref":29791},[25022,105425,105427],{"dataMmlNode":25031,"transform":105426},"translate(1877.7,0)",[25033,105428],{"dataC":27679,"xLinkHref":105418},[25022,105430,105432],{"dataMmlNode":25942,"transform":105431},"translate(2377.9,0)",[25033,105433],{"dataC":27596,"xLinkHref":105434},"#MJX-52-TEX-I-1D452",[25022,105436,105438],{"dataMmlNode":25942,"transform":105437},"translate(2843.9,0)",[25033,105439],{"dataC":25966,"xLinkHref":46647},[25022,105441,105443],{"dataMmlNode":25942,"transform":105442},"translate(3294.9,0)",[25033,105444],{"dataC":25966,"xLinkHref":46647},[25022,105446,105448],{"dataMmlNode":25942,"transform":105447},"translate(3745.9,0)",[25033,105449],{"dataC":25973,"xLinkHref":105450},"#MJX-52-TEX-I-1D45C",[25022,105452,105454],{"dataMmlNode":25942,"transform":105453},"translate(4230.9,0)",[25033,105455],{"dataC":25966,"xLinkHref":46647},[25022,105457,105459],{"dataMmlNode":25031,"transform":105458},"translate(4959.7,0)",[25033,105460],{"dataC":25078,"xLinkHref":90518},[25022,105462,105464],{"dataMmlNode":26031,"transform":105463},"translate(6015.4,0)",[25033,105465],{"dataC":26103,"xLinkHref":29791},[16,105467],{},[106,105469,105470],{},"結果：",[24923,105472,105474],{"className":105473,"jax":24927},[24926],[24929,105475,105476,105483],{"style":61403,"xmlns":24932,"width":103170,"height":103171,"role":1930,"focusable":8058,"viewBox":103172,"xmlnsXLink":24936},[24938,105477,105478,105480],{},[24941,105479],{"id":51178,"d":37254},[24941,105481],{"id":105482,"d":26455},"MJX-53-TEX-N-33",[25022,105484,105485],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105486,105487],{"dataMmlNode":25028},[25022,105488,105489,105493],{"dataMmlNode":26023},[25022,105490,105491],{"dataMmlNode":25942},[25033,105492],{"dataC":37310,"xLinkHref":51197},[25022,105494,105495],{"dataMmlNode":26031,"transform":43003},[25033,105496],{"dataC":26486,"xLinkHref":105497},"#MJX-53-TEX-N-33"," 完全不會被修改。",[16,105500],{},[106,105502,89395],{},"「這局上面根本沒有黑棋國王，所以這次預測失準絕對不是黑棋國王害的！不要動它的旋鈕！」",[26,105505,105506,105536,105538,105539,105578,105579,86843,105646,105648,7206,105650,105677,105678,105680,105682,105683,105710],{},[106,105507,105508,105509,105331],{},"黑棋的旋鈕 (",[24923,105510,105512],{"className":105511,"jax":24927},[24926],[24929,105513,105516,105522],{"style":61340,"xmlns":24932,"width":103170,"height":105514,"role":1930,"focusable":8058,"viewBox":105515,"xmlnsXLink":24936},"1.342ex","0 -443 1152.6 593",[24938,105517,105518,105520],{},[24941,105519],{"id":37543,"d":37254},[24941,105521],{"id":30074,"d":25914},[25022,105523,105524],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105525,105526],{"dataMmlNode":25028},[25022,105527,105528,105532],{"dataMmlNode":26023},[25022,105529,105530],{"dataMmlNode":25942},[25033,105531],{"dataC":37310,"xLinkHref":37561},[25022,105533,105534],{"dataMmlNode":26031,"transform":43003},[25033,105535],{"dataC":26035,"xLinkHref":30144},[16,105537],{},"盤面上有 5 顆黑棋 (",[24923,105540,105542],{"className":105541,"jax":24927},[24926],[24929,105543,105544,105555],{"style":29805,"xmlns":24932,"width":104850,"height":29807,"role":1930,"focusable":8058,"viewBox":104851,"xmlnsXLink":24936},[24938,105545,105546,105549,105551,105553],{},[24941,105547],{"id":105548,"d":29821},"MJX-55-TEX-I-1D453",[24941,105550],{"id":30179,"d":25914},[24941,105552],{"id":30176,"d":24972},[24941,105554],{"id":90747,"d":26660},[25022,105556,105557],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105558,105559,105570,105574],{"dataMmlNode":25028},[25022,105560,105561,105566],{"dataMmlNode":26023},[25022,105562,105563],{"dataMmlNode":25942},[25033,105564],{"dataC":29864,"xLinkHref":105565},"#MJX-55-TEX-I-1D453",[25022,105567,105568],{"dataMmlNode":26031,"transform":43996},[25033,105569],{"dataC":26035,"xLinkHref":30206},[25022,105571,105572],{"dataMmlNode":25031,"transform":54498},[25033,105573],{"dataC":25078,"xLinkHref":30200},[25022,105575,105576],{"dataMmlNode":26031,"transform":54504},[25033,105577],{"dataC":26717,"xLinkHref":90815},")，數字很大。公式後半段會變成：",[24923,105580,105582],{"className":105581,"jax":24927},[24926],[24929,105583,105586,105603],{"style":27778,"xmlns":24932,"width":105584,"height":35776,"role":1930,"focusable":8058,"viewBox":105585,"xmlnsXLink":24936},"10.593ex","0 -666 4681.9 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會被大幅度修改。",[16,105679],{},[106,105681,89395],{},"「畫面上有一大堆黑棋，結果我們分數還估錯，那『黑棋分數 (",[24923,105684,105686],{"className":105685,"jax":24927},[24926],[24929,105687,105688,105695],{"style":61340,"xmlns":24932,"width":103170,"height":105514,"role":1930,"focusable":8058,"viewBox":105515,"xmlnsXLink":24936},[24938,105689,105690,105693],{},[24941,105691],{"id":105692,"d":37254},"MJX-58-TEX-I-1D464",[24941,105694],{"id":30467,"d":25914},[25022,105696,105697],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105698,105699],{"dataMmlNode":25028},[25022,105700,105701,105706],{"dataMmlNode":26023},[25022,105702,105703],{"dataMmlNode":25942},[25033,105704],{"dataC":37310,"xLinkHref":105705},"#MJX-58-TEX-I-1D464",[25022,105707,105708],{"dataMmlNode":26031,"transform":43003},[25033,105709],{"dataC":26035,"xLinkHref":30499},")』這個旋鈕肯定設錯了，趕快用力調一下！」",[12,105712,105713,105714,7206,105743,105746],{},"總結來說，",[106,105715,105716],{},[24923,105717,105719],{"className":105718,"jax":24927},[24926],[24929,105720,105721,105728],{"style":29805,"xmlns":24932,"width":45482,"height":29807,"role":1930,"focusable":8058,"viewBox":45483,"xmlnsXLink":24936},[24938,105722,105723,105726],{},[24941,105724],{"id":105725,"d":29821},"MJX-59-TEX-I-1D453",[24941,105727],{"id":47396,"d":27108},[25022,105729,105730],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105731,105732],{"dataMmlNode":25028},[25022,105733,105734,105739],{"dataMmlNode":26023},[25022,105735,105736],{"dataMmlNode":25942},[25033,105737],{"dataC":29864,"xLinkHref":105738},"#MJX-59-TEX-I-1D453",[25022,105740,105741],{"dataMmlNode":25942,"transform":43996},[25033,105742],{"dataC":27210,"xLinkHref":47423},[106,105744,105745],{},"就是告訴電腦：「在這個回合裡，這項特徵參與度有多高。」"," 參與度越高的特徵，如果最後發現預測錯誤，它的權重就會被修正得越明顯。",[12,105748,105749],{},"直覺上很簡單：",[23,105751,105752,105755,105758],{},[26,105753,105754],{},"如果預測正確 → 不改變權重",[26,105756,105757],{},"如果預測太高 → 按比例減少權重",[26,105759,105760],{},"如果預測太低 → 按比例增加權重",[12,105762,105763,105764,105767],{},"在合理的情況下，LMS 保證能慢慢收斂，找到能讓",[106,105765,105766],{},"均方誤差 (Mean Squared Error, MSE)"," 降到最低的那組權重：",[12,105769,105770],{},[24923,105771,105773],{"className":105772,"jax":24927},[24926],[24929,105774,105778,105813],{"style":47283,"xmlns":24932,"width":105775,"height":105776,"role":1930,"focusable":8058,"viewBox":105777,"xmlnsXLink":24936},"25.71ex","3.176ex","0 -1051 11364 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",[24923,105935,105937],{"className":105936,"jax":24927},[24926],[24929,105938,105941,105948],{"style":27744,"xmlns":24932,"width":105939,"height":44389,"role":1930,"focusable":8058,"viewBox":105940,"xmlnsXLink":24936},"2.054ex","0 -683 908 840.8",[24938,105942,105943,105946],{},[24941,105944],{"id":105945,"d":47123},"MJX-61-TEX-I-1D44C",[24941,105947],{"id":51737,"d":27108},[25022,105949,105950],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,105951,105952],{"dataMmlNode":25028},[25022,105953,105954,105959],{"dataMmlNode":26023},[25022,105955,105956],{"dataMmlNode":25942},[25033,105957],{"dataC":47216,"xLinkHref":105958},"#MJX-61-TEX-I-1D44C",[25022,105960,105961],{"dataMmlNode":25942,"transform":47220},[25033,105962],{"dataC":27210,"xLinkHref":51753}," 就是我們剛剛作為目標的 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1208.8",[24938,106056,106057,106060,106063],{},[24941,106058],{"id":106059,"d":47123},"MJX-63-TEX-I-1D44C",[24941,106061],{"id":106062,"d":47129},"MJX-63-TEX-N-5E",[24941,106064],{"id":30870,"d":27108},[25022,106066,106067],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,106068,106069],{"dataMmlNode":25028},[25022,106070,106071,106085],{"dataMmlNode":26023},[25022,106072,106073],{"dataMmlNode":27203,"dataMjxTexclass":27205},[25022,106074,106075,106080],{"dataMmlNode":47235},[25022,106076,106077],{"dataMmlNode":25942},[25033,106078],{"dataC":47216,"xLinkHref":106079},"#MJX-63-TEX-I-1D44C",[25022,106081,106082],{"dataMmlNode":25031,"transform":47242},[25033,106083],{"dataC":47245,"xLinkHref":106084},"#MJX-63-TEX-N-5E",[25022,106086,106087],{"dataMmlNode":25942,"transform":47220},[25033,106088],{"dataC":27210,"xLinkHref":30937}," 則是我們當前的預測值 ",[24923,106091,106093],{"className":106092,"jax":24927},[24926],[24929,106094,106095,106107],{"style":24931,"xmlns":24932,"width":42536,"height":24934,"role":1930,"focusable":8058,"viewBox":42537,"xmlnsXLink":24936},[24938,106096,106097,106100,106102,106105],{},[24941,106098],{"id":106099,"d":27385},"MJX-64-TEX-I-1D449",[24941,106101],{"id":31298,"d":24944},[24941,106103],{"id":106104,"d":42548},"MJX-64-TEX-I-1D44F",[24941,106106],{"id":31317,"d":24988},[25022,106108,106109],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,106110,106111,106116,106120,106125],{"dataMmlNode":25028},[25022,106112,106113],{"dataMmlNode":25942},[25033,106114],{"dataC":27570,"xLinkHref":106115},"#MJX-64-TEX-I-1D449",[25022,106117,106118],{"dataMmlNode":25031,"transform":42562},[25033,106119],{"dataC":25035,"xLinkHref":31376},[25022,106121,106122],{"dataMmlNode":25942,"transform":42567},[25033,106123],{"dataC":42570,"xLinkHref":106124},"#MJX-64-TEX-I-1D44F",[25022,106126,106127],{"dataMmlNode":25031,"transform":42574},[25033,106128],{"dataC":25108,"xLinkHref":31412},[24923,106130,106132],{"className":106131,"jax":24927},[24926],[24929,106133,106136,106146],{"style":27778,"xmlns":24932,"width":106134,"height":27780,"role":1930,"focusable":8058,"viewBox":106135,"xmlnsXLink":24936},"5.566ex","0 -705 2460 727",[24938,106137,106138,106141,106143],{},[24941,106139],{"id":106140,"d":31314},"MJX-65-TEX-I-1D440",[24941,106142],{"id":51990,"d":25900},[24941,106144],{"id":106145,"d":25869},"MJX-65-TEX-I-1D438",[25022,106147,106148],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,106149,106150,106155,106159],{"dataMmlNode":25028},[25022,106151,106152],{"dataMmlNode":25942},[25033,106153],{"dataC":31405,"xLinkHref":106154},"#MJX-65-TEX-I-1D440",[25022,106156,106157],{"dataMmlNode":25942,"transform":47145},[25033,106158],{"dataC":26000,"xLinkHref":52022},[25022,106160,106161],{"dataMmlNode":25942,"transform":47151},[25033,106162],{"dataC":25945,"xLinkHref":106163},"#MJX-65-TEX-I-1D438"," 其實就是把每次的誤差平方後加總平均。",[13438,106166,106168,106171],{"className":106167},[13441,13442],[12,106169,106170],{},"想像你面前有一台電腦，上面有七個旋鈕（分別對應黑棋數量、紅棋數量等等）。",[12,106172,106173],{},"電腦一開始是瞎猜的。我們讓電腦給旋鈕亂填幾個數字。假設電腦預測這盤棋的局勢能拿 80 分，結果實際打完發現只有 50 分（預測太高了）。LMS 演算法其實就是一個**「旋鈕微調機制」**：它會告訴電腦，「你分數估太高了，去把剛才轉太高的那幾個旋鈕往回轉一點點」。每一次下棋都在微調旋鈕，下了幾萬盤之後，旋鈕就會被調到一個「預測最準」的位置。這就是 LMS 在做的事。",[6964,106175,106177],{"id":106176},"訓練資料從哪來","訓練資料從哪來？",[12,106179,106180],{},"訓練資料的來源和品質非常重要，常見的來源有這幾種：",[23,106182,106183,106189,106195,106201],{},[26,106184,106185,106188],{},[106,106186,106187],{},"環境提供的隨機範例","（實務上最常見）",[26,106190,106191,106194],{},[106,106192,106193],{},"教師挑選的範例","，專門挑最有學習價值的資料（例如「差一點就猜對」的例子）",[26,106196,106197,106200],{},[106,106198,106199],{},"主動學習 (Active learning)","，模型遇到不確定的題目時，主動去問人類專家標準答案",[26,106202,106203,106206],{},[106,106204,106205],{},"自主實驗 (Self-directed experimentation)","，學習系統自己設計實驗來找答案",[12,106208,106209,106210,106213,106214,106217,106218,106221],{},"做機器學習時有一個很大的假設，就是訓練資料和未來要預測的測試資料必須是",[106,106211,106212],{},"獨立同分布 (Independently and Identically Distributed, IID)"," 的——白話文就是「它們必須來自同一個同溫層」。如果這個假設不成立，你就得用到像",[106,106215,106216],{},"遷移學習 (Transfer learning)"," 或",[106,106219,106220],{},"集體分類 (Collective Classification)"," 等更進階的技巧了。",[25631,106223,106225,106228],{"className":106224},[13441,13442],[25635,106226,106227],{},"補充説明：什麽是IID？",[13438,106229,106231,106240,106242,106247,106250,106253,106267,106270,106275,106277,106280,106282,106287,106290,106293,106295,106300,106303],{"className":106230},[25641],[12,106232,106233,106234,87213,106237,86843],{},"我們把這個詞拆成兩半來看：",[106,106235,106236],{},"「獨立 (Independent)」",[106,106238,106239],{},"「同分布 (Identically Distributed)」",[36,106241],{},[12,106243,106244],{},[106,106245,106246],{},"同分布 (Identically Distributed)",[12,106248,106249],{},"打個比方：訓練題庫和最終考試的「範圍與難度」必須一樣。",[12,106251,106252],{},"機器學習就像是一個準備大考的學生。",[23,106254,106255,106261],{},[26,106256,106257,106260],{},[106,106258,106259],{},"訓練資料"," = 你平常寫的模擬考題。",[26,106262,106263,106266],{},[106,106264,106265],{},"測試資料"," = 你最終去考的學測或統測。",[12,106268,106269],{},"如果你的模擬考題都是「高中數學」，而最終的大考也是考「高中數學」，我們就說這兩批資料是**「同分布」**的。因為它們背後出題的規則、範圍、難度都是一致的。學生（機器學習模型）只要平常練習得夠好，大考就能考高分。",[12,106271,106272],{},[106,106273,106274],{},"什麼情況下是「不同分布」？",[16,106276],{},[12,106278,106279],{}," 假設你平常寫的模擬考題都是「小學算術」（訓練資料），結果大考當天發下來的考卷是「大學微積分」（測試資料）。 這時學生絕對會考零分，因為考題的「分布（規則與環境）」完全改變了。",[16,106281],{},[12,106283,106284],{},[106,106285,106286],{},"獨立 (Independent)",[12,106288,106289],{},"每一筆資料都是相互獨立的，不會互相抄襲或影響。",[12,106291,106292],{},"想像一下你丟硬幣。你前一次丟出正面，會影響下一次丟出反面的機率嗎？不會。每一次丟硬幣都是一個全新的幾率，這就叫做**「獨立」**。",[16,106294],{},[12,106296,106297],{},[106,106298,106299],{},"什麼情況下是「不獨立」？",[12,106301,106302],{},"想像你在預測明天的天氣或明天的股票價格。 明天的天氣，絕對跟「今天的天氣」有高度關聯；明天的台積電股價，絕對跟「今天的台積電股價」息息相關。",[12,106304,106305],{},"在這種情況下，資料點和資料點之間是「互相影響」的。如果你把這些資料打亂順序隨便餵給 AI，AI 會學得一塌糊塗，因為它忽略了資料之間「時間上的因果關係」。",[6964,106307,106309],{"id":106308},"目標函數可以長什麼樣","目標函數可以長什麼樣？",[12,106311,106312],{},"不同的表達方式各有優缺點，主要就是在「表達能力」跟「好不好學」之間做取捨：",[23,106314,106315,106321,106327,106333],{},[26,106316,106317,106320],{},[106,106318,106319],{},"數值函數"," ： 線性迴歸、神經網路、支援向量機 (SVM)。這些非常擅長捕捉平滑、連續的模式。",[26,106322,106323,106326],{},[106,106324,106325],{},"符號函數"," ： 決策樹、邏輯規則。這種生出來的模型人類比較看得懂。",[26,106328,106329,106332],{},[106,106330,106331],{},"基於實例的函數"," ： 最近鄰演算法 (Nearest-neighbor)、基於案例的推理。它遇到新問題時，會去翻舊筆記，找最像的範例來參考。",[26,106334,106335,106338],{},[106,106336,106337],{},"機率圖模型"," ： 貝氏分類器 (Naïve Bayes)、貝氏網路、隱馬可夫模型。這類方法可以直接把變數之間的不確定性跟依賴關係建進模型裡。",[12,106340,106341],{},"一個模型的表達能力越強，能處理的問題就越複雜，但也代表你需要餵給它更多的資料，它才能學得準確。這是機器學習裡永遠逃不掉的取捨。",[6964,106343,106344],{"id":106344},"評估學習系統",[12,106346,106347],{},"訓練完一個模型之後，你怎麼知道它到底行不行？主要有兩種方法：",[23,106349,106350,106356],{},[26,106351,106352,106355],{},[106,106353,106354],{},"實驗評估"," ： 拿一組標準的資料集，用交叉驗證 (Cross-validation) 來做對照實驗。看的指標包括測試準確率、訓練時間、預測速度等等。然後用統計檢定來確認不同方法之間的差異到底是真的還是碰巧。",[26,106357,106358,106361],{},[106,106359,106360],{},"理論分析"," ： 用數學去證明這個演算法的能耐。例如它的運算複雜度、擬合能力、樣本複雜度 (Sample complexity)（到底需要多少資料才能學得夠好） 等等。",[12,106363,106364],{},"實務上，大部分 ML 工作兩種都會用到。",[6964,106366,106367],{"id":106367},"重點回顧",[3603,106369,106370,106376,106382,106388,106391],{},[26,106371,106372,106375],{},[106,106373,106374],{},"機器學習是關於從經驗中學習"," ： 使用資料來逼近函數，而不是靠人工編寫規則。",[26,106377,106378,106381],{},[106,106379,106380],{},"所有機器學習問題都能拆成 T、P、E","：任務、效能指標、經驗。",[26,106383,106384,106387],{},[106,106385,106386],{},"設計系統的四個選擇","：訓練經驗、目標函數、表達方式、學習演算法。",[26,106389,106390],{},"模型的表達能力越強，需要的資料量就越大。",[26,106392,106393,106396],{},[106,106394,106395],{},"評估是關鍵","：模型到底好不好，取決於它面對「沒看過的資料」時表現得怎樣。",[9332,106398,31966],{},{"title":180,"searchDepth":181,"depth":181,"links":106400},[106401,106402,106403,106404,106405,106409,106410,106411,106417,106418,106419,106420],{"id":102197,"depth":181,"text":102197},{"id":102209,"depth":181,"text":102210},{"id":102240,"depth":181,"text":102241},{"id":102287,"depth":181,"text":102287},{"id":102319,"depth":181,"text":102319,"children":106406},[106407,106408],{"id":102322,"depth":7430,"text":102322},{"id":102360,"depth":7430,"text":102361},{"id":102381,"depth":181,"text":102382},{"id":102478,"depth":181,"text":102478},{"id":102510,"depth":181,"text":102511,"children":106412},[106413,106414,106415,106416],{"id":102517,"depth":7430,"text":102518},{"id":102820,"depth":7430,"text":102821},{"id":103525,"depth":7430,"text":103525},{"id":103548,"depth":7430,"text":103549},{"id":106176,"depth":181,"text":106177},{"id":106308,"depth":181,"text":106309},{"id":106344,"depth":181,"text":106344},{"id":106367,"depth":181,"text":106367},"2026-04-02","基於 WIA1006\u002FWID3006 機器學習 — 馬來亞大學，2025\u002F2026 學年第 2 學期","2026-04-04","\u002Fimages\u002Fuploads\u002F1775143473434-G-LPXbYa0AMCyl5.jpg",{},"\u002Fblog\u002F機器學習導論",{"title":102192,"description":106422},{"loc":106426},"blog\u002F機器學習導論","8au4At6DZoyxPpUqidYCXyiAXS-ExBgUqCnvIo6LXPg",{"id":106432,"title":106433,"author":7,"body":106434,"date":106533,"description":106534,"draft":185,"edited_at":106533,"extension":186,"featured_image":106535,"meta":106536,"navigation":189,"path":106537,"pinned":185,"seo":106538,"sitemap":106539,"stem":106540,"tags":194,"__hash__":106541},"blog\u002Fblog\u002FCVA（三）血管症候群-i.md","CVA（三）血管症候群 (i)",{"type":9,"value":106435,"toc":106531},[106436,106445,106454,106472,106490,106506,106509,106514],[12,106437,106438,106439,106441,106442,106444],{},"決定受損程度有兩個因素",[16,106440],{},"\n第一 受傷區域的大小還有位置，等下會重點說明這一塊",[16,106443],{},"\n第二 側支循環的量：就是備用小路，假如這條路徑良好 可以減少神經系統受損程度",[12,106446,106447,106448,106450,106451,106453],{},"假如以血管來看",[16,106449],{},"\n頸動脈堵住的話，會呈現單側偏癱，因為頸動脈分左右，堵住一邊，另一住還能流通",[16,106452],{},"\n但基底動脈的話，因為它是由兩條椎動脈匯流而程一條 再從這裡分出包括PCA在內的一堆血管 所以會呈現雙側偏癱",[12,106455,106456,106457,106459,106460,106462,106463,106465,106466,106468,106469,106471],{},"接下來講一下威氏環",[16,106458],{},"\n先來講由頸內動脈分出來的血管它們分別是",[16,106461],{},"\nMCA、ACA、還有連接ACA之間的前交通支、最後就是後交通支",[16,106464],{},"\n接著來講椎動脈出來的，包括了",[16,106467],{},"\n基底動脈及PCA",[16,106470],{},"\n（僅講義重點標注 不代表沒有其他）",[12,106473,106474,106475,106477,106478,106480,106481,106483,106484,106486,106487,106489],{},"那從以上這些血管假如出狀況了，會演變到以下的症候群",[16,106476],{},"\nACA症候群",[16,106479],{},"\nMCA症候群",[16,106482],{},"\n內頸動脈症候群",[16,106485],{},"\nPCA症候群",[16,106488],{},"\n椎動脈症候群",[12,106491,106492,106493,106495,106496,106498,106499,106502,106503,106505],{},"先來講ACA Syndromes ",[16,106494],{},"\nACA的血主要供應兩個區域",[16,106497],{},"\n首先是大腦半球內側，會供應額葉（",[106,106500,106501],{},"初級運動皮層","）以及頂葉（初級本體感覺皮質區）",[16,106504],{},"\n接著也會往裡面跑，供應基底核（前內囊以及尾狀核下部）、穹隆前部、胼胝體前4\u002F5",[12,106507,106508],{},"那假如ACA出狀況了 會有以下癥狀",[23,106510,106511],{},[26,106512,106513],{},"對側偏癱（下肢比上肢嚴重）————初級運動區域受影響",[23,106515,106516,106519,106522,106525,106528],{},[26,106517,106518],{},"對側感覺喪失（下肢比下肢嚴重）——初級感覺區域受影響",[26,106520,106521],{},"尿失禁————額葉的後內側受影響",[26,106523,106524],{},"失用症及雙手協調問題————胼胝體受影響",[26,106526,106527],{},"意志喪失\u002F動作遲緩（冷漠）————額葉內側受影響",[26,106529,106530],{},"一些原始反射如抓握\u002F吸吮反射再現——額葉受影響",{"title":180,"searchDepth":181,"depth":181,"links":106532},[],"2026-04-01","臨床表現與病灶位置的關係~ACA","\u002Fimages\u002Fuploads\u002F1775007735475-127888519_p0_master1200.jpg",{},"\u002Fblog\u002FCVA（三）血管症候群-i",{"title":106433,"description":106534},{"loc":106537},"blog\u002FCVA（三）血管症候群-i","aIiQIjhfqz8s6fj3maeOUNODLbO0hRK-c4us0W15xUs",{"id":106543,"title":106544,"author":7,"body":106545,"date":106533,"description":106719,"draft":185,"edited_at":101919,"extension":186,"featured_image":106720,"meta":106721,"navigation":189,"path":106722,"pinned":185,"seo":106723,"sitemap":106724,"stem":106725,"tags":194,"__hash__":106726},"blog\u002Fblog\u002FCVA（四）血管症候群-(ii).md","CVA（四）血管症候群 (ii)",{"type":9,"value":106546,"toc":106717},[106547,106566,106569,106601,106610,106616,106639,106642,106662,106668,106674,106680,106700,106703],[12,106548,106549,106550,106552,106553,106555,106556,106560,106562,106563,106565],{},"接著來講MCA Syndromes",[16,106551],{},"\n一樣先講他的供應部位",[16,106554],{},"\n它一部份會跑到大腦表層，供應",[75,106557,106559],{"className":106558},[79],"前、頂還有顳葉",[16,106561],{},"\n而另一部則會跑到基底核供應裏面的結構包括了",[16,106564],{},"\n後內囊、放射冠、殼核、尾狀核，以及蒼白球外側",[12,106567,106568],{},"假如MCA出狀況 就會有以下癥狀",[23,106570,106571,106574,106577,106580,106583,106586,106589,106592,106595,106598],{},[26,106572,106573],{},"對側偏癱（上肢比下肢嚴重）———初級運動皮質區受影響",[26,106575,106576],{},"對側感覺喪失（上肢比下肢嚴重）——初級感覺皮質區受影響",[26,106578,106579],{},"運動型失語症（吞吞吐吐，能理解）——Broca's 區受影響（優勢大腦半球）",[26,106581,106582],{},"理解型失語症（講話流暢 ，但答非所問）——Wernicke's 區受影響（優勢大腦半球）",[26,106584,106585],{},"全面性失語症（既說不了也理解不了）———額葉以及顳葉大面積受損（優勢大腦半球）",[26,106587,106588],{},"知覺障礙（失認症、空間感喪失、半側忽略）——右側頂葉受影響",[26,106590,106591],{},"對側同側偏盲（左腦受損，那兩隻眼的右半邊都看不見）——內囊的視角放射纖維受影響",[26,106593,106594],{},"喪失對側共軛注視（左腦損傷，無法看向右側）———額葉眼動區受影響",[26,106596,106597],{},"肢體運動性失用症（無法精細動作，肌力尚在）———後內囊上部受影響",[26,106599,106600],{},"感覺性運動失調（因失去本體感覺而運動不穩）———頂葉受影響",[12,106602,106603,106604,106606,106607,106609],{},"然後來講PCA Syndromes",[16,106605],{},"\n它會供應到",[16,106608],{},"\n枕葉（視皮質）、內側及下部顳葉、1\u002F5胼胝體、上腦幹、中腦、後間腦及大部份的視丘",[12,106611,106612,106613,106615],{},"假如PCA出狀況 ",[16,106614],{},"\n它分成了兩個區域 首先是周邊區域就會有以下癥狀",[23,106617,106618,106621,106624,106627,106630,106633,106636],{},[26,106619,106620],{},"對側同側偏盲（左腦受損，那兩隻眼的右半邊都看不見）——初級視覺皮質區或者視放射受影響",[26,106622,106623],{},"雙側偏盲但黃斑部(中心)不受影響（兩邊都盲但只有中間看得到）——禽距皮質受影響（但黃斑部由MCA供血 不受影響）",[26,106625,106626],{},"視覺失認症（知覺看到，但到了認知理解不了）————左枕葉受影響",[26,106628,106629],{},"臉盲症（無法用眼分辨人臉）————視覺聯合皮質區受損",[26,106631,106632],{},"失讀症（沒有失寫症）（會寫字但不知道自己寫甚麼）———優勢腦（左腦）禽距或後胼胝體受影響",[26,106634,106635],{},"記憶缺損（失憶）————顳葉內（包括海馬體）受影響",[26,106637,106638],{},"地形定向障礙（失去地圖感，再熟悉也會迷路）————非優勢腦（右腦）初級視覺皮質區受影響",[12,106640,106641],{},"接著是中心區域就會有以下癥狀",[23,106643,106644,106650,106653,106656,106659],{},[26,106645,106646,106647,106649],{},"視丘疼痛症候群、感覺缺失、感覺障礙————視丘的腹後外側核(VPL)受影響",[16,106648],{},"\n視丘疼痛症候群有（自發性對側疼痛、灼燒感、輕輕碰到也會很痛）",[26,106651,106652],{},"不自主運動（舞蹈症、意向顫抖、半身不自主亂揮）———下視丘核受影響",[26,106654,106655],{},"對側偏癱————中腦的腦角受影響",[26,106657,106658],{},"韋伯氏症候群（眼睛無法轉向同側+對側半身偏癱）————第三對腦神經和大腦同時受影響",[26,106660,106661],{},"眼睛無法垂直轉動、瞳孔反應遲鈍、眼瞼下垂————第三對腦神經的核上纖維受影響",[12,106663,106664,106665,106667],{},"最後講到椎基底動脈症候群",[16,106666],{},"\n將兩個血管分開來看它們分別的供應區域",[12,106669,106670,106671,106673],{},"首先是椎動脈 它分別供應了小腦跟延腦",[16,106672],{},"\n接著是基底動脈 它分別供應了橋腦、內耳以及小腦",[12,106675,106676,106677,106679],{},"接著分別來講椎動脈和基底動腦出狀況會有哪些癥狀",[16,106678],{},"\n先來講椎動脈 有延腦外側症候群 包括以下癥狀",[23,106681,106682,106685,106688,106691,106694,106697],{},[26,106683,106684],{},"同側臉部溫覺以痛覺————三叉神經受影響",[26,106686,106687],{},"小腦型共濟失調（走路不穩，肢體不協調）————小腦受影響",[26,106689,106690],{},"眩暈、噁心、嘔吐、眼球震顫————前庭神經核受影響",[26,106692,106693],{},"霍納氏症候群（瞳孔縮小、眼瞼下垂、無汗）———下行交感神經纖維受影響",[26,106695,106696],{},"吞嚥與發聲困難（嘔吐反射喪失）————第九第十對腦神經受影響",[26,106698,106699],{},"對側身體超過50%痛溫覺受損————脊髓視丘束受影響",[12,106701,106702],{},"接著講基底動脈，有閉鎖症候群（LIS），包括以下癥狀",[23,106704,106705,106708,106711,106714],{},[26,106706,106707],{},"四肢癱瘓————雙側皮質脊髓束受影響",[26,106709,106710],{},"雙側腦神經痲痺（眼睛能上下動及眨眼）————雙側腦神經的長束（Long tract）",[26,106712,106713],{},"昏迷————網狀活化系統（假如這裡沒有受傷就是清醒的LIS）",[26,106715,106716],{},"認知功能保留（知道發生了甚麼，但無法回應）",{"title":180,"searchDepth":181,"depth":181,"links":106718},[],"MCA PCA 椎基底動脈 症候群","\u002Fimages\u002Fuploads\u002F1775182252893-HE0KsPOaYAAB8aI.jpg",{},"\u002Fblog\u002FCVA（四）血管症候群-(ii)",{"title":106544,"description":106719},{"loc":106722},"blog\u002FCVA（四）血管症候群-(ii)","tffLbyOs4jqV6xMktkS4Kqc3h9q8svjEgnN-S5mwhqg",{"id":106728,"title":106729,"author":7,"body":106730,"date":107190,"description":107191,"draft":185,"edited_at":98838,"extension":186,"featured_image":107192,"meta":107193,"navigation":189,"path":107194,"pinned":185,"seo":107195,"sitemap":107196,"stem":107197,"tags":194,"__hash__":107198},"blog\u002Fblog\u002FMotor-Learning.md","Motor Learning小兒（一）",{"type":9,"value":106731,"toc":107188},[106732,106735,106744,106753,106771,106780,106789,106805,106814,106842,106916,106931,106934,106976,106982,107030,107076,107117,107120,107157,107172],[12,106733,106734],{},"那這一節課著重在動態模組理論，跟黃老師教的是互通的",[12,106736,106737,106738,106740,106741,106743],{},"那首先是講感覺統合的部份",[16,106739],{},"\n意思是指 知覺+大腦以往的經歷 結合成為認知的這麼一個過程",[16,106742],{},"\n而感覺統合就是結合的過程 認知是結果這樣(這跟神經物治講的動作控制學雷同)",[12,106745,106746,106747,106749,106750,106752],{},"而接下來講到動作的發展",[16,106748],{},"\n那老師用的理論是動態模組理論",[16,106751],{},"\n假如還記得黃老師的Motor Control最後一課講的應該不難吃",[12,106754,106755,106756,106758,106759,106761,106762,106764,106765,106767,106768,106770],{},"不過我還是先列出來老師PPT裡的五個角度",[16,106757],{},"\n第一 動作技巧的顯現",[16,106760],{},"\n第二 動作型態的精準度",[16,106763],{},"\n第三 動作成果(可從評估表推估) ",[16,106766],{},"\n第四 組合技巧的獲得(次系統組合的結果)",[16,106769],{},"\n第五 對環境適應力的增強",[12,106772,106773,106774,106776,106777,106779],{},"我們現在來講平衡系統的發展",[16,106775],{},"\n1)平衡系統有三個人一起負責 迷路系統 本體覺與觸覺 視角",[16,106778],{},"\n以站立來形容的話",[12,106781,106782,106783,106785,106786,106788],{},"2)人類的重心(COM)基本會在S2 ",[16,106784],{},"\n那壓力中心(COP)是動態的 他會因應COM的改變而改變",[16,106787],{},"\n那底面積(BOS)就是兩隻腳踩在地面的距離",[12,106790,106791,106792,106796,106797,106799,106800,106804],{},"3)那平衡的演化 在早期 小孩會",[75,106793,106795],{"className":106794},[79,78],"極度依賴視角","來判斷",[16,106798],{},"\n但在成熟的過程中 我們會",[75,106801,106803],{"className":106802},[5387],"越來越傾向用本體感覺","去判斷",[12,106806,106807,106808,106810,106811,106813],{},"4)穩定極限(LOS) 他把我們人類想象成一個倒立的三角錐",[16,106809],{},"\n在不移動的情況下身體能偏多少(前6度 側8度 後4度)",[16,106812],{},"\n因為前面有腳趾輔助 而且側邊是比較寛 但後面甚麼都沒有 所以角度就變成這樣了(可以看游老師的講義)",[12,106815,106816,106817,106819,106820,106824,106825,106829,106830,106832,106833,106837,106838],{},"5)承受外力時的反應時 人類會有兩種變化",[16,106818],{},"\n第一種就是假如這",[75,106821,106823],{"className":106822},[78,173],"外力少到一定範圍"," 就只會用",[75,106826,106828],{"className":106827},[6760],"姿勢穩定反應","來做調整",[16,106831],{},"\n另一種則是假如外力",[75,106834,106836],{"className":106835},[78,79],"超過姿勢穩定反應可以調整的程度"," 就會產生",[75,106839,106841],{"className":106840},[5387],"保護性姿勢",[12,106843,106844,106845,106847,106848,106850,106851,106853,106854,106858,106860,106861,106865,106869,106871,106872,106874,106876,106877,106881,106882,106884,106885,4995,106889,106891,106892,106894,106896,106897,106899,106900,106902,106903,106907,106908,106910,106912,106913,106915],{},"6)",[16,106846],{},"\ni) 我們在平衡反應上還有兩種策略可以選 一種是固定BOS的 一種是改變BOS的",[16,106849],{},"\n首先是固定的一類 最熟悉的就是我們的踝策略和髖策略",[16,106852],{},"\n那一般而言 越成熟的人 會越偏好用",[75,106855,106857],{"className":106856},[79],"踝策略",[16,106859],{},"\n而目前比較新的提出 就是",[75,106862,106864],{"className":106863},[78392,78],"抓握",[75,106866,106868],{"className":106867},[78392],"是一種不算策略(Strategy)的方法",[16,106870],{},"\n例如老人在快摔倒時握緊拐仗穩定好自己 這也是一種功能性向的反應",[16,106873],{},[16,106875],{},"\nii) 而另一點則是改變BOS的 例子就有跨步策略 身體是",[75,106878,106880],{"className":106879},[79],"被迫做出","的 伸手出去支撐也是如此",[16,106883],{},"\n那在發展過程中 而在",[75,106886,106888],{"className":106887},[79],"方向性上也是有順序",[16,106890],{},"\n往前大概在6個月大才學會 而側向是8個月大 最後才是後向10個月大",[16,106893],{},[16,106895],{},"\niii) 那在臨床上 有些病人可能會因為關節或肌肉力量的原因 ",[16,106898],{},"\n我們會教導他們去利用拐仗 來增加BOS的大小 這樣他的LOS也會變大",[16,106901],{},"\n我們",[75,106904,106906],{"className":106905},[78],"可以通過適度的給他們外力","訓練他們",[16,106909],{},[16,106911],{},"\niv) 目前 reach test 是最好用來測病人靜態以及半靜態平衡的方法",[16,106914],{},"\n盡可能讓病人手伸出去多一點 那假如伸得遠出去 平衡就越好",[12,106917,106918,106919,106921,106922,106924,106925,106927,106928,106930],{},"那對於小兒的發展議題，主要有這幾個",[16,106920],{},"\ni)——新形式的起源(The origin of new form)",[16,106923],{},"\nii)——連續性與非連續性(Continuity vs discontinuity)",[16,106926],{},"\niii)——變異性(Variability)",[16,106929],{},"\niv)——生物與環境的影響(Biological vs. environmental influence)",[12,106932,106933],{},"那動態模組理論是經歷過以下發展而來的",[3603,106935,106936,106943,106963],{},[26,106937,106938,106939],{},"那首先是小兒痲痺(1910s)——當年的理念是哪邊無力練哪邊",[75,106940,106942],{"className":106941},[78,5387],"(肌肉再教育 muscle re-education)",[26,106944,106945,106946,106950,106951,106953,106954,106956,106957,106959,106960,106962],{},"接著是1950s——當年他們開始著重以神經成熟理論為基礎學派，著重在",[75,106947,106949],{"className":106948},[5387,78],"感覺輸入","促進正常的動作以及動作發展",[16,106952],{},"\n神經成熟理論(1945s)由Gesell和McGraw提出，他們的關鍵理念有如下",[16,106955],{},"\nCNS髓鞘化(Increased myelination of CNS)",[16,106958],{},"\n皮質抑制皮質下中心(Inhibition of the subcortical centers by the higher cerebral cortex)",[16,106961],{},"\n層級控制(Hierarchical control)",[26,106964,106965,106966,106970,106971,106975],{},"最後來講到階層理論的雛型(1980s)——當年他們覺得動作發展",[75,106967,106969],{"className":106968},[78],"不只單一的神經支配","，把",[75,106972,106974],{"className":106973},[78,79],"其他的系統、環境","一併整合進來",[12,106977,106978,106979,106981],{},"那神經成熟理論貢獻有兩點，正常的發展與病理的進程，以及反射到動作里程碑的評估",[16,106980],{},"\n那它有三個假設",[3603,106983,106984,106995,107014],{},[26,106985,106986,106987,106989,106990,106994],{},"從反射到隨意動作——意思是剛出生只有原始反射，但隨著大腦成熟這些反射會被抑制",[16,106988],{},"\n但在1979s有學者提出說，其實原始反射對於每個寶寶來說",[75,106991,106993],{"className":106992},[79],"變異性","也很大",[26,106996,106997,106998,107000,107001,107005,107006,98980,107010],{},"頭尾走向——意思是身體的發展一定是從頭到胸到骨盤再到下肢的發展",[16,106999],{},"\n但在1976s有學者發現到，南非的寶寶被家長",[75,107002,107004],{"className":107003},[78],"直立抱法或者練習坐","，會比美國寶寶",[75,107007,107009],{"className":107008},[79],"更快發展出下肢動作",[75,107011,107013],{"className":107012},[173],"爬和翻身較慢",[26,107015,107016,107017,107019,107020,107024,107025,107029],{},"最後是近端到遠端——意思是身體發展要先學會控制近端(肩、軀幹)，而後才能控制遠端(手指)",[16,107018],{},"\n但在1988s有學者提出，寶寶的",[75,107021,107023],{"className":107022},[173],"伸手以及操作","的發展在近端遠端上是",[75,107026,107028],{"className":107027},[78,79],"同時存在","的(同步進行發展，並沒有明確上下關係)",[12,107031,107032,107033,107035,107036,107040,107041,107045,107047,107048,107052,107053,107055,107056,107060,107061,107065,107066,107070,107072,107073,107075],{},"那每個人在做動作時一定會受到自身物理特性影響例如關節角度、肌肉長度等因素",[16,107034],{},"\n那協調就是將這些複雜的東西，透過",[75,107037,107039],{"className":107038},[79],"降低自由度","，來",[75,107042,107044],{"className":107043},[173],"產生平滑的動作",[16,107046],{},"\n那降低自由度的策略會牽涉到一個叫",[75,107049,107051],{"className":107050},[5387],"自我組織","的東西，那它的核心理念是說",[16,107054],{},"\n動作是有",[75,107057,107059],{"className":107058},[5387],"適應性","的，次系統會根據",[75,107062,107064],{"className":107063},[78,79],"環境任務","等因素做調整，",[75,107067,107069],{"className":107068},[173],"再拼湊起來",[16,107071],{},"\n(舉例來說，即使一樣是走路，在光亮的環境和在昏暗的環境下，也會做一些微調)（Bernstein提出）",[16,107074],{},"\n有提到三個重點",[3603,107077,107078,107084,107100],{},[26,107079,107080,107081,107083],{},"次序參數——它探討的是關節之間的協同",[16,107082],{},"\n例如在同一個動作型態上，做不同的任務，力度或關節角度也稍微不同，但假如相對時間相同，就能利用次序參數說明",[26,107085,107086,107087,107091,107092,107094,107095,107099],{},"那個體在發展過程中會因應不同的任務去做動作調整，而通常會採取",[75,107088,107090],{"className":107089},[78,5387],"最省力、自然以及方便","的動作型態",[16,107093],{},"\n那假如這個動作達到了上述說的型態，便是",[75,107096,107098],{"className":107097},[79,78],"最穩定型態","，即使會因應任務去做改變也是十分穩定的",[26,107101,107102,107103,107105,107106,107110,107111,107113,107114,107116],{},"相移現象——意思就是說原本在做著一個穩定的動作型態",[16,107104],{},"\n因為一些",[75,107107,107109],{"className":107108},[79],"因素所干擾","，就會變成另一個動作型態",[16,107112],{},"\n(例如速度的改變從走路變成跑步，相對時間不同，所以是另一個型態，不適應次序參數)",[16,107115],{},"\n（對 黃老師教動態系統理論也提到過這個）",[12,107118,107119],{},"最後講兩個實驗",[3603,107121,107122,107143],{},[26,107123,107124,107125,107127,107128,107130,107131,92559,107135,107139,107140,107142],{},"第一個是關於踏步反射的實驗，那踏步反射在作原始反射，在第四周會消失，但在一歲時又會再次出現",[16,107126],{},"\n而神經成熟理論認為，是因為大腦成熟了，讓高階抑制了低階",[16,107129],{},"\n但動態模組理論發現，其實是因為在第四周寶寶",[75,107132,107134],{"className":107133},[78,79],"長脂肪比長肌肉快",[75,107136,107138],{"className":107137},[173],"力量不足","做不出來",[16,107141],{},"\n將寶寶放在水中利用浮力降低脂肪重量，踏步反射就可以又被誘發出來",[26,107144,107145,107146,107148,107149,107151,107152,107156],{},"第二個實驗，是觀察青蛙生長的環境與發育的關係",[16,107147],{},"\n青蛙需要水跟泥土的環境才能發育健全，假如只給水不給泥土的環境，就會讓發育有缺陷",[16,107150],{},"\n所以神經成熟是可以讓動作型態形成，但必須得加上",[75,107153,107155],{"className":107154},[78,79],"環境，時間以及個體本身條件","才能讓型態表現更好",[12,107158,107159,107160,107162,107163,107165,107166,107168,107169,107171],{},"那在臨床意義上有以下幾點",[16,107161],{},"\n分散式控制(Distributed control)",[16,107164],{},"\n控制參數可能來自兒童或者環境(Control parameter may be from the child or from the environment)",[16,107167],{},"\n由兒童主導(Active role of the child)",[16,107170],{},"\n個體差異的訊息(variability provides important information)",[12,107173,107174,107175,107179,107180,107184,107185,107187],{},"評估",[75,107176,107178],{"className":107177},[78,6760],"不應只侷限在探討神經發展過程","，而是要根據",[75,107181,107183],{"className":107182},[5387,78],"多個系統，家庭因素","等問題",[16,107186],{},"\n(之前聽到的例子，有一個治療師說家長假如一直給小兒玩手機沒讓他們多動，確實會讓發展遲緩)",{"title":180,"searchDepth":181,"depth":181,"links":107189},[],"2026-03-31","感覺統合~三個協調的重點","\u002Fimages\u002Fuploads\u002F1775795013607-141589321_p0.jpg",{},"\u002Fblog\u002FMotor-Learning",{"title":106729,"description":107191},{"loc":107194},"blog\u002FMotor-Learning","EmOKsVRjY6Fz-rmd5PiPG83wEP6j59SCfF75Y1WnFNU",{"id":107200,"title":107201,"author":7,"body":107202,"date":107457,"description":107458,"draft":185,"edited_at":107457,"extension":186,"featured_image":107459,"meta":107460,"navigation":189,"path":107461,"pinned":185,"seo":107462,"sitemap":107463,"stem":107464,"tags":107465,"__hash__":107467},"blog\u002Fblog\u002F腦性痲痺-CP.md","腦性痲痺 CP",{"type":9,"value":107203,"toc":107455},[107204,107207,107244,107256,107259,107311,107343,107372,107384,107410,107436,107441,107446],[12,107205,107206],{},"以下根據余老師做的總結來編寫",[12,107208,107209,107210,107212,107213,107217,107218,107222,107223,107225,107226,107230,107231,107235,107237,107238,107240,107241,107243],{},"首先是定義 ",[16,107211],{},"\n在大腦",[75,107214,107216],{"className":107215},[5387],"未成熟","的階段 因為",[75,107219,107221],{"className":107220},[5387],"腦傷而導致的動作障礙","為主的症候群",[16,107224],{},"\n而CP有",[75,107227,107229],{"className":107228},[79],"三個非","合併導致",[75,107232,107234],{"className":107233},[79],"發展遲緩",[16,107236],{},"\n非進行性(傷口不會再惡化) ",[16,107239],{},"\n非暫時性(腦傷不會好)",[16,107242],{},"\n非單純性(有機會響影到多個區域 症狀會有例如智能不足等)",[12,107245,107246,107247,107249,107250,107252,107253,107255],{},"影響區域有三種區分",[16,107248],{},"\n單側麻痺(Hemiplegia)(患側手腳無力)",[16,107251],{},"\n雙側麻痺(Diplegia)(通常雙腿影響比雙手明顯)",[16,107254],{},"\n四肢麻痺(Quadriplegia)(四肢 但上肢影響程度比下肢大)",[12,107257,107258],{},"我們現在來講三大分類",[12,107260,107261,107262,107264,107265,107269,107270,107272,107273,107277,107278,107280,107284,107285,107287,107288,107293,107294,107298,107299,107301,107302,107304,107305,107307,107308,107310],{},"首先是痙攣型(Spasticity) 也是最大宗的一類(70%)",[16,107263],{},"\n傷到了",[75,107266,107268],{"className":107267},[5387],"錐狀束","(錐體路徑 Pyramidal Tract)",[16,107271],{},"\n而四肢張力都會變大 ",[75,107274,107276],{"className":107275},[5387],"上肢比下肢","還要嚴重",[16,107279],{},[75,107281,107283],{"className":107282},[78],"上肢為Flex協同動作 下肢為Exten協同動作","(詳細請見李老師CVA中的表)",[16,107286],{},"\n併發症有 ",[75,107289,107292],{"className":107290},[107291],"green-5","斜視","(70%) ",[75,107295,107297],{"className":107296},[107291],"癲癇","(50%)",[16,107300],{},"\n治療重點有",[16,107303],{},"\n增加中軸張力(核心穩定)",[16,107306],{},"\n降低肢體張力",[16,107309],{},"\n提升抗重力能力",[12,107312,107313,107314,107264,107316,107320,107321,107323,107324,107328,107329,107331,107332,107334,107335,107301,107337,107339,107340,107342],{},"接著講徐動型(Athethoid)",[16,107315],{},[75,107317,107319],{"className":107318},[5387],"基底核黃核","部位",[16,107322],{},"\n張力",[75,107325,107327],{"className":107326},[79],"時大時小","(有時候很硬 有時候很軟)",[16,107330],{},"\n而頭部控制差 上肢較下肢嚴重 且有不隨意扭動",[16,107333],{},"\n併發症有 聽覺障礙(高頻聽不見) 語言障礙",[16,107336],{},[16,107338],{},"\n對稱協調控制",[16,107341],{},"\n雙側控制",[12,107344,107345,107346,107264,107348,107351,107353,107354,107356,107357,107359,107360,107362,107363,107365,107366,107368,107369,107371],{},"最後來講失調型(Ataxia)",[16,107347],{},[75,107349,77994],{"className":107350},[5387],[16,107352],{},"\n張力小",[16,107355],{},"\n特徵有 步幅大(怕跌倒) 意向性震顫(手主動靠近目標 手越抖)",[16,107358],{},"\n併發有 平衡 感覺統合 知覺都會較差",[16,107361],{},"\n在治療重點上",[16,107364],{},"\n會給他們穿加壓衣(增加感統知覺) ",[16,107367],{},"\n穩定與張力提升",[16,107370],{},"\n承重(增加感統知覺)",[12,107373,107374,107375,107377,107378,107380,107381,107383],{},"還有一些小分類 簡單帶過一下",[16,107376],{},"\n首先是僵直型 這個是比痙攣型張力還要再強",[16,107379],{},"\n接著是顫抖型 這個與失調型的意向性震顫不同 在靜止時也會自己節律的抖動",[16,107382],{},"\n最後是低張型 名稱軟寶寶 多為一歲前會觀察到 到兩三歲會再發展成痙攣型抑或者徐動型",[12,107385,107386,107387,107389,107390,107392,107393,107395,107396,107398,107399,107401,107402,107404,107405,107409],{},"接著來講治療原則 主要有五點",[16,107388],{},"\n早期教育(早療 黃金期)",[16,107391],{},"\n個別化(畢竟每一個孩子的症狀都不盡相同 要針對他們的症狀計劃)",[16,107394],{},"\n治療項目盡量以ADL為主",[16,107397],{},"\n日常活動(結合孩子一天的作息去制定訓練 回家訓練)",[16,107400],{},"\n感官輸入(如本體感覺 觸覺)",[16,107403],{},"\n老師有圈了個重點 ",[75,107406,107408],{"className":107407},[79],"張力 並不等於 肌力","(張力很大 但肌力通常低下 要控制張力同時訓練肌力)",[12,107411,107412,107413,107415,107416,107420,107421,107415,107423,107427,107428,107430,107431,107435],{},"最後來講預後",[16,107414],{},"\n假如 ",[75,107417,107419],{"className":107418},[78],"2歲前","的寶寶能自行坐起來 未來100%能夠自行走動",[16,107422],{},[75,107424,107426],{"className":107425},[78],"4歲的","寶寶還不會自行坐起來 未來不會走",[16,107429],{},"\n假如 是",[75,107432,107434],{"className":107433},[78],"偏癱","的寶寶 未來100%能夠自行走動",[12,107437,107438],{},[1930,107439],{"alt":1932,"src":107440},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774841979537-Screenshot_20260330_100529_Gallery.png",[12,107442,107443],{},[1930,107444],{"alt":1932,"src":107445},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774842001784-Screenshot_20260330_100531_Gallery.png",[12,107447,107448,107449,107451],{},"一些講義的內容 我隨後會在另一篇裡加上 ",[16,107450],{},[75,107452,107454],{"className":107453},[5387],"讀英文！！！",{"title":180,"searchDepth":181,"depth":181,"links":107456},[],"2026-03-30","按余老師給的總結懶人包做填寫","\u002Fimages\u002Fuploads\u002F1774842345403-129630217_p0_master1200.jpg",{},"\u002Fblog\u002F腦性痲痺-CP",{"title":107201,"description":107458},{"loc":107461},"blog\u002F腦性痲痺-CP",[107466],"小兒PT","Oa9L6eaO1VoQayD-z7Id8c20dDInqUv2J9f4wkCkC9g",{"id":107469,"title":107470,"author":7,"body":107471,"date":107631,"description":107632,"draft":185,"edited_at":106533,"extension":186,"featured_image":107633,"meta":107634,"navigation":189,"path":107635,"pinned":185,"seo":107636,"sitemap":107637,"stem":107638,"tags":194,"__hash__":107639},"blog\u002Fblog\u002FCVA（二）.md","CVA（二）",{"type":9,"value":107472,"toc":107629},[107473,107485,107506,107509,107534,107549,107567,107582,107585,107597,107613],[12,107474,107475,107476,107478,107479,107481,107482,107484],{},"現在我們來講分類 從病理上分成了三種",[16,107477],{},"\n血栓型(60%) 栓塞型(20%) 出血型(5%)",[16,107480],{},"\n而從發展進程來看又分成了三種(15%)",[16,107483],{},"\n完全型 發展型 短暫性腦缺血發作(TIA)",[12,107486,107487,107488,107490,107491,107493,107494,107496,107497,107499,107500,107502,107503,107505],{},"我們首先來講血栓型(Thrombotic Infarction)",[16,107489],{},"\n主要由粥狀動脈硬化斑塊和高血壓交互形成",[16,107492],{},"\n多發生在血管彎曲處積累班塊(多數為腦動脈第一主要分支處)",[16,107495],{},"\n而它是進行式的 所以在中風之前 多數患者都不清楚斑塊積累已深",[16,107498],{},"\n關於TIA這個前置警訊 它會誘發兩個機制",[16,107501],{},"\n腦血管痙攣--因血管突然收縮 導致原本就狹窄的血管完全被阻斷",[16,107504],{},"\n暫時性全身血壓過低--原本就狹窄 血壓還突然下降 導致血液供給不足",[12,107507,107508],{},"那來講機制 簡單就是血塊在斑塊中破開 觸發凝血因子 導致血栓(新民老師有專門講過 maybe重要)",[12,107510,107511,107512,107514,107515,107517,107518,107520,107521,107523,107524,107526,107527,7206,107531,107533],{},"接著講栓塞型(Embolic Infarction)",[16,107513],{},"\n栓塞型是說有栓子在血液中流血最終塞到血管",[16,107516],{},"\n通常栓子有三種來源",[16,107519],{},"\n心臟 頸內動脈血栓 頸動脈竇的粥狀斑塊",[16,107522],{},"\n而最常發生在MCA處",[16,107525],{},"\n而比起血栓型 栓塞型",[75,107528,107530],{"className":107529},[5387],"更加致命",[16,107532],{},"\n因為血栓型是發展型的(慢慢堵住) 而栓塞型是完全型的(突然有一塊東西堵住)",[12,107535,107536,107537,107539,107540,107542,107543,107545,107546,107548],{},"然後講有一個比較特殊的變體 叫空隙性中風(小洞中風 Lacunar Infarction)",[16,107538],{},"\n是一些近乎90度分出來的微小管血處發生 通常影響這三個部位",[16,107541],{},"\n基底核--在MCA的M1處 ",[16,107544],{},"\n腦幹--在基底動脈處",[16,107547],{},"\n視丘--在PCA的P1\u002FP2處",[12,107550,107551,107552,107554,107555,107557,107558,107560,107561,107563,107564,107566],{},"最後我們來講出血型(Hemorrhage)",[16,107553],{},"\n分成腦內出血跟蜘蛛膜下腔出血",[16,107556],{},"\n腦內出血就是指血管破裂 血跑到並壓迫腦實質空間",[16,107559],{},"\n而蜘蛛膜下腔出血通常是腦動脈瘤破裂",[16,107562],{},"\n而患者通常會說有以下徵狀",[16,107565],{},"\n劇烈頭痛(這輩子最痛的) 噁心嘔吐 脖子僵硬 意識昏迷",[12,107568,107569,107570,107572,107573,107575,107576,107578,107579,107581],{},"而出血型通常有這四種損傷機制",[16,107571],{},"\n第一點 血會被噴出來 擠壓到周邊神經組織",[16,107574],{},"\n第二點 腦內壓升高(ICP)",[16,107577],{},"\n第三點 腦疝脫 是指血塊搶佔了空間 把正常腦組織推走(mass effect)",[16,107580],{},"\n第四點 周圍血流減少 意思是不只佔了腦組織空間 還從血管外壓到血管 導致供血不能",[12,107583,107584],{},"通常有這三種原因導致腦中風",[12,107586,107587,107588,107590,107591,107593,107594,107596],{},"第一種 高血壓 分成了三個要點",[16,107589],{},"\n長期的高血壓會對血管管壁造成損傷(因長期承受高壓力沖刷導致管壁失去彈性)",[16,107592],{},"\n跟小洞中風一樣 都是微小血管問題(區別在一個堵住一個破開)",[16,107595],{},"\n高血壓性小腦出血",[12,107598,107599,107600,107602,107603,107605,107606,107610,107612],{},"第二種 囊狀動脈瘤",[16,107601],{},"\n也是因為長期被沖刷 導致內壁被過度拉伸",[16,107604],{},"\n好發在威氏環",[75,107607,107609],{"className":107608},[78],"前半部",[16,107611],{},"\n通常為8~10mm",[12,107614,107615,107616,107618,107619,107621,107622,107624,107625],{},"最後一種是 AVM 動靜脈畸型",[16,107617],{},"\n這一個是天生的 比較難辦 意思是動靜脈直接對接 中間並沒有微血管",[16,107620],{},"\n而靜脈沒有辦法承受來自動脈的血壓 導致破裂",[16,107623],{},"\n好發在大腦",[75,107626,107628],{"className":107627},[78],"後半部",{"title":180,"searchDepth":181,"depth":181,"links":107630},[],"2026-03-26","CVA分類","\u002Fimages\u002Fuploads\u002F1774515663816-IMG_20200709_193107.jpg",{},"\u002Fblog\u002FCVA（二）",{"title":107470,"description":107632},{"loc":107635},"blog\u002FCVA（二）","0XDkR6RCTktZl4TtW0mVMvgZKB74JCKuVMlzqb5NoMQ",{"id":107641,"title":107642,"author":7,"body":107643,"date":107739,"description":107740,"draft":185,"edited_at":107631,"extension":186,"featured_image":107741,"meta":107742,"navigation":189,"path":107743,"pinned":185,"seo":107744,"sitemap":107745,"stem":107746,"tags":194,"__hash__":107747},"blog\u002Fblog\u002F腦血管病變（CVA）（中風）(一).md","腦血管病變（CVA）（中風）(一)",{"type":9,"value":107644,"toc":107737},[107645,107648,107669,107690,107696,107698,107701,107710,107713,107716,107722,107728,107731,107734],[12,107646,107647],{},"好了同學們 我們來讀CVA",[12,107649,107650,107651,107653,107654,107656,107657,107659,107660,107662,107663,107665,107666,107668],{},"從成因上能分成 梗塞性(Infarction) 和出血性(hemorrhage)",[16,107652],{},"\n那主要徵狀有",[16,107655],{},"\n偏癱",[16,107658],{},"\n感覺異常",[16,107661],{},"\n失語症\u002F構音困難",[16,107664],{},"\n視野缺損",[16,107667],{},"\n認知及智力受損",[12,107670,107671,107672,107674,107675,107677,107678,107680,107681,107683,107684,107686,107687,107689],{},"流行病學上 ",[16,107673],{},"\nCVA在美國都排第三第四名 台灣排第二第三名",[16,107676],{},"\n年齡也是重要因素 大多數CVA都在65歲以上",[16,107679],{},"\n而到了55歲 那麼每十年 患CVA的機率就會翻一倍",[16,107682],{},"\n男性比女性更容易患上",[16,107685],{},"\n黑人比白人更容易患上",[16,107688],{},"\n梗塞性佔所有類型70~85%",[12,107691,107692,107693,107695],{},"然後有三種高風險因素",[16,107694],{},"\n高血壓 糖尿病 心臟病",[36,107697],{},[12,107699,107700],{},"以下補充一下來自物治精選神經的內容 可看可不看",[12,107702,107703,107704,107706,107707,107709],{},"除了這些還有其他高危因子",[16,107705],{},"\n膽固醇過高 運動量不足 過重 抽煙 或有CVA家族史",[16,107708],{},"\n上述八種有其三 屬高危族群",[12,107711,107712],{},"在國民健康署給了六項建議預防中風",[12,107714,107715],{},"定期健檢",[12,107717,107718,107719,107721],{},"血壓測量722原則--18歲以上每年至少在家量一次血壓 並按照以下組數測量",[16,107720],{},"\n7天連續量 一天早晚量一次共2次 每次量2遍取平均值 (建議血壓值在120\u002F80mmHg)",[12,107723,107724,107725,107727],{},"三高異常辨識--血壓在120\u002F80mmHg以下 ",[16,107726],{},"\n空腹血糖值在70~99mg\u002FdL LDL(低密度膽固醇)維持在130mg\u002FdL以下",[12,107729,107730],{},"科學算病管評估中心--利用 科學算病館 網站",[12,107732,107733],{},"維持良好習慣--如作息 運動 飲食(三低一高) 及情緒穩定",[12,107735,107736],{},"中風跡象--看微笑 看舉手 說你好",{"title":180,"searchDepth":181,"depth":181,"links":107738},[],"2026-03-24","CVACVACVACVA~(路人可只看底下橫線的內容)","\u002Fimages\u002Fuploads\u002F1774504345861-IMG_20200706_025803.jpg",{},"\u002Fblog\u002F腦血管病變（CVA）（中風）(一)",{"title":107642,"description":107740},{"loc":107743},"blog\u002F腦血管病變（CVA）（中風）(一)","40yKgAyqPlHgDoUlh3UA5TAEj97nPaw4hyO4mbERsx4",{"id":107749,"title":107750,"author":7,"body":107751,"date":107917,"description":107918,"draft":185,"edited_at":107739,"extension":186,"featured_image":107919,"meta":107920,"navigation":189,"path":107921,"pinned":185,"seo":107922,"sitemap":107923,"stem":107924,"tags":194,"__hash__":107925},"blog\u002Fblog\u002F動作學習（十）回饋-2-擴增性(KR).md","動作學習（十）回饋-2-擴增性(KR)",{"type":9,"value":107752,"toc":107915},[107753,107759,107797,107819,107825,107837,107852,107861,107879,107897,107903,107909],[12,107754,107755,107756,107758],{},"接下來到了動作過程中~動作結束這兩個階段的感覺回饋",[16,107757],{},"\n分成了固有回饋(內發 個體本身的) 和擴增性回饋(外在的 旁人給的)",[12,107760,107761,107762,107764,107765,107767,107768,107772,107773,107777,107778,107780,107781,107785,107786,107790,107791,107793,107794,107796],{},"我們來講擴增性回饋 ",[16,107763],{},"\n時間點上也是分成在動作過程中給他回饋 跟結束給他回饋",[16,107766],{},"\n而結束的時間點上還能分成",[75,107769,107771],{"className":107770},[5387],"立刻","給他回饋 和",[75,107774,107776],{"className":107775},[5387],"過一陣子","再給他回饋",[16,107779],{},"\n再頻率上還能分成 ",[75,107782,107784],{"className":107783},[5387],"做完一次","給一次回饋(單次) 和",[75,107787,107789],{"className":107788},[5387],"做完幾次","後才給一次回饋(累積)",[16,107792],{},"\n例如 心率監測器(一直跳多少一直變化一直調整給你看)",[16,107795],{},"\n抑或者跑步APP裡記錄的全程心跳過程(整組一次過統計平均給你看)",[12,107798,107799,107800,107802,107803,107805,107806,107810,107811,107815,107816,107818],{},"剛說完時間上 接著講特質上",[16,107801],{},"\n分成了口語化跟非口語化",[16,107804],{},"\n而口語化上又分成了 ",[75,107807,107809],{"className":107808},[5387],"講結果","(KR) 跟",[75,107812,107814],{"className":107813},[5387],"講表現","(KP)",[16,107817],{},"\n而非口語化上意思是例如給一些特定的訊號讓他知道(會意)自己剛剛的行為",[12,107820,107821,107822,107824],{},"KR像剛說的直接講結果 從結果上讓接收者去做調整",[16,107823],{},"\n例如說 投那顆球 差兩公分就進了 抑或者 沒進 沒中",[12,107826,107827,107828,107830,107831,107833,107834,107836],{},"KR在回饋的內容上 像剛剛的例子來看",[16,107829],{},"\n直接數值(差公分)的叫量化回饋 而講二分法(中 沒中)的叫質性回饋",[16,107832],{},"\n僅從表現結果來看",[16,107835],{},"\n量化＞質性＞不給回饋＞亂給回饋",[12,107838,107839,107840,107842,107843,107845,107846,107848,107849,107851],{},"在回饋頻率上也分成了 絕對回饋 跟相對回饋",[16,107841],{},"\n絕對回饋意思是 在一整組的動作裡 給了多少次回饋",[16,107844],{},"\n而相對回饋 就是用絕對回饋 除以練習次數",[16,107847],{},"\n例子 在一整組動作裡 給了50次回饋 那絕對回饋頻率=50",[16,107850],{},"\n而假如這整組動作裡做了100次動作 那相對回饋頻率就是50\u002F100=50%",[12,107853,107854,107855,107857,107858,107860],{},"而在KR上 相對回饋頻率的%數減少 能有效促進學習",[16,107856],{},"\n分別有以下做法",[16,107859],{},"\n帶狀回饋 消褪回饋 總結回饋 平均回饋",[12,107862,107863,107864,107866,107867,107869,107870,107872,107873,107875,107876,107878],{},"首先來講帶狀回饋 意思是假如病人在做動作的過程中",[16,107865],{},"\n設定一個範圍 給他空間自己調整",[16,107868],{},"\n假如說 太嚴苛(精準到0.幾公分那種) 比較不適合他去學習",[16,107871],{},"\n(過於精確的回饋對學習無益)",[16,107874],{},"\n比較建議是錯的太離譜才去給回饋",[16,107877],{},"\n(我自己理解--可能可以因應表現隨時調整範圍 讓病人的動作品質上升?)",[12,107880,107881,107882,107884,107885,107887,107888,107890,107891,107893,107894,107896],{},"而接下來消褪回饋 跟剛剛帶狀回饋一樣設一個範圍",[16,107883],{},"\n也是不到範圍內去給回饋",[16,107886],{},"\n而消褪比較偏向是頻率減少的部份 讓我自己去抓那個正確(動作及格)的範圍",[16,107889],{},"\n例如我第一次做 四次不達標 給我四次回饋-->下次做兩次不達標 給兩次回饋",[16,107892],{},"\n最後全部達標 不給回饋這樣",[16,107895],{},"\n(我自己理解--目的應該是更偏向病人自己抓感覺?)",[12,107898,107899,107900,107902],{},"接下來講結回饋 就是一組結束後再給他講分數 屬於是累積回饋的分類",[16,107901],{},"\n頻率可以因應難度而定(例如簡單的動作可以給更少的回饋)",[12,107904,107905,107906,107908],{},"最後講平均回饋 就是給他講一組的平均分 一樣是累積回饋",[16,107907],{},"\n例如 這組比上組平均高了幾分",[12,107910,107911],{},[75,107912,107914],{"className":107913},[5387],"而研究指出 學習者主動要求何時獲得回饋可有利學習",{"title":180,"searchDepth":181,"depth":181,"links":107916},[],"2026-03-23","擴增性KR","\u002Fimages\u002Fuploads\u002F1774235040613-IMG_20201025_201029.jpg",{},"\u002Fblog\u002F動作學習（十）回饋-2-擴增性(KR)",{"title":107750,"description":107918},{"loc":107921},"blog\u002F動作學習（十）回饋-2-擴增性(KR)","ZzMln4Vm6Ua3jNQdAp2tEruLv7ljm1uoXTPN0TokPVE",{"id":107927,"title":107928,"author":7,"body":107929,"date":107917,"description":108075,"draft":185,"edited_at":107739,"extension":186,"featured_image":108076,"meta":108077,"navigation":189,"path":108078,"pinned":185,"seo":108079,"sitemap":108080,"stem":108081,"tags":194,"__hash__":108082},"blog\u002Fblog\u002F動作學習（十一）回饋-3-擴增性(KP).md","動作學習（十一）回饋-3-擴增性(KP)",{"type":9,"value":107930,"toc":108073},[107931,107937,107943,107975,107989,108015,108030,108055],[12,107932,107933,107934,107936],{},"我們接著講KP 它是從動作的形態上 動作層面上的 偏向接收者要去腦內想畫面",[16,107935],{},"\n例如 你剛剛手抬不夠高 你剛剛投籃出手可以直一點",[12,107938,107939,107940,107942],{},"而KP也有一下四種回饋做法",[16,107941],{},"\n影像示範回饋 動態學回饋 生物回饋 動力學回饋",[12,107944,107945,107946,107948,107949,107953,107954,107956,107957,107959,107960,107964,107965,107967,107968,107972,107974],{},"先來講示範回饋 ",[16,107947],{},"\n示範回饋比不是",[75,107950,107952],{"className":107951},[5387],"漫無目的","地做整套動作讓病人自己吸收",[16,107955],{},"\n在這個基礎上還可以分成兩個時間段 ",[16,107958],{},"\n第一是示範前 引導病人去看 ",[75,107961,107963],{"className":107962},[5387],"局部的 關鍵的"," 動作形態",[16,107966],{},"\n第二則是示範過程中~結束 偏向讓病人去看哪裡有不對 提示他下次",[75,107969,107971],{"className":107970},[5387],"糾正",[16,107973],{},"\n有研究指示 有糾正的看示範會比只引導的看示範來的更有效",[12,107976,107977,107978,107980,107981,107983,107984,7963],{},"這裡有兩點是之前提到的 再申一次",[16,107979],{},"\n第一點 因為病人的注意力有限 所以得讓病的注意力放在局部的 關鍵的地方上",[16,107982],{},"\n第二點 就是看別人做錯的事能促進學習(",[2330,107985,107988],{"href":107986,"rel":107987},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E5%8B%95%E4%BD%9C%E5%AD%B8%E7%BF%92%EF%BC%88%E4%B9%9D%EF%BC%89%E5%9B%9E%E9%A5%8B-1",[2334],"也是之前提到的",[12,107990,107991,107992,107994,107995,107997,107998,108000,108001,108003,108004,108008,108009,108011,108012,108014],{},"接著來講動態學回饋",[16,107993],{},"\n是著重在動作軌跡 形態上的 而不是關注出多少力 ",[16,107996],{},"\n關注的點可以從角度 時間 速度 協調等 ",[16,107999],{},"\n例如 你手可以抬高點\u002F抬高30度",[16,108002],{},"\n(這個跟",[75,108005,108007],{"className":108006},[5387],"KR的直講結果上的數據","不一樣哦 這是在動作層面上的 不要搞混了)",[16,108010],{},"\n這裡也說了假如我們給甚麼樣的回饋 病人的成果會更偏向達成我們給予的回饋",[16,108013],{},"\n例如 只跟他講注意步距 步距的改善就會比協調的改善還要來得更好",[12,108016,108017,108018,108020,108021,108023,108024,108026,108027,108029],{},"然後來講生物回饋",[16,108019],{},"\n他會偏向讓病人看到生理上數值的變化 例如肌電圖 ",[16,108022],{},"\n他就按著肌電圖的改變 去調整自己的動作",[16,108025],{},"\n另一種則是給予聾啞人看聲波圖 也可以讓他們注意到聲波的變化",[16,108028],{},"\n讓他們盡量以達到跟聲波圖上的圖形重合 來讓他們嘗試發音",[12,108031,108032,108033,108035,108036,108040,108041,108043,108044,108048,108049,108051,108052,108054],{},"最後來講動力學回饋",[16,108034],{},"\n他跟動態動講動作軌跡 形態上有所不同 他關注的是",[75,108037,108039],{"className":108038},[5387],"肌肉使用上","的多少",[16,108042],{},"\n但也指出 實際讓他知道出力的",[75,108045,108047],{"className":108046},[6760],"具體參數","會不好",[16,108050],{},"\n直接在他出力的同時跟他說 大力了一點 小力了一點 這種讓他自己意會 動腦去想的 會更優",[16,108053],{},"\n而降低回饋頻率也能促進學習",[12,108056,108057,108058,108060,108061,108063,108064,108066,108067,108069,108070,108072],{},"那麼最後總結一下 臨床上按照以上回饋方式可以得出以下總結",[16,108059],{},"\n第一 在學新動作之前 可以示範一次給病人看 有利於在腦中構築動作型態",[16,108062],{},"\n第二 指導 示範 回饋的資訊不是越多越好",[16,108065],{},"\n第三 示範不必是正確的",[16,108068],{},"\n第四 假如病人在學習過程中能自行找出錯誤 可以減少回饋",[16,108071],{},"\n第五 回饋的頻率不要過高",{"title":180,"searchDepth":181,"depth":181,"links":108074},[],"擴增性KP","\u002Fimages\u002Fuploads\u002F1774238705780-IMG_20201030_204929.jpg",{},"\u002Fblog\u002F動作學習（十一）回饋-3-擴增性(KP)",{"title":107928,"description":108075},{"loc":108078},"blog\u002F動作學習（十一）回饋-3-擴增性(KP)","SiOk-dlj9g_UOnNQPqP-2GCWYSHsW5kIv45x4J4hJAY",{"id":108084,"title":108085,"author":7,"body":108086,"date":107917,"description":108219,"draft":185,"edited_at":107739,"extension":186,"featured_image":108220,"meta":108221,"navigation":189,"path":108222,"pinned":185,"seo":108223,"sitemap":108224,"stem":108225,"tags":194,"__hash__":108226},"blog\u002Fblog\u002F動作學習（十一）回饋-4-(間隔+物理引導).md","動作學習（十二）回饋-4-(間隔+物理引導)",{"type":9,"value":108087,"toc":108217},[108088,108091,108105,108127,108136,108153,108156,108162,108168,108177,108183,108192,108214],[12,108089,108090],{},"終於要結束回饋了",[12,108092,108093,108094,108096,108097,108099,108100,108104],{},"在回饋的延時上有兩種區分",[16,108095],{},"\n一種是回饋間隔 就是動作結束後 隔多久才給予回饋",[16,108098],{},"\n另一種是回饋後間隔 就是給完回饋後 隔了多久",[75,108101,108103],{"className":108102},[5387],"消化","完開始下一次動作的時間段",[12,108106,108107,108108,108112,108113,108115,108116,7963,108121,108123,108124,108126],{},"而在回饋間隔上 ",[75,108109,108111],{"className":108110},[5387],"不宜在動作結束","立即給予回饋 ",[16,108114],{},"\n因為會中斷基模自行修正 不利於基模的建立 (",[2330,108117,108120],{"href":108118,"rel":108119},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E5%8B%95%E4%BD%9C%E5%AD%B8%E7%BF%92%EF%BC%88%E4%BA%94%EF%BC%89%E7%90%86%E8%AB%96-2",[2334],"引用基模理論",[16,108122],{},"\n所以應當延時3~8秒才給予回饋",[16,108125],{},"\n(應當讓內在基模修正 而後以外在回饋方式進一步改進基模)",[12,108128,108129,108130,108132,108133,108135],{},"接著來講練習間隔 練習間隔是把 回饋間隔 與回饋後間隔 組合在一起",[16,108131],{},"\n而多數學者認為這個練習間隔越長 效果越佳",[16,108134],{},"\n(因為間隔長了 代表不記得先前的動作方式 讓大腦重新想一次動作程式)",[12,108137,108138,108139,108143,108144,108146,108147,108149],{},"而在動作練習組數之間還可以給予",[75,108140,108142],{"className":108141},[5387],"空白間隔"," (比較適用在組內在做同一種動作)",[16,108145],{},"\n例如第一組練習後給了回饋 在做二~五組時不給回饋 在做完第六組後再給回饋",[16,108148],{},[75,108150,108152],{"className":108151},[5387],"可以減少依賴 強化內在基模",[12,108154,108155],{},"在回饋間隔裡的活動 大概有兩種 一種是干擾活動 一種是主觀判斷",[12,108157,108158,108159,108161],{},"干擾活動中 在後續測驗表現會比較差 ",[16,108160],{},"\n因為大腦處理能力有限 假如還要給病人加重負擔 會不利於學習",[12,108163,108164,108165,108167],{},"而主觀判斷的意思是 在回饋的間隔中 ",[16,108166],{},"\n提醒病人以主觀判斷自己做的動作對不對(主動強化修正基模)",[12,108169,108170,108171,108173,108174,108176],{},"接著來講物理引導上 對於學習上也是有一定幫助的",[16,108172],{},"\n比較多用在一些很需要抓感覺的動作",[16,108175],{},"\n例如騎腳踏車 裝輔助輪跟我扶著 都是物理引導 在學會之前也是要抓平衡感(感覺)",[12,108178,108179,108180,108182],{},"而怎麼利用物理引導呢 像很常說的 手把手教真的學不會",[16,108181],{},"\n那麼有以下幾點",[12,108184,108185,108186,108188,108189,108191],{},"第一 穿插練習 例如10次動作裡 第 1 4 7次做引導 其他七次讓他自己做(減少依賴)",[16,108187],{},"\n第二 在最開始 陌生的時候給予引導 讓他先在大腦裡構建自己的模版",[16,108190],{},"\n第三 而在速度方面 指導的越慢 效果越好 反之亦然",[12,108193,108194,108195,108197,108198,108200,108201,108203,108204,108206,108207,108209,108210],{},"總結一下四種核心",[16,108196],{},"\n在最開始給予觀念",[16,108199],{},"\n慢速指導會更好",[16,108202],{},"\n可以預防受傷",[16,108205],{},"\n降低恐懼感",[16,108208],{},"\n但是最主要的 ",[75,108211,108213],{"className":108212},[5387],"不能代替主動學習",[12,108215,108216],{},"哇哈哈 還有100頁就結束啦",{"title":180,"searchDepth":181,"depth":181,"links":108218},[],"間隔+物理引導","\u002Fimages\u002Fuploads\u002F1774240031787-IMG_20201101_110456.jpg",{},"\u002Fblog\u002F動作學習（十一）回饋-4-(間隔+物理引導)",{"title":108085,"description":108219},{"loc":108222},"blog\u002F動作學習（十一）回饋-4-(間隔+物理引導)","zz94v9AMd7luMquc-2rTGiBAZFtvmF_8C8_bV-rwMlg",{"id":108228,"title":108229,"author":7,"body":108230,"date":107917,"description":108594,"draft":185,"edited_at":107739,"extension":186,"featured_image":108595,"meta":108596,"navigation":189,"path":108597,"pinned":185,"seo":108598,"sitemap":108599,"stem":108600,"tags":194,"__hash__":108601},"blog\u002Fblog\u002F動作學習（十三）臨床應用.md","動作學習（十五）臨床應用",{"type":9,"value":108231,"toc":108592},[108232,108235,108250,108283,108302,108305,108314,108344,108373,108413,108434,108439,108442,108474,108486,108489,108511,108516,108519,108550,108571,108589],[12,108233,108234],{},"加油 剩80頁PPT不到了",[12,108236,108237,108238,108240,108241,108243,108244,108246,108247,108249],{},"接下來講臨床應用部份",[16,108239],{},"\n分成三個大方向 ",[16,108242],{},"\n肌肉再教育",[16,108245],{},"\n神經誘發",[16,108248],{},"\n任務取向模式",[12,108251,108252,108253,108257,108258,108260,108261,108263,108264,108266,108267,108269,108270,108272,108273,108277,108279,108280,108282],{},"先來講肌肉再教育 背景是因應",[75,108254,108256],{"className":108255},[5387],"小兒麻痺大流行","而受推廣",[16,108259],{},"\n治療重點包括",[16,108262],{},"\n評估:徒手肌力測試",[16,108265],{},"\n處置:對較弱的肌肉做訓練 維持ROM完整 避免固定不動",[16,108268],{},"\n強調病人主動參與 預防併發症",[16,108271],{},"\n必要時",[75,108274,108276],{"className":108275},[5387],"可用輔具協助",[16,108278],{},"\n限制:因為日常生活並非由單一肌肉達成 而是靠協同 只專注在單一肌肉上成效不佳",[16,108281],{},"\nUMN lesion患者無法做出單關節動作",[12,108284,108285,108286,108260,108288,108290,108291,108293,108294,7963,108299,108301],{},"接著來講神經誘發 針對剛提到的UMNL病人發展而來",[16,108287],{},[16,108289],{},"\n評估:反射動作",[16,108292],{},"\n處置:以感覺回饋修改反射動作 亦鼓勵高階控制動作練習 重複練習正確動作(無意外應該是",[2330,108295,108298],{"href":108296,"rel":108297},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E5%91%A8%E5%9C%8D%E7%90%86%E8%AB%96%E5%AD%B8%E6%B4%BE%EF%BC%88%E4%BA%8C%EF%BC%89%E9%9A%8E%E5%B1%A4%E7%90%86%E8%AB%96",[2334],"階層理論",[16,108300],{},"\n強調抑制異常動作與痙攣 誘發正常動作 並透過反覆練習使動作自動化與習慣化",[12,108303,108304],{},"神經誘發還分成六個學派...........",[12,108306,108307,108308,108310,108311,108313],{},"首先來講第一個......感覺統合(SI)",[16,108309],{},"\n它認為 是感知或知覺受影響 而不是動作上有問題",[16,108312],{},"\n所以會結合多感觀刺激輸入以訓練 強調感覺回饋及重複練習",[12,108315,108316,108317,108260,108319,108321,108322,108324,108325,108327,108328,108330,108331,108333,108334,108336,108337,108341,108343],{},"接著講.......Brunnstorm的理論 由治療偏癱發展而來",[16,108318],{},[16,108320],{},"\n初期可利用反射誘發動作發生",[16,108323],{},"\n依患者恢復程度加以訓練",[16,108326],{},"\n逐一打破協同動作引導自主性動作(Brunnstorm分期表 可見新民老師講義裡的表)",[16,108329],{},"\n促進手部功能",[16,108332],{},"\n行走前需要準備運動的練習",[16,108335],{},"\n治療目的:以感覺刺激輸入訊息給患者在訓練過程中正向加強 提升",[75,108338,108340],{"className":108339},[5387],"主動控制協同動作的能力",[16,108342],{},"\n強調站及步態訓練",[12,108345,108346,108347,108349,108350,108352,108353,108358,108359,108361,108362,108364,108365,108369,108370,108372],{},"接著來講Bobath的NDT...... 他是階層理論來解釋嬰兒CP的不正常姿勢反射(較適合在嬰兒評估治療)",[16,108348],{},"\n治療重點:使用擺位產生正常 從而抑制不正常的動作 也靠抑制不正常動作抑制異常張力",[16,108351],{},"\n而且按照",[75,108354,108357],{"className":108355},[108356],"bg-orange-3","正常發展程序","(近端到遠近的控制順序)",[16,108360],{},"\n與關鍵點控制(近端keypoint控制張力 遠端keypoint控制精細動作)",[16,108363],{},"\n而Bobath認為不該以使用",[75,108366,108368],{"className":108367},[5387],"反射及聯合動作","(會加重痙攣)訓練及不做肌肉訓練(讓肢體變硬)",[16,108371],{},"\n在動作控制時不應過度用力(他認為會引起不正常反射)",[12,108374,108375,108376,108378,108379,108381,108382,108386,108387,108389,108390,108392,108393,108397,108398,108400,108401,108405,108406,108408,108409],{},"而在最新的Bobath理論又更新出五點面向",[16,108377],{},"\n第一點 多重任務導向 不只是身體本身 還探討了任務本身及環境因素",[16,108380],{},"\n第二點 功能導向 以改善",[75,108383,108385],{"className":108384},[6760],"功能性動作","為主(以達成任務為主) 並更貼合ADL ",[16,108388],{},"\n第三點 動作學習 意思是除了在治療過程中學到的 ",[16,108391],{},"\n_在",[75,108394,108396],{"className":108395},[6760],"治療外的時間點","會不會讓學習更有益(即日常主動解決事情) 望求達到最佳效",[16,108399],{},"\n第四點 動作分析 這邊就是",[75,108402,108404],{"className":108403},[6760],"考我們PT的功力","了 不只要看出動作上有啥問題 更要看仔細一點 例如肌肉力量",[16,108407],{},"\n第五點 附加治療 需要時可與各種療法並用 例如副木 跑步機 ",[75,108410,108412],{"className":108411},[6760],"肌力訓練",[12,108414,108415,108416,108418,108419,108423,108425,108426,108430,108431,108433],{},"還有三個......理論 接著講Johnstone的 他更偏向治療CVA",[16,108417],{},"\n他也是用階層理論 亦支持剛剛講Bobath時提到的 ",[75,108420,108422],{"className":108421},[108356],"正常發展程式",[16,108424],{},"\n他提出可以用",[75,108427,108429],{"className":108428},[5387],"填充空氣的塑料副木","控制病人的肢體姿勢 以達到正常擺位和避免肩關節疼痛等",[16,108432],{},"\n(可以參考一下Gemini)",[12,108435,108436],{},[1930,108437],{"alt":1932,"src":108438},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774260332783-Screenshot_2026-03-23_180500.png",[12,108440,108441],{},"還有 兩個 理論(咳血)",[12,108443,108444,108445,108447,108448,4995,108451,108453,108454,108456,108457,108461,108462,108464,108465,108467,108468,108470,108471,108473],{},"接著講PNF 大二下基物二都跑過台的了 對角線動作跟訓練 ",[16,108446],{},"\n他也是強調",[75,108449,108422],{"className":108450},[108356],[16,108452],{},"\n(但不要因為PNF D1 D2先從遠端開始動就覺得他是反過來的 ",[16,108455],{},"\n只是因為",[75,108458,108460],{"className":108459},[5387],"旋轉能給予本體感覺"," 可帶動整段肢體)",[16,108463],{},"\n而他治療重點包括",[16,108466],{},"\n本體感覺刺激 只需要給一些小阻力或者牽拉 誘發動作",[16,108469],{},"\n神經肌肉控制正常化 引導肌肉主動或被動收縮或放鬆 讓神經和肌肉控制機制變正常",[16,108472],{},"\n功能改善:由上述過程 增加肌力 增加ROM",[12,108475,108476,108477,108479,108480,108482,108483,108485],{},"他提到治療師應靈活應用多重感覺刺激輸入引導或強化動作",[16,108478],{},"\n觸覺:治療師的接觸與引導(例 教導病人PNF的二指握法)",[16,108481],{},"\n聽覺:聲音指令和音量控制",[16,108484],{},"\n本體覺:誘發牽張反射",[12,108487,108488],{},"他還提到了善用擴散現象 意思是對健側邊施最大阻力 可誘發其他的肌肉收縮與動作產生",[12,108490,108491,108492,108494,108495,108499,108501,108502,108504,108505,108507,108508,108510],{},"結束 理論 啦......我們最後來講ROOD",[16,108493],{},"\n他一樣強調感覺輸入 而且是 ",[75,108496,108498],{"className":108497},[5387],"外在的感覺輸入直接導入動作輸出",[16,108500],{},"\n且動作本身也是感覺 會形成一個感知迴路",[16,108503],{},"\n治療重點十分重視感覺輸入(輕撫 刷擦 冷刺激 按壓 牽張 輕叩)",[16,108506],{},"\n而正常的發展進行這一點在小兒余亮老師有提到就是",[16,108509],{},"\n先有動作 再穩定 接著在穩定的姿勢下轉移重心 最後是重心轉移下做動作",[12,108512,108513],{},[1930,108514],{"alt":1932,"src":108515},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774263529039-Screenshot_2026-03-23_185830.png",[12,108517,108518],{},"我們來講最後一個大方向(沒錯 是不是忘記了是三種教法)",[12,108520,108521,108522,108524,108525,108529,108531,108532,108534,108535,108537,108538,108540,108541,108543,108544,108546,108547,108549],{},"最後一個是任務導向模式 它認為動作是靠 個人 任務 環境三者互動起來自然呈現的結果",[16,108523],{},"\n動作由目標而發 且感覺系統在動作控上不僅是反射刺激的關係 而",[75,108526,108528],{"className":108527},[5387],"用於動作之預測和適應",[16,108530],{},"\n治療重點如下",[16,108533],{},"\n任務評估跟目標設定",[16,108536],{},"\n強調患者主動參與及自我修正",[16,108539],{},"\n特定任務訓練與高劑量練習",[16,108542],{},"\n善用練習的變異性",[16,108545],{},"\n多感覺擴增性回饋",[16,108548],{},"\n個別化設計",[12,108551,108552,108553,108555,108556,108558,108559,108561,108562,108564,108565,108567,108568,108570],{},"而他認為 任務導向是唯一 專一的",[16,108554],{},"\n即要學開車就在車上練開車 而不是踩腳踏車",[16,108557],{},"\n在步態學習上",[16,108560],{},"\n應該以最小化損傷 是身體限制 那就修好它 例如牽拉 肌肉訓練",[16,108563],{},"\n最大化步態策略 練出最省力 有效的做法(CVA病人比較常見到甩腿這麼費力的動作)",[16,108566],{},"\n和環境適應力 應對各種環境",[16,108569],{},"\n這三做結合",[12,108572,108573,108574,108576,108577,108579,108580,108582,108583,108585,108586,108588],{},"而有人提出動作再學習模式",[16,108575],{},"\n有四步",[16,108578],{},"\n第一步 評估分析任務",[16,108581],{},"\n第二步 練習缺乏的動作成份",[16,108584],{},"\n第三步 直接練習該任務",[16,108587],{},"\n第四步 訓練不同情境下的動作",[12,108590,108591],{},"剩不到 五十頁 加 油.....",{"title":180,"searchDepth":181,"depth":181,"links":108593},[],"第四份PPT","\u002Fimages\u002Fuploads\u002F1774265882810-10575241i.jpg",{},"\u002Fblog\u002F動作學習（十三）臨床應用",{"title":108229,"description":108594},{"loc":108597},"blog\u002F動作學習（十三）臨床應用","xiYDXET39VdyF8nIp3fQZmsXGShto4FaqLo5NHDoclw",{"id":108603,"title":108604,"author":7,"body":108605,"date":107917,"description":108788,"draft":185,"edited_at":107739,"extension":186,"featured_image":108789,"meta":108790,"navigation":189,"path":108791,"pinned":185,"seo":108792,"sitemap":108793,"stem":108794,"tags":194,"__hash__":108795},"blog\u002Fblog\u002F動作學習（十二）神經可塑性.md","動作學習（十三）神經可塑性",{"type":9,"value":108606,"toc":108786},[108607,108610,108636,108658,108661,108664,108713,108725,108728,108755,108761,108764,108770,108775,108781],[12,108608,108609],{},"我們現在來講在學習時神經會有啥變化",[12,108611,108612,108613,108615,108616,108620,108621,108623,108624,108626,108627,108629,108630,108632,108633,108635],{},"先來描述一下神經系統 它是可以被修改的 ",[16,108614],{},"\n而神經系統的改變 ",[75,108617,108619],{"className":108618},[5387],"約等於","行為\u002F心理上的改變",[16,108622],{},"\n假如我們看到有一個人的行為發生改變 那他大概有以一下幾點物理變化",[16,108625],{},"\n學習\u002F記憶",[16,108628],{},"\n發育",[16,108631],{},"\n成癮",[16,108634],{},"\n恢復",[12,108637,108638,108639,108641,108642,108646,108647,108649,108650,108654,108655,108657],{},"而假如將神經從生物體觀察上 有兩種分期",[16,108640],{},"\n一種是短期功能性上的塑性 簡單來說就是",[75,108643,108645],{"className":108644},[5387],"神經元傳導","變得更有效更高效了",[16,108648],{},"\n而另一種則是結構上的塑性 意思是從",[75,108651,108653],{"className":108652},[6760],"數量或結構","上發生了改變",[16,108656],{},"\n而有一個重點 動作的學習是以上兩種情況加在一起 是自然呈現的結果",[12,108659,108660],{},"而在神經塑性有分了三個時期",[12,108662,108663],{},"首先是發展時期的神經塑性",[12,108665,108666,108667,108669,108670,108672,108673,108677,108678,6756,108680,108684,108685,108687,108688,108690,108691,108695,108696,108698,108699,108701,108702,108704,108705,108709,108710,108712],{},"先來講標的細胞 這個東西會決定神細元存活的數目",[16,108668],{},"\n有人做過研究 分別對雞胚胎移植移除標的細胞",[16,108671],{},"\n發現",[75,108674,108676],{"className":108675},[5387],"移除","標的細胞的胚胎那一個區域的神經元明顯下降",[16,108679],{},[75,108681,108683],{"className":108682},[6760],"移植","了標的細胞的胚胎神經元則明顯增加",[16,108686],{},"\n所以讓神經元存活的關鍵",[16,108689],{},"\n就是看",[75,108692,108694],{"className":108693},[5387],"突觸前神經元與標的細胞","有沒有互動",[16,108697],{},"\n假如沒有跟標的細胞互動的 就會流失掉",[16,108700],{},"\n而在發展期 一個標的細胞會同時連接多個神經元",[16,108703],{},"\n但在成熟後 會變成",[75,108706,108708],{"className":108707},[5387],"一個標的細胞連接一個神經元"," 其他會被淘汰",[16,108711],{},"\n(競爭是重要的)",[12,108714,108715,108716,108718,108719,108721,108722,108724],{},"接著來講受傷時期的神經塑性",[16,108717],{},"\n在受傷後 運動功能多少會被影響",[16,108720],{},"\n而神經重塑的程度 決定了受傷後復原的運動能力",[16,108723],{},"\n而重塑也受環境與經驗影響",[12,108726,108727],{},"那麼先來講PNS的神經受傷生理變化的過程",[12,108729,108730,108731,108733,108734,108738,108740,108741,108745,108746,108748,108749,108751,108752,108754],{},"那先來講受傷處的近端",[16,108732],{},"\n細胞本體會腫大 細胞核移到周邊位置並發生",[75,108735,108737],{"className":108736},[1149],"溶解現象",[16,108739],{},"\n然後講遠端 會發生一種叫 ",[75,108742,108744],{"className":108743},[5387],"瓦勒式變性現象"," 的改變",[16,108747],{},"\n意思是細胞核遠端的軸突跟髓鞘發生溶解 破碎 最後被吞噬",[16,108750],{},"\n最後 受傷的上下游也會有兩種變化",[16,108753],{},"\n受傷部位的上游會發生 而受傷部位的下遊則發生前溯變化",[12,108756,108757,108758,108760],{},"而受傷之後他們會開始復原 而且通常復原機會良好 稱為重生",[16,108759],{},"\n而通常會在軸突再次出芽生長",[12,108762,108763],{},"首先第一種叫再生性出芽生長 直在在軸突末端長出新的軸突",[12,108765,108766,108767,108769],{},"第二種叫代償性出芽生長",[16,108768],{},"\n意思是有標的細胞因神經受損空著沒人佔據 而附近沒有受傷的神經元軸突會生長並佔據",[12,108771,108772],{},[1930,108773],{"alt":1932,"src":108774},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774251258882-Screenshot_2026-03-23_153408.png",[12,108776,108777,108778,108780],{},"最後代償性出芽生長 簡單來說就是遠端軸突受傷 ",[16,108779],{},"\n那麼近端假如還能長那就在近端長出來(都是在同一個神經元上)",[12,108782,108783],{},[1930,108784],{"alt":1932,"src":108785},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774251534288-Screenshot_2026-03-23_153759.png",{"title":180,"searchDepth":181,"depth":181,"links":108787},[],"第五份PPT 介紹~CNS","\u002Fimages\u002Fuploads\u002F1774251705899-IMG_20200616_205324.jpg",{},"\u002Fblog\u002F動作學習（十二）神經可塑性",{"title":108604,"description":108788},{"loc":108791},"blog\u002F動作學習（十二）神經可塑性","7GhX2gBVVqtQXf9KDdSFTtWLZmH2XWkBFyh2jQg1XEw",{"id":108797,"title":108798,"author":7,"body":108799,"date":107917,"description":108898,"draft":185,"edited_at":107739,"extension":186,"featured_image":108899,"meta":108900,"navigation":189,"path":108901,"pinned":185,"seo":108902,"sitemap":108903,"stem":108904,"tags":194,"__hash__":108905},"blog\u002Fblog\u002F動作學習（十二）神經可塑性2.md","動作學習（十四）神經可塑性2",{"type":9,"value":108800,"toc":108896},[108801,108804,108832,108838,108844,108850,108856,108878,108884,108887],[12,108802,108803],{},"我們接著來講CNS上的重塑",[12,108805,108806,108807,108809,108810,108814,108816,108817,108819,108820,108822,108823,108825,108826,108828,108829,108831],{},"而CNS上的受損 通常無法像PNS那樣重生 ",[16,108808],{},"\n但是復原後 還是被受損時期功能性還要好 不是因為",[75,108811,108813],{"className":108812},[5387],"生長了啥 重生了啥",[16,108815],{},"\n而是因為以下幾點",[16,108818],{},"\n水腫消除",[16,108821],{},"\n殘存的細胞恢復活力",[16,108824],{},"\n休克過後恢復",[16,108827],{},"\n原本意義不明的區域被徵召",[16,108830],{},"\n使用某些路線平替",[12,108833,108834,108835,108837],{},"雖然CNS沒有重生 但還是會有出芽生長的情形 形成新的軸突聯結",[16,108836],{},"\n但目前臨床應用上還不夠明顯",[12,108839,108840,108841,108843],{},"而CNS受損後 神經會有三種恢復時間段",[16,108842],{},"\n神經休克-->去神經化過敏感性-->側枝生長",[12,108845,108846,108847,108849],{},"去神經化過敏感性 意思是受傷後 剩下的神經纖維會對神經傳導物質變得敏感",[16,108848],{},"\n激活相關通路 讓功能恢復",[12,108851,108852,108853,108855],{},"最後我們來講學習跟經驗引起的神經塑性",[16,108854],{},"\n會圍繞這四點來討論",[12,108857,108858,108859,108861,108862,108866,108867,108869,108870,108874,108875,108877],{},"首先是習慣化",[16,108860],{},"\n習慣化分成兩種 一種是",[75,108863,108865],{"className":108864},[5387],"功能性(短期習慣化)","(降低感覺) ",[16,108868],{},"\n一種是",[75,108871,108873],{"className":108872},[5387],"結構化(長期習慣化)","(直接讓通路消失)",[16,108876],{},"\n習慣化會使多數突觸喪失功能(突觸數量減少)",[12,108879,108880,108881,108883],{},"第二種 敏感化 就是習慣化反過來 ",[16,108882],{},"\n突觸傳導變多 會導致突觸數量變多",[12,108885,108886],{},"第三 重複刺激 意思是反複的刺激同一個區域 他在腦部的皮質區會增大",[12,108888,108889,108890,108892,108893,108895],{},"第四 減少使用 會讓腦部的皮質區變小",[16,108891],{},"\n以研究為例 把猴子的三四指縫起來成同一隻以減少使用",[16,108894],{},"\n過了一段時間觀察腦部皮質區變更模糊",{"title":180,"searchDepth":181,"depth":181,"links":108897},[],"CNS ~第五份PPT 完","\u002Fimages\u002Fuploads\u002F1774254235999-IMG_20200616_205906.jpg",{},"\u002Fblog\u002F動作學習（十二）神經可塑性2",{"title":108798,"description":108898},{"loc":108901},"blog\u002F動作學習（十二）神經可塑性2","h1KUAhBjhiWGXqtJ4mnlK_obIKYWo7J8zKuNzL10WN0",{"id":108907,"title":108908,"author":7,"body":108909,"date":107917,"description":109142,"draft":185,"edited_at":107739,"extension":186,"featured_image":109143,"meta":109144,"navigation":189,"path":109145,"pinned":185,"seo":109146,"sitemap":109147,"stem":109148,"tags":194,"__hash__":109149},"blog\u002Fblog\u002F動作學習（十五）影響表現的因素.md","動作學習（十七）影響表現的因素",{"type":9,"value":108910,"toc":109140},[108911,108914,108920,108926,108935,108944,108953,108968,108977,108995,108998,109004,109009,109018,109027,109033,109036,109039,109044,109050,109053,109071,109074,109092,109119,109137],[12,108912,108913],{},"同學 撐過這裡 就結束了 小考加油",[12,108915,108916,108917,108919],{},"那表現之前說過了 是來推估動作學到了多少",[16,108918],{},"\n但這個表現 會被暫時性的因素所影響 例如疲勞",[12,108921,108922,108923,108925],{},"那在動作學習上 就是相當永久的的改變",[16,108924],{},"\n那麼影響動作學習的因素就有 像練習 或者回饋",[12,108927,108928,108929,108931,108932,108934],{},"而這兩個分別代表暫時性跟永久性的被影響的差異",[16,108930],{},"\n那這裡有共通的被影響特點",[16,108933],{},"\n個人能力 注意力 警醒程度與焦慮 動機",[12,108936,108937,108938,108940,108941,108943],{},"個人能力呢 也可以叫說資質 也可以用悟性來講",[16,108939],{},"\n基本上屬於先天性的",[16,108942],{},"\n他不太會因為被學習跟經驗而改變",[12,108945,108946,108947,108949,108950,108952],{},"那在能力上也分成兩種 一種叫通用動作能力",[16,108948],{},"\n它認為 假如你平衡感好 那你協調跟反應也一定很快",[16,108951],{},"\n不過很多研究都證明說它還是不對的",[12,108954,108955,108956,108958,108959,108961,108962,108964,108965,108967],{},"而另一個 動作能力特效性假說",[16,108957],{},"\n他則因為 能力不只這麼大範疇就能被概括(The number of motor abilities was very large)",[16,108960],{},"\n像是手部的控制就能被拆分成精細度 協調能力 靈巧度 手指速度等等",[16,108963],{},"\n然後 每種動作能力都是相互獨立的",[16,108966],{},"\n最後 任務是由大量的能力組合在一起的",[12,108969,108970,108971,108973,108974,108976],{},"接著來講注意力 他有以下特點",[16,108972],{},"\n串聯 容量有限 費力的 ",[16,108975],{},"\n還有限制同時進行多種任務的能力 會干援訊息處理的反應選擇(怎麼動)跟反應程式期(下命令)",[12,108978,108979,108980,108982,108983,108985,108986,108988,108989,108991,108992,108994],{},"而下命令也分了五種類型",[16,108981],{},"\n第一種 集中注意力 比較偏向反應一類",[16,108984],{},"\n第二種 持續注意力 在警醒度上比較有關聯",[16,108987],{},"\n第三種 選擇注意力 就是能不能不被某些事干擾的能力",[16,108990],{},"\n第四種 交替注意力 就是將原本的注意力轉移到新的事情上 也比較屬於反應一類",[16,108993],{},"\n第五種 分隔注意力 就是同時處理幾件不同範疇的能力 類似邊開車邊聽電話",[12,108996,108997],{},"而注意力有四種理論模型",[12,108999,109000,109001,109003],{},"第一種 濾管理論 他的理論是人的注意力有限的 像東西穿過瓶口",[16,109002],{},"\n所以也須要選擇甚麼東西進去(選擇注意力)",[12,109005,109006],{},[1930,109007],{"alt":1932,"src":109008},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774282019738-Screenshot_2026-03-24_000646.png",[12,109010,109011,109012,109014,109015,109017],{},"第二種 注意力容量彈性論 他認為注意力不是固定容量",[16,109013],{},"\n而是會根據任務的難度 增加容量 以應付",[16,109016],{},"\n但假如當任務太難 那麼會導致過載 表現變差",[12,109019,109020,109021,109023,109024,109026],{},"第三種 多階段注意理論",[16,109022],{},"\n他認為假如是兩種不同範疇任務上 例如邊織毛衣邊說話",[16,109025],{},"\n抑或者在兩種不同階段的認為上 例如看著琴譜彈鋼琴",[12,109028,109029,109030,109032],{},"而行動選擇論 他認為 注意力的分配 是我個人想優先選擇做哪些任務先",[16,109031],{},"\n所以注意力分配看的是結果而不是原因",[12,109034,109035],{},"而在注意力上還有這些現象可以看到這些",[12,109037,109038],{},"像下圖 比較反直覺(非認明刺激)可直接理解 這個叫Stroop effect",[12,109040,109041],{},[1930,109042],{"alt":1932,"src":109043},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774282805876-Screenshot_2026-03-24_001953.png",[12,109045,109046,109047,109049],{},"第二個 雞尾酒現象 就像是你在跟別人專心聊天時 ",[16,109048],{},"\n即時背景有三到四個人聊天 你也聽不出來背景的人說甚麼(解釋了選擇注意力)",[12,109051,109052],{},"第三個 訊息越有意義 注意力就越集中(Von restorff effect)",[12,109054,109055,109056,109058,109059,109061,109062,109064,109065,109067,109068,109070],{},"接著講第三個特點 警醒程度跟焦慮",[16,109057],{},"\n警醒程度是偏向生理上的(或多或少會有指標參數) 簡單來說是我們有多活躍",[16,109060],{},"\n然後是焦慮 他是偏向心因性 那焦慮還分成兩種",[16,109063],{},"\n一種是天生的(特質焦慮)",[16,109066],{},"\n一種是因為某些事導致的(情境焦慮)",[16,109069],{},"\n而研究中表示 覺醒程度過高也不好 有可能會是極端焦慮 過低也不好(嗜睡)",[12,109072,109073],{},"最後是動機 就是讓他做的任務的驅動力",[12,109075,109076,109077,109079,109080,109082,109083,109085,109086,109088,109089,109091],{},"動機有以下理論",[16,109078],{},"\n第一種 因為需要 所以想要",[16,109081],{},"\n第二種 想活下去 那就去工作(被接納 認可) 那工作需要知識(學習) (Tolman三階層需要)",[16,109084],{},"\n第三種 為了爽跟避免痛苦是最大動機(享樂理論)",[16,109087],{},"\n第四種 人類有與環境互動的最基本需求(容量取決理論)",[16,109090],{},"\n第五點 人就是為了獲得成就感跟避免失誤(成就需求理論)",[12,109093,109094,109095,109097,109098,109100,109101,109103,109104,109106,109107,109109,109110,109112,109113,109115,109116,109118],{},"那增加動機有這些方法",[16,109096],{},"\n本身就有趣",[16,109099],{},"\n社會的期許(被壓力)",[16,109102],{},"\n鼓勵性質給分",[16,109105],{},"\n給錢獎勵",[16,109108],{},"\n表現的重要性",[16,109111],{},"\n相互競爭的社會動機",[16,109114],{},"\n進步本身有獎賞作用",[16,109117],{},"\n退步會有罰",[12,109120,109121,109122,109124,109125,109127,109128,109130,109131,109133,109134,109136],{},"而臨床上增加動機的方法有",[16,109123],{},"\n注意病人的警醒度適中",[16,109126],{},"\n配合疲勞調整",[16,109129],{},"\n提起 可能失敗 引起動機",[16,109132],{},"\n給口頭或實際的獎懲",[16,109135],{},"\n告知任務有何實際幫助",[12,109138,109139],{},"晚安各位孩子 該睡了",{"title":180,"searchDepth":181,"depth":181,"links":109141},[],"恭喜同學結束了 小考加油","\u002Fimages\u002Fuploads\u002F1774274522604-10576526i.jpg",{},"\u002Fblog\u002F動作學習（十五）影響表現的因素",{"title":108908,"description":109142},{"loc":109145},"blog\u002F動作學習（十五）影響表現的因素","StGxZlrKFnsOb0qeM00Ll_WWS6C7WzBBjn5gp220gbE",{"id":109151,"title":109152,"author":7,"body":109153,"date":107917,"description":109398,"draft":185,"edited_at":107739,"extension":186,"featured_image":109399,"meta":109400,"navigation":189,"path":109401,"pinned":185,"seo":109402,"sitemap":109403,"stem":109404,"tags":194,"__hash__":109405},"blog\u002Fblog\u002F動作學習（十四）記憶.md","動作學習（十六）記憶",{"type":9,"value":109154,"toc":109394},[109155,109158,109176,109188,109193,109214,109219,109241,109250,109281,109289,109294,109304,109319,109325,109343,109355,109364,109388],[12,109156,109157],{},"加油加油一起加油",[12,109159,109160,109161,109163,109164,109166,109167,109169,109170,109172,109173,109175],{},"這下我們來探討一下記憶",[16,109162],{},"\n先來講記憶的種類",[16,109165],{},"\n分成了斜述性(外顯性)記憶 即可以說出來 ",[16,109168],{},"\n又可以分成了情節記憶(自傳式) 跟語意知識(一般性知識)",[16,109171],{},"\n還有程式性(內隱性)記憶 即技巧",[16,109174],{},"\n在腦子裡知道怎麼去做的記憶",[12,109177,109178,109179,109181,109182,109184,109185,109187],{},"那記憶以時間段又分成了三種",[16,109180],{},"\n立刻記憶 又稱感觀記憶 以秒為單位",[16,109183],{},"\n短期記憶 又稱工作記憶 以分鐘到幾天為單位 ",[16,109186],{},"\n長期記憶 以日~年為單位 經定型的過程 在神經網絡上會有結構的改變",[12,109189,109190],{},[1930,109191],{"alt":1932,"src":109192},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774268317554-Screenshot_2026-03-23_201822.png",[12,109194,109195,109196,109198,109199,109201,109202,109204,109205,109207,109208,109210,109211,109213],{},"而記憶會經歷這三種歷程",[16,109197],{},"\n首先是登錄 立刻記憶被轉為短期記憶",[16,109200],{},"\n然後是留存 是發生在記短期記憶的過程",[16,109203],{},"\n一般會進行歸檔 覆誦 譯碼 和決定等處理 保存下來",[16,109206],{},"\n最後是提取 是指經過登錄 譯碼過後的訊息 會被與長期記憶做比較的過程",[16,109209],{},"\n包括回憶 起來某些事(主動)",[16,109212],{},"\n和再認 當時情重現時 會被想起來(被動)",[12,109215,109216],{},[1930,109217],{"alt":1932,"src":109218},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774268686265-Screenshot_2026-03-23_202432.png",[12,109220,109221,109222,109224,109225,109229,109230,109232,109233,109237,109238,109240],{},"接下來講記憶元素理論",[16,109223],{},"\n它提到 大腦會依據",[75,109226,109228],{"className":109227},[5387],"重要性","來選擇 像是考試重點 你就記得很清楚",[16,109231],{},"\n記憶不是全有全無 當某個意識積累到到了",[75,109234,109236],{"className":109235},[5387],"最低門檻","就會被想起來",[16,109239],{},"\n而競爭機制就是說 意識A比意識B要深刻 那在想起來的速度肯定是A先出來",[12,109242,109243,109244,109246,109247,109249],{},"那失憶的部份可以用時間來區分兩種類別",[16,109245],{},"\n一種是記不住新的東西 叫前向性失憶",[16,109248],{},"\n一種是記不起來以前的東西 叫回溯性記憶",[12,109251,109252,109253,109255,109256,109258,109259,109261,109262,109264,109265,109267,109268,109270,109271,109275,109276,109280],{},"而遺忘有分成這幾個理論",[16,109254],{},"\n第一種是記憶轉變論 因為長期沒用 導致變形 再忘掉(哎喲真的太久了我都給忘記具體東西了)",[16,109257],{},"\n第二種是干擾理論 還分成了兩種時間性",[16,109260],{},"\n一種是舊的東西干擾到新的東西 叫前向干擾",[16,109263],{},"\n另一種是新的東西干擾到舊的東西 叫回溯干擾",[16,109266],{},"\n最後一種是永久記憶及附屬理論",[16,109269],{},"\n他提出記憶形成就會是",[75,109272,109274],{"className":109273},[1149],"永久"," 只是變得",[75,109277,109279],{"className":109278},[1149],"次要","才不容易被想起來",[7166,109282,109284,109285],{"id":109283},"注意-右上角的是要改成阿茲海默症","注意 右上角的是要改成",[75,109286,109288],{"className":109287},[5387],"阿茲海默症",[12,109290,109291],{},[1930,109292],{"alt":1932,"src":109293},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774269914781-Screenshot_2026-03-23_204430.png",[12,109295,109296,109297,109299,109300],{},"在記憶網絡裡頭",[16,109298],{},"\n記憶不只存在於單一區域 而是",[75,109301,109303],{"className":109302},[5387],"遍佈整個大腦",[12,109305,109306,109307,109309,109310,109312,109313,109315,109316,109318],{},"接著來講兩種記憶類型會在哪裡建立",[16,109308],{},"\n首先來講敍述性記憶 那新的敍述性記憶會產生至海馬迴",[16,109311],{},"\n而像是辨識人臉 事件 空間等敍述性記憶 高度依賴內側顳葉(海馬迴與杏仁體)",[16,109314],{},"\n接著講程序性記憶 因為他偏向技巧 所以是發生在大腦皮層跟基底核",[16,109317],{},"\n而傳統制約反應就與小腦有關",[12,109320,109321,109322,109324],{},"記憶從短期記憶變成長期記憶也需要用到海馬回",[16,109323],{},"\n而這個過程需要時間 被稱作定型或記憶固化",[12,109326,109327,109328,109330,109331,109333,109334,109336,109337,109339,109340,109342],{},"而促進學習有這幾種",[16,109329],{},"\n第一是適合在腦傷病人上的",[16,109332],{},"\n主要看他左腦還是右腦受傷",[16,109335],{},"\n左腦 是邏輯腦 但是受傷了 所以要用右腦來訓練 ",[16,109338],{},"\n而右腦是圖像腦 抽象腦 可以多給他視覺輸入 或者心像練習",[16,109341],{},"\n反過來是右腦受傷了的話 那就用節拍 口訣一類的去練 用語言邏輯補回來",[12,109344,109345,109346,109348,109349,109351,109352,109354],{},"第二 鼓勵過度學習 即使學會了也要再額外練",[16,109347],{},"\n第三 事件分類記憶",[16,109350],{},"\n第四 利用近期-初始效應 因為干擾少 便方便記",[16,109353],{},"\n第五 回饋給予原則",[12,109356,109357,109358,109360,109361,109363],{},"展開一下來講近期-初始效應",[16,109359],{},"\n初始效應可以利用在剛開始學 干擾不多的情況下 這樣更容易形成記憶",[16,109362],{},"\n而近期效應 就是說剛做完這件事 我過了半會還記得",[12,109365,109366,109367,109369,109370,109372,109373,109375,109376,109378,109379,109381,109382,109384,109385,109387],{},"目前有七種學習方式來增強記憶形成",[16,109368],{},"\n無錯練習",[16,109371],{},"\n比喻練習",[16,109374],{},"\n雙項式練習",[16,109377],{},"\n發現練習",[16,109380],{},"\n觀察練習",[16,109383],{},"\n試行錯誤練習",[16,109386],{},"\n心像練習",[12,109389,109390,109391,109393],{},"那外顯性可以促記憶的 可以試講把東西講出來 自己把東西講細節 順序都講出來",[16,109392],{},"\n那內隱式可以用 無錯練習的方式去練 因為這東西基本上是技巧 那用無錯練習可以把它練更優",{"title":180,"searchDepth":181,"depth":181,"links":109395},[109396],{"id":109283,"depth":7430,"text":109397},"注意 右上角的是要改成阿茲海默症","還有不到五十頁 同學加油","\u002Fimages\u002Fuploads\u002F1774272957036-10477521i.jpg",{},"\u002Fblog\u002F動作學習（十四）記憶",{"title":109152,"description":109398},{"loc":109401},"blog\u002F動作學習（十四）記憶","AklZNcjNRuR93rHbK_aFXJWbVAL_qDhjceP_g9LxxB0",{"id":109407,"title":109408,"author":7,"body":109409,"date":109581,"description":109582,"draft":185,"edited_at":109581,"extension":186,"featured_image":109583,"meta":109584,"navigation":189,"path":109585,"pinned":185,"seo":109586,"sitemap":109587,"stem":109588,"tags":194,"__hash__":109589},"blog\u002Fblog\u002F動作學習（七）練習方式-1.md","動作學習（七）練習方式-1",{"type":9,"value":109410,"toc":109579},[109411,109414,109417,109444,109462,109477,109492,109495,109501,109513,109522,109531,109543,109552,109558],[12,109412,109413],{},"好好好 終於要輪到怎麼學習了",[12,109415,109416],{},"我先在這一篇講 集中分散 固定變異 還隨機段落",[12,109418,109419,109420,109422,109423,109425,109426,109428,109429,109431,109432,109434,109435,109437,109438,109440,109441,109443],{},"先來比較集中練習跟分散練習",[16,109421],{},"\n集中練習是練習時間＞休息時間 而分散練是休息時間＞練習時間",[16,109424],{},"\n集中練習比較適用在 ",[16,109427],{},"\n簡單的內容",[16,109430],{},"\n快速 離散的動作 意思是 動作開始跟結束有明顯段落 像投籃 打哥爾夫球",[16,109433],{},"\n那分散練習比較適用在",[16,109436],{},"\n危險的任務(例如需要高度集中的任務)",[16,109439],{},"\n複雜的內容(訊息量過多)",[16,109442],{},"\n長時間 連續性的動作(容易疲勞) 例如游泳 跑步等",[12,109445,109446,109447,109449,109450,109452,109453,109455,109456,109458,109459,109461],{},"而第二類 固定練習跟變異練習做比較",[16,109448],{},"\n固定練習就是指參數全部一致的情況下做重複的訓練",[16,109451],{},"\n而變異練習則是指在一種或以上的參數不同情況下做訓練",[16,109454],{},"\n拿籃球做例子 你當然可以一個一直站在三分線上投籃",[16,109457],{},"\n但實際出來的效果可能不如 我在跟別人實際比賽練出來的本領強",[16,109460],{},"\n而變異練習還有一個特點 就是可以訓練概化能力(即之前提到的舉一反三能力)",[12,109463,109464,109465,109467,109468,109470,109471,109473,109474,109476],{},"第三類來到了 段落練習跟隨機練習",[16,109466],{},"\n段落練習在練習上沒有變化 例如  AA BB CC 這樣練",[16,109469],{},"\n相對單調且簡單 適合新人 但效果沒有隨機練習好",[16,109472],{},"\n而隨機練習就是混著來 ACB BAC ABC CBA  這樣練",[16,109475],{},"\n組合複雜且隨機 相對進階 且時間會相較段落練習高 但在運用上會更優",[12,109478,109479,109480,109482,109483,109485,109486,109488,109489,109491],{},"在情境干擾(CI)上也能解釋段落練習跟隨機練習的差異",[16,109481],{},"\n那low CI代表基本上沒有變化 基本上可以重複的做同一件事(計算low)",[16,109484],{},"\n(只有一種任務重複實行)",[16,109487],{},"\n而higi CI代表著一直變 開始需要隨機應變了(計算up)",[16,109490],{},"\n(即多種任務需要交替進行)",[12,109493,109494],{},"而這裡有五種假說分別說出為甚麼隨機練習比段落練習好",[12,109496,109497,109498,109500],{},"第一種處理階層假說 他意思是段落思考不需要動腦想 只需要從短期記憶裡翻記錄就好",[16,109499],{},"\n而隨機練習因為所有預設都不一樣 所以不用直接沿用短期記憶 得自己動腦得出結果",[12,109502,109503,109504,109506,109507,109509,109510,109512],{},"第二種遺忘學說 他們認為學習就是一直在做運算才能變好",[16,109505],{},"\n而隨機練習的過程 每一次做事都不一樣 ",[16,109508],{},"\n做新的事就會忘記舊的事 且重新運算 如此類推 所以學得更好",[16,109511],{},"\n段落練習則相反 每次做得事都一樣 沒有重新運算 所以學不好",[12,109514,109515,109516,109518,109519,109521],{},"第三種基模假說 就是認為任務像公式",[16,109517],{},"\n段落練習只是用一種公式一直練 例如1+1不會解微積分",[16,109520],{},"\n而隨機練習則代表一直變換公式 這個公式裡面的參數會不斷被調整 這樣你會學得更多",[12,109523,109524,109525,109527,109528,109530],{},"第四種詳述假說 他說大腦一定要多分析才能有效學習",[16,109526],{},"\n而段落練習只能分析一件事 重複的",[16,109529],{},"\n而隨機練習是同時分析不同的單項任務 而不同的任務間還可以做歸納 比較 更有利學習",[12,109532,109533,109534,109536,109537,109539,109540,109542],{},"第五種回溯抑制假說 他的意思是後面學的會抑制(稀釋掉)前面學到的東西",[16,109535],{},"\n段落練習為例 AAABBBCCC 而到了C之後就開始抑制A的練習",[16,109538],{},"\n但隨機練習因為隨機不重複的運算 不太容易被相互抑制導致忘掉",[16,109541],{},"\n而他這個假說可以解釋為甚麼在保留測驗過程中段落練習出來的分數較差",[12,109544,109545,109546,109548,109549,109551],{},"但隨機學很慢很痛苦 段落太簡單又學不到甚麼 在介於兩者之間出現了序列訓練",[16,109547],{},"\n結合了可預測性跟不重複性 即 ABC ABC ABC ABC",[16,109550],{},"\n(有研究指出序列練習跟隨機練習在得分上幾乎相同)",[12,109553,109554,109555,109557],{},"他再指出 練習表現好 不代表 你真的學得好",[16,109556],{},"\n段落練習學得快 精準度高 但在隨後的保留測驗中其實不如隨機測驗",[12,109559,109560,109561,109563,109564,109566,109567,109569,109570,109572,109573,109575,109576,109578],{},"最後提到隨機練習的具以下特性",[16,109562],{},"\n第一 隨機練具備更強的適應性",[16,109565],{},"\n第二 困難的任務用隨機練效果更好",[16,109568],{},"\n簡單的任務用隨機練沒有明顯優勢",[16,109571],{},"\n第三 因應技巧程度定制",[16,109574],{},"\n假如是技巧不多的人 可以利用段落跟序列練習",[16,109577],{},"\n而當技巧十分高 則可以利用隨機練習",{"title":180,"searchDepth":181,"depth":181,"links":109580},[],"2026-03-22","集中分散 固定變異 隨機段落","\u002Fimages\u002Fuploads\u002F1774164465962-IMG_20200617_181520.jpg",{},"\u002Fblog\u002F動作學習（七）練習方式-1",{"title":109408,"description":109582},{"loc":109585},"blog\u002F動作學習（七）練習方式-1","FVh2e6eGwG6-W9rznbFr95AUKxPlqtfQmBgkp4vwoq4",{"id":109591,"title":109592,"author":7,"body":109593,"date":109581,"description":109739,"draft":185,"edited_at":109581,"extension":186,"featured_image":109740,"meta":109741,"navigation":189,"path":109742,"pinned":185,"seo":109743,"sitemap":109744,"stem":109745,"tags":194,"__hash__":109746},"blog\u002Fblog\u002F動作學習（三）Measurement-of-learning.md","動作學習（三）Measurement of learning",{"type":9,"value":109594,"toc":109737},[109595,109610,109620,109638,109641,109650,109656,109662,109675,109690,109708,109717],[12,109596,109597,109598,109600,109601,109603,109604,109606,109607,109609],{},"我們的學習成果 可以被兩種表現曲線所推估",[16,109599],{},"\n一種是成就曲線 (Achievement Curve)",[16,109602],{},"\n簡單來說就是得分計量 你學了一天的得分 會跟你學了十天的得分不一樣",[16,109605],{},"\n而另一種是錯誤曲線 (Error Curve)",[16,109608],{},"\n以剛剛的例子來看 你第一天學錯的多 但你學了十天錯得會比第一天少",[12,109611,109612,109613,109617,109619],{},"但這裡有個大誤區 ",[75,109614,109616],{"className":109615},[5387],"表現曲線 不代表 學習曲線",[16,109618],{},"\n因為學習沒有辦法被直接測量 所以只能靠對錯的程度來猜測 進步了多少",[12,109621,109622,109623,109625,109626,109628,109629,109631,109632,109634,109635,109637],{},"而在測量學習的過程中有兩個容易失真的情況",[16,109624],{},"\n一個是天花板效應 (Ceiling effects)",[16,109627],{},"\n意思就是 今天你的能力明顯比測驗內容還要高出好多 你也拿到了滿分 但不代表你進步了",[16,109630],{},"\n另一個是地板效應 (Floor effects)",[16,109633],{},"\n就是剛剛的反過來 測驗難度明顯比你的能力還要高 ",[16,109636],{},"\n那你會拿到很低分 也測不出進步 但不代表你實際沒有任何進步",[12,109639,109640],{},"雖然學習沒有辦法直接被測量 但還是有判斷的標準",[12,109642,109643,109644,109646,109647,109649],{},"第一是留存現象 像第一節課講到的相當永久的特性",[16,109645],{},"\n這代表著假如有學到點甚麼 就一定會稍微存起來 ",[16,109648],{},"\n可以隔一天再讓你做測驗來判斷有沒有學起來\u002F進步",[12,109651,109652,109653,109655],{},"第二是概化現象 相當於舉一反三",[16,109654],{},"\n簡單來說今天我學到了扔棒球 那你可能也會用差不多的姿勢來扔其他球類例如壘球",[12,109657,109658,109659,109661],{},"第三是改變了情境還能做出差不多表現動作的能力",[16,109660],{},"\n例如我在室內學會了走路 來到戶外人行道 我也能走",[12,109663,109664,109665,109667,109668,109672,109674],{},"還有一點可以用來測量的是 轉移(Transfer)",[16,109666],{},"\n意思是今天學到的一樣東西 或多或少會對另外一些技巧有 ",[75,109669,109671],{"className":109670},[5387],"好 或者 不好 的影響",[16,109673],{},"\n而兩樣東西越像 關聯越多 轉移就更多",[12,109676,109677,109678,109680,109681,109683,109684,109686,109687,109689],{},"先來介紹第一種轉移 正轉移跟負轉移",[16,109679],{},"\n正轉移意思 學到了A 在做B的時候多少有些幫助 ",[16,109682],{},"\n例如 會騎腳踏車的人 更容易學會騎機車",[16,109685],{},"\n而負轉移就反過來 例如我本身就會打網球 但學了打羽球後 我打網球的實力下降",[16,109688],{},"\n因為手感不一樣",[12,109691,109692,109693,109695,109696,109698,109699,109701,109702,109704,109705,109707],{},"而第二種轉移 近轉移跟遠轉移",[16,109694],{},"\n先來講近轉移 意思是訓練內容跟實際的情況相差不大",[16,109697],{},"\n在室內訓練他 他學會了 把他扔到平原上 他也能做出差不多表現",[16,109700],{},"\n而另一種遠轉移 則是說雖然練的跟實際情況看似很大 但本質上是雷同的",[16,109703],{},"\n例如 今天教他站起身的同時教他重心轉移的技巧",[16,109706],{},"\n那麼他今天向前彎伸手抓取東西時的重心轉移 也會有所提升",[12,109709,109710,109711,109713,109714,109716],{},"而第三種轉移 對測轉移(交叉轉移)",[16,109712],{},"\n意思是你右手學會了某個字 左手也能寫出同一個字",[16,109715],{},"\n就像之前講通用動作程式時提到的一樣 不同的器官能夠做出同樣的動作 差別在完美度",[12,109718,109719,109720,109722,109723,109727,109729,109730,109732,109733],{},"最後用來測量學習表現的指標 背景轉換阻⼒(resistance to contextual change)",[16,109721],{},"\n我把他理解成 換了環境 動作表現有沒有大的改變 用來測量",[75,109724,109726],{"className":109725},[5387],"下限值",[16,109728],{},"\n因為學到的東西不能只對單一環境有表現 ",[16,109731],{},"\n所以要靠變換不同的變數來實際驗證 靠的就是",[75,109734,109736],{"className":109735},[5387],"適應性跟轉移",{"title":180,"searchDepth":181,"depth":181,"links":109738},[],"在嘗試成為知識焚化爐的路上","\u002Fimages\u002Fuploads\u002F1774146343774-20201202_121856.jpg",{},"\u002Fblog\u002F動作學習（三）Measurement-of-learning",{"title":109592,"description":109739},{"loc":109742},"blog\u002F動作學習（三）Measurement-of-learning","QA5iZSNc7FcYL0a7LkSFITy2Jhp4SAOuvQV5P3r1JI0",{"id":109748,"title":109749,"author":7,"body":109750,"date":109581,"description":109870,"draft":185,"edited_at":107917,"extension":186,"featured_image":109871,"meta":109872,"navigation":189,"path":109873,"pinned":185,"seo":109874,"sitemap":109875,"stem":109876,"tags":194,"__hash__":109877},"blog\u002Fblog\u002F動作學習（九）回饋-1.md","動作學習（九）回饋-1",{"type":9,"value":109751,"toc":109868},[109752,109755,109758,109763,109766,109775,109784,109792,109801,109807,109825,109841,109850],[12,109753,109754],{},"讓我盡量有三篇內容解決你吧 哈哈哈哈哈哈",[12,109756,109757],{},"感覺的訊息如下圖",[12,109759,109760],{},[1930,109761],{"alt":1932,"src":109762},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774168154011-Screenshot_2026-03-22_162846.png",[12,109764,109765],{},"先來講動作前給的感覺訊息 包括了口語教導跟口語提示",[12,109767,109768,109769,109771,109772,109774],{},"口語教導(Verbal instruction)是動作出來前給予的",[16,109770],{},"\n教怎麼做這個動作(描述特性 順序 目標) ",[16,109773],{},"\n但不要描述具體至參數等 會使動作變得遲緩僵硬",[12,109776,109777,109778,109780,109781,109783],{},"口語提示 (Verbal Cue)是在動作過程中給予的",[16,109779],{},"\n只需要短又精準的語句引導即可(以讓整體動作變得更省力\u002F效率為主)",[16,109782],{},"\n假如像最常聽到的挺胸收腹",[12,109785,109786,109787,109791],{},"假如在",[75,109788,109790],{"className":109789},[5387],"探索式情境學習","中 不給指導會比起給予指導來得更要好",[12,109793,109794,109795,109797,109798,109800],{},"而在給口頭語句時也更偏向讓一些情境給他 而不是直接講怎麼動",[16,109796],{},"\n例如說 大腿抬起起來幾度 這樣病人可能會要想一下",[16,109799],{},"\n但假如直接講 用膝蓋往上頂 他會更加快的出來",[12,109802,109803,109804,109806],{},"而且人的注意力是有限的",[16,109805],{},"\n在講的時候 講一兩個簡單直接的指令 會比你直接講一大串文字教他怎麼動會更好",[12,109808,109809,109810,109812,109813,109815,109816,109818,109819,109821],{},"在學習剛開始 先看到榜樣的動作會比聽別人口述標準還要有效",[16,109811],{},"\n口述是從語言轉化成動作 會相較於看別人的視覺轉化成動作還要來得慢",[16,109814],{},"\n而且讓他先看一套榜樣動作 可以在腦內先有個大概",[16,109817],{},"\n這樣即使手還沒有開始動之前 腦內也可以先進行模擬",[16,109820],{},[75,109822,109824],{"className":109823},[5387],"可以不用做最完美的動作來做榜樣",[12,109826,109827,109828,109830,109831,109833,109834,109836,109837],{},"而有一位學者做了研究指出 新手之間互相觀察彼此動作會比新手看專家還要有效",[16,109829],{},"\n因為在AB都新手的情況下 B起拿對照表看著A做得如何",[16,109832],{},"\n這樣A會被B嚴格觀察並且記錄下來(B也會相對重視起A會錯的那些)",[16,109835],{},"\n這樣交互觀察 ",[75,109838,109840],{"className":109839},[5387],"能有效促進主動解決問題的決策過程",[12,109842,109843,109844,109846,109847,109849],{},"說完了視覺回饋 接下來講聽覺回饋",[16,109845],{},"\n聽覺回饋有一個特點 就是他很能抓時間 ",[16,109848],{},"\n假如那個動作對時間精準有要求 那就盡量用聽覺回饋",[12,109851,109852,109853,109855,109856,109858,109859,109861,109862,109864,109865,109867],{},"而聽覺回饋有兩種",[16,109854],{},"\n第一種是音樂動作回饋(MMF)",[16,109857],{},"\n意思是可以將音樂與動作組合在一起 ",[16,109860],{},"\n例如跳舞 聽著歌來跳會比不聽歌跳來得要有趣跟投入(情緒上)",[16,109863],{},"\n第二種是規律聽覺刺激(RAS)",[16,109866],{},"\n即聽著節拍讓動作組合在一起",{"title":180,"searchDepth":181,"depth":181,"links":109869},[],"口頭+視覺回饋","\u002Fimages\u002Fuploads\u002F1774175583479-IMG_20200617_181537.jpg",{},"\u002Fblog\u002F動作學習（九）回饋-1",{"title":109749,"description":109870},{"loc":109873},"blog\u002F動作學習（九）回饋-1","2fYhsaEeT7oT5MxiayunzYb6zq6ePVWrpt3jaASy8YA",{"id":109879,"title":109880,"author":7,"body":109881,"date":109581,"description":110032,"draft":185,"edited_at":107739,"extension":186,"featured_image":110033,"meta":110034,"navigation":189,"path":110035,"pinned":185,"seo":110036,"sitemap":110037,"stem":110038,"tags":194,"__hash__":110039},"blog\u002Fblog\u002F動作學習（二）form-of-learning.md","動作學習（二）forms of learning",{"type":9,"value":109882,"toc":110030},[109883,109889,109903,109912,109924,109930,109936,109941,109947,109962,109967,109973,109994,110004],[12,109884,109885,109886,109888],{},"現在來講動作學習的形式",[16,109887],{},"\n首先來介紹關聯性跟非關聯性",[12,109890,109891,109892,109894,109895,109897,109899,109900,109902],{},"非關聯性 意思是 只針對單一刺激輸入的部份 從而對那個刺激的接收強度發生變化",[16,109893],{},"\n因為他只針對單一刺激輸入 假如換成別的刺激輸入就要重新適應(無法舉一反三)",[16,109896],{},[16,109898],{},"\n可以讓刺激輸入分辨有沒有害",[16,109901],{},"\n使神經系統對這刺激做兩種演化 一種是習慣化 一種是敏感化",[12,109904,109905,109906,109908,109909,109911],{},"先來講習慣化 就是習慣了 讓那個刺激輸入對我而言變得平淡(被人打多了習慣痛了的意思)",[16,109907],{},"\n在臨床應用上 前庭病變的病人會因為頭部轉動而發生不適 ",[16,109910],{},"\n而我們可以多給他刺激到會產生不適的地方 讓他習慣 讓不適感降低(脫敏訓練)",[12,109913,109914,109915,109917,109918,109920,109921,109923],{},"而敏感化 就是更加有感覺 對我而言刺激輸入比以往來得更明顯(剛剛出完車禍 後面會更加注意駕駛)",[16,109916],{},"\n在臨床應用上 跌倒對PT來說是大忌 我們要訓練讓病人不容易跌倒",[16,109919],{},"\n在訓練的過程中加入一些模仿跌倒的環節 讓他的刺激輸入更加敏感 ",[16,109922],{},"\n以求日後他能知道跌倒的前兆 減少跌倒的風險",[12,109925,109926,109927,109929],{},"接著來到關聯性 意思是看到前兆就知道肯定後發生甚麼(因果關係)(舉一反三)",[16,109928],{},"\n這個也有兩種演化 一種是傳統制約反應 一種是操作制約反應",[12,109931,109932,109933,109935],{},"傳統制約反應 就是將刺激輸入跟結果緊緊捆綁(有前兆就有結果)(A是刺激輸入)",[16,109934],{},"\n最典型的例子 有A+B就有得吃 假如只有A 那他也以為有得吃",[12,109937,109938],{},[1930,109939],{"alt":1932,"src":109940},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774089362522-Screenshot_2026-03-21_183544.png",[12,109942,109943,109944,109946],{},"在臨床應用上 可以給病人先對刺激輸入與結果做連結",[16,109945],{},"\n例如 口頭提示(起來)+操作(拉他起來)=引導他站起來 假若沒有操作介入他也能站起來",[12,109948,109949,109950,109952,109953,109955,109956,109958,109959,109961],{},"那麼操作制約反應 就是做A才會有B B才是刺激輸入 ",[16,109951],{},"\n這裡有兩種刺激輸入可以做改變 一個是正向 一個是負面",[16,109954],{},"\n例子 開車沒扣安全帶 會有警示燈 ",[16,109957],{},"\n在臨床應用上 可以利用生物回饋(Biofeedback)",[16,109960],{},"\n這裡我直接偷懶引用gemini了",[12,109963,109964],{},[1930,109965],{"alt":1932,"src":109966},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774091802836-Screenshot_2026-03-21_191304.png",[12,109968,109969,109970,109972],{},"接下來講第二類學習的形式",[16,109971],{},"\n敍述性跟非敍述性",[12,109974,109975,109976,109978,109979,109981,109982,109984,109985,109987,109988,109990,109991,109993],{},"先來講敍述性 又叫做外顯性的認知",[16,109977],{},"\n就像是你玩教別人玩遊戲 變強的竅門 可以被說出來的就是敍述性",[16,109980],{},"\n而敍述性也代表著需要專心 要意識參與",[16,109983],{},"\n也可以被描述 ",[16,109986],{},"\n可以因被別人重複描述而轉化成自己的程序性記憶",[16,109989],{},"\n但有些腦傷的病人可能在認知方面有缺失 所以沒有辦法利用這點",[16,109992],{},"\n而心像練習也算是一種敍述性",[12,109995,109996,109997,109999,110000],{},"而非敍述性 又叫做內隱性的認知",[16,109998],{},"\n就像是你不太可能教別人玩遊戲 把一些肌肉記憶 而不用",[75,110001,110003],{"className":110002},[5387],"刻意回想",[12,110005,110006,110007,110012,110014,110015,110017,110018,110020,110021,110023,110024,110026,110027,110029],{},"而非敍述性有一個小分類叫做",[75,110008,110011],{"className":110009},[110010],"bg-green-4","程式學習",[16,110013],{},"\n意思就是一直練一直練練到會",[16,110016],{},"\n這點玩遊戲的同學就很有感受了 菜就多練 但這種沒辦法教給別人 ",[16,110019],{},"\n有這些特徵",[16,110022],{},"\n因重複學習而獲得行為表現的進步",[16,110025],{},"\n不需要經過思考就能做出來(像是所謂的肌肉記憶)",[16,110028],{},"\n在臨床上可以練像是轉移位的動作",{"title":180,"searchDepth":181,"depth":181,"links":110031},[],"絕讚啃書中","\u002Fimages\u002Fuploads\u002F1774142429929-20201202_121927.jpg",{},"\u002Fblog\u002F動作學習（二）form-of-learning",{"title":109880,"description":110032},{"loc":110035},"blog\u002F動作學習（二）form-of-learning","Cs501hS6WjyS3sBEmUDDirkmrv8zgEYSZklWmV6HStI",{"id":110041,"title":110042,"author":7,"body":110043,"date":109581,"description":110189,"draft":185,"edited_at":107917,"extension":186,"featured_image":110190,"meta":110191,"navigation":189,"path":110192,"pinned":185,"seo":110193,"sitemap":110194,"stem":110195,"tags":194,"__hash__":110196},"blog\u002Fblog\u002F動作學習（五）理論-2.md","動作學習（五）理論-2",{"type":9,"value":110044,"toc":110187},[110045,110048,110074,110092,110115,110120,110125,110127,110139,110157,110181],[12,110046,110047],{},"阿巴阿巴阿巴 我們繼續來講理論",[12,110049,110050,110051,110053,110054,110056,110057,110061,110062,110067,110068,110070,110071,110073],{},"現在我們來講基模理論",[16,110052],{},"\n他認為 學到的經驗 是概念 是規則 是關係 ",[16,110055],{},"\n而學習是具",[75,110058,110060],{"className":110059},[5387],"開放鏈","性質(",[2330,110063,110066],{"href":110064,"rel":110065},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E9%80%B2%E9%9A%8E%E6%8E%A7%E5%88%B6%E5%88%B6%E5%88%B6%E5%88%B6",[2334],"引見此篇內的開放式回路",")(會自己修正)",[16,110069],{},"\n動作程式理面的東西是抽象的 不是模版 也並不唯一",[16,110072],{},"\n而我們學一個新的程式 學的不是細節 學的的本質",[12,110075,110076,110077,110079,110080,110082,110083,110085,110086,110088,110089,110091],{},"而基模包含以下四點內容 ",[16,110078],{},"\n起始狀態",[16,110081],{},"\n預期的動作結果",[16,110084],{},"\n動作變數",[16,110087],{},"\n感覺結果",[16,110090],{},"\n這四點組合在一起變成抽象的概念或者記憶",[12,110093,110094,110095,110097,110098,110100,110101,110105,110107,110108,110110,110111],{},"而這四種內容又會組合成兩種基模",[16,110096],{},"\n召還基模 即知道做一個動作需要甚麼參數",[16,110099],{},"\n構成召還基模的有 起始狀態 預期的動作結果 ",[75,110102,110104],{"className":110103},[6760],"動作變數",[16,110106],{},"\n再認基模 意思是還沒做出來之前就猜到了會有啥結果",[16,110109],{},"\n構成召還基模的則有 起始狀態 預期的動作結果 ",[75,110112,110114],{"className":110113},[5387],"感覺結果",[12,110116,110117],{},[1930,110118],{"alt":1932,"src":110119},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774152546064-Screenshot_2026-03-22_120449.png",[12,110121,110122],{},[1930,110123],{"alt":1932,"src":110124},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774242658032-Screenshot_2026-03-23_131009.png",[16,110126],{},[12,110128,110129,110130,110132,110133,110135,110136,110138],{},"接下來講生態學習論",[16,110131],{},"\n提到 學習這個過程包含三點 練習 環境限制 為了完成任務而尋找最佳策略",[16,110134],{},"\n而在尋找最佳的過程中包含 感知線索(perceptual cues) 及 動作反應(motor response)",[16,110137],{},"\n當認知與動作結合的過程 是找出完成任務的最佳解",[12,110140,110141,110142,110144,110145,110147,110148,110150,110151,110153,110154,110156],{},"先來講感知線索他分成了三個要素",[16,110143],{},"\nPrescriptive role(處方的角色) 那我理解成理解題目並構思答案",[16,110146],{},"\n(understanding the goal of the task and the movements to be learned)",[16,110149],{},"\nFeedback role(回饋的角色) ",[16,110152],{},"\n在動作過程中(KP)和動作結果(KR) 都有回饋以修正",[16,110155],{},"\n組織的角色 他可以將認知跟動作組合在一起 而這個組合經過篩選 已是最佳解",[12,110158,110159,110160,110162,110163,110165,110166,110168,110169,7206,110173,7206,110177,7963],{},"他還認為 學習的這個過程 不是重複單一學習一種技巧",[16,110161],{},"\n是在特定工作目標下 將感知跟動作結合在一起不斷修正優化 ",[16,110164],{},"\n而不是基模理論的增強動作的規則",[16,110167],{},"\n所以這個理論會更加注重在我們動作控制的前三堂講(",[2330,110170,98736],{"href":110171,"rel":110172},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E6%8E%A7%E5%88%B6%E8%87%AA%E5%B7%B1-Be-water-my-friend",[2334],[2330,110174,98740],{"href":110175,"rel":110176},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E6%8E%A7%E5%88%B6%E8%87%AA%E5%B7%B1-Be-water-my-friend%EF%BC%88%E4%BA%8C%EF%BC%89",[2334],[2330,110178,98790],{"href":110179,"rel":110180},"https:\u002F\u002Fblog.chinono.dev\u002Fblog\u002F%E6%8E%A7%E5%88%B6%E8%87%AA%E5%B7%B1-Be-water-my-friend%EF%BC%88%E4%B8%89%EF%BC%89",[2334],[12,110182,110183,110184,110186],{},"看下來 生態學習論一樣強調感覺刺激的回饋",[16,110185],{},"\n但他應該是更偏向主動對刺激的應對 而不是像之前提到的因應刺激輸入做改變",{"title":180,"searchDepth":181,"depth":181,"links":110188},[],"基模理論和生態學習論","\u002Fimages\u002Fuploads\u002F1774151870253-IMG_20200616_205821.jpg",{},"\u002Fblog\u002F動作學習（五）理論-2",{"title":110042,"description":110189},{"loc":110192},"blog\u002F動作學習（五）理論-2","A9jlSp5KRM3tuCcHzIVZtjoHP10AhfSEWpsA51UlZ58",{"id":110198,"title":110199,"author":7,"body":110200,"date":109581,"description":110303,"draft":185,"edited_at":107739,"extension":186,"featured_image":110304,"meta":110305,"navigation":189,"path":110306,"pinned":185,"seo":110307,"sitemap":110308,"stem":110309,"tags":194,"__hash__":110310},"blog\u002Fblog\u002F動作學習（八）練習方式-2.md","動作學習（八）練習方式-2",{"type":9,"value":110201,"toc":110301},[110202,110220,110229,110232,110241,110253,110256,110265,110271,110280,110283,110286,110292,110298],[12,110203,110204,110205,110207,110208,110210,110211,110213,110214,110216,110217,110219],{},"到了心象練習 在內心想 結合自己平時的經驗\u002F記憶 在腦海裡優化 這樣下一次做就會更好",[16,110206],{},"\n不管是正常人 還是病人(中風 SCI 帕金森氏症等)都能這樣重複在腦海裡練",[16,110209],{},"\n而這只是輔助 是互補 而不是替代 ",[16,110212],{},"\n他不能實際讓你動作一下子只腦海模擬那樣做完美 ",[16,110215],{},"\n但例如用的力度 肢體角度做優化 讓整套動作來得有效率 ",[16,110218],{},"\n(這是真的有用 因為我玩快打旋風之前腦海裡平時沒事做都在想怎麼搓招接招 實際玩下來是有用的)",[12,110221,110222,110223,110225,110226,110228],{},"最後到了全部練習跟部份練習",[16,110224],{},"\n全部練習意思是從頭學到尾 沒有停不會變 ",[16,110227],{},"\n部份練習也叫分解練習 就是把一段東西拆開了練",[12,110230,110231],{},"他有兩種情境讓我們區分怎麼用全部或者部份練習",[12,110233,110234,110235,110237,110238,110240],{},"首先是任務的本質 ",[16,110236],{},"\n假如整套任務的動作都是合在一起才能做到那就不要拆開 例如投籃",[16,110239],{},"\n但像是整套任務的動作可以分開互不干預那就可以拆開看 例如穿衣服+扣扭扣",[12,110242,110243,110244,110246,110247,110249,110250,110252],{},"其次是時間跟動作類型",[16,110245],{},"\n再拿投籃為例 舉手再到投出去不到1秒時間可以做完那就不要拆開來練 用全部練習",[16,110248],{},"\n抑或者像走路那樣 只練一隻腳伸直腳屈曲腳是沒意義 只能用全部練習",[16,110251],{},"\n而像是跳舞 由多個動作組合的 那就適合部份練習",[12,110254,110255],{},"而分解練習也分成三種形式",[12,110257,110258,110259,110261,110262,110264],{},"第一種單獨分解練習",[16,110260],{},"\n他的意思是將複雜的動作拆分成兩部份或以上單獨訓練 ",[16,110263],{},"\n例如剛學泳游的我們會用浮板代替手部划水 而腳在後面蹬腳踩水這樣",[12,110266,110267,110268,110270],{},"第二種漸進分解練習 由簡單的部份遞進 會比先從困難部份開始學好",[16,110269],{},"\n例如籃球運球上籃 從運球 變成運球+拿起籃球走兩步 再變成運球+拿起籃球走兩步+跳起來出手",[12,110272,110273,110274,110276,110277,110279],{},"第三種則是簡化分解練習 例如說讓籃球在指尖上轉很難 那我就先利用書本來轉 學習找中心點",[16,110275],{},"\n也可以放慢速度來練 例如像剛剛運球上籃為例 假如走兩步沒辦法快速走 那就從慢走兩步練",[16,110278],{},"\n在室內練習也相當於是戶外任務的簡化",[12,110281,110282],{},"最後轉到了轉移效果 把動作分解成零件來看 怎麼拆才合理",[12,110284,110285],{},"他用互動性來區分",[12,110287,110288,110289,110291],{},"假設前者零件跟後者零件要一起運作才是連貫的動作 那就代表他這互動性強",[16,110290],{},"\n例子有棒球 你可以純空揮 但對於你擊球效果來說肯定沒這麼有效 要把揮球+擊球才有用",[12,110293,110294,110295,110297],{},"而互動性弱 就代表零件之間拆開來也不唐突",[16,110296],{},"\n例如跳舞 我轉體後再跳躍 很明顯可以拆開來 先練轉體 再練跳躍",[12,110299,110300],{},"好耶 終於可以下一章了 還有150頁 加油加油",{"title":180,"searchDepth":181,"depth":181,"links":110302},[],"心像練習 全部部份","\u002Fimages\u002Fuploads\u002F1774167633805-IMG_20200617_182035.jpg",{},"\u002Fblog\u002F動作學習（八）練習方式-2",{"title":110199,"description":110303},{"loc":110306},"blog\u002F動作學習（八）練習方式-2","JcPSdzMyU_HGqOk_w40r_o5lxnmINVMCI7xjO1fUggs",{"id":110312,"title":110313,"author":7,"body":110314,"date":109581,"description":110383,"draft":185,"edited_at":109581,"extension":186,"featured_image":110384,"meta":110385,"navigation":189,"path":110386,"pinned":185,"seo":110387,"sitemap":110388,"stem":110389,"tags":194,"__hash__":110390},"blog\u002Fblog\u002F動作學習（六）理論-3.md","動作學習（六）理論-3",{"type":9,"value":110315,"toc":110381},[110316,110319,110331,110334,110343,110349,110352,110357,110366,110378],[12,110317,110318],{},"阿巴阿巴 理論到頭了",[12,110320,110321,110322,110324,110325,110327,110328,110330],{},"現在來講 Bernstein 提出的動作學習三階段論",[16,110323],{},"\n他把動作學習的過程用身體自由度來解釋動作學習的進程",[16,110326],{},"\n他提出有三個過程",[16,110329],{},"\n凍結自由度 釋放與重組 善⽤機械特性",[12,110332,110333],{},"我們來滑雪做例子",[12,110335,110336,110337,110339,110340,110342],{},"先來講凍結自由度 意思是在學滑雪的開始 ",[16,110338],{},"\n我們會先學好站穩 再看能不能動起來 ",[16,110341],{},"\n這時候我們的關緊肯定是鎖最緊的",[12,110344,110345,110346,110348],{},"第二點 釋放與重組 這時候我們滑雪可以滑一段距離且不太會跌",[16,110347],{},"\n關節比起剛學要更輕鬆 開始有協調動作",[12,110350,110351],{},"第三點 善用機械特性 就是已經會了滑雪了 利用物理特性來滑得更好更快",[12,110353,110354],{},[1930,110355],{"alt":1932,"src":110356},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774158176692-Screenshot_2026-03-22_134239.png",[12,110358,110359,110360,110362,110363,110365],{},"最後我們來講 訊息理論之最佳挑戰觀點",[16,110361],{},"\n他認為 學習過程伴隨訊息消化 假如訊息複雜或者大量 會影響消化",[16,110364],{},"\n而這個消化的能力是跟 我們的技巧跟任務的難度 掛勾",[12,110367,110368,110369,110371,110372,110374,110375,110377],{},"但他也同時說了訊息的內容決定了學習時長 假設如下",[16,110370],{},"\n沒有訊息就沒有辦法產生學習效應",[16,110373],{},"\n太難或太簡單會影響學習速度",[16,110376],{},"\n任何動作技巧學習都有相對應的最佳訊息量",[12,110379,110380],{},"理論 結束啦 啊嗚嗚嗚",{"title":180,"searchDepth":181,"depth":181,"links":110382},[],"Bernstein的動作學習三階段 及 訊息理論之最佳挑戰觀點","\u002Fimages\u002Fuploads\u002F1774159201152-IMG_20200616_205416.jpg",{},"\u002Fblog\u002F動作學習（六）理論-3",{"title":110313,"description":110383},{"loc":110386},"blog\u002F動作學習（六）理論-3","FlJuDSMZYa_JJjwDyzshCzZ6fAInyKn6KVjJedrlFiU",{"id":110392,"title":110393,"author":7,"body":110394,"date":109581,"description":110549,"draft":185,"edited_at":109581,"extension":186,"featured_image":110550,"meta":110551,"navigation":189,"path":110552,"pinned":185,"seo":110553,"sitemap":110554,"stem":110555,"tags":194,"__hash__":110556},"blog\u002Fblog\u002F動作學習（四）理論.md","動作學習（四）理論",{"type":9,"value":110395,"toc":110547},[110396,110399,110447,110452,110457,110463,110478,110487,110498,110511,110528],[12,110397,110398],{},"好好好 終於要來講理論了",[12,110400,110401,110402,110404,110405,110407,110408,110410,110411,110413,110414,110416,110417,110419,110420,110424,110425,110429,110430,110432,110433,110437,110438,110440,110441,110446],{},"先來講 Three-stage model (動作分期說)",[16,110403],{},"\n這個理論動作的學習分成三個時期",[16,110406],{},"\n認知期 (此階段停留時間短) ",[16,110409],{},"\n連結期 (此階段停留時間中等)",[16,110412],{},"\n自動化期 (此階段因已學會所以停留最長)",[16,110415],{},"\n這邊直接上圖表並且講解",[16,110418],{},"\n我們可以看到 認知期 就是剛開始學習 這時候動作表現",[75,110421,110423],{"className":110422},[5387],"最差"," 而且需要",[75,110426,110428],{"className":110427},[5387],"高度注意力 無法分心"," 依賴回饋",[16,110431],{},"\n而到了連結期 算是對學到的動作理解了 開始",[75,110434,110436],{"className":110435},[6760],"精進的過程"," 錯誤比前者少 注意力也可以少一點 分心一點",[16,110439],{},"\n最後到了自動化期 這時候已經到達了",[75,110442,110445],{"className":110443},[110444],"bg-blue-3","專精"," 錯誤極少 也可以分心 例如聊天或者想其他事",[12,110448,110449],{},[1930,110450],{"alt":1932,"src":110451},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774146845133-Screenshot_2026-03-22_103304.png",[12,110453,110454],{},[1930,110455],{"alt":1932,"src":110456},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774147172273-Screenshot_2026-03-22_103918.png",[12,110458,110459,110460,110462],{},"接下來講另一個動作分期說 由Gentile提出 叫Gentile’s two-stage model",[16,110461],{},"\n把動作學習分成兩種時期",[12,110464,110465,110466,110468,110469,110471,110472,110474,110475,110477],{},"第一期是 調控跟非調控 就像是Motor control 裡頭講的須調整跟不須調整環境",[16,110467],{},"\n就像杯子的大小 會影響我手張多開 這個叫調控",[16,110470],{},"\n但就像標子的顏色就算五顏六色 也不會影響我手張多開 這就是非調控",[16,110473],{},"\n而第一期他提到有三個重要的特徵",[16,110476],{},"\n理解任務要求 發展基本的動作協調模式 區辨環境特徵",[12,110479,110480,110481,110483,110484,110486],{},"第二期 他覺得學到的東西會有兩種變體",[16,110482],{},"\n一種是專注化 意思是同一件事得分率提升",[16,110485],{},"\n一種是多樣化 在地鐵站上的逆著人流走 跟在廣場上逆著人流走 使用的動作大差不差",[12,110488,110489,110490,110492,110493,110497],{},"這篇最後教的理論是閉鎖式動作學習",[16,110491],{},"\n他主張 大腦必須要有一個",[75,110494,110496],{"className":110495},[5387],"正確的標準"," 這樣在學習的過程中才會按照這個標準作修正",[12,110499,110500,110501,110503,110504,110506,110507,7963],{},"他表示學的過程中有兩個重要的特徵",[16,110502],{},"\n第一是記憶痕跡 動作開始之前 選一個之前學到的正確的標準 做準備",[16,110505],{},"\n第二是感知痕跡 在做動作的時候 會比對那個正確的標準 隨時修正(即",[75,110508,110510],{"className":110509},[5387],"強調回饋",[12,110512,110513,110514,110516,110517,110519,110520,110522,110523,110527],{},"他的理論強調這三點",[16,110515],{},"\n第一 感知痕跡可以被訓練",[16,110518],{},"\n第二 感知痕跡會因為訓練次數會越來越敏感(練到第一百次 即使你閉著眼彈錯一個琴鍵 你也感覺得到",[16,110521],{},"\n第三 強調",[75,110524,110526],{"className":110525},[5387],"重複練習相同且正確","的動作",[12,110529,110530,110531,110533,110534,110538,110539,110541,110542,110546],{},"但他的理論也有限制",[16,110532],{},"\n首先是回饋部份 像",[75,110535,110537],{"className":110536},[6760],"CPG","裡頭就有提到 在沒有回饋的情況下 也能做得到動作 甚至能學到某些內容",[16,110540],{},"\n第二點 根據現在的研究指出 ",[75,110543,110545],{"className":110544},[6760],"變異多樣化","的學習 比起只學正確單一重複 還要更有效果",{"title":180,"searchDepth":181,"depth":181,"links":110548},[],"動作分期說到閉鏈式學習理論","\u002Fimages\u002Fuploads\u002F1774150308835-IMG_20200616_135236.jpg",{},"\u002Fblog\u002F動作學習（四）理論",{"title":110393,"description":110549},{"loc":110552},"blog\u002F動作學習（四）理論","WBvSuJvaXWInhFm416tA68l2z-SBv2679qo9o1C3wLQ",{"id":110558,"title":110559,"author":7,"body":110560,"date":110700,"description":110701,"draft":185,"edited_at":107457,"extension":186,"featured_image":110702,"meta":110703,"navigation":189,"path":110704,"pinned":185,"seo":110705,"sitemap":110706,"stem":110707,"tags":194,"__hash__":110708},"blog\u002Fblog\u002F中樞理論學派（二）Motor-Program.md","中樞理論學派（二）Motor Program",{"type":9,"value":110561,"toc":110698},[110562,110585,110588,110594,110606,110621,110633,110636,110639,110645,110648,110654,110663,110672,110677,110695],[12,110563,110564,110565,110567,110568,110570,110571,110575,110576,110578,110579,110581,110582,110584],{},"現在來到了Motor Program的部份 大家看到Program這個詞很容易聯想到 程式 代碼之類",[16,110566],{},"\n沒錯 他這個就是指說在動作產生前 腦子已經想好一套流程 它就是Motor Program",[16,110569],{},"\n而因為這一套流程是先想好的 一樣",[75,110572,110574],{"className":110573},[79],"不需要透過感覺回饋"," 就能產生動作",[16,110577],{},"\n他這個Program包括了 肌肉收縮順序 產生的力量等等 ",[16,110580],{},"\n而感覺回饋僅用來糾錯調整 不是用來刺激動作產生的",[16,110583],{},"\n而Program一但被執行 就必須被完成",[12,110586,110587],{},"以下將用動作程式代指Motor Program",[12,110589,110590,110591,110593],{},"那通用動作有三個理論 ",[16,110592],{},"\n1. 開鏈式系統 2. 通用動作程式 3. 階層特性",[12,110595,110596,110597,110599,110600,110602,110603,110605],{},"先來講開鏈式系統 像剛描述到的 一被執行則必須被完成 ",[16,110598],{},"\n而他的特性就是 在動作結束前 都沒有回饋 達成後才有回饋",[16,110601],{},"\n對 這裡的特性就跟 \u003C\u003C第六課>> 文中的開放式回路相似",[16,110604],{},"\n而這個系統在穩定且可預測的環境中最有效",[12,110607,110608,110609,110611,110612,110614,110615,110617,110618,110620],{},"接下來到通用動作程式 顧名思義 就是他把動作拆分成各個參數 ",[16,110610],{},"\n而有些是不可變的參數 裡面有1. 順序 2. 相對間隔\u002F相對時間(比例) 3. 相對力量(比例)",[16,110613],{},"\n這三個不可變的參數構成通用動作程式",[16,110616],{},"\n假若有兩種動作的這三個不可變參數皆相同 那他們所使用的通用動作程式就是同一個",[16,110619],{},"\n即使有任一不可變參數不同 那兩者的通用動作程式也不一樣",[12,110622,110623,110624,110626,110627,110629,110630,110632],{},"舉一個例子 土跟士這兩個字 別看這兩字僅上下橫線長度不一樣",[16,110625],{},"\n但土字在寫字裡上橫比下橫用力 士則相反 下橫比上橫用力 ",[16,110628],{},"\n即使順序(筆畫)跟相對間隔一樣 但兩者相對力量也不一樣",[16,110631],{},"\n所以是兩種通用動作程式",[12,110634,110635],{},"來突擊問一下 跑步跟走路兩種通用動作程式一樣嗎",[12,110637,110638],{},"而在通用動作程式裡可以改的參數",[12,110640,110641,110642,110644],{},"第一點 肌肉(部位) 意思就是 右撇子的你用左手寫同一個字 雖然字是比較醜 ",[16,110643],{},"\n但同一個字的不變參數相同 不影響同一個通用動作程式在不同部位的應用",[12,110646,110647],{},"第二點 總力量 今天你拿毛筆寫字 你寫的總力度變大 但同一個字的相對力量比例不變 亦適用通用動作程式",[12,110649,110650,110651,110653],{},"第三點 總時間間隔 今天寫同一個字 把他寫大一點佔多點位置 但時長用更長",[16,110652],{},"\n而寫小一個字 佔的位置小 整體時間比例也會縮短",[12,110655,110656,110657,110659,110660,110662],{},"而最後講到階層特性 他跟前面周圍理論學派的階層理論不同",[16,110658],{},"\n這邊的階層是指說 今天我要做一個任務例如拿水杯 就會讓小的動作程式組合成拿水杯的任務",[16,110661],{},"\n動作程式如下",[12,110664,110665,110666,110668,110669,110671],{},"1)身體前彎的動作程式 2) 手臂往前伸的動作程式 3) 手指打開的動作程式 ",[16,110667],{},"\n4) 抓握水杯的動作程式 5) 手臂回來的動作程式 ",[16,110670],{},"\n這樣把一套動作所需的動作程式串聯在一起 就是階層特性",[12,110673,110674],{},[1930,110675],{"alt":1932,"src":110676},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774062985971-Screenshot_2026-03-21_111610.png",[12,110678,110679,110680,110682,110683,110685,110686,110688,110689,110691,110692,110694],{},"臨床上 會注意不要打斷病人的動作 尤其是快速的動作",[16,110681],{},"\n且鼓勵病人自動的做出來(利用開鏈式)",[16,110684],{},"\n但當預到環境會變化的情境 則鼓勵利用閉鏈式(即\u003C\u003C第六課>>的閉鎖式回路 注重感覺回饋以調整糾錯",[16,110687],{},"\n假若病人因中風偏癱 動作程式沒有缺失情況下 可訓練病人用另一側肢體肌肉以執行動作程式",[16,110690],{},"\n可以在原本的動作程式中修改 不需要重新構築新的動作程式",[16,110693],{},"\n鼓勵善用階層特性",[12,110696,110697],{},"還有 三個理論...就結束...動作控制學了.....(吐血",{"title":180,"searchDepth":181,"depth":181,"links":110699},[],"2026-03-21","越讀越爽了 哈嘶哈嘶哈嘶","\u002Fimages\u002Fuploads\u002F1774063674180-EhtmpujU4AAFKcy.jpg",{},"\u002Fblog\u002F中樞理論學派（二）Motor-Program",{"title":110559,"description":110701},{"loc":110704},"blog\u002F中樞理論學派（二）Motor-Program","w_fr5Twnh_FzKNMAYDaPyiNwjhahG7n9Jm_V5HwuDN0",{"id":110710,"title":110711,"author":7,"body":110712,"date":110700,"description":110793,"draft":185,"edited_at":107457,"extension":186,"featured_image":110794,"meta":110795,"navigation":189,"path":110796,"pinned":185,"seo":110797,"sitemap":110798,"stem":110799,"tags":194,"__hash__":110800},"blog\u002Fblog\u002F動作學習（一）介紹.md","動作學習（一）介紹",{"type":9,"value":110713,"toc":110791},[110714,110742],[12,110715,110716,110717,110719,110720,110722,110723,110727,110728,110730,110731,110733,110734,110738,110739,110741],{},"動作學習 它的定義是 動作 藉由學習或者經驗產生的這個能力 是相當之永久的改變過程",[16,110718],{},"\n而我們把他拆分成三段",[16,110721],{},"\n手段: 靠",[75,110724,110726],{"className":110725},[5387],"學習或經驗","產生的",[16,110729],{},"\n結果: 學到了=變成本領",[16,110732],{},"\n特性: ",[75,110735,110737],{"className":110736},[5387],"相當永久","的改變過程-->意思練完之後 回到家忘記怎麼做了 那就不算是學",[16,110740],{},"\n而當他真的學會了 假如許久不未碰 讓他再做同樣的動作 他一樣能完成(只是可能跟以前比不算完美)",[12,110743,110744,110745,110747,110748,110752,110753,110755,110756,110758,110760,110761,110763,110765,110766,110768,110769,110771,110773,110774,110776,110777,110779,110780,110782,110783,110787,110788,110790],{},"而動作學習有四大特徵",[16,110746],{},"\n第一點 學習是獲得 ",[75,110749,110751],{"className":110750},[5387],"執行技巧的能力"," 的過程 ",[16,110754],{},"\n是一種能力上的改變 是內在的 可以理解成下限",[16,110757],{},[16,110759],{},"\n第二點 學習意味著經驗與練習",[16,110762],{},[16,110764],{},"\n第三點 學習無法被測量 只能靠行為上推演",[16,110767],{},"\n例如 我今天耍雜技比昨天好 那只能代表我昨天有好好學習\u002F或學到甚麼小竅門",[16,110770],{},[16,110772],{},"\n第四點 學習到的技巧是 相對永久 的變化",[16,110775],{},"\n因為提到學習到的技巧是內在的 是下限",[16,110778],{},"\n但假如因為其他因素存在 而做得出來 這不叫做學會",[16,110781],{},"\n所以",[75,110784,110786],{"className":110785},[5387],"排除","一些其他的因素 依然做得出來 這個才叫學會",[16,110789],{},"\n(其他因素包括但不限於 疲勞 情緒 動機 干擾等)",{"title":180,"searchDepth":181,"depth":181,"links":110792},[],"好了 這下連教人類怎麼學習動作都出來了","\u002Fimages\u002Fuploads\u002F1774140403310-EnL1WTpVEAAuJgW.jpg",{},"\u002Fblog\u002F動作學習（一）介紹",{"title":110711,"description":110793},{"loc":110796},"blog\u002F動作學習（一）介紹","2ccLs_CRG2tYt0eVKeFKLEbO9-HTG8BuSCJld4Np18w",{"id":110802,"title":110803,"author":7,"body":110804,"date":110700,"description":110887,"draft":185,"edited_at":107457,"extension":186,"featured_image":110888,"meta":110889,"navigation":189,"path":110890,"pinned":185,"seo":110891,"sitemap":110892,"stem":110893,"tags":194,"__hash__":110894},"blog\u002Fblog\u002F動態模組理論.md","動態模組理論",{"type":9,"value":110805,"toc":110885},[110806,110815,110818,110838,110847,110862,110868,110877,110882],[12,110807,110808,110809,110811,110812,110814],{},"動態系統理論認為 動作的產生經由次系統",[16,110810],{},"\n而次系統負責選用徵召哪些部位實行 哪些部位限制 再協同完成",[16,110813],{},"\n次系統就是剛剛系統理論提到的中樞系統 肌肉骨骼系統 感覺系統 外部因素",[12,110816,110817],{},"次系統有以下幾點特徵",[12,110819,110820,110821,110825,110826,110830,110832,110833,110837],{},"自我重組 他描述著 只要一個動作包含的",[75,110822,110824],{"className":110823},[5387],"次系統(零件)","達到了",[75,110827,110829],{"className":110828},[5387],"臨界點(critical level of skill)",[16,110831],{},"\n那麼 動作就會",[75,110834,110836],{"className":110835},[5387],"自然產生出來"," 不用被教會 亦適於用嬰兒",[12,110839,110840,110841,110843,110844,110846],{},"非線性發展 每個次系統發展都不一樣 且不是線性 ",[16,110842],{},"\n所以每個個體達成某些動作的時間都不一樣 ",[16,110845],{},"\n例如嬰兒獨立行走出現的時間 發展較慢的個體會影響獨立行走出現的時間",[12,110848,110849,110850,110852,110853,110855,110856,110858,110859,110861],{},"相移現象",[16,110851],{},"\n動作系統結構的改變非線性",[16,110854],{},"\n假如當變換某些次系統的控制參數 到達臨界",[16,110857],{},"\n動作將會以新的形態出現",[16,110860],{},"\n且不會出現介於兩個動作形態之間不穩定動作形態",[12,110863,110864,110865,110867],{},"而這套理論講述著 某一動作 所有的次系統皆成熟起來 動作才會發生 ",[16,110866],{},"\n所以穩定性極佳 且生物體會偏向做出 最省能量且效率最高的動作形態",[12,110869,110870,110871,110873,110874,110876],{},"而臨床應用上 應為他把動作描述為 次系統到達臨界點並組合而成的產物",[16,110872],{},"\n所以可以觀察病人 若某些動作無法達成時 注意他們的次系統 並加以訓練 已達成臨界點產生動作",[16,110875],{},"\n而速度也是一種動作形態的變因",[12,110878,110879],{},[1930,110880],{"alt":1932,"src":110881},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774079521551-Screenshot_2026-03-21_155143.png",[12,110883,110884],{},"很好 動作控制終於讀完了 該開始下一堂課了",{"title":180,"searchDepth":181,"depth":181,"links":110886},[],"結束 啦！！！！！！！！！","\u002Fimages\u002Fuploads\u002F1774079542057-10558485i.jpg",{},"\u002Fblog\u002F動態模組理論",{"title":110803,"description":110887},{"loc":110890},"blog\u002F動態模組理論","5u42Evmeegk2Tvy9XEhMv3VUuU-7xTBqEE5_hHtqjbU",{"id":110896,"title":110897,"author":7,"body":110898,"date":110700,"description":110990,"draft":185,"edited_at":107457,"extension":186,"featured_image":110991,"meta":110992,"navigation":189,"path":110993,"pinned":185,"seo":110994,"sitemap":110995,"stem":110996,"tags":194,"__hash__":110997},"blog\u002Fblog\u002F模組理論.md","模組理論",{"type":9,"value":110899,"toc":110988},[110900,110906,110914,110930,110936,110948,110967,110970,110979,110985],[12,110901,110902,110903,110905],{},"模組理論是把腦子切分成不同區域 不同區域負責計算自己內容 例如 ",[16,110904],{},"\n負責杯子在多遠的距離的部門 杯子多大手要張開多大部門等等",[12,110907,110908,110910,110911,110913],{},[16,110909],{},"\n而不同模組計算出來的東西能彼此利用 例如",[16,110912],{},"\n我A區計算完杯子的距離 可以傳給B區口頭講給別人聽 不需要重新再計算",[12,110915,110916,110917,110919,110920,110922,110923,110925,110926],{},"在不同的任務裡 都會去計算他們共同的參數 裡面包括",[16,110918],{},"\n選擇(選擇的部位) 順序 時間 力量",[16,110921],{},"\n這裡有點像中樞理論系統的通用動作程式",[16,110924],{},"\n還是有不同的地方是 ",[75,110927,110929],{"className":110928},[5387],"有沒有硬性要求部位",[12,110931,110932,110933,110935],{},"模組理論是探究大腦分部門後 每個部門得出的共同數值會與互相傳達 節省算力",[16,110934],{},"\n而通用動作程式是探究動作本身的不可變參數 使動作可以在不同部位裡做出相約的事情",[12,110937,110938,110939,110941,110942,110944,110945,110947],{},"而這邊提到了動作處理有某些重要的部門 ",[16,110940],{},"\n副皮質運動區(SMA) 處理動作的順序計算",[16,110943],{},"\n小腦 計算動作的時長",[16,110946],{},"\n基底核 計算力的分配",[12,110949,110950,110951,110955,110957,110958,110960,110961,110963,110964,110966],{},"小腦的計算部門分成了內外小腦 ",[75,110952,110954],{"className":110953},[5387],"這兩個部門算力是獨立進行的",[16,110956],{},"\n外小腦是用來計算時間的 中樞系統裡頭的時鐘",[16,110959],{},"\n內小腦則是運動部門 負責將指令傳達給肌肉",[16,110962],{},"\n因為兩個區域是獨立的 假若外小腦受損而內小腦完整 則會影響動作的節奏 抑或分便不出誰長短(時間上",[16,110965],{},"\n反之 動作整體時間相同 但因為運動部門受損 導致動作整體不佳(如抖動 不協調)",[12,110968,110969],{},"臨床上 可以靠這個理論來評估病人為哪個區域受損",[12,110971,110972,110973,110975,110976,110978],{},"而治療應用上 假如小腦受損 可以利用身體其他運動器官隨節拍拍打 而後彼此傳輸參數",[16,110974],{},"\n例如 手跟著拍節拍打 然後讓腳試著跟著同樣的節拍 最後再訓練說話的節奏",[16,110977],{},"\n用這種不同器官互相傳遞共用數據的方式訓練",[12,110980,110981,110982,110984],{},"但仍有些許限制",[16,110983],{},"\n例如無法解釋小腦病變病患者同時有低張力問題",[12,110986,110987],{},"要到頭了... 剩30頁就結束這節課了...",{"title":180,"searchDepth":181,"depth":181,"links":110989},[],"啊！！！！！！！快 結束 一節課啦","\u002Fimages\u002Fuploads\u002F1774068417258-10543846i.jpg",{},"\u002Fblog\u002F模組理論",{"title":110897,"description":110990},{"loc":110993},"blog\u002F模組理論","lkUCIRDuMhkpedCDL5EJ-E-IRzLl3RO0gbXhQCzzdXk",{"id":110999,"title":111000,"author":7,"body":111001,"date":110700,"description":111151,"draft":185,"edited_at":107457,"extension":186,"featured_image":111152,"meta":111153,"navigation":189,"path":111154,"pinned":185,"seo":111155,"sitemap":111156,"stem":111157,"tags":194,"__hash__":111158},"blog\u002Fblog\u002F系統理論.md","系統理論",{"type":9,"value":111002,"toc":111149},[111003,111017,111022,111060,111063,111078],[12,111004,111005,111006,111008,111009,111011,111012,111016],{},"系統理論認為動作不只是單一系統控制\u002F產生的  ",[16,111007],{},"\n中樞系統 肌肉骨骼系統 感覺系統 外部因素 ",[16,111010],{},"\n這四點一起作用下的結果",[75,111013,111015],{"className":111014},[5387],"浮現","(意指共同作用交織下自然發生)",[12,111018,111019],{},[1930,111020],{"alt":1932,"src":111021},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774069493665-Screenshot_2026-03-21_130425.png",[12,111023,111024,111025,111027,111028,111030,111032,111033,111035,111037,111038,111040,111041,111043,111044,111046,111048,111049,111051,111052,111054,111056,111057,111059],{},"此理論主要觀點有",[16,111026],{},"\n第一點 將身體作用在機械系統 那麼神經系統就不可以想出違反物理原則的動作",[16,111029],{},[16,111031],{},"\n第二點 動作產生不是反應的結果 而是我們主動活動而產生的現象 所以動物才要靠持續感知而修改行為",[16,111034],{},[16,111036],{},"\n第三點 就像是做一些很多個關節一起動的動作 假如腦子要每個獨立思考運算會很佔算力",[16,111039],{},"\n所以他提出來協調結構(coordinative structure) ",[16,111042],{},"\n意思是大腦會讓肌肉組成一個小組 對小組下達指令 讓他們共同作用",[16,111045],{},[16,111047],{},"\n第四點 生物體能活用多餘的自由度",[16,111050],{},"\n意思就是 將多個單關節的組合在一起 高效且流暢的完成動作",[16,111053],{},[16,111055],{},"\n第五點 大腦會把要做的動作做好參數 然後發到脊髓處 讓脊髓來完成動作 ",[16,111058],{},"\n那大腦只需要負責計算 脊髓則實行起來 減少大腦的負荷 稱最小互動原則",[12,111061,111062],{},"也有一個特性是分佈控制 意思就是做一件動作 同時發到給不同的小組 讓他們彼此做自己的份內事",[12,111064,111065,111066,111068,111069,111071,111072,111074,111075,111077],{},"系統模式有兩種假設",[16,111067],{},"\n第一種 神經系統要做的不是對小組指導他們做甚麼 ",[16,111070],{},"\n而是讓他們以完成工作為目的 他們自行分配身體的資源達成目的",[16,111073],{},"\n第二種 神經系統會依照肌肉骨骼系統所擁有的資源 預測動作的方式 ",[16,111076],{},"\n並且在過程中不斷適合(優化資源分配)已達成最有效工作方式",[12,111079,111080,111081,111083,111084,111086,111087,111089,111091,111092,111094,111096,111097,111099,111100,111102,111104,111105,111107,111108,111110,111111,111113,111115,111116,111118,111119,111121,111122,111124,111125,111127,111128,111130,111131,111133,111134,111136,111137,111139,111140,111142,111143,111145,111146,111148],{},"臨床應用上 主要有以下幾點",[16,111082],{},"\n第一點 動作是為了達成目的而產生 所以盡量訓練功能性動作主為 ",[16,111085],{},"\n避免單純的練神經反射或者動作的型態(動作怎麼動)",[16,111088],{},[16,111090],{},"\n第二點 可以加入不同環境 姿勢起始 視覺環境來訓練預測和適應性",[16,111093],{},[16,111095],{},"\n第三點 教導病人以不同小組為主進行訓練",[16,111098],{},"\n假若有些慣用小組受損 那麼我們應該引導病人嘗試他們目前最有效的小組 並且加以訓練",[16,111101],{},[16,111103],{},"\n第四點 他強調 腦傷的病人的動作障礙雖有別於常人 ",[16,111106],{},"\n但仍然是大腦已結合自身僅有的東西組合而成的結果",[16,111109],{},"\n例如中風病人步態雖一拐一拐 但為了極力保持能動跟相對不容易跌倒 這是好的結果",[16,111112],{},[16,111114],{},"\n第五點 他強調動作的形成可以是動態的 並不單一",[16,111117],{},"\n動作會因為目標和環境而改變",[16,111120],{},"\n例如 今天打球 ",[16,111123],{},"\n有人傳球給你(環境刺激)",[16,111126],{},"\n目標A: 投籃",[16,111129],{},"\n目標B: 運球",[16,111132],{},"\n這兩個動作在刺激一樣上 得到兩個不同的結果 原因在目標上 ",[16,111135],{},"\n而另一個例子 今天我在公車上",[16,111138],{},"\n要站穩(目的)",[16,111141],{},"\n刺激A: 公車煞車",[16,111144],{},"\n刺激B: 有人推撞",[16,111147],{},"\n而這兩個刺激在這個動作的產生上是相同的 都是抓住扶手或者踏步來穩住身體",{"title":180,"searchDepth":181,"depth":181,"links":111150},[],"要到頭 啦！！！！！！！！！！！！","\u002Fimages\u002Fuploads\u002F1774074083123-10496727i.jpg",{},"\u002Fblog\u002F系統理論",{"title":111000,"description":111151},{"loc":111154},"blog\u002F系統理論","s6JSzWo6pIssnGzsZhmviOpDivIY3A5Ap12lsS3uQzQ",{"id":111160,"title":111161,"author":7,"body":111162,"date":111251,"description":111252,"draft":185,"edited_at":107457,"extension":186,"featured_image":111253,"meta":111254,"navigation":189,"path":111255,"pinned":185,"seo":111256,"sitemap":111257,"stem":111258,"tags":194,"__hash__":111259},"blog\u002Fblog\u002F中樞理論學派（一）CPG.md","中樞理論學派（一）CPG",{"type":9,"value":111163,"toc":111249},[111164,111176,111185,111197,111209,111212,111221,111226,111231,111234,111243,111246],[12,111165,111166,111167,111169,111170,111172,111173,111175],{},"中樞理論與剛剛的周圍理論是相反的",[16,111168],{},"\n他們主張動作只由大腦傳達指令 不需要感覺刺激 自發的完成動作",[16,111171],{},"\n而他們有兩種理論",[16,111174],{},"\n1. CPG(中央行式產生器) 2. Motor Program",[12,111177,111178,111179,111181,111182,111184],{},"首先來講CPG",[16,111180],{},"\nCPG就是能在沒有周圍感覺輸入狀況下 規律或交替的完成動作型態的神經網絡",[16,111183],{},"\n存在於脊髓之中",[12,111186,111187,111188,111190,111191,111193,111194,111196],{},"有兩種特點",[16,111189],{},"\n第一種就是與生俱來 因為CPG是存在於脊髓 不用教也不用刻意管控",[16,111192],{},"\n第二種是Phasic(陣發性) 以走路為例 以周圍理論來說就是因感覺輸入後才會產生交替動作",[16,111195],{},"\n但CPG則認為即使沒有感覺輸入 也能一步一步走",[12,111198,111199,111200,111202,111203,111205,111206,111208],{},"而CPG理論有三種模型 ",[16,111201],{},"\n雙半中心模型 ",[16,111204],{},"\n閉鏈式迴路模型 ",[16,111207],{},"\n節律器模型",[12,111210,111211],{},"雙半串心模型簡單來說就是一組半中心組在一起 而這個半中心互為拮抗肌 以達成自動交替動作",[12,111213,111214,111215,111217,111218,111220],{},"而閉鎖是具校正功能 路徑是先從脊髓發出指令 ",[16,111216],{},"\n而一部份則會回傳回脊髓 用下圖來看就是中途拐彎  拐彎回去過程再加入其他感覺輸入 繼續迴圈",[16,111219],{},"\n其中就包括了 肌梭在走過的過程中會一直回傳感覺輸入給CPG讓他可以自行調整",[12,111222,111223],{},[1930,111224],{"alt":1932,"src":111225},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774058007726-Screenshot_2026-03-21_095315.png",[12,111227,111228],{},[1930,111229],{"alt":1932,"src":111230},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774056138600-Screenshot_2026-03-21_092144.png",[12,111232,111233],{},"最後節律器模型就是認為 神經元可以像心臟一樣自律跳動 活化且產生肌肉收縮後 還能再被自動活化",[12,111235,111236,111237,111239,111240,111242],{},"在臨床的應用上 因為CPG是存在於脊髓上 而SCI病人是脊髓路徑受損 ",[16,111238],{},"\n不存在像階層理論說的傳不上去高中階無法被抑制\u002F誘發 ",[16,111241],{},"\n所以CPG還是能被自動誘發做節律交替作(如行走)",[12,111244,111245],{},"簡單來說 CPG的特性可以幫助大腦達成自主自動 且節律交替有規則的動作 為大腦節省算力(如走路)",[12,111247,111248],{},"越讀越爽 我是不是這輩子有了",{"title":180,"searchDepth":181,"depth":181,"links":111250},[],"2026-03-20","啊~~~~~殺了我吧","\u002Fimages\u002Fuploads\u002F1774056557316-EhtmqMpUcAIrzEc.jpg",{},"\u002Fblog\u002F中樞理論學派（一）CPG",{"title":111161,"description":111252},{"loc":111255},"blog\u002F中樞理論學派（一）CPG","PJlOC9e8GBZz9-R6tVQCldt49j3soOWStseuotGbCk0",{"id":111261,"title":111262,"author":7,"body":111263,"date":111251,"description":111370,"draft":185,"edited_at":107457,"extension":186,"featured_image":111371,"meta":111372,"navigation":189,"path":111373,"pinned":185,"seo":111374,"sitemap":111375,"stem":111376,"tags":194,"__hash__":111377},"blog\u002Fblog\u002F周圍理論學派（一）反射理論.md","周圍理論學派（一）反射理論",{"type":9,"value":111264,"toc":111368},[111265,111271,111283,111292,111298,111303,111309,111321,111324,111327,111330,111339,111357,111362],[12,111266,111267,111268,111270],{},"周圍理論學派他們主張感覺是動作啟動到完成最重要 必要的條件 ",[16,111269],{},"\n而裡面有兩個理論 第一個是反射鏈理論 而另一個則是階層理論",[12,111272,111273,111274,111276,111277,111279,111280,111282],{},"先來講反射鏈理論 誰提出我不想鳥了",[16,111275],{},"\n他們主張動作是由感覺引發肌肉產生的 不論是外在還是內發的",[16,111278],{},"\n而這些動作包括了 視覺 前庭覺 體感",[16,111281],{},"\n複雜的動作由多個簡單的反射組成 故稱為反射鏈",[12,111284,111285,111286,111288,111289,111291],{},"反射會有閾值 意思是到達最基本刺激點會起反應 ",[16,111287],{},"\n例如 閾值是3 即使在怎麼往上給刺激 也只會到3的程度 往上加不會反應形態",[16,111290],{},"\n但假如只給到了2以則不會產生任何反應",[12,111293,111294,111295,111297],{},"而身體上會有感覺接收區 現稱為皮質感覺區(Dermatome)",[16,111296],{},"\n運動神經元只會對他該出的感覺刺激做出反應",[12,111299,111300],{},[1930,111301],{"alt":1932,"src":111302},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773995822036-Screenshot_2026-03-20_163641.png",[12,111304,111305,111306,111308],{},"而所有動作指令會被傳送到脊髓前角的alpha-motor neuron才會被引起肌肉收縮",[16,111307],{},"\n而alpha-motor neuron被稱為最終共同路徑",[12,111310,111311,111312,111314,111315,111317,111318,111320],{},"感覺訊息可以分為",[16,111313],{},"\n內在的訊息 即臟器的感覺",[16,111316],{},"\n身體表面感覺到的訊息 如體感覺(觸 壓 溫度) 和 前庭覺",[16,111319],{},"\n身體外的事物感受到的訊息 如視覺",[12,111322,111323],{},"而反射之間也有三個作用",[12,111325,111326],{},"第一個就是在促進 我給予病人外在的刺激例如拍打該區以強調力量產生能促進肌力增加",[12,111328,111329],{},"第二是互相抑制 只是利用了拮抗肌的效果 例如病人肱二頭肌力量過大 我便拍打肱三減少肱二用力",[12,111331,111332,111333,111335,111336,111338],{},"第三是發散作用 這種大家比較有感覺 踢到桌角或者踩到釘子 我們都會把受傷腳提起來 而另一隻角就伸直",[16,111334],{},"\n受傷腳踩到釘子 受傷腳是主要感覺的區域 ",[16,111337],{},"\n但非受傷腳沒有感覺也會自己有動作(伸直) 非受傷腳採取的的動作就是發散作用",[12,111340,111341,111342,111344,111345,111347,111348,111350,111351,111353,111354,111356],{},"但反射理論也有他解釋不到的地方",[16,111343],{},"\n首先是他沒有辦法解釋自發(從思考 內心而發 有感而發)的動作",[16,111346],{},"\n第二點就是沒有感覺刺激下也能產生動作 例如把感覺路徑去掉 動作雖然變得笨拙但不代表做不出來",[16,111349],{},"\n第三點是快速的動作無法解釋 例如打字1秒十下 但按照反射從感覺到動作產生只能做到1秒三下(生理)",[16,111352],{},"\n第四點是無法解釋同樣刺激會有多種\u002F不同的動作反應 像剛剛發散作用的例子另一隻腳會被伸直 但當假如另一隻腳是殘廢的 則會抑制伸直的動作",[16,111355],{},"\n第五點就是沒有辦法解釋動作怎麼學習的",[12,111358,111359],{},[1930,111360],{"alt":1932,"src":111361},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773998047912-Screenshot_2026-03-20_171344.png",[12,111363,111364,111365,111367],{},"臨床上我們可以透過反射理論來評估病人預後",[16,111366],{},"\n可以在訓練期間使用反射促進或抑制使病人訓練更佳",{"title":180,"searchDepth":181,"depth":181,"links":111369},[],"開始理論啦！！！","\u002Fimages\u002Fuploads\u002F1774000375308-EjZb4n1U8AAOBtz.jpg",{},"\u002Fblog\u002F周圍理論學派（一）反射理論",{"title":111262,"description":111370},{"loc":111373},"blog\u002F周圍理論學派（一）反射理論","BTRc77dJ8DQwSQChvIRm-EdV43-ht585NTKc8KtSJ2Y",{"id":111379,"title":111380,"author":7,"body":111381,"date":111251,"description":111444,"draft":185,"edited_at":107457,"extension":186,"featured_image":111445,"meta":111446,"navigation":189,"path":111447,"pinned":185,"seo":111448,"sitemap":111449,"stem":111450,"tags":194,"__hash__":111451},"blog\u002Fblog\u002F周圍理論學派（二）階層理論.md","周圍理論學派（二）階層理論",{"type":9,"value":111382,"toc":111442},[111383,111404,111409,111415,111424,111436],[12,111384,111385,111386,111388,111389,111391,111392,111394,111395,111397,111398,111400,111401,111403],{},"階層理論最核心的點是 自主動作與自動動作是被各個階層所管理",[16,111387],{},"\n簡單講解一下 自主動作是我想個人出來的想去做的",[16,111390],{},"\n而自動動作是由不需要我去想就能自己做出來的動作 例如呼吸 走路等 不用自己去想怎麼提腳那種",[16,111393],{},"\n存在三種層級",[16,111396],{},"\n高階中心(聯合皮質區)",[16,111399],{},"\n中階中心(運動皮質區)",[16,111402],{},"\n初級中心(脊髓)",[12,111405,111406],{},[1930,111407],{"alt":1932,"src":111408},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1774001045510-Screenshot_2026-03-20_180349.png",[12,111410,111411,111412,111414],{},"高階可以誘發或抑制低階的動作",[16,111413],{},"\n且同樣注重周圍的感覺回饋",[12,111416,111417,111418,111420,111421,111423],{},"階層理論有以下特點",[16,111419],{},"\n1. 上往下的控制 意思就是 低一階只可以被它的上一階所控制 而高一階的也只能控制它的下一階",[16,111422],{},"\n2. 反向演化 正常而言 反射的動作會因為被其高一階所抑制 但假若高階受損 低階的反射動作將會出現",[12,111425,111426,111427,111429,111430,111432,111433,111435],{},"當牽張反射產生 會先引起低階動作 如膝跳反射 即初階中心 ",[16,111428],{},"\n會慢慢回傳到中階中心 最後才會到大腦皮質區如運動及聯合皮質區",[16,111431],{},"\n但假若病人為SCI(脊髓損傷病人) 則只有低階產生 不會傳至中高級",[16,111434],{},"\n亦會使低階變強 因沒有中階所抑制",[12,111437,111438,111439,111441],{},"臨床應用上 因為這個理論提及高階受損低階被誘發或被增強 ",[16,111440],{},"\n所以可以用來評估腦損傷程度",{"title":180,"searchDepth":181,"depth":181,"links":111443},[],"繼續理論！！","\u002Fimages\u002Fuploads\u002F1774002644179-IMG_20201115_202140.jpg",{},"\u002Fblog\u002F周圍理論學派（二）階層理論",{"title":111380,"description":111444},{"loc":111447},"blog\u002F周圍理論學派（二）階層理論","udVV0CFXfKyrnixZ7EwmiVg0-5X23XqkFv6fxHfNNRM",{"id":111453,"title":111454,"author":7,"body":111455,"date":111251,"description":111557,"draft":185,"edited_at":107457,"extension":186,"featured_image":111558,"meta":111559,"navigation":189,"path":111560,"pinned":185,"seo":111561,"sitemap":111562,"stem":111563,"tags":194,"__hash__":111564},"blog\u002Fblog\u002F控制自己-Be-water-my-friend.md","Motor Control（一）",{"type":9,"value":111456,"toc":111555},[111457,111460,111463,111466,111469,111472,111475,111481,111486,111489,111492,111498,111501,111507,111513,111519,111525,111534,111549],[12,111458,111459],{},"早安各位孩子 我是你們的黑奴前領導人之一",[12,111461,111462],{},"今天我是來為大家講解我們PT的其中一個負責領域 神經領域的入門",[12,111464,111465],{},"大家可能會好奇動作控制不是人類隨隨便便都能做到的嗎 研究來幹嘛",[12,111467,111468],{},"是沒錯但 當假如你今天被上了多個debuff再加上一堆場地陷阱卡時做著雜技 你也能隨便實現嗎",[12,111470,111471],{},"很顯然不行 就算今天只是發個燒也能讓你頭昏眼花躺死在床",[12,111473,111474],{},"所以我就來簡單講述一下動作科學裡的第一大類————動作控制",[12,111476,111477,111478,111480],{},"動作控制最基本也分了三個大類",[16,111479],{},"\n1. 個體(Individual) 2. 任務(Task) 3. 環境(Environment)",[12,111482,111483],{},[1930,111484],{"alt":1932,"src":111485},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773978076463-movernment.png",[12,111487,111488],{},"這三個會構成我們的動作的產生條件\u002F變數",[12,111490,111491],{},"我先來描述個體 就是自己個體本身在決定動作內容的內在變數",[12,111493,111494,111495,111497],{},"有三種狀況會影響自己在決定動作內容的變數",[16,111496],{},"\n1. 知覺(Perception) 2.認知(Cognition) 3. 行動(Action or Activities)",[12,111499,111500],{},"我來用群友最愛的例子來解釋這三個狀況",[12,111502,111503,111504,111506],{},"今天你被蒙上眼睛",[16,111505],{},"\n被人用某種棒狀物觸碰著並有人在你耳邊說話",[12,111508,111509,111510,111512],{},"知覺就是用身體上所有的感覺辨識 包括視嗅聽味觸",[16,111511],{},"\n你知道被碰觸 知道這東西的質感 大小等一切條件",[12,111514,111515,111516,111518],{},"認知就是結合腦中整合知覺產生的東西 你突然驚覺這硬硬的 長長的 但沒有溫度 表面平滑 ",[16,111517],{},"\n有人拿著棍子在碰我 或者想對我做些甚麼",[12,111520,111521,111522,111524],{},"並且認知也是想做甚麼並且下達指令給行動產生動作的一環",[16,111523],{},"\n欸 有人拿著棍子蒙著我眼睛想幹嘛 有種不好的預感 我想要脫下眼罩 站起來跟他對峙",[12,111526,111527,111528,111530,111531,111533],{},"而行動就是實行認知給的指令的環節",[16,111529],{},"\n我伸手把眼罩拿下來 並且站起身看起來張開眼看向他 ",[16,111532],{},"\n的這系列所有肌肉調動 全都是行動做的事",[12,111535,111536,111537,111539,111540,111542,111543,111545,111546,111548],{},"好 這裡給同學提問",[16,111538],{},"\n當我假如用眼睛目視到那根棍子跟一個男人 是哪一類",[16,111541],{},"\n我分辨出來那跟棍子是球棒跟他是群友 是哪一類",[16,111544],{},"\n我看到這般情景想要懇求群友不要把我抓回去逆當黑奴 是哪一類",[16,111547],{},"\n我實際說出懇求的話 是哪一類",[12,111550,111551,111552,111554],{},"同學也可以自行給出例子寫到下面來",[16,111553],{},"\n有補充或者有大佬留言糾正就不吝留言",{"title":180,"searchDepth":181,"depth":181,"links":111556},[],"來給對PT有興趣的路人們簡單講解一下學渣對這門課的理解","\u002Fimages\u002Fuploads\u002F1773978280816-20201130_205814.jpg",{},"\u002Fblog\u002F控制自己-Be-water-my-friend",{"title":111454,"description":111557},{"loc":111560},"blog\u002F控制自己-Be-water-my-friend","-MinLavxE12JubhqJ5Pgq73DqmWL1E6rpdcAF3XCoIk",{"id":111566,"title":111567,"author":7,"body":111568,"date":111251,"description":111557,"draft":185,"edited_at":107457,"extension":186,"featured_image":111611,"meta":111612,"navigation":189,"path":111613,"pinned":185,"seo":111614,"sitemap":111615,"stem":111616,"tags":194,"__hash__":111617},"blog\u002Fblog\u002F控制自己-Be-water-my-friend（三）.md","Motor Control（三）",{"type":9,"value":111569,"toc":111609},[111570,111576,111591,111603,111606],[12,111571,111572,111573,111575],{},"終於來到了最後的環境一類了",[16,111574],{},"\n而這裡的環境是指會不會影響我的動作的部份",[12,111577,111578,111579,111581,111582,111584,111585,111587,111588,111590],{},"分為了須調整和不須調整",[16,111580],{},"\n須調整就是 他一定會影響你動作變化",[16,111583],{},"\n例如握住杯子 用的是手指 手指在剛剛說了是小的 算精細動作",[16,111586],{},"\n但杯子的大小也會影響手指張開多大",[16,111589],{},"\n這種不是自己能改變的就叫須調整",[12,111592,111593,111594,111596,111597,111599,111600,111602],{},"不須調整就是怎麼影響你也不會影響做事情本身的動作",[16,111595],{},"\n例如 噪音 有人正在G8你 或者光線的顏色 環境音",[16,111598],{},"\n這些在去掉心理因素時所影響的變量本身對我握住杯子沒有任何影響",[16,111601],{},"\n這種叫在不須調整",[12,111604,111605],{},"以上包括前兩篇提到的都是 調控任何動作行為任務的一切參數",[12,111607,111608],{},"這篇先寫短一點 等下還有一長串東西呢 草",{"title":180,"searchDepth":181,"depth":181,"links":111610},[],"\u002Fimages\u002Fuploads\u002F1773982753027-IMG_20200623_142022.jpg",{},"\u002Fblog\u002F控制自己-Be-water-my-friend（三）",{"title":111567,"description":111557},{"loc":111613},"blog\u002F控制自己-Be-water-my-friend（三）","NaWpAn9OLyMOVkdZUI1K-TkyEv0v51AZnQ9ZuUMxEQU",{"id":111619,"title":111620,"author":7,"body":111621,"date":111251,"description":111557,"draft":185,"edited_at":107457,"extension":186,"featured_image":111758,"meta":111759,"navigation":189,"path":111760,"pinned":185,"seo":111761,"sitemap":111762,"stem":111763,"tags":194,"__hash__":111764},"blog\u002Fblog\u002F控制自己-Be-water-my-friend（二）.md","Motor Control（二）",{"type":9,"value":111622,"toc":111756},[111623,111626,111629,111641,111646,111652,111661,111670,111682,111700,111712,111718,111724,111730,111747],[12,111624,111625],{},"上回說了個體本身對動作產生的影響會怎樣",[12,111627,111628],{},"而接下來便到了任務(Task)的部份",[12,111630,111631,111632,111634,111635,111637,111638,111640],{},"任務也分成了三種大變數",[16,111633],{},"\n1. 封閉或開放式動作",[16,111636],{},"\n2. 連續不連續串聯動作",[16,111639],{},"\n3. 粗大動作\u002F精細動作",[12,111642,111643],{},[1930,111644],{"alt":1932,"src":111645},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773981198506-Screenshot_2026-03-20_123256.png",[12,111647,111648,111649,111651],{},"那我先來講第一點",[16,111650],{},"\nWhat is封閉 Whats is 開放",[12,111653,111654,111655,111657,111658,111660],{},"封閉意思是指任務本身所有變量不變",[16,111656],{},"\n重複的定點投籃 在家中拿起在桌上的杯子 在只有自己的房間扣鈕扣",[16,111659],{},"\n這種只需要專注在一件事情上的我們叫在封閉",[12,111662,111663,111664,111666,111667,111669],{},"而開放式代表著一堆變化的參數影響著做任務本身的你",[16,111665],{},"\n踢球 打網球 擊打投手投來的棒球 想要接住被擊飛在空中準備落下的棒球的你",[16,111668],{},"\n這種無法自己全部掌握需要一直變化著思緒策略或者力度的行為叫做開放式",[12,111671,111672,111673,111675,111676,111678,111679,111681],{},"到了連續跟不連續動作與串聯動作",[16,111674],{},"\n連續是指沒有明確段落或開始結束的行為",[16,111677],{},"\n例子有走路 跑步 遊泳 ",[16,111680],{},"\n你可以隨意的變換目的類型",[12,111683,111684,111685,111687,111688,111690,111691,111693,111694,111696,111697,111699],{},"可能這邊開始有些抽象 那我拿不連續當反例",[16,111686],{},"\n不連續就指說有明確段落或開始結束",[16,111689],{},"\n例子有開關燈 敲打一個鋼琴鍵",[16,111692],{},"\n像是開關 按到一半鬆開手 然後燈沒開或者沒關 一直這樣按一半鬆手這種毫無意思的抽象行為沒意義",[16,111695],{},"\n只有讓開關整個按下去 觸發電流 讓燈開起來\u002F關燈",[16,111698],{},"\n所以不連續就是達成某些段落才有意義的行為",[12,111701,111702,111703,111705,111706,111708,111709,111711],{},"而串聯性動作就是指多個不連續的動作組合在一起的行為",[16,111704],{},"\n例如彈鋼琴一首曲子 用鍵盤打字",[16,111707],{},"\n鋼琴也是一樣要按鋼琴鍵到聽到聲音才有意義 ",[16,111710],{},"\n那麼我要彈一首曲子就是我按著譜子去敲打鋼琴鍵 也是把不連續串起來",[12,111713,111714,111715,111717],{},"好了好了知道有人快要睡著了最後了",[16,111716],{},"\n最後一個任務本身會影響的參數是粗大動作和精細動作",[12,111719,111720,111721,111723],{},"粗大動作就是 對 大的肌肉動起來的動作 動起來很明顯的動作",[16,111722],{},"\n走跑跳 揮拳伸懶腰那些",[12,111725,111726,111727,111729],{},"反過來精細動作就是小的動作 整體動起來不會有明顯驅體變化的動作",[16,111728],{},"\n講話 手指動來動去 眨眼 咀嚼 這些都是精細動作",[12,111731,111732,111733,111735,111736,111738,111740,111741,111743,111744,111746],{},"好 提問環節到了",[16,111734],{},"\n分辨以下動作是封閉開放 是精細粗大 是連續還不連續還串聯",[16,111737],{},[16,111739],{},"\n我對著琴譜彈鋼琴",[16,111742],{},"\n運球上籃",[16,111745],{},"\n對著節拍哼聲",[12,111748,111749,111750,111752,111753,111755],{},"還有大概一百五十頁PPT的內容呢 尼馬",[16,111751],{},"\n一周能噴出200頁PPT 一次考要諗約600~900頁PPT",[16,111754],{},"\n我以後還碰國戰遊戲被推上當領導我原地自刎歸天",{"title":180,"searchDepth":181,"depth":181,"links":111757},[],"\u002Fimages\u002Fuploads\u002F1773981453970-IMG_20200616_135133.jpg",{},"\u002Fblog\u002F控制自己-Be-water-my-friend（二）",{"title":111620,"description":111557},{"loc":111760},"blog\u002F控制自己-Be-water-my-friend（二）","32oOpUo7MyoMKg0f5PI6ewUFopNm4yRctbRmpminMyE",{"id":111766,"title":111767,"author":7,"body":111768,"date":111251,"description":111557,"draft":185,"edited_at":107457,"extension":186,"featured_image":111838,"meta":111839,"navigation":189,"path":111840,"pinned":185,"seo":111841,"sitemap":111842,"stem":111843,"tags":194,"__hash__":111844},"blog\u002Fblog\u002F控制自己-Be-water-my-friend（五）.md","Motor Control（五）",{"type":9,"value":111769,"toc":111836},[111770,111785,111800,111812,111815,111820,111825,111828,111833],[12,111771,111772,111773,111775,111776,111778,111779,111781,111782,111784],{},"書接上回",[16,111774],{},"\n現在來到了功能分類 是指說我們人在執行動作時有沒有動",[16,111777],{},"\n一樣分了兩類",[16,111780],{},"\n固定不變穩定的還是會動來動去的",[16,111783],{},"\n有操作就是沒操作的",[12,111786,111787,111788,111790,111791,111793,111794,111796,111797,111799],{},"那先來第一類 Stability 和 Transport",[16,111789],{},"\nStability我就把他理解成固定的不動的",[16,111792],{},"\n顧名思義 站立不動 坐著不動 躺著不動 這些都叫固定",[16,111795],{},"\n那Transport我就解釋成移動",[16,111798],{},"\n走路 跑步 泳遊 跳",[12,111801,111802,111803,111805,111806,111808,111809,111811],{},"而第二類有沒有操作",[16,111804],{},"\n最基本理解就是手部有沒有被分心出去做事",[16,111807],{},"\n射箭 有操作 滑手機 有操作 擦槍 有操作 ",[16,111810],{},"\n而啥也沒幹的就是無操作",[12,111813,111814],{},"所以 對 又被分成了這兩表格",[12,111816,111817],{},[1930,111818],{"alt":1932,"src":111819},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773985015794-Screenshot_2026-03-20_133628.png",[12,111821,111822],{},[1930,111823],{"alt":1932,"src":111824},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773985083041-Screenshot_2026-03-20_133751.png",[12,111826,111827],{},"結果 按照環境的兩大類跟身體動作的兩大類 我們就能變出上一節課講的4X4表格",[12,111829,111830],{},[1930,111831],{"alt":1932,"src":111832},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773985213471-Screenshot_2026-03-20_133959.png",[12,111834,111835],{},"歡迎發揮你們抽象的想法在最後一張圖挑幾個留言分享",{"title":180,"searchDepth":181,"depth":181,"links":111837},[],"\u002Fimages\u002Fuploads\u002F1773985655349-___.jpg",{},"\u002Fblog\u002F控制自己-Be-water-my-friend（五）",{"title":111767,"description":111557},{"loc":111840},"blog\u002F控制自己-Be-water-my-friend（五）","UdFkvCwkyaaXPpmvTPoh5uAfKRgr9oFi8ZR-i7rKqTg",{"id":111846,"title":111847,"author":7,"body":111848,"date":111251,"description":111557,"draft":185,"edited_at":107457,"extension":186,"featured_image":111937,"meta":111938,"navigation":189,"path":111939,"pinned":185,"seo":111940,"sitemap":111941,"stem":111942,"tags":194,"__hash__":111943},"blog\u002Fblog\u002F控制自己-Be-water-my-friend（四）.md","Motor Control（四）",{"type":9,"value":111849,"toc":111935},[111850,111856,111874,111886,111895,111910,111922,111925,111930],[12,111851,111852,111853,111855],{},"很好各位同學 假如你們看完剛剛三篇一定是對動作控制稍微理解",[16,111854],{},"\n也相信你們不是對PT感興趣就是抖M才點進來這裡對吧",[12,111857,111858,111859,111861,111862,111864,111865,111867,111868,111870,111871,111873],{},"而這裡有個大聰明把剛剛所有的東西用一個4X4的大表格列完出來",[16,111860],{},"\n對 沒想到居然有這麼大個表格是吧",[16,111863],{},"\n由這三點變出來的不同任務描寫對我們自己身體調控參數的影響",[16,111866],{},"\n• 依據環境 (environment) 分類 ",[16,111869],{},"\n• 依據行動的功能 (function of action) 分類 ",[16,111872],{},"\n• 任務的綜合分類",[12,111875,111876,111877,111879,111880,111882,111883,111885],{},"那先來講述環境分類",[16,111878],{},"\n環境分成兩類",[16,111881],{},"\n1. 動態與靜態",[16,111884],{},"\n2. 一致與變化",[12,111887,111888,111889,111891,111892,111894],{},"先來講環境的第一類",[16,111890],{},"\n動態靜態就是指說場地本身會不會動",[16,111893],{},"\n例如搭電扶梯 電扶梯是動的 走在平原 平原是靜的 搭電梯 動的 坐在咖啡廳裡 靜的",[12,111896,111897,111898,111900,111901,111903,111904,111906,111907,111909],{},"而第二類一致與變化",[16,111899],{},"\n一致是指做每一次動作都不會有環境變化 可預測的地方",[16,111902],{},"\n在啥也沒有的平原上來回走動",[16,111905],{},"\n按著距離等長的三角椎繞行來回",[16,111908],{},"\n這些本身不會被動的 做幾次都不會有環境變化",[12,111911,111912,111913,111915,111916,111918,111919,111921],{},"而變化就是每做一次都會隨機變化 不可預測的地方",[16,111914],{},"\n擁擠人潮裡走動",[16,111917],{},"\n凹突不平的路面走動",[16,111920],{},"\n這些每次都有變化抑或者不可預測的情形就是變化的",[12,111923,111924],{},"然後整合這兩類就會得到這兩個表",[12,111926,111927],{},[1930,111928],{"alt":1932,"src":111929},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773983922403-Screenshot_2026-03-20_131746.png",[12,111931,111932],{},[1930,111933],{"alt":1932,"src":111934},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773984031992-Screenshot_2026-03-20_132002.png",{"title":180,"searchDepth":181,"depth":181,"links":111936},[],"\u002Fimages\u002Fuploads\u002F1773983954366-IMG_20200722_231036.jpg",{},"\u002Fblog\u002F控制自己-Be-water-my-friend（四）",{"title":111847,"description":111557},{"loc":111939},"blog\u002F控制自己-Be-water-my-friend（四）","wvQwit-t5AEuz9WpQObO1SbCmIZZ_kqQQE3ih7__D-4",{"id":111945,"title":111946,"author":7,"body":111947,"date":111251,"description":112145,"draft":185,"edited_at":107457,"extension":186,"featured_image":112146,"meta":112147,"navigation":189,"path":112148,"pinned":185,"seo":112149,"sitemap":112150,"stem":112151,"tags":194,"__hash__":112152},"blog\u002Fblog\u002F進階控制制制制.md","Motor Control（六）",{"type":9,"value":111948,"toc":112143},[111949,111964,111979,111984,111996,111999,112004,112016,112022,112047,112056,112059,112086,112089,112092,112098,112103,112106,112111,112117,112123,112128,112134,112140],[12,111950,111951,111952,111954,111955,111957,111958,111960,111961,111963],{},"關於在我們動作產生到完成時 我們的大腦一定不是空的 ",[16,111953],{},"\n而是聚焦在這三點",[16,111956],{},"\n訊息理論 ",[16,111959],{},"\n反應時間 ",[16,111962],{},"\n閉鎖式回路與開放式回路",[12,111965,111966,111967,111969,111970,111972,111973,111975,111976,111978],{},"首先來講訊息理論 由一個荷蘭人提出",[16,111968],{},"\n他說我們在做動作時會經歷這三個階段",[16,111971],{},"\n認明刺激(Stimulus Identification)",[16,111974],{},"\n反應選擇(Response Selection) ",[16,111977],{},"\n反應程式組合(Response Programming)",[12,111980,111981],{},[1930,111982],{"alt":1932,"src":111983},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773990603831-Screenshot_2026-03-20_150938.png",[12,111985,111986,111987,111989,111990,111992,111993,111995],{},"其實內容與第一~三堂講的內容相似",[16,111988],{},"\n認明刺激就是認知在對知覺加工的過程",[16,111991],{},"\n反應選擇就是認知",[16,111994],{},"\n反應程式組合就是生成整套動作的內容 例如拿起前方50cm杯子 要身體前彎 伸出手 張開手 握住杯 拿回來",[12,111997,111998],{},"而反應時間就是知覺偵測到開始變成認知再產生動作產生的過程",[12,112000,112001],{},[1930,112002],{"alt":1932,"src":112003},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773991127457-Screenshot_2026-03-20_151747.png",[12,112005,112006,112007,112009,112010,112012,112013,112015],{},"賽車的五盞紅燈亮起來是預警期 ",[16,112008],{},"\n轉綠燈再到你開始踩油門的瞬間 這麼短的過程就是反應時間",[16,112011],{},"\n動作時間就是你開始踩到真踩到油門的時長",[16,112014],{},"\n而回應時間就是將反應時間和動作時間加在一起",[12,112017,112018,112019,112021],{},"這種反應時間會因為知覺認知配對的組合增加而變長 且呈線性增長 這就是亥可定律",[16,112020],{},"\n給你個例子 講出以下顏色",[12,112023,112024,112027,112028,112032,112033,112037,112038,112032,112043],{},[75,112025,89936],{"className":112026},[107291],"                      ",[75,112029,112031],{"className":112030},[79],"黃","                       ",[75,112034,89976],{"className":112035},[112036],"purple-3","                        ",[75,112039,112042],{"className":112040},[112041],"yellow-1","紫",[75,112044,112046],{"className":112045},[81363],"綠",[12,112048,112049,112050,112052,112053,112055],{},"這種就是比較典型的反應時間會增",[16,112051],{},"\n原因為是因為刺激與反應配對數目增加",[16,112054],{},"\n(代表知覺被觸發到認知的辨認結束 因為不是很直覺得辨認需要被拖長而理解)",[12,112057,112058],{},"但是這不符合亥可定律",[12,112060,112061,112064,112027,112067,112070,112037,112073,112076,112079,112080,112079,112083],{},[75,112062,89936],{"className":112063},[79],[75,112065,180],{"className":112066},[107291],[75,112068,112031],{"className":112069},[112041],[75,112071,180],{"className":112072},[79],[75,112074,89976],{"className":112075},[81363],[75,112077,180],{"className":112078},[112036],"                         ",[75,112081,112042],{"className":112082},[112036],[75,112084,112046],{"className":112085},[173],[12,112087,112088],{},"因為他並沒有讓配對組合增加(沒有反直覺)",[12,112090,112091],{},"而最後就是閉鎖式跟開放式回路的對比",[12,112093,112094,112095,112097],{},"先來講閉鎖式的回路",[16,112096],{},"\n簡單來說就是在持續的動作中不斷回傳讓自己可以持續調整",[12,112099,112100],{},[1930,112101],{"alt":1932,"src":112102},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773992386445-Screenshot_2026-03-20_153850.png",[12,112104,112105],{},"而我們PT可以在這時給予例如像口頭的回饋給病人聽讓他改進動作內容",[12,112107,112108],{},[1930,112109],{"alt":1932,"src":112110},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773992691800-Screenshot_2026-03-20_154139.png",[12,112112,112113,112114,112116],{},"到了開放式回路",[16,112115],{},"\n就是讓整套動作做完結束之後才得到回饋 與閉鏈式相比無法在得知事情結束前更改",[12,112118,112119,112120,112122],{},"但開放式回路卻有一種叫做輸出副本的東西",[16,112121],{},"\n他可以在認知將想做的傳達到給動作前一刻 先傳回去給腦子 讓我們有個預期可以會變成怎樣",[12,112124,112125],{},[1930,112126],{"alt":1932,"src":112127},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1773993075653-Screenshot_2026-03-20_155057.png",[12,112129,112130,112131,112133],{},"再申 這個輸出副本只是在腦子裡有個模擬事情會怎樣過程怎樣完成",[16,112132],{},"\n而與閉鎖式回饋不同是在動作進行中持續調整修改",[12,112135,112136,112137,112139],{},"很好 這麼多才16頁內容 ",[16,112138],{},"\n卒",[12,112141,112142],{},"歡迎大佬補充糾正",{"title":180,"searchDepth":181,"depth":181,"links":112144},[],"以達進階 小心食用 噎到不會幫你叫救護車","\u002Fimages\u002Fuploads\u002F1773993528425-77896396_p0_master1200.jpg",{},"\u002Fblog\u002F進階控制制制制",{"title":111946,"description":112145},{"loc":112148},"blog\u002F進階控制制制制","TNLZw0f3RUfcOQpjCJ_Hxum8P8fYgb8ZwahF6sj3960",{"id":112154,"title":112155,"author":112156,"body":112157,"date":112179,"description":112180,"draft":185,"edited_at":112179,"extension":186,"featured_image":112181,"meta":112182,"navigation":189,"path":112183,"pinned":185,"seo":112184,"sitemap":112185,"stem":112186,"tags":194,"__hash__":112187},"blog\u002Fblog\u002F一隻小花妖.md","一隻小花妖","花夜",{"type":9,"value":112158,"toc":112177},[112159,112162,112165,112168,112171,112174],[12,112160,112161],{},"啊......已經開始了嗎?",[12,112163,112164],{},"大家好我是花夜，是一隻小花妖，已成功地融入現代社會之中，或許某一天你們能在街頭的某個角落看到我?",[12,112166,112167],{},"喜歡的東西有點多，例如電動、漫畫、動畫、歷史等等，涉獵範圍廣但不深，拜託了不要嗆我我只是一個小菜雞｡ﾟヽ(ﾟ´Д`)ﾉﾟ｡",[12,112169,112170],{},"順帶一提，本人是遊戲王愛好者，之後會在這裡做各種雜七雜八的遊戲王OCG介紹，歡迎大家一起討論",[12,112172,112173],{},"無論你們身在何處，我永遠在這裡靜候你們的到來",[12,112175,112176],{},"寫這篇的時候眼睛受傷，盯著螢幕好不舒服，祝你們一輩子健康無慮",{"title":180,"searchDepth":181,"depth":181,"links":112178},[],"2026-03-05","讓你們淺淺的認識我","\u002Fimages\u002Fuploads\u002F1772722320083-791_20260218161129.png",{},"\u002Fblog\u002F一隻小花妖",{"title":112155,"description":112180},{"loc":112183},"blog\u002F一隻小花妖","h2FLJr4xW1vHokPgPrwlpMOXkN9WwZfTdYz-3-pLVVY",{"id":112189,"title":112190,"author":6946,"body":112191,"date":113002,"description":113003,"draft":185,"edited_at":194,"extension":186,"featured_image":113004,"meta":113005,"navigation":189,"path":113006,"pinned":185,"seo":113007,"sitemap":113008,"stem":113009,"tags":194,"__hash__":113010},"blog\u002Fblog\u002F紀錄一次Discord機器人的開發過程.md","紀錄一次Discord機器人的開發過程",{"type":9,"value":112192,"toc":112997},[112193,112200,112203,112206,112213,112222,112226,112229,112259,112262,112282,112286,112292,112309,112312,112316,112319,112326,112332,112335,112338,112343,112354,112363,112982,112985,112991,112994],[112194,112195,112197],"h6",{"id":112196},"每次都要找些好看的封面圖還挺麻煩的就當是美圖分享吧",[5610,112198,112199],{},"每次都要找些好看的封面圖還挺麻煩的，就當是美圖分享吧",[12,112201,112202],{},"最近假期在家真的閒到發慌。家裡管得嚴，加上妹妹正在準備升學考試，為了不吵到她，我在家也不太好意思一直打遊戲。結果呢，唯一的娛樂活動反而變成了寫程式~~（蛤~~",[12,112204,112205],{},"剛好，把我這個小破站翻新完之後，突然覺得有點空虛，沒有什麼別的事要做了。於是，在發呆了一下午之後還是決定：不如再來試著重新寫一次 Discord 音樂機器人好了。",[12,112207,112208,112209,112212],{},"寫音樂機器人這件事我在中學時就做過一次了。當時花了我好幾個月的時間，寫了一個能24\u002F7在線的音樂機器人。",[5610,112210,112211],{},"至今還記得音樂開始播放的時候有多興奮","。所以這一次不是簡單的做一個會播歌機器人就好，那也太無聊了。我想要來寫點好玩的：給我的機器人加一個可以互動的網頁介面。",[13438,112214,112216],{"className":112215},[13441,53830],[12,112217,112218,112219],{},"請注意：這篇不是那種按部就班的「保姆級教學文」喔...我只想在這裡紀錄開發歷程，並且盡我所能解釋其背後的工作原理。",[5610,112220,112221],{},"想找教程可以出門左轉去谷歌上找(x",[6964,112223,112225],{"id":112224},"_1-介紹","1. 介紹",[12,112227,112228],{},"如果你在 2026 年還想弄一個能在 Discord 語音頻道裡播歌的機器人，查完資料你就會發現，這是一件苦差事。",[12,112230,112231,112232,112242,112243,112250,112251,112258],{},"自從 Groovy 和 Rythm 在 2021 年被 YouTube 官方強制下線後，開發音樂機器人就成了一場開發者與YouTube 的軍備競賽。以前方便好用的 ",[106,112233,112234,112241],{},[2330,112235,112238],{"href":112236,"rel":112237},"https:\u002F\u002Fgithub.com\u002Ffent\u002Fnode-ytdl-core",[2334],[6979,112239,112240],{},"node-ytdl-core","在2024年就掛掉了","，而其他的替代品 ：",[2330,112244,112247],{"href":112245,"rel":112246},"https:\u002F\u002Fgithub.com\u002Fplay-dl\u002Fplay-dl",[2334],[6979,112248,112249],{},"play-dl","和",[2330,112252,112255],{"href":112253,"rel":112254},"https:\u002F\u002Fgithub.com\u002Fdistubejs\u002Fytdl-core",[2334],[6979,112256,112257],{},"ytdl-core","也紛紛停止開發了。",[12,112260,112261],{},"想用以前的老套路，硬找個還活著的 package 來下載 YouTube 音源再塞給 Discord，你會遇到一堆問題：",[23,112263,112264,112270,112276],{},[26,112265,112266,112269],{},[106,112267,112268],{},"極度脆弱："," YouTube 現在會主動打擊爬蟲。他們會頻繁更改客戶端、加密演算法，甚至對資料中心的 IP 進行嚴格的限流。這意味著機器人可能昨天還好好的，今天早上起來就滿地報錯。",[26,112271,112272,112275],{},[106,112273,112274],{},"效能問題："," Node.js 是一個單執行緒（Single-threaded）的執行環境，不是設計來做繁重媒體處理的。讓它一邊下載音訊、解碼，還要重新編碼並即時推送到 Discord，只要同時服務幾個不同的伺服器，CPU 就要撐不住開始哀嚎了。",[26,112277,112278,112281],{},[106,112279,112280],{},"職責過載："," 機器人要負責監聽 Discord 訊息事件、爬取 YouTube、處理音軌，還要維持語音頻道的串流。只要其中一個環節崩潰，整個機器人就會直接斷線，聽歌聽到一半的朋友大概會氣死。",[6964,112283,112285],{"id":112284},"_2-lavalink","2. Lavalink",[12,112287,112288,112289],{},"如果 Node.js 處理音訊這麼痛苦，那答案很簡單：",[106,112290,112291],{},"不要讓你的機器人碰音訊。",[12,112293,112294,112295,112304,112305,112308],{},"解決這個問題的其中一個方法就是使用 ",[106,112296,112297],{},[2330,112298,112301],{"href":112299,"rel":112300},"https:\u002F\u002Fgithub.com\u002Flavalink-devs\u002FLavalink",[2334],[6979,112302,112303],{},"Lavalink","。它是一個專門為 Discord 機器人打造的獨立 Java 音訊伺服器。機器人只負責接收使用者的指令（例如 ",[6979,112306,112307],{},"\u002Fplay","），Lavalink 則扛下所有髒活：解析 YouTube 網址、繞過限制、解碼音訊，並直接把串流送到 Discord 的語音頻道裡。兩者之間只透過 WebSocket 和 REST API 進行溝通。",[12,112310,112311],{},"這樣一來，如果機器人因為某些 Bug 當機重啟了，Lavalink 依然會在背景繼續播歌，完全不影響正在聽音樂的使用者。",[6964,112313,112315],{"id":112314},"_3-代碼結構","3. 代碼結構",[12,112317,112318],{},"所以說了這麼多，機器人應該長什麼樣？",[12,112320,112321,112322,112325],{},"我這裡選用的是 ",[106,112323,112324],{},"Deno + Svelte + Lavalink"," 的組合。整個專案包含了三個伺服器，資料夾結構大概長這樣：",[6972,112327,112330],{"className":112328,"code":112329,"language":6977},[6975],"Music-bot\u002F\n├── deno.json                  # 配置文件\n├── .gitignore\n├── bot-backend\u002F               # 機器人本體\n│   ├── main.ts                \n│   ├── deploy-commands.ts     \n│   └── .env\n├── lavalink-engine\u002F           # Lavalink\n│   ├── Lavalink.jar           \n│   ├── application.yml        \n│   └── plugins\u002F               \n└── web-dashboard\u002F             # SvelteKit 網頁前端\n",[6979,112331,112329],{"__ignoreMap":180},[12,112333,112334],{},"這個專案裡包含了三個伺服器：網頁後端，Lavalink伺服器和機器人。",[12,112336,112337],{},"運行邏輯大約是這樣的：",[12,112339,112340],{},[1930,112341],{"alt":1932,"src":112342},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1771778271569-Screenshot_2026-02-23_at_12.36.38_AM.png",[12,112344,112345,112346,112349,112350,112353],{},"機器人代碼意外的非常簡單。只需要用到 ",[6979,112347,112348],{},"Shoukaku","（負責跟 Lavalink 溝通）和 ",[6979,112351,112352],{},"discord.js","（負責跟 Discord 溝通），就可以開始動工了。",[12,112355,112356,112357,87213,112359,112362],{},"一個最基本~~（陽春）~~的機器人代碼（只有",[6979,112358,112307],{},[6979,112360,112361],{},"\u002Fstop","指令）：",[6972,112364,112368],{"className":112365,"code":112366,"language":112367,"meta":180,"style":180},"language-JavaScript shiki shiki-themes github-light github-dark","import {\n  Client,\n  GatewayIntentBits,\n  type ChatInputCommandInteraction,\n} from \"discord.js\";\nimport { Shoukaku, Connectors, LoadType } from \"shoukaku\";\nimport \"@std\u002Fdotenv\u002Fload\";\n\nconst client = new Client({\n  intents: [\n    GatewayIntentBits.Guilds,\n    GatewayIntentBits.GuildVoiceStates,\n  ],\n});\n\n\u002F\u002F Shoukaku 初始化\n\nconst nodes = [\n  {\n    name: \"main\",\n    url: `${Deno.env.get(\"LAVALINK_HOST\")}:${Deno.env.get(\"LAVALINK_PORT\")}`,\n    auth: Deno.env.get(\"LAVALINK_PASSWORD\")!,\n  },\n];\n\nconst shoukaku = new Shoukaku(new Connectors.DiscordJS(client), nodes, {\n  reconnectTries: 10,\n  reconnectInterval: 3000,\n});\n\nshoukaku.on(\"error\", (_, error) => console.error(\"Shoukaku error:\", error));\nshoukaku.on(\"ready\", (name) => console.log(`Lavalink node \"${name}\" connected`));\nshoukaku.on(\"disconnect\", (name, reason) =>\n  console.warn(`Lavalink node \"${name}\" disconnected — ${reason ?? \"unknown reason\"}. Reconnecting...`)\n);\n\n\u002F\u002F ====================\n\nclient.once(\"clientReady\", () => {\n  console.log(`Logged in as ${client.user?.tag}`);\n});\n\nclient.on(\"interactionCreate\", async (interaction) => {\n  if (!interaction.isChatInputCommand()) return;\n\n  switch (interaction.commandName) {\n    case \"play\":\n      await handlePlay(interaction);\n      break;\n    case \"stop\":\n      await handleStop(interaction);\n      break;\n  }\n});\n\nasync function handlePlay(interaction: ChatInputCommandInteraction) {\n  const member = await interaction.guild?.members.fetch(interaction.user.id);\n  const voiceChannel = member?.voice.channel;\n\n  if (!voiceChannel) {\n    await interaction.reply({\n      content: \"需要加入語音頻道才能播放音樂。\",\n      ephemeral: true,\n    });\n    return;\n  }\n\n  await interaction.deferReply();\n\n  const query = interaction.options.getString(\"query\", true);\n  const node = shoukaku.options.nodeResolver(shoukaku.nodes);\n\n  if (!node) {\n    await interaction.editReply(\"找不到Lavalink節點。\");\n    return;\n  }\n\n  const isUrl = query.startsWith(\"http:\u002F\u002F\") || query.startsWith(\"https:\u002F\u002F\");\n  const result = await node.rest.resolve(isUrl ? query : `ytsearch:${query}`);\n\n  if (!result || result.loadType === LoadType.EMPTY || result.loadType === LoadType.ERROR) {\n    await interaction.editReply(\"找不到任何結果。\");\n    return;\n  }\n\n  const track = result.loadType === LoadType.SEARCH || result.loadType === LoadType.PLAYLIST\n    ? result.data[0]\n    : result.data;\n\n  let player = shoukaku.players.get(interaction.guildId!);\n  if (!player) {\n    player = await shoukaku.joinVoiceChannel({\n      guildId: interaction.guildId!,\n      channelId: voiceChannel.id,\n      shardId: 0,\n    });\n  }\n\n  await player.playTrack({ track: { encoded: track.encoded } });\n\n  await interaction.editReply(`Now playing: **${track.info.title}**`);\n}\n\nasync function handleStop(interaction: ChatInputCommandInteraction) {\n  const player = shoukaku.players.get(interaction.guildId!);\n\n  if (!player) {\n    await interaction.reply({\n      content: \"目前沒有播放音樂。\",\n      ephemeral: true,\n    });\n    return;\n  }\n\n  await shoukaku.leaveVoiceChannel(interaction.guildId!);\n  await interaction.reply(\"停止播放並退出語音頻道。\");\n}\n\nclient.login(Deno.env.get(\"DISCORD_TOKEN\"));\n","JavaScript",[6979,112369,112370,112375,112380,112385,112390,112395,112400,112405,112410,112415,112420,112425,112430,112435,112440,112444,112449,112453,112458,112463,112468,112473,112478,112483,112487,112491,112496,112501,112506,112510,112514,112519,112524,112529,112534,112538,112542,112547,112551,112556,112561,112565,112569,112574,112579,112583,112588,112593,112598,112603,112608,112613,112617,112621,112625,112629,112634,112640,112646,112651,112656,112662,112668,112674,112680,112686,112691,112696,112702,112707,112713,112719,112724,112730,112736,112741,112746,112751,112757,112763,112768,112774,112780,112785,112790,112795,112801,112807,112813,112818,112824,112830,112836,112842,112848,112854,112859,112864,112869,112875,112880,112886,112891,112896,112902,112908,112913,112918,112923,112929,112934,112939,112944,112949,112954,112960,112966,112971,112976],{"__ignoreMap":180},[75,112371,112372],{"class":7383,"line":7384},[75,112373,112374],{},"import {\n",[75,112376,112377],{"class":7383,"line":181},[75,112378,112379],{},"  Client,\n",[75,112381,112382],{"class":7383,"line":7430},[75,112383,112384],{},"  GatewayIntentBits,\n",[75,112386,112387],{"class":7383,"line":7436},[75,112388,112389],{},"  type ChatInputCommandInteraction,\n",[75,112391,112392],{"class":7383,"line":7458},[75,112393,112394],{},"} from \"discord.js\";\n",[75,112396,112397],{"class":7383,"line":7474},[75,112398,112399],{},"import { Shoukaku, Connectors, LoadType } from \"shoukaku\";\n",[75,112401,112402],{"class":7383,"line":7480},[75,112403,112404],{},"import \"@std\u002Fdotenv\u002Fload\";\n",[75,112406,112407],{"class":7383,"line":7485},[75,112408,112409],{"emptyLinePlaceholder":189},"\n",[75,112411,112412],{"class":7383,"line":7498},[75,112413,112414],{},"const client = new Client({\n",[75,112416,112417],{"class":7383,"line":7516},[75,112418,112419],{},"  intents: [\n",[75,112421,112422],{"class":7383,"line":7521},[75,112423,112424],{},"    GatewayIntentBits.Guilds,\n",[75,112426,112427],{"class":7383,"line":7526},[75,112428,112429],{},"    GatewayIntentBits.GuildVoiceStates,\n",[75,112431,112432],{"class":7383,"line":7539},[75,112433,112434],{},"  ],\n",[75,112436,112437],{"class":7383,"line":7552},[75,112438,112439],{},"});\n",[75,112441,112442],{"class":7383,"line":7557},[75,112443,112409],{"emptyLinePlaceholder":189},[75,112445,112446],{"class":7383,"line":7562},[75,112447,112448],{},"\u002F\u002F Shoukaku 初始化\n",[75,112450,112451],{"class":7383,"line":7571},[75,112452,112409],{"emptyLinePlaceholder":189},[75,112454,112455],{"class":7383,"line":7584},[75,112456,112457],{},"const nodes = [\n",[75,112459,112460],{"class":7383,"line":7602},[75,112461,112462],{},"  {\n",[75,112464,112465],{"class":7383,"line":7607},[75,112466,112467],{},"    name: \"main\",\n",[75,112469,112470],{"class":7383,"line":8360},[75,112471,112472],{},"    url: `${Deno.env.get(\"LAVALINK_HOST\")}:${Deno.env.get(\"LAVALINK_PORT\")}`,\n",[75,112474,112475],{"class":7383,"line":8374},[75,112476,112477],{},"    auth: Deno.env.get(\"LAVALINK_PASSWORD\")!,\n",[75,112479,112480],{"class":7383,"line":8387},[75,112481,112482],{},"  },\n",[75,112484,112485],{"class":7383,"line":8400},[75,112486,23314],{},[75,112488,112489],{"class":7383,"line":8413},[75,112490,112409],{"emptyLinePlaceholder":189},[75,112492,112493],{"class":7383,"line":8418},[75,112494,112495],{},"const shoukaku = new Shoukaku(new Connectors.DiscordJS(client), nodes, {\n",[75,112497,112498],{"class":7383,"line":8438},[75,112499,112500],{},"  reconnectTries: 10,\n",[75,112502,112503],{"class":7383,"line":8456},[75,112504,112505],{},"  reconnectInterval: 3000,\n",[75,112507,112508],{"class":7383,"line":8476},[75,112509,112439],{},[75,112511,112512],{"class":7383,"line":8481},[75,112513,112409],{"emptyLinePlaceholder":189},[75,112515,112516],{"class":7383,"line":8487},[75,112517,112518],{},"shoukaku.on(\"error\", (_, error) => console.error(\"Shoukaku error:\", error));\n",[75,112520,112521],{"class":7383,"line":8493},[75,112522,112523],{},"shoukaku.on(\"ready\", (name) => console.log(`Lavalink node \"${name}\" connected`));\n",[75,112525,112526],{"class":7383,"line":15119},[75,112527,112528],{},"shoukaku.on(\"disconnect\", (name, reason) =>\n",[75,112530,112531],{"class":7383,"line":15124},[75,112532,112533],{},"  console.warn(`Lavalink node \"${name}\" disconnected — ${reason ?? \"unknown reason\"}. Reconnecting...`)\n",[75,112535,112536],{"class":7383,"line":15143},[75,112537,7738],{},[75,112539,112540],{"class":7383,"line":15151},[75,112541,112409],{"emptyLinePlaceholder":189},[75,112543,112544],{"class":7383,"line":15171},[75,112545,112546],{},"\u002F\u002F ====================\n",[75,112548,112549],{"class":7383,"line":15188},[75,112550,112409],{"emptyLinePlaceholder":189},[75,112552,112553],{"class":7383,"line":15199},[75,112554,112555],{},"client.once(\"clientReady\", () => {\n",[75,112557,112558],{"class":7383,"line":15204},[75,112559,112560],{},"  console.log(`Logged in as ${client.user?.tag}`);\n",[75,112562,112563],{"class":7383,"line":15214},[75,112564,112439],{},[75,112566,112567],{"class":7383,"line":15219},[75,112568,112409],{"emptyLinePlaceholder":189},[75,112570,112571],{"class":7383,"line":15224},[75,112572,112573],{},"client.on(\"interactionCreate\", async (interaction) => {\n",[75,112575,112576],{"class":7383,"line":15252},[75,112577,112578],{},"  if (!interaction.isChatInputCommand()) return;\n",[75,112580,112581],{"class":7383,"line":15263},[75,112582,112409],{"emptyLinePlaceholder":189},[75,112584,112585],{"class":7383,"line":15272},[75,112586,112587],{},"  switch (interaction.commandName) {\n",[75,112589,112590],{"class":7383,"line":15281},[75,112591,112592],{},"    case \"play\":\n",[75,112594,112595],{"class":7383,"line":15310},[75,112596,112597],{},"      await handlePlay(interaction);\n",[75,112599,112600],{"class":7383,"line":15334},[75,112601,112602],{},"      break;\n",[75,112604,112605],{"class":7383,"line":15358},[75,112606,112607],{},"    case \"stop\":\n",[75,112609,112610],{"class":7383,"line":15382},[75,112611,112612],{},"      await handleStop(interaction);\n",[75,112614,112615],{"class":7383,"line":15409},[75,112616,112602],{},[75,112618,112619],{"class":7383,"line":15433},[75,112620,8490],{},[75,112622,112623],{"class":7383,"line":15438},[75,112624,112439],{},[75,112626,112627],{"class":7383,"line":15445},[75,112628,112409],{"emptyLinePlaceholder":189},[75,112630,112631],{"class":7383,"line":15450},[75,112632,112633],{},"async function handlePlay(interaction: ChatInputCommandInteraction) {\n",[75,112635,112637],{"class":7383,"line":112636},57,[75,112638,112639],{},"  const member = await interaction.guild?.members.fetch(interaction.user.id);\n",[75,112641,112643],{"class":7383,"line":112642},58,[75,112644,112645],{},"  const voiceChannel = member?.voice.channel;\n",[75,112647,112649],{"class":7383,"line":112648},59,[75,112650,112409],{"emptyLinePlaceholder":189},[75,112652,112653],{"class":7383,"line":27671},[75,112654,112655],{},"  if (!voiceChannel) {\n",[75,112657,112659],{"class":7383,"line":112658},61,[75,112660,112661],{},"    await interaction.reply({\n",[75,112663,112665],{"class":7383,"line":112664},62,[75,112666,112667],{},"      content: \"需要加入語音頻道才能播放音樂。\",\n",[75,112669,112671],{"class":7383,"line":112670},63,[75,112672,112673],{},"      ephemeral: true,\n",[75,112675,112677],{"class":7383,"line":112676},64,[75,112678,112679],{},"    });\n",[75,112681,112683],{"class":7383,"line":112682},65,[75,112684,112685],{},"    return;\n",[75,112687,112689],{"class":7383,"line":112688},66,[75,112690,8490],{},[75,112692,112694],{"class":7383,"line":112693},67,[75,112695,112409],{"emptyLinePlaceholder":189},[75,112697,112699],{"class":7383,"line":112698},68,[75,112700,112701],{},"  await interaction.deferReply();\n",[75,112703,112705],{"class":7383,"line":112704},69,[75,112706,112409],{"emptyLinePlaceholder":189},[75,112708,112710],{"class":7383,"line":112709},70,[75,112711,112712],{},"  const query = interaction.options.getString(\"query\", true);\n",[75,112714,112716],{"class":7383,"line":112715},71,[75,112717,112718],{},"  const node = shoukaku.options.nodeResolver(shoukaku.nodes);\n",[75,112720,112722],{"class":7383,"line":112721},72,[75,112723,112409],{"emptyLinePlaceholder":189},[75,112725,112727],{"class":7383,"line":112726},73,[75,112728,112729],{},"  if (!node) {\n",[75,112731,112733],{"class":7383,"line":112732},74,[75,112734,112735],{},"    await interaction.editReply(\"找不到Lavalink節點。\");\n",[75,112737,112739],{"class":7383,"line":112738},75,[75,112740,112685],{},[75,112742,112744],{"class":7383,"line":112743},76,[75,112745,8490],{},[75,112747,112749],{"class":7383,"line":112748},77,[75,112750,112409],{"emptyLinePlaceholder":189},[75,112752,112754],{"class":7383,"line":112753},78,[75,112755,112756],{},"  const isUrl = query.startsWith(\"http:\u002F\u002F\") || query.startsWith(\"https:\u002F\u002F\");\n",[75,112758,112760],{"class":7383,"line":112759},79,[75,112761,112762],{},"  const result = await node.rest.resolve(isUrl ? query : `ytsearch:${query}`);\n",[75,112764,112766],{"class":7383,"line":112765},80,[75,112767,112409],{"emptyLinePlaceholder":189},[75,112769,112771],{"class":7383,"line":112770},81,[75,112772,112773],{},"  if (!result || result.loadType === LoadType.EMPTY || result.loadType === LoadType.ERROR) {\n",[75,112775,112777],{"class":7383,"line":112776},82,[75,112778,112779],{},"    await interaction.editReply(\"找不到任何結果。\");\n",[75,112781,112783],{"class":7383,"line":112782},83,[75,112784,112685],{},[75,112786,112788],{"class":7383,"line":112787},84,[75,112789,8490],{},[75,112791,112793],{"class":7383,"line":112792},85,[75,112794,112409],{"emptyLinePlaceholder":189},[75,112796,112798],{"class":7383,"line":112797},86,[75,112799,112800],{},"  const track = result.loadType === LoadType.SEARCH || result.loadType === LoadType.PLAYLIST\n",[75,112802,112804],{"class":7383,"line":112803},87,[75,112805,112806],{},"    ? result.data[0]\n",[75,112808,112810],{"class":7383,"line":112809},88,[75,112811,112812],{},"    : result.data;\n",[75,112814,112816],{"class":7383,"line":112815},89,[75,112817,112409],{"emptyLinePlaceholder":189},[75,112819,112821],{"class":7383,"line":112820},90,[75,112822,112823],{},"  let player = shoukaku.players.get(interaction.guildId!);\n",[75,112825,112827],{"class":7383,"line":112826},91,[75,112828,112829],{},"  if (!player) {\n",[75,112831,112833],{"class":7383,"line":112832},92,[75,112834,112835],{},"    player = await shoukaku.joinVoiceChannel({\n",[75,112837,112839],{"class":7383,"line":112838},93,[75,112840,112841],{},"      guildId: interaction.guildId!,\n",[75,112843,112845],{"class":7383,"line":112844},94,[75,112846,112847],{},"      channelId: voiceChannel.id,\n",[75,112849,112851],{"class":7383,"line":112850},95,[75,112852,112853],{},"      shardId: 0,\n",[75,112855,112857],{"class":7383,"line":112856},96,[75,112858,112679],{},[75,112860,112862],{"class":7383,"line":112861},97,[75,112863,8490],{},[75,112865,112867],{"class":7383,"line":112866},98,[75,112868,112409],{"emptyLinePlaceholder":189},[75,112870,112872],{"class":7383,"line":112871},99,[75,112873,112874],{},"  await player.playTrack({ track: { encoded: track.encoded } });\n",[75,112876,112878],{"class":7383,"line":112877},100,[75,112879,112409],{"emptyLinePlaceholder":189},[75,112881,112883],{"class":7383,"line":112882},101,[75,112884,112885],{},"  await interaction.editReply(`Now playing: **${track.info.title}**`);\n",[75,112887,112889],{"class":7383,"line":112888},102,[75,112890,7610],{},[75,112892,112894],{"class":7383,"line":112893},103,[75,112895,112409],{"emptyLinePlaceholder":189},[75,112897,112899],{"class":7383,"line":112898},104,[75,112900,112901],{},"async function handleStop(interaction: ChatInputCommandInteraction) {\n",[75,112903,112905],{"class":7383,"line":112904},105,[75,112906,112907],{},"  const player = shoukaku.players.get(interaction.guildId!);\n",[75,112909,112911],{"class":7383,"line":112910},106,[75,112912,112409],{"emptyLinePlaceholder":189},[75,112914,112916],{"class":7383,"line":112915},107,[75,112917,112829],{},[75,112919,112921],{"class":7383,"line":112920},108,[75,112922,112661],{},[75,112924,112926],{"class":7383,"line":112925},109,[75,112927,112928],{},"      content: \"目前沒有播放音樂。\",\n",[75,112930,112932],{"class":7383,"line":112931},110,[75,112933,112673],{},[75,112935,112937],{"class":7383,"line":112936},111,[75,112938,112679],{},[75,112940,112942],{"class":7383,"line":112941},112,[75,112943,112685],{},[75,112945,112947],{"class":7383,"line":112946},113,[75,112948,8490],{},[75,112950,112952],{"class":7383,"line":112951},114,[75,112953,112409],{"emptyLinePlaceholder":189},[75,112955,112957],{"class":7383,"line":112956},115,[75,112958,112959],{},"  await shoukaku.leaveVoiceChannel(interaction.guildId!);\n",[75,112961,112963],{"class":7383,"line":112962},116,[75,112964,112965],{},"  await interaction.reply(\"停止播放並退出語音頻道。\");\n",[75,112967,112969],{"class":7383,"line":112968},117,[75,112970,7610],{},[75,112972,112974],{"class":7383,"line":112973},118,[75,112975,112409],{"emptyLinePlaceholder":189},[75,112977,112979],{"class":7383,"line":112978},119,[75,112980,112981],{},"client.login(Deno.env.get(\"DISCORD_TOKEN\"));\n",[12,112983,112984],{},"如你所見，複雜的處理音頻 \u002F 連接語音頻道代碼都被外包給了 Shoukaku 和 Lavalink。整個機器人的代碼非常簡潔，只有一百多行。當然，這仍不包含更複雜的其他音樂控制指令。但是，有了這個基本框架，想要繼續擴展控制指令就很方便了。",[12,112986,112987,112988],{},"而關於網頁控制頁面，目前我也還沒有開發完成前端，也不知道具體會長什麼樣。",[5610,112989,112990],{},"不如就把它留到下一篇文章再聊吧（挖坑",[12,112992,112993],{},"感謝你看到這裡！趁著假期我想多用部落格練練手，多寫點文章紀錄學習過程，順便練習一下如何把問題給講解清楚。如果你有任何問題，或者是文中有出錯的地方，都歡迎留言告訴我！",[9332,112995,112996],{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":180,"searchDepth":181,"depth":181,"links":112998},[112999,113000,113001],{"id":112224,"depth":181,"text":112225},{"id":112284,"depth":181,"text":112285},{"id":112314,"depth":181,"text":112315},"2026-02-22","閒著沒事幹當作練習～","\u002Fimages\u002Fuploads\u002F1771752251524-108199050_p0.jpg",{},"\u002Fblog\u002F紀錄一次Discord機器人的開發過程",{"title":112190,"description":113003},{"loc":113006},"blog\u002F紀錄一次Discord機器人的開發過程","sAYvzHOgGtMOQujO0wYQ6p4GLZ9-_MuR9xNTBYOcJ84",{"id":113012,"title":113013,"author":6946,"body":113014,"date":113002,"description":113226,"draft":185,"edited_at":113227,"extension":186,"featured_image":113228,"meta":113229,"navigation":189,"path":113230,"pinned":185,"seo":113231,"sitemap":113232,"stem":113233,"tags":194,"__hash__":113234},"blog\u002Fblog\u002F紀錄一次Discord機器人的開發過程（2）.md","紀錄一次Discord機器人的開發過程（2）",{"type":9,"value":113015,"toc":113216},[113016,113021,113027,113030,113033,113036,113042,113045,113051,113054,113068,113077,113083,113086,113103,113109,113115,113118,113121,113129,113136,113142,113145,113148,113151,113157,113160,113163,113166,113173,113176,113179,113182,113185,113190,113193,113203,113206,113209],[112194,113017,113019],{"id":113018},"就說我很認真在選封面圖片嘛",[5610,113020,113018],{},[6964,113022,113024],{"id":113023},"_4-網頁控制面板",[106,113025,113026],{},"4. 網頁控制面板",[12,113028,113029],{},"上一篇文章結尾我自己挖的坑：「網頁控制面板留到下一篇」，現在該來乖乖填坑了。",[12,113031,113032],{},"我想要做的很簡單：打開網頁，用 Discord 帳號登入，選一個機器人所在的伺服器，然後就可以在網頁上搜尋歌曲、暫停、跳過、查看播放佇列——所有你在 Discord 裡用斜槓指令能做的事，都能在網頁上完成，而且是即時同步的。",[12,113034,113035],{},"既然是自己做著好玩的專案，我索性選了個沒用過的新玩具：SvelteKit 來當前端框架，順便學點新東西。",[7166,113037,113039],{"id":113038},"_41-sveltekit的路由",[106,113040,113041],{},"4.1 SvelteKit 的路由",[12,113043,113044],{},"SvelteKit 的路由系統挺好懂的，基本上就是資料夾長怎樣，網址就長怎樣。舉個例子：",[6972,113046,113049],{"className":113047,"code":113048,"language":6977},[6975],"src\u002Froutes\u002F\n├── +page.svelte                          → \u002F\n├── dashboard\u002F\n│   ├── +page.svelte                      → \u002Fdashboard\n│   └── [guildId]\u002F\n│       └── +page.svelte                  → \u002Fdashboard\u002Fabc123\n└── auth\u002F\n    ├── login\u002F+server.ts                  → \u002Fauth\u002Flogin\n    ├── callback\u002F+server.ts               → \u002Fauth\u002Fcallback\n    └── logout\u002F+server.ts                 → \u002Fauth\u002Flogout\n",[6979,113050,113048],{"__ignoreMap":180},[12,113052,113053],{},"每個路由可以有兩個搭檔文件：",[23,113055,113056,113062],{},[26,113057,113058,113061],{},[6979,113059,113060],{},"+page.server.ts","：在伺服器端執行，負責載入資料（例如查詢資料庫、呼叫 API）。它回傳的資料會自動傳給對應的 .svelte 頁面。",[26,113063,113064,113067],{},[6979,113065,113066],{},"+page.svelte","：在瀏覽器端渲染的 UI 元件，透過 $props() 就能拿到伺服器端載入的資料。",[12,113069,113070,113071,113076],{},"另外還有一種叫 ",[106,113072,113073],{},[6979,113074,113075],{},"+server.ts","，它不負責畫畫面，純粹用來處理後端的 API 請求。我們接下來要講的Discord 登入，就是靠它來搞定的。",[7166,113078,113080],{"id":113079},"_42-discordoauth2-登入",[106,113081,113082],{},"4.2 Discord OAuth2 登入",[12,113084,113085],{},"要讓大家在網頁上控制機器人，第一步總得先知道「你是誰」對吧？這時候就遇到了第一個大難題：標準的 OAuth2 登入流程。聽起來好像很複雜，其實就是三步：",[3603,113087,113088,113094,113100],{},[26,113089,113090,113093],{},[106,113091,113092],{},"點擊「用 Discord 登入」","，網頁把你重新導向到 Discord 的授權頁面，問你：「這個應用程式想看你的大頭貼和伺服器清單，可以嗎？」",[26,113095,113096,113099],{},[106,113097,113098],{},"使用者按下同意","，Discord 就會把你送回我的網站，並且在網址上塞給你一個「一次性授權碼」。",[26,113101,113102],{},"我的網頁後端拿到這個碼，跑去跟 Discord 換真正的「Access Token」，然後查出你的名字和頭像，最後發一個簽名過的 Cookie 給你，這樣就算登入成功啦！",[12,113104,113105,113106],{},"這裡所有的 Token 交換都發生在伺服器端，瀏覽器永遠不會碰到 Access Token。",[5610,113107,113108],{},"文檔看了好久才搞定這個破登入",[7166,113110,113112],{"id":113111},"_43-伺服器選擇器",[106,113113,113114],{},"4.3 伺服器選擇器",[12,113116,113117],{},"登入之後，使用者會看到一個伺服器選擇頁面。但這裡有個問題：我總不能把你加入的所有私人伺服器都列出來吧？我只能顯示你跟機器人同時都在的伺服器。",[12,113119,113120],{},"所以載入這個頁面時，後端會做兩件事：",[3603,113122,113123,113126],{},[26,113124,113125],{},"用你的的 Access Token 去問 Discord：「他加入了哪些伺服器？」",[26,113127,113128],{},"透過 WebSocket 去問機器人：「你現在在哪些伺服器裡？」",[12,113130,113131,113132,113135],{},"然後取兩個清單的",[106,113133,113134],{},"交集","——只顯示使用者和機器人同時都在的伺服器。點擊任意一個伺服器卡片，就會進入該伺服器的音樂控制頁面。",[7166,113137,113139],{"id":113138},"_44-即時音樂控制",[106,113140,113141],{},"4.4 即時音樂控制",[12,113143,113144],{},"這才是整個網頁面板最好玩的部分。",[12,113146,113147],{},"進入控制頁面之後，頁面會直接從瀏覽器跟機器人的 WebSocket 伺服器建立連線，並且「訂閱」這個伺服器的狀態更新。一旦訂閱成功，機器人就會把當前的播放狀態推送過來：正在播什麼歌、佇列裡有什麼、有沒有暫停、播到哪了。",[12,113149,113150],{},"之後，不管狀態怎麼變化——不管是有人在 Discord 裡打了 \u002Fskip，還是有人在網頁上按了暫停，機器人都會把最新的狀態推送給所有訂閱了這個伺服器的瀏覽器。",[7166,113152,113154],{"id":113153},"_45-事件匯流排event-bus",[106,113155,113156],{},"4.5 事件匯流排（Event Bus）",[12,113158,113159],{},"那怎麼讓 Discord 指令和網頁控制互相同步？",[12,113161,113162],{},"舉個例子：我在 Discord 打了 \u002Fpause，網頁上的暫停按鈕也應該要跟著變。反過來，我在網頁上按了跳過，Discord 裡的播放佇列也要跟著更新。",[12,113164,113165],{},"我一開始忘了處理這個同步問題，導致在 Discord 裡暫停音樂後，網頁完全沒反應——因為斜槓指令只是直接操作了播放器，沒有通知任何人「我改了狀態」。",[12,113167,113168,113169,113172],{},"解決方法是加一個簡單的",[106,113170,113171],{},"事件匯流排（Event Bus）","。不管狀態是從哪裡被改變的（Discord 指令、網頁按鈕、歌曲自然播完），都會經過同一個通知機制。WebSocket 伺服器監聽這個事件，然後把最新狀態廣播給所有訂閱中的瀏覽器。",[12,113174,113175],{},"Discord 指令 ──→ 改變佇列狀態 ──→ 發送事件 ──→ 廣播給網頁",[12,113177,113178],{},"網頁控制按鈕 ──→ 改變佇列狀態 ──→ 發送事件 ──→ 廣播給網頁",[12,113180,113181],{},"歌曲自然播完 ──→ 改變佇列狀態 ──→ 發送事件 ──→ 廣播給網頁",[12,113183,113184],{},"所有的路徑最終都匯聚到同一個出口，這就保證了不管變化從哪裡來，所有的客戶端都會同步更新。",[6964,113186,113188],{"id":113187},"結語",[106,113189,113187],{},[12,113191,113192],{},"到這裡，一個帶有網頁控制面板的 Discord 音樂機器人就基本成形了。",[12,113194,113195],{},[32509,113196,113197,113200],{},[1930,113198],{"alt":1932,"src":113199},"https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1771778325963-Screenshot_2026-02-22_at_10.03.25_PM.webp",[32516,113201,113202],{},"給你看一眼這個超醜的介面～",[12,113204,113205],{},"回頭看，其實整個專案最有趣的部分不是寫 UI~~（其實也沒寫多少）~~，而是搞清楚這些不同的系統之間要怎麼溝通：機器人怎麼跟 Lavalink 溝通、網頁怎麼跟機器人溝通、Discord 指令和網頁操作怎麼互相同步。",[12,113207,113208],{},"目前的介面還很陽春，功能上也還有不少可以改進的地方（例如搜尋結果列表、拖曳排序佇列、音量調整等等）。也許之後有興趣了，可以再回來改進一下UI介面設計。",[12,113210,113211,113212,113215],{},"能有耐心看到這裡的話...",[5610,113213,113214],{},"我真的好愛你","！感謝您看到這裡！如果你也在折騰類似的，希望這篇文章能給你一些靈感。有任何問題都歡迎留言！",{"title":180,"searchDepth":181,"depth":181,"links":113217},[113218,113225],{"id":113023,"depth":181,"text":113026,"children":113219},[113220,113221,113222,113223,113224],{"id":113038,"depth":7430,"text":113041},{"id":113079,"depth":7430,"text":113082},{"id":113111,"depth":7430,"text":113114},{"id":113138,"depth":7430,"text":113141},{"id":113153,"depth":7430,"text":113156},{"id":113187,"depth":181,"text":113187},"講講寫網頁控制面板的那些事","2026-02-24","\u002Fimages\u002Fuploads\u002F1771775949773-69618131_p0.jpg",{},"\u002Fblog\u002F紀錄一次Discord機器人的開發過程（2）",{"title":113013,"description":113226},{"loc":113230},"blog\u002F紀錄一次Discord機器人的開發過程（2）","AWkz54IxmZyCRZQGe99kM7V34358eqrPKxOr_mprIT0",{"id":113236,"title":113237,"author":6946,"body":113238,"date":113474,"description":113475,"draft":185,"edited_at":194,"extension":186,"featured_image":113476,"meta":113477,"navigation":189,"path":113478,"pinned":185,"seo":113479,"sitemap":113480,"stem":113481,"tags":194,"__hash__":113482},"blog\u002Fblog\u002F部落格架站指南.md","部落格架站指南",{"type":9,"value":113239,"toc":113463},[113240,113245,113247,113250,113254,113257,113260,113266,113270,113273,113277,113284,113287,113290,113294,113297,113300,113303,113306,113310,113313,113323,113326,113329,113334,113338,113341,113351,113354,113357,113359,113362,113457,113460],[112194,113241,113243],{"id":113242},"配上好看的圖把人騙進來再說",[5610,113244,113242],{},[6964,113246,102197],{"id":102197},[12,113248,113249],{},"這是一篇寫給對架設、設計部落格有興趣的朋友的文章，希望能幫到未來的你。這篇文章會把重點放在「設計」與「框架選擇」的層面，帶你了解市場上的不同方案該如何挑選。如果有說錯的地方，也請多多指教！",[6964,113251,113253],{"id":113252},"部落格是怎麼運作的","部落格是怎麼運作的？",[12,113255,113256],{},"歸根結底，部落格就是一個非常簡單的網站。你完全可以用最基礎的HTML，CSS，JavaScript三件套來手刻自己的部落格。",[12,113258,113259],{},"但現實是，一個完整的部落格不僅僅有文章，還要有橫幅、目錄、頂部導航、留言區等固定的介面格式。寫部落格的時候，人們當然首先要考慮的是文章內容，而不是網站的排版。試想一下：當你終於擠出一點時間，想坐下來好好寫一篇文章時，如果還要分心去搞定網頁排版、文字大小這些瑣事，等一切調整完畢，寫作靈感大概也消失殆盡了。",[12,113261,113262,113263,86843],{},"為了讓創作者能專注於「內容」本身，這時候我們就要請出另外一個非常好用的工具：",[106,113264,113265],{},"部落格框架",[6964,113267,113269],{"id":113268},"部落格框架該怎麼選","部落格框架該怎麼選？",[12,113271,113272],{},"根據用戶需求與技術背景的不同，市場上有幾種主流的選擇：",[7166,113274,113276],{"id":113275},"wordpress傳統動態-cms","WordPress（傳統動態 CMS）",[12,113278,113279,113280,113283],{},"作為世界上最受歡迎的部落格系統，WordPress的優點非常明顯：",[106,113281,113282],{},"開箱即用","。它擁有海量的外掛和佈景主題。你幾乎可以不寫一行程式碼，就把網站架起來。但隨著時間推移，它的缺點也會慢慢浮現。它是一個相對龐大且「笨重」的系統，一旦你對排版有強迫症，比如想要像素級地修改某個區塊的設計，你會發現自己必須鑽進一堆複雜的 PHP 程式碼裡。對於喜歡完全掌控視覺設計的人來說，這有時候會讓人感到相當綁手綁腳。",[12,113285,113286],{},"雖然 WordPress 號稱免費開源，但對於我們這種想白嫖到底的「免費仔」來說，它其實隱藏著不少成本。首先，它需要傳統的伺服器（Hosting）和 MySQL 資料庫來運行，這意味著你每個月都要掏出真金白銀來付主機費。",[12,113288,113289],{},"其次，就是著名的外掛地獄。想要SEO好一點？裝個外掛；想要網站跑快一點？裝個快取外掛；想要防止被駭客攻擊？再裝個資安外掛。這些好用的外掛往往都是採用訂閱制收費的。當你把網站拼湊成你想要的樣子時，不僅網站變得臃腫無比，載入速度變慢，每個月的開銷也可能是一筆不小的數目。這也是為什麼，很多懂一點技術的人，最終都會選擇逃離WordPress。",[7166,113291,113293],{"id":113292},"靜態網站生成器ssg","靜態網站生成器（SSG）",[12,113295,113296],{},"於是，這些受夠了笨重系統的開發者們，轉向了另一個極致：像Hexo、Hugo或Astro這樣的「靜態網站生成器」。",[12,113298,113299],{},"SSG允許你自己設計部落格的外觀。如果覺得自己設計不好，社區也有大量的主題可供使用。它的運作邏輯非常純粹：你只需要在本地端用Markdown寫作，然後在終端機敲幾行指令，框架就會把你的文字編譯成輕量、響應速度快的純HTML網頁。",[12,113301,113302],{},"這樣做的一大好處是，你可以免費部署在很多的網頁託管服務上。大多數的託管服務（如 GitHub Pages、Cloudflare Pages、Vercel 等）針對純靜態網頁都有很寬裕的免費額度。你甚至可以設定自動化腳本（CI\u002FCD），只要推送 Markdown 檔案上去，伺服器就會自動幫你編譯更新。這種網頁速度快、沒有資料庫被駭的風險，省心又省事。",[12,113304,113305],{},"但痛點來了： 每次寫文章都要打開VS Code，沒有那個我們熟悉好用的視覺化後台。如果你今天出門在外，突然想用手機改文章裡的一個錯字；又或者是你想邀請不懂 Markdown 的朋友一起共筆......那就麻煩了。",[7166,113307,113309],{"id":113308},"headless-cms無頭內容管理系統","Headless CMS（無頭內容管理系統）",[12,113311,113312],{},"難道我們不能同時擁有靜態網站的自由與速度加上好用的後台管理介面嗎？這時候，時代的解藥出現了——Headless CMS。",[12,113314,113315,113316,87213,113319,113322],{},"什麼是「無頭」？簡單來說，它把網站的 ",[106,113317,113318],{},"前台視覺設計",[106,113320,113321],{},"後台內容管理"," 徹底切開了。後台只負責儲存你的文章和圖片資料，而網站要長成圓的還是扁的，完全是你自己說了算。Headless CMS會提供三樣東西：內容管理（新增\u002F刪除文章）、文字編輯器，以及內容儲存API。",[12,113324,113325],{},"當然，天下沒有白吃的午餐。如果你想當個免費仔，通常會選擇GitHub作為文章的儲存庫（Git-based CMS）。但 GitHub 作為儲存方案有個致命傷：如果兩個人同時編輯同一篇文章，缺乏即時同步機制的 Git 就會發生衝突（Git conflict），不僅可能導致心血白費，網站管理員處理起來更是頭痛。為了解決這個痛點，許多好用的 CMS 都會改用傳統的雲端資料庫來支援多人即時協作——當然，這也意味著它們必須開始向你收費了。",[12,113327,113328],{},"另一方面，完全免費的開源 CMS（如 Decap CMS、Sveltia CMS），其內建的編輯器又著實不太好用。CMS 為了兼顧所有用戶，通常會做出許多妥協：塞滿你不需要的功能，但你想客製化的特殊功能它又做不到。",[7351,113330,113331],{},[12,113332,113333],{},"私心插一嘴：SveltiaCMS其實是個很優秀的開源專案，介面現代且完全免費。但致命傷在於，它不支援客製化的實時預覽（WYSIWYG）。如果我想在編輯器裡直接預覽LaTeX數學公式或是特殊的自定義組件，它完全做不到。這對我來說是個硬傷。或許未來它更新後會是個完美的解決方案，但現在，我只能繼續尋找別的出路。",[7166,113335,113337],{"id":113336},"全部客製化web框架-編輯器","全部客製化：Web框架 + 編輯器",[12,113339,113340],{},"前面的那麼多選擇，對於我這個免費仔來說都有各種各樣的缺點~~（要錢！）~~。在經過了很長一段時間的開發後，我終於受不了了，決定自己從頭做一個後台。",[12,113342,113343,113344,87213,113347,113350],{},"我此前最大的抱怨就是編輯器實在是不好用，但很幸運的是，開源界有很多超好用的神級編輯器框架，例如 ",[106,113345,113346],{},"TipTap",[106,113348,113349],{},"Milkdown","。 他們最大的好處在，只提供插件。你可以根據自己的需求，拿著這些插件隨心所欲的組裝自己想要的編輯器。想要什麼客製化的功能？想要適配特定的格式？沒問題，自己動手就好了！",[12,113352,113353],{},"一般很少會有人走到像我這一步：又要多人協作，又要GitHub儲存，又要有預覽功能的編輯器，還不能收費。我早期開發部落格，在上網研究的時候，完全沒見到有這種組合方法，只能想盡辦法繞過各種CMS的限制。直到偶然發現可以單獨把Milkdown這類編輯器抽出來，結合Web框架自己寫介面，才讓我重新把擱置了幾個月的部落格拿出來大改。",[12,113355,113356],{},"雖然這條路一開始最難走，但看著一個完全免費、有著好用實時預覽，且100%自己設計的部落格後台在手中誕生，那種成就感，絕對是任何現成框架都給不了的。",[6964,113358,98493],{"id":98493},[12,113360,113361],{},"如果看到這裡，你還是不太確定應該怎麼選，可以參考以下表格：",[7085,113363,113364,113388],{},[7088,113365,113366],{},[7091,113367,113368,113373,113378,113383],{},[7094,113369,113370],{"align":47883},[106,113371,113372],{},"方案",[7094,113374,113375],{"align":47883},[106,113376,113377],{},"最大優點",[7094,113379,113380],{"align":47883},[106,113381,113382],{},"致命痛點",[7094,113384,113385],{"align":47883},[106,113386,113387],{},"適合對象",[7104,113389,113390,113406,113422,113438],{},[7091,113391,113392,113397,113400,113403],{},[7109,113393,113394],{"align":47883},[106,113395,113396],{},"WordPress",[7109,113398,113399],{"align":47883},"開箱即用、生態系龐大",[7109,113401,113402],{"align":47883},"笨重、主機成本、外掛地獄",[7109,113404,113405],{"align":47883},"想快速建站、無技術背景者",[7091,113407,113408,113413,113416,113419],{},[7109,113409,113410],{"align":47883},[106,113411,113412],{},"SSG 靜態生成",[7109,113414,113415],{"align":47883},"免費託管、載入快",[7109,113417,113418],{"align":47883},"缺乏視覺化後台、需依賴終端機",[7109,113420,113421],{"align":47883},"習慣本地Markdown寫作的開發者",[7091,113423,113424,113429,113432,113435],{},[7109,113425,113426],{"align":47883},[106,113427,113428],{},"Headless CMS",[7109,113430,113431],{"align":47883},"前後端分離、自訂性高",[7109,113433,113434],{"align":47883},"多人協作與複雜功能多需付費",[7109,113436,113437],{"align":47883},"需要管理介面且有預算的團隊",[7091,113439,113440,113445,113448,113451],{},[7109,113441,113442],{"align":47883},[106,113443,113444],{},"自製後台",[7109,113446,113447],{"align":47883},"完全免費、100%客製化",[7109,113449,113450],{"align":47883},"開發成本高、需從零打造",[7109,113452,113453,113454,2589],{"align":47883},"追求自由的開發者（",[5610,113455,113456],{},"或者像我一樣的免費仔",[12,113458,113459],{},"希望這篇文章，能為正在猶豫如何架設部落格的你帶來一些收穫！也許之後我會再開一個新的文章，聊聊設計心得、或是開發教程之類的東西。但是嘛...我也還不清楚應該要從哪裡開始說就是了。",[12,113461,113462],{},"非常感謝你願意花時間看到這裡！",{"title":180,"searchDepth":181,"depth":181,"links":113464},[113465,113466,113467,113473],{"id":102197,"depth":181,"text":102197},{"id":113252,"depth":181,"text":113253},{"id":113268,"depth":181,"text":113269,"children":113468},[113469,113470,113471,113472],{"id":113275,"depth":7430,"text":113276},{"id":113292,"depth":7430,"text":113293},{"id":113308,"depth":7430,"text":113309},{"id":113336,"depth":7430,"text":113337},{"id":98493,"depth":181,"text":98493},"2026-02-21","這麼多框架應該怎麼選？","\u002Fimages\u002Fuploads\u002F1771647859909-99136266_p9.png",{},"\u002Fblog\u002F部落格架站指南",{"title":113237,"description":113475},{"loc":113478},"blog\u002F部落格架站指南","dPWxsOx0V6K72-XMDVAGDgy92KHHWuiXj86QL6JgKdQ",{"id":113484,"title":113485,"author":113486,"body":113487,"date":113528,"description":113529,"draft":185,"edited_at":194,"extension":186,"featured_image":113530,"meta":113531,"navigation":189,"path":113532,"pinned":185,"seo":113533,"sitemap":113534,"stem":113535,"tags":113536,"__hash__":113537},"blog\u002Fblog\u002F北冰洋漂流記.md","北冰洋漂流記","灰海獅",{"type":9,"value":113488,"toc":113526},[113489,113499,113502,113505,113509,113519],[12,113490,113491,113494,113495,113498],{},[5610,113492,113493],{},"在經歷了一段時間的拖稿後決定寫出這篇了","，首先呢，做個簡單的自我介紹，這裡是灰海獅，正在北冰洋漂流中，也有人會叫我寒潮或",[5610,113496,113497],{},"蒙蒙","，總之呢，情況就是這麼個情況。",[12,113500,113501],{},"本人是動漫宅，同時也涉獵一些雜七雜八的東西，包括但不限於:布袋戲、歷史、宗教等等的。",[12,113503,113504],{},"平時閒來無事也會寫小說，把我腦中那些天馬行空的想法寫出來，雖然常常因為懶得動而無限期擱置(至於文筆稀爛又是另一回事了)，但我還是很享受寫小說的時光的。",[113506,113507,113508],"h1",{"id":113508},"名字的由來",[12,113510,113511,113512,113515,113516],{},"好吧，名字好像沒有甚麼特別的由來XD，會有這名字是因為我小時候有隻很喜歡的布偶，就是隻灰色的海豹，沒錯，海豹，",[5610,113513,113514],{},"只是我當初不知道它是海豹，還以為是海獅","，於是這名字就一路用到現在了，寒潮的話，是我在自己小說裡給自己的名字，而蒙蒙...",[5610,113517,113518],{},"這名字的由來改天再分享。",[12,113520,113521,113522,113525],{},"大致就是這樣了，謝謝願意在這忙碌的世界留下一點時間看這篇文章的你，",[5610,113523,113524],{},"也許改天我會開始分享我寫的小說(?","，總之就是這樣~ 我要繼續回去漂流了(x",{"title":180,"searchDepth":181,"depth":181,"links":113527},[],"2026-02-06","淺淺的自我介紹","\u002Fimages\u002Fuploads\u002Fimg_4487.jpg",{},"\u002Fblog\u002F北冰洋漂流記",{"title":113485,"description":113529},{"loc":113532},"blog\u002F北冰洋漂流記",[],"cCGozAjcdbK3IHCBwmBULBDvwiaBXp1LNVtACLDHdy8",{"id":113539,"title":113540,"author":113541,"body":113542,"date":117118,"description":117119,"draft":185,"edited_at":113474,"extension":186,"featured_image":117120,"meta":117121,"navigation":189,"path":117122,"pinned":185,"seo":117123,"sitemap":117124,"stem":117125,"tags":117126,"__hash__":117132},"blog\u002Fblog\u002F清華大學電機系-大一雜魚修課心得-總結.md","清華大學電機系 - 大一雜魚修課心得 & 總結","mahiro",{"type":9,"value":113543,"toc":117099},[113544,113549,113551,113554,113557,113560,113562,113567,113571,113575,113578,113581,113584,113587,113590,113780,113783,114104,114107,114110,114143,114154,114158,114161,114166,114174,114177,114180,114184,114216,114219,114222,114225,114246,114250,114254,114257,114263,114267,114270,114273,114277,114280,114283,114286,114289,114293,114296,114300,114304,114308,114312,114315,114317,114320,114348,114386,114390,114393,114396,114400,114403,114406,114410,114414,114421,114425,114429,114432,114435,114437,114443,114446,114455,114459,114490,114494,114497,114500,114503,114506,114510,114513,114516,114519,114539,114542,114545,114590,114594,114598,114601,114605,114608,114611,114615,114618,114622,114625,114629,114634,114638,114647,114650,114656,114658,114661,114690,114742,114744,114747,114750,114754,114757,114760,114762,114765,114768,114771,114773,114776,114779,114781,114788,114796,114799,114802,114805,114808,114811,114851,114854,114924,114928,114931,114935,114939,114942,114945,114948,114951,114955,114958,114961,114965,114969,114972,114975,114977,114980,115014,115075,115078,115085,115096,115101,115104,115109,115114,115117,115120,115125,115134,115143,115146,115151,115154,115160,115163,115199,115203,115206,115225,115228,115232,115235,115239,115242,115246,115249,115253,115257,115268,115274,115278,115282,115285,115292,115297,115299,115302,115329,115333,115336,115339,115402,115405,115419,115422,115433,115441,115449,115452,115455,115458,115460,115466,115469,115490,115493,115496,115499,115502,115506,115509,115513,115516,115520,115523,115526,115530,115533,115536,115540,115544,115547,115549,115552,115588,115609,115613,115616,115619,115622,115625,115628,115631,115634,115637,115643,115647,115689,115693,115745,115748,115751,115754,115757,115762,115766,115769,115774,115778,115781,115808,115811,115814,115844,115848,115851,115886,115918,115921,115929,115933,115937,115940,115943,115946,115949,115952,115955,115958,115961,115964,115968,115972,115975,115977,115980,116010,116015,116025,116028,116031,116034,116037,116040,116043,116046,116049,116052,116055,116061,116064,116067,116093,116096,116099,116102,116105,116108,116112,116115,116119,116122,116125,116128,116131,116134,116138,116141,116144,116148,116151,116154,116158,116161,116164,116167,116170,116173,116176,116179,116182,116191,116193,116196,116222,116244,116247,116250,116253,116256,116259,116262,116265,116268,116271,116274,116281,116285,116288,116323,116326,116353,116357,116360,116395,116398,116401,116411,116414,116428,116431,116434,116438,116465,116468,116471,116475,116479,116482,116485,116488,116491,116494,116497,116501,116505,116508,116512,116515,116518,116521,116523,116526,116551,116554,116557,116559,116562,116565,116567,116570,116573,116576,116578,116581,116584,116587,116589,116592,116595,116598,116601,116604,116607,116613,116616,116620,116623,116628,116631,116656,116659,116673,116676,116690,116693,116707,116710,116713,116715,116718,116721,116724,116727,116730,116733,116736,116739,116743,116747,116751,116754,116759,116762,116765,116767,116770,116794,116798,116801,116805,116811,116815,116821,116825,116829,116835,116838,116844,116848,116890,116894,116898,116901,116904,116907,116910,116914,116918,116923,116925,116928,116950,116953,116956,116959,116962,116965,116968,116974,116978,116981,117028,117030,117033,117036,117040,117043,117046,117049,117053,117056,117063,117067,117070,117073,117076,117079,117082,117085,117088,117091,117094,117097],[113545,113546,113548],"h5",{"id":113547},"歌愛雪鎮樓求你保佑我這樣的小雜魚","歌愛雪鎮樓~求你保佑我這樣的小雜魚。",[6964,113550,102197],{"id":102197},[12,113552,113553],{},"不管是什麼風、還是演算法的小巧思帶你來到這邊。總之，很感謝你點開這篇文章。",[12,113555,113556],{},"這裡是撐了一個學期後的我，在寒假利用空餘時間寫的、本學期的選課紀錄與心得。",[12,113558,113559],{},"如果你正苦苦尋求著什麼關鍵字、或哪堂課的評價，希望這篇能幫到你~",[36,113561],{},[12,113563,113566],{"className":113564},[113565],"indent","高中在鄉下小學校廢了三年後，到大學馬上迎來的是一週接一週、不間斷的月考、各種報告及考試，以及一次比一次差的段考成績。尤其是我又很不知死活地參考卷哥心得選了25學分，搞得這半年十分甚至九分地累，甚至還遠超我準備學測的那半年，真的不知道我怎麼活下來的…\n",[12,113568,113570],{"className":113569},[113565],"總之現在撐過來、撐到寒假了，考慮到過去受前人修課心得獲益良多，但電機系這邊的選課心得真的特別少，因此決定也來試試寫一份自己的修課心得吧。除了方便未來回憶我這學期修25學分有多累人之外，也能作為未來學生的選課參考~雖然可能沒人會看到就是。\n",[12,113572,113574],{"className":113573},[113565],"本來打算在二選之前打完的，沒想到一拖就拖到了三選之後.....拖延症真糟糕啊我。\n",[12,113576,113577],{},"另外，本學期選課選得有億點多，甚至還不知死活地選了第二外語，導致代價就是累成狗，成績單也不好看。所以千萬建議學弟妹選課須謹慎…當然你是卷哥卷姊的話就另當別論了。",[6964,113579,113580],{"id":113580},"選課列表",[12,113582,113583],{},"基本上，大一課表上是以必修課為主，沒有什麼加選修課的空間",[12,113585,113586],{},"通識的部分因為一選二選大多都被搶完了，輪到新生選課時名額也沒剩多少，因此加退選通常才是主要戰場。大一上的體育課則不能選，而由班級決定。",[12,113588,113589],{},"(可點擊藍色字體，跳轉到該章節)",[7085,113591,113592,113608],{},[7088,113593,113594],{},[7091,113595,113596,113599,113602,113605],{},[7094,113597,113598],{},"學分類型",[7094,113600,113601],{},"科目名稱",[7094,113603,113604],{},"授課教師",[7094,113606,113607],{},"學分",[7104,113609,113610,113626,113641,113656,113671,113686,113702,113718,113734,113749,113765],{},[7091,113611,113612,113615,113621,113624],{},[7109,113613,113614],{},"EE必修（大一上）",[7109,113616,113617],{},[2330,1136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231001",[7109,113820,113821,113823],{},[2330,113822,113665],{"href":113664},"PHYS113304",[7109,113825,113826],{},[16,113827],{},[7109,113829,113830,113818],{},[2330,113831,113620],{"href":113619},[7109,113833,113834,113823],{},[2330,113835,113665],{"href":113664},[7091,113837,113838,113840,113844,113848,113852,113856],{},[7109,113839,8879],{},[7109,113841,113842,113818],{},[2330,113843,113620],{"href":113619},[7109,113845,113846,113823],{},[2330,113847,113665],{"href":113664},[7109,113849,113850],{},[16,113851],{},[7109,113853,113854,113818],{},[2330,113855,113620],{"href":113619},[7109,113857,113858,113823],{},[2330,113859,113665],{"href":113664},[7091,113861,113862,113864,113868,113873,113878,113882],{},[7109,113863,8830],{},[7109,113865,113866],{},[16,113867],{},[7109,113869,113870,113872],{},[2330,113871,113650],{"href":113649},"MATH101007",[7109,113874,113875,113877],{},[2330,113876,113696],{"href":113695},"LANG103025",[7109,113879,113880,113872],{},[2330,113881,113650],{"href":113649},[7109,113883,113884,113886],{},[2330,113885,113743],{"href":113742},"GE 144300",[7091,113888,113889,113891,113895,113899,113903,113907],{},[7109,113890,8348],{},[7109,113892,113893],{},[16,113894],{},[7109,113896,113897,113872],{},[2330,113898,113650],{"href":113649},[7109,113900,113901,113877],{},[2330,113902,113696],{"href":113695},[7109,113904,113905,113872],{},[2330,113906,113650],{"href":113649},[7109,113908,113909,113886],{},[2330,113910,113743],{"href":113742},[7091,113912,113913,113915,113920,113925,113930,113934],{},[7109,113914,15531],{},[7109,113916,113917,113919],{},[2330,113918,113680],{"href":113679},"PHYS101001",[7109,113921,113922,113924],{},[2330,113923,113635],{"href":113634},"EECS101002",[7109,113926,113927,113929],{},[2330,113928,113728],{"href":113727},"GE 158400",[7109,113931,113932,113924],{},[2330,113933,113635],{"href":113634},[7109,113935,113936,113938],{},[2330,113937,113759],{"href":113758},"CL 101051",[7091,113940,113941,113943,113947,113951,113955,113959],{},[7109,113942,86109],{},[7109,113944,113945,113919],{},[2330,113946,113680],{"href":113679},[7109,113948,113949,113924],{},[2330,113950,113635],{"href":113634},[7109,113952,113953,113929],{},[2330,113954,113728],{"href":113727},[7109,113956,113957,113924],{},[2330,113958,113635],{"href":113634},[7109,113960,113961,113938],{},[2330,113962,113759],{"href":113758},[7091,113964,113965,113967,113971,113975,113979,113984],{},[7109,113966,80330],{},[7109,113968,113969,113919],{},[2330,113970,113680],{"href":113679},[7109,113972,113973],{},[16,113974],{},[7109,113976,113977],{},[16,113978],{},[7109,113980,113981,113983],{},[2330,113982,113712],{"href":113711},"FL 204107",[7109,113985,113986,113988],{},[2330,113987,113774],{"href":113773},"PE 111039",[7091,113990,113991,113993,113997,114001,114005,114009],{},[7109,113992,31184],{},[7109,113994,113995,113919],{},[2330,113996,113680],{"href":113679},[7109,113998,113999],{},[16,114000],{},[7109,114002,114003],{},[16,114004],{},[7109,114006,114007,113983],{},[2330,114008,113712],{"href":113711},[7109,114010,114011,113988],{},[2330,114012,113774],{"href":113773},[7091,114014,114015,114017,114021,114025,114029,114033],{},[7109,114016,21841],{},[7109,114018,114019],{},[16,114020],{},[7109,114022,114023],{},[16,114024],{},[7109,114026,114027],{},[16,114028],{},[7109,114030,114031,113983],{},[2330,114032,113712],{"href":113711},[7109,114034,114035],{},[16,114036],{},[7091,114038,114039,114041,114044,114048,114051,114055],{},[7109,114040,2330],{},[7109,114042,114043],{},"邏輯設計（TA hour）",[7109,114045,114046],{},[16,114047],{},[7109,114049,114050],{},"微積分Ａ一（演習課）",[7109,114052,114053],{},[16,114054],{},[7109,114056,114057],{},[16,114058],{},[7091,114060,114061,114064,114066,114070,114072,114076],{},[7109,114062,114063],{},"b",[7109,114065,114043],{},[7109,114067,114068],{},[16,114069],{},[7109,114071,114050],{},[7109,114073,114074],{},[16,114075],{},[7109,114077,114078],{},[16,114079],{},[7091,114081,114082,114084,114088,114092,114096,114100],{},[7109,114083,8682],{},[7109,114085,114086],{},[16,114087],{},[7109,114089,114090],{},[16,114091],{},[7109,114093,114094],{},[16,114095],{},[7109,114097,114098],{},[16,114099],{},[7109,114101,114102],{},[16,114103],{},[6964,114105,114106],{"id":114106},"修課心得",[7166,114108,114109],{"id":113620},"計算機程式設計 Introduction to Programming",[13438,114111,114113,114122,114125,114128,114131,114134,114140],{"className":114112},[13441,13442],[12,114114,114115,114118,114119],{},[106,114116,114117],{},"科號："," 11410EE 231001\n開課教師：張彌彰  \u002F\u002F ",[5610,114120,114121],{},"名字可以倒過來唸",[12,114123,114124],{},"課程學分：3",[12,114126,114127],{},"課程類型： 電機系必修（大一上）",[12,114129,114130],{},"上課時間：M1M2 R1R2 \u002F\u002F 大一的第一堂課…",[12,114132,114133],{},"修課人數：76 -> 64 \u002F20(新生20) 容量 139  ",[12,114135,114136,114137],{},"上課教材：C Programming: A Modern Approach\n",[1930,114138],{"alt":180,"src":114139},"\u002Fimages\u002Fuploads\u002F%E5%9C%96%E7%89%871.jpg",[12,114141,114142],{},"需要原文書嗎？ 上機考可以翻，所以建議買，不過我期中之後就沒再翻過就是。",[13438,114144,114146,114151],{"className":114145},[13441,53830],[12,114147,114148],{},[106,114149,114150],{},"範圍：",[12,114152,114153],{},"C語言，從 Ch 1 ~ Ch 20：Input\u002Foutput; Variables and data types; expressions & operators;\nConditional (if-else\u002Fswitch-case); Iteration(for\u002Fwhile); Arrays; Functions;\nPointers; String; Bitwise, Link list…等",[26236,114155,114157],{"id":114156},"心得待補充","心得：(待補充)",[12,114159,114160],{},"我覺得這堂課完全值得我另外開一篇文介紹。",[12,114162,114163],{},[1930,114164],{"alt":1932,"src":114165},"https:\u002F\u002Ftruth.bahamut.com.tw\u002Fs01\u002F202601\u002Fc3fe429536e05b78d51e2c4447eaee42.PNG",[26236,114167,114169,114170],{"id":114168},"點名方式-每堂點名","點名方式： ",[75,114171,114173],{"className":114172},[29982],"每堂點名",[12,114175,114176],{},"上課時講到一半會隨機點人回答問題，發現有人沒到就會全班點名。就算沒有也可能會點。",[12,114178,114179],{},"每缺席一次-1分，事前須請假寄信，事後只接受病假，且須附上相關證明。",[26236,114181,114183],{"id":114182},"成績算法","成績算法：",[7085,114185,114186,114202],{},[7088,114187,114188],{},[7091,114189,114190,114193,114196,114199],{},[7094,114191,114192],{},"作業（Lab）（每次4% 共12次）",[7094,114194,114195],{},"Quiz （每次4% 共3次）",[7094,114197,114198],{},"Midterm（上機10% + 筆試10%）",[7094,114200,114201],{},"Final（上機10% + 筆試10%）",[7104,114203,114204],{},[7091,114205,114206,114209,114212,114214],{},[7109,114207,114208],{},"48%",[7109,114210,114211],{},"12%",[7109,114213,78936],{},[7109,114215,78936],{},[12,114217,114218],{},"小考(Quiz 1 ~ Quiz 3)和段考的滿分皆為110分。",[12,114220,114221],{},"期末 20% (皆為上機10% + 筆試10%)",[12,114223,114224],{},"另外，今年有額外的加分小考。",[7085,114226,114227,114237],{},[7088,114228,114229],{},[7091,114230,114231,114234],{},[7094,114232,114233],{},"Quiz 4（極難，滿分100）",[7094,114235,114236],{},"Survey（期末問卷）",[7104,114238,114239],{},[7091,114240,114241,114244],{},[7109,114242,114243],{},"+4",[7109,114245,114243],{},[26236,114247,114249],{"id":114248},"調分嗎不調別想了孩子","調分嗎？不調，別想了孩子。",[26236,114251,114253],{"id":114252},"給加簽嗎部分加簽","給加簽嗎？部分加簽：",[12,114255,114256],{},"限電機\u002F電資院大一學生，學期前寄信加簽",[12,114258,114259,114260],{},"沒上就只能去隔壁班",[5610,114261,114262],{},"逃過一劫",[26236,114264,114266],{"id":114265},"涼度","涼度：★★☆☆☆",[12,114268,114269],{},"如果0基礎來上的話，雖然應該能聽到會，但肯定得更吃力些",[12,114271,114272],{},"此外後期作業難度相當大也極度耗時，debug容易會瘋掉",[26236,114274,114276],{"id":114275},"甜度","甜度：★☆☆☆☆",[12,114278,114279],{},"隔壁班62人裡面20個A+，將近30%，",[12,114281,114282],{},"這邊我期末上機全對被手改後的分數跟期中錯一題一樣低",[12,114284,114285],{},"我真的是跟我大學生活過不去才會選這班",[12,114287,114288],{},"推薦度：",[26236,114290,114292],{"id":114291},"想學好程式尤其c","想學好程式(尤其C) ★★★★⯪ ",[12,114294,114295],{},"(上課很扎實，不過比較趕的關係，有些像file read\u002Fwrite的部分就沒講)",[26236,114297,114299],{"id":114298},"有考古或學長姐","有考古或學長姐：★★★★★",[26236,114301,114303],{"id":114302},"看重gpa","看重GPA：★☆☆☆☆",[26236,114305,114307],{"id":114306},"總結如果時間倒轉我想重上一次因為我作業分數被扣爛幹","總結：如果時間倒轉，我想重上一次！因為我作業分數被扣爛幹",[26236,114309,114311],{"id":114310},"成績分布-待補","成績分布： (待補)",[12,114313,114314],{},"μ = 77.14 σ  = 10.84",[36,114316],{},[7166,114318,114319],{"id":113635},"邏輯設計 Logic Design",[13438,114321,114323,114328,114331,114333,114336,114339,114342,114345],{"className":114322},[13441,13442],[12,114324,114325],{},[106,114326,114327],{},"科號：11410EECS101002\t",[12,114329,114330],{},"  \u003Cp>\n開課教師：馬席彬",[12,114332,114124],{},[12,114334,114335],{},"課程類型：電機系必修（大一上）",[12,114337,114338],{},"上課時間：T5T6 R5R6",[12,114340,114341],{},"修課人數：177 -> 167 \u002F20 (新生20) 容量170 (旺宏懷樸)",[12,114343,114344],{},"上課教材：自編講義",[12,114346,114347],{},"需要原文書嗎？ 完全不用",[13438,114349,114351,114355,114358],{"className":114350},[13441,53830],[12,114352,114353],{},[106,114354,114150],{},[12,114356,114357],{},"  \u003Cp>",[3603,114359,114360,114363,114366,114369,114372,114378,114381],{},[26,114361,114362],{},"數位系統與2\u002F8\u002F16進位",[26,114364,114365],{},"Boolean Algebra, Boolean Function, Logic Gates(AND OR)",[26,114367,114368],{},"Simplify Boolean Function (用K-Map)",[26,114370,114371],{},"Combinational Logic ----一段範圍(~乘法器)----",[26,114373,114374,114375,114377],{},"Synchronous Sequential Logic (有Feedback)",[16,114376],{},"----二段範圍----",[26,114379,114380],{},"Registers & Counter",[26,114382,114383,114384],{},"Memory & Programmable Logic",[16,114385],{},[26236,114387,114389],{"id":114388},"心得","心得：",[12,114391,114392],{},"前排提醒：據說跟歷屆相比，今年題目特別簡單又過於甜。",[12,114394,114395],{},"由於第一堂課隔壁班英文授課聽不太懂，長跑到懷樸後聽課有種豁然開朗的感覺，於是果斷加簽這班。",[12,114397,114399],{"className":114398},[113565],"這堂課不點名，自從113年改16週開始也沒作業了，不過eeclass仍會附上前四屆的作業，不計分，但小考題型幾乎都來自這些作業的題目(有時甚至數值都一樣)。\n",[12,114401,114402],{},"而教授在課後會把每一堂課錄影放到eeclass，因此理論上你也可以考試到場就好，其他時候看線上影片+講義自己讀或補眠。",[12,114404,114405],{},"而教室也的確只有在考試時才會坐滿人，有一次下大雨時教室甚至空了一半以上。",[12,114407,114409],{"className":114408},[113565],"不過因為課綱差異，有些作業題目換到這屆並不會考(或教授會在課上強調不考)，這點可以問教授或TA hour問助教確認。\n",[12,114411,114413],{"className":114412},[113565],"上課方式是照著自己編的講義教(所以不用原文書)，從介紹數位與進位系統、Boolean代數等，一路組合、進階到後面回授邏輯等，每一章節都是前一章的延伸。\n",[12,114415,114417,114418,114420],{"className":114416},[113565],"1~5章的前半段通常是廣泛的入門介紹，主要考點則大部分在後半段。\n教授課上會把各個概念介紹得很詳細，如果都聽懂的話作業題目應該是都沒問題的(不過有些題目倒是要看一遍解答才會)。\n",[16,114419],{},"\n ",[12,114422,114424],{"className":114423},[113565],"值得注意的是：從第五章涉及feedback開始，總體課程難度會陡升，而Quiz 5也似乎是唯一沒人提前交卷的小考(不過反而不是平均最低分的一次)。\n",[12,114426,114428],{"className":114427},[113565],"儘管如此，如果對上課內容、作業有疑問或不清楚的話，可以下課後去問教授，也可以每週一晚上7:00~9:00的TA hour去問助教(但我通常只有考試那個禮拜才去啦)。\n",[12,114430,114431],{},"這堂課最大的難點大概就是它排在中午後一節，剛吃完飯血糖上升之外，因為懷樸不通風、冷氣又強，教授聲音還十分有磁性很催眠，因此很容易就恍神…然後一回神就突然跟不上進度了。",[12,114433,114434],{},"這點教授本人也有提過他上課聲音會讓人想睡就是。好在恍神的話還有上課影片可以補進度。",[16,114436],{},[12,114438,114439,114440],{},"課堂上不能發出聲響、聊天及睡覺，教授很看重這些。\n",[5610,114441,114442],{},"不過我還是看過有在課堂上開筆電看棒球的。",[12,114444,114445],{},"這堂課的內容將會在下學期的邏輯設計實驗中付諸實作，屆時再補充吧(前提是我有選到)。",[26236,114447,114449,114450],{"id":114448},"點名方式不點名","點名方式：",[75,114451,114454],{"className":114452},[114453],"green","不點名",[26236,114456,114458],{"id":114457},"成績算法考多少拿多少","成績算法：考多少拿多少",[7085,114460,114461,114477],{},[7088,114462,114463],{},[7091,114464,114465,114468,114471,114474],{},[7094,114466,114467],{},"Quiz （7次）",[7094,114469,114470],{},"Midterm 1（1~4章）",[7094,114472,114473],{},"Midterm 2（4~5章）",[7094,114475,114476],{},"Final（6~7章）",[7104,114478,114479],{},[7091,114480,114481,114484,114486,114488],{},[7109,114482,114483],{},"25%",[7109,114485,114483],{},[7109,114487,114483],{},[7109,114489,114483],{},[26236,114491,114493],{"id":114492},"小考形式","小考形式：",[12,114495,114496],{},"預先公布在eeclass，範圍是前一個章節。",[12,114498,114499],{},"一般排在章節教完後的下堂第二節課小考，如果進度落後則會順延。",[12,114501,114502],{},"由於大多都是作業題型(甚至可能數值都沒換)，講義+4屆作業做完就差不多了。",[12,114504,114505],{},"分數就是很暴力的七次取平均。七次Quiz平均如圖",[26236,114507,114509],{"id":114508},"段考形式","段考形式：",[12,114511,114512],{},"相比起小考的作業題型，會多一些活用題，部分題目的解題思路也會比較刁鑽一些，但總體還是上課時的概念。",[12,114514,114515],{},"由於各章節是層層堆疊，以及有時段考出前面單元的題目(如期末直接出)，因此要確保前面的單元能讀懂。",[12,114517,114518],{},"小考與段考的時間基本上都偏充裕，最多壓線完成，不會寫不完。",[23,114520,114521,114524,114527,114530,114533,114536],{},[26,114522,114523],{},"段考似乎跟劉班是共同出題？",[26,114525,114526],{},"三次平均80.67\u002F91.02\u002F83.30",[26,114528,114529],{},"標準差17.16\u002F12.74\u002F18.25",[26,114531,114532],{},"Midterm 2滿分105，其餘滿分100",[26,114534,114535],{},"一段原本平均好像67，題目爭議的關係有重改",[26,114537,114538],{},"Quiz考卷可保留，但期中\u002F期末考卷會收回",[12,114540,114541],{},"值得注意的是，往年期中平均才約50，甚至有一半二退的紀錄...(見補充)",[12,114543,114544],{},"但抱怨一下，馬班助教改考卷都特別慢，一段考卷在二段考前才出來、而隔壁班二段一週後就出來。然後Quiz 5期末當週才出來，而二段、Quiz 6、Quiz 7 跟Final甚至都到期末後一個禮拜才改好...\n不過助教人很好，改考卷也不刁難人，所以...好啦，可以原諒。",[12,114546,114547,114548,114589],{},"另外建議not A寫成",[24923,114549,114551],{"className":114550,"jax":24927},[24926],[24929,114552,114555,114564],{"style":27802,"xmlns":24932,"width":56642,"height":114553,"role":1930,"focusable":8058,"viewBox":114554,"xmlnsXLink":24936},"2.337ex","0 -1033 750 1033",[24938,114556,114557,114560],{},[24941,114558],{"id":114559,"d":41102},"MJX-1-TEX-I-1D434",[24941,114561],{"id":114562,"d":114563},"MJX-1-TEX-S4-2013","M0 248V285H499V248H0Z",[25022,114565,114566],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,114567,114568],{"dataMmlNode":25028},[25022,114569,114570,114575],{"dataMmlNode":47235},[25022,114571,114572],{"dataMmlNode":25942},[25033,114573],{"dataC":41149,"xLinkHref":114574},"#MJX-1-TEX-I-1D434",[25022,114576,114578],{"dataMmlNode":25031,"transform":114577},"translate(0,648)",[24929,114579,114584],{"width":114580,"height":114581,"x":9180,"y":114582,"viewBox":114583},750,237,"148","187.5 148 750 237",[25033,114585],{"dataC":114586,"xLinkHref":114587,"transform":114588},"2013","#MJX-1-TEX-S4-2013","scale(2.25,1)","  ( A上一橫線)而不是A’，避免計算時漏看(血的教訓…)",[26236,114591,114593],{"id":114592},"調分嗎a太多就沒調","調分嗎？A+太多就沒調",[26236,114595,114597],{"id":114596},"給加簽嗎全簽","給加簽嗎？全簽：",[12,114599,114600],{},"第一堂到課加簽，電機\u002F電資院優先。不過最後有座位就給簽就是",[26236,114602,114604],{"id":114603},"涼度-1","涼度：★★★★⯪",[12,114606,114607],{},"懷樸冷氣每次都開超強",[12,114609,114610],{},"沒作業，成績完全取決於小考。不過小考題基本上都跟歷屆作業差不多就是",[26236,114612,114614],{"id":114613},"甜度-1","甜度：★★★★★",[12,114616,114617],{},"53.9% A+，平均85.9，邏設開通識課了？\n聽說歷屆平均沒那麼高的說...",[26236,114619,114621],{"id":114620},"推薦度推-見補充","推薦度：推？   見補充",[12,114623,114624],{},"總結：感謝教授這屆大發慈悲、感謝今年助教大發慈悲",[26236,114626,114628],{"id":114627},"成績分布-μ-8590-σ-1275","成績分布： μ = 85.90 σ  = 12.75",[12,114630,114631],{},[1930,114632],{"alt":1932,"src":114633},"https:\u002F\u002Fmegapx-assets.dcard.tw\u002Fimages\u002Fb2950511-f4d1-43f1-86fb-a454645229f2\u002F1280.webp",[26236,114635,114637],{"id":114636},"補充","補充：",[12,114639,114640,114641,114646],{},"根據",[2330,114642,114645],{"href":114643,"rel":114644},"https:\u002F\u002Fwww.dcard.tw\u002Ff\u002Fnthu\u002Fp\u002F260615470",[2334],"前幾屆大二、大三的學長姐所述的慘況","，去年(113)同教授的邏設期中平均僅40~50分，二退人數更達到70人(快一半)；兩年前 (112) 調分(好像調3~5分)前甚至不到10個A+、三年前(111)期中考平均50，最終成績分布如下方附圖。",[12,114648,114649],{},"相比起來，今年完全通識課難度，最後也只退10人，幾乎可以說是邏輯設計趣談了。我本來還想推薦資工系來修，但現在我完全不敢保證下一屆是否仍舊這種甜涼度...",[12,114651,114652,114653],{},"附上三年前(111)同教授的成績分布(來源dcard)給未來選課的參考一下\n",[1930,114654],{"alt":180,"src":114655},"https:\u002F\u002Fmegapx-assets.dcard.tw\u002Fimages\u002F80028095-9c12-4cea-9d58-dee39d50aece\u002F1280.webp",[36,114657],{},[7166,114659,114660],{"id":113650},"微積分Ａ一 Calculus I",[13438,114662,114664,114669,114670,114673,114676,114678,114681,114684,114687],{"className":114663},[13441,13442],[12,114665,114666],{},[106,114667,114668],{},"科號：11410MATH101007","  ",[12,114671,114672],{},"開課教師：江金城 ",[12,114674,114675],{},"課程學分：4",[12,114677,114335],{},[12,114679,114680],{},"上課時間：T3T4 R3R4 + WaWb（演習課）",[12,114682,114683],{},"修課人數：117 -> 116 \u002F120 (新生95)  容量139",[12,114685,114686],{},"上課教材：Calculus 9\u002Fe Stewart (我不確定是哪個封面)",[12,114688,114689],{},"需要原文書嗎？上課基本用不到，但課本習題對小考很有用。",[13438,114691,114693,114698,114700,114703,114706,114709,114712,114715,114719,114722,114728,114733,114736,114739],{"className":114692},[13441,53830],[12,114694,114695],{},[106,114696,114697],{},"範圍： ",[12,114699,114357],{},[12,114701,114702],{},"1.\tfunctions and limits;",[12,114704,114705],{},"2.\tderivatives; ----一段範圍(~2-4)----",[12,114707,114708],{},"4.\tintegrals;",[12,114710,114711],{},"5.\tapplications of integration; (旋轉體體積)",[12,114713,114714],{},"6.\tinverse, logarithmic, exponential ----二段範圍(~6-3)----",[12,114716,114718],{"className":114717},[113565],"and inverse trigonometric functions; ",[12,114720,114721],{},"7.\ttechniques of integration;",[12,114723,114724],{},[75,114725,114727],{"className":114726},[29982],"9.\tDifferential Equations;",[3603,114729,114730],{},[26,114731,114732],{},"Parametric Equations and Polar Coordinates",[12,114734,114735],{},"範圍理論上應該只到積分應用，但後面為了趕課在下學期早點進入多變數",[12,114737,114738],{},"從原定第7章飆去10章參數式…",[12,114740,114741],{},"\u003C\u002Fp>",[12,114743,114389],{},[12,114745,114746],{},"江教授以往大多都是在數學系開課，教數學系的微積分、高等微積分為主。這次算是難得開微積分A的課，更難得來電機系開課。不過我是選了課之後才知道他是數學系系主任...",[12,114748,114749],{},"上課方式是依照原文書內容寫英文版書、以及講解部分例題。基本上全部抄下來後，再寫一些原文書後面的例題就足夠了。",[12,114751,114753],{"className":114752},[113565],"前期主要考 epsilon – delta (極限證明)的部分，也有大量的證明題以及定理推導，這也是前期微積分困難的地方。只是畢竟不是數學系，教授也提過證明題比例不會超過三分之一。\n",[12,114755,114756],{},"而且這些證明大多僅限於第一次段考範圍 (1~2章) ，從第五、六章開始證明題就少很多了，取而代之的是計算為主的題目。",[12,114758,114759],{},"雖然這麼說，但中後期課上有些定義(如微積分基本定理、黎曼和與積分定義、曲線算法)可能會直接作為段考考題要你寫出來，有背就有分(偏偏我沒背到，導致分數直接跳水)",[16,114761],{},[12,114763,114764],{},"前4章基礎的極限、微分、積分部分還可以吃高中的老本，越往後就越多新東西，難度也會逐漸升高。",[12,114766,114767],{},"中期開始課程進度相當之快，幾乎一週一個新章節。後期為了讓我們在下學期進早點進入多變數函數的部分，因此雖然課綱只到第七章(積分方法)，但後來還是趕著教到微分方程與參數式，使得最後一次段考的範圍相當之大。",[12,114769,114770],{},"不過教授人很好，二段的時候直接在課上讓我們舉手決定考試範圍到哪裡。最後我們為了壓證明題的比例，決定到6-3，而期末的範圍似乎也凸顯了這是正確的決定；後來二段出題不小心總分只配90分時，那10分更直接白送。",[16,114772],{},[12,114774,114775],{},"另外，每週三19:00會有微積分演習課+小考，講課部分是助教講解一些課本例題(當然有不少人都在讀自己的書)，小考難度稍高，方式在下方。",[12,114777,114778],{},"而段考部分題目大都不太會出過於刁鑽的題目；但只要有認真聽課，基本上都可以有足夠扎實的微積分基本功(一些證明除外，我恨 epsilon – delta)",[16,114780],{},[12,114782,114783,114784,114787],{},"因為個人微積分因為某些奇怪的原因",[5610,114785,114786],{},"其實就是太菜","幾乎都沒考到很好，教授真的救了我的微積分成績QQ",[26236,114789,114169,114791],{"id":114790},"點名方式-偶爾點名",[75,114792,114795],{"className":114793},[114794],"orange","偶爾點名",[12,114797,114798],{},"點名沒到會扣出席成績。",[12,114800,114801],{},"前期點過兩次名，中間為了趕課沒點名，後面近學期末又點了兩次",[12,114803,114804],{},"不過因為每次都只點一部分人(估計20~30個)，而且優先點上次缺席的人",[12,114806,114807],{},"所以我直到學期末都沒被點到就是",[26236,114809,114183],{"id":114810},"成績算法-1",[7085,114812,114813,114835],{},[7088,114814,114815],{},[7091,114816,114817,114820,114823,114826,114829,114832],{},[7094,114818,114819],{},"Quiz（共12次）",[7094,114821,114822],{},"作業（共14次）",[7094,114824,114825],{},"出席（點名）",[7094,114827,114828],{},"Midterm 1（1 ~ 2-4）",[7094,114830,114831],{},"Midterm 2（2-5 ~ 6-3）",[7094,114833,114834],{},"Final（6-4 ~ 10-3）",[7104,114836,114837],{},[7091,114838,114839,114841,114843,114845,114847,114849],{},[7109,114840,78951],{},[7109,114842,78951],{},[7109,114844,78951],{},[7109,114846,114483],{},[7109,114848,114483],{},[7109,114850,114483],{},[12,114852,114853],{},"(沒錯，滿分105)",[23,114855,114856,114870,114878,114904],{},[26,114857,114858,114859],{},"作業：在線上答題平台上，當週(週二、週四)上課內容。",[23,114860,114861,114864,114867],{},[26,114862,114863],{},"通常20~30題計算題，填答題為主，難度不一。",[26,114865,114866],{},"最多曾出到57題，有時則是20題但計算量大。",[26,114868,114869],{},"另外線上平台要額外付700~900。記得付錢，否則到時做不了作業，還得自行連絡書商購買授權。",[26,114871,114872,114873],{},"出席：被點名缺席就會扣分",[23,114874,114875],{},[26,114876,114877],{},"如果全學期沒被點過名，也是10%全拿",[26,114879,114880,114881],{},"小考方式：每週三7:00~8:00演習課時小考，共12次(排除段考後)。",[23,114882,114883,114886,114889,114892,114895,114898,114901],{},[26,114884,114885],{},"但比重不重，目的也是以弄清概念為主。",[26,114887,114888],{},"每次3題，滿分30，題型大都來自課本習題",[26,114890,114891],{},"捨去最低兩次後，以滿分300分計算。",[26,114893,114894],{},"範圍通常是上週上課內容。",[26,114896,114897],{},"題目比段考難，常考些較刁鑽的題目，時間也相對較趕。",[26,114899,114900],{},"部分題目跟課本習題差不多。",[26,114902,114903],{},"不要太混的話最低拿18分以上(60%)應該還是可以的",[26,114905,114906,114907],{},"段考共三次",[23,114908,114909,114912,114915,114918,114921],{},[26,114910,114911],{},"每次7~8大題(其中1~3小題)。",[26,114913,114914],{},"時間100分鐘",[26,114916,114917],{},"時間跟難度都比小考寬裕許多，寫完一次大概還會剩20分鐘可以檢查。",[26,114919,114920],{},"三次平均73.36\u002F70.3=>80.3\u002F80",[26,114922,114923],{},"標準差14.3\u002F14.4\u002F(未知)\n小考不要太混基本上都能拿到6分以上，作業出席的部分更可說是白送\n可以發現：比起一些小考段考各占25%的教授，這邊相當於直接白送20分，總分還溢出5分。所以",[26236,114925,114927],{"id":114926},"調分嗎-不調分這給分還不夠甜嗎","調分嗎？ 不調分，這給分還不夠甜嗎",[12,114929,114930],{},"(據說出席夠高的話快被當教授會撈...不過微A這邊要被當應該很難)",[26236,114932,114934],{"id":114933},"給加簽嗎系統寫不接受但第一堂下課後讓加簽的找他我猜是給簽吧","給加簽嗎？系統寫不接受，但第一堂下課後讓加簽的找他，我猜是給簽…吧？",[26236,114936,114938],{"id":114937},"涼度-2","涼度：★★★☆☆",[12,114940,114941],{},"相比隔壁韓國教授10題證明 + 負向調分到沒人A+",[12,114943,114944],{},"排除掉微積分這科本身的難度來看，這邊簡直是新手村",[26236,114946,114614],{"id":114947},"甜度-2",[12,114949,114950],{},"我沒拿到A+是我太混，不是教授給分不夠甜，我愧對教授",[26236,114952,114954],{"id":114953},"推薦度","推薦度：★★★★★",[12,114956,114957],{},"不過電機微積分教授每年輪換，明年應該也選不到他了",[12,114959,114960],{},"如果之後他回去教數學系，系上微A又都是大刀的話或許可以考慮？",[26236,114962,114964],{"id":114963},"總結-江大佛名不虛傳","總結： 江大佛名不虛傳",[26236,114966,114968],{"id":114967},"成績分布不公開","成績分布：不公開",[12,114970,114971],{},"不過54個A+，共46.6%，",[12,114973,114974],{},"μ = 86.55， σ  = 11.5",[36,114976],{},[7166,114978,114979],{"id":113665},"普通物理Ｂ一 General Physics B I",[13438,114981,114983,114988,114991,114993,114995,114998,115001,115004,115009,115012],{"className":114982},[13441,13442],[12,114984,114985],{},[106,114986,114987],{},"科號：11410PHYS113304\t",[12,114989,114990],{},"  \u003Cp>開課教師：鄭弘泰",[12,114992,114124],{},[12,114994,114335],{},[12,114996,114997],{},"上課時間：T1T2 F1F2",[12,114999,115000],{},"修課人數：173 -> 161 \u002F140 (新生120)  容量150",[12,115002,115003],{},"上課教材：Essential University Physics (4th Global Edition), Richard Wolfson",[12,115005,115006],{},[1930,115007],{"alt":1932,"src":115008},"\u002Fimages\u002Fuploads\u002F%E5%9C%96%E7%89%874.jpg",[12,115010,115011],{},"需要原文書嗎？搞個林志明的講義比較重要。",[12,115013,114741],{},[13438,115015,115017,115022,115025,115028,115031,115034,115037,115040,115043,115046,115049,115052,115055,115058,115060,115063,115066,115069,115072],{"className":115016},[13441,53830],[12,115018,115019],{},[106,115020,115021],{},"範圍",[12,115023,115024],{},"  \u003Cp>Ch 05 - Newton's Laws Applications 牛頓定律的應用",[12,115026,115027],{},"Ch 06 - Work, Energy & Power 功，能和功率",[12,115029,115030],{},"Ch 07 - Conservation of Energy 能量守恆",[12,115032,115033],{},"Ch 08 - Gravity 重力",[12,115035,115036],{},"Ch 09 - Systems of Particles 多質點系统",[12,115038,115039],{},"----一段範圍----",[12,115041,115042],{},"Ch 10 - Rotational Motion 轉動",[12,115044,115045],{},"Ch 11 - Rotational Vectors & Angular Momentum 轉動向量和角動量",[12,115047,115048],{},"Ch 12 - Static Equilibrium 靜力平衡",[12,115050,115051],{},"Ch 13 - Oscillatory Motion 振盪性運動 (振動)",[12,115053,115054],{},"Ch 14 - Wave Motion 波動",[12,115056,115057],{},"Ch 32 - Interference & Diffraction 干涉和繞射",[12,115059,114377],{},[12,115061,115062],{},"Ch 15 - Fluid Motion 流體力學",[12,115064,115065],{},"Ch 16 - Temperature and Heat 温度和熱量",[12,115067,115068],{},"Ch 17 - The Thermal Behavior of Matter 物質的熱行為",[12,115070,115071],{},"Ch 18 - Heat, Work, and the First Law of Thermodynamics 熱，功，和熱力學第一定律",[12,115073,115074],{},"Ch 19 - The Second Law of Thermodynamics 熱力學第二定律",[26236,115076,114389],{"id":115077},"心得-1",[12,115079,115081,115082,115084],{"className":115080},[113565],"上課方法基本上就是照著課本念與寫板書，以及寫一些例題的方式。出於這個原因，如果有林志明講義的話，會發現他的上課順序和講義完全相同。\n",[16,115083],{},"\n\t",[12,115086,115088,115089,115091,115092,115094],{"className":115087},[113565],"相比其他教授，這堂課沒有小考也不點名，平時分數也完全來自elearn上面的題目。考慮到這堂是早八，所以可以明顯看見每過一堂課，到課的人會越來越少...\n",[16,115090],{},"\n儘管loading小，但這堂課(據說)是普物調分數一數二多的，學期末更直接平調15分。\n",[16,115093],{},[16,115095],{},[12,115097,115098,115100],{},[16,115099],{},"\n上課是先從1~4章開始講，而1~4章幾乎全是高中內容，會讓人產生普物難度很低的錯覺，使得許多人前3天才開始讀一段範圍。",[12,115102,115103],{},"但因為1~4章不考(不考幹嘛教...)，而5~8的難度又跟1~4差上不少，導致最後一群不努力的人被第一次段考送下去，包括我。",[12,115105,115106,115108],{},[16,115107],{},"\n考題是所有教授共同命題，因此可以自由選普物B教授旁聽。雖說都是特定的例題，不過範圍很廣、難度也不低。好就在幾乎每屆的普物都是差不多的題型，因此建議備考的時候多寫幾屆普物考古題(可以不用計時，熟悉題型為優先)。",[12,115110,115111,115113],{},[16,115112],{},"\n第一次段考力學的部分跟高中實際上有非常多重複，只是要改成用微積分的形式(但這時候也還沒教積分，所以積分題目大多都是舉例帶過而已)；",[12,115115,115116],{},"二段範圍同時包含轉動、簡協、波動、光學等範圍，難度也是這幾次裡面最高的。",[12,115118,115119],{},"期末熱力學就相對仁慈許多了，難度回歸正常水準。",[12,115121,115122,115124],{},[16,115123],{},"\n第一次段考的平均偏低(這邊後來有51，不過我看其他班好像是45左右)，第二次因為範圍極大(加上轉動慣量這個全新的概念)平均也差不多，期末熱力學難度則有明顯的降低。",[12,115126,115127,115129,115130,115133],{},[16,115128],{},"\n相比起普A，普物B會更看重在用公式的部分，因此只要公式全背起來、題型熟悉就行。",[5610,115131,115132],{},"也可以叫AI幫你為這本書生一份公式整理","；然後推導的部分不要記，什麼偏微分體積分環場積都看過去就好，微積分沒教就別理他。",[12,115135,115136,115138,115139,115142],{},[16,115137],{},"\n而複習的時候比起啃原文書，拿個林志明的講義自己念會有效率不少。也可以買一本",[106,115140,115141],{},"大學物理學精要","來讀(考研究所用的參考書，被稱為物理科聖經。)",[12,115144,115145],{},"不過這些講義的內容不一定是普物B會考的的內容，像二段我還以為要算面積分\u002F體積分，浪費了四天讀... (實際上會旋轉體就差不多了)",[12,115147,115148,115150],{},[16,115149],{},"\n也聽說很多電機系的同學會自己回家啃原文書\u002F拿林志明講義來讀。不過我資質愚鈍所以最後還是早八乖乖爬起來。",[12,115152,115153],{},"不過如果要聽普物，可以考慮去校門口化工館旁聽劉怡維上課，概念講得十分清楚(但他較少講例題，講課順序也跟林志明講義不同。)",[26236,115155,114449,115157],{"id":115156},"點名方式不點名-1",[75,115158,114454],{"className":115159},[114453],[26236,115161,114183],{"id":115162},"成績算法-2",[7085,115164,115165,115184],{},[7088,115166,115167],{},[7091,115168,115169,115172,115175,115178,115181],{},[7094,115170,115171],{},"平時 （作業）",[7094,115173,115174],{},"Midterm 1",[7094,115176,115177],{},"Midterm 2",[7094,115179,115180],{},"Final",[7094,115182,115183],{},"調分",[7104,115185,115186],{},[7091,115187,115188,115190,115192,115194,115196],{},[7109,115189,114483],{},[7109,115191,114483],{},[7109,115193,114483],{},[7109,115195,114483],{},[7109,115197,115198],{},"+15",[26236,115200,115202],{"id":115201},"作業在elearn上每次段考範圍一份","作業：在elearn上，每次段考範圍一份。",[12,115204,115205],{},"因為沒有小考，所以平時成績完全來自作業",[23,115207,115208,115219,115222],{},[26,115209,115210,115211],{},"每章6題，每題可以作答3次。",[23,115212,115213,115216],{},[26,115214,115215],{},"建議算錯一次之後就拍下來丟AI了，否則要是答錯，分數影響不小。",[26,115217,115218],{},"時間從上次段考截止到下次段考前一天晚上11:50，但有時會更早，這點要注意。",[26,115220,115221],{},"如果調完差5分以內及格的話，教授會仁慈幫你撈到60。",[26,115223,115224],{},"要拿到A+的話需要三次段考總和198~200+",[12,115226,115227],{},"三次段考平均：51.14\u002F40.85\u002F58.37",[26236,115229,115231],{"id":115230},"調分嗎-爆調15分","調分嗎？   爆調15分",[12,115233,115234],{},"似乎是調最多的，調分完之後距離及格縣5分內會撈。",[26236,115236,115238],{"id":115237},"給加簽嗎第一堂課到課加簽","給加簽嗎？第一堂課到課加簽",[12,115240,115241],{},"好像簽20個左右吧，但要簽的人非常多。",[26236,115243,115245],{"id":115244},"涼度-3","涼度：★★★★★",[12,115247,115248],{},"不用點名、不用小考，也不用到課",[26236,115250,115252],{"id":115251},"甜度-3","甜度：★★★⯪☆",[26236,115254,115256],{"id":115255},"推薦度無腦推","推薦度：無腦推",[23,115258,115259,115262,115265],{},[26,115260,115261],{},"想衝GPA：★★★★★",[26,115263,115264],{},"求輕鬆過：★★★★★",[26,115266,115267],{},"想睡掉早八：★★★★★",[12,115269,115270,115271,115273],{},"想早八爬起來、認真在課上學懂普物：★★★★★",[16,115272],{},"\n（不點名、不小考所以自己也能自己選教授旁聽。但這種人存在嗎?）",[26236,115275,115277],{"id":115276},"總結調分魔術師","總結：調分魔術師",[26236,115279,115281],{"id":115280},"成績分布不公開但會附上全班成績excel","成績分布：不公開，但會附上全班成績Excel",[12,115283,115284],{},"μ = 78.61 σ = 14.00",[32509,115286,115287,115288],{},"\n  ",[1930,115289],{"src":115290,"alt":115291},"\u002Fimages\u002Fuploads\u002F圖片6.png","undefined",[32509,115293,115287,115294],{},[1930,115295],{"src":115296,"alt":115291},"\u002Fimages\u002Fuploads\u002F圖片5.png",[36,115298],{},[7166,115300,115301],{"id":113680},"普通物理實驗一 General Physics Laboratory I",[13438,115303,115305,115310,115313,115316,115318,115321,115324,115327],{"className":115304},[13441,13442],[12,115306,115307],{},[106,115308,115309],{},"科號：11410PHYS101002",[12,115311,115312],{},"  \u003Cp>開課教師：戴明鳳 （實際上多是助教帶）",[12,115314,115315],{},"課程學分：1  \u002F\u002F 幹 破課",[12,115317,114127],{},[12,115319,115320],{},"上課時間： M5M6M7M8",[12,115322,115323],{},"修課人數：53  \u002F58 （新生58）",[12,115325,115326],{},"範圍：同普通物理B一（力學&熱力學）",[12,115328,114741],{},[26236,115330,115332],{"id":115331},"心得-excel應用實驗一","心得：   Excel應用實驗一",[12,115334,115335],{},"電機系（其實好像全校都這樣）人見人幹的破課。",[12,115337,115338],{},"分為講解課（L+講）、實驗課與演示實驗（報告）。",[23,115340,115341,115352,115363,115374,115382,115399],{},[26,115342,115343,115344],{},"講解課就是解釋實驗原理",[23,115345,115346,115349],{},[26,115347,115348],{},"但對實驗的幫助不大，原因見下。",[26,115350,115351],{},"好處就在講解課通常下午三點半就下課，以及回答問題可以加分。",[26,115353,115354,115355],{},"實驗課內容通常會配合當學期的普通物理範圍",[23,115356,115357,115360],{},[26,115358,115359],{},"所以普物實驗一就是力學、轉動、波動、熱力學",[26,115361,115362],{},"是的，全是這些電機系完全用不到的東西。",[26,115364,115365,115366],{},"實驗前會先有一次 open-book 課前小考，範圍就是當次實驗內容",[23,115367,115368,115371],{},[26,115369,115370],{},"可以自己印預報來看。",[26,115372,115373],{},"個人建議將講義內容貼到預報裡面，考題重複性很高。",[26,115375,115376,115377],{},"考完後，講師會講解實驗內容與操作",[23,115378,115379],{},[26,115380,115381],{},"然後你會發現實驗內容跟你預報看到的八竿子打不著，所以實驗都以現場講師為主。",[26,115383,115384,115385],{},"做完實驗收集完數據、在Excel算出要求的數值後給實驗助教檢查，才能離開。",[23,115386,115387,115390,115393,115396],{},[26,115388,115389],{},"數據的部分據說有些助教會要求誤差需要小於5%\u002F20%之類才能離開，不然就需要重做；",[26,115391,115392],{},"不過每次實驗的助教都不同人，所以標準也各有不同",[26,115394,115395],{},"但我們這屆的助教對這方面要求都寬鬆許多，實驗數據合理、做得出結報就行。",[26,115397,115398],{},"真的要感謝這一屆的實驗助教，謝謝你們救了我們寶貴的肝。",[26,115400,115401],{},"之後再依照這些數據做一份結報出來，下周一之前交，這次實驗才算完成。",[12,115403,115404],{},"出於上述原因、以及實驗光是器材架設就很耗時，實驗大部分都做不完：",[23,115406,115407,115410,115413,115416],{},[26,115408,115409],{},"第一次實驗滑車做到快7點",[26,115411,115412],{},"第二次因為各組排隊等蓋格計數器，三點做完實驗還排隊等到六點。",[26,115414,115415],{},"不過之後實驗室都改為6點就清場了，只是後期實驗大多也得做到5點半左右。",[26,115417,115418],{},"但做實驗的速度是會隨著操作逐漸變快的，到後面幾次最快4點就全員離場了，難得做完實驗天還是亮的。",[12,115420,115421],{},"演示實驗課則是助教會給你一個實驗器材，你需要提前做好ppt介紹器材、以及它的原理及應用，並在演示實驗課當天上台報告，報告完根據建議改ppt後再交一份修改版。",[23,115423,115424,115427,115430],{},[26,115425,115426],{},"但我們一班大概有23組，所以一般建議壓在5~10分鐘內。",[26,115428,115429],{},"只是講到一半戴教授常常補充一些阿里阿雜的東西，導致每次演示實驗都拖到6:30左右才能離場==；",[26,115431,115432],{},"通常前幾組報告還會有人專心聽，到後面幾乎都死一片了。",[12,115434,115435,115436,115438,115440],{},"出於以上原因，這堂課離場時間平均至少5:30，最晚6:30~7:00，所以實驗課當天晚上還是千萬不要排課（但是戴明鳳開在當天晚上的通識可能可以啦）。",[16,115437],{},[16,115439],{},"\n要早點離場的話，個人建議一定要搞到家傳：因為實驗內容每屆幾乎都大同小異， Excel 裡面的公式改一改，套實驗的數據進就能用，可以讓你省下不少寫公式、測數據的時間；",[12,115442,115443,115444,115446,115448],{},"而測出來的數據不準的時候，也有補救方法，不用花半天重建那一堆實驗器材；而結報也有一個格式可以參考，更不容易需要熬夜。",[16,115445],{},[16,115447],{},"\n預報的部分基本上都是去普物實驗室網站複製貼上，結報要不要參考家傳就見仁見智了；",[12,115450,115451],{},"但說真的，除非你是天生不用讀書的卷哥卷姊，否則我建議能混就混、別花太多時間熬夜在這1學分的破課上面，因為會壓縮到睡眠以及讀其他科目的時間。",[12,115453,115454],{},"加上課程內容以及實驗做的力學、熱力學對未來幫助也不大（普物實驗二的電磁、電路學反而比較相關，但（幸好）必修沒物理實驗二）、甚至對學普物B一也沒太大幫助（頂多就是學到些Excel技巧）。",[12,115456,115457],{},"喔對，這學期周一放假放掉兩次實驗課，還把碰撞實驗給放掉了，結報-2，爽。",[16,115459],{},[26236,115461,114169,115463],{"id":115462},"點名方式-每堂點名-1",[75,115464,114173],{"className":115465},[29982],[26236,115467,114183],{"id":115468},"成績算法-3",[7085,115470,115471,115480],{},[7088,115472,115473],{},[7091,115474,115475,115478],{},[7094,115476,115477],{},"平時成績",[7094,115479,115180],{},[7104,115481,115482],{},[7091,115483,115484,115487],{},[7109,115485,115486],{},"70%",[7109,115488,115489],{},"30%",[12,115491,115492],{},"平時成績包含每次實驗報告、實驗前考試、演示實驗報告等分數，以及講解課回答加分",[12,115494,115495],{},"結報可以把問題與討論、心得都寫一寫，但也不用太努力拚加分啦，因為怎麼加都加不到期末的那30%裡面==",[12,115497,115498],{},"期末前半部分是考實驗操作器材、用途與方法（如游標卡尺、向心力機器）跟實驗內容，跟實驗前小考的內容有些重疊；後半部分則是講解實驗課的器材，比例分別60%跟40%。我是建議專心讀前者就好。",[12,115500,115501],{},"另外戴班不會有助教連連看，所以別想背助教了。",[26236,115503,115505],{"id":115504},"調分嗎大調","調分嗎？大調",[12,115507,115508],{},"最後班平均調到82多，戴班最後沒意外都A\u002FA+",[26236,115510,115512],{"id":115511},"給加簽嗎-好問題","給加簽嗎？ 好問題",[12,115514,115515],{},"第一節課有要求工學院去其他班，所以這班應該是只留給電機系",[26236,115517,115519],{"id":115518},"涼度-4","涼度：⯪☆☆☆☆",[12,115521,115522],{},"一學分的破課，5學分的loading",[12,115524,115525],{},"做到半夜，帶你熬夜",[26236,115527,115529],{"id":115528},"甜度-4","甜度：★★★☆☆",[12,115531,115532],{},"考前30分鐘才發現可以帶大抄==",[26236,115534,115535],{"id":114953},"推薦度：   ☆☆☆☆☆",[26236,115537,115539],{"id":115538},"總結-我很感謝我的組員帶飛","總結：  我很感謝我的組員帶飛",[26236,115541,115543],{"id":115542},"成績分布不公開-1","成績分布：不公開。",[12,115545,115546],{},"μ = 82.13 σ  = 6.26  ，",[36,115548],{},[7166,115550,115551],{"id":113696},"中高級英文三-閱讀 Upper-Intermediate English III：Reading",[13438,115553,115555,115560,115563,115566,115569,115572,115575,115578,115583,115586],{"className":115554},[13441,13442],[12,115556,115557],{},[106,115558,115559],{},"科號：11410LANG103025\t",[12,115561,115562],{},"  \u003Cp>開課教師：劉文貞",[12,115564,115565],{},"課程學分：2",[12,115567,115568],{},"課程類型：校定必修（英文）",[12,115570,115571],{},"上課時間： W3W4",[12,115573,115574],{},"修課人數：AA -> BB \u002F30 （新生30）, 容量50",[12,115576,115577],{},"上課教材：自製簡報 + Pathways：Reading，Writing，and CriticalThinking （4） 3\u002Fe",[12,115579,115580],{},[1930,115581],{"alt":1932,"src":115582},"\u002Fimages\u002Fuploads\u002F%E5%9C%96%E7%89%877.jpg",[12,115584,115585],{},"需要原文書嗎？   必需（水木書店有賣）",[12,115587,114741],{},[13438,115589,115591,115595,115598,115601,115604,115607],{"className":115590},[13441,53830],[12,115592,115593],{},[106,115594,114150],{},[12,115596,115597],{},"  \u003Cp>\nUnit 1 The Changing Workplace",[12,115599,115600],{},"Unit 4 Rethinking Transport",[12,115602,115603],{},"Unit 8 Living Longer",[12,115605,115606],{},"Unit 6 Why Language Matters (須自己讀)",[12,115608,114741],{},[26236,115610,115612],{"id":115611},"心得-2","心得：  ",[12,115614,115615],{},"就...很標準的高中英文課，例如隨堂點名、分組加分、不准用電子產品等等。大概就只跟高中差在全英授課，文章比較長、難度比較高（C1難度）、以及要求上課都用全英對話這點了。",[12,115617,115618],{},"上課會先分組，然後從組別間英文自我介紹開始講起。開始上課後，回答問題搶答可以加分，有時上課也會有些小組團隊合作回答問題、或玩kahoot!的環節。最後學期末似乎會算小組總分。",[12,115620,115621],{},"上課內容則是從閱讀技巧講起，然後用自製的簡報來帶課文，並有時會發學習單，讓學生以教過的閱讀技巧（Summarize 、Outline、picturing等）去分析、理解該篇比高中課本還長上數倍的課文；",[12,115623,115624],{},"回家作業則是將單字與解釋連接起來，上課檢討，不過不算分。",[12,115626,115627],{},"全學期有算分的回家作業大概只有一次，在運動會週。那週有傳影片要求自學。不過作業也是很簡單的文章分析而已，eeclass上繳交。",[12,115629,115630],{},"如果高中英文課就有學過相關閱讀技巧的話，由於幾乎沒有回家作業，這堂課頂多就是Quiz與小組報告前會稍忙一些，其餘應該沒太多課後負擔要做。",[12,115632,115633],{},"另外，雖然是全英上課，不過老師的英文沒有什麼口音，反而聽到她講中文時會有外國人講中文的感覺；",[12,115635,115636],{},"此外，上課時後老師會時不時補充些在國外的生活經驗、或引用生活例子來解釋課文內容。",[26236,115638,114169,115640],{"id":115639},"點名方式-每堂點名-2",[75,115641,114173],{"className":115642},[29982],[26236,115644,115646],{"id":115645},"成績算法總分可達105","成績算法：總分可達105",[7085,115648,115649,115670],{},[7088,115650,115651],{},[7091,115652,115653,115656,115659,115662,115665,115668],{},[7094,115654,115655],{},"出席（平時）",[7094,115657,115658],{},"上課作業（學習單）",[7094,115660,115661],{},"回家作業（只有一次）",[7094,115663,115664],{},"分組報告",[7094,115666,115667],{},"Quiz（多次）",[7094,115669,115180],{},[7104,115671,115672],{},[7091,115673,115674,115677,115679,115682,115685,115687],{},[7109,115675,115676],{},"15~20%",[7109,115678,78936],{},[7109,115680,115681],{},"5%",[7109,115683,115684],{},"15%",[7109,115686,115684],{},[7109,115688,115489],{},[26236,115690,115692],{"id":115691},"出席分數算法","出席分數算法：",[7085,115694,115695,115705],{},[7088,115696,115697],{},[7091,115698,115699,115702],{},[7094,115700,115701],{},"評分項目",[7094,115703,115704],{},"分數",[7104,115706,115707,115714,115722,115730,115738],{},[7091,115708,115709,115712],{},[7109,115710,115711],{},"初始分數",[7109,115713,21841],{},[7091,115715,115716,115719],{},[7109,115717,115718],{},"全勤",[7109,115720,115721],{},"+2",[7091,115723,115724,115727],{},[7109,115725,115726],{},"遲到",[7109,115728,115729],{},"-2",[7091,115731,115732,115735],{},[7109,115733,115734],{},"缺席",[7109,115736,115737],{},"-5",[7091,115739,115740,115743],{},[7109,115741,115742],{},"在課堂上從事無關活動",[7109,115744,115729],{},[12,115746,115747],{},"若為負分，將併入學期成績計算；若累積達 20 分，則學期成績額外加 5 分，也就是說總成績的出席項會拿到20分。",[12,115749,115750],{},"這一項我估計是以課堂參與度下去評分的，小組分數似乎也是記到這邊。",[12,115752,115753],{},"不確定前半學期跟後半的出席分是算總和還是算平均。",[12,115755,115756],{},"上課作業：發學習單，以教過的技巧分組去分析文章。",[23,115758,115759],{},[26,115760,115761],{},"有時也會玩kahoot!",[26236,115763,115765],{"id":115764},"回家作業","回家作業：",[12,115767,115768],{},"只有一份。是運動會那週的閱讀分析作業",[23,115770,115771],{},[26,115772,115773],{},"內容和上課學習單大同小異，eeclass線上繳交。",[26236,115775,115777],{"id":115776},"分組報告-pair-article-presentation","分組報告( Pair Article Presentation )：",[12,115779,115780],{},"期初二人分組，選擇給定文章中的一篇，並在指定的時段以英文上台報告該指定長文。",[23,115782,115783,115786,115789,115797,115800],{},[26,115784,115785],{},"分析的部分不難，比較難的是口說的部份。",[26,115787,115788],{},"這部分建議選擇前期一點的組別（最好在10月初以前），這樣期末就可以少一項負擔；",[26,115790,115791,115792],{},"加上分組報告後提出問題是可以加分的。",[23,115793,115794],{},[26,115795,115796],{},"我塞在期末趕太多報告，在這部分沒準備好直接炸開...",[26,115798,115799],{},"報告完後，老師會給一份挺詳細的報告分析表以及改善建議，十分用心~",[26,115801,115802,115803],{},"然後絕對不要帶僥倖心態用AI生成報告，期末有不少人被抓包。",[23,115804,115805],{},[26,115806,115807],{},"僅允許AI部分潤飾報告文筆及抓錯，不過有許多特殊規定要遵守。",[26236,115809,114493],{"id":115810},"小考形式-1",[12,115812,115813],{},"有兩次在表上的小考，一次範圍通常是一篇課文",[23,115815,115816,115819,115836],{},[26,115817,115818],{},"兩次表上的小考各5%，共占10%，可以當作兩次期中。",[26,115820,115821,115822],{},"分為三個部分：單字與片語、課文閱讀理解、閱讀技巧",[23,115823,115824,115827,115830,115833],{},[26,115825,115826],{},"會提前給單字表，通常20~30個，建議考前狂抄單字來練習。",[26,115828,115829],{},"閱讀理解部分為open-book，範圍是關於上課內容的課文",[26,115831,115832],{},"基本上內容都跟之前學習單帶過的題目大同小異",[26,115834,115835],{},"閱讀技巧則是給一篇新的文章，要用課上教的技巧去分析該文章。",[26,115837,115838,115839],{},"此外，通常在分組報告前，課堂上可能有報告範圍的pop-Quiz。",[23,115840,115841],{},[26,115842,115843],{},"這部分共占5%",[26236,115845,115847],{"id":115846},"final形式","Final形式：",[12,115849,115850],{},"英文課共同命題，難度較高；範圍則是上課教過的課文。",[23,115852,115853,115867,115875,115883],{},[26,115854,115855,115856],{},"題型跟小考大同小異，只是還多一份申論（作文）",[23,115857,115858,115861,115864],{},[26,115859,115860],{},"題目是關於段考的文章內容",[26,115862,115863],{},"要求180字以上",[26,115865,115866],{},"寫起來感覺比學測簡單",[26,115868,115869,115870],{},"閱讀理解部分同樣是問課文，但close-book。",[23,115871,115872],{},[26,115873,115874],{},"不過也不會像小考問得那麼細節。",[26,115876,115877,115878],{},"單字有將近240個，可能要花點時間準備",[23,115879,115880],{},[26,115881,115882],{},"可以提前整理一遍單字-字意表，再重點標記不熟的單字",[26,115884,115885],{},"估計Final平均分應該也沒到太好。",[7085,115887,115888,115904],{},[7088,115889,115890],{},[7091,115891,115892,115895,115898,115901],{},[7094,115893,115894],{},"單字",[7094,115896,115897],{},"閱讀理解（課文）",[7094,115899,115900],{},"申論（作文）",[7094,115902,115903],{},"長文閱讀",[7104,115905,115906],{},[7091,115907,115908,115910,115913,115915],{},[7109,115909,115489],{},[7109,115911,115912],{},"20~30%",[7109,115914,78951],{},[7109,115916,115917],{},"30~40%",[12,115919,115920],{},"BONUS：參加培力英文測驗：共+4",[23,115922,115923,115926],{},[26,115924,115925],{},"報名聽+讀 or 說+寫，可各拿到一分(共+2)",[26,115927,115928],{},"聽+讀 or 說+寫 達到B2等級，可再各拿一分(共+2)。",[26236,115930,115932],{"id":115931},"調分嗎-全班平調3分","調分嗎？   全班平調+3分。",[26236,115934,115936],{"id":115935},"給加簽嗎-似乎不給","給加簽嗎？    似乎不給",[26236,115938,114938],{"id":115939},"涼度-5",[12,115941,115942],{},"上課規矩限制、以及要做的事情不少，還挺忙的",[12,115944,115945],{},"不過幾乎沒回家作業，不會占花太多課後時間。",[12,115947,115948],{},"一半一半吧。",[26236,115950,115252],{"id":115951},"甜度-5",[12,115953,115954],{},"加分機會很多，建議全部拿滿",[26236,115956,114288],{"id":115957},"推薦度-1",[12,115959,115960],{},"因為新生保留，新生可以選，如果缺英文學分的話也很推薦選這堂；",[12,115962,115963],{},"但如果要考免修的話不建議花時間來修",[26236,115965,115967],{"id":115966},"總結-中高英裡面非常推薦的一堂課","總結： 中高英裡面非常推薦的一堂課！",[26236,115969,115971],{"id":115970},"成績分布不公開-2","成績分布：不公開，",[12,115973,115974],{},"μ = 84.13  σ  = 8.54，",[36,115976],{},[7166,115978,115979],{"id":113712},"初級德語一 Beginning German I",[13438,115981,115983,115988,115991,115993,115996,115999,116002,116005,116008],{"className":115982},[13441,13442],[12,115984,115985],{},[106,115986,115987],{},"科號：11410FL 204107\t",[12,115989,115990],{},"  \u003Cp>開課教師：韓佩雯",[12,115992,114124],{},[12,115994,115995],{},"課程類型：第二外語（抵英文學分）",[12,115997,115998],{},"上課時間：R7R8R9",[12,116000,116001],{},"修課人數：40 -> 36 \u002F42",[12,116003,116004],{},"上課教材：Das Leben - Kurs- und Übungsbuch - A1",[12,116006,116007],{},"需要原文書嗎？ 必需（水木書店有賣）",[12,116009,114741],{},[12,116011,116012],{},[1930,116013],{"alt":1932,"src":116014},"\u002Fimages\u002Fuploads\u002F%E5%9C%96%E7%89%878.jpg",[13438,116016,116018,116022],{"className":116017},[13441,53830],[12,116019,116020],{},[106,116021,114150],{},[12,116023,116024],{},"  \u003Cp>德文字母、課本1~6課。\u003C\u002Fp>",[26236,116026,114389],{"id":116027},"心得-3",[12,116029,116030],{},"第一堂課會先從德文的26+4個（三個變音+ ß ）德文字母發音開始教，之後開始教單字。",[12,116032,116033],{},"由於老師幾乎不用電子產品（不過不像程設課那樣完全禁用），因此筆記部分需要上課時自己抄下來（或拍下來）。而習題、考卷解答也是完全手寫、傳下去給同學自行拍照。",[12,116035,116036],{},"上課會先從單字開始講起，之後進到課本課文，從課文帶到各個單字與例句，並依照課本例題去講解德文的一些文法與用例。老師基本上會把每個用例都講清楚講明白。",[12,116038,116039],{},"雖然這麼說，不過德文文法部分真的特別搞剛：除了動詞現在式有六種變化態、受詞（目前）有兩種等等規則之外，名詞不只分陰陽性，連複數型也是不規則的。因此就算英文底子好，學德文也難免會被這些規則卡住。",[12,116041,116042],{},"我是修了之後，才知道原來德文是文法最複雜的語言之一...這點老師也承認德文真的很難、學習門檻不低。",[12,116044,116045],{},"不過德文發音是完全按照字母念的，這點算是德文比英文好的一點就是，一些外來字也是按讀音改寫法。只是因為德文字母與英文重疊，但又常有很多念法上的異同（聽看看das Restaurant就知道了），這點頗需要適應。",[12,116047,116048],{},"老師有時也會分享一些以前她在德文系就讀、以及之前在歐洲、德國的許多生活習慣與經驗，再加上歐洲的一些節慶、歷史故事、王室或介紹一座城市。",[12,116050,116051],{},"也因此，這堂課上不只可以學到德文，也能學到不少課本上讀不到的，德國與歐洲的人文歷史小知識。因為我個人暑假剛去過德國與瑞士，所以聽到老師分享這些在歐洲的生活經驗時，總有種親切感。",[12,116053,116054],{},"雖然第一堂課老師看起來十分嚴格，不過隨著課程進行，會發現其實老師其實蠻和善的、也蠻可愛的ww，有德文的相關問題也可以下課去請教，雖然好像因為班上都是電神很少人會提問就是。",[26236,116056,114449,116058],{"id":116057},"點名方式偶爾點名",[75,116059,114795],{"className":116060},[114794],[12,116062,116063],{},"全學期只點過三次，兩次在前幾堂課，中期趕課，剩下一次在最後一堂",[26236,116065,114183],{"id":116066},"成績算法-4",[7085,116068,116069,116082],{},[7088,116070,116071],{},[7091,116072,116073,116076,116079],{},[7094,116074,116075],{},"Quiz （1次）",[7094,116077,116078],{},"Midterm（1~2課）",[7094,116080,116081],{},"Final（3~6課）",[7104,116083,116084],{},[7091,116085,116086,116088,116091],{},[7109,116087,115684],{},[7109,116089,116090],{},"35%",[7109,116092,78920],{},[12,116094,116095],{},"小考形式是聽寫單字與句子，基本上只是為了熟悉考試方法而已。",[12,116097,116098],{},"之後與下學期也不會有其他次小考。",[12,116100,116101],{},"不過後來有允許用期末的分數補期中。",[12,116103,116104],{},"BONUS：全勤+1分",[12,116106,116107],{},"有抓到教授上課講錯\u002F寫錯+1分（不過最後好像兩三個人拿到而已...）",[26236,116109,116111],{"id":116110},"調分嗎不調","調分嗎？不調",[12,116113,116114],{},"不過這次有改成績算法（我直接被救起來）",[26236,116116,116118],{"id":116117},"給加簽嗎不接受加簽不過加退選上的機率很高","給加簽嗎？不接受加簽，不過加退選上的機率很高",[12,116120,116121],{},"第一節課後的加退選，通常會有名額出來",[12,116123,116124],{},"因此想選可以第一堂就開始旁聽，兩週內應該會有名額。",[26236,116126,114266],{"id":116127},"涼度-6",[12,116129,116130],{},"這是我期末花最多時間讀的科目。",[12,116132,116133],{},"到底誰修課心得填說這是五星輕鬆甜涼課…",[26236,116135,116137],{"id":116136},"甜度-6","甜度：★★★★☆",[12,116139,116140],{},"考試每格扣分大概只扣0.幾",[12,116142,116143],{},"上課聽得懂+有認真讀，應該是不難拿分的（這不是我）",[26236,116145,116147],{"id":116146},"推薦度不建議大一修","推薦度：不建議大一修",[12,116149,116150],{},"推薦給想花時間學好德文、或有德文底子的",[12,116152,116153],{},"但如果時間不多、還不適應大學生活、或語言底子弱的話，就建議大二再說",[26236,116155,116157],{"id":116156},"總結-德文真的很難但肯努力學肯定能學到不少東西not-me","總結：  德文真的很難，但肯努力學，肯定能學到不少東西（not me）。",[26236,116159,114968],{"id":116160},"成績分布不公開-3",[12,116162,116163],{},"μ = 85.03 σ  = 11.76",[26236,116165,114637],{"id":116166},"補充-1",[12,116168,116169],{},"我是怎麼敢在日文平假名都花一年才背起來的情況下來修德文的啊...",[12,116171,116172],{},"因為輕忽語言課，看有空堂就選了，使得最後把這堂課排在有早八的週四，跑去挑戰人家外語系的必修。偏偏當天從1~6節都是主科（程設+邏設+微積分...），使得我整個學期上課幾乎恍神、直到最後一個月才聽懂課上內容...最後在這堂課的表現真的不太好，學習成果也不佳。",[12,116174,116175],{},"大一對環境不熟悉的情況下，除了卷哥卷姊，否則真的完全不建議修第二外語，尤其是在對一個語言完全不熟悉的情況下：因為相關單字、語法是很大的額外負擔。",[12,116177,116178],{},"除非是本來就有該語言基礎（如日文），不然在需要占用大量專業科目複習時間的情況下，語言課修下去真的可說是成績炸彈。",[12,116180,116181],{},"總之，考慮到下學期專業科目較多，我自己是學乖了，初德二大概得等二年級課業較輕的時候再繼續修了...",[12,116183,116184,116185,116190],{},"估計",[2330,116186,116189],{"href":116187,"rel":116188},"https:\u002F\u002Fblog.chinono.dev\u002F?author=%E6%B3%95%E6%B3%95",[2334],"法法","的德文都比混了一個學期的我還強了。",[36,116192],{},[7166,116194,116195],{"id":113728},"當代國際議題研究 Current important issues in international Relations",[13438,116197,116199,116204,116207,116209,116212,116215,116218,116220],{"className":116198},[13441,13442],[12,116200,116201],{},[106,116202,116203],{},"科號：11410GE 158400",[12,116205,116206],{},"  \u003Cp>開課教師：方天賜",[12,116208,115565],{},[12,116210,116211],{},"課程類型：一般通識（社會領域）",[12,116213,116214],{},"上課時間：W5W6",[12,116216,116217],{},"修課人數：57 \u002F60 容量77",[12,116219,114344],{},[12,116221,114741],{},[13438,116223,116225,116230,116233,116236,116239,116242],{"className":116224},[13441,53830],[12,116226,116227],{},[106,116228,116229],{},"內容：",[12,116231,116232],{},"  \u003Cp>國際關係的意義（基本政治概念）、大國角力、",[12,116234,116235],{},"領土爭端、民族主義、宗教衝突、外交實務（演講）、",[12,116237,116238],{},"民主發展、軍備管制、恐怖主義、小國外交、人類安全、",[12,116240,116241],{},"中小國外交。",[12,116243,114741],{},[26236,116245,114389],{"id":116246},"心得-4",[12,116248,116249],{},"算是十分搶手的一堂輕鬆通識，我自己也是在加退選才偶然選上，加上本來就對國際議題有點研究才選。",[12,116251,116252],{},"上課一開始會先從一些基礎的政治學概念講起，之後的上課內容則藉著近代的大國政治與時事，藉以講解主題以及複雜的國際關係。不過今年因為川普上台的關係，美國的部分占的篇幅相當大，而教授也有對川普的外交手法進行分析。",[12,116254,116255],{},"而教授作為前駐印度外交官，不只有時會分享過去派駐印度的經驗，課上有時也會拿印度來當例子。像是前期介紹大國政治的時候，教授就有拿印度總理莫迪與川普、對台灣的互動做舉例。也因為他在印度待了好幾年，教授更是強調別請他吃咖哩或印度料理ww",[12,116257,116258],{},"除此之外，每堂課剛開始上課時，會鼓勵同學上台發表、分享當週的國際大事。而教授也會自製ppt，分享該週有哪些國際大事。不過可能大學生都不太敢發表，全學期有上台過的大概就3~4個，而且大多都在前半學期。",[12,116260,116261],{},"也因為教授曾在外交部工作過，每個學期會邀請一位外交相關的專家來演講，像是這學期就邀請了以前台灣駐沙烏地阿拉伯的大使（過去有正式邦交關係時的那種）來分享外交工作以及派駐經驗。",[12,116263,116264],{},"另外，到最後一堂課，教授還會做一部影片來回顧整學期的上課內容、帶過上課的每一個主題，可以說十分用心，甚至讓大家不自覺全場鼓掌~而學期也有邀請大家加入時事與印度議題的兩個討論群組。",[12,116266,116267],{},"教授雖然看起來很嚴肅，實際上人很好。上課時要求不多，期末作業前也是開放大家跟他討論題目。除此之外，教授更可說是冷面笑匠，雖然總是一臉正經，講話卻有時很好笑，使得雖然國際關係很複雜，上課卻一點也不枯燥。",[12,116269,116270],{},"儘管教授對於國際關係與政治有著自己的一套看法，但他也很十分歡迎不同意見，在這方面上不會太刁難學生。所以作業的部分不必太拘謹或擔心需要討好，專心發表自己的想法即可~",[12,116272,116273],{},"正如第一堂課所說，教授希望讓每個人能培養「國際觀」；而這堂課整學期下來，不論本來是否有在關注時事議題，對於國際議題肯定都會有新的一層理解。",[26236,116275,114169,116277,116280],{"id":116276},"點名方式-每堂點名簽到單",[75,116278,114173],{"className":116279},[29982],"（簽到單）",[26236,116282,116284],{"id":116283},"成績算法不考試只有作業與期末報告未來可能更動","成績算法：不考試，只有作業與期末報告（未來可能更動！）",[12,116286,116287],{},"備註：似乎因為AI問題，這堂課之後應該也會全面改成考試形式了。",[7085,116289,116290,116309],{},[7088,116291,116292],{},[7091,116293,116294,116297,116300,116303,116306],{},[7094,116295,116296],{},"出席",[7094,116298,116299],{},"國際大事分享",[7094,116301,116302],{},"綜合表現",[7094,116304,116305],{},"平時作業",[7094,116307,116308],{},"期末報告",[7104,116310,116311],{},[7091,116312,116313,116315,116317,116319,116321],{},[7109,116314,78951],{},[7109,116316,115681],{},[7109,116318,78951],{},[7109,116320,78943],{},[7109,116322,114483],{},[12,116324,116325],{},"國際大事分享：可選擇在課程開頭口頭報告，或在LINE群組內分享時事新聞。",[23,116327,116328,116336,116339,116342,116350],{},[26,116329,116330,116331],{},"分享「過去一週內」發生的國際新聞大事",[23,116332,116333],{},[26,116334,116335],{},"不一定限於國際政治，像是Gemini 3剛推出的時候，教授就有介紹",[26,116337,116338],{},"口頭報告大概2~3分鐘，並大致介紹事件背景與經過",[26,116340,116341],{},"傳在LINE群組，則是上傳相關新聞與約100字的短評與心得分析",[26,116343,116344,116345],{},"前8週（半學期）繳交就能5分拿滿，之後會以2週為單位遞減。",[23,116346,116347],{},[26,116348,116349],{},"畢竟之前教授發現同學會堆在期末才發表...",[26,116351,116352],{},"分享一次即可。",[26236,116354,116356],{"id":116355},"平時作業45","平時作業：45%",[12,116358,116359],{},"裡面含5項作業 ；此外還有期末報告一份，皆為個人作業：",[7085,116361,116362,116381],{},[7088,116363,116364],{},[7091,116365,116366,116369,116372,116375,116378],{},[7094,116367,116368],{},"選課調查",[7094,116370,116371],{},"大國政治",[7094,116373,116374],{},"領土爭議",[7094,116376,116377],{},"演講心得",[7094,116379,116380],{},"軍備管制",[7104,116382,116383],{},[7091,116384,116385,116387,116389,116391,116393],{},[7109,116386,115681],{},[7109,116388,78951],{},[7109,116390,78951],{},[7109,116392,78951],{},[7109,116394,78951],{},[12,116396,116397],{},"BONUS： 演講提問有加分",[12,116399,116400],{},"平時作業的話共有4次，每次10%，通常是關於當週上課內容的主題",[23,116402,116403],{},[26,116404,116405,116406],{},"沒有限定字數，闡述觀點即可",[23,116407,116408],{},[26,116409,116410],{},"我自己是寫少則600，多則1500字都有",[12,116412,116413],{},"四項平時作業的主題分別是：",[3603,116415,116416,116419,116422,116425],{},[26,116417,116418],{},"大國政治（Q：請問你認為美中衝突是否不可避免?）",[26,116420,116421],{},"領土爭議（Q：台灣應該放棄對南海的主權主張（十一段線、U型線）嗎?  ）",[26,116423,116424],{},"演講心得（沙烏地阿拉伯大使演講）",[26,116426,116427],{},"軍備管制（Q：台灣應該發展核子武器嗎?）",[12,116429,116430],{},"（第四次作業出這樣的主題，真的蠻佩服教授的...）",[12,116432,116433],{},"儘管教授會在課堂上對該議題發表自己的看法，但每一項作業都沒有標準答案。回答的觀點與教授不同更完全ok，只要論述充足就行。像是據教授說，第四次作業最後結果就是各半。",[26236,116435,116437],{"id":116436},"期末報告至少3000字以上","期末報告：至少3000字以上",[23,116439,116440,116443,116448,116451,116454,116457,116460],{},[26,116441,116442],{},"距離期限兩週之前可以找教授討論題目。",[26,116444,116445],{},[5610,116446,116447],{},"如果你想挑戰自我的話，可以寫印度，挑戰教授最擅長的領域",[26,116449,116450],{},"期末當週同時 eeclass上繳交word檔以及課上繳交紙本。",[26,116452,116453],{},"參考來源不可引自維基（但可以用上面的CC共享圖片），通常參考國內外的新聞報導即可",[26,116455,116456],{},"3000字看起來很多，實際寫起來會發現很輕鬆就能達標。像我本來花了幾小時寫本文，結果後來發現文句潤飾一下、把心得結語加上去後，再把註腳跟參考資料加完，就已經超過8000了...",[26,116458,116459],{},"至於格式的部分，剛好直接把大學中文的格式規範拿來用就好了。總體看下來可能我這門課的報告完成度還比中文課高...",[26,116461,116462],{},[5610,116463,116464],{},"然後我現在一想，幹早知道就一份報告兩邊交==",[12,116466,116467],{},"另外，有做課上筆記、或聽相關演講的話，也可以在期末週給教授查閱，聽說也有加分。",[12,116469,116470],{},"不過每項評分細項、作業批改結果全都不公開，所以也不知道具體拿多少。只是最後看下來，評分應該都給得不錯。",[26236,116472,116474],{"id":116473},"給加簽嗎應該給簽但最後也沒滿","給加簽嗎？應該給簽，但最後也沒滿。",[26236,116476,116478],{"id":116477},"涼度-7","涼度：",[12,116480,116481],{},"常關注國際議題者：★★★★★\n除了做期末報告時特別忙，可能會花好幾天，其他時間都挺閒的\n不過期末報告再麻煩也沒中文報告難寫又耗時",[12,116483,116484],{},"沒接觸過、想了解國際議題者：★★★⯪☆\n如果都沒接觸過相關概念，那可能要花點時間研究上課內容以及作業之類",[12,116486,116487],{},"甜度：★★★★☆？\n不清楚成績分布，也不知道給分，估計期末報告跟課堂參與占大項",[26236,116489,114288],{"id":116490},"推薦度-2",[12,116492,116493],{},"對國際政治感興趣者：必須選",[12,116495,116496],{},"沒接觸過、想了解國際議題者：大推，不過可能得花點時間去理解",[26236,116498,116500],{"id":116499},"總結-負擔很輕又能學到東西對國際政治有興趣的可謂必選","總結：  負擔很輕又能學到東西，對國際政治有興趣的可謂必選",[26236,116502,116504],{"id":116503},"成績分布-不公開作業分數也是不公開","成績分布：  不公開。（作業分數也是不公開）",[12,116506,116507],{},"μ = 86.25 σ  = 5.27",[26236,116509,116511],{"id":116510},"備註","備註：",[12,116513,116514],{},"儘管教授有政黨背景，上課也會對各項議題發表些自己的觀點，",[12,116516,116517],{},"不過他不會強求你接受他的觀點，作業也不會因此差別給分(像是核武議題)，課堂包容性非常高。",[12,116519,116520],{},"但如果對這方面還是有點芥蒂的話，選課時可能就得多加考慮。",[36,116522],{},[7166,116524,116525],{"id":113743},"恐怖主義與反恐 Terrorism and Anti-Terrorism",[13438,116527,116529,116534,116537,116539,116541,116544,116546,116549],{"className":116528},[13441,13442],[12,116530,116531],{},[106,116532,116533],{},"科號：11410GE 144300",[12,116535,116536],{},"  \u003Cp>開課教師：侍建宇",[12,116538,115565],{},[12,116540,116211],{},[12,116542,116543],{},"上課時間：F3F4",[12,116545,116217],{},[12,116547,116548],{},"上課教材：自編講義（會放在eeclass）",[12,116550,114741],{},[26236,116552,114389],{"id":116553},"心得-5",[12,116555,116556],{},"課上會先從恐怖主義開始講解，一路往下講解到一些恐怖主義相關的概念，以及用以色列&巴勒斯坦、蓋達組織、ISIS、以及中國新疆等議題做舉例講解。",[16,116558],{},[12,116560,116561],{},"課上講義會放在eeclass，用的是全英文，所以如果不熟悉相關概念的話，可能會比較看不懂。不過教授上課是用中文講解，因此問題不大。（學期結束會取消分享，要存的記得。）",[12,116563,116564],{},"上課的時候不太點名，也很少管底下在幹嘛，加上或許是上課語調比較平的關係，所以可以看到很多人都沒在專心聽，甚至在打報告或滑手機...到後面很多人甚至沒來上課，使得50多人的教室只剩下20多人，而教授這時候就常常直接點名了。",[16,116566],{},[12,116568,116569],{},"有時教授會在上課找紀錄片來播，並在播完後問大家有沒有心得。不過就如上面所述的，大學大多數時候都是沒人敢舉手（或是都在滑手機），然後教授就會稍微對參與度的部分發牢騷而已。",[12,116571,116572],{},"教授對新疆、維吾爾議題特別有研究，曾在數十年前去過新疆當地田野調查，也常去參加相關議題訪談。而教授過去也是在香港大學任教，他也常常分享他以前在香港的教書經歷，尤其是在看到我們上課幾乎都沒互動的時候（不過好像香港大學那邊也是同樣的情況。）",[12,116574,116575],{},"不過教授有時補充會講嗨忘記時間，像是講香港大學的經歷或提巴勒斯坦問題的時候，常常一講就是十幾分鐘。這也讓課程內容有時會有點跳tone難以聽懂。",[16,116577],{},[12,116579,116580],{},"如上面所述，教授有時在到課人數太少時會突襲點名。所以期末的時候，估計大家很怕突然點名，才全部到課（難得看到教室坐滿），結果教授反而看大家都來了就沒點名了（教授也知道大家是來期末點名的）",[12,116582,116583],{},"另外學期中，教授有邀請一位與他同屬國安研究院的專家來演講國家監控與個人（恐怖組織）逃脫相關議題，而其中一份作業也是寫演講心得。但作業中除了學到的東西之外，如果有，教授也鼓勵我們指出該講師的演講不足之處。",[12,116585,116586],{},"不過可能因為講者是國安研究院的，有點講嗨了，演講內容更多是聚焦在國安議題，反而恐怖主義的部分講蠻少...",[16,116588],{},[12,116590,116591],{},"上課作業很少，除了一份上述演講心得外，其餘作業內容通常依照給定的一段訪談或新聞，並回答申論題，大概1000~2000字。",[12,116593,116594],{},"不過整學期只派過四次作業，其中兩次還是在期末之後。所以換算下來學期內只有兩份作業，涼到怕。",[12,116596,116597],{},"而且作業有交就有分，所以教授鼓勵就算遲交也盡量交。",[12,116599,116600],{},"而教授也鼓勵我們在群組內分享恐怖主義發展相關新聞，這也會記入平時考量之內。\n雖然教授有時在課上會小小抱怨我們貼的內容，但也不會因此刁難人或不給分。",[12,116602,116603],{},"值得注意的是：這堂課不下課，是直接1小時50分鐘不中斷的。不過也沒啥特別勞累的活動，所以差別不大就是。",[12,116605,116606],{},"另外，這學期因為週五放兩次假，加上教授有時又會去美國演講的關係，換算下來大概少了四堂課。",[26236,116608,114449,116610],{"id":116609},"點名方式偶爾點名-1",[75,116611,114795],{"className":116612},[114794],[12,116614,116615],{},"人太少的時候會點名，總共點過兩三次而已",[26236,116617,116619],{"id":116618},"成績算法好吧其實我有點忘記算法了","成績算法：好吧，其實我有點忘記算法了",[12,116621,116622],{},"只有4份作業，沒有考試，沒有期末報告",[12,116624,116625],{},[106,116626,116627],{},"但是這堂是16 + 2，代表有期末後作業。",[12,116629,116630],{},"根據課綱：",[7085,116632,116633,116645],{},[7088,116634,116635],{},[7091,116636,116637,116639,116642],{},[7094,116638,116296],{},[7094,116640,116641],{},"平時",[7094,116643,116644],{},"作業",[7104,116646,116647],{},[7091,116648,116649,116651,116653],{},[7109,116650,78951],{},[7109,116652,115489],{},[7109,116654,116655],{},"60%",[12,116657,116658],{},"平時部分：",[23,116660,116661,116664,116667,116670],{},[26,116662,116663],{},"可在LINE群組分享關於恐怖主義發展的新聞",[26,116665,116666],{},"時間不限定",[26,116668,116669],{},"每次1分，至多15分",[26,116671,116672],{},"其餘應該是依照發言、上課參與度去評",[12,116674,116675],{},"作業部份：依據恐怖主義相關內容（新聞\u002F訪談\u002F紀錄片），寫課後心得報告。 -",[23,116677,116678,116681,116684,116687],{},[26,116679,116680],{},"長度為1000-2000字。",[26,116682,116683],{},"每次作業期限大概給到20天以上，也寬限遲交（會不會扣分就不知道了）",[26,116685,116686],{},"有時會需要看1小時以上的訪談或紀錄片，這點須注意",[26,116688,116689],{},"一般大概一天內可以完成（但最後一項作業花我三天）",[12,116691,116692],{},"四次主題分別是：",[3603,116694,116695,116698,116701,116704],{},[26,116696,116697],{},"定義恐怖主義（從以哈戰爭，探討恐怖主義的定義）",[26,116699,116700],{},"國家恐怖主義與中國（從新疆問題探討國家恐怖主義的定義）",[26,116702,116703],{},"演講心得（演講完好像一兩週後才派這份作業...）",[26,116705,116706],{},"穆斯林兄弟會紀錄片",[12,116708,116709],{},"（5%去哪了？？？？）",[12,116711,116712],{},"後兩次作業在期末後2週各一份。",[12,116714,116470],{},[12,116716,116717],{},"給加簽嗎？應該全簽，但最後也沒滿",[26236,116719,115245],{"id":116720},"涼度-8",[12,116722,116723],{},"上課很輕鬆",[12,116725,116726],{},"前兩次作業一天內能完成，最後一次作業花了我三天...",[12,116728,116729],{},"不過期限十分長，教授也有把最後兩次作業的時間也有延長到期末之後",[26236,116731,116137],{"id":116732},"甜度-7",[12,116734,116735],{},"課堂參與度高點、上課多發言，教授給分也不會太刁難。",[12,116737,116738],{},"不過這學期這堂課平均比歷屆高了快5分，似乎是課上參與度比較好的關係？",[26236,116740,116742],{"id":116741},"推薦度-甜涼好課對相關議題有興趣更推薦","推薦度： 甜涼好課，對相關議題有興趣更推薦",[26236,116744,116746],{"id":116745},"總結-教授人很好但多給教授點回應真的不要去教室滑手機啦","總結：  教授人很好~但多給教授點回應，真的不要去教室滑手機啦",[26236,116748,116750],{"id":116749},"成績分布歷年好像比較多都81左右-但標準差大","成績分布：（歷年好像比較多都81左右 但標準差大）",[12,116752,116753],{},"μ =85.32 σ  = 3.96",[12,116755,116756],{},[1930,116757],{"alt":1932,"src":116758},"\u002Fimages\u002Fuploads\u002F%E5%9C%96%E7%89%879.png",[26236,116760,114637],{"id":116761},"補充-2",[12,116763,116764],{},"沒想到這學期搶到的兩堂通識都剛好是政治類的...不知道該說是有緣嗎？雖然兩堂還是都修得很快樂、成績也不低就是。",[36,116766],{},[7166,116768,116769],{"id":113759},"大學中文 \t College Chinese",[13438,116771,116773,114669,116778,116781,116783,116786,116789,116792],{"className":116772},[13441,13442],[12,116774,116775],{},[106,116776,116777],{},"科號：11410CL 101051\t",[12,116779,116780],{},"開課教師：王達甫 ",[12,116782,115565],{},[12,116784,116785],{},"課程類型：校定必修",[12,116787,116788],{},"上課時間：F5F6",[12,116790,116791],{},"修課人數：25->24 \u002F25（新生保留25）",[12,116793,116548],{},[26236,116795,116797],{"id":116796},"心得懲罰你國文沒考15級分的課","心得：懲罰你國文沒考15級分的課。",[12,116799,116800],{},"學測國文沒有15級分的學生都得修這堂課。如果說普通物理實驗是Excel使用訓練課，那麼這堂課就是Word實作練習課。",[12,116802,116804],{"className":116803},[113565],"這堂課跟高中的國文課十分不一樣。上課最開始會先講一些有趣的古文，以及討論一些相關的論文研究，之後帶到一些論文寫作規範、寫作技巧。然後則需要依照這些技巧，一步一步根據課上的論文格式教學，從前言、大綱到內容、結論逐漸寫出一篇完整的、5000字以上的個人報告。如果高中有寫過小論文的話，這堂或許會比較輕鬆。\n",[12,116806,116808,116809],{"className":116807},[113565],"雖然教授常常嚇人，讓人以為他很嚴厲、甚至第一堂課還鼓勵大家退掉選其他教授；不過實際上教授人很nice，文章有時找不到方向時也開放和他討論。中間有幾次發現作業太趕的時候，更是開放延長期限到假日，讚啦。\n",[16,116810],{},[12,116812,116814],{"className":116813},[113565],"此外，開學的時候就會分小組，並且古文介紹相關的課時會有一些小組活動、互相提問與討論。而後面也會有一次小組報告，讓小組做一篇簡報去講解一份論文（剛好跟物理段考撞在同一週，裂開），並且互相提問。不過也只是用來算小組參與度分數，最後個人報告還是各做各的。\n",[12,116816,116818,116819],{"className":116817},[113565],"上課討論的加分機會千萬要拿好，另外組長分數會往上調一個級距（A變A+等），建議主動爭取，尤其是大一的時候、大部分人還不熟環境而不敢自告奮勇時。\n",[16,116820],{},[12,116822,116824],{"className":116823},[113565],"這一班大學中文報告的題目是自己定，但可以找教授討論、以抓一個大致的方向。有時候有幾節課是這樣子的一對一討論課，上課前寫黑板決定討論順序，討論完就可以回去打報告了（但沒用，下一節還是有體育課...）\n",[12,116826,116828],{"className":116827},[113565],"中後期會有一次格式教學課，需要按照格式規範將文章的標點符號、參考資料、縮排、字體與格式在上課時改好。不過每一屆的結果都是上課時進度爆炸改不完，最後放學生回家改就是了，所以這點不必太擔心。由於太繁雜，基本上建議把格式規定跟文章丟給AI幫你抓。\t\n",[12,116830,116832,116833],{"className":116831},[113565],"除此之外，後期文章將近完成時，也會有一次類似口試的、約6分鐘的上台報告。這點千萬要注意別超時。\n",[16,116834],{},[12,116836,116837],{},"雖然講了這麼多，教授人也很nice，但他課上也提到之後應該也不會帶大學中文課了，實在可惜。",[26236,116839,114169,116841],{"id":116840},"點名方式-每堂點名-3",[75,116842,114173],{"className":116843},[29982],[26236,116845,116847],{"id":116846},"成績算法-期末報告5000字以上沒交死當","成績算法： 期末報告5000字以上，沒交死當。",[7085,116849,116850,116872],{},[7088,116851,116852],{},[7091,116853,116854,116856,116859,116862,116865,116868,116870],{},[7094,116855,102197],{},[7094,116857,116858],{},"期中大綱",[7094,116860,116861],{},"小組導讀",[7094,116863,116864],{},"稿件修改（格式）",[7094,116866,116867],{},"個人簡報（口試）",[7094,116869,116308],{},[7094,116871,116641],{},[7104,116873,116874],{},[7091,116875,116876,116878,116880,116882,116884,116886,116888],{},[7109,116877,78951],{},[7109,116879,78936],{},[7109,116881,78951],{},[7109,116883,115681],{},[7109,116885,78951],{},[7109,116887,78936],{},[7109,116889,115684],{},[26236,116891,116893],{"id":116892},"調分嗎-不調因為我就差一分","調分嗎？ 不調，因為我就差一分==",[26236,116895,116897],{"id":116896},"給加簽嗎-不給簽","給加簽嗎？ 不給簽",[26236,116899,116900],{"id":114265},"涼度： ★★⯪☆☆",[12,116902,116903],{},"上課負擔不重，基本上負擔幾乎全部來自於個人報告",[26236,116905,115529],{"id":116906},"甜度-8",[12,116908,116909],{},"差一分A+ （我算了一下，好像甚至是零點多分） ，幹",[26236,116911,116913],{"id":116912},"總結excel有普物實驗word有大學中文","總結：Excel有普物實驗，Word有大學中文",[26236,116915,116917],{"id":116916},"成績分布-μ-8304-σ-1770","成績分布： μ = 83.04 σ  = 17.70",[12,116919,116920],{},[1930,116921],{"alt":1932,"src":116922},"\u002Fimages\u002Fuploads\u002F%E5%9C%96%E7%89%8710.png",[36,116924],{},[7166,116926,116927],{"id":113774},"大一體育 Physical Education",[13438,116929,116931,116936,116939,116942,116944,116947],{"className":116930},[13441,13442],[12,116932,116933],{},[106,116934,116935],{},"11410PE 111039",[12,116937,116938],{},"  \u003Cp>開課教師：戴偉國",[12,116940,116941],{},"課程學分：0",[12,116943,116785],{},[12,116945,116946],{},"上課時間：F7F8",[12,116948,116949],{},"修課人數：56 ",[26236,116951,114389],{"id":116952},"心得-6",[12,116954,116955],{},"上課地點會前堂課email或體幹公布。基本上同一班（清班\u002F華班）會是同一個體育老師。",[12,116957,116958],{},"每堂課都是不同的運動內容，會一堂一堂帶我們造訪全校的各個體育設施，帶同學熟悉環境的同時，也讓大家能為未來五次的體育必修志願填寫做準備。",[12,116960,116961],{},"另外，雖然體育老師人看起來很嚴肅，不過其實人很好，有時上課講話也常逗笑大家。",[12,116963,116964],{},"雖然0學分，電機系的同學真的得好好珍惜這堂課，因為從期中開始，這大概是唯一可以活動筋骨的時間了。",[12,116966,116967],{},"期末測驗就是一些關於學校設施&運動姿勢的選擇題，5分鐘就能完成，比重也很輕，這大概是我上大學後第一次寫到選擇題...",[26236,116969,114169,116971],{"id":116970},"點名方式-每堂點名-4",[75,116972,114173],{"className":116973},[29982],[26236,116975,116977],{"id":116976},"成績算法憑記憶","成績算法：(憑記憶)",[12,116979,116980],{},"每個大一需要參加運動會項目",[7085,116982,116983,117004],{},[7088,116984,116985],{},[7091,116986,116987,116990,116993,116996,116999,117001],{},[7094,116988,116989],{},"體適能",[7094,116991,116992],{},"運動會（個人）",[7094,116994,116995],{},"運動會（團體）",[7094,116997,116998],{},"游泳（6~7章）",[7094,117000,116296],{},[7094,117002,117003],{},"Midterm（4~5章）",[7104,117005,117006],{},[7091,117007,117008,117010,117015,117019,117021,117026],{},[7109,117009,78951],{},[7109,117011,117012,117014],{},[5610,117013,78951],{}," （取消）",[7109,117016,117017,117014],{},[5610,117018,78951],{},[7109,117020,78951],{},[7109,117022,117023,117025],{},[5610,117024,78920],{}," 70%",[7109,117027,78951],{},[16,117029],{},[12,117031,117032],{},"由於運動會取消（感謝颱風），原本運動會分數併入出席分。",[12,117034,117035],{},"不過有參加預賽仍然每個項目加2~3分",[26236,117037,117039],{"id":117038},"給加簽嗎不確定","給加簽嗎？不確定",[12,117041,117042],{},"不過有4~5個本系大二生（似乎是轉學生）   ",[26236,117044,115245],{"id":117045},"涼度-9",[12,117047,117048],{},"有到就會過，取消運動會之後就沒啥要做的事情了  ",[26236,117050,117052],{"id":117051},"甜度-9","甜度：★★★★☆   ",[12,117054,117055],{},"不過0學分也沒差  ",[26236,117057,117059,117060],{"id":117058},"總結-健身熟悉環境-讓你沒辦法三點回家","總結：  健身+熟悉環境+ ",[5610,117061,117062],{},"讓你沒辦法三點回家",[26236,117064,117066],{"id":117065},"成績分布-不公開","成績分布：  不公開   ",[12,117068,117069],{},"μ = 4.08(GPA，對應91.93) ， σ  = 0.24",[6964,117071,117072],{"id":117072},"小結",[12,117074,117075],{},"修完有獎勵，修不完有懲罰。",[12,117077,117078],{},"選課須適度，參考須謹慎。尤其是千萬別盡信GPA 4.3的卷哥卷姐的心得，很容易一不小心就翻車。",[12,117080,117081],{},"這學期從微積分A開頭之後，從9\u002F30~11\u002F20，段考幾乎就沒有中斷過，一週接一週，我的活動範圍也幾乎只剩下教室與圖書館。",[12,117083,117084],{},"一堂課究竟如何、loading多重，還是要自己去修才知道。有時看著前人修著輕鬆，但自己去選也可能完全不是那樣。",[12,117086,117087],{},"另外，這學期各科的給分，真的無處不反映了「選擇大於努力」：不論是程式設計、微積分A、邏輯設計甚至是線性代數，兩位教授的給分真的可說天壤之別。希望下學期能保持這樣的選課運吧。",[12,117089,117090],{},"嘛，不過對於這學期，講的還是太多了，未來估計開別篇文再詳細講講吧。",[12,117092,117093],{},"至於未來還會不會寫大一下、大二大三......那就得看我經過三電的洗禮後，還有沒有餘力了。但首先，拜託保佑我選到電路學。",[12,117095,117096],{},"不管如何，如果你有幸看到這篇文，感謝願意你看到這邊。有閒之時，也歡迎去看看本部落格的其他文章~",[9332,117098,31966],{},{"title":180,"searchDepth":181,"depth":181,"links":117100},[117101,117102,117103,117104,117117],{"id":102197,"depth":181,"text":102197},{"id":113580,"depth":181,"text":113580},{"id":113782,"depth":181,"text":113782},{"id":114106,"depth":181,"text":114106,"children":117105},[117106,117107,117108,117109,117110,117111,117112,117113,117114,117115,117116],{"id":113620,"depth":7430,"text":114109},{"id":113635,"depth":7430,"text":114319},{"id":113650,"depth":7430,"text":114660},{"id":113665,"depth":7430,"text":114979},{"id":113680,"depth":7430,"text":115301},{"id":113696,"depth":7430,"text":115551},{"id":113712,"depth":7430,"text":115979},{"id":113728,"depth":7430,"text":116195},{"id":113743,"depth":7430,"text":116525},{"id":113759,"depth":7430,"text":116769},{"id":113774,"depth":7430,"text":116927},{"id":117072,"depth":181,"text":117072},"2026-01-21","一個雜魚，在充滿怪物的學校中前半年的生存狀況","https:\u002F\u002Ftruth.bahamut.com.tw\u002Fs01\u002F202601\u002Ff2c1c67681cab28e27fe24104ceae821.JPG",{},"\u002Fblog\u002F清華大學電機系-大一雜魚修課心得-總結",{"title":113540,"description":117119},{"loc":117122},"blog\u002F清華大學電機系-大一雜魚修課心得-總結",[114106,117127,117128,117129,117130,117131,34660],"清華大學","電機","反思","大一","新生","wB5AdcuvU61Bwqaw_jDiMD5e5l1gaDIQ-x12Tiasx7U",{"id":117134,"title":117135,"author":117136,"body":117137,"date":117198,"description":117199,"draft":185,"edited_at":194,"extension":186,"featured_image":117200,"meta":117201,"navigation":189,"path":117202,"pinned":185,"seo":117203,"sitemap":117204,"stem":117205,"tags":117206,"__hash__":117207},"blog\u002Fblog\u002F小紅書被封禁究竟是濫權還是無視法規呢.md","小紅書被封禁究竟是濫權還是無視法規呢?","osborrrrn",{"type":9,"value":117138,"toc":117196},[117139,117180,117189],[12,117140,117141,117142,117144,117145,117147,117148,117150,117151,117153,117154,117156,117157,117159,117160,117162,117164,117165,117167,117168,117171,117172,117174,117176,117177,117179],{},"小紅書為近年風靡於台灣女性與學生團體，但在2025年12月4日由內政部與刑事警察局聯合通過記者會公布為期一年的禁令，同時要求網路業者立即封鎖小紅書的域名和伺服器。",[16,117143],{},"\n小紅書在台擁有超過300萬人使用然而經國家安全局資安檢測，15項指標均未符合標準，且該軟體自113年起涉入1706件詐騙案件造成國民損失約為2億4,768萬餘元\n執法機關偵辦犯罪時面臨重大困難，形成實質法律真空。而內政部為保障台灣用戶隱私安全於2025年10月14日海基會函請小紅書母公司中國行吟信息科技（上海）有限公司依法提出具體改善作為，但迄今業者無任何回應。試圖迴避中華民國法律管轄。且放任詐騙與其他違法行為置之不理故依照《詐欺犯罪危害防制條例》第42條「詐欺犯罪防制緊急事件」規定，將對「小紅書」APP發布網際網路停止解析或限制接取之命令，暫定期間一年；後續將視該公司是否善意回應、主動配合我國相關法規並確保國人數位安全，再研議後續處置。",[16,117146],{},"\n(節取於內政部守護民眾資訊安全 對涉詐高風險APP發布暫行措施114-12-04)",[16,117149],{},"\n對於此事件許多民眾在網路反映：",[16,117152],{},"\n1.是否違反憲法",[16,117155],{},"\n2.政府為何不去封禁FACEBOOK(Meta)、LINE 、TikTok",[16,117158],{},"\n3.政治因素。",[16,117161],{},[16,117163],{},"\n而我對於此事件的想法",[16,117166],{},"\n1.來討論是否違反憲法：網民認為說，直接封禁過度限制了人民言論與資訊自由。然而此次封禁由內政部新聞稿知是違反了《詐欺犯罪危害防制條例》，而內政部依法行政並要求ISP也就是",[106,117169,117170],{},"網際網路服務供應商","封鎖小紅書的域名和伺服器，此APP造成許多危害且不願意在台設立據點提供檢方調查犯罪，而現今我國與中國沒有合適的共同反詐管道，因此無法有效限制詐騙故直接封禁。可能符合憲法中的必要性與比例原則。",[16,117173],{},[16,117175],{},"\n2.政府為何不去封禁FACEBOOK(Meta)和LINE、TikTok等等國際平台：上述平台同樣充斥著詐騙如以Meta為例：Meta2025一月至4月共撤銷了430萬則詐騙廣告數億新台幣的損失，為何Meta不必被封禁呢因為Meta在台北南山廣場大樓具有營運地點，也有提供過詐騙分子IP與行為訊號協助警方打擊詐騙並快速移除冒名詐騙貼文，然而Meta仍然充斥著詐騙因此被數發部多次開罰。而LINE在台同樣擁有據點為在台北市宏匯瑞光大樓。TikTok雖並未在台有據點然而平台在台灣有配合法規設置法律代表人也推出防詐宣導活動，並且有提供資料給予警方調查詐騙。\n",[16,117178],{},"\n3.政治因素：名面上政府是以反詐為由來封禁小紅書然而是否因政治因素無從得知故本文不多做評論。",[12,117181,117182,117183,117185,117186,117188],{},"結論",[16,117184],{},"\n這篇文章僅為個人薄見若有錯誤歡迎討論。而這次的行動也是台灣首次大規模封禁社群平台，看到了許多人的討論和觀點，是否違憲我也說不準只能表達我個人的觀點以及等待憲法法庭釋憲。而許多人利用這平台來學習美妝、調查觀光景點，學習知識甚至是家居裝潢是個具有多功能的平台失去了這個程式對於部分人來說十分重大，而後續的處理辦法也值得我們持續觀察。",[16,117187],{},"\n若釋憲最終推翻了我所描述的內容我將刪除此篇文章，感謝你看到這裡。",[12,117190,117191,117192],{},"封面來自於中央社2025 12\u002F6\u002F19:50 ",[2330,117193,117194],{"href":117194,"rel":117195},"https:\u002F\u002Fwww.cna.com.tw\u002Fnews\u002Faipl\u002F202512060192.aspx",[2334],{"title":180,"searchDepth":181,"depth":181,"links":117197},[],"2025-12-07","小紅書被台灣政府暫時封禁一年，其原由是否合乎法規與具備正當理由，並反思其所造成之影響。","\u002Fimages\u002Fuploads\u002F小紅書封禁.jpg",{},"\u002Fblog\u002F小紅書被封禁究竟是濫權還是無視法規呢",{"title":117135,"description":117199},{"loc":117202},"blog\u002F小紅書被封禁究竟是濫權還是無視法規呢",[117129],"KYVoO3s0YDjW1icbpaccFUQQ_BNjR5Ppm5st5NjvjQg",{"id":117209,"title":117210,"author":116189,"body":117211,"date":117245,"description":117246,"draft":185,"edited_at":194,"extension":186,"featured_image":194,"meta":117247,"navigation":189,"path":117248,"pinned":185,"seo":117249,"sitemap":117250,"stem":117251,"tags":194,"__hash__":117252},"blog\u002Fblog\u002F2025數學回顧.md","2025數學回顧",{"type":9,"value":117212,"toc":117243},[117213,117216,117219,117222,117225,117228,117231,117234,117237,117240],[12,117214,117215],{},"2025對我來說可能是挑戰特別多的一年，很不幸的，這些挑戰都是來自生活、家人之類的，和數學無關，所以自然也壓縮到我讀書的時間，所以在年初時我直接開啟了gap year的模式，雖然學到的東西也蠻多的。",[12,117217,117218],{},"今年主要的讀書方向還是代數幾何，但是因為興趣的關係，我並沒有走向複幾何跟雙有理之類的偏代幾自身的研究方向，主要還是關注特定的問題所需要的代幾工具，就像仔細run一次étale cohomology是我今年的主要目標。這部分我是先讀了Poonen的Rational points on variety，大概在年初花了1個月看了第一次全書，最近準備看第二次跟習題，這也很符合我的讀書習慣，我常常用最短時間先抓各章的大致目標、重點以及難度，這樣可以方便我真的需要時比較輕鬆的掌握，在一些論文需要時也能很直接的接受那些我不知道怎麼證的結論。在這本書中，能看到蠻多的老朋友，比如fppf fpqc morphism之類的，之後還有Weil conjecture的一些東西之類的，最後一章是一些曲面理論，目前還沒用到，但是我預計會讀一下，最近和蠻多人討論還有在知乎逛到一些雙有理的在講一些曲面分類的東西，感覺就特別有趣。講到fppf fpqc morphism就會想到年初我也讀了FGA Explain 裡面很多目前被我當作black box的東西，比如Hilbert scheme，我是預計明年上半年就解決這本書，目前的記憶是它沒有習題，可能要找一些習題來練筆。",[12,117220,117221],{},"講到代幾就想到今年雙十一買了中國那邊給大學生的代數幾何書，主要內容還是老生常談的variety跟曲線論之類的，以Riemann-Roch 作為最終目標，有趣的是裡面蠻多的結果是我沒看過的，這讓我想起去年讀Weil的代幾書，展示了Grothendieck scheme理論之前的代幾模樣，也就是義大利學派他們的一些代數曲面分類的一些基礎跟相關的結果(至於核心的分類結果我不知道，所以就不多說什麼了XD)。今年也看了陳志杰的模、範疇、同調代數與層，這本書我認為是最棒的給要讀代幾的人的同調代數教材，一方面我們並不需要太多的同調代數知識，比如Kan decomposition，另一方面這本書用相對薄的厚度把代幾需要的同調代數都包含了進來，是一本我難得會翻下去想一看再看的中文數學書！",[12,117223,117224],{},"講到這裡就想到最近剛出版的Rising sea，這本書可以說是非常多人推薦的一本教科書，雖然我不是用這本書入門的，但是因為很多人都引用了這本前身的講義，所以就花了1800左右買下了。我目前仔細看了5章，也就是到Scheme自身的一些property，沒記錯的話應該是只有local的部分，整體來講沒有說特別符合我的胃口，誠然裡面的敘述是那種很苦口婆心式的敘述方法，但是裡面很多定義其實蠻口語化的，強調幾何的直觀。可能是習慣GTM52那種formal definition後再直觀，所以跟我不合，但是根據大家都看到一定程度，我自己還是會找時間看完它ww",[12,117226,117227],{},"和代幾相關但是也沒有太多相關的是暑假讀的K theory，我讀的是黎景輝的書，主要是代數詮釋的K theory，裡面的內容我查不多忘光了，但是最有趣的還是一些K group的認識，比如Grothendieck group跟Milner group，如果明年有空應該會再看一次，目標是把scheme的K theory給吃掉。",[12,117229,117230],{},"和自己方向相關還要讀的書大概是代數數論的部分，因為自己其實不需要特別多這方面的知識，因此看的書偏向一些比較基礎的書，比如Neukirch Marcus之類的書，Marcus是年初買的書，在最近才把它大致看完跟寫習題，除此之外我還買了馮克勤的代數數論，只剩下最後的Class field theory因為可能不需要就放著不看了。其實在讀馮的書到解析理論時，我突然發現那些東西特別好玩，所以現在我的購物車裡面放了很多解析數論的書，而且這些書都好貴，果然是高貴的數論(X",[12,117232,117233],{},"再來是一些被我當作”閒書“的數學書，今年的這類閒書真的蠻多的簡單講幾本好了。",[12,117235,117236],{},"我第一個想到的是一位跟我不錯的幾何分析教授推薦給我的一本GSM的extrinsic geometric flows，他給我的意見是可以看看前三章，也就是一些PDE的結果跟curve shortening flow，這本書很符合我對幾何分析的刻板印象，內容一堆不等式估計，有些upper bound的結果蠻醜的w 不過裡面的一些漂亮結果是書的後面才證的，主要是mean curvature flow的結果，至於有沒有機會繼續讀基本是看緣分了。",[12,117238,117239],{},"再來是Spectral geometry，也是上述教授推薦給我的，他自己也想要嘗試往這個方向，所以丟了一個40幾頁的note給我看。這個方向主要是考慮manifold上的laplacian operator，有些問題比如根據旋律去判斷鼓的形狀也是這個方向的問題之一，這部分的推廣形式會用到moduli space，是我第一次看見微幾根代幾的相交部分，雖然有聽過spectral algebraic geometry，但是這部分好像是代數拓撲那邊的東西，所以關於spectral geometry這個方向，我應該會找時間和教授討論一下要不要一起讀。",[12,117241,117242],{},"最後看了一下自己打的東西，注意到基本上都是講書，沒有論文之類的。我自己的理由應該是因為書給我的啟發比較多，因為目前的階段看書還是比較重要的一環，論文的話，我目前應該只能看出作者做了啥，沒有一個完整的圖像在我腦海裡，應該再幾年就能講一篇論文然後結合看過的相關資料講一些東西了XD",{"title":180,"searchDepth":181,"depth":181,"links":117244},[],"2025-12-05","上課無聊來隨便寫一下",{},"\u002Fblog\u002F2025數學回顧",{"title":117210,"description":117246},{"loc":117248},"blog\u002F2025數學回顧","HgI0glEIXL8H0oODA-g_MiCTJyncfDx9LQh4femnkaY",{"id":117254,"title":117255,"author":6946,"body":117256,"date":118704,"description":118705,"draft":185,"edited_at":194,"extension":186,"featured_image":118706,"meta":118707,"navigation":189,"path":118708,"pinned":185,"seo":118709,"sitemap":118710,"stem":118711,"tags":118712,"__hash__":118713},"blog\u002Fblog\u002Flatex-example.md","LaTeX Example",{"type":9,"value":117257,"toc":118702},[117258,117311,117318,117456,117462,117535,117542,117704,117711,117870,118106,118116,118369,118375,118559,118568,118700],[12,117259,117260,117261,117310],{},"Here is some inline math: ",[24923,117262,117264],{"className":117263,"jax":24927},[24926],[24929,117265,117268,117281],{"style":26868,"xmlns":24932,"width":117266,"height":38070,"role":1930,"focusable":8058,"viewBox":117267,"xmlnsXLink":24936},"8.699ex","0 -833.9 3845.1 915.9",[24938,117269,117270,117272,117274,117276,117279],{},[24941,117271],{"id":86909,"d":25869},[24941,117273],{"id":24971,"d":24972},[24941,117275],{"id":32154,"d":28310},[24941,117277],{"id":117278,"d":27045},"MJX-1-TEX-I-1D450",[24941,117280],{"id":76019,"d":25928},[25022,117282,117283],{"stroke":25024,"fill":25024,"stroke-width":9180,"transform":25025},[25022,117284,117285,117289,117293,117297],{"dataMmlNode":25028},[25022,117286,117287],{"dataMmlNode":25942},[25033,117288],{"dataC":25945,"xLinkHref":86962},[25022,117290,117291],{"dataMmlNode":25031,"transform":30197},[25033,117292],{"dataC":25078,"xLinkHref":25079},[25022,117294,117295],{"dataMmlNode":25942,"transform":30203},[25033,117296],{"dataC":28479,"xLinkHref":32194},[25022,117298,117300,117305],{"dataMmlNode":31565,"transform":117299},"translate(2975.6,0)",[25022,117301,117302],{"dataMmlNode":25942},[25033,117303],{"dataC":27054,"xLinkHref":117304},"#MJX-1-TEX-I-1D450",[25022,117306,117308],{"dataMmlNode":26031,"transform":117307},"translate(466,363) scale(0.707)",[25033,117309],{"dataC":26058,"xLinkHref":76028},". This is a classic.",[12,117312,117313,117314,117317],{},"Now, let's look at the ",[106,117315,117316],{},"Quadratic Formula"," as a display equation:",[12,117319,117320],{},[24923,117321,117323],{"className":117322,"jax":24927},[24926],[24929,117324,117329,117357],{"style":117325,"xmlns":24932,"width":117326,"height":117327,"role":1930,"focusable":8058,"viewBox":117328,"xmlnsXLink":24936},"vertical-align: -0.797ex;","14.815ex","3.262ex","0 -1089.5 6548.3 1441.6",[24938,117330,117331,117333,117335,117337,117339,117343,117347,117349,117352,117354],{},[24941,117332],{"id":32285,"d":32148},[24941,117334],{"id":42603,"d":24972},[24941,117336],{"id":32288,"d":25910},[24941,117338],{"id":42598,"d":42548},[24941,117340],{"id":117341,"d":117342},"MJX-2-TEX-N-B1","M56 320T56 333T70 353H369V502Q369 651 371 655Q376 666 388 666Q402 666 405 654T409 596V500V353H707Q722 345 722 333Q722 320 707 313H409V40H707Q722 32 722 20T707 0H70Q56 7 56 20T70 40H369V313H70Q56 320 56 333Z",[24941,117344],{"id":117345,"d":117346},"MJX-2-TEX-N-221A","M95 178Q89 178 81 186T72 200T103 230T169 280T207 309Q209 311 212 311H213Q219 311 227 294T281 177Q300 134 312 108L397 -77Q398 -77 501 136T707 565T814 786Q820 800 834 800Q841 800 846 794T853 782V776L620 293L385 -193Q381 -200 366 -200Q357 -200 354 -197Q352 -195 256 15L160 225L144 214Q129 202 113 190T95 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example on how to write latex on this blog.","\u002Fimages\u002Fuploads\u002Flatex.png",{},"\u002Fblog\u002Flatex-example",{"title":117255,"description":118705},{"loc":118708},"blog\u002Flatex-example",[25028],"Um1SZ-xPIu95MQNFoYI97Jmi98XTTv03VGWXpQeU8tM",{"id":118715,"title":118716,"author":116189,"body":118717,"date":118704,"description":118846,"draft":185,"edited_at":194,"extension":186,"featured_image":118847,"meta":118848,"navigation":189,"path":118849,"pinned":185,"seo":118850,"sitemap":118851,"stem":118852,"tags":118853,"__hash__":118855},"blog\u002Fblog\u002F微積分隨筆-未完成版.md","微積分隨筆",{"type":9,"value":118718,"toc":118840},[118719,118722,118730,118734,118737,118740,118743,118747,118750,118755,118758,118761,118766,118769,118772,118775,118778,118781,118784,118787,118790,118793,118796,118799,118802,118805,118810,118814,118819,118822,118825,118828,118831,118834,118837],[12,118720,118721],{},"這篇我應該主要講的是一些回頭看自己學習微積分or to say 分析的歷程，講一些自己想講的東西。而因為自己基本上都是在讀代數相關的東西，分析很久沒有碰，所以可能講的不是特別好。",[7351,118723,118724,118727],{},[12,118725,118726],{},"首先，這裡的微積分的定義有點模糊，因為我自己是直接看Baby Rudin學微積分的，然後再在大一的微積分課程補了一下自己的計算能力不足，所以我打算講到啥就講啥，不想太嚴格。",[12,118728,118729],{},"再來，以下可能會出現3種語言，中文、英文 and 台語。",[6964,118731,118733],{"id":118732},"why-we-learn-calculus","Why we learn Calculus",[12,118735,118736],{},"對非數學系的人來說，微積分是一個語言，它是用來描述那些連續相關性質的語言，比如自然界中的種種現象，我們可以用微分方程去描述，而微分方程的基礎語言就是微積分，而對於統計機率來說，微積分也很重要，這是因為考慮連續的分佈(distribution)，我們要看那些random variable是遵循什麼樣的分佈時，就會利用微積分的語言了。",[12,118738,118739],{},"在數學系的視角來看，微積分就是一個很基礎的理論，它可以讓你第一次從高中那種比較不嚴格的論述走向嚴格論述，這也是數學系新生可能第一次面對到的困難–epsilon-delta定義。",[12,118741,118742],{},"在這之後的章節大概就是微積分的幾個topic了，定義連續、連續函數、連續函數的幾大定理、定義微分及微分的性質blabla，其中我覺得比較重要的是隱函數定理(Implicit Function Theorem)。在幾何中，隱函數定理很常出現，我記得在去年讀Loring W. Tu的那本manifold中，突然出現了隱函數定理，然後去問了系上做幾何分析的教授，才發現隱函數定理大有學問，這也讓我想到自己在代數路上走太深了，沒有怎麼注意其他方向的有趣東西，真是慚愧。",[6964,118744,118746],{"id":118745},"some-topics-of-calculus","Some topics of Calculus",[12,118748,118749],{},"這邊我想要講一下微積分的看點，會粗略分為計算跟理論兩個部分。",[23,118751,118752],{},[26,118753,118754],{},"計算",[12,118756,118757],{},"計算這部分真的算老生常談，首先是計算能力。計算能力需要一次又一次的練習去練起來，根據我的經驗，無論是教授或是身邊同學，大家都這樣講的，所以還是不要像我一樣，都看題目然後想到方法就不動手算。",[12,118759,118760],{},"第二部分是積分法則。這部分我還記得大一時我覺得非常無聊，因為我實在懶得因為考試去背這些東西ww，但是和同學討論後，他給我了一個我覺得很棒的理由，他叫我給出有理函數都是可積的證明，而這個就是你可以在微積分課本上看到那些長長的公式的理論基礎。而這些積分法則在計算上真的很方便，這在有上到常微分方程課程的科系就能知道的事實。",[23,118762,118763],{},[26,118764,118765],{},"理論",[12,118767,118768],{},"這部分我覺得可以講幾個有趣的看點",[12,118770,118771],{},"首先是實數，這部分在大一時基本不會遇到，除非你是數學系，尤其是國外的數學系學生。實數定義本身就很有趣，在分析上，我有看過用公理直接定義的(比如Zorich那本數學分析書)、用Dedekind cut定義(大部分書籍都這樣定義)以及用Cantor sequence定義(Tao以及Amann -Escher裡面有）。不同方法代表了作者的品味，比如公理法就是簡單粗暴，我很喜歡這種風格，另外兩種是比較符合數學那種定義定理證明的調調，但是都需要花一個大篇幅去講，除非像Rudin直接塞在習題那就沒差了XD",[12,118773,118774],{},"第二個看點我覺得是微積分基本定理",[12,118776,118777],{},"微積分基本定理作為一個現在高中生都知道的定理，但是它背後藏了很多很深的東西，這些東西可以顯現數學家為了一般化，所以去設計出後續的很多理論，而微積分基本定理是一個引入Lebesgue Integral的一個很好的motivation。",[12,118779,118780],{},"在一般的書籍中，我們能看見這樣的微積分基本定理的條件：",[12,118782,118783],{},"F(x)可導、f(x)可積、F'(x)=f(x)",[12,118785,118786],{},"這個形式我記得跟高中不同，因為F的導數不一定可基以及f不一定可導，所以我們不要求f一定是F的導數，也就是這個形式是比高中時還要強的。",[12,118788,118789],{},"除此之外我們還有其他更強的形式，但是都得不到像高中那種簡潔的形式，這是因為Riemann積分的侷限性，而為了解決這個問題，我們可以引入Lebesgue積分，而這部分就是其他故事了，以後有機會再講。",[12,118791,118792],{},"第三個看點是隱函數定理，而它的重要性前面已經講過了，這邊稍微講一下證明它的幾個方法。",[12,118794,118795],{},"第一個方法是傳統的解方程，也就是做消元，之後做歸納，這個方法你能在100多年前的分析書找到，真的很傳統。第二個方法是找extreme value，這個方法我瞭解的不多，因為這是當初同學報告時給的酷東西，大概的難點是要找到一個函數去幫助你做出一些東西以及這個方法不能推廣到更一般的space，只能在歐式空間。第三個是現代最常見的方法，大家是叫它不動點法，它只需要一些前置知識比如metric space跟Banach不動點定理，就可以很漂亮的解決，而且在這個證明下，我們可以得知隱函數定理在一般的Banach space也是valid的。",[12,118797,118798],{},"最後一個看點我想要講的是一些作者會把拓撲、實分析、複分析以及流形之類的知識塞進課本中。",[12,118800,118801],{},"這部分有的人會覺得很討厭，因為課程安排可能會重疊，比如Courant-John的那三本Introduction to Calculus and Analysis中，塞了非常多東西，導致在之後的課程要上分析時，會重疊到，而且在那三本中的敘述就已經很嚴格了，並不只是單純的topic重疊。",[12,118803,118804],{},"對我來說，我覺得把一些高級的知識放進大一大二的課程中是可行的，因為那些知識雖然說是高級，只不過是一些常見的東西抽象化之後的產物，而且這個抽象化沒有特別抽象。比如metric space，只要對real spce有最簡單的理解，就可以用類比的方式去看它。",[7351,118806,118807],{},[12,118808,118809],{},"這邊還有一個看點超級重要，這邊因為我不想講所以跳過。如果說看一本分析書要看哪裡，就是要看它對於重積分換元定理的證明長什麼樣。這個定理要完完全全的證明可能要7 8頁正文，有些書是很含糊的就帶過。",[6964,118811,118813],{"id":118812},"ways-to-learn-calculus","Ways to learn Calculus",[12,118815,118816],{},[106,118817,118818],{},"實踐是檢驗真理的唯一標準",[12,118820,118821],{},"數學這個科目不可避免的要做題目，但是要根據自己的胃口去寫多少以及怎麼樣的題目。這部分我的大致建議是一個章節盡量不要超過10題，選題方法可以去問看看自己的老師或是教授，甚至是去問AI，主要是問哪些題目是經典的題目就行了。",[12,118823,118824],{},"再來是選書，以非數學系的學生來講，只要找到一本最適合自己的書就好了，這部分可以去網路找看看自己相同科系或者背景的人的心得。",[12,118826,118827],{},"而對於數學系的學生，我認為除了一本最適合自己的書外，還要找好幾本參考書做參考。以我為例子好了，我當初是Baby Rudin當精讀，Zorich以及Amann-Escher當作參考書，而其中後者第三卷也是我精讀的，這是因為Baby Rudin最後講了Lebesgue theory ，但是大家的共識跟我自己也覺得寫的不是特別好，所以就用其他書了。",[12,118829,118830],{},"最後是心態，學習數學本身就是挫折滿滿，這是不管在哪個階段都會遇到的，對於挫折這件事，應該是要根據個人而去找方法解決，我自己是本身就習慣了這種挫折，所以才一直學數學到現在並且未來還想繼續走這條路下去。",[6964,118832,118833],{"id":118833},"後記",[12,118835,118836],{},"這是我第一次寫這種文章，歡迎指出缺點。",[12,118838,118839],{},"以前都會覺得這種科普文章沒什麼，但是現在發現要給一些不是數學領域的人知道數學的一些東西，不是特別容易的事哈哈哈。",{"title":180,"searchDepth":181,"depth":181,"links":118841},[118842,118843,118844,118845],{"id":118732,"depth":181,"text":118733},{"id":118745,"depth":181,"text":118746},{"id":118812,"depth":181,"text":118813},{"id":118833,"depth":181,"text":118833},"應該高中生能看懂吧","\u002Fimages\u002Fuploads\u002Fscreenshot_20251104_160859_onedrive.jpg",{},"\u002Fblog\u002F微積分隨筆-未完成版",{"title":118716,"description":118846},{"loc":118849},"blog\u002F微積分隨筆-未完成版",[118854],"隨筆","U2e6vXmnlrnTxLcmhQYo-028yrMQF1MQSJKwBVWL-rY",{"id":118857,"title":118858,"author":118859,"body":118860,"date":119090,"description":119091,"draft":185,"edited_at":194,"extension":186,"featured_image":119092,"meta":119093,"navigation":189,"path":119094,"pinned":185,"seo":119095,"sitemap":119096,"stem":119097,"tags":119098,"__hash__":119100},"blog\u002Fblog\u002Fptau217-marks-tau-seeding-hotspots-in-early-alzheimer’s-cortex.md","pTau217 Marks Tau Seeding Hotspots in Early Alzheimer’s Cortex","automata",{"type":9,"value":118861,"toc":119079},[118862,118864,118867,118875,118878,118889,118892,118897,118901,118906,118910,118913,118916,118925,118929,118932,118935,118944,118948,118951,118959,118965,118974,118977,118988,118993,118997,119003,119006,119026,119032,119037,119041,119044,119047,119049,119055,119060,119074],[6964,118863,82448],{"id":82447},[12,118865,118866],{},"阿茲海默症（Alzheimer’s disease, AD）是一種神經退行性疾病，其典型特徵包含 β-類澱粉蛋白（amyloid-β, Aβ）沉積與 tau 病理的累積。tau 病理不僅表現為神經元內神經纖維纏結（neurofibrillary tangles, NFTs），也出現為可驅動聚集的種子型 tau（seeding-competent tau），而這種 seeding activity 被認為與病理向不同腦區擴散相關。",[12,118868,118869,118872,118874],{},[106,118870,118871],{},"臨床與病理研究長期觀察到一個現象：AD 並非所有腦區同時、等量地受影響。",[16,118873],{},"\n相反地，一些區域（例如 entorhinal cortex, ENT 和 inferior temporal gyrus, ITG）在疾病早期就累積 tau 病理並表現強烈的退化跡象，而其他區域（例如cerebellum, CER）相對保留較長的時間。這種「區域差異」（regional variation）可能反映了兩件事：1. 某些腦區較早、較強地承受 tau 病理壓力。2. 某些 tau 分子型態（例如特定位點的p-tau）或 seeding activity，在不同腦區中本身就不一樣。",[12,118876,118877],{},"本研究的核心目標，是同時比較多個腦區的 tau 特性，包括：",[23,118879,118880,118883,118886],{},[26,118881,118882],{},"tau 播種活性（seeding activity）",[26,118884,118885],{},"特定位點磷酸化 tau（例如 pTau217, pTau262, pTau396 等）",[26,118887,118888],{},"tau 蛋白的整體含量",[12,118890,118891],{},"並將這些分子層級指標，對照這些腦區在阿茲海默症病程中所扮演的角色。",[7351,118893,118894],{},[12,118895,118896],{},"換句話說，問題不是單純「AD 腦有沒有 tau」，而是「哪一塊腦最早出現具播種能力的 tau？那一塊腦的 tau 長什麼生化樣態？這種 tau 樣態是否對應到可臨床量測的生物標記（例如 pTau217）？」",[6964,118898,118900],{"id":118899},"background-and-rationale","Background and Rationale",[7351,118902,118903],{},[12,118904,118905],{},"這邊在論文架構上是不存在的，但我覺得需要補充一下資訊，以防有人看不懂 w",[7166,118907,118909],{"id":118908},"_1-tau-and-phosphorylated-tau","1. Tau and phosphorylated tau",[12,118911,118912],{},"Tau 是一種 microtubule-associated protein，在成熟神經元特別是 axon 中，協助穩定 microtubules ，維持細胞骨架與軸突內物質運輸的完整性（Hardy and Selkoe 2002）。在阿茲海默症中，tau 分子會發生過度磷酸化（hyperphosphorylation），使 tau 從微管上解離，並逐步聚集成高度有序的 β-摺疊纖維，最終形成 NFTs，而 NFTs 是阿茲海默症病理診斷的經典標誌之一（Hardy and Selkoe 2002）",[12,118914,118915],{},"所謂磷酸化 tau（phosphorylated tau, p-tau）指的是 tau 在特定胺基酸殘基（通常是 Ser 或 Thr）上被磷酸化後的特定構型。不同位點的磷酸化定義出不同的 p-tau ，如 pTau217（Thr217 磷酸化）、pTau181（Thr181）、pTau262（Ser262 附近）、pTau396（Ser396\u002FSer404 區域）等。這些位點並非可互相取代：以 pTau217 為例，它在臨床研究中被證明在 AD 患者腦部和血漿中都顯著上升，並能將 AD 與其他失智症形式區隔，甚至在臨床症狀尚不嚴重的階段亦有區辨力（Palmqvist et al. 2020）",[32509,118917,115287,118918,115287,118922],{},[1930,118919],{"src":118920,"alt":118921},"\u002Fimages\u002Fuploads\u002F螢幕擷取畫面-2025-11-02-163341.png","Intracellular neuronal aggregates in tauopathies",[32516,118923,118924],{},"Figure1. Tau phosphorylation, detachment from microtubules, and aggregation into neurofibrillary pathology. (a) In healthy neurons, tau binds to and stabilizes axonal microtubules, which are cylindrical polymers of α- and β-tubulin. Tau contains microtubule-binding repeats and additional domains (proline-rich region, projection domain) that help maintain microtubule structure. (b) Hyperphosphorylation of tau at multiple serine\u002Fthreonine sites by kinases such as GSK3β, CDK5, and ERK2 reduces tau’s affinity for microtubules, destabilizing the microtubule network. (c) Abnormally phosphorylated tau dissociates from microtubules and forms paired helical filaments (PHFs), which are heavily phosphorylated and often ubiquitinated. (d) Further processing and aggregation of tau yields oligomers and neurofibrillary tangles (NFTs), changes that are associated with neuronal dysfunction and ultimately cell death. Source: Mazanetz MP & Fischer PM, 2007, \\\\\\\\\\\\\\\\\\*Nature Reviews Drug Discovery\\\\\\\\\\\\\\\\\\* 6:464–479.",[7166,118926,118928],{"id":118927},"_2-tau-seeding-activity","2. Tau seeding activity",[12,118930,118931],{},"Tau seeding activity 為描述一種「模板驅動聚集」的行為：已經錯誤折疊、並形成聚集核心的 tau（seed），能夠促使原本可溶性的 tau 轉換為相同β-摺疊聚集構型，並往外延伸成纖維狀聚集體。這代表 tau 病理不只是「自己累積變多」，而是具有可擴散的構形訊號（Kraus et al. 2019; Orrù et al. 2017）",[12,118933,118934],{},"在實驗上，這項播種活性可用 real-time quaking-induced conversion (RT-QuIC) 測量。RT-QuIC 將可能含有播種活性的腦組織均質液（brain homogenate）與重組 tau 底物共同孵育，並在週期性震盪條件下促進聚集。如果樣本中真的含有播種活性，則聚集會加速發生，並可用螢光染劑 Thioflavin T (ThT) 的訊號上升速率、終點強度、lag time（起始延遲時間）等指標來量化。此法也可計算 SD50（引發 50% 反應孔出現陽性訊號所需的稀釋度），作為播種活性強度的功能性指標（Kraus et al. 2019; Orrù et al. 2017）",[32509,118936,115287,118937,115287,118941],{},[1930,118938],{"src":118939,"alt":118940},"\u002Fimages\u002Fuploads\u002F螢幕擷取畫面-2025-11-02-164733.png","The most promising techniques for the quantitation of tau seeding activity in human biofluid",[32516,118942,118943],{},"Figure 2. Quantifying tau seeding activity using biosensor cells and RT-QuIC. (A) In a cell-based biosensor assay, a sample containing tau seeds is applied to a reporter cell line overexpressing tau repeat domains fused to fluorescent proteins (e.g., CFP\u002FYFP). Seed-competent tau from the sample is taken up by the cells and induces intracellular tau aggregation. Aggregation brings the tagged tau molecules into close proximity, producing a FRET signal that can be quantified by flow cytometry as integrated FRET density (combining %FRET-positive cells and median FRET intensity). (B) In RT-QuIC, a seed-containing sample is incubated with excess recombinant tau substrate and thioflavin T (ThT) under shaking\u002Fincubation cycles. Seed-competent tau drives templated aggregation of the substrate, generating a characteristic ThT fluorescence trace: an initial lag phase followed by a rapid aggregation phase and plateau. The ThT kinetics provide a quantitative readout of tau seeding activity in biofluids such as CSF. Source: Lathuiliere A. & Hyman B.T., 2021, Frontiers in Neuroscience 15:654176.",[7166,118945,118947],{"id":118946},"_3-braak-staging-thal-staging","3. Braak staging \u002F Thal staging",[12,118949,118950],{},"阿茲海默症病理的「空間 - 時間順序」可以用兩個常用分期系統來描述 :",[7351,118952,118953],{},[12,118954,118955,118956,118958],{},"這邊我們著重於 Braak staging，因為它在描述 tau上與本研究比較有關連 ; ",[16,118957],{},"\n有興趣可以去查另一個 Thal staging，描述 β-amyloid 沉積的範圍與分布廣度",[12,118960,118961,118962,118964],{},"Braak staging：描述 tau 病理（特別是 NFTs）如何在大腦中依序出現與擴散。",[16,118963],{},"\n典型順序是：最早在 entorhinal cortex 與內側顳葉（包括海馬相關迴路）觀察到高度 tau 病理；接著外側顳葉皮質，包括 inferior temporal gyrus (ITG)、middle temporal gyrus (MTG)、superior temporal gyrus (STG)；最後擴散到更廣泛的新皮質聯合區域。cerebellum (CER, 小腦) 在典型 AD 病程中通常相對少見明顯 tau 纖維性病變（Braak and Braak 1991; Braak and Braak 1995; Braak et al. 2006）。",[32509,118966,115287,118967,115287,118971],{},[1930,118968],{"src":118969,"alt":118970},"\u002Fimages\u002Fuploads\u002Ftau-tracer-uptake-patterns-resemble-ex-vivo-braak-stages-a-schematic-display-of-braak.png","Figure 3. Braak stages of tau pathology in Alzheimer’s disease and their in vivo PET correlates.",[32516,118972,118973],{},"Figure 3. Braak stages of tau pathology in Alzheimer’s disease and their in vivo PET correlates. (A) Schematic post mortem staging of Alzheimer’s disease tau pathology (neurofibrillary tangles composed of hyperphosphorylated tau). Earliest involvement (Braak stages I\u002FII) is localized to the transentorhinal \u002F entorhinal cortex and adjacent medial temporal structures. Intermediate stages (III\u002FIV) involve limbic regions, including medial temporal lobe and parts of medial parietal cortex. Late stages (V\u002FVI) show widespread neocortical involvement. (B) Tau PET tracer uptake patterns in vivo (e.g., \\\\\\\\\\\\\\[¹⁸F]Flortaucipir) recapitulate this hierarchical progression: signal first appears in entorhinal \u002F medial temporal cortex, then spreads to inferior and lateral temporal cortex and medial parietal regions, and finally becomes widespread across association neocortex. These PET patterns provide living-brain support for Braak’s ex vivo staging of tau pathology. Source: Adapted from “Biomarkers for tau pathology,” Schöll M. et al., 2018, Molecular and Cellular Neuroscience.",[12,118975,118976],{},"本研究特別挑選多個腦區（ENT、ITG、MTG、STG、CER 等）來比較播種活性與 tau 分子狀態，這些腦區並不是隨機抽樣，而是涵蓋：",[23,118978,118979,118982,118985],{},[26,118980,118981],{},"典型 AD 病程中「早期高風險、最先累積 tau 病理」的區域（ENT、ITG）",[26,118983,118984],{},"進一步受累的外側顳葉皮質（MTG、STG）",[26,118986,118987],{},"以及對照用的、在典型 Braak 分期中較少 tau 聚集的小腦（CER）",[7351,118989,118990],{},[12,118991,118992],{},"透過橫跨這些腦區，本研究在測試「Braak staging 的時序 ≈ seeding activity 的梯度」",[7166,118994,118996],{"id":118995},"_4-rt-quic","4. RT-QuIC",[12,118998,118999,119000,119002],{},"Real-time quaking-induced conversion (RT-QuIC) 是一個高靈敏度的體外偵測方法。",[16,119001],{},"\n最初用於朊毒蛋白（prion）類疾病，後來擴展至 α-synuclein、tau 等可產生 seeding activity 的錯誤折疊蛋白（Orrù et al. 2017）。本研究利用 RT-QuIC 來問：不同腦區的 AD 樣本，其播種活性是否同樣強？是否和 Braak staging 順序吻合？是否與特定 p-tau 物種（例如 pTau217）對應？",[12,119004,119005],{},"RT-QuIC 的量測重點指標包含：",[23,119007,119008,119014,119020],{},[26,119009,119010,119013],{},[106,119011,119012],{},"lag time","：螢光訊號（ThT）開始上升前的延遲時間，越短代表播種引發聚集的速度越快",[26,119015,119016,119019],{},[106,119017,119018],{},"end-point ThT fluorescence","：反應終點的螢光強度，代表最終聚集程度",[26,119021,119022,119025],{},[106,119023,119024],{},"SD50","：引發 50% 反應孔出現陽性的劑量",[12,119027,119028,119029,119031],{},"重點是，RT-QuIC 測到的是「功能性 seeding activity」，而不是單純的「total tau 」。",[16,119030],{},"\n也就是說，就算兩個腦區的 tau 含量類似，它們的 seeding activity 仍可能不同。",[7351,119033,119034],{},[12,119035,119036],{},"如果對於 RT-QuIC 有興趣可自行上網搜尋相關步驟，這邊偷懶省略方法學",[7166,119038,119040],{"id":119039},"_5-clinical-biomarker-relevance-of-ptau217","5. Clinical biomarker relevance of pTau217",[12,119042,119043],{},"臨床上，pTau217 已被提出為具高潛力的 AD marker，包含腦脊髓液與血漿檢測。研究顯示，血漿 pTau217 在區分典型 AD 與其他失智症型態時具高敏感度與高特異度，甚至在臨床症狀尚未完全明顯之前已可異常（Palmqvist et al. 2020；Jack et al. 2024）。",[12,119045,119046],{},"本研究將這個臨床層訊號往回對照腦內：pTau217 是否在 AD 早期最脆弱、最早受累積的皮質區域（例如 ITG、ENT）顯著集中？這些同時又是否是 seeding activity 最強的腦區？如果答案是肯定的，那麼 pTau217可能反映了特定腦區內、 tau 本身的狀態。",[6964,119048,23810],{"id":21312},[12,119050,119051,119052,119054],{},"對於方法學有興趣的可以去讀一下原論文，它放在 Discussion下面。",[16,119053],{},"\n這邊我將 Methods 提前說明，配合 Results 的實驗結果。",[12,119056,119057],{},[106,119058,119059],{},"超簡單版方法學 :",[3603,119061,119062,119065,119068,119071],{},[26,119063,119064],{},"使用已確診阿茲海默症病例的死後人類腦組織，切片+免疫染色後掃描",[26,119066,119067],{},"將各腦區的腦漿稀釋後，加入含有 tau 的多孔板，在震盪\u002F孵育條件下進行 RT-QuIC",[26,119069,119070],{},"跑 Western blotting，採用密度分析法測定 tau protein 強度，進行定量分析",[26,119072,119073],{},"將 AD 患者和非 AD 對照組的腦漿拿去跑液相層析質譜 (LC-MS) 分析",[7351,119075,119076],{},[12,119077,119078],{},"ps. 這篇的大腦來源是北京和湖南長沙，看起來是南昌大學醫學院主導的",{"title":180,"searchDepth":181,"depth":181,"links":119080},[119081,119082,119089],{"id":82447,"depth":181,"text":82448},{"id":118899,"depth":181,"text":118900,"children":119083},[119084,119085,119086,119087,119088],{"id":118908,"depth":7430,"text":118909},{"id":118927,"depth":7430,"text":118928},{"id":118946,"depth":7430,"text":118947},{"id":118995,"depth":7430,"text":118996},{"id":119039,"depth":7430,"text":119040},{"id":21312,"depth":181,"text":23810},"2025-11-02","論文整理小筆記 (可能適合?對於生物醫學有興趣的人閱讀)","\u002Fimages\u002Fuploads\u002F螢幕擷取畫面-2025-11-02-145011.png",{},"\u002Fblog\u002Fptau217-marks-tau-seeding-hotspots-in-early-alzheimer’s-cortex",{"title":118858,"description":119091},{"loc":119094},"blog\u002Fptau217-marks-tau-seeding-hotspots-in-early-alzheimer’s-cortex",[119099],"Research notes","QX7r8PsxNfNTIH-ZHzyKHwToyv4O46juN0Ba4rLHHwQ",{"id":119102,"title":119103,"author":118859,"body":119104,"date":119725,"description":119726,"draft":185,"edited_at":194,"extension":186,"featured_image":119727,"meta":119728,"navigation":189,"path":119729,"pinned":185,"seo":119730,"sitemap":119731,"stem":119732,"tags":119733,"__hash__":119735},"blog\u002Fblog\u002Fstability-instead-of-revenge-a-reconstruction-strategy-for-post-world-war-i-germany-1919-1935.md","Stability Instead of Revenge: A Reconstruction Strategy for Post-World War I Germany, 1919-1935",{"type":9,"value":119105,"toc":119708},[119106,119108,119111,119113,119121,119124,119127,119130,119133,119137,119140,119143,119146,119149,119152,119156,119159,119162,119165,119168,119171,119175,119180,119183,119186,119189,119192,119195,119203,119207,119210,119215,119218,119221,119224,119228,119231,119236,119239,119247,119252,119255,119260,119263,119268,119271,119274,119278,119281,119284,119292,119295,119298,119302,119305,119310,119313,119318,119321,119326,119329,119332,119336,119339,119342,119345,119349,119352,119355,119358,119361,119365,119368,119371,119660,119664,119670,119676,119679,119685,119691,119693],[6964,119107,15812],{"id":9514},[12,119109,119110],{},"本文將論證，德國在 World War I 戰敗後 (1919-1935) 並非只能在被凡爾賽條約長期壓制與走向法西斯主義擴張之間二選一。相反的，1919年至1929年之間確實存在一條穩定化重建路線：以跨派系的文人聯合政府、賠款-貨幣重組與外資帶動工業復甦、防禦導向而受政府監督的軍隊、社會安全網與公共工程吸收失業人口、以及將德國高技術工業定位為歐洲復興工廠，而非侵略工具。¹ 本文指出，這套路線在1924-1929年的德國局部實現過，並一度成功讓德國恢復貨幣穩定、重返國際金融、獲得部分外交正常化，甚至暫時壓低極端主義吸引力。² 然而，該路線並未在1920年代以制度（社會保險、政府節制軍權、危機時的公共就業機制）形式被保留；因此，當1929年全球經濟崩盤、外資抽離、德國失業激增至數百萬時，政治中心沒有緩衝能力，極端民族主義遂乘虛而入，最終在1933年奪權。³ 本文主張，威瑪共和國的悲劇不在於從未有機會，而在於曾經接近成功，卻未能將短暫的穩定變成結構性的常態。",[6964,119112,82448],{"id":82447},[32509,119114,115287,119115,115287,119118],{},[1930,119116],{"src":119117,"alt":115291},"\u002Fimages\u002Fuploads\u002Fsignatures_du_traité_de_versailles_28_juin_1919_-_page_213.jpg",[32516,119119,119120],{},"fac-simile des trois pages de signature du traité de Versailles ,Amalgamated press Ltd",[12,119122,119123],{},"1918年11月，德國在 WW1 中求和，帝國倒台，威瑪共和國成立。⁴ 隨後的《凡爾賽條約》強制德國接受全部「戰爭罪責」、割讓領土、喪失殖民地，並將其陸軍限制在10萬人，同時禁止坦克、重砲、潛艦與軍用飛機，要求解散總參謀部。此外，條約在賠款問題上採取懲罰性設計，最終以1320億金馬克的天文數額呈現；協約國亦保留以軍事占領方式強制徵收的權利。⁵ 這些條件不只重創德國經濟結構與軍事威信，也使新生的議會政體在國內缺乏正當性，被廣泛視為受監管的戰敗政權，而非能保護國家利益的主權政府。⁶",[12,119125,119126],{},"傳統論述往往把德國的命運描述為宿命式的二擇一：第一是屈從於凡爾賽秩序、在外力操控下的共和國；第二是走向民族復仇主義與極端威權統治，即納粹主導的軍事再擴張帝國。⁷ 本文將提出第三條路線的存在：1919-1929年間，德國在結構上其實有一個政治與經濟的穩定化方案，能在不立即求戰的前提下，逐步恢復主權談判能力、貨幣信用、社會秩序與產業競爭力。這條路線的本質不是引起戰爭轉移國內焦點，也不是肅清左右極端份子，而是以國家主導的秩序與社會安全作為生存工具。",[12,119128,119129],{},"本文的核心論點如下：德國在戰後真正需要、而且在部分時期曾經短暫擁有的，是一套以五大支柱構成的穩定化戰略：(1) 跨派系的重建型文人內閣，壓制左右極端武裝、確立文人節制武力；(2) 將賠款由懲罰機制轉化為外資、工業復甦的長期重建機制，同時提前穩定貨幣；(3) 以防禦導向、專業化、政府監督的小型軍隊作為外交籌碼，而非立即擴張；(4) 把失業保險、公共工程與工人制度化參與視為國安緩衝，而非慈善；(5) 把德國的高端化工、機械、電氣工程產業視為德國整體重建不可或缺的關鍵點，以產業互賴取代領土擴張。",[12,119131,119132],{},"本文將依照時間脈絡與主題層面證明，這套方案在1924-1929年曾局部落地，卻未能制度化，因此在1929年全球經濟衝擊下崩解，讓極端主義接收一切。",[6964,119134,119136],{"id":119135},"historiographical-background","Historiographical Background",[12,119138,119139],{},"戰後德國的崩壞與納粹上台，長久以來在史學上有幾種主要解讀。",[12,119141,119142],{},"Richard J. Evans 代表的解釋，強調威瑪共和國一出生便被內部暴力政治侵蝕：政府在1919-1920年仰賴右翼準軍事部隊（自由軍團）鎮壓左翼革命，但事後並未真正將武力壟斷收回國家，反而讓軍國民族主義勢力保留武裝自主性，進一步破壞了議會體制的正當性。依此觀點，威瑪的軟弱、菁英階層對民主的半心半意、以及1930年代初期的大規模失業，塑造了極端民族主義成為唯一有力方案的氛圍。⁸",[12,119144,119145],{},"Adam Tooze 則將經濟－金融結構視為關鍵：德國的貨幣信用在1922-1923年間幾近毀滅，1923年魯爾工業區被法國與比利時佔領，柏林政府以消極抵抗配合印鈔補貼工人，導致惡性通膨摧毀了中產階級的儲蓄與對政府的信任；德國後來雖靠《道威斯計畫》(1924) 和《楊格計畫》(1929) 爭取重新融入國際金融體系，卻因此對外資與出口過度依賴。當美國在1929年後抽走資金時，德國經濟瞬間崩盤，失業暴增，政治因此坍塌。⁹",[12,119147,119148],{},"Margaret MacMillan 則從國際秩序角度強調，《凡爾賽條約》不只是處罰性的和平條件，而是一套長期監管機制：它限制德國軍力，保留盟國對德國工業心臟地帶動武索賠的正當性，並透過賠款架構將戰敗德國鎖在一個受監督的次等國地位。凡爾賽因此在心理與制度層面同時引發國恥與必須突破的國內敘事。¹⁰",[12,119150,119151],{},"本文介入的角度是：上述觀點對威瑪為何失敗的描述相當有力，但它們往往把失敗看成一條直線。本文將主張，1924-1929年間其實出現過一個「幾乎可行」的折衷模式：外交上用合作談判取代立即復仇、金融上用國際貸款換取賠款分期、軍事上用防禦導向取信鄰國、國內用公共工程與社會安全網抑制失業激進化。而這套模式之所以沒能避免1933年的極權化，不是因為它過於理想化，而是因為它沒有被及時制度化成持久的政策。¹¹",[6964,119153,119155],{"id":119154},"methodology-and-scope","Methodology and Scope",[12,119157,119158],{},"本文的論證結合兩種層次。",[12,119160,119161],{},"第一，本文採用傳統敘事史料與權威次級史料（Evans、Tooze、MacMillan 等）來重構1918-1935年間的德國：惡性通膨、魯爾佔領、《道威斯計畫》與《楊格計畫》、1925年《洛迦諾條約》、德國1926年進入國際聯盟、1929年美資抽離後的失業潮與政治極端化。¹²",[12,119163,119164],{},"第二，本文採用反事實測試（counterfactual testing），但以非常有限且可驗證的方式進行。本文並不空想一個完全不同的德國，而是以假設：「如果已在1924-1929年短暫出現的做法（例如：用外資重組賠款、穩定貨幣並恢復工業、以合作外交換取撤軍、在城市層級實施公共工程）能夠在1919-1923年更早被啟動，並在1920年代中期就被立法固化成制度，那麼1929-1932年全球衰退是否仍會必然導向極端民族主義的勝利？」這種推論並非「許願」，而是把歷史上確實存在的政策片段往前推幾年，並問它們是否有現實的政治－外交條件可以提前出現。¹³",[12,119166,119167],{},"本研究的侷限也在此，1929年的全球金融崩潰與美資撤離並非德國可以單方面避免的衝擊；即使德國在1920年代中期就完成經濟制度與軍隊管理，也不能保證納粹不會崛起。",[12,119169,119170],{},"本文的主張較為溫和：不是「希特勒一定消失」，而是威瑪共和國其實有一個窄而真實的生存窗口，且它幾乎成功了，但沒能建立起制度。",[6964,119172,119174],{"id":119173},"i-the-postwar-crisis-structural-constraints","I. The Postwar Crisis: Structural Constraints",[12,119176,119177],{},[106,119178,119179],{},"1918-1923年的德國同時面臨四個危機，而且這四個危機互相強化彼此。",[12,119181,119182],{},"首先是財政與貨幣的崩潰。戰爭主要靠舉債支撐的德意志帝國在戰敗後不但要面對本身龐大債務，還要負擔賠款。1923年，法國與比利時為強行收取煤炭和工業產出，直接佔領魯爾工業區；柏林政府呼籲工人消極抵抗，並以印鈔補貼他們不工作的收入。結果是惡性通膨在1922-1923年全面爆發，馬克幾乎失去價值，中產階級的畢生積蓄在數月間蒸發。這一事件摧毀了社會對威瑪政府「至少能保護你的財產」的信任。¹⁴",[12,119184,119185],{},"第二是內部武裝政治。1919年1月柏林的左翼革命嘗試建立評議會式政權，卻被政府授權的右翼自由軍團（Freikorps）所鎮壓。這些自由軍團並不效忠議會民主，而是民族主義、反共主義的半獨立武裝。結果，威瑪政府一開始就把國家軍事外包給一群其實也想推翻體制的人，等於告訴社會：「政治靠槍，不靠憲法。」¹⁵",[12,119187,119188],{},"第三是外交上的受監管地位。《凡爾賽條約》不但限制軍備，還授權協約國在德國境內以占領、查驗裁軍等手段執行戰後和平。德國失去殖民地、重裝軍力與威懾力，變成一個可以被隨時施壓的工業國。¹⁶",[12,119190,119191],{},"第四是大規模失業與社會挫敗。戰後復員釋放出大量失業退伍軍人，魯爾停工又使經濟中樞停擺。年輕勞工、退伍士兵、低階中產失去收入與尊嚴，開始把希望投向極端勢力—左派是革命化的工人組織，右派是民族羞辱敘事下的準軍事團體。¹⁷",[12,119193,119194],{},"這四重危機構成一個惡性迴路：經濟毀滅感、外來占領的羞辱、政治暴力的常態化、與失業的絕望感互相疊加，讓威瑪政府的正常議會政治顯得無能而軟弱。¹⁸",[32509,119196,115287,119197,115287,119200],{},[1930,119198],{"src":119199,"alt":180},"\u002Fimages\u002Fuploads\u002Fgermany_hyperinflation.svg.png",[32516,119201,119202],{},"Logarithmic chart of German Hyperinflation. Based on the values in Table IV (page 441) of The Economics of Inflation by Costantino Bresciani-Turroni, published 1937.",[6964,119204,119206],{"id":119205},"ii-political-consolidation-a-reconstruction-cabinet","II. Political Consolidation: A Reconstruction Cabinet",[12,119208,119209],{},"若要讓德國不在1919-1920年就走向內戰，需要立即讓國內停火，也就是建立一個跨派系的重建內閣。這個內閣必須同時吸納溫和社會民主派（SPD）、天主教中間政黨（Zentrum）、自由派資產階級與務實保守派官僚／工業界代表，並對外聲明：此政府的優先目標是國家穩定，而非階級清算或民族復仇。¹⁹",[12,119211,119212],{},[106,119213,119214],{},"這樣的重建內閣必須同時完成三個條件。",[12,119216,119217],{},"第一個交換：彼此承諾不打內戰。 溫和左派同意暫不推行全面評議會革命，而接受議會制度；溫和保守派與工商菁英同意不再縱容自由軍團等右翼私兵作為警察替代品。換句話說，政府不得再用「一邊的民兵」去屠殺「另一邊的民兵」，卻又不把武力收回國家。Evans 指出，威瑪在1919-1920年的最大錯誤，正是用右翼私軍去鎮壓左翼，卻沒有在事後國家化右翼武力，導致雙方都不再信任共和國的中立性。²⁰",[12,119219,119220],{},"第二個交換：把全部武裝力量國家化並置於政府監督。 所有準軍事組織：左翼工人武裝、右翼自由軍團，都應在1919-1920年被編入「國家治安與邊境防衛部隊」，同時在法理上接受國會監督，仿照軍事專業加政治監督的雙軌模式。這樣的目的，是阻止「平行軍閥國」在德國內部持續存在。²¹",[12,119222,119223],{},"第三個交換：對外給出一致、務實、可談判的訊號。 重建內閣必須在1919-1920年對英國與美國釋放明確訊號：「德國希望以談判重訂賠款的支付方式與占領條件，而非以短期軍事報復行動推翻凡爾賽秩序。」這個訊號並非投降，而是一種拉攏：英國與美國本身就擔心法國對魯爾的強硬占領會導致全歐洲不穩，而不穩會威脅英美資本的安全。若柏林能從一開始就主打「我要求生存空間，而不是立刻打仗」，它在國際金融與外交上更容易獲得緩衝，換取之後的賠款重談。²²",[6964,119225,119227],{"id":119226},"iii-reparations-currency-and-industrial-recovery","III. Reparations, Currency, and Industrial Recovery",[12,119229,119230],{},"若無法處理賠款制度、貨幣信用與工業復甦，德國無法持續存在為一個可治理的現代國家。",[12,119232,119233],{},[106,119234,119235],{},"賠款重組：從國際懲罰轉為分期＋外資＋工業復甦。",[12,119237,119238],{},"1924年的《道威斯計畫》將原本幾乎不可支付的賠款重新包裝：美國與其他貸款向德國注入資本，以恢復其工業生產；德國則以恢復後的出口收入，按年、分期支付較低的賠款，並以此換取法國1925年逐步撤出魯爾。到1929年，《楊格計畫》更嘗試將賠款總額下修、付款期延長，並逐漸撤除對德國內政的直接監管。這相當於把懲罰性賠款變成分期付款的迴路。²³",[32509,119240,115287,119241,115287,119244],{},[1930,119242],{"src":119243,"alt":115291},"\u002Fimages\u002Fuploads\u002Fdawesplan.webp",[32516,119245,119246],{},"The Dawes Plan-Chapter 15 – Years of Crisis. Mrs. Tucker World History Victor Valley High School. Main Points. Postwar Uncertainty Worldwide Depression Fascism Rising in Europe Aggressors Invade Nations.",[12,119248,119249],{},[106,119250,119251],{},"(1) 貨幣穩定化（1923 年後的倫滕馬克改革）",[12,119253,119254],{},"惡性通膨是威瑪合法性的致命傷。1922-1923年間，政府為了補貼魯爾工業區的消極抵抗，用印鈔撐住工人的生活，結果貨幣價值崩毀，德國中產階級的存款化為廢紙。這讓數百萬人相信：共和國無法保護你的基本財產。真正可行的穩定方案是停止用印鈔當社會安撫工具，改用稅收與國內公債、讓魯爾重新運轉來恢復稅基、並在1923年底以倫滕馬克（以工業資產為擔保）重建貨幣信用。這些措施在史實中確實止住了惡性通膨，但它們是在經濟已經全毀後才被迫上線，而不是在1921-1922年就被主動採取。²⁴",[12,119256,119257],{},[106,119258,119259],{},"(2) 把德國的高技術工業當成談判籌碼",[12,119261,119262],{},"德國在化工、染料、藥品、精密機械、電氣工程、光學儀器等領域的技術力，在1920年代中後期迅速恢復並成為歐洲重建所需要的關鍵供應來源。德國可以（在1924-1929年部分確實這樣做）向英法表示：「你們想要恢復自己的現代工業，就需要穩定、運作中的德國重工與化工。要達到這點，你們必須允許我脫離被占領狀態，並把賠款轉成可行的長期分期。」把工業復甦轉換成外交籌碼，而非軍事威脅。²⁵",[12,119264,119265],{},[106,119266,119267],{},"(3) 風險：對外資的高度依賴",[12,119269,119270],{},"這條重建路線有一個致命弱點：它嚴重依賴美國資本。當美國在1929年後金融崩盤並抽回資金時，德國外匯斷裂，工廠倒閉，失業數百萬人，整個社會再次陷入絕望，而極端政黨正是在這個真空期迅速坐大。²⁶",[12,119272,119273],{},"統整以上資料分析，賠款重組＋工業復甦買到的其實是一段黃金緩衝期，而不是永久解答。這段緩衝期只對德國有用，前提是它用來重建社會安全網、政府節制軍權、公共工程就業保險化，讓金融衝擊不會直接使德國崩盤。²⁷",[6964,119275,119277],{"id":119276},"iv-military-and-diplomacy-using-a-defensive-army-as-a-negotiating-tool","IV. Military and Diplomacy: Using a Defensive Army as a Negotiating Tool",[12,119279,119280],{},"《凡爾賽條約》將德國陸軍限縮為10萬人的國防軍，禁止重裝武器、坦克、潛艦、軍機，並解散總參謀部，目的就是讓德國喪失重新發動大規模歐陸戰爭的能力。²⁸ 德國菁英在1920年代有兩種本能反應：一是祕密規避條約（例如在蘇俄境內試驗裝甲戰術與航空訓練）；二是鼓吹「凡爾賽就是國恥，必須以武力打破」。²⁹",[12,119282,119283],{},"穩定化策略的替代方案，則是把軍事限制本身轉化成外交籌碼。也就是說：德國公開承諾維持小型、專業、明確防禦導向、受政府監督的軍隊，並以此要求法國撤出魯爾工業區、要求停止「佔領式討債」、要求德國被承認為「正常歐洲國家」，而不再是永久的戰犯監管對象。這個邏輯在1925年《洛迦諾條約》中開始具體化：德國與法國、比利時互保西部邊界的現狀，英國與義大利作為保證國，德國藉此逐步恢復外交主體資格，並在1926年加入國際聯盟。³⁰",[32509,119285,115287,119286,115287,119289],{},[1930,119287],{"src":119288,"alt":115291},"\u002Fimages\u002Fuploads\u002Fwar_ensign_of_germany_-1921–1933-.svg.png",[32516,119290,119291],{},"War flag of Weimar Republic with iron cross (1921–1933) ,Jwnabd",[12,119293,119294],{},"然而，這要求國防軍（Reichswehr）接受兩件在普魯士軍事文化裡並不自然的事：(1) 服從文人政府，而不是把自己視為「真正的國家」；(2) 把任務定義為防禦與內部秩序維持，而非準備下一場復仇戰爭。³¹",[12,119296,119297],{},"Evans 指出，1920年代的國防軍仍保留高度專業自豪感，並非納粹式黨軍；它轉向侵略與意識形態戰爭的徹底化，是1933年後才發生的事。也就是說，如果威瑪政府能在1920年代中期就將防禦導向、政府監督的軍隊用憲政形式限制，國防軍可能成為秩序維持者，而非未來極權政權的工具。³²",[6964,119299,119301],{"id":119300},"v-social-stability-as-national-security-unemployment-insurance-public-works-and-worker-participation","V. Social Stability as National Security: Unemployment Insurance, Public Works, and Worker Participation",[12,119303,119304],{},"1932年時，德國失業人口被估計在六百萬上下，約接近總勞動力三成，這種規模的失業讓無數工人、店員、店主、退伍兵相信國家已經徹底失能。在這樣的心理氣候下，任何承諾立刻採取強力行動、恢復秩序與尊嚴的勢力，無論是革命左派或極端民族主義右派，都會看起來比沉悶的議會政黨更現實。穩定化路線因此把社會政策視為穩定國安，而不是慈善事業。³³",[12,119306,119307],{},[106,119308,119309],{},"(1) 失業保險與最低生存線。",[12,119311,119312],{},"1920年代的德國政府若能提早（在景氣尚可的1924-1929年間）用稅收與長期內債建立全國性的失業保險、基本糧食與住房補助制度，失業者在景氣下滑時就不會瞬間掉到今天沒飯吃的深淵。這會直接降低因失業而轉變為暴力動員的效率。失業保險在此是國家安全閥，而非單純社福。³⁴",[12,119314,119315],{},[106,119316,119317],{},"(2) 公共工程作為就業緩衝池。",[12,119319,119320],{},"威瑪時期的地方政府其實已經在做市政現代化、公共住宅、都市基礎建設、電氣化、自來水與下水道。如果中央政府在1920年代中後期就把這些工程制度化，那麼1929年全球衰退來臨時，數十萬甚至百萬失業者就可以被短期吸收進去，而不是被丟回街頭流浪。這一步等於把「經濟危機 → 大規模失業 → 極端政黨壯大」的鏈條切斷。³⁵",[12,119322,119323],{},[106,119324,119325],{},"(3) 制度化工人參與，讓衝突走法律管道而非街頭武裝。",[12,119327,119328],{},"若能讓工廠層級的「勞資協商委員會」取得法定地位（工時、安全、解僱、資訊透明），工人便不必把希望全部押在街頭暴力或全面革命；同時，資方可以在法律程序中處理勞資爭議，而非每逢罷工就請右翼武裝回廠維持秩序。³⁶",[12,119330,119331],{},"對1920年代的工業資本家來說，這聽起來像把企業主權稀釋。但以歷史角度來看會發現：拒絕讓步的後果，就是1933年之後的軍備國家。名義上企業仍屬私人，但實質生產、投資方向、勞動結構全由極權國家決定。從長期風險角度，1920年代就接受「福利國家＋工人制度化參與」對資本家反而是較小的痛苦。³⁷",[6964,119333,119335],{"id":119334},"vi-technology-and-industry-making-europe-dependent-on-germany-instead-of-germany-conquering-europe","VI. Technology and Industry: Making Europe “Dependent on Germany” Instead of “Germany Conquering Europe”",[12,119337,119338],{},"德國在戰前即是世界級的化工、藥品、合成染料、精密光學、機械加工、電氣工程強國。在1920年代後半，隨著貨幣穩定與外資注入，這些高附加價值產業快速恢復出口，重新成為歐洲現代化所仰賴的供應來源。³⁸",[12,119340,119341],{},"穩定化戰略的第五支柱，就是把這件事升級成國家級外交框架：「歐洲的再工業化需要德國產能；要獲得這個產能，歐洲列強必須停止把德國當作永久罪犯並解除佔領式壓制，轉而接受一個合作式的、長期的賠款與安全架構。」換句話說，德國不是用坦克，而是用機械、電力系統、化學肥料與工業零件來綁住對手的經濟復甦。這種「產業互賴」的邏輯，在1924-1929年已經隱約出現。³⁹",[12,119343,119344],{},"此路線的弱點很明顯：它將德國的命運綁在國際經濟循環上，因此在1929年全球金融崩盤時，德國首當其衝。⁴⁰",[6964,119346,119348],{"id":119347},"vii-feasibility-and-structural-fragility-the-19241929-golden-window-and-its-collapse","VII. Feasibility and Structural Fragility: The 1924–1929 Golden Window and Its Collapse",[12,119350,119351],{},"1924-1929年的德國，事實上已經短暫展現了這條穩定化路線的雛形。史特雷澤曼的外交推動《道威斯計畫》（1924），促使法國1925年撤出魯爾；《洛迦諾條約》（1925）讓德國－西歐邊界關係走向互保；德國於1926年加入國際聯盟。惡性通膨在1923年底後被壓制，貨幣重新取得信用；外資貸款湧入工業與市政建設；出口產業（化工、機械、電氣）恢復動能；城市生活與文化活力在黃金二十年代中繁榮起來。這一切說明：德國確實可以在不立刻求戰的前提下，逐步恢復外交主體、經濟活力與社會穩定。⁴¹",[12,119353,119354],{},"然而，這個黃金窗口並未被用來建立穩定制度。失業保險並未全面法制化成聯邦級安全網；公共工程沒有被寫成「一旦景氣下跌就自動啟動」的緩衝機制；工人參與和勞資協商權並未以國家層級保障；國防軍的政府監督與防禦定位也沒有被憲政化，而仍保留「我們才是國家」的菁英自我意識。⁴²",[12,119356,119357],{},"因此，當1929年全球大蕭條爆發，美國資金抽離，德國出口需求斷崖式下跌，失業者在短時間內暴增至數百萬，國家並沒有現成的安全閥。群眾對議會民主的信任在數季之內瓦解，轉而支持那些承諾「立刻行動、恢復尊嚴、打破凡爾賽」的極端民族主義者。1933年，極端民族主義政權正式奪權。⁴³",[12,119359,119360],{},"換句話說，不是穩定化路線從來不存在，而是路線存在並曾運作，但它被當成過渡性權宜之計，沒有被制度化；因此一遇到全球級外生衝擊，它瞬間瓦解。",[6964,119362,119364],{"id":119363},"conclusion","Conclusion",[12,119366,119367],{},"本文已論證，威瑪共和國的失敗並非單純注定，而是因為一個幾乎完成、卻沒完成的穩定化工程。1919-1929年之間，德國其實有機會在五個面向上打造一個不以侵略為前提的復甦方案：(1) 政治上，以跨派系的重建內閣終止內戰化，並把所有武裝力量國家化，置於政府監督；(2) 經濟與金融上，以《道威斯計畫》式的模式，將賠款轉為長期、外資支撐的重建迴路，並以魯爾工業區生產恢復稅基、穩定貨幣；(3) 軍事與外交上，將小而防禦導向的國防軍本身當成外交籌碼，透過《洛迦諾條約》與進入國際聯盟換取撤軍與地位正常化，而不是立刻重整大軍；(4) 社會結構上，把失業保險、公共工程與工人制度化參與當作國安閥，以阻止失業潮直接投向極端主義；(5) 產業與科技上，把德國在化工、機械、電氣等高附加價值出口部門視為歐洲整體復甦的必要條件，以「產業互賴」取代「領土擴張」作為國力來源。這套方案並非烏托邦，它在1924-1929年其實大半存在；但它沒有被法制化成「自動啟動的安全網」，因此在1929年衝擊下瓦解，為1933年的奪權鋪路。⁴⁴",[6964,119369,100546],{"id":119370},"notes",[3603,119372,119373,119384,119394,119404,119409,119416,119423,119430,119435,119441,119446,119454,119466,119469,119480,119486,119490,119497,119505,119510,119514,119518,119525,119533,119540,119544,119548,119555,119559,119563,119567,119576,119583,119591,119598,119604,119613,119620,119624,119628,119632,119636,119646,119654],{},[26,119374,119375,119376,119379,119380,119383],{},"Margaret MacMillan, ",[7289,119377,119378],{},"Paris 1919: Six Months That Changed the World"," (New York: Random House, 2001), 389–391, 404–406, 480; Richard J. Evans, ",[7289,119381,119382],{},"The Coming of the Third Reich"," (New York: Penguin, 2004), 13–17.",[26,119385,119386,119387,119390,119391,119393],{},"Adam Tooze, ",[7289,119388,119389],{},"The Wages of Destruction: The Making and Breaking of the Nazi Economy"," (London: Allen Lane, 2006), 37–38, 43–44, 95–107; Evans, ",[7289,119392,119382],{},", 280–284.",[26,119395,119396,119397,119399,119400,119403],{},"Evans, ",[7289,119398,119382],{},", 80–84, 88; Tooze, ",[7289,119401,119402],{},"The Wages of Destruction",", 173–174, 189–190.",[26,119405,119396,119406,119408],{},[7289,119407,119382],{},", 13–17.",[26,119410,119411,119412,119415],{},"MacMillan, ",[7289,119413,119414],{},"Paris 1919",", 389–391, 404–406.",[26,119417,119411,119418,119420,119421,119408],{},[7289,119419,119414],{},", 404–406, 480; Evans, ",[7289,119422,119382],{},[26,119424,119396,119425,119399,119427,119429],{},[7289,119426,119382],{},[7289,119428,119402],{},", 95–107.",[26,119431,119396,119432,119434],{},[7289,119433,119382],{},", 52–53, 80–84, 88.",[26,119436,119437,119438,119440],{},"Tooze, ",[7289,119439,119402],{},", 37–38, 43–44, 95–107, 173–174, 189–190.",[26,119442,119411,119443,119445],{},[7289,119444,119414],{},", 389–391, 404–406, 480.",[26,119447,119396,119448,119450,119451,119453],{},[7289,119449,119382],{},", 280–284; Tooze, ",[7289,119452,119402],{},", 37–38, 43–44, 95–107.",[26,119455,119396,119456,119458,119459,119461,119462,119465],{},[7289,119457,119382],{},", 13–17, 280–284; MacMillan, ",[7289,119460,119414],{},", 404–406; Tooze, ",[7289,119463,119464],{},"The Wages of Destruction,"," 37–38, 43–44, 95–107, 173–174, 189–190.",[26,119467,119468],{},"Hamilton College Writing Center, “Writing a Good History Paper” (Clinton, NY: Hamilton College Writing Center, n.d.).",[26,119470,119437,119471,119473,119474,119476,119477,119479],{},[7289,119472,119402],{},", 37–38, 43–44; MacMillan, ",[7289,119475,119414],{},", 404–406; Evans, ",[7289,119478,119382],{},", 52–53.",[26,119481,119482,119483,119485],{},"Richard J. Evans, ",[7289,119484,119382],{}," (New York: Penguin, 2004), 52–53.",[26,119487,119411,119488,119445],{},[7289,119489,119414],{},[26,119491,119386,119492,119494,119495,119479],{},[7289,119493,119389],{}," (London: Allen Lane, 2006), 37–38, 43–44; Evans, ",[7289,119496,119382],{},[26,119498,119396,119499,119501,119502,119504],{},[7289,119500,119382],{},", 13–17; Tooze, ",[7289,119503,119402],{},", 37–38, 43–44.",[26,119506,119396,119507,119509],{},[7289,119508,119382],{},", 13–17, 52–53.",[26,119511,119396,119512,119479],{},[7289,119513,119382],{},[26,119515,119396,119516,119479],{},[7289,119517,119382],{},[26,119519,119411,119520,119522,119523,119504],{},[7289,119521,119414],{},", 404–406, 480; Tooze, ",[7289,119524,119402],{},[26,119526,119437,119527,119529,119530,119532],{},[7289,119528,119402],{},", 37–38, 43–44, 95–107; Margaret MacMillan, ",[7289,119531,119378],{}," (New York: Random House, 2001), 404–406.",[26,119534,119437,119535,119537,119538,119479],{},[7289,119536,119402],{},", 37–38, 43–44; Evans, ",[7289,119539,119382],{},[26,119541,119437,119542,119429],{},[7289,119543,119402],{},[26,119545,119437,119546,119403],{},[7289,119547,119402],{},[26,119549,119437,119550,119552,119553,119393],{},[7289,119551,119402],{},", 95–107; Evans, ",[7289,119554,119382],{},[26,119556,119411,119557,119415],{},[7289,119558,119414],{},[26,119560,119396,119561,119393],{},[7289,119562,119382],{},[26,119564,119396,119565,119393],{},[7289,119566,119382],{},[26,119568,119396,119569,119571,119572,119575],{},[7289,119570,119382],{},", 280–284; Richard J. Evans, ",[7289,119573,119574],{},"The Third Reich in Power",", 1933–1939 (New York: Penguin, 2005), 18–21.",[26,119577,119396,119578,119580,119581,119393],{},[7289,119579,119574],{},", 1933–1939, 18–21; Evans, ",[7289,119582,119382],{},[26,119584,119437,119585,119587,119588,119590],{},[7289,119586,119402],{},", 173–174, 189–190; Evans, ",[7289,119589,119382],{},", 80–84, 88.",[26,119592,119437,119593,119537,119595,119597],{},[7289,119594,119402],{},[7289,119596,119382],{},", 80–84.",[26,119599,119396,119600,119450,119602,119429],{},[7289,119601,119382],{},[7289,119603,119402],{},[26,119605,119396,119606,119608,119609,119612],{},[7289,119607,119382],{},", 52–53, 80–84; Evans, ",[7289,119610,119611],{},"The Third Reich in Power,"," 1933–1939, 18–21.",[26,119614,119396,119615,119617,119618,119429],{},[7289,119616,119611],{}," 1933–1939, 18–21; Tooze, ",[7289,119619,119402],{},[26,119621,119437,119622,119429],{},[7289,119623,119402],{},[26,119625,119437,119626,119429],{},[7289,119627,119402],{},[26,119629,119437,119630,119403],{},[7289,119631,119402],{},[26,119633,119437,119634,119453],{},[7289,119635,119402],{},[26,119637,119396,119638,119640,119641,119643,119644,119429],{},[7289,119639,119382],{},", 280–284; Evans, ",[7289,119642,119574],{},", 1933–1939, 18–21; Tooze, ",[7289,119645,119402],{},[26,119647,119648,119649,119587,119652,119590],{},"Tooze, T",[7289,119650,119651],{},"he Wages of Destruction",[7289,119653,119382],{},[26,119655,119396,119656,119450,119658,119429],{},[7289,119657,119382],{},[7289,119659,119402],{},[6964,119661,119663],{"id":119662},"bibliography","Bibliography",[12,119665,119666,119667,119669],{},"Evans, Richard J. ",[7289,119668,119382],{},". New York: Penguin, 2004.",[12,119671,119666,119672,119675],{},[7289,119673,119674],{},"The Third Reich in Power, 1933–1939",". New York: Penguin, 2005.",[12,119677,119678],{},"Hamilton College Writing Center. “Writing a Good History Paper.” Clinton, NY: Hamilton College Writing Center, n.d.",[12,119680,119681,119682,119684],{},"MacMillan, Margaret. ",[7289,119683,119378],{},". New York: Random House, 2001.",[12,119686,119687,119688,119690],{},"Tooze, Adam. ",[7289,119689,119389],{},". London: Allen Lane, 2006.",[6964,119692,118833],{"id":118833},[12,119694,119695,119696,119698,119699,119701,119702,119704,119705,119707],{},"這是一篇莫名其妙就生成出來的...偽歷史論文?",[16,119697],{},"\n我也不知道為何一頓飯的時間就討論出這些奇妙東西，人生真是奇妙對吧。",[16,119700],{},"\n如有任何錯誤或遺漏，請寄email到我的信箱~也歡迎指教~",[16,119703],{},"\n(註 : 此文有使用chatGTP進行文獻搜索與文字潤飾)",[16,119706],{},"\n感謝您長達30分鐘的觀看XD",{"title":180,"searchDepth":181,"depth":181,"links":119709},[119710,119711,119712,119713,119714,119715,119716,119717,119718,119719,119720,119721,119722,119723,119724],{"id":9514,"depth":181,"text":15812},{"id":82447,"depth":181,"text":82448},{"id":119135,"depth":181,"text":119136},{"id":119154,"depth":181,"text":119155},{"id":119173,"depth":181,"text":119174},{"id":119205,"depth":181,"text":119206},{"id":119226,"depth":181,"text":119227},{"id":119276,"depth":181,"text":119277},{"id":119300,"depth":181,"text":119301},{"id":119334,"depth":181,"text":119335},{"id":119347,"depth":181,"text":119348},{"id":119363,"depth":181,"text":119364},{"id":119370,"depth":181,"text":100546},{"id":119662,"depth":181,"text":119663},{"id":118833,"depth":181,"text":118833},"2025-10-31","一篇在吃飯時莫名討論出的偽論文 (Chicago Manual of Style)\n備註 : 目前引用文獻頁碼錯誤極多，但觀點大致無誤，修正中","\u002Fimages\u002Fuploads\u002Fatlas_van_der_hagen-kw1049b10_047-s._imperium_romano-germanicum_oder_deutschland_mit_seinen_angräntzenden_königreichen_und_provincien_neulich_entworffen_und_theils_gezeichnet_durch_iulium_reichelt_chur_pfal.jpg",{},"\u002Fblog\u002Fstability-instead-of-revenge-a-reconstruction-strategy-for-post-world-war-i-germany-1919-1935",{"title":119103,"description":119726},{"loc":119729},"blog\u002Fstability-instead-of-revenge-a-reconstruction-strategy-for-post-world-war-i-germany-1919-1935",[119734],"World War II conjecture","ZTFCFE8taIIfYGhlAfppsY8s1Acy-QJky_9u2ZT8c6c",{"id":119737,"title":119738,"author":6946,"body":119739,"date":119725,"description":119841,"draft":185,"edited_at":119842,"extension":186,"featured_image":119843,"meta":119844,"navigation":189,"path":119845,"pinned":185,"seo":119846,"sitemap":119847,"stem":119848,"tags":119849,"__hash__":119851},"blog\u002Fblog\u002F根本沒人在乎你的部落格.md","根本沒人在乎你的部落格",{"type":9,"value":119740,"toc":119837},[119741,119743,119746,119753,119755,119758,119784,119787,119831,119834],[6964,119742,102197],{"id":102197},[12,119744,119745],{},"今天在網絡上無意間翻到了這篇文章。十分認同作者的觀點。我嘗試建立部落格已經一個多月，和大部分我見到的人一樣，部落格建立好之後草草寫一個Hello World就放著好幾天沒動了。僅僅是在想著各種設計、外觀、參考別人的部落格就花費了許多時間，彷彿這才是部落格的真正意義。當終於設計好部落格，架在伺服器上了，反而不知道要寫什麼。直到我看了這篇文章，才明白部落格的意義何在，於是決定翻譯分享出來.",[12,119747,119748,119749],{},"文章來自Alex Molas，連接：",[2330,119750,119751],{"href":119751,"rel":119752},"https:\u002F\u002Fwww.alexmolas.com\u002F2023\u002F07\u002F15\u002Fnobody-cares-about-your-blog.html",[2334],[6964,119754,119738],{"id":119738},[12,119756,119757],{},"我幾年前開始寫部落格，從那時起，我就有過無數個想放棄的理由。以下就來列舉，為什麼根本沒人在乎你的部落格：",[3603,119759,119760,119766,119772,119778],{},[26,119761,119762,119765],{},[106,119763,119764],{},"你的部落格毫無原創性。"," 外面有成千上萬的部落格，你憑什麼覺得自己的與眾不同？你很可能只是在拾人牙慧，重複你在別處看來的東西。",[26,119767,119768,119771],{},[106,119769,119770],{},"你根本不是你那個領域的專家。"," 否則，你早就去寫論文、接受採訪了，哪還會在這裡發部落格。",[26,119773,119774,119777],{},[106,119775,119776],{},"你只是在向全世界秀下限，顯得你有多蠢。"," 如果你說的話還不如保持沉默，那你最好閉嘴。",[26,119779,119780,119783],{},[106,119781,119782],{},"就算哪天真的有人在乎你的部落格，那也只是為了來批評你。"," 你的作品就是垃圾，把它公諸於世只會讓人們發現，你本人也是個垃圾。",[12,119785,119786],{},"但這一切根本不是問題，因為你壓根就不該在乎別人是怎麼想的。以下才是我認為你應該在乎（你自己的）部落格的真正理由：",[3603,119788,119789,119795,119801,119807,119813,119819,119825],{},[26,119790,119791,119794],{},[106,119792,119793],{},"你可以把它當成寫給未來自己的筆記。"," 幾年後，你就會擁有一本很棒的日誌，記錄著你這些年來的進化。重讀你過去的文字，就像在與過去的自己對話。",[26,119796,119797,119800],{},[106,119798,119799],{},"釋放你腦中那些需要傾瀉而出的想法。"," 即使沒有人在乎這些想法，把它們寫下來本身就是一種心靈的宣洩過程。",[26,119802,119803,119806],{},[106,119804,119805],{},"學習如何寫作和表達複雜的概念。"," 最好的學習方式就是教學（即便根本沒有讀者）。正如保羅·格雷厄姆 (Paul Graham) 所說，親自動筆去寫一件事，即便是你自認很熟的事。通常你會發現，你其實並不像自己想像的那麼了解它。",[26,119808,119809,119812],{},[106,119810,119811],{},"就算你分享給世界的觀點並非原創，你也可以用你個人的視角來豐富它們。"," 如同比爾·瑟斯頓 (Bill Thurson) 所言，我們每個人都只對少數事物有清晰的理解，而對更多的概念則模糊不清。這世上永遠不缺需要被釐清的想法。",[26,119814,119815,119818],{},[106,119816,119817],{},"如果你曾花費大把時間解決某個超級特定的問題，那你更該把它寫下來。"," 說不定哪天，你就會因此成為別人的英雄。也可能，未來你會再次遇到同樣的問題，到那時，你就是你自己的英雄。",[26,119820,119821,119824],{},[106,119822,119823],{},"維護一個部落格是件很酷的事，哪怕只是從技術角度來看。"," 這種對某樣東西擁有完全所有權的感覺真的很有成就感，即使它只是這個虛無飄渺的世界中，遠端伺服器上的幾個位元組。",[26,119826,119827,119830],{},[106,119828,119829],{},"你想他媽的寫什麼都行。"," 這是你的部落格，你不需要遵守任何規則。我剛才就飆髒話了，而你拿我一點辦法也沒有，因為這是我的部落格，老子愛怎樣就怎樣。這種自由感真的超酷（恕我直言）。",[12,119832,119833],{},"P.S. 我本來正要發布這篇文章，但那時我突然開始想：「我為什麼要發這篇文？誰會浪費時間來讀這堆狗屁？」",[12,119835,119836],{},"但你知道嗎？我才不在乎你怎麼想，文章我照發，你照樣拿我沒轍 :)",{"title":180,"searchDepth":181,"depth":181,"links":119838},[119839,119840],{"id":102197,"depth":181,"text":102197},{"id":119738,"depth":181,"text":119738},"看到了一篇寫的很棒的文章，決定翻譯之後轉發一下。","2026-02-16","\u002Fimages\u002Fuploads\u002F__plana_and_arona_blue_archive__sample-3e0b6caf7810576f0fd6c98c77565758.jpg",{},"\u002Fblog\u002F根本沒人在乎你的部落格",{"title":119738,"description":119841},{"loc":119845},"blog\u002F根本沒人在乎你的部落格",[119850],"日誌觀點","f1Jl14sKKmvKrUmTdiS8xJz-iZTaQ90Mt_VzzIpIgTE",{"id":119853,"title":119854,"author":116189,"body":119855,"date":119950,"description":119951,"draft":185,"edited_at":194,"extension":186,"featured_image":119952,"meta":119953,"navigation":189,"path":119954,"pinned":185,"seo":119955,"sitemap":119956,"stem":119957,"tags":119958,"__hash__":119960},"blog\u002Fblog\u002F早安-午安-晚安.md","早安 午安 晚安",{"type":9,"value":119856,"toc":119946},[119857,119860,119863,119867,119870,119873,119876,119879,119882,119885,119889,119892,119918,119921,119928,119936,119939],[12,119858,119859],{},"您好、Hello、Guten Tag。",[12,119861,119862],{},"我是法法，很高興有人打開這個文章。先祝你今天平安，have a nice day.",[6964,119864,119866],{"id":119865},"who-am-i","Who Am I?",[12,119868,119869],{},"我是台灣嘉義人，是個分享欲有點重的人，目前在跟數學展開一個愛恨情仇的故事，希望這個故事有個美好結局XD",[12,119871,119872],{},"我的名字一開始叫做大法，是一個遊戲我取的網名，後面被當時的熟人稱為法法，因此就用到現在了。",[12,119874,119875],{},"為什麼不用大法呢? 對我來說，那個遊戲是一個敏感遊戲，因此我不可避免地做了一些去識別化，而在這邊我想要多做一點真正的自己，所以就選擇了法法這個名字了w",[12,119877,119878],{},"興趣的話，由於我是那種天然對知識就有興趣的人，所以我自己涉略的領域有一點多。不過! 數學是永遠的愛(雖然它也給我了很多挫折ww",[12,119880,119881],{},"簡單來講，我看了一些心理學和社會學等的教科書，也看了一些哲學和女性主義的書籍，這些都是當時的我因為某些契機讀的。因為說到底還是一個門外漢，所以如果你對以上領域有興趣的話，歡迎跟我交流喔!",[12,119883,119884],{},"除了這些硬硬的知識之外，我也喜歡自己一個人旅行跟吃美食。",[6964,119886,119888],{"id":119887},"something-i-can-bring-here","Something I Can Bring Here",[12,119890,119891],{},"既然跟糯糯拿到了權限，我覺得自己應該要做出一些東西才是XD",[23,119893,119894,119900,119906,119912],{},[26,119895,119896,119897,119899],{},"一些數學筆記",[16,119898],{},"這部分本來是我平常讀書或是paper時隨筆寫下的一些東西，我覺得可以整理一下，然後利用純文字的方式給出自己的理解。",[26,119901,119902,119903,119905],{},"日常生活",[16,119904],{},"比如一些美食照之類的w",[26,119907,119908,119909,119911],{},"人類觀察",[16,119910],{},"自從在今年把部分把精力放在線上遊戲之後，發現網路正常人真的是不可多得，因此想寫一些東西當作給自己的警戒或者是發牢騷。",[26,119913,119914,119915,119917],{},"書籍分享",[16,119916],{},"正如我在前面講的，我看了很多領域的書籍，而在打到這裡的當下，我身邊還有很多非數學書在讀，這些書感覺是可以分享的。",[12,119919,119920],{},"最後來簡單介紹圖片好了",[12,119922,119923,119924,119927],{},"This diagram is the expression of ",[106,119925,119926],{},"snake lemma."," Snake lemma is valid in every abelian category, and it's a important tool  in homological algebra.",[12,119929,119930,119935],{},[7289,119931,119932],{},[106,119933,119934],{},"Snake lemma."," See diagram above, consider second and third row, they composed a diagram which is commutative. Moreover, they are exact and 0 is zero object. Then there exists a morphism d such that d connects first row and forth row is exact.",[12,119937,119938],{},"The rest of the proof is left as an exercise to the reader XDDD.",[12,119940,119941,119942],{},"The diagram comes from ",[2330,119943,119944],{"href":119944,"rel":119945},"https:\u002F\u002Fupload.wikimedia.org\u002Fwikipedia\u002Fcommons\u002Fthumb\u002F9\u002F9d\u002FSnake_lemma_complete.svg\u002F1920px-Snake_lemma_complete.svg.png",[2334],{"title":180,"searchDepth":181,"depth":181,"links":119947},[119948,119949],{"id":119865,"depth":181,"text":119866},{"id":119887,"depth":181,"text":119888},"2025-10-28","想說的一些話","\u002Fimages\u002Fuploads\u002Fsnake_lemma_complete.svg.png",{},"\u002Fblog\u002F早安-午安-晚安",{"title":119854,"description":119951},{"loc":119954},"blog\u002F早安-午安-晚安",[119959],"介紹","ByBFclfoD0Ypz6_BPxPJCMVdue4N88E8dtpcaK_Ei4A",{"id":119962,"title":119963,"author":117136,"body":119964,"date":119950,"description":120017,"draft":185,"edited_at":194,"extension":186,"featured_image":120018,"meta":120019,"navigation":189,"path":120020,"pinned":185,"seo":120021,"sitemap":120022,"stem":120023,"tags":120024,"__hash__":120025},"blog\u002Fblog\u002F走進、走近、走盡.md","走進、走近、走盡",{"type":9,"value":119965,"toc":120012},[119966,119969,119972,119975,119978,119981,119985,119988,119991],[12,119967,119968],{},"Hello 你好 Bonjour\n我是風玄也可以叫我Osborn，感謝你點開這篇文章，希望你今天過得愉快",[6964,119970,119971],{"id":119865},"Who am I",[12,119973,119974],{},"我是風玄來自台灣新北人幹話特別多",[12,119976,119977],{},"網名一開始叫風影但由於太過中二所以改名風玄雖然還是一樣中二w，Osborn則是來自於之前玩遊戲跟朋友互取英文名子時想到，其來自於北歐神話涵義神聖的武士或是神熊!?",[12,119979,119980],{},"興趣是打遊戲和看書涉及的種類蠻多元的有關法律，歷史和人文類的也會看些輕小說。打遊戲喜歡fps和養成類遊戲，非常喜歡賽馬娘裡的東海帝王，原型是來自日本1990年代的賽馬真實故事十分精采，馬生有過三次骨折，最後成功康復並奪得1993有馬紀念頭馬，故事讓人憐惜與敬佩。對於歷史與法律有所興趣。歷史對於我來說是個警惕自我的鏡子，法律則是從小對於不公正的事會想要試圖導正，然而因為能力不足無法有所作為。我很喜歡一句話：法律是保護懂法律的人。我希望自己能夠利用個人的能力來幫助他人。",[6964,119982,119984],{"id":119983},"why-am-i-here","Why am I here",[12,119986,119987],{},"聽說七糯糯建設了一個有趣的網站，若能成為我幫助他人的第一步就太有意思了，時間會沖淡記憶但文字卻是永恆的，我或許不能幫助到100%的人但如果可以幫助到那1%的人也不虛此行，雖然不是最專業但會盡量保持正確度，若有錯誤歡迎指正與討論。",[6964,119989,74376],{"id":119990},"program",[23,119992,119993,120000,120006],{},[26,119994,119995,119999],{},[7166,119996,119998],{"id":119997},"thread上或是新聞上的法律謬誤","Thread上或是新聞上的法律謬誤","時常看到一些錯誤的法律言論誤導大眾，希望能透過棉薄之力來導正",[26,120001,120002,120005],{},[7166,120003,120004],{"id":120004},"書籍閱讀心得","去分享一些有趣的書與分享內心的感悟",[26,120007,120008,120011],{},[7166,120009,120010],{"id":120010},"生活瑣事","可能是美食照或是風景照!?",{"title":180,"searchDepth":181,"depth":181,"links":120013},[120014,120015,120016],{"id":119865,"depth":181,"text":119971},{"id":119983,"depth":181,"text":119984},{"id":119990,"depth":181,"text":74376},"希望能讓大家了解我","\u002Fimages\u002Fuploads\u002Fimg_3171.jpeg",{},"\u002Fblog\u002F走進、走近、走盡",{"title":119963,"description":120017},{"loc":120020},"blog\u002F走進、走近、走盡",[119959],"dfwjVVa7PrpYl4UtL1lyDtzndxEjP4n7DhFFxOiBz4k",{"id":120027,"title":120028,"author":118859,"body":120029,"date":120096,"description":120097,"draft":185,"edited_at":194,"extension":186,"featured_image":120098,"meta":120099,"navigation":189,"path":120100,"pinned":185,"seo":120101,"sitemap":120102,"stem":120103,"tags":120104,"__hash__":120105},"blog\u002Fblog\u002F一個漂流到地球的故事.md","一個漂流到地球的故事",{"type":9,"value":120030,"toc":120091},[120031,120040,120043,120050,120057,120066,120069,120072,120075,120077,120088],[12,120032,120033,120034,120036,120037,120039],{},"Hello、你好、Salvēte、こんにちは。",[16,120035],{},"\n在82億人之中，我們相遇了，或許認識，或許不是很熟，但歡迎你的到來。",[16,120038],{},"\nI am Automata，你也可以叫我天喵，我們有很多時間能互相認識。",[6964,120041,120042],{"id":119865},"Who Am I ?",[12,120044,120045,120046,120049],{},"產地 : 臺灣、屬名 : Homo sapiens、性別 : ",[106,120047,120048],{},"_"," (自由填入)",[12,120051,120052,120053,120056],{},"天喵起源於一款幻想歷史遊戲，當時想叫天可汗 (因為他打跑蒙古人XD)，但因為這名字太無聊了，想多一點變化，便加入了貓咪元素，於是搖身一變成就了\"天喵汗\" ",[106,120054,120055],{},";"," 而 Automata則是來源於 NieR:Automata 這部作品，我很喜歡那種破碎、荒廢、未來都市的感覺。",[12,120058,120059,120060,120062,120063,120065],{},"興趣 : 醫學、旅遊、滑雪、美食、酒精、Formula One (害我想買方向盤)",[16,120061],{},"\n正在坐牢 : Immunology、Embryology (後悔中)",[16,120064],{},"\ncancer research (終極大坐牢)",[12,120067,120068],{},"喜歡冬天去日本自由旅行，享受滑雪時在雪上飛的感受，拉麵、溫泉、雪地都是我的最愛。對於酒精也略有研究，紅酒、清酒、琴酒、威士忌、蘭姆酒，有時會去酒展和酒吧逛逛~ (註 : 理性飲酒，理性醉)",[6964,120070,120071],{"id":119983},"Why am I here ?",[12,120073,120074],{},"當時聽說七糯糯想要創立blog，臨機一動就想說可以一起寫寫看，分享一些生活中的趣事、知識、或單純練習一下文筆，讓生活能留下一些痕跡，讓一些想法能以文筆互相交流，友誼可能會隨著時間沖淡，但遺留在這邊的回憶永存 (直到網站被關掉為止XD)，或許回頭一看，會覺得當年的自己很酷呢~",[6964,120076,74376],{"id":119990},[3603,120078,120079,120082,120085],{},[26,120080,120081],{},"生活趣事 (不定時更新，可能是研究生小八卦，或是美食評價)",[26,120083,120084],{},"生醫筆記 (可能會更新論文review)",[26,120086,120087],{},"滑雪日記 (或許哪天可以跟七糯糯一起去滑雪? )",[12,120089,120090],{},"可能就這樣吧XD，畢竟寫文章也是需要精力的~感謝你願意花時間閱讀啦~",{"title":180,"searchDepth":181,"depth":181,"links":120092},[120093,120094,120095],{"id":119865,"depth":181,"text":120042},{"id":119983,"depth":181,"text":120071},{"id":119990,"depth":181,"text":74376},"2025-10-26","Toast、biomedicine and snowboard","\u002Fimages\u002Fuploads\u002Fhasan-almasi-5ddkjb6fgtq-unsplash.jpg",{},"\u002Fblog\u002F一個漂流到地球的故事",{"title":120028,"description":120097},{"loc":120100},"blog\u002F一個漂流到地球的故事",[119959],"pHB8Mg--UUPaNcpq71OTWiYNOArcjWxHYWpCR04MyMk",{"id":120107,"title":120108,"author":6946,"body":120109,"date":120240,"description":120241,"draft":185,"edited_at":120242,"extension":186,"featured_image":120243,"meta":120244,"navigation":189,"path":120245,"pinned":189,"seo":120246,"sitemap":120247,"stem":120248,"tags":120249,"__hash__":120250},"blog\u002Fblog\u002F你好，世界。.md","你好，世界。",{"type":9,"value":120110,"toc":120231},[120111,120114,120117,120120,120134,120137,120141,120144,120151,120154,120157,120160,120163,120166,120169,120172,120175,120178,120181,120184,120187,120190,120193,120196,120199,120202,120205,120208,120211,120214,120217,120220,120222,120225,120228],[12,120112,120113],{},"首先，致所有讀者：歡迎來到「星谷雜貨店」！",[12,120115,120116],{},"經過一番（慘痛的）折騰和無數次的「為什麼這破程式碼又壞了」之後，這個專屬於我們的小角落終於正式上線了。",[12,120118,120119],{},"在這裡，你可以：",[23,120121,120122,120125,120128,120131],{},[26,120123,120124],{},"分享你最近在追的劇、聽的歌、玩的遊戲。",[26,120126,120127],{},"寫下你在生活中遇到的趣事或牢騷。",[26,120129,120130],{},"貼一張你拍到（或在網路上看到）的圖片。",[26,120132,120133],{},"分享你任何的奇思妙想！",[12,120135,120136],{},"總之，這是一個沒有壓力、沒有主題限制的共享空間。",[6964,120138,120140],{"id":120139},"我是誰","我是誰？",[12,120142,120143],{},"天天躲在家裡的宅，馬來亞大學軟件工程系學生。名字的話，我也忘記是誰給我取的了。我以前叫智乃乃，因為自己很喜歡「請問您今天要來點兔子嗎？」裡面的香風智乃（超級可愛啊啊啊啊），又想要一個可愛一點的名字，就這麼取了。英文自然就選了Kafuu Chino裡的Chino，所以叫Chinono. 後來有個友人玩笑間把我叫成了七糯糯。我一下子就愛上了這個超可愛的名字。",[12,120145,120146,120147,120150],{},"之前是讀醫學系的",[5610,120148,120149],{},"不過被當掉了","。然後因為對電腦超級感興趣，所以有的時候癮上來，考試前連休息都不用了，讀完書就去寫電腦程式，寫了幾個小時才發現好像要來不及讀書了。",[12,120152,120153],{},"愛好的話，當然是電腦程式啦！在這個之外我也有很多愛好，比如我是小小的手辦收藏家，之前也有迷戀過機械鍵盤，耳機，模型車之類的。所以房間裡堆著很多其實我都用不太到的東西。因為有好的耳機，所以我也喜歡經常聽歌，經常會一個人安靜的坐在房間裡閉上眼專心聽歌。",[12,120155,120156],{},"我大部分的電腦程式技能都是靠著自學來的。我不覺得自己的技術很好，也不知道自己學的方法對不對。但是當我看著自己寫了好幾個星期的作品運行起來，那種成就感真的是太棒了。因為是自學的，我對於很多技術原理，專有名詞並不熟悉。我的學習方法都是想好自己想做什麼，閱讀文檔，查詢資料，然後就直接下場開始實作了。",[6964,120158,120159],{"id":120159},"創作部落格的目的",[12,120161,120162],{},"我從來沒有寫過部落格，也不知道寫這個好不好玩。但是，我有過幾次自己架設網站的經驗，也希望能通過架設部落格學習新的東西。",[12,120164,120165],{},"部落格的大部分代碼，都是我自己慢慢查文檔，排查錯誤，設計網站，慢慢做出來的。過程很漫長，但是成功之後也很開心！這個網站也是我自費（其實大部分都是免費服務）架起來的。這裡是一個真正屬於我們的空間，不受平台政策，廣告的煩擾、可以隨心所欲客製化的小空間。",[12,120167,120168],{},"最近我見到我的一位網友，退出了discord平台，剛剛用WordPress製作了一個新的個人部落格。我覺得，這未嘗也不是一種能自由表達自己的方法。我在這裡可以不受任何約束，不顧任何人的眼光，寫我想寫的，說我想說的話。",[12,120170,120171],{},"我開始研究架設部落格幾天後，分享給了我的朋友天喵。他當時就覺得說，那不如你做一個部落格，讓我也能加入一起寫吧。我覺得這也是個好主意，就採納了建議。",[12,120173,120174],{},"所以嘛，這個部落格才會用到“我們”，因為我打算開放給我的幾個朋友一起編輯，體驗一下寫部落格的感受。如果你也想加入我們，歡迎來我的discord找我提出來。只不過，如果我不認識你的話，很抱歉我可能不會接受，所以不妨先讓我認識你！",[6964,120176,120177],{"id":120177},"盯著螢幕的空虛感",[12,120179,120180],{},"不知道你們有沒有這種體驗：有些天，在夜深人靜的時候，我經常會盯著手機螢幕，打開各種社交軟件，看看有沒有人在線、有沒有人看過我分享的東西、有沒有人想找我...即使是半夜兩三點，我的朋友們都不在線，我仍然會拿著我的手機，到處翻找。",[12,120182,120183],{},"即使我很不想承認，但是有的時候，我會躺在床上，盯著手機螢幕，讀完網友們的每一條訊息紀錄，把自己身邊發生的所有事情都分享出來，只想得到朋友們的一點點關注，害怕自己落下什麼沒跟上...時間久了，真的會感覺很空虛。",[6964,120185,120186],{"id":120186},"網路與現實的失衡",[12,120188,120189],{},"用了太多的社交媒體，也讓我的社交關係變得很糟糕。前年，我剛剛進入大學的時候，很怕生，不知道該找誰說話。社交媒體給了我很大的安慰。我無聊的時候就可以打開dc，看看朋友們又說了些啥。",[12,120191,120192],{},"但是...時間久了，我發現，我越來越依賴著社交媒體。我沒有幾個現實的朋友，因為我幾乎都不會去主動找人交談。我的時間都花在了社交媒體上。我不知道這是不是一件好事，但是我其實內心不敢和我的家人和現實朋友們分享我和網友們的事。在網絡上的我，其實和現實的我相差很大。",[12,120194,120195],{},"我越孤單，就越會盯著手機看。",[12,120197,120198],{},"我沒有朋友的情況，也間接導致了我的成績一落千丈。",[12,120200,120201],{},"醫學，靠自己一個人真的讀不過來。",[6964,120203,120204],{"id":120204},"從醫學到軟件工程",[12,120206,120207],{},"在今年，我從醫學系轉到了軟件工程。原因有很多，但是最主要的還是...我對醫學其實並不是很感興趣。",[12,120209,120210],{},"我身邊的同學們感覺都超級熱愛醫學，但是我不知道為什麼我要讀醫學。我讀下去只是為了考得好，以後找一份安穩的，高薪的工作。",[12,120212,120213],{},"很多人大概會指責我的想法，但我確實是這麼想的。但不一樣的是，我真的真的很喜歡軟件工程。當我寫的代碼在電腦上，經過無數次debug，終於運行起來的時候，那種成就感是無法比擬的。所以，去年得知我考試考不過的時候，我就和父母提出想要轉系了。父母大致上還是理解我的，雖然覺得很可惜，但我至少沒有選擇躺平。能讀自己真正喜歡的科系也好，而且未來工作應該也不會太難找。",[6964,120215,120216],{"id":120216},"我的希望",[12,120218,120219],{},"今年，我已經嘗試在減少discord的使用時間了。我開始和室友們相處、交談、打遊戲、出去玩。我希望能在網絡世界和現實世界找到一個新的平衡點。我希望的是，我可以養成一個寫部落格的習慣，讓我在空閒的時候，不要一直想著去看手機，打開discord毫無目的的到處翻找。這部落格對我來說，就像是一個日記，只不過更加先進。",[6964,120221,118833],{"id":118833},[12,120223,120224],{},"不知不覺就寫了一個半小時的文章。",[12,120226,120227],{},"我還是第一次，花了這麼長的時間，把自己最真誠的一面展現出來。希望我的朋友們，有看完這篇文章的話，能更好地認識我！",[12,120229,120230],{},"那麼，今天就寫這麼多了，謝謝你讀到這裡，也再一次歡迎各位！",{"title":180,"searchDepth":181,"depth":181,"links":120232},[120233,120234,120235,120236,120237,120238,120239],{"id":120139,"depth":181,"text":120140},{"id":120159,"depth":181,"text":120159},{"id":120177,"depth":181,"text":120177},{"id":120186,"depth":181,"text":120186},{"id":120204,"depth":181,"text":120204},{"id":120216,"depth":181,"text":120216},{"id":118833,"depth":181,"text":118833},"2025-10-24","關於我和這個部落格，以及一些自己的反思。","2026-02-18","\u002Fimages\u002Fuploads\u002F84017469_p0.jpg",{},"\u002Fblog\u002F你好，世界。",{"title":120108,"description":120241},{"loc":120245},"blog\u002F你好，世界。",[119959,117129],"2bZ0YGK-ZV7ohFUd-Ur5AG4MSrFjsWop62WvyIaHmWs",{"id":120252,"extension":120253,"meta":120254,"series":120255,"stem":76546,"__hash__":120328},"series\u002Fseries.json","json",{},{"微積分教學":120256,"生活紀錄":120258,"Motor Control":120259,"生活隨筆":120267,"Motor learning":120269,"小兒物治":120278,"中風":120288,"平衡":120295,"Network Communication":120297,"CSA":120304,"機器學習":120310,"小腦":120311,"SCI脊髓損傷":120312,"Machine Learning":120317,"Data 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