[{"data":1,"prerenderedAt":6284},["ShallowReactive",2],{"\u002Fblog\u002FArtificial-Neural-Networks-and-Backpropagation":3,"post-count":5876,"series-global-data":5877,"authors-all":5987,"series-sidebar-none":6182,"sidebar-authors":6183},{"id":4,"title":5,"author":6,"body":7,"date":5865,"description":5833,"draft":5866,"edited_at":5865,"extension":5867,"featured_image":5868,"meta":5869,"navigation":5870,"path":5871,"pinned":5866,"seo":5872,"sitemap":5873,"stem":5874,"tags":5868,"__hash__":5875},"blog\u002Fblog\u002FArtificial-Neural-Networks-and-Backpropagation.md","Artificial Neural Networks and Backpropagation","chinono",{"type":8,"value":9,"toc":5832},"minimark",[10,15,24,46,53,58,61,271,275,278,282,285,312,316,331,629,633,636,639,643,657,661,685,689,692,712,716,723,730,734,742,746,749,1637,1906,1910,1913,2515,2519,2649,2792,2855,2859,2889,2958,3158,4035,4102,4238,4803,5271,5519,5523,5526,5546,5550,5553,5557,5632,5636,5719,5723,5726,5730,5756,5760,5828],[11,12,14],"h2",{"id":13},"_1-the-paradigm-shift-why-artificial-neural-networks","1. The Paradigm Shift: Why Artificial Neural Networks?",[16,17,18,19,23],"p",{},"For decades, computing has been dominated by the ",[20,21,22],"strong",{},"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:",[25,26,27,34,40],"ul",{},[28,29,30,33],"li",{},[20,31,32],{},"Pattern Recognition:"," Identifying faces, deciphering handwritten characters, or processing natural speech.",[28,35,36,39],{},[20,37,38],{},"Content-Addressable Recall:"," Retrieving complex memories based on partial cues rather than explicit memory addresses.",[28,41,42,45],{},[20,43,44],{},"Approximate Reasoning:"," Making common-sense decisions in ambiguous, ill-defined environments (e.g., driving a car, playing a sport).",[16,47,48,49,52],{},"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 ",[20,50,51],{},"Artificial Neural Networks (ANNs)",".",[54,55,57],"h3",{"id":56},"biological-neural-networks-bnn-vs-von-neumann-machines","Biological Neural Networks (BNN) vs. Von Neumann Machines",[16,59,60],{},"The fundamental difference between biological brains and traditional computers lies in their architecture and processing methods.",[62,63,64,81],"table",{},[65,66,67],"thead",{},[68,69,70,75,78],"tr",{},[71,72,74],"th",{"align":73},"left","Feature",[71,76,77],{"align":73},"Von Neumann Machine",[71,79,80],{"align":73},"Human Brain (Biological Neural Network)",[82,83,84,164,177,232,245,258],"tbody",{},[68,85,86,92,95],{},[87,88,89],"td",{"align":73},[20,90,91],{},"Processors",[87,93,94],{"align":73},"One or a few high-speed processors (nanosecond operations).",[87,96,97,98,163],{"align":73},"Billions (",[99,100,104],"mjx-container",{"className":101,"jax":103},[102],"MathJax","SVG",[105,106,115,127],"svg",{"style":107,"xmlns":108,"width":109,"height":110,"role":111,"focusable":112,"viewBox":113,"xmlnsXLink":114},"vertical-align: -0.05ex;","http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","4.05ex","2.005ex","img","false","0 -864 1790.1 886","http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxlink",[116,117,118,123],"defs",{},[119,120],"path",{"id":121,"d":122},"MJX-1-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",[119,124],{"id":125,"d":126},"MJX-1-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",[128,129,133],"g",{"stroke":130,"fill":130,"stroke-width":131,"transform":132},"currentColor","0","scale(1,-1)",[128,134,136],{"dataMmlNode":135},"math",[128,137,139,152],{"dataMmlNode":138},"msup",[128,140,142,147],{"dataMmlNode":141},"mn",[143,144],"use",{"dataC":145,"xLinkHref":146},"31","#MJX-1-TEX-N-31",[143,148],{"dataC":149,"xLinkHref":150,"transform":151},"30","#MJX-1-TEX-N-30","translate(500,0)",[128,153,157],{"dataMmlNode":154,"transform":155,"dataMjxTexclass":156},"TeXAtom","translate(1033,393.1) scale(0.707)","ORD",[128,158,159,161],{"dataMmlNode":141},[143,160],{"dataC":145,"xLinkHref":146},[143,162],{"dataC":145,"xLinkHref":146,"transform":151},") of low-speed processors (millisecond operations).",[68,165,166,171,174],{},[87,167,168],{"align":73},[20,169,170],{},"Computing Power",[87,172,173],{"align":73},"Massive power concentrated in localized CPU units.",[87,175,176],{"align":73},"Limited individual power, relying on massive collective parallelization.",[68,178,179,184,187],{},[87,180,181],{"align":73},[20,182,183],{},"Connectivity",[87,185,186],{"align":73},"Shared, high-speed buses routing data sequentially.",[87,188,189,190,231],{"align":73},"Massive interconnections (",[99,191,193],{"className":192,"jax":103},[102],[105,194,195,207],{"style":107,"xmlns":108,"width":109,"height":110,"role":111,"focusable":112,"viewBox":113,"xmlnsXLink":114},[116,196,197,200,203],{},[119,198],{"id":199,"d":122},"MJX-2-TEX-N-31",[119,201],{"id":202,"d":126},"MJX-2-TEX-N-30",[119,204],{"id":205,"d":206},"MJX-2-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 157Z",[128,208,209],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,210,211],{"dataMmlNode":135},[128,212,213,221],{"dataMmlNode":138},[128,214,215,218],{"dataMmlNode":141},[143,216],{"dataC":145,"xLinkHref":217},"#MJX-2-TEX-N-31",[143,219],{"dataC":149,"xLinkHref":220,"transform":151},"#MJX-2-TEX-N-30",[128,222,223],{"dataMmlNode":154,"transform":155,"dataMjxTexclass":156},[128,224,225,227],{"dataMmlNode":141},[143,226],{"dataC":145,"xLinkHref":217},[143,228],{"dataC":229,"xLinkHref":230,"transform":151},"35","#MJX-2-TEX-N-35"," synapses) operating concurrently.",[68,233,234,239,242],{},[87,235,236],{"align":73},[20,237,238],{},"Memory Access",[87,240,241],{"align":73},"Sequential access via explicit physical addresses.",[87,243,244],{"align":73},"Content-addressable recall (retrieval by association).",[68,246,247,252,255],{},[87,248,249],{"align":73},[20,250,251],{},"Knowledge Storage",[87,253,254],{"align":73},"Knowledge and problem-solving logic are explicitly separated from the computing hardware.",[87,256,257],{"align":73},"Knowledge is distributed and physically resides in the synaptic connectivity between neurons.",[68,259,260,265,268],{},[87,261,262],{"align":73},[20,263,264],{},"Adaptability",[87,266,267],{"align":73},"Hard-coded and rigid; highly susceptible to catastrophic failure if hardware is damaged.",[87,269,270],{"align":73},"Highly adaptive; learns by altering network connectivity. Exhibits graceful degradation if partially damaged.",[11,272,274],{"id":273},"_2-from-biological-to-artificial-neurons","2. From Biological to Artificial Neurons",[16,276,277],{},"To engineer an ANN, we must abstract the biological mechanisms of the brain into mathematical models.",[54,279,281],{"id":280},"the-biological-neuron","The Biological Neuron",[16,283,284],{},"A biological neuron consists of four primary components:",[286,287,288,294,300,306],"ol",{},[28,289,290,293],{},[20,291,292],{},"Dendrites:"," Branching structures that receive incoming signals from other neurons.",[28,295,296,299],{},[20,297,298],{},"Soma (Cell Body):"," Accumulates the incoming signals.",[28,301,302,305],{},[20,303,304],{},"Axon:"," A long fiber that transmits the signal outward if the accumulated stimulus exceeds a certain threshold (action potential).",[28,307,308,311],{},[20,309,310],{},"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.",[54,313,315],{"id":314},"the-artificial-neuron","The Artificial Neuron",[16,317,318,319,322,323,326,327,330],{},"The artificial neuron (often called a ",[20,320,321],{},"node",", ",[20,324,325],{},"unit",", or ",[20,328,329],{},"perceptron",") mirrors this biological structure mathematically:",[286,332,333,445,543,574],{},[28,334,335,444],{},[20,336,337,338,443],{},"Inputs (",[99,339,341],{"className":340,"jax":103},[102],[105,342,347,372],{"style":343,"xmlns":108,"width":344,"height":345,"role":111,"focusable":112,"viewBox":346,"xmlnsXLink":114},"vertical-align: -0.439ex;","13.212ex","1.984ex","0 -683 5839.6 877",[116,348,349,353,356,360,364,368],{},[119,350],{"id":351,"d":352},"MJX-3-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",[119,354],{"id":355,"d":122},"MJX-3-TEX-N-31",[119,357],{"id":358,"d":359},"MJX-3-TEX-N-2C","M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z",[119,361],{"id":362,"d":363},"MJX-3-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",[119,365],{"id":366,"d":367},"MJX-3-TEX-N-2E","M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z",[119,369],{"id":370,"d":371},"MJX-3-TEX-I-1D45A","M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z",[128,373,374],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,375,376,392,400,413,420,425,430],{"dataMmlNode":135},[128,377,379,386],{"dataMmlNode":378},"msub",[128,380,382],{"dataMmlNode":381},"mi",[143,383],{"dataC":384,"xLinkHref":385},"1D44B","#MJX-3-TEX-I-1D44B",[128,387,389],{"dataMmlNode":141,"transform":388},"translate(861,-150) scale(0.707)",[143,390],{"dataC":145,"xLinkHref":391},"#MJX-3-TEX-N-31",[128,393,396],{"dataMmlNode":394,"transform":395},"mo","translate(1264.6,0)",[143,397],{"dataC":398,"xLinkHref":399},"2C","#MJX-3-TEX-N-2C",[128,401,403,407],{"dataMmlNode":378,"transform":402},"translate(1709.2,0)",[128,404,405],{"dataMmlNode":381},[143,406],{"dataC":384,"xLinkHref":385},[128,408,409],{"dataMmlNode":141,"transform":388},[143,410],{"dataC":411,"xLinkHref":412},"32","#MJX-3-TEX-N-32",[128,414,416],{"dataMmlNode":394,"transform":415},"translate(2973.8,0)",[143,417],{"dataC":418,"xLinkHref":419},"2E","#MJX-3-TEX-N-2E",[128,421,423],{"dataMmlNode":394,"transform":422},"translate(3418.4,0)",[143,424],{"dataC":418,"xLinkHref":419},[128,426,428],{"dataMmlNode":394,"transform":427},"translate(3863.1,0)",[143,429],{"dataC":418,"xLinkHref":419},[128,431,433,437],{"dataMmlNode":378,"transform":432},"translate(4307.8,0)",[128,434,435],{"dataMmlNode":381},[143,436],{"dataC":384,"xLinkHref":385},[128,438,439],{"dataMmlNode":381,"transform":388},[143,440],{"dataC":441,"xLinkHref":442},"1D45A","#MJX-3-TEX-I-1D45A","):"," Represent the incoming signals (dendrites).",[28,446,447,542],{},[20,448,449,450,443],{},"Weights (",[99,451,453],{"className":452,"jax":103},[102],[105,454,457,478],{"style":343,"xmlns":108,"width":455,"height":345,"role":111,"focusable":112,"viewBox":456,"xmlnsXLink":114},"13.999ex","0 -683 6187.6 877",[116,458,459,463,466,469,472,475],{},[119,460],{"id":461,"d":462},"MJX-4-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 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 674T327 656Q322 641 318 637H297Q236 634 232 620Q262 160 266 136L501 550L499 587Q496 629 489 632Q483 636 447 637Q428 637 422 639T416 648Q416 650 418 660Q419 664 420 669T421 676T424 680T428 682T436 683Z",[119,464],{"id":465,"d":122},"MJX-4-TEX-N-31",[119,467],{"id":468,"d":359},"MJX-4-TEX-N-2C",[119,470],{"id":471,"d":363},"MJX-4-TEX-N-32",[119,473],{"id":474,"d":367},"MJX-4-TEX-N-2E",[119,476],{"id":477,"d":371},"MJX-4-TEX-I-1D45A",[128,479,480],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,481,482,496,502,514,520,525,530],{"dataMmlNode":135},[128,483,484,490],{"dataMmlNode":378},[128,485,486],{"dataMmlNode":381},[143,487],{"dataC":488,"xLinkHref":489},"1D44A","#MJX-4-TEX-I-1D44A",[128,491,493],{"dataMmlNode":141,"transform":492},"translate(977,-150) scale(0.707)",[143,494],{"dataC":145,"xLinkHref":495},"#MJX-4-TEX-N-31",[128,497,499],{"dataMmlNode":394,"transform":498},"translate(1380.6,0)",[143,500],{"dataC":398,"xLinkHref":501},"#MJX-4-TEX-N-2C",[128,503,505,509],{"dataMmlNode":378,"transform":504},"translate(1825.2,0)",[128,506,507],{"dataMmlNode":381},[143,508],{"dataC":488,"xLinkHref":489},[128,510,511],{"dataMmlNode":141,"transform":492},[143,512],{"dataC":411,"xLinkHref":513},"#MJX-4-TEX-N-32",[128,515,517],{"dataMmlNode":394,"transform":516},"translate(3205.8,0)",[143,518],{"dataC":418,"xLinkHref":519},"#MJX-4-TEX-N-2E",[128,521,523],{"dataMmlNode":394,"transform":522},"translate(3650.4,0)",[143,524],{"dataC":418,"xLinkHref":519},[128,526,528],{"dataMmlNode":394,"transform":527},"translate(4095.1,0)",[143,529],{"dataC":418,"xLinkHref":519},[128,531,533,537],{"dataMmlNode":378,"transform":532},"translate(4539.8,0)",[128,534,535],{"dataMmlNode":381},[143,536],{"dataC":488,"xLinkHref":489},[128,538,539],{"dataMmlNode":381,"transform":492},[143,540],{"dataC":441,"xLinkHref":541},"#MJX-4-TEX-I-1D45A"," Represent the synaptic strength. A high weight amplifies an input; a low weight diminishes it.",[28,544,545,573],{},[20,546,547,548,443],{},"Summation Function (",[99,549,551],{"className":550,"jax":103},[102],[105,552,557,563],{"style":553,"xmlns":108,"width":554,"height":555,"role":111,"focusable":112,"viewBox":556,"xmlnsXLink":114},"vertical-align: 0;","1.633ex","1.545ex","0 -683 722 683",[116,558,559],{},[119,560],{"id":561,"d":562},"MJX-5-TEX-N-3A3","M666 247Q664 244 652 126T638 4V0H351Q131 0 95 0T57 5V6Q54 12 57 17L73 36Q89 54 121 90T182 159L305 299L56 644L55 658Q55 677 60 681Q63 683 351 683H638V679Q640 674 652 564T666 447V443H626V447Q618 505 604 543T559 605Q529 626 478 631T333 637H294H189L293 494Q314 465 345 422Q400 346 400 340Q400 338 399 337L154 57Q407 57 428 58Q476 60 508 68T551 83T575 103Q595 125 608 162T624 225L626 251H666V247Z",[128,564,565],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,566,567],{"dataMmlNode":135},[128,568,569],{"dataMmlNode":381},[143,570],{"dataC":571,"xLinkHref":572},"3A3","#MJX-5-TEX-N-3A3"," Acts as the soma, computing the weighted sum of all incoming signals.",[28,575,576,604,605,628],{},[20,577,578,579,443],{},"Activation Function (",[99,580,582],{"className":581,"jax":103},[102],[105,583,588,594],{"style":584,"xmlns":108,"width":585,"height":586,"role":111,"focusable":112,"viewBox":587,"xmlnsXLink":114},"vertical-align: -0.464ex;","1.244ex","2.059ex","0 -705 550 910",[116,589,590],{},[119,591],{"id":592,"d":593},"MJX-6-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",[128,595,596],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,597,598],{"dataMmlNode":135},[128,599,600],{"dataMmlNode":381},[143,601],{"dataC":602,"xLinkHref":603},"1D453","#MJX-6-TEX-I-1D453"," Acts as the axon, determining the final output (",[99,606,608],{"className":607,"jax":103},[102],[105,609,612,618],{"style":553,"xmlns":108,"width":610,"height":555,"role":111,"focusable":112,"viewBox":611,"xmlnsXLink":114},"1.726ex","0 -683 763 683",[116,613,614],{},[119,615],{"id":616,"d":617},"MJX-7-TEX-I-1D44C","M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z",[128,619,620],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,621,622],{"dataMmlNode":135},[128,623,624],{"dataMmlNode":381},[143,625],{"dataC":626,"xLinkHref":627},"1D44C","#MJX-7-TEX-I-1D44C",") based on the aggregated signal. It evaluates whether the \"neuron\" should fire.",[11,630,632],{"id":631},"_3-neural-network-architecture-and-topology","3. Neural Network Architecture and Topology",[16,634,635],{},"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.",[16,637,638],{},"Networks are classified by several attributes:",[54,640,642],{"id":641},"_1-connection-type","1. Connection Type",[25,644,645,651],{},[28,646,647,650],{},[20,648,649],{},"Static (Feedforward):"," Signals travel in strictly one direction—from input to output. There are no loops.",[28,652,653,656],{},[20,654,655],{},"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.",[54,658,660],{"id":659},"_2-topology","2. Topology",[25,662,663,669,679],{},[28,664,665,668],{},[20,666,667],{},"Single-layer:"," Inputs map directly to a single layer of output nodes.",[28,670,671,674,675,678],{},[20,672,673],{},"Multi-layer:"," Contains one or more ",[20,676,677],{},"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.",[28,680,681,684],{},[20,682,683],{},"Self-organized:"," Networks that autonomously organize their topology based on data patterns (often used in unsupervised learning).",[11,686,688],{"id":687},"_4-learning-paradigms","4. Learning Paradigms",[16,690,691],{},"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.",[286,693,694,700,706],{},[28,695,696,699],{},[20,697,698],{},"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.",[28,701,702,705],{},[20,703,704],{},"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.",[28,707,708,711],{},[20,709,710],{},"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.",[11,713,715],{"id":714},"_5-the-multilayer-perceptron-mlp-and-backpropagation","5. The Multilayer Perceptron (MLP) and Backpropagation",[16,717,718,719,722],{},"The most fundamental architecture for supervised learning is the ",[20,720,721],{},"Multilayer Perceptron (MLP)",". An MLP consists of an input layer, at least one hidden layer, and an output layer.",[16,724,725,726,729],{},"To train an MLP, we use the ",[20,727,728],{},"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.",[54,731,733],{"id":732},"the-challenge-of-hidden-layers","The Challenge of Hidden Layers",[16,735,736,737,741],{},"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 ",[738,739,740],"em",{},"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.",[54,743,745],{"id":744},"mathematical-notation-for-backpropagation","Mathematical Notation for Backpropagation",[16,747,748],{},"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):",[25,750,751,1049,1256,1294,1332,1413],{},[28,752,753,756],{},[20,754,755],{},"Weights:",[25,757,758,983],{},[28,759,760,838,839,936,937,959,960,52],{},[99,761,763],{"className":762,"jax":103},[102],[105,764,769,792],{"style":765,"xmlns":108,"width":766,"height":767,"role":111,"focusable":112,"viewBox":768,"xmlnsXLink":114},"vertical-align: -0.025ex;","5.097ex","2.046ex","0 -893.3 2252.8 904.3",[116,770,771,775,779,782,785,788],{},[119,772],{"id":773,"d":774},"MJX-8-TEX-I-1D464","M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z",[119,776],{"id":777,"d":778},"MJX-8-TEX-N-28","M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z",[119,780],{"id":781,"d":122},"MJX-8-TEX-N-31",[119,783],{"id":784,"d":359},"MJX-8-TEX-N-2C",[119,786],{"id":787,"d":126},"MJX-8-TEX-N-30",[119,789],{"id":790,"d":791},"MJX-8-TEX-N-29","M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z",[128,793,794],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,795,796],{"dataMmlNode":135},[128,797,798,804],{"dataMmlNode":138},[128,799,800],{"dataMmlNode":381},[143,801],{"dataC":802,"xLinkHref":803},"1D464","#MJX-8-TEX-I-1D464",[128,805,807,813,819,825,831],{"dataMmlNode":154,"transform":806,"dataMjxTexclass":156},"translate(749,363) scale(0.707)",[128,808,809],{"dataMmlNode":394},[143,810],{"dataC":811,"xLinkHref":812},"28","#MJX-8-TEX-N-28",[128,814,816],{"dataMmlNode":141,"transform":815},"translate(389,0)",[143,817],{"dataC":145,"xLinkHref":818},"#MJX-8-TEX-N-31",[128,820,822],{"dataMmlNode":394,"transform":821},"translate(889,0)",[143,823],{"dataC":398,"xLinkHref":824},"#MJX-8-TEX-N-2C",[128,826,828],{"dataMmlNode":141,"transform":827},"translate(1167,0)",[143,829],{"dataC":149,"xLinkHref":830},"#MJX-8-TEX-N-30",[128,832,834],{"dataMmlNode":394,"transform":833},"translate(1667,0)",[143,835],{"dataC":836,"xLinkHref":837},"29","#MJX-8-TEX-N-29",": Weight matrix from the input layer to the hidden layer. Specifically, ",[99,840,842],{"className":841,"jax":103},[102],[105,843,847,875],{"style":844,"xmlns":108,"width":766,"height":845,"role":111,"focusable":112,"viewBox":846,"xmlnsXLink":114},"vertical-align: -0.991ex;","3.391ex","0 -1060.7 2252.8 1498.8",[116,848,849,852,855,858,861,864,867,871],{},[119,850],{"id":851,"d":774},"MJX-9-TEX-I-1D464",[119,853],{"id":854,"d":778},"MJX-9-TEX-N-28",[119,856],{"id":857,"d":122},"MJX-9-TEX-N-31",[119,859],{"id":860,"d":359},"MJX-9-TEX-N-2C",[119,862],{"id":863,"d":126},"MJX-9-TEX-N-30",[119,865],{"id":866,"d":791},"MJX-9-TEX-N-29",[119,868],{"id":869,"d":870},"MJX-9-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 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z",[119,872],{"id":873,"d":874},"MJX-9-TEX-I-1D456","M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z",[128,876,877],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,878,879],{"dataMmlNode":135},[128,880,882,887,915],{"dataMmlNode":881},"msubsup",[128,883,884],{"dataMmlNode":381},[143,885],{"dataC":802,"xLinkHref":886},"#MJX-9-TEX-I-1D464",[128,888,890,895,900,905,910],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},"translate(749,530.4) scale(0.707)",[128,891,892],{"dataMmlNode":394},[143,893],{"dataC":811,"xLinkHref":894},"#MJX-9-TEX-N-28",[128,896,897],{"dataMmlNode":141,"transform":815},[143,898],{"dataC":145,"xLinkHref":899},"#MJX-9-TEX-N-31",[128,901,902],{"dataMmlNode":394,"transform":821},[143,903],{"dataC":398,"xLinkHref":904},"#MJX-9-TEX-N-2C",[128,906,907],{"dataMmlNode":141,"transform":827},[143,908],{"dataC":149,"xLinkHref":909},"#MJX-9-TEX-N-30",[128,911,912],{"dataMmlNode":394,"transform":833},[143,913],{"dataC":836,"xLinkHref":914},"#MJX-9-TEX-N-29",[128,916,918,924,929],{"dataMmlNode":154,"transform":917,"dataMjxTexclass":156},"translate(749,-293.8) scale(0.707)",[128,919,920],{"dataMmlNode":381},[143,921],{"dataC":922,"xLinkHref":923},"1D457","#MJX-9-TEX-I-1D457",[128,925,927],{"dataMmlNode":394,"transform":926},"translate(412,0)",[143,928],{"dataC":398,"xLinkHref":904},[128,930,932],{"dataMmlNode":381,"transform":931},"translate(690,0)",[143,933],{"dataC":934,"xLinkHref":935},"1D456","#MJX-9-TEX-I-1D456"," is the weight connecting input node ",[99,938,940],{"className":939,"jax":103},[102],[105,941,945,950],{"style":765,"xmlns":108,"width":942,"height":943,"role":111,"focusable":112,"viewBox":944,"xmlnsXLink":114},"0.781ex","1.52ex","0 -661 345 672",[116,946,947],{},[119,948],{"id":949,"d":874},"MJX-10-TEX-I-1D456",[128,951,952],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,953,954],{"dataMmlNode":135},[128,955,956],{"dataMmlNode":381},[143,957],{"dataC":934,"xLinkHref":958},"#MJX-10-TEX-I-1D456"," to hidden node ",[99,961,963],{"className":962,"jax":103},[102],[105,964,969,974],{"style":965,"xmlns":108,"width":966,"height":967,"role":111,"focusable":112,"viewBox":968,"xmlnsXLink":114},"vertical-align: -0.462ex;","0.932ex","1.957ex","0 -661 412 865",[116,970,971],{},[119,972],{"id":973,"d":870},"MJX-11-TEX-I-1D457",[128,975,976],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,977,978],{"dataMmlNode":135},[128,979,980],{"dataMmlNode":381},[143,981],{"dataC":922,"xLinkHref":982},"#MJX-11-TEX-I-1D457",[28,984,985,1048],{},[99,986,988],{"className":987,"jax":103},[102],[105,989,990,1010],{"style":765,"xmlns":108,"width":766,"height":767,"role":111,"focusable":112,"viewBox":768,"xmlnsXLink":114},[116,991,992,995,998,1001,1004,1007],{},[119,993],{"id":994,"d":774},"MJX-12-TEX-I-1D464",[119,996],{"id":997,"d":778},"MJX-12-TEX-N-28",[119,999],{"id":1000,"d":363},"MJX-12-TEX-N-32",[119,1002],{"id":1003,"d":359},"MJX-12-TEX-N-2C",[119,1005],{"id":1006,"d":122},"MJX-12-TEX-N-31",[119,1008],{"id":1009,"d":791},"MJX-12-TEX-N-29",[128,1011,1012],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1013,1014],{"dataMmlNode":135},[128,1015,1016,1021],{"dataMmlNode":138},[128,1017,1018],{"dataMmlNode":381},[143,1019],{"dataC":802,"xLinkHref":1020},"#MJX-12-TEX-I-1D464",[128,1022,1023,1028,1033,1038,1043],{"dataMmlNode":154,"transform":806,"dataMjxTexclass":156},[128,1024,1025],{"dataMmlNode":394},[143,1026],{"dataC":811,"xLinkHref":1027},"#MJX-12-TEX-N-28",[128,1029,1030],{"dataMmlNode":141,"transform":815},[143,1031],{"dataC":411,"xLinkHref":1032},"#MJX-12-TEX-N-32",[128,1034,1035],{"dataMmlNode":394,"transform":821},[143,1036],{"dataC":398,"xLinkHref":1037},"#MJX-12-TEX-N-2C",[128,1039,1040],{"dataMmlNode":141,"transform":827},[143,1041],{"dataC":145,"xLinkHref":1042},"#MJX-12-TEX-N-31",[128,1044,1045],{"dataMmlNode":394,"transform":833},[143,1046],{"dataC":836,"xLinkHref":1047},"#MJX-12-TEX-N-29",": Weight matrix from the hidden layer to the output layer.",[28,1050,1051,1054,1055,1078,1079,52],{},[20,1052,1053],{},"Training Samples:"," A set of ",[99,1056,1058],{"className":1057,"jax":103},[102],[105,1059,1062,1068],{"style":553,"xmlns":108,"width":1060,"height":555,"role":111,"focusable":112,"viewBox":1061,"xmlnsXLink":114},"1.699ex","0 -683 751 683",[116,1063,1064],{},[119,1065],{"id":1066,"d":1067},"MJX-13-TEX-I-1D443","M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z",[128,1069,1070],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1071,1072],{"dataMmlNode":135},[128,1073,1074],{"dataMmlNode":381},[143,1075],{"dataC":1076,"xLinkHref":1077},"1D443","#MJX-13-TEX-I-1D443"," samples, denoted as ",[99,1080,1082],{"className":1081,"jax":103},[102],[105,1083,1088,1136],{"style":1084,"xmlns":108,"width":1085,"height":1086,"role":111,"focusable":112,"viewBox":1087,"xmlnsXLink":114},"vertical-align: -0.65ex;","23.386ex","2.347ex","0 -750 10336.5 1037.2",[116,1089,1090,1094,1097,1101,1105,1108,1112,1115,1119,1123,1126,1129,1132],{},[119,1091],{"id":1092,"d":1093},"MJX-14-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 -231Z",[119,1095],{"id":1096,"d":778},"MJX-14-TEX-N-28",[119,1098],{"id":1099,"d":1100},"MJX-14-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",[119,1102],{"id":1103,"d":1104},"MJX-14-TEX-I-1D45D","M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z",[119,1106],{"id":1107,"d":359},"MJX-14-TEX-N-2C",[119,1109],{"id":1110,"d":1111},"MJX-14-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",[119,1113],{"id":1114,"d":791},"MJX-14-TEX-N-29",[119,1116],{"id":1117,"d":1118},"MJX-14-TEX-N-2223","M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z",[119,1120],{"id":1121,"d":1122},"MJX-14-TEX-N-3D","M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z",[119,1124],{"id":1125,"d":122},"MJX-14-TEX-N-31",[119,1127],{"id":1128,"d":367},"MJX-14-TEX-N-2E",[119,1130],{"id":1131,"d":1067},"MJX-14-TEX-I-1D443",[119,1133],{"id":1134,"d":1135},"MJX-14-TEX-N-7D","M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z",[128,1137,1138],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1139,1140,1146,1151,1166,1172,1186,1192,1199,1204,1211,1217,1222,1228,1233,1238,1243,1249],{"dataMmlNode":135},[128,1141,1142],{"dataMmlNode":394},[143,1143],{"dataC":1144,"xLinkHref":1145},"7B","#MJX-14-TEX-N-7B",[128,1147,1148],{"dataMmlNode":394,"transform":151},[143,1149],{"dataC":811,"xLinkHref":1150},"#MJX-14-TEX-N-28",[128,1152,1153,1159],{"dataMmlNode":378,"transform":821},[128,1154,1155],{"dataMmlNode":381},[143,1156],{"dataC":1157,"xLinkHref":1158},"1D465","#MJX-14-TEX-I-1D465",[128,1160,1162],{"dataMmlNode":381,"transform":1161},"translate(605,-150) scale(0.707)",[143,1163],{"dataC":1164,"xLinkHref":1165},"1D45D","#MJX-14-TEX-I-1D45D",[128,1167,1169],{"dataMmlNode":394,"transform":1168},"translate(1899.7,0)",[143,1170],{"dataC":398,"xLinkHref":1171},"#MJX-14-TEX-N-2C",[128,1173,1175,1181],{"dataMmlNode":378,"transform":1174},"translate(2344.3,0)",[128,1176,1177],{"dataMmlNode":381},[143,1178],{"dataC":1179,"xLinkHref":1180},"1D451","#MJX-14-TEX-I-1D451",[128,1182,1184],{"dataMmlNode":381,"transform":1183},"translate(553,-150) scale(0.707)",[143,1185],{"dataC":1164,"xLinkHref":1165},[128,1187,1189],{"dataMmlNode":394,"transform":1188},"translate(3303,0)",[143,1190],{"dataC":836,"xLinkHref":1191},"#MJX-14-TEX-N-29",[128,1193,1195],{"dataMmlNode":394,"transform":1194},"translate(3969.8,0)",[143,1196],{"dataC":1197,"xLinkHref":1198},"2223","#MJX-14-TEX-N-2223",[128,1200,1202],{"dataMmlNode":381,"transform":1201},"translate(4525.6,0)",[143,1203],{"dataC":1164,"xLinkHref":1165},[128,1205,1207],{"dataMmlNode":394,"transform":1206},"translate(5306.3,0)",[143,1208],{"dataC":1209,"xLinkHref":1210},"3D","#MJX-14-TEX-N-3D",[128,1212,1214],{"dataMmlNode":141,"transform":1213},"translate(6362.1,0)",[143,1215],{"dataC":145,"xLinkHref":1216},"#MJX-14-TEX-N-31",[128,1218,1220],{"dataMmlNode":394,"transform":1219},"translate(6862.1,0)",[143,1221],{"dataC":398,"xLinkHref":1171},[128,1223,1225],{"dataMmlNode":394,"transform":1224},"translate(7306.8,0)",[143,1226],{"dataC":418,"xLinkHref":1227},"#MJX-14-TEX-N-2E",[128,1229,1231],{"dataMmlNode":394,"transform":1230},"translate(7751.5,0)",[143,1232],{"dataC":418,"xLinkHref":1227},[128,1234,1236],{"dataMmlNode":394,"transform":1235},"translate(8196.1,0)",[143,1237],{"dataC":418,"xLinkHref":1227},[128,1239,1241],{"dataMmlNode":394,"transform":1240},"translate(8640.8,0)",[143,1242],{"dataC":398,"xLinkHref":1171},[128,1244,1246],{"dataMmlNode":381,"transform":1245},"translate(9085.5,0)",[143,1247],{"dataC":1076,"xLinkHref":1248},"#MJX-14-TEX-I-1D443",[128,1250,1252],{"dataMmlNode":394,"transform":1251},"translate(9836.5,0)",[143,1253],{"dataC":1254,"xLinkHref":1255},"7D","#MJX-14-TEX-N-7D",[28,1257,1258,1293],{},[20,1259,1260,1261,443],{},"Input Pattern (",[99,1262,1264],{"className":1263,"jax":103},[102],[105,1265,1269,1277],{"style":1084,"xmlns":108,"width":1266,"height":1267,"role":111,"focusable":112,"viewBox":1268,"xmlnsXLink":114},"2.287ex","1.65ex","0 -442 1010.7 729.2",[116,1270,1271,1274],{},[119,1272],{"id":1273,"d":1100},"MJX-15-TEX-I-1D465",[119,1275],{"id":1276,"d":1104},"MJX-15-TEX-I-1D45D",[128,1278,1279],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1280,1281],{"dataMmlNode":135},[128,1282,1283,1288],{"dataMmlNode":378},[128,1284,1285],{"dataMmlNode":381},[143,1286],{"dataC":1157,"xLinkHref":1287},"#MJX-15-TEX-I-1D465",[128,1289,1290],{"dataMmlNode":381,"transform":1161},[143,1291],{"dataC":1164,"xLinkHref":1292},"#MJX-15-TEX-I-1D45D"," A vector of inputs for a given sample.",[28,1295,1296,1331],{},[20,1297,1298,1299,443],{},"Desired Output (",[99,1300,1302],{"className":1301,"jax":103},[102],[105,1303,1307,1315],{"style":1084,"xmlns":108,"width":1304,"height":1305,"role":111,"focusable":112,"viewBox":1306,"xmlnsXLink":114},"2.169ex","2.22ex","0 -694 958.7 981.2",[116,1308,1309,1312],{},[119,1310],{"id":1311,"d":1111},"MJX-16-TEX-I-1D451",[119,1313],{"id":1314,"d":1104},"MJX-16-TEX-I-1D45D",[128,1316,1317],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1318,1319],{"dataMmlNode":135},[128,1320,1321,1326],{"dataMmlNode":378},[128,1322,1323],{"dataMmlNode":381},[143,1324],{"dataC":1179,"xLinkHref":1325},"#MJX-16-TEX-I-1D451",[128,1327,1328],{"dataMmlNode":381,"transform":1183},[143,1329],{"dataC":1164,"xLinkHref":1330},"#MJX-16-TEX-I-1D45D"," The ground truth target vector.",[28,1333,1334,1371,1372,1371,1375,1412],{},[20,1335,1336,1337],{},"Actual Output (",[99,1338,1340],{"className":1339,"jax":103},[102],[105,1341,1344,1353],{"style":1084,"xmlns":108,"width":1342,"height":1267,"role":111,"focusable":112,"viewBox":1343,"xmlnsXLink":114},"2.101ex","0 -442 928.7 729.2",[116,1345,1346,1350],{},[119,1347],{"id":1348,"d":1349},"MJX-17-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",[119,1351],{"id":1352,"d":1104},"MJX-17-TEX-I-1D45D",[128,1354,1355],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1356,1357],{"dataMmlNode":135},[128,1358,1359,1365],{"dataMmlNode":378},[128,1360,1361],{"dataMmlNode":381},[143,1362],{"dataC":1363,"xLinkHref":1364},"1D466","#MJX-17-TEX-I-1D466",[128,1366,1368],{"dataMmlNode":381,"transform":1367},"translate(523,-150) scale(0.707)",[143,1369],{"dataC":1164,"xLinkHref":1370},"#MJX-17-TEX-I-1D45D"," ",[20,1373,1374],{},"or",[20,1376,1377,443],{},[99,1378,1380],{"className":1379,"jax":103},[102],[105,1381,1385,1394],{"style":1084,"xmlns":108,"width":1382,"height":1383,"role":111,"focusable":112,"viewBox":1384,"xmlnsXLink":114},"2.09ex","1.647ex","0 -441 923.7 728.2",[116,1386,1387,1391],{},[119,1388],{"id":1389,"d":1390},"MJX-18-TEX-I-1D45C","M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z",[119,1392],{"id":1393,"d":1104},"MJX-18-TEX-I-1D45D",[128,1395,1396],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1397,1398],{"dataMmlNode":135},[128,1399,1400,1406],{"dataMmlNode":378},[128,1401,1402],{"dataMmlNode":381},[143,1403],{"dataC":1404,"xLinkHref":1405},"1D45C","#MJX-18-TEX-I-1D45C",[128,1407,1409],{"dataMmlNode":381,"transform":1408},"translate(518,-150) scale(0.707)",[143,1410],{"dataC":1164,"xLinkHref":1411},"#MJX-18-TEX-I-1D45D"," The network's calculated output vector.",[28,1414,1415,1474,1475,1494,1495,1517,1518,52],{},[20,1416,1417,1418,443],{},"Error (",[99,1419,1421],{"className":1420,"jax":103},[102],[105,1422,1427,1442],{"style":1423,"xmlns":108,"width":1424,"height":1425,"role":111,"focusable":112,"viewBox":1426,"xmlnsXLink":114},"vertical-align: -0.666ex;","2.771ex","2.236ex","0 -694 1224.6 988.2",[116,1428,1429,1433,1436,1439],{},[119,1430],{"id":1431,"d":1432},"MJX-19-TEX-I-1D459","M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z",[119,1434],{"id":1435,"d":1104},"MJX-19-TEX-I-1D45D",[119,1437],{"id":1438,"d":359},"MJX-19-TEX-N-2C",[119,1440],{"id":1441,"d":870},"MJX-19-TEX-I-1D457",[128,1443,1444],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1445,1446],{"dataMmlNode":135},[128,1447,1448,1454],{"dataMmlNode":378},[128,1449,1450],{"dataMmlNode":381},[143,1451],{"dataC":1452,"xLinkHref":1453},"1D459","#MJX-19-TEX-I-1D459",[128,1455,1457,1462,1468],{"dataMmlNode":154,"transform":1456,"dataMjxTexclass":156},"translate(331,-150) scale(0.707)",[128,1458,1459],{"dataMmlNode":381},[143,1460],{"dataC":1164,"xLinkHref":1461},"#MJX-19-TEX-I-1D45D",[128,1463,1465],{"dataMmlNode":394,"transform":1464},"translate(503,0)",[143,1466],{"dataC":398,"xLinkHref":1467},"#MJX-19-TEX-N-2C",[128,1469,1471],{"dataMmlNode":381,"transform":1470},"translate(781,0)",[143,1472],{"dataC":922,"xLinkHref":1473},"#MJX-19-TEX-I-1D457"," The difference between the desired and actual output for node ",[99,1476,1478],{"className":1477,"jax":103},[102],[105,1479,1480,1485],{"style":965,"xmlns":108,"width":966,"height":967,"role":111,"focusable":112,"viewBox":968,"xmlnsXLink":114},[116,1481,1482],{},[119,1483],{"id":1484,"d":870},"MJX-20-TEX-I-1D457",[128,1486,1487],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1488,1489],{"dataMmlNode":135},[128,1490,1491],{"dataMmlNode":381},[143,1492],{"dataC":922,"xLinkHref":1493},"#MJX-20-TEX-I-1D457"," given sample ",[99,1496,1498],{"className":1497,"jax":103},[102],[105,1499,1503,1508],{"style":343,"xmlns":108,"width":1500,"height":1501,"role":111,"focusable":112,"viewBox":1502,"xmlnsXLink":114},"1.138ex","1.439ex","0 -442 503 636",[116,1504,1505],{},[119,1506],{"id":1507,"d":1104},"MJX-21-TEX-I-1D45D",[128,1509,1510],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1511,1512],{"dataMmlNode":135},[128,1513,1514],{"dataMmlNode":381},[143,1515],{"dataC":1164,"xLinkHref":1516},"#MJX-21-TEX-I-1D45D",". ",[99,1519,1521],{"className":1520,"jax":103},[102],[105,1522,1525,1552],{"style":1423,"xmlns":108,"width":1523,"height":1425,"role":111,"focusable":112,"viewBox":1524,"xmlnsXLink":114},"15.031ex","0 -694 6643.7 988.2",[116,1526,1527,1530,1533,1536,1539,1542,1545,1549],{},[119,1528],{"id":1529,"d":1432},"MJX-22-TEX-I-1D459",[119,1531],{"id":1532,"d":1104},"MJX-22-TEX-I-1D45D",[119,1534],{"id":1535,"d":359},"MJX-22-TEX-N-2C",[119,1537],{"id":1538,"d":870},"MJX-22-TEX-I-1D457",[119,1540],{"id":1541,"d":1122},"MJX-22-TEX-N-3D",[119,1543],{"id":1544,"d":1111},"MJX-22-TEX-I-1D451",[119,1546],{"id":1547,"d":1548},"MJX-22-TEX-N-2212","M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z",[119,1550],{"id":1551,"d":1349},"MJX-22-TEX-I-1D466",[128,1553,1554],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1555,1556,1580,1586,1608,1615],{"dataMmlNode":135},[128,1557,1558,1563],{"dataMmlNode":378},[128,1559,1560],{"dataMmlNode":381},[143,1561],{"dataC":1452,"xLinkHref":1562},"#MJX-22-TEX-I-1D459",[128,1564,1565,1570,1575],{"dataMmlNode":154,"transform":1456,"dataMjxTexclass":156},[128,1566,1567],{"dataMmlNode":381},[143,1568],{"dataC":1164,"xLinkHref":1569},"#MJX-22-TEX-I-1D45D",[128,1571,1572],{"dataMmlNode":394,"transform":1464},[143,1573],{"dataC":398,"xLinkHref":1574},"#MJX-22-TEX-N-2C",[128,1576,1577],{"dataMmlNode":381,"transform":1470},[143,1578],{"dataC":922,"xLinkHref":1579},"#MJX-22-TEX-I-1D457",[128,1581,1583],{"dataMmlNode":394,"transform":1582},"translate(1502.4,0)",[143,1584],{"dataC":1209,"xLinkHref":1585},"#MJX-22-TEX-N-3D",[128,1587,1589,1594],{"dataMmlNode":378,"transform":1588},"translate(2558.1,0)",[128,1590,1591],{"dataMmlNode":381},[143,1592],{"dataC":1179,"xLinkHref":1593},"#MJX-22-TEX-I-1D451",[128,1595,1596,1600,1604],{"dataMmlNode":154,"transform":1183,"dataMjxTexclass":156},[128,1597,1598],{"dataMmlNode":381},[143,1599],{"dataC":1164,"xLinkHref":1569},[128,1601,1602],{"dataMmlNode":394,"transform":1464},[143,1603],{"dataC":398,"xLinkHref":1574},[128,1605,1606],{"dataMmlNode":381,"transform":1470},[143,1607],{"dataC":922,"xLinkHref":1579},[128,1609,1611],{"dataMmlNode":394,"transform":1610},"translate(4226.9,0)",[143,1612],{"dataC":1613,"xLinkHref":1614},"2212","#MJX-22-TEX-N-2212",[128,1616,1618,1623],{"dataMmlNode":378,"transform":1617},"translate(5227.2,0)",[128,1619,1620],{"dataMmlNode":381},[143,1621],{"dataC":1363,"xLinkHref":1622},"#MJX-22-TEX-I-1D466",[128,1624,1625,1629,1633],{"dataMmlNode":154,"transform":1367,"dataMjxTexclass":156},[128,1626,1627],{"dataMmlNode":381},[143,1628],{"dataC":1164,"xLinkHref":1569},[128,1630,1631],{"dataMmlNode":394,"transform":1464},[143,1632],{"dataC":398,"xLinkHref":1574},[128,1634,1635],{"dataMmlNode":381,"transform":1470},[143,1636],{"dataC":922,"xLinkHref":1579},[16,1638,1639,1640,1643,1644,1647,1648,1667,1668,1691,1692],{},"The overall ",[20,1641,1642],{},"Objective Function"," is to minimize the ",[20,1645,1646],{},"Sum Squared Error (E)"," across all ",[99,1649,1651],{"className":1650,"jax":103},[102],[105,1652,1653,1658],{"style":553,"xmlns":108,"width":1060,"height":555,"role":111,"focusable":112,"viewBox":1061,"xmlnsXLink":114},[116,1654,1655],{},[119,1656],{"id":1657,"d":1067},"MJX-23-TEX-I-1D443",[128,1659,1660],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1661,1662],{"dataMmlNode":135},[128,1663,1664],{"dataMmlNode":381},[143,1665],{"dataC":1076,"xLinkHref":1666},"#MJX-23-TEX-I-1D443"," training samples and all ",[99,1669,1671],{"className":1670,"jax":103},[102],[105,1672,1675,1681],{"style":553,"xmlns":108,"width":1673,"height":555,"role":111,"focusable":112,"viewBox":1674,"xmlnsXLink":114},"2.011ex","0 -683 889 683",[116,1676,1677],{},[119,1678],{"id":1679,"d":1680},"MJX-24-TEX-I-1D43E","M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",[128,1682,1683],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1684,1685],{"dataMmlNode":135},[128,1686,1687],{"dataMmlNode":381},[143,1688],{"dataC":1689,"xLinkHref":1690},"1D43E","#MJX-24-TEX-I-1D43E"," output nodes:\n",[99,1693,1695],{"className":1694,"jax":103},[102],[105,1696,1701,1751],{"style":1697,"xmlns":108,"width":1698,"height":1699,"role":111,"focusable":112,"viewBox":1700,"xmlnsXLink":114},"vertical-align: -0.956ex;","28.333ex","3.128ex","0 -960 12523 1382.5",[116,1702,1703,1707,1710,1714,1717,1720,1723,1726,1730,1733,1736,1739,1742,1745,1748],{},[119,1704],{"id":1705,"d":1706},"MJX-25-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",[119,1708],{"id":1709,"d":1122},"MJX-25-TEX-N-3D",[119,1711],{"id":1712,"d":1713},"MJX-25-TEX-SO-2211","M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z",[119,1715],{"id":1716,"d":1067},"MJX-25-TEX-I-1D443",[119,1718],{"id":1719,"d":1104},"MJX-25-TEX-I-1D45D",[119,1721],{"id":1722,"d":122},"MJX-25-TEX-N-31",[119,1724],{"id":1725,"d":1680},"MJX-25-TEX-I-1D43E",[119,1727],{"id":1728,"d":1729},"MJX-25-TEX-I-1D458","M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z",[119,1731],{"id":1732,"d":778},"MJX-25-TEX-N-28",[119,1734],{"id":1735,"d":1111},"MJX-25-TEX-I-1D451",[119,1737],{"id":1738,"d":359},"MJX-25-TEX-N-2C",[119,1740],{"id":1741,"d":1548},"MJX-25-TEX-N-2212",[119,1743],{"id":1744,"d":1390},"MJX-25-TEX-I-1D45C",[119,1746],{"id":1747,"d":791},"MJX-25-TEX-N-29",[119,1749],{"id":1750,"d":363},"MJX-25-TEX-N-32",[128,1752,1753],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1754,1755,1761,1767,1803,1835,1841,1864,1870,1892],{"dataMmlNode":135},[128,1756,1757],{"dataMmlNode":381},[143,1758],{"dataC":1759,"xLinkHref":1760},"1D438","#MJX-25-TEX-I-1D438",[128,1762,1764],{"dataMmlNode":394,"transform":1763},"translate(1041.8,0)",[143,1765],{"dataC":1209,"xLinkHref":1766},"#MJX-25-TEX-N-3D",[128,1768,1771,1777,1785],{"dataMmlNode":1769,"transform":1770},"munderover","translate(2097.6,0)",[128,1772,1773],{"dataMmlNode":394},[143,1774],{"dataC":1775,"xLinkHref":1776},"2211","#MJX-25-TEX-SO-2211",[128,1778,1780],{"dataMmlNode":154,"transform":1779,"dataMjxTexclass":156},"translate(1089,477.1) scale(0.707)",[128,1781,1782],{"dataMmlNode":381},[143,1783],{"dataC":1076,"xLinkHref":1784},"#MJX-25-TEX-I-1D443",[128,1786,1788,1793,1797],{"dataMmlNode":154,"transform":1787,"dataMjxTexclass":156},"translate(1089,-285.4) scale(0.707)",[128,1789,1790],{"dataMmlNode":381},[143,1791],{"dataC":1164,"xLinkHref":1792},"#MJX-25-TEX-I-1D45D",[128,1794,1795],{"dataMmlNode":394,"transform":1464},[143,1796],{"dataC":1209,"xLinkHref":1766},[128,1798,1800],{"dataMmlNode":141,"transform":1799},"translate(1281,0)",[143,1801],{"dataC":145,"xLinkHref":1802},"#MJX-25-TEX-N-31",[128,1804,1806,1810,1817],{"dataMmlNode":1769,"transform":1805},"translate(4662.6,0)",[128,1807,1808],{"dataMmlNode":394},[143,1809],{"dataC":1775,"xLinkHref":1776},[128,1811,1812],{"dataMmlNode":154,"transform":1779,"dataMjxTexclass":156},[128,1813,1814],{"dataMmlNode":381},[143,1815],{"dataC":1689,"xLinkHref":1816},"#MJX-25-TEX-I-1D43E",[128,1818,1819,1825,1830],{"dataMmlNode":154,"transform":1787,"dataMjxTexclass":156},[128,1820,1821],{"dataMmlNode":381},[143,1822],{"dataC":1823,"xLinkHref":1824},"1D458","#MJX-25-TEX-I-1D458",[128,1826,1828],{"dataMmlNode":394,"transform":1827},"translate(521,0)",[143,1829],{"dataC":1209,"xLinkHref":1766},[128,1831,1833],{"dataMmlNode":141,"transform":1832},"translate(1299,0)",[143,1834],{"dataC":145,"xLinkHref":1802},[128,1836,1838],{"dataMmlNode":394,"transform":1837},"translate(7073.7,0)",[143,1839],{"dataC":811,"xLinkHref":1840},"#MJX-25-TEX-N-28",[128,1842,1844,1849],{"dataMmlNode":378,"transform":1843},"translate(7462.7,0)",[128,1845,1846],{"dataMmlNode":381},[143,1847],{"dataC":1179,"xLinkHref":1848},"#MJX-25-TEX-I-1D451",[128,1850,1851,1855,1860],{"dataMmlNode":154,"transform":1183,"dataMjxTexclass":156},[128,1852,1853],{"dataMmlNode":381},[143,1854],{"dataC":1164,"xLinkHref":1792},[128,1856,1857],{"dataMmlNode":394,"transform":1464},[143,1858],{"dataC":398,"xLinkHref":1859},"#MJX-25-TEX-N-2C",[128,1861,1862],{"dataMmlNode":381,"transform":1470},[143,1863],{"dataC":1823,"xLinkHref":1824},[128,1865,1867],{"dataMmlNode":394,"transform":1866},"translate(9208.5,0)",[143,1868],{"dataC":1613,"xLinkHref":1869},"#MJX-25-TEX-N-2212",[128,1871,1873,1878],{"dataMmlNode":378,"transform":1872},"translate(10208.8,0)",[128,1874,1875],{"dataMmlNode":381},[143,1876],{"dataC":1404,"xLinkHref":1877},"#MJX-25-TEX-I-1D45C",[128,1879,1880,1884,1888],{"dataMmlNode":154,"transform":1408,"dataMjxTexclass":156},[128,1881,1882],{"dataMmlNode":381},[143,1883],{"dataC":1164,"xLinkHref":1792},[128,1885,1886],{"dataMmlNode":394,"transform":1464},[143,1887],{"dataC":398,"xLinkHref":1859},[128,1889,1890],{"dataMmlNode":381,"transform":1470},[143,1891],{"dataC":1823,"xLinkHref":1824},[128,1893,1895,1900],{"dataMmlNode":138,"transform":1894},"translate(11697.4,0)",[128,1896,1897],{"dataMmlNode":394},[143,1898],{"dataC":836,"xLinkHref":1899},"#MJX-25-TEX-N-29",[128,1901,1903],{"dataMmlNode":141,"transform":1902},"translate(422,363) scale(0.707)",[143,1904],{"dataC":411,"xLinkHref":1905},"#MJX-25-TEX-N-32",[54,1907,1909],{"id":1908},"step-1-forward-computing-the-forward-pass","Step 1: Forward Computing (The Forward Pass)",[16,1911,1912],{},"Data propagates forward from inputs to outputs.",[286,1914,1915,2236],{},[28,1916,1917,1920,1921,1943,1944,1968,1969],{},[20,1918,1919],{},"Hidden Layer Calculation:"," The input vector ",[99,1922,1924],{"className":1923,"jax":103},[102],[105,1925,1929,1934],{"style":765,"xmlns":108,"width":1926,"height":1927,"role":111,"focusable":112,"viewBox":1928,"xmlnsXLink":114},"1.294ex","1.025ex","0 -442 572 453",[116,1930,1931],{},[119,1932],{"id":1933,"d":1100},"MJX-26-TEX-I-1D465",[128,1935,1936],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1937,1938],{"dataMmlNode":135},[128,1939,1940],{"dataMmlNode":381},[143,1941],{"dataC":1157,"xLinkHref":1942},"#MJX-26-TEX-I-1D465"," is multiplied by the input-to-hidden weights to generate a net input, which is then passed through an activation function ",[99,1945,1947],{"className":1946,"jax":103},[102],[105,1948,1952,1958],{"style":107,"xmlns":108,"width":1949,"height":1950,"role":111,"focusable":112,"viewBox":1951,"xmlnsXLink":114},"1.459ex","1.645ex","0 -705 645 727",[116,1953,1954],{},[119,1955],{"id":1956,"d":1957},"MJX-27-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",[128,1959,1960],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,1961,1962],{"dataMmlNode":135},[128,1963,1964],{"dataMmlNode":381},[143,1965],{"dataC":1966,"xLinkHref":1967},"1D446","#MJX-27-TEX-I-1D446",".\n",[99,1970,1972],{"className":1971,"jax":103},[102],[105,1973,1978,2036],{"style":1974,"xmlns":108,"width":1975,"height":1976,"role":111,"focusable":112,"viewBox":1977,"xmlnsXLink":114},"vertical-align: -1.469ex;","34.733ex","4.07ex","0 -1149.5 15352.2 1799",[116,1979,1980,1983,1986,1989,1992,1995,1998,2001,2005,2009,2013,2017,2021,2024,2027,2030,2033],{},[119,1981],{"id":1982,"d":1100},"MJX-28-TEX-I-1D465",[119,1984],{"id":1985,"d":778},"MJX-28-TEX-N-28",[119,1987],{"id":1988,"d":122},"MJX-28-TEX-N-31",[119,1990],{"id":1991,"d":791},"MJX-28-TEX-N-29",[119,1993],{"id":1994,"d":870},"MJX-28-TEX-I-1D457",[119,1996],{"id":1997,"d":1122},"MJX-28-TEX-N-3D",[119,1999],{"id":2000,"d":1957},"MJX-28-TEX-I-1D446",[119,2002],{"id":2003,"d":2004},"MJX-28-TEX-LO-28","M180 96T180 250T205 541T266 770T353 944T444 1069T527 1150H555Q561 1144 561 1141Q561 1137 545 1120T504 1072T447 995T386 878T330 721T288 513T272 251Q272 133 280 56Q293 -87 326 -209T399 -405T475 -531T536 -609T561 -640Q561 -643 555 -649H527Q483 -612 443 -568T353 -443T266 -270T205 -41Z",[119,2006],{"id":2007,"d":2008},"MJX-28-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",[119,2010],{"id":2011,"d":2012},"MJX-28-TEX-I-1D452","M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z",[119,2014],{"id":2015,"d":2016},"MJX-28-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",[119,2018],{"id":2019,"d":2020},"MJX-28-TEX-LO-29","M35 1138Q35 1150 51 1150H56H69Q113 1113 153 1069T243 944T330 771T391 541T416 250T391 -40T330 -270T243 -443T152 -568T69 -649H56Q43 -649 39 -647T35 -637Q65 -607 110 -548Q283 -316 316 56Q324 133 324 251Q324 368 316 445Q278 877 48 1123Q36 1137 35 1138Z",[119,2022],{"id":2023,"d":1713},"MJX-28-TEX-SO-2211",[119,2025],{"id":2026,"d":874},"MJX-28-TEX-I-1D456",[119,2028],{"id":2029,"d":774},"MJX-28-TEX-I-1D464",[119,2031],{"id":2032,"d":359},"MJX-28-TEX-N-2C",[119,2034],{"id":2035,"d":126},"MJX-28-TEX-N-30",[128,2037,2038],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,2039,2040,2071,2077,2083,2142,2147,2152],{"dataMmlNode":135},[128,2041,2042,2047,2065],{"dataMmlNode":881},[128,2043,2044],{"dataMmlNode":381},[143,2045],{"dataC":1157,"xLinkHref":2046},"#MJX-28-TEX-I-1D465",[128,2048,2050,2055,2060],{"dataMmlNode":154,"transform":2049,"dataMjxTexclass":156},"translate(605,530.4) scale(0.707)",[128,2051,2052],{"dataMmlNode":394},[143,2053],{"dataC":811,"xLinkHref":2054},"#MJX-28-TEX-N-28",[128,2056,2057],{"dataMmlNode":141,"transform":815},[143,2058],{"dataC":145,"xLinkHref":2059},"#MJX-28-TEX-N-31",[128,2061,2062],{"dataMmlNode":394,"transform":821},[143,2063],{"dataC":836,"xLinkHref":2064},"#MJX-28-TEX-N-29",[128,2066,2068],{"dataMmlNode":381,"transform":2067},"translate(605,-293.8) scale(0.707)",[143,2069],{"dataC":922,"xLinkHref":2070},"#MJX-28-TEX-I-1D457",[128,2072,2074],{"dataMmlNode":394,"transform":2073},"translate(1836.5,0)",[143,2075],{"dataC":1209,"xLinkHref":2076},"#MJX-28-TEX-N-3D",[128,2078,2080],{"dataMmlNode":381,"transform":2079},"translate(2892.2,0)",[143,2081],{"dataC":1966,"xLinkHref":2082},"#MJX-28-TEX-I-1D446",[128,2084,2087,2093,2100,2107,2136],{"dataMmlNode":2085,"transform":2086},"mrow","translate(3703.9,0)",[128,2088,2090],{"dataMmlNode":394,"transform":2089},"translate(0 -0.5)",[143,2091],{"dataC":811,"xLinkHref":2092},"#MJX-28-TEX-LO-28",[128,2094,2096],{"dataMmlNode":381,"transform":2095},"translate(597,0)",[143,2097],{"dataC":2098,"xLinkHref":2099},"1D45B","#MJX-28-TEX-I-1D45B",[128,2101,2103],{"dataMmlNode":381,"transform":2102},"translate(1197,0)",[143,2104],{"dataC":2105,"xLinkHref":2106},"1D452","#MJX-28-TEX-I-1D452",[128,2108,2110,2116,2131],{"dataMmlNode":881,"transform":2109},"translate(1663,0)",[128,2111,2112],{"dataMmlNode":381},[143,2113],{"dataC":2114,"xLinkHref":2115},"1D461","#MJX-28-TEX-I-1D461",[128,2117,2119,2123,2127],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},"translate(394,530.4) scale(0.707)",[128,2120,2121],{"dataMmlNode":394},[143,2122],{"dataC":811,"xLinkHref":2054},[128,2124,2125],{"dataMmlNode":141,"transform":815},[143,2126],{"dataC":145,"xLinkHref":2059},[128,2128,2129],{"dataMmlNode":394,"transform":821},[143,2130],{"dataC":836,"xLinkHref":2064},[128,2132,2134],{"dataMmlNode":381,"transform":2133},"translate(394,-293.8) scale(0.707)",[143,2135],{"dataC":922,"xLinkHref":2070},[128,2137,2139],{"dataMmlNode":394,"transform":2138},"translate(3010.7,0) translate(0 -0.5)",[143,2140],{"dataC":836,"xLinkHref":2141},"#MJX-28-TEX-LO-29",[128,2143,2145],{"dataMmlNode":394,"transform":2144},"translate(7589.4,0)",[143,2146],{"dataC":1209,"xLinkHref":2076},[128,2148,2150],{"dataMmlNode":381,"transform":2149},"translate(8645.1,0)",[143,2151],{"dataC":1966,"xLinkHref":2082},[128,2153,2155,2159,2174,2220,2231],{"dataMmlNode":2085,"transform":2154},"translate(9456.8,0)",[128,2156,2157],{"dataMmlNode":394,"transform":2089},[143,2158],{"dataC":811,"xLinkHref":2092},[128,2160,2162,2167],{"dataMmlNode":2161,"transform":2095},"munder",[128,2163,2164],{"dataMmlNode":394},[143,2165],{"dataC":1775,"xLinkHref":2166},"#MJX-28-TEX-SO-2211",[128,2168,2169],{"dataMmlNode":154,"transform":1787,"dataMjxTexclass":156},[128,2170,2171],{"dataMmlNode":381},[143,2172],{"dataC":934,"xLinkHref":2173},"#MJX-28-TEX-I-1D456",[128,2175,2177,2182,2206],{"dataMmlNode":881,"transform":2176},"translate(2146.6,0)",[128,2178,2179],{"dataMmlNode":381},[143,2180],{"dataC":802,"xLinkHref":2181},"#MJX-28-TEX-I-1D464",[128,2183,2184,2188,2192,2197,2202],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,2185,2186],{"dataMmlNode":394},[143,2187],{"dataC":811,"xLinkHref":2054},[128,2189,2190],{"dataMmlNode":141,"transform":815},[143,2191],{"dataC":145,"xLinkHref":2059},[128,2193,2194],{"dataMmlNode":394,"transform":821},[143,2195],{"dataC":398,"xLinkHref":2196},"#MJX-28-TEX-N-2C",[128,2198,2199],{"dataMmlNode":141,"transform":827},[143,2200],{"dataC":149,"xLinkHref":2201},"#MJX-28-TEX-N-30",[128,2203,2204],{"dataMmlNode":394,"transform":833},[143,2205],{"dataC":836,"xLinkHref":2064},[128,2207,2208,2212,2216],{"dataMmlNode":154,"transform":917,"dataMjxTexclass":156},[128,2209,2210],{"dataMmlNode":381},[143,2211],{"dataC":922,"xLinkHref":2070},[128,2213,2214],{"dataMmlNode":394,"transform":926},[143,2215],{"dataC":398,"xLinkHref":2196},[128,2217,2218],{"dataMmlNode":381,"transform":931},[143,2219],{"dataC":934,"xLinkHref":2173},[128,2221,2223,2227],{"dataMmlNode":378,"transform":2222},"translate(4399.4,0)",[128,2224,2225],{"dataMmlNode":381},[143,2226],{"dataC":1157,"xLinkHref":2046},[128,2228,2229],{"dataMmlNode":381,"transform":1161},[143,2230],{"dataC":934,"xLinkHref":2173},[128,2232,2234],{"dataMmlNode":394,"transform":2233},"translate(5298.4,0) translate(0 -0.5)",[143,2235],{"dataC":836,"xLinkHref":2141},[28,2237,2238,2241,2242],{},[20,2239,2240],{},"Output Layer Calculation:"," The hidden layer's output becomes the input for the output layer.\n",[99,2243,2245],{"className":2244,"jax":103},[102],[105,2246,2249,2308],{"style":1974,"xmlns":108,"width":2247,"height":1976,"role":111,"focusable":112,"viewBox":2248,"xmlnsXLink":114},"40.072ex","0 -1149.5 17712 1799",[116,2250,2251,2254,2257,2260,2263,2266,2269,2272,2275,2278,2281,2284,2287,2290,2293,2296,2299,2302,2305],{},[119,2252],{"id":2253,"d":1349},"MJX-29-TEX-I-1D466",[119,2255],{"id":2256,"d":1729},"MJX-29-TEX-I-1D458",[119,2258],{"id":2259,"d":1122},"MJX-29-TEX-N-3D",[119,2261],{"id":2262,"d":1390},"MJX-29-TEX-I-1D45C",[119,2264],{"id":2265,"d":1957},"MJX-29-TEX-I-1D446",[119,2267],{"id":2268,"d":2004},"MJX-29-TEX-LO-28",[119,2270],{"id":2271,"d":2008},"MJX-29-TEX-I-1D45B",[119,2273],{"id":2274,"d":2012},"MJX-29-TEX-I-1D452",[119,2276],{"id":2277,"d":2016},"MJX-29-TEX-I-1D461",[119,2279],{"id":2280,"d":778},"MJX-29-TEX-N-28",[119,2282],{"id":2283,"d":363},"MJX-29-TEX-N-32",[119,2285],{"id":2286,"d":791},"MJX-29-TEX-N-29",[119,2288],{"id":2289,"d":2020},"MJX-29-TEX-LO-29",[119,2291],{"id":2292,"d":1713},"MJX-29-TEX-SO-2211",[119,2294],{"id":2295,"d":870},"MJX-29-TEX-I-1D457",[119,2297],{"id":2298,"d":774},"MJX-29-TEX-I-1D464",[119,2300],{"id":2301,"d":359},"MJX-29-TEX-N-2C",[119,2303],{"id":2304,"d":122},"MJX-29-TEX-N-31",[119,2306],{"id":2307,"d":1100},"MJX-29-TEX-I-1D465",[128,2309,2310],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,2311,2312,2324,2330,2342,2347,2353,2405,2410,2415],{"dataMmlNode":135},[128,2313,2314,2319],{"dataMmlNode":378},[128,2315,2316],{"dataMmlNode":381},[143,2317],{"dataC":1363,"xLinkHref":2318},"#MJX-29-TEX-I-1D466",[128,2320,2321],{"dataMmlNode":381,"transform":1367},[143,2322],{"dataC":1823,"xLinkHref":2323},"#MJX-29-TEX-I-1D458",[128,2325,2327],{"dataMmlNode":394,"transform":2326},"translate(1219.2,0)",[143,2328],{"dataC":1209,"xLinkHref":2329},"#MJX-29-TEX-N-3D",[128,2331,2333,2338],{"dataMmlNode":378,"transform":2332},"translate(2275,0)",[128,2334,2335],{"dataMmlNode":381},[143,2336],{"dataC":1404,"xLinkHref":2337},"#MJX-29-TEX-I-1D45C",[128,2339,2340],{"dataMmlNode":381,"transform":1408},[143,2341],{"dataC":1823,"xLinkHref":2323},[128,2343,2345],{"dataMmlNode":394,"transform":2344},"translate(3489.1,0)",[143,2346],{"dataC":1209,"xLinkHref":2329},[128,2348,2350],{"dataMmlNode":381,"transform":2349},"translate(4544.9,0)",[143,2351],{"dataC":1966,"xLinkHref":2352},"#MJX-29-TEX-I-1D446",[128,2354,2356,2361,2366,2371,2400],{"dataMmlNode":2085,"transform":2355},"translate(5356.6,0)",[128,2357,2358],{"dataMmlNode":394,"transform":2089},[143,2359],{"dataC":811,"xLinkHref":2360},"#MJX-29-TEX-LO-28",[128,2362,2363],{"dataMmlNode":381,"transform":2095},[143,2364],{"dataC":2098,"xLinkHref":2365},"#MJX-29-TEX-I-1D45B",[128,2367,2368],{"dataMmlNode":381,"transform":2102},[143,2369],{"dataC":2105,"xLinkHref":2370},"#MJX-29-TEX-I-1D452",[128,2372,2373,2378,2395],{"dataMmlNode":881,"transform":2109},[128,2374,2375],{"dataMmlNode":381},[143,2376],{"dataC":2114,"xLinkHref":2377},"#MJX-29-TEX-I-1D461",[128,2379,2380,2385,2390],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},[128,2381,2382],{"dataMmlNode":394},[143,2383],{"dataC":811,"xLinkHref":2384},"#MJX-29-TEX-N-28",[128,2386,2387],{"dataMmlNode":141,"transform":815},[143,2388],{"dataC":411,"xLinkHref":2389},"#MJX-29-TEX-N-32",[128,2391,2392],{"dataMmlNode":394,"transform":821},[143,2393],{"dataC":836,"xLinkHref":2394},"#MJX-29-TEX-N-29",[128,2396,2398],{"dataMmlNode":381,"transform":2397},"translate(394,-317.1) scale(0.707)",[143,2399],{"dataC":1823,"xLinkHref":2323},[128,2401,2402],{"dataMmlNode":394,"transform":2138},[143,2403],{"dataC":836,"xLinkHref":2404},"#MJX-29-TEX-LO-29",[128,2406,2408],{"dataMmlNode":394,"transform":2407},"translate(9242,0)",[143,2409],{"dataC":1209,"xLinkHref":2329},[128,2411,2413],{"dataMmlNode":381,"transform":2412},"translate(10297.8,0)",[143,2414],{"dataC":1966,"xLinkHref":2352},[128,2416,2418,2422,2436,2484,2510],{"dataMmlNode":2085,"transform":2417},"translate(11109.5,0)",[128,2419,2420],{"dataMmlNode":394,"transform":2089},[143,2421],{"dataC":811,"xLinkHref":2360},[128,2423,2424,2429],{"dataMmlNode":2161,"transform":2095},[128,2425,2426],{"dataMmlNode":394},[143,2427],{"dataC":1775,"xLinkHref":2428},"#MJX-29-TEX-SO-2211",[128,2430,2431],{"dataMmlNode":154,"transform":1787,"dataMjxTexclass":156},[128,2432,2433],{"dataMmlNode":381},[143,2434],{"dataC":922,"xLinkHref":2435},"#MJX-29-TEX-I-1D457",[128,2437,2439,2444,2468],{"dataMmlNode":881,"transform":2438},"translate(2194,0)",[128,2440,2441],{"dataMmlNode":381},[143,2442],{"dataC":802,"xLinkHref":2443},"#MJX-29-TEX-I-1D464",[128,2445,2446,2450,2454,2459,2464],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,2447,2448],{"dataMmlNode":394},[143,2449],{"dataC":811,"xLinkHref":2384},[128,2451,2452],{"dataMmlNode":141,"transform":815},[143,2453],{"dataC":411,"xLinkHref":2389},[128,2455,2456],{"dataMmlNode":394,"transform":821},[143,2457],{"dataC":398,"xLinkHref":2458},"#MJX-29-TEX-N-2C",[128,2460,2461],{"dataMmlNode":141,"transform":827},[143,2462],{"dataC":145,"xLinkHref":2463},"#MJX-29-TEX-N-31",[128,2465,2466],{"dataMmlNode":394,"transform":833},[143,2467],{"dataC":836,"xLinkHref":2394},[128,2469,2471,2475,2479],{"dataMmlNode":154,"transform":2470,"dataMjxTexclass":156},"translate(749,-317.1) scale(0.707)",[128,2472,2473],{"dataMmlNode":381},[143,2474],{"dataC":1823,"xLinkHref":2323},[128,2476,2477],{"dataMmlNode":394,"transform":1827},[143,2478],{"dataC":398,"xLinkHref":2458},[128,2480,2482],{"dataMmlNode":381,"transform":2481},"translate(799,0)",[143,2483],{"dataC":922,"xLinkHref":2435},[128,2485,2487,2492,2506],{"dataMmlNode":881,"transform":2486},"translate(4446.8,0)",[128,2488,2489],{"dataMmlNode":381},[143,2490],{"dataC":1157,"xLinkHref":2491},"#MJX-29-TEX-I-1D465",[128,2493,2494,2498,2502],{"dataMmlNode":154,"transform":2049,"dataMjxTexclass":156},[128,2495,2496],{"dataMmlNode":394},[143,2497],{"dataC":811,"xLinkHref":2384},[128,2499,2500],{"dataMmlNode":141,"transform":815},[143,2501],{"dataC":145,"xLinkHref":2463},[128,2503,2504],{"dataMmlNode":394,"transform":821},[143,2505],{"dataC":836,"xLinkHref":2394},[128,2507,2508],{"dataMmlNode":381,"transform":2067},[143,2509],{"dataC":922,"xLinkHref":2435},[128,2511,2513],{"dataMmlNode":394,"transform":2512},"translate(6005.5,0) translate(0 -0.5)",[143,2514],{"dataC":836,"xLinkHref":2404},[54,2516,2518],{"id":2517},"step-2-the-activation-function-sigmoid","Step 2: The Activation Function (Sigmoid)",[16,2520,2521,2522,2525,2526],{},"For backpropagation to work, the activation function must be non-linear and differentiable. The most common classical choice is the ",[20,2523,2524],{},"Logistic Sigmoid Function",":\n",[99,2527,2529],{"className":2528,"jax":103},[102],[105,2530,2535,2565],{"style":2531,"xmlns":108,"width":2532,"height":2533,"role":111,"focusable":112,"viewBox":2534,"xmlnsXLink":114},"vertical-align: -0.912ex;","12.976ex","2.869ex","0 -864.9 5735.4 1267.9",[116,2536,2537,2540,2543,2546,2549,2552,2555,2559,2562],{},[119,2538],{"id":2539,"d":1957},"MJX-30-TEX-I-1D446",[119,2541],{"id":2542,"d":778},"MJX-30-TEX-N-28",[119,2544],{"id":2545,"d":1100},"MJX-30-TEX-I-1D465",[119,2547],{"id":2548,"d":791},"MJX-30-TEX-N-29",[119,2550],{"id":2551,"d":1122},"MJX-30-TEX-N-3D",[119,2553],{"id":2554,"d":122},"MJX-30-TEX-N-31",[119,2556],{"id":2557,"d":2558},"MJX-30-TEX-N-2B","M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z",[119,2560],{"id":2561,"d":2012},"MJX-30-TEX-I-1D452",[119,2563],{"id":2564,"d":1548},"MJX-30-TEX-N-2212",[128,2566,2567],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,2568,2569,2574,2580,2586,2592,2598],{"dataMmlNode":135},[128,2570,2571],{"dataMmlNode":381},[143,2572],{"dataC":1966,"xLinkHref":2573},"#MJX-30-TEX-I-1D446",[128,2575,2577],{"dataMmlNode":394,"transform":2576},"translate(645,0)",[143,2578],{"dataC":811,"xLinkHref":2579},"#MJX-30-TEX-N-28",[128,2581,2583],{"dataMmlNode":381,"transform":2582},"translate(1034,0)",[143,2584],{"dataC":1157,"xLinkHref":2585},"#MJX-30-TEX-I-1D465",[128,2587,2589],{"dataMmlNode":394,"transform":2588},"translate(1606,0)",[143,2590],{"dataC":836,"xLinkHref":2591},"#MJX-30-TEX-N-29",[128,2593,2595],{"dataMmlNode":394,"transform":2594},"translate(2272.8,0)",[143,2596],{"dataC":1209,"xLinkHref":2597},"#MJX-30-TEX-N-3D",[128,2599,2602,2608,2642],{"dataMmlNode":2600,"transform":2601},"mfrac","translate(3328.6,0)",[128,2603,2605],{"dataMmlNode":141,"transform":2604},"translate(1026.7,394) scale(0.707)",[143,2606],{"dataC":145,"xLinkHref":2607},"#MJX-30-TEX-N-31",[128,2609,2611,2615,2621],{"dataMmlNode":2085,"transform":2610},"translate(220,-345) scale(0.707)",[128,2612,2613],{"dataMmlNode":141},[143,2614],{"dataC":145,"xLinkHref":2607},[128,2616,2617],{"dataMmlNode":394,"transform":151},[143,2618],{"dataC":2619,"xLinkHref":2620},"2B","#MJX-30-TEX-N-2B",[128,2622,2624,2629],{"dataMmlNode":138,"transform":2623},"translate(1278,0)",[128,2625,2626],{"dataMmlNode":381},[143,2627],{"dataC":2105,"xLinkHref":2628},"#MJX-30-TEX-I-1D452",[128,2630,2632,2637],{"dataMmlNode":154,"transform":2631,"dataMjxTexclass":156},"translate(499,289) scale(0.707)",[128,2633,2634],{"dataMmlNode":394},[143,2635],{"dataC":1613,"xLinkHref":2636},"#MJX-30-TEX-N-2212",[128,2638,2640],{"dataMmlNode":381,"transform":2639},"translate(778,0)",[143,2641],{"dataC":1157,"xLinkHref":2585},[2643,2644],"rect",{"width":2645,"height":2646,"x":2647,"y":2648},2166.9,60,"120","220",[16,2650,2651,2652],{},"The sigmoid function maps any real-valued number into a smooth range between 0 and 1. Its most crucial mathematical property is its easily computable derivative, which is vital for the chain rule:\n",[99,2653,2655],{"className":2654,"jax":103},[102],[105,2656,2661,2688],{"style":2657,"xmlns":108,"width":2658,"height":2659,"role":111,"focusable":112,"viewBox":2660,"xmlnsXLink":114},"vertical-align: -0.566ex;","22.959ex","2.283ex","0 -759 10148.1 1009",[116,2662,2663,2666,2670,2673,2676,2679,2682,2685],{},[119,2664],{"id":2665,"d":1957},"MJX-31-TEX-I-1D446",[119,2667],{"id":2668,"d":2669},"MJX-31-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",[119,2671],{"id":2672,"d":778},"MJX-31-TEX-N-28",[119,2674],{"id":2675,"d":1100},"MJX-31-TEX-I-1D465",[119,2677],{"id":2678,"d":791},"MJX-31-TEX-N-29",[119,2680],{"id":2681,"d":1122},"MJX-31-TEX-N-3D",[119,2683],{"id":2684,"d":122},"MJX-31-TEX-N-31",[119,2686],{"id":2687,"d":1548},"MJX-31-TEX-N-2212",[128,2689,2690],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,2691,2692,2706,2712,2718,2724,2730,2735,2740,2745,2750,2755,2761,2767,2772,2777,2782,2787],{"dataMmlNode":135},[128,2693,2694,2699],{"dataMmlNode":138},[128,2695,2696],{"dataMmlNode":381},[143,2697],{"dataC":1966,"xLinkHref":2698},"#MJX-31-TEX-I-1D446",[128,2700,2702],{"dataMmlNode":394,"transform":2701},"translate(729.6,363) scale(0.707)",[143,2703],{"dataC":2704,"xLinkHref":2705},"2032","#MJX-31-TEX-V-2032",[128,2707,2709],{"dataMmlNode":394,"transform":2708},"translate(974.1,0)",[143,2710],{"dataC":811,"xLinkHref":2711},"#MJX-31-TEX-N-28",[128,2713,2715],{"dataMmlNode":381,"transform":2714},"translate(1363.1,0)",[143,2716],{"dataC":1157,"xLinkHref":2717},"#MJX-31-TEX-I-1D465",[128,2719,2721],{"dataMmlNode":394,"transform":2720},"translate(1935.1,0)",[143,2722],{"dataC":836,"xLinkHref":2723},"#MJX-31-TEX-N-29",[128,2725,2727],{"dataMmlNode":394,"transform":2726},"translate(2601.8,0)",[143,2728],{"dataC":1209,"xLinkHref":2729},"#MJX-31-TEX-N-3D",[128,2731,2733],{"dataMmlNode":381,"transform":2732},"translate(3657.6,0)",[143,2734],{"dataC":1966,"xLinkHref":2698},[128,2736,2738],{"dataMmlNode":394,"transform":2737},"translate(4302.6,0)",[143,2739],{"dataC":811,"xLinkHref":2711},[128,2741,2743],{"dataMmlNode":381,"transform":2742},"translate(4691.6,0)",[143,2744],{"dataC":1157,"xLinkHref":2717},[128,2746,2748],{"dataMmlNode":394,"transform":2747},"translate(5263.6,0)",[143,2749],{"dataC":836,"xLinkHref":2723},[128,2751,2753],{"dataMmlNode":394,"transform":2752},"translate(5652.6,0)",[143,2754],{"dataC":811,"xLinkHref":2711},[128,2756,2758],{"dataMmlNode":141,"transform":2757},"translate(6041.6,0)",[143,2759],{"dataC":145,"xLinkHref":2760},"#MJX-31-TEX-N-31",[128,2762,2764],{"dataMmlNode":394,"transform":2763},"translate(6763.8,0)",[143,2765],{"dataC":1613,"xLinkHref":2766},"#MJX-31-TEX-N-2212",[128,2768,2770],{"dataMmlNode":381,"transform":2769},"translate(7764.1,0)",[143,2771],{"dataC":1966,"xLinkHref":2698},[128,2773,2775],{"dataMmlNode":394,"transform":2774},"translate(8409.1,0)",[143,2776],{"dataC":811,"xLinkHref":2711},[128,2778,2780],{"dataMmlNode":381,"transform":2779},"translate(8798.1,0)",[143,2781],{"dataC":1157,"xLinkHref":2717},[128,2783,2785],{"dataMmlNode":394,"transform":2784},"translate(9370.1,0)",[143,2786],{"dataC":836,"xLinkHref":2723},[128,2788,2790],{"dataMmlNode":394,"transform":2789},"translate(9759.1,0)",[143,2791],{"dataC":836,"xLinkHref":2723},[16,2793,2794,2797,2798,2854],{},[738,2795,2796],{},"Note on Saturation:"," If the net input ",[99,2799,2801],{"className":2800,"jax":103},[102],[105,2802,2807,2821],{"style":2803,"xmlns":108,"width":2804,"height":2805,"role":111,"focusable":112,"viewBox":2806,"xmlnsXLink":114},"vertical-align: -0.564ex;","4.486ex","2.26ex","0 -749.5 1983 999",[116,2808,2809,2812,2815,2818],{},[119,2810],{"id":2811,"d":1118},"MJX-32-TEX-N-7C",[119,2813],{"id":2814,"d":2008},"MJX-32-TEX-I-1D45B",[119,2816],{"id":2817,"d":2012},"MJX-32-TEX-I-1D452",[119,2819],{"id":2820,"d":2016},"MJX-32-TEX-I-1D461",[128,2822,2823],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,2824,2825,2831,2837,2843,2849],{"dataMmlNode":135},[128,2826,2827],{"dataMmlNode":394,"transform":2089},[143,2828],{"dataC":2829,"xLinkHref":2830},"7C","#MJX-32-TEX-N-7C",[128,2832,2834],{"dataMmlNode":381,"transform":2833},"translate(278,0)",[143,2835],{"dataC":2098,"xLinkHref":2836},"#MJX-32-TEX-I-1D45B",[128,2838,2840],{"dataMmlNode":381,"transform":2839},"translate(878,0)",[143,2841],{"dataC":2105,"xLinkHref":2842},"#MJX-32-TEX-I-1D452",[128,2844,2846],{"dataMmlNode":381,"transform":2845},"translate(1344,0)",[143,2847],{"dataC":2114,"xLinkHref":2848},"#MJX-32-TEX-I-1D461",[128,2850,2852],{"dataMmlNode":394,"transform":2851},"translate(1705,0) translate(0 -0.5)",[143,2853],{"dataC":2829,"xLinkHref":2830}," 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.",[54,2856,2858],{"id":2857},"step-3-backpropagating-the-error-the-backward-pass","Step 3: Backpropagating the Error (The Backward Pass)",[16,2860,2861,2862,2884,2885,2888],{},"We must update the weights to minimize the error ",[99,2863,2865],{"className":2864,"jax":103},[102],[105,2866,2870,2875],{"style":553,"xmlns":108,"width":2867,"height":2868,"role":111,"focusable":112,"viewBox":2869,"xmlnsXLink":114},"1.729ex","1.538ex","0 -680 764 680",[116,2871,2872],{},[119,2873],{"id":2874,"d":1706},"MJX-33-TEX-I-1D438",[128,2876,2877],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,2878,2879],{"dataMmlNode":135},[128,2880,2881],{"dataMmlNode":381},[143,2882],{"dataC":1759,"xLinkHref":2883},"#MJX-33-TEX-I-1D438",". We do this using ",[20,2886,2887],{},"Gradient Descent",", updating each weight by an amount proportional to the negative gradient of the error with respect to that weight.",[2890,2891,2893,2894,2957],"h4",{"id":2892},"updating-output-layer-weights","Updating Output Layer Weights (",[99,2895,2897],{"className":2896,"jax":103},[102],[105,2898,2899,2919],{"style":765,"xmlns":108,"width":766,"height":767,"role":111,"focusable":112,"viewBox":768,"xmlnsXLink":114},[116,2900,2901,2904,2907,2910,2913,2916],{},[119,2902],{"id":2903,"d":774},"MJX-34-TEX-I-1D464",[119,2905],{"id":2906,"d":778},"MJX-34-TEX-N-28",[119,2908],{"id":2909,"d":363},"MJX-34-TEX-N-32",[119,2911],{"id":2912,"d":359},"MJX-34-TEX-N-2C",[119,2914],{"id":2915,"d":122},"MJX-34-TEX-N-31",[119,2917],{"id":2918,"d":791},"MJX-34-TEX-N-29",[128,2920,2921],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,2922,2923],{"dataMmlNode":135},[128,2924,2925,2930],{"dataMmlNode":138},[128,2926,2927],{"dataMmlNode":381},[143,2928],{"dataC":802,"xLinkHref":2929},"#MJX-34-TEX-I-1D464",[128,2931,2932,2937,2942,2947,2952],{"dataMmlNode":154,"transform":806,"dataMjxTexclass":156},[128,2933,2934],{"dataMmlNode":394},[143,2935],{"dataC":811,"xLinkHref":2936},"#MJX-34-TEX-N-28",[128,2938,2939],{"dataMmlNode":141,"transform":815},[143,2940],{"dataC":411,"xLinkHref":2941},"#MJX-34-TEX-N-32",[128,2943,2944],{"dataMmlNode":394,"transform":821},[143,2945],{"dataC":398,"xLinkHref":2946},"#MJX-34-TEX-N-2C",[128,2948,2949],{"dataMmlNode":141,"transform":827},[143,2950],{"dataC":145,"xLinkHref":2951},"#MJX-34-TEX-N-31",[128,2953,2954],{"dataMmlNode":394,"transform":833},[143,2955],{"dataC":836,"xLinkHref":2956},"#MJX-34-TEX-N-29",")",[16,2959,2960,2961],{},"Applying the chain rule to find how an output weight affects the total error:\n",[99,2962,2964],{"className":2963,"jax":103},[102],[105,2965,2970,3014],{"style":2966,"xmlns":108,"width":2967,"height":2968,"role":111,"focusable":112,"viewBox":2969,"xmlnsXLink":114},"vertical-align: -2.073ex;","17.264ex","4.473ex","0 -1060.7 7630.6 1977",[116,2971,2972,2976,2979,2982,2985,2988,2991,2994,2997,3000,3004,3007,3011],{},[119,2973],{"id":2974,"d":2975},"MJX-35-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",[119,2977],{"id":2978,"d":774},"MJX-35-TEX-I-1D464",[119,2980],{"id":2981,"d":778},"MJX-35-TEX-N-28",[119,2983],{"id":2984,"d":363},"MJX-35-TEX-N-32",[119,2986],{"id":2987,"d":359},"MJX-35-TEX-N-2C",[119,2989],{"id":2990,"d":122},"MJX-35-TEX-N-31",[119,2992],{"id":2993,"d":791},"MJX-35-TEX-N-29",[119,2995],{"id":2996,"d":1729},"MJX-35-TEX-I-1D458",[119,2998],{"id":2999,"d":870},"MJX-35-TEX-I-1D457",[119,3001],{"id":3002,"d":3003},"MJX-35-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",[119,3005],{"id":3006,"d":1548},"MJX-35-TEX-N-2212",[119,3008],{"id":3009,"d":3010},"MJX-35-TEX-I-1D715","M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z",[119,3012],{"id":3013,"d":1706},"MJX-35-TEX-I-1D438",[128,3015,3016],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,3017,3018,3024,3075,3082,3088],{"dataMmlNode":135},[128,3019,3020],{"dataMmlNode":381},[143,3021],{"dataC":3022,"xLinkHref":3023},"394","#MJX-35-TEX-N-394",[128,3025,3027,3032,3059],{"dataMmlNode":881,"transform":3026},"translate(833,0)",[128,3028,3029],{"dataMmlNode":381},[143,3030],{"dataC":802,"xLinkHref":3031},"#MJX-35-TEX-I-1D464",[128,3033,3034,3039,3044,3049,3054],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,3035,3036],{"dataMmlNode":394},[143,3037],{"dataC":811,"xLinkHref":3038},"#MJX-35-TEX-N-28",[128,3040,3041],{"dataMmlNode":141,"transform":815},[143,3042],{"dataC":411,"xLinkHref":3043},"#MJX-35-TEX-N-32",[128,3045,3046],{"dataMmlNode":394,"transform":821},[143,3047],{"dataC":398,"xLinkHref":3048},"#MJX-35-TEX-N-2C",[128,3050,3051],{"dataMmlNode":141,"transform":827},[143,3052],{"dataC":145,"xLinkHref":3053},"#MJX-35-TEX-N-31",[128,3055,3056],{"dataMmlNode":394,"transform":833},[143,3057],{"dataC":836,"xLinkHref":3058},"#MJX-35-TEX-N-29",[128,3060,3061,3066,3070],{"dataMmlNode":154,"transform":2470,"dataMjxTexclass":156},[128,3062,3063],{"dataMmlNode":381},[143,3064],{"dataC":1823,"xLinkHref":3065},"#MJX-35-TEX-I-1D458",[128,3067,3068],{"dataMmlNode":394,"transform":1827},[143,3069],{"dataC":398,"xLinkHref":3048},[128,3071,3072],{"dataMmlNode":381,"transform":2481},[143,3073],{"dataC":922,"xLinkHref":3074},"#MJX-35-TEX-I-1D457",[128,3076,3078],{"dataMmlNode":394,"transform":3077},"translate(3363.6,0)",[143,3079],{"dataC":3080,"xLinkHref":3081},"221D","#MJX-35-TEX-N-221D",[128,3083,3085],{"dataMmlNode":394,"transform":3084},"translate(4419.4,0)",[143,3086],{"dataC":1613,"xLinkHref":3087},"#MJX-35-TEX-N-2212",[128,3089,3091,3106,3155],{"dataMmlNode":2600,"transform":3090},"translate(5197.4,0)",[128,3092,3094,3100],{"dataMmlNode":2085,"transform":3093},"translate(746.4,394) scale(0.707)",[128,3095,3096],{"dataMmlNode":381},[143,3097],{"dataC":3098,"xLinkHref":3099},"1D715","#MJX-35-TEX-I-1D715",[128,3101,3103],{"dataMmlNode":381,"transform":3102},"translate(566,0)",[143,3104],{"dataC":1759,"xLinkHref":3105},"#MJX-35-TEX-I-1D438",[128,3107,3109,3113],{"dataMmlNode":2085,"transform":3108},"translate(220,-590) scale(0.707)",[128,3110,3111],{"dataMmlNode":381},[143,3112],{"dataC":3098,"xLinkHref":3099},[128,3114,3115,3119,3141],{"dataMmlNode":881,"transform":3102},[128,3116,3117],{"dataMmlNode":381},[143,3118],{"dataC":802,"xLinkHref":3031},[128,3120,3121,3125,3129,3133,3137],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,3122,3123],{"dataMmlNode":394},[143,3124],{"dataC":811,"xLinkHref":3038},[128,3126,3127],{"dataMmlNode":141,"transform":815},[143,3128],{"dataC":411,"xLinkHref":3043},[128,3130,3131],{"dataMmlNode":394,"transform":821},[143,3132],{"dataC":398,"xLinkHref":3048},[128,3134,3135],{"dataMmlNode":141,"transform":827},[143,3136],{"dataC":145,"xLinkHref":3053},[128,3138,3139],{"dataMmlNode":394,"transform":833},[143,3140],{"dataC":836,"xLinkHref":3058},[128,3142,3143,3147,3151],{"dataMmlNode":154,"transform":2470,"dataMjxTexclass":156},[128,3144,3145],{"dataMmlNode":381},[143,3146],{"dataC":1823,"xLinkHref":3065},[128,3148,3149],{"dataMmlNode":394,"transform":1827},[143,3150],{"dataC":398,"xLinkHref":3048},[128,3152,3153],{"dataMmlNode":381,"transform":2481},[143,3154],{"dataC":922,"xLinkHref":3074},[2643,3156],{"width":3157,"height":2646,"x":2647,"y":2648},2193.2,[16,3159,3160,3161,3180,3181,3214,3215,3292,3293,2525,3381,3728],{},"Since ",[99,3162,3164],{"className":3163,"jax":103},[102],[105,3165,3166,3171],{"style":553,"xmlns":108,"width":2867,"height":2868,"role":111,"focusable":112,"viewBox":2869,"xmlnsXLink":114},[116,3167,3168],{},[119,3169],{"id":3170,"d":1706},"MJX-36-TEX-I-1D438",[128,3172,3173],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,3174,3175],{"dataMmlNode":135},[128,3176,3177],{"dataMmlNode":381},[143,3178],{"dataC":1759,"xLinkHref":3179},"#MJX-36-TEX-I-1D438"," depends on the output ",[99,3182,3184],{"className":3183,"jax":103},[102],[105,3185,3190,3198],{"style":3186,"xmlns":108,"width":3187,"height":3188,"role":111,"focusable":112,"viewBox":3189,"xmlnsXLink":114},"vertical-align: -0.357ex;","2.119ex","1.355ex","0 -441 936.4 598.8",[116,3191,3192,3195],{},[119,3193],{"id":3194,"d":1390},"MJX-37-TEX-I-1D45C",[119,3196],{"id":3197,"d":1729},"MJX-37-TEX-I-1D458",[128,3199,3200],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,3201,3202],{"dataMmlNode":135},[128,3203,3204,3209],{"dataMmlNode":378},[128,3205,3206],{"dataMmlNode":381},[143,3207],{"dataC":1404,"xLinkHref":3208},"#MJX-37-TEX-I-1D45C",[128,3210,3211],{"dataMmlNode":381,"transform":1408},[143,3212],{"dataC":1823,"xLinkHref":3213},"#MJX-37-TEX-I-1D458",", which depends on ",[99,3216,3218],{"className":3217,"jax":103},[102],[105,3219,3224,3247],{"style":3220,"xmlns":108,"width":3221,"height":3222,"role":111,"focusable":112,"viewBox":3223,"xmlnsXLink":114},"vertical-align: -0.735ex;","5.461ex","3.135ex","0 -1060.7 2413.7 1385.6",[116,3225,3226,3229,3232,3235,3238,3241,3244],{},[119,3227],{"id":3228,"d":2008},"MJX-38-TEX-I-1D45B",[119,3230],{"id":3231,"d":2012},"MJX-38-TEX-I-1D452",[119,3233],{"id":3234,"d":2016},"MJX-38-TEX-I-1D461",[119,3236],{"id":3237,"d":778},"MJX-38-TEX-N-28",[119,3239],{"id":3240,"d":363},"MJX-38-TEX-N-32",[119,3242],{"id":3243,"d":791},"MJX-38-TEX-N-29",[119,3245],{"id":3246,"d":1729},"MJX-38-TEX-I-1D458",[128,3248,3249],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,3250,3251,3256,3262],{"dataMmlNode":135},[128,3252,3253],{"dataMmlNode":381},[143,3254],{"dataC":2098,"xLinkHref":3255},"#MJX-38-TEX-I-1D45B",[128,3257,3259],{"dataMmlNode":381,"transform":3258},"translate(600,0)",[143,3260],{"dataC":2105,"xLinkHref":3261},"#MJX-38-TEX-I-1D452",[128,3263,3265,3270,3287],{"dataMmlNode":881,"transform":3264},"translate(1066,0)",[128,3266,3267],{"dataMmlNode":381},[143,3268],{"dataC":2114,"xLinkHref":3269},"#MJX-38-TEX-I-1D461",[128,3271,3272,3277,3282],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},[128,3273,3274],{"dataMmlNode":394},[143,3275],{"dataC":811,"xLinkHref":3276},"#MJX-38-TEX-N-28",[128,3278,3279],{"dataMmlNode":141,"transform":815},[143,3280],{"dataC":411,"xLinkHref":3281},"#MJX-38-TEX-N-32",[128,3283,3284],{"dataMmlNode":394,"transform":821},[143,3285],{"dataC":836,"xLinkHref":3286},"#MJX-38-TEX-N-29",[128,3288,3289],{"dataMmlNode":381,"transform":2397},[143,3290],{"dataC":1823,"xLinkHref":3291},"#MJX-38-TEX-I-1D458",", which in turn depends on the weight ",[99,3294,3296],{"className":3295,"jax":103},[102],[105,3297,3301,3327],{"style":3298,"xmlns":108,"width":766,"height":3299,"role":111,"focusable":112,"viewBox":3300,"xmlnsXLink":114},"vertical-align: -1.044ex;","3.444ex","0 -1060.7 2252.8 1522.1",[116,3302,3303,3306,3309,3312,3315,3318,3321,3324],{},[119,3304],{"id":3305,"d":774},"MJX-39-TEX-I-1D464",[119,3307],{"id":3308,"d":778},"MJX-39-TEX-N-28",[119,3310],{"id":3311,"d":363},"MJX-39-TEX-N-32",[119,3313],{"id":3314,"d":359},"MJX-39-TEX-N-2C",[119,3316],{"id":3317,"d":122},"MJX-39-TEX-N-31",[119,3319],{"id":3320,"d":791},"MJX-39-TEX-N-29",[119,3322],{"id":3323,"d":1729},"MJX-39-TEX-I-1D458",[119,3325],{"id":3326,"d":870},"MJX-39-TEX-I-1D457",[128,3328,3329],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,3330,3331],{"dataMmlNode":135},[128,3332,3333,3338,3365],{"dataMmlNode":881},[128,3334,3335],{"dataMmlNode":381},[143,3336],{"dataC":802,"xLinkHref":3337},"#MJX-39-TEX-I-1D464",[128,3339,3340,3345,3350,3355,3360],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,3341,3342],{"dataMmlNode":394},[143,3343],{"dataC":811,"xLinkHref":3344},"#MJX-39-TEX-N-28",[128,3346,3347],{"dataMmlNode":141,"transform":815},[143,3348],{"dataC":411,"xLinkHref":3349},"#MJX-39-TEX-N-32",[128,3351,3352],{"dataMmlNode":394,"transform":821},[143,3353],{"dataC":398,"xLinkHref":3354},"#MJX-39-TEX-N-2C",[128,3356,3357],{"dataMmlNode":141,"transform":827},[143,3358],{"dataC":145,"xLinkHref":3359},"#MJX-39-TEX-N-31",[128,3361,3362],{"dataMmlNode":394,"transform":833},[143,3363],{"dataC":836,"xLinkHref":3364},"#MJX-39-TEX-N-29",[128,3366,3367,3372,3376],{"dataMmlNode":154,"transform":2470,"dataMjxTexclass":156},[128,3368,3369],{"dataMmlNode":381},[143,3370],{"dataC":1823,"xLinkHref":3371},"#MJX-39-TEX-I-1D458",[128,3373,3374],{"dataMmlNode":394,"transform":1827},[143,3375],{"dataC":398,"xLinkHref":3354},[128,3377,3378],{"dataMmlNode":381,"transform":2481},[143,3379],{"dataC":922,"xLinkHref":3380},"#MJX-39-TEX-I-1D457",[99,3382,3384],{"className":3383,"jax":103},[102],[105,3385,3389,3440],{"style":2966,"xmlns":108,"width":3386,"height":3387,"role":111,"focusable":112,"viewBox":3388,"xmlnsXLink":114},"26.715ex","5.059ex","0 -1319.8 11807.9 2236.1",[116,3390,3391,3394,3397,3400,3403,3406,3409,3412,3415,3418,3421,3424,3427,3431,3434,3437],{},[119,3392],{"id":3393,"d":3010},"MJX-40-TEX-I-1D715",[119,3395],{"id":3396,"d":1706},"MJX-40-TEX-I-1D438",[119,3398],{"id":3399,"d":774},"MJX-40-TEX-I-1D464",[119,3401],{"id":3402,"d":778},"MJX-40-TEX-N-28",[119,3404],{"id":3405,"d":363},"MJX-40-TEX-N-32",[119,3407],{"id":3408,"d":359},"MJX-40-TEX-N-2C",[119,3410],{"id":3411,"d":122},"MJX-40-TEX-N-31",[119,3413],{"id":3414,"d":791},"MJX-40-TEX-N-29",[119,3416],{"id":3417,"d":1729},"MJX-40-TEX-I-1D458",[119,3419],{"id":3420,"d":870},"MJX-40-TEX-I-1D457",[119,3422],{"id":3423,"d":1122},"MJX-40-TEX-N-3D",[119,3425],{"id":3426,"d":1390},"MJX-40-TEX-I-1D45C",[119,3428],{"id":3429,"d":3430},"MJX-40-TEX-N-22C5","M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z",[119,3432],{"id":3433,"d":2008},"MJX-40-TEX-I-1D45B",[119,3435],{"id":3436,"d":2012},"MJX-40-TEX-I-1D452",[119,3438],{"id":3439,"d":2016},"MJX-40-TEX-I-1D461",[128,3441,3442],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,3443,3444,3516,3522,3557,3564,3630,3635],{"dataMmlNode":135},[128,3445,3446,3458,3514],{"dataMmlNode":2600},[128,3447,3448,3453],{"dataMmlNode":2085,"transform":3093},[128,3449,3450],{"dataMmlNode":381},[143,3451],{"dataC":3098,"xLinkHref":3452},"#MJX-40-TEX-I-1D715",[128,3454,3455],{"dataMmlNode":381,"transform":3102},[143,3456],{"dataC":1759,"xLinkHref":3457},"#MJX-40-TEX-I-1D438",[128,3459,3460,3464],{"dataMmlNode":2085,"transform":3108},[128,3461,3462],{"dataMmlNode":381},[143,3463],{"dataC":3098,"xLinkHref":3452},[128,3465,3466,3471,3498],{"dataMmlNode":881,"transform":3102},[128,3467,3468],{"dataMmlNode":381},[143,3469],{"dataC":802,"xLinkHref":3470},"#MJX-40-TEX-I-1D464",[128,3472,3473,3478,3483,3488,3493],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,3474,3475],{"dataMmlNode":394},[143,3476],{"dataC":811,"xLinkHref":3477},"#MJX-40-TEX-N-28",[128,3479,3480],{"dataMmlNode":141,"transform":815},[143,3481],{"dataC":411,"xLinkHref":3482},"#MJX-40-TEX-N-32",[128,3484,3485],{"dataMmlNode":394,"transform":821},[143,3486],{"dataC":398,"xLinkHref":3487},"#MJX-40-TEX-N-2C",[128,3489,3490],{"dataMmlNode":141,"transform":827},[143,3491],{"dataC":145,"xLinkHref":3492},"#MJX-40-TEX-N-31",[128,3494,3495],{"dataMmlNode":394,"transform":833},[143,3496],{"dataC":836,"xLinkHref":3497},"#MJX-40-TEX-N-29",[128,3499,3500,3505,3509],{"dataMmlNode":154,"transform":2470,"dataMjxTexclass":156},[128,3501,3502],{"dataMmlNode":381},[143,3503],{"dataC":1823,"xLinkHref":3504},"#MJX-40-TEX-I-1D458",[128,3506,3507],{"dataMmlNode":394,"transform":1827},[143,3508],{"dataC":398,"xLinkHref":3487},[128,3510,3511],{"dataMmlNode":381,"transform":2481},[143,3512],{"dataC":922,"xLinkHref":3513},"#MJX-40-TEX-I-1D457",[2643,3515],{"width":3157,"height":2646,"x":2647,"y":2648},[128,3517,3519],{"dataMmlNode":394,"transform":3518},"translate(2711,0)",[143,3520],{"dataC":1209,"xLinkHref":3521},"#MJX-40-TEX-N-3D",[128,3523,3525,3536,3554],{"dataMmlNode":2600,"transform":3524},"translate(3766.8,0)",[128,3526,3528,3532],{"dataMmlNode":2085,"transform":3527},"translate(281,394) scale(0.707)",[128,3529,3530],{"dataMmlNode":381},[143,3531],{"dataC":3098,"xLinkHref":3452},[128,3533,3534],{"dataMmlNode":381,"transform":3102},[143,3535],{"dataC":1759,"xLinkHref":3457},[128,3537,3539,3543],{"dataMmlNode":2085,"transform":3538},"translate(220,-345.6) scale(0.707)",[128,3540,3541],{"dataMmlNode":381},[143,3542],{"dataC":3098,"xLinkHref":3452},[128,3544,3545,3550],{"dataMmlNode":378,"transform":3102},[128,3546,3547],{"dataMmlNode":381},[143,3548],{"dataC":1404,"xLinkHref":3549},"#MJX-40-TEX-I-1D45C",[128,3551,3552],{"dataMmlNode":381,"transform":1408},[143,3553],{"dataC":1823,"xLinkHref":3504},[2643,3555],{"width":3556,"height":2646,"x":2647,"y":2648},1262.4,[128,3558,3560],{"dataMmlNode":394,"transform":3559},"translate(5491.3,0)",[143,3561],{"dataC":3562,"xLinkHref":3563},"22C5","#MJX-40-TEX-N-22C5",[128,3565,3567,3584,3627],{"dataMmlNode":2600,"transform":3566},"translate(5991.6,0)",[128,3568,3570,3574],{"dataMmlNode":2085,"transform":3569},"translate(742.3,451.6) scale(0.707)",[128,3571,3572],{"dataMmlNode":381},[143,3573],{"dataC":3098,"xLinkHref":3452},[128,3575,3576,3580],{"dataMmlNode":378,"transform":3102},[128,3577,3578],{"dataMmlNode":381},[143,3579],{"dataC":1404,"xLinkHref":3549},[128,3581,3582],{"dataMmlNode":381,"transform":1408},[143,3583],{"dataC":1823,"xLinkHref":3504},[128,3585,3586,3590,3595,3601],{"dataMmlNode":2085,"transform":3108},[128,3587,3588],{"dataMmlNode":381},[143,3589],{"dataC":3098,"xLinkHref":3452},[128,3591,3592],{"dataMmlNode":381,"transform":3102},[143,3593],{"dataC":2098,"xLinkHref":3594},"#MJX-40-TEX-I-1D45B",[128,3596,3598],{"dataMmlNode":381,"transform":3597},"translate(1166,0)",[143,3599],{"dataC":2105,"xLinkHref":3600},"#MJX-40-TEX-I-1D452",[128,3602,3604,3609,3623],{"dataMmlNode":881,"transform":3603},"translate(1632,0)",[128,3605,3606],{"dataMmlNode":381},[143,3607],{"dataC":2114,"xLinkHref":3608},"#MJX-40-TEX-I-1D461",[128,3610,3611,3615,3619],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},[128,3612,3613],{"dataMmlNode":394},[143,3614],{"dataC":811,"xLinkHref":3477},[128,3616,3617],{"dataMmlNode":141,"transform":815},[143,3618],{"dataC":411,"xLinkHref":3482},[128,3620,3621],{"dataMmlNode":394,"transform":821},[143,3622],{"dataC":836,"xLinkHref":3497},[128,3624,3625],{"dataMmlNode":381,"transform":2397},[143,3626],{"dataC":1823,"xLinkHref":3504},[2643,3628],{"width":3629,"height":2646,"x":2647,"y":2648},2307,[128,3631,3633],{"dataMmlNode":394,"transform":3632},"translate(8760.7,0)",[143,3634],{"dataC":3562,"xLinkHref":3563},[128,3636,3638,3677,3726],{"dataMmlNode":2600,"transform":3637},"translate(9261,0)",[128,3639,3641,3645,3649,3653],{"dataMmlNode":2085,"transform":3640},"translate(220,569.7) scale(0.707)",[128,3642,3643],{"dataMmlNode":381},[143,3644],{"dataC":3098,"xLinkHref":3452},[128,3646,3647],{"dataMmlNode":381,"transform":3102},[143,3648],{"dataC":2098,"xLinkHref":3594},[128,3650,3651],{"dataMmlNode":381,"transform":3597},[143,3652],{"dataC":2105,"xLinkHref":3600},[128,3654,3655,3659,3673],{"dataMmlNode":881,"transform":3603},[128,3656,3657],{"dataMmlNode":381},[143,3658],{"dataC":2114,"xLinkHref":3608},[128,3660,3661,3665,3669],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},[128,3662,3663],{"dataMmlNode":394},[143,3664],{"dataC":811,"xLinkHref":3477},[128,3666,3667],{"dataMmlNode":141,"transform":815},[143,3668],{"dataC":411,"xLinkHref":3482},[128,3670,3671],{"dataMmlNode":394,"transform":821},[143,3672],{"dataC":836,"xLinkHref":3497},[128,3674,3675],{"dataMmlNode":381,"transform":2397},[143,3676],{"dataC":1823,"xLinkHref":3504},[128,3678,3680,3684],{"dataMmlNode":2085,"transform":3679},"translate(276.9,-590) scale(0.707)",[128,3681,3682],{"dataMmlNode":381},[143,3683],{"dataC":3098,"xLinkHref":3452},[128,3685,3686,3690,3712],{"dataMmlNode":881,"transform":3102},[128,3687,3688],{"dataMmlNode":381},[143,3689],{"dataC":802,"xLinkHref":3470},[128,3691,3692,3696,3700,3704,3708],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,3693,3694],{"dataMmlNode":394},[143,3695],{"dataC":811,"xLinkHref":3477},[128,3697,3698],{"dataMmlNode":141,"transform":815},[143,3699],{"dataC":411,"xLinkHref":3482},[128,3701,3702],{"dataMmlNode":394,"transform":821},[143,3703],{"dataC":398,"xLinkHref":3487},[128,3705,3706],{"dataMmlNode":141,"transform":827},[143,3707],{"dataC":145,"xLinkHref":3492},[128,3709,3710],{"dataMmlNode":394,"transform":833},[143,3711],{"dataC":836,"xLinkHref":3497},[128,3713,3714,3718,3722],{"dataMmlNode":154,"transform":2470,"dataMjxTexclass":156},[128,3715,3716],{"dataMmlNode":381},[143,3717],{"dataC":1823,"xLinkHref":3504},[128,3719,3720],{"dataMmlNode":394,"transform":1827},[143,3721],{"dataC":398,"xLinkHref":3487},[128,3723,3724],{"dataMmlNode":381,"transform":2481},[143,3725],{"dataC":922,"xLinkHref":3513},[2643,3727],{"width":3629,"height":2646,"x":2647,"y":2648},[99,3729,3731],{"className":3730,"jax":103},[102],[105,3732,3736,3807],{"style":2966,"xmlns":108,"width":3733,"height":3734,"role":111,"focusable":112,"viewBox":3735,"xmlnsXLink":114},"37.794ex","4.674ex","0 -1149.5 16705 2065.8",[116,3737,3738,3741,3744,3747,3750,3753,3756,3759,3762,3765,3768,3771,3774,3777,3780,3783,3786,3789,3792,3795,3798,3801,3804],{},[119,3739],{"id":3740,"d":3010},"MJX-41-TEX-I-1D715",[119,3742],{"id":3743,"d":1706},"MJX-41-TEX-I-1D438",[119,3745],{"id":3746,"d":774},"MJX-41-TEX-I-1D464",[119,3748],{"id":3749,"d":778},"MJX-41-TEX-N-28",[119,3751],{"id":3752,"d":363},"MJX-41-TEX-N-32",[119,3754],{"id":3755,"d":359},"MJX-41-TEX-N-2C",[119,3757],{"id":3758,"d":122},"MJX-41-TEX-N-31",[119,3760],{"id":3761,"d":791},"MJX-41-TEX-N-29",[119,3763],{"id":3764,"d":1729},"MJX-41-TEX-I-1D458",[119,3766],{"id":3767,"d":870},"MJX-41-TEX-I-1D457",[119,3769],{"id":3770,"d":1122},"MJX-41-TEX-N-3D",[119,3772],{"id":3773,"d":1548},"MJX-41-TEX-N-2212",[119,3775],{"id":3776,"d":1111},"MJX-41-TEX-I-1D451",[119,3778],{"id":3779,"d":1390},"MJX-41-TEX-I-1D45C",[119,3781],{"id":3782,"d":3430},"MJX-41-TEX-N-22C5",[119,3784],{"id":3785,"d":1957},"MJX-41-TEX-I-1D446",[119,3787],{"id":3788,"d":2669},"MJX-41-TEX-V-2032",[119,3790],{"id":3791,"d":2004},"MJX-41-TEX-LO-28",[119,3793],{"id":3794,"d":2008},"MJX-41-TEX-I-1D45B",[119,3796],{"id":3797,"d":2012},"MJX-41-TEX-I-1D452",[119,3799],{"id":3800,"d":2016},"MJX-41-TEX-I-1D461",[119,3802],{"id":3803,"d":2020},"MJX-41-TEX-LO-29",[119,3805],{"id":3806,"d":1100},"MJX-41-TEX-I-1D465",[128,3808,3809],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,3810,3811,3883,3888,3893,3898,3903,3915,3920,3932,3937,3943,3956,4004,4009],{"dataMmlNode":135},[128,3812,3813,3825,3881],{"dataMmlNode":2600},[128,3814,3815,3820],{"dataMmlNode":2085,"transform":3093},[128,3816,3817],{"dataMmlNode":381},[143,3818],{"dataC":3098,"xLinkHref":3819},"#MJX-41-TEX-I-1D715",[128,3821,3822],{"dataMmlNode":381,"transform":3102},[143,3823],{"dataC":1759,"xLinkHref":3824},"#MJX-41-TEX-I-1D438",[128,3826,3827,3831],{"dataMmlNode":2085,"transform":3108},[128,3828,3829],{"dataMmlNode":381},[143,3830],{"dataC":3098,"xLinkHref":3819},[128,3832,3833,3838,3865],{"dataMmlNode":881,"transform":3102},[128,3834,3835],{"dataMmlNode":381},[143,3836],{"dataC":802,"xLinkHref":3837},"#MJX-41-TEX-I-1D464",[128,3839,3840,3845,3850,3855,3860],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,3841,3842],{"dataMmlNode":394},[143,3843],{"dataC":811,"xLinkHref":3844},"#MJX-41-TEX-N-28",[128,3846,3847],{"dataMmlNode":141,"transform":815},[143,3848],{"dataC":411,"xLinkHref":3849},"#MJX-41-TEX-N-32",[128,3851,3852],{"dataMmlNode":394,"transform":821},[143,3853],{"dataC":398,"xLinkHref":3854},"#MJX-41-TEX-N-2C",[128,3856,3857],{"dataMmlNode":141,"transform":827},[143,3858],{"dataC":145,"xLinkHref":3859},"#MJX-41-TEX-N-31",[128,3861,3862],{"dataMmlNode":394,"transform":833},[143,3863],{"dataC":836,"xLinkHref":3864},"#MJX-41-TEX-N-29",[128,3866,3867,3872,3876],{"dataMmlNode":154,"transform":2470,"dataMjxTexclass":156},[128,3868,3869],{"dataMmlNode":381},[143,3870],{"dataC":1823,"xLinkHref":3871},"#MJX-41-TEX-I-1D458",[128,3873,3874],{"dataMmlNode":394,"transform":1827},[143,3875],{"dataC":398,"xLinkHref":3854},[128,3877,3878],{"dataMmlNode":381,"transform":2481},[143,3879],{"dataC":922,"xLinkHref":3880},"#MJX-41-TEX-I-1D457",[2643,3882],{"width":3157,"height":2646,"x":2647,"y":2648},[128,3884,3885],{"dataMmlNode":394,"transform":3518},[143,3886],{"dataC":1209,"xLinkHref":3887},"#MJX-41-TEX-N-3D",[128,3889,3890],{"dataMmlNode":394,"transform":3524},[143,3891],{"dataC":1613,"xLinkHref":3892},"#MJX-41-TEX-N-2212",[128,3894,3896],{"dataMmlNode":141,"transform":3895},"translate(4544.8,0)",[143,3897],{"dataC":411,"xLinkHref":3849},[128,3899,3901],{"dataMmlNode":394,"transform":3900},"translate(5044.8,0)",[143,3902],{"dataC":811,"xLinkHref":3844},[128,3904,3906,3911],{"dataMmlNode":378,"transform":3905},"translate(5433.8,0)",[128,3907,3908],{"dataMmlNode":381},[143,3909],{"dataC":1179,"xLinkHref":3910},"#MJX-41-TEX-I-1D451",[128,3912,3913],{"dataMmlNode":381,"transform":1183},[143,3914],{"dataC":1823,"xLinkHref":3871},[128,3916,3918],{"dataMmlNode":394,"transform":3917},"translate(6627.4,0)",[143,3919],{"dataC":1613,"xLinkHref":3892},[128,3921,3923,3928],{"dataMmlNode":378,"transform":3922},"translate(7627.6,0)",[128,3924,3925],{"dataMmlNode":381},[143,3926],{"dataC":1404,"xLinkHref":3927},"#MJX-41-TEX-I-1D45C",[128,3929,3930],{"dataMmlNode":381,"transform":1408},[143,3931],{"dataC":1823,"xLinkHref":3871},[128,3933,3935],{"dataMmlNode":394,"transform":3934},"translate(8564,0)",[143,3936],{"dataC":836,"xLinkHref":3864},[128,3938,3940],{"dataMmlNode":394,"transform":3939},"translate(9175.2,0)",[143,3941],{"dataC":3562,"xLinkHref":3942},"#MJX-41-TEX-N-22C5",[128,3944,3946,3951],{"dataMmlNode":138,"transform":3945},"translate(9675.5,0)",[128,3947,3948],{"dataMmlNode":381},[143,3949],{"dataC":1966,"xLinkHref":3950},"#MJX-41-TEX-I-1D446",[128,3952,3953],{"dataMmlNode":394,"transform":2701},[143,3954],{"dataC":2704,"xLinkHref":3955},"#MJX-41-TEX-V-2032",[128,3957,3959,3964,3969,3974,3999],{"dataMmlNode":2085,"transform":3958},"translate(10816.2,0)",[128,3960,3961],{"dataMmlNode":394,"transform":2089},[143,3962],{"dataC":811,"xLinkHref":3963},"#MJX-41-TEX-LO-28",[128,3965,3966],{"dataMmlNode":381,"transform":2095},[143,3967],{"dataC":2098,"xLinkHref":3968},"#MJX-41-TEX-I-1D45B",[128,3970,3971],{"dataMmlNode":381,"transform":2102},[143,3972],{"dataC":2105,"xLinkHref":3973},"#MJX-41-TEX-I-1D452",[128,3975,3976,3981,3995],{"dataMmlNode":881,"transform":2109},[128,3977,3978],{"dataMmlNode":381},[143,3979],{"dataC":2114,"xLinkHref":3980},"#MJX-41-TEX-I-1D461",[128,3982,3983,3987,3991],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},[128,3984,3985],{"dataMmlNode":394},[143,3986],{"dataC":811,"xLinkHref":3844},[128,3988,3989],{"dataMmlNode":141,"transform":815},[143,3990],{"dataC":411,"xLinkHref":3849},[128,3992,3993],{"dataMmlNode":394,"transform":821},[143,3994],{"dataC":836,"xLinkHref":3864},[128,3996,3997],{"dataMmlNode":381,"transform":2397},[143,3998],{"dataC":1823,"xLinkHref":3871},[128,4000,4001],{"dataMmlNode":394,"transform":2138},[143,4002],{"dataC":836,"xLinkHref":4003},"#MJX-41-TEX-LO-29",[128,4005,4007],{"dataMmlNode":394,"transform":4006},"translate(14646.1,0)",[143,4008],{"dataC":3562,"xLinkHref":3942},[128,4010,4012,4017,4031],{"dataMmlNode":881,"transform":4011},"translate(15146.3,0)",[128,4013,4014],{"dataMmlNode":381},[143,4015],{"dataC":1157,"xLinkHref":4016},"#MJX-41-TEX-I-1D465",[128,4018,4019,4023,4027],{"dataMmlNode":154,"transform":2049,"dataMjxTexclass":156},[128,4020,4021],{"dataMmlNode":394},[143,4022],{"dataC":811,"xLinkHref":3844},[128,4024,4025],{"dataMmlNode":141,"transform":815},[143,4026],{"dataC":145,"xLinkHref":3859},[128,4028,4029],{"dataMmlNode":394,"transform":821},[143,4030],{"dataC":836,"xLinkHref":3864},[128,4032,4033],{"dataMmlNode":381,"transform":2067},[143,4034],{"dataC":922,"xLinkHref":3880},[2890,4036,4038,4039,2957],{"id":4037},"updating-hidden-layer-weights","Updating Hidden Layer Weights (",[99,4040,4042],{"className":4041,"jax":103},[102],[105,4043,4044,4064],{"style":765,"xmlns":108,"width":766,"height":767,"role":111,"focusable":112,"viewBox":768,"xmlnsXLink":114},[116,4045,4046,4049,4052,4055,4058,4061],{},[119,4047],{"id":4048,"d":774},"MJX-42-TEX-I-1D464",[119,4050],{"id":4051,"d":778},"MJX-42-TEX-N-28",[119,4053],{"id":4054,"d":122},"MJX-42-TEX-N-31",[119,4056],{"id":4057,"d":359},"MJX-42-TEX-N-2C",[119,4059],{"id":4060,"d":126},"MJX-42-TEX-N-30",[119,4062],{"id":4063,"d":791},"MJX-42-TEX-N-29",[128,4065,4066],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,4067,4068],{"dataMmlNode":135},[128,4069,4070,4075],{"dataMmlNode":138},[128,4071,4072],{"dataMmlNode":381},[143,4073],{"dataC":802,"xLinkHref":4074},"#MJX-42-TEX-I-1D464",[128,4076,4077,4082,4087,4092,4097],{"dataMmlNode":154,"transform":806,"dataMjxTexclass":156},[128,4078,4079],{"dataMmlNode":394},[143,4080],{"dataC":811,"xLinkHref":4081},"#MJX-42-TEX-N-28",[128,4083,4084],{"dataMmlNode":141,"transform":815},[143,4085],{"dataC":145,"xLinkHref":4086},"#MJX-42-TEX-N-31",[128,4088,4089],{"dataMmlNode":394,"transform":821},[143,4090],{"dataC":398,"xLinkHref":4091},"#MJX-42-TEX-N-2C",[128,4093,4094],{"dataMmlNode":141,"transform":827},[143,4095],{"dataC":149,"xLinkHref":4096},"#MJX-42-TEX-N-30",[128,4098,4099],{"dataMmlNode":394,"transform":833},[143,4100],{"dataC":836,"xLinkHref":4101},"#MJX-42-TEX-N-29",[16,4103,4104,4105,4124,4125,4128,4129,4151,4152,4237],{},"This is the core of backpropagation. A hidden node ",[99,4106,4108],{"className":4107,"jax":103},[102],[105,4109,4110,4115],{"style":965,"xmlns":108,"width":966,"height":967,"role":111,"focusable":112,"viewBox":968,"xmlnsXLink":114},[116,4111,4112],{},[119,4113],{"id":4114,"d":870},"MJX-43-TEX-I-1D457",[128,4116,4117],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,4118,4119],{"dataMmlNode":135},[128,4120,4121],{"dataMmlNode":381},[143,4122],{"dataC":922,"xLinkHref":4123},"#MJX-43-TEX-I-1D457"," connects to ",[738,4126,4127],{},"multiple"," output nodes ",[99,4130,4132],{"className":4131,"jax":103},[102],[105,4133,4137,4142],{"style":765,"xmlns":108,"width":4134,"height":4135,"role":111,"focusable":112,"viewBox":4136,"xmlnsXLink":114},"1.179ex","1.595ex","0 -694 521 705",[116,4138,4139],{},[119,4140],{"id":4141,"d":1729},"MJX-44-TEX-I-1D458",[128,4143,4144],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,4145,4146],{"dataMmlNode":135},[128,4147,4148],{"dataMmlNode":381},[143,4149],{"dataC":1823,"xLinkHref":4150},"#MJX-44-TEX-I-1D458",". Therefore, the error derivative with respect to a hidden layer weight ",[99,4153,4155],{"className":4154,"jax":103},[102],[105,4156,4157,4183],{"style":844,"xmlns":108,"width":766,"height":845,"role":111,"focusable":112,"viewBox":846,"xmlnsXLink":114},[116,4158,4159,4162,4165,4168,4171,4174,4177,4180],{},[119,4160],{"id":4161,"d":774},"MJX-45-TEX-I-1D464",[119,4163],{"id":4164,"d":778},"MJX-45-TEX-N-28",[119,4166],{"id":4167,"d":122},"MJX-45-TEX-N-31",[119,4169],{"id":4170,"d":359},"MJX-45-TEX-N-2C",[119,4172],{"id":4173,"d":126},"MJX-45-TEX-N-30",[119,4175],{"id":4176,"d":791},"MJX-45-TEX-N-29",[119,4178],{"id":4179,"d":870},"MJX-45-TEX-I-1D457",[119,4181],{"id":4182,"d":874},"MJX-45-TEX-I-1D456",[128,4184,4185],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,4186,4187],{"dataMmlNode":135},[128,4188,4189,4194,4221],{"dataMmlNode":881},[128,4190,4191],{"dataMmlNode":381},[143,4192],{"dataC":802,"xLinkHref":4193},"#MJX-45-TEX-I-1D464",[128,4195,4196,4201,4206,4211,4216],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,4197,4198],{"dataMmlNode":394},[143,4199],{"dataC":811,"xLinkHref":4200},"#MJX-45-TEX-N-28",[128,4202,4203],{"dataMmlNode":141,"transform":815},[143,4204],{"dataC":145,"xLinkHref":4205},"#MJX-45-TEX-N-31",[128,4207,4208],{"dataMmlNode":394,"transform":821},[143,4209],{"dataC":398,"xLinkHref":4210},"#MJX-45-TEX-N-2C",[128,4212,4213],{"dataMmlNode":141,"transform":827},[143,4214],{"dataC":149,"xLinkHref":4215},"#MJX-45-TEX-N-30",[128,4217,4218],{"dataMmlNode":394,"transform":833},[143,4219],{"dataC":836,"xLinkHref":4220},"#MJX-45-TEX-N-29",[128,4222,4223,4228,4232],{"dataMmlNode":154,"transform":917,"dataMjxTexclass":156},[128,4224,4225],{"dataMmlNode":381},[143,4226],{"dataC":922,"xLinkHref":4227},"#MJX-45-TEX-I-1D457",[128,4229,4230],{"dataMmlNode":394,"transform":926},[143,4231],{"dataC":398,"xLinkHref":4210},[128,4233,4234],{"dataMmlNode":381,"transform":931},[143,4235],{"dataC":934,"xLinkHref":4236},"#MJX-45-TEX-I-1D456"," must sum the gradients from all subsequent output nodes it influences.",[16,4239,4240],{},[99,4241,4243],{"className":4242,"jax":103},[102],[105,4244,4249,4322],{"style":4245,"xmlns":108,"width":4246,"height":4247,"role":111,"focusable":112,"viewBox":4248,"xmlnsXLink":114},"vertical-align: -2.148ex;","50.671ex","5.428ex","0 -1449.5 22396.5 2399",[116,4250,4251,4254,4257,4260,4263,4266,4269,4272,4275,4278,4281,4284,4287,4290,4293,4297,4300,4303,4306,4309,4312,4315,4318],{},[119,4252],{"id":4253,"d":3010},"MJX-46-TEX-I-1D715",[119,4255],{"id":4256,"d":1706},"MJX-46-TEX-I-1D438",[119,4258],{"id":4259,"d":774},"MJX-46-TEX-I-1D464",[119,4261],{"id":4262,"d":778},"MJX-46-TEX-N-28",[119,4264],{"id":4265,"d":122},"MJX-46-TEX-N-31",[119,4267],{"id":4268,"d":359},"MJX-46-TEX-N-2C",[119,4270],{"id":4271,"d":126},"MJX-46-TEX-N-30",[119,4273],{"id":4274,"d":791},"MJX-46-TEX-N-29",[119,4276],{"id":4277,"d":870},"MJX-46-TEX-I-1D457",[119,4279],{"id":4280,"d":874},"MJX-46-TEX-I-1D456",[119,4282],{"id":4283,"d":1122},"MJX-46-TEX-N-3D",[119,4285],{"id":4286,"d":1713},"MJX-46-TEX-SO-2211",[119,4288],{"id":4289,"d":1680},"MJX-46-TEX-I-1D43E",[119,4291],{"id":4292,"d":1729},"MJX-46-TEX-I-1D458",[119,4294],{"id":4295,"d":4296},"MJX-46-TEX-S3-28","M701 -940Q701 -943 695 -949H664Q662 -947 636 -922T591 -879T537 -818T475 -737T412 -636T350 -511T295 -362T250 -186T221 17T209 251Q209 962 573 1361Q596 1386 616 1405T649 1437T664 1450H695Q701 1444 701 1441Q701 1436 681 1415T629 1356T557 1261T476 1118T400 927T340 675T308 359Q306 321 306 250Q306 -139 400 -430T690 -924Q701 -936 701 -940Z",[119,4298],{"id":4299,"d":1390},"MJX-46-TEX-I-1D45C",[119,4301],{"id":4302,"d":3430},"MJX-46-TEX-N-22C5",[119,4304],{"id":4305,"d":2008},"MJX-46-TEX-I-1D45B",[119,4307],{"id":4308,"d":2012},"MJX-46-TEX-I-1D452",[119,4310],{"id":4311,"d":2016},"MJX-46-TEX-I-1D461",[119,4313],{"id":4314,"d":363},"MJX-46-TEX-N-32",[119,4316],{"id":4317,"d":1100},"MJX-46-TEX-I-1D465",[119,4319],{"id":4320,"d":4321},"MJX-46-TEX-S3-29","M34 1438Q34 1446 37 1448T50 1450H56H71Q73 1448 99 1423T144 1380T198 1319T260 1238T323 1137T385 1013T440 864T485 688T514 485T526 251Q526 134 519 53Q472 -519 162 -860Q139 -885 119 -904T86 -936T71 -949H56Q43 -949 39 -947T34 -937Q88 -883 140 -813Q428 -430 428 251Q428 453 402 628T338 922T245 1146T145 1309T46 1425Q44 1427 42 1429T39 1433T36 1436L34 1438Z",[128,4323,4324],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,4325,4326,4398,4403,4432],{"dataMmlNode":135},[128,4327,4328,4340,4396],{"dataMmlNode":2600},[128,4329,4330,4335],{"dataMmlNode":2085,"transform":3093},[128,4331,4332],{"dataMmlNode":381},[143,4333],{"dataC":3098,"xLinkHref":4334},"#MJX-46-TEX-I-1D715",[128,4336,4337],{"dataMmlNode":381,"transform":3102},[143,4338],{"dataC":1759,"xLinkHref":4339},"#MJX-46-TEX-I-1D438",[128,4341,4342,4346],{"dataMmlNode":2085,"transform":3108},[128,4343,4344],{"dataMmlNode":381},[143,4345],{"dataC":3098,"xLinkHref":4334},[128,4347,4348,4353,4380],{"dataMmlNode":881,"transform":3102},[128,4349,4350],{"dataMmlNode":381},[143,4351],{"dataC":802,"xLinkHref":4352},"#MJX-46-TEX-I-1D464",[128,4354,4355,4360,4365,4370,4375],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,4356,4357],{"dataMmlNode":394},[143,4358],{"dataC":811,"xLinkHref":4359},"#MJX-46-TEX-N-28",[128,4361,4362],{"dataMmlNode":141,"transform":815},[143,4363],{"dataC":145,"xLinkHref":4364},"#MJX-46-TEX-N-31",[128,4366,4367],{"dataMmlNode":394,"transform":821},[143,4368],{"dataC":398,"xLinkHref":4369},"#MJX-46-TEX-N-2C",[128,4371,4372],{"dataMmlNode":141,"transform":827},[143,4373],{"dataC":149,"xLinkHref":4374},"#MJX-46-TEX-N-30",[128,4376,4377],{"dataMmlNode":394,"transform":833},[143,4378],{"dataC":836,"xLinkHref":4379},"#MJX-46-TEX-N-29",[128,4381,4382,4387,4391],{"dataMmlNode":154,"transform":917,"dataMjxTexclass":156},[128,4383,4384],{"dataMmlNode":381},[143,4385],{"dataC":922,"xLinkHref":4386},"#MJX-46-TEX-I-1D457",[128,4388,4389],{"dataMmlNode":394,"transform":926},[143,4390],{"dataC":398,"xLinkHref":4369},[128,4392,4393],{"dataMmlNode":381,"transform":931},[143,4394],{"dataC":934,"xLinkHref":4395},"#MJX-46-TEX-I-1D456",[2643,4397],{"width":3157,"height":2646,"x":2647,"y":2648},[128,4399,4400],{"dataMmlNode":394,"transform":3518},[143,4401],{"dataC":1209,"xLinkHref":4402},"#MJX-46-TEX-N-3D",[128,4404,4405,4410,4417],{"dataMmlNode":1769,"transform":3524},[128,4406,4407],{"dataMmlNode":394},[143,4408],{"dataC":1775,"xLinkHref":4409},"#MJX-46-TEX-SO-2211",[128,4411,4412],{"dataMmlNode":154,"transform":1779,"dataMjxTexclass":156},[128,4413,4414],{"dataMmlNode":381},[143,4415],{"dataC":1689,"xLinkHref":4416},"#MJX-46-TEX-I-1D43E",[128,4418,4419,4424,4428],{"dataMmlNode":154,"transform":1787,"dataMjxTexclass":156},[128,4420,4421],{"dataMmlNode":381},[143,4422],{"dataC":1823,"xLinkHref":4423},"#MJX-46-TEX-I-1D458",[128,4425,4426],{"dataMmlNode":394,"transform":1827},[143,4427],{"dataC":1209,"xLinkHref":4402},[128,4429,4430],{"dataMmlNode":141,"transform":1832},[143,4431],{"dataC":145,"xLinkHref":4364},[128,4433,4435,4440,4472,4478,4541,4546,4621,4626,4700,4705,4797],{"dataMmlNode":2085,"transform":4434},"translate(6344.5,0)",[128,4436,4437],{"dataMmlNode":394,"transform":2089},[143,4438],{"dataC":811,"xLinkHref":4439},"#MJX-46-TEX-S3-28",[128,4441,4443,4453,4470],{"dataMmlNode":2600,"transform":4442},"translate(736,0)",[128,4444,4445,4449],{"dataMmlNode":2085,"transform":3527},[128,4446,4447],{"dataMmlNode":381},[143,4448],{"dataC":3098,"xLinkHref":4334},[128,4450,4451],{"dataMmlNode":381,"transform":3102},[143,4452],{"dataC":1759,"xLinkHref":4339},[128,4454,4455,4459],{"dataMmlNode":2085,"transform":3538},[128,4456,4457],{"dataMmlNode":381},[143,4458],{"dataC":3098,"xLinkHref":4334},[128,4460,4461,4466],{"dataMmlNode":378,"transform":3102},[128,4462,4463],{"dataMmlNode":381},[143,4464],{"dataC":1404,"xLinkHref":4465},"#MJX-46-TEX-I-1D45C",[128,4467,4468],{"dataMmlNode":381,"transform":1408},[143,4469],{"dataC":1823,"xLinkHref":4423},[2643,4471],{"width":3556,"height":2646,"x":2647,"y":2648},[128,4473,4475],{"dataMmlNode":394,"transform":4474},"translate(2460.6,0)",[143,4476],{"dataC":3562,"xLinkHref":4477},"#MJX-46-TEX-N-22C5",[128,4479,4481,4497,4539],{"dataMmlNode":2600,"transform":4480},"translate(2960.8,0)",[128,4482,4483,4487],{"dataMmlNode":2085,"transform":3569},[128,4484,4485],{"dataMmlNode":381},[143,4486],{"dataC":3098,"xLinkHref":4334},[128,4488,4489,4493],{"dataMmlNode":378,"transform":3102},[128,4490,4491],{"dataMmlNode":381},[143,4492],{"dataC":1404,"xLinkHref":4465},[128,4494,4495],{"dataMmlNode":381,"transform":1408},[143,4496],{"dataC":1823,"xLinkHref":4423},[128,4498,4499,4503,4508,4513],{"dataMmlNode":2085,"transform":3108},[128,4500,4501],{"dataMmlNode":381},[143,4502],{"dataC":3098,"xLinkHref":4334},[128,4504,4505],{"dataMmlNode":381,"transform":3102},[143,4506],{"dataC":2098,"xLinkHref":4507},"#MJX-46-TEX-I-1D45B",[128,4509,4510],{"dataMmlNode":381,"transform":3597},[143,4511],{"dataC":2105,"xLinkHref":4512},"#MJX-46-TEX-I-1D452",[128,4514,4515,4520,4535],{"dataMmlNode":881,"transform":3603},[128,4516,4517],{"dataMmlNode":381},[143,4518],{"dataC":2114,"xLinkHref":4519},"#MJX-46-TEX-I-1D461",[128,4521,4522,4526,4531],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},[128,4523,4524],{"dataMmlNode":394},[143,4525],{"dataC":811,"xLinkHref":4359},[128,4527,4528],{"dataMmlNode":141,"transform":815},[143,4529],{"dataC":411,"xLinkHref":4530},"#MJX-46-TEX-N-32",[128,4532,4533],{"dataMmlNode":394,"transform":821},[143,4534],{"dataC":836,"xLinkHref":4379},[128,4536,4537],{"dataMmlNode":381,"transform":2397},[143,4538],{"dataC":1823,"xLinkHref":4423},[2643,4540],{"width":3629,"height":2646,"x":2647,"y":2648},[128,4542,4544],{"dataMmlNode":394,"transform":4543},"translate(5730,0)",[143,4545],{"dataC":3562,"xLinkHref":4477},[128,4547,4549,4587,4619],{"dataMmlNode":2600,"transform":4548},"translate(6230.2,0)",[128,4550,4551,4555,4559,4563],{"dataMmlNode":2085,"transform":3640},[128,4552,4553],{"dataMmlNode":381},[143,4554],{"dataC":3098,"xLinkHref":4334},[128,4556,4557],{"dataMmlNode":381,"transform":3102},[143,4558],{"dataC":2098,"xLinkHref":4507},[128,4560,4561],{"dataMmlNode":381,"transform":3597},[143,4562],{"dataC":2105,"xLinkHref":4512},[128,4564,4565,4569,4583],{"dataMmlNode":881,"transform":3603},[128,4566,4567],{"dataMmlNode":381},[143,4568],{"dataC":2114,"xLinkHref":4519},[128,4570,4571,4575,4579],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},[128,4572,4573],{"dataMmlNode":394},[143,4574],{"dataC":811,"xLinkHref":4359},[128,4576,4577],{"dataMmlNode":141,"transform":815},[143,4578],{"dataC":411,"xLinkHref":4530},[128,4580,4581],{"dataMmlNode":394,"transform":821},[143,4582],{"dataC":836,"xLinkHref":4379},[128,4584,4585],{"dataMmlNode":381,"transform":2397},[143,4586],{"dataC":1823,"xLinkHref":4423},[128,4588,4590,4594],{"dataMmlNode":2085,"transform":4589},"translate(522.3,-590) scale(0.707)",[128,4591,4592],{"dataMmlNode":381},[143,4593],{"dataC":3098,"xLinkHref":4334},[128,4595,4596,4601,4615],{"dataMmlNode":881,"transform":3102},[128,4597,4598],{"dataMmlNode":381},[143,4599],{"dataC":1157,"xLinkHref":4600},"#MJX-46-TEX-I-1D465",[128,4602,4603,4607,4611],{"dataMmlNode":154,"transform":2049,"dataMjxTexclass":156},[128,4604,4605],{"dataMmlNode":394},[143,4606],{"dataC":811,"xLinkHref":4359},[128,4608,4609],{"dataMmlNode":141,"transform":815},[143,4610],{"dataC":145,"xLinkHref":4364},[128,4612,4613],{"dataMmlNode":394,"transform":821},[143,4614],{"dataC":836,"xLinkHref":4379},[128,4616,4617],{"dataMmlNode":381,"transform":2067},[143,4618],{"dataC":922,"xLinkHref":4386},[2643,4620],{"width":3629,"height":2646,"x":2647,"y":2648},[128,4622,4624],{"dataMmlNode":394,"transform":4623},"translate(8999.4,0)",[143,4625],{"dataC":3562,"xLinkHref":4477},[128,4627,4629,4660,4698],{"dataMmlNode":2600,"transform":4628},"translate(9499.6,0)",[128,4630,4632,4636],{"dataMmlNode":2085,"transform":4631},"translate(522.3,649.7) scale(0.707)",[128,4633,4634],{"dataMmlNode":381},[143,4635],{"dataC":3098,"xLinkHref":4334},[128,4637,4638,4642,4656],{"dataMmlNode":881,"transform":3102},[128,4639,4640],{"dataMmlNode":381},[143,4641],{"dataC":1157,"xLinkHref":4600},[128,4643,4644,4648,4652],{"dataMmlNode":154,"transform":2049,"dataMjxTexclass":156},[128,4645,4646],{"dataMmlNode":394},[143,4647],{"dataC":811,"xLinkHref":4359},[128,4649,4650],{"dataMmlNode":141,"transform":815},[143,4651],{"dataC":145,"xLinkHref":4364},[128,4653,4654],{"dataMmlNode":394,"transform":821},[143,4655],{"dataC":836,"xLinkHref":4379},[128,4657,4658],{"dataMmlNode":381,"transform":2067},[143,4659],{"dataC":922,"xLinkHref":4386},[128,4661,4662,4666,4670,4674],{"dataMmlNode":2085,"transform":3108},[128,4663,4664],{"dataMmlNode":381},[143,4665],{"dataC":3098,"xLinkHref":4334},[128,4667,4668],{"dataMmlNode":381,"transform":3102},[143,4669],{"dataC":2098,"xLinkHref":4507},[128,4671,4672],{"dataMmlNode":381,"transform":3597},[143,4673],{"dataC":2105,"xLinkHref":4512},[128,4675,4676,4680,4694],{"dataMmlNode":881,"transform":3603},[128,4677,4678],{"dataMmlNode":381},[143,4679],{"dataC":2114,"xLinkHref":4519},[128,4681,4682,4686,4690],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},[128,4683,4684],{"dataMmlNode":394},[143,4685],{"dataC":811,"xLinkHref":4359},[128,4687,4688],{"dataMmlNode":141,"transform":815},[143,4689],{"dataC":145,"xLinkHref":4364},[128,4691,4692],{"dataMmlNode":394,"transform":821},[143,4693],{"dataC":836,"xLinkHref":4379},[128,4695,4696],{"dataMmlNode":381,"transform":2133},[143,4697],{"dataC":922,"xLinkHref":4386},[2643,4699],{"width":3629,"height":2646,"x":2647,"y":2648},[128,4701,4703],{"dataMmlNode":394,"transform":4702},"translate(12268.8,0)",[143,4704],{"dataC":3562,"xLinkHref":4477},[128,4706,4708,4747,4795],{"dataMmlNode":2600,"transform":4707},"translate(12769,0)",[128,4709,4711,4715,4719,4723],{"dataMmlNode":2085,"transform":4710},"translate(220,649.7) scale(0.707)",[128,4712,4713],{"dataMmlNode":381},[143,4714],{"dataC":3098,"xLinkHref":4334},[128,4716,4717],{"dataMmlNode":381,"transform":3102},[143,4718],{"dataC":2098,"xLinkHref":4507},[128,4720,4721],{"dataMmlNode":381,"transform":3597},[143,4722],{"dataC":2105,"xLinkHref":4512},[128,4724,4725,4729,4743],{"dataMmlNode":881,"transform":3603},[128,4726,4727],{"dataMmlNode":381},[143,4728],{"dataC":2114,"xLinkHref":4519},[128,4730,4731,4735,4739],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},[128,4732,4733],{"dataMmlNode":394},[143,4734],{"dataC":811,"xLinkHref":4359},[128,4736,4737],{"dataMmlNode":141,"transform":815},[143,4738],{"dataC":145,"xLinkHref":4364},[128,4740,4741],{"dataMmlNode":394,"transform":821},[143,4742],{"dataC":836,"xLinkHref":4379},[128,4744,4745],{"dataMmlNode":381,"transform":2133},[143,4746],{"dataC":922,"xLinkHref":4386},[128,4748,4749,4753],{"dataMmlNode":2085,"transform":3679},[128,4750,4751],{"dataMmlNode":381},[143,4752],{"dataC":3098,"xLinkHref":4334},[128,4754,4755,4759,4781],{"dataMmlNode":881,"transform":3102},[128,4756,4757],{"dataMmlNode":381},[143,4758],{"dataC":802,"xLinkHref":4352},[128,4760,4761,4765,4769,4773,4777],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,4762,4763],{"dataMmlNode":394},[143,4764],{"dataC":811,"xLinkHref":4359},[128,4766,4767],{"dataMmlNode":141,"transform":815},[143,4768],{"dataC":145,"xLinkHref":4364},[128,4770,4771],{"dataMmlNode":394,"transform":821},[143,4772],{"dataC":398,"xLinkHref":4369},[128,4774,4775],{"dataMmlNode":141,"transform":827},[143,4776],{"dataC":149,"xLinkHref":4374},[128,4778,4779],{"dataMmlNode":394,"transform":833},[143,4780],{"dataC":836,"xLinkHref":4379},[128,4782,4783,4787,4791],{"dataMmlNode":154,"transform":917,"dataMjxTexclass":156},[128,4784,4785],{"dataMmlNode":381},[143,4786],{"dataC":922,"xLinkHref":4386},[128,4788,4789],{"dataMmlNode":394,"transform":926},[143,4790],{"dataC":398,"xLinkHref":4369},[128,4792,4793],{"dataMmlNode":381,"transform":931},[143,4794],{"dataC":934,"xLinkHref":4395},[2643,4796],{"width":3629,"height":2646,"x":2647,"y":2648},[128,4798,4800],{"dataMmlNode":394,"transform":4799},"translate(15316,0) translate(0 -0.5)",[143,4801],{"dataC":836,"xLinkHref":4802},"#MJX-46-TEX-S3-29",[16,4804,4805,4806],{},"Expanding this partial derivative mathematically:\n",[99,4807,4809],{"className":4808,"jax":103},[102],[105,4810,4815,4906],{"style":4811,"xmlns":108,"width":4812,"height":4813,"role":111,"focusable":112,"viewBox":4814,"xmlnsXLink":114},"vertical-align: -2.036ex;","64.26ex","4.636ex","0 -1149.5 28403.1 2049.3",[116,4816,4817,4820,4823,4826,4829,4832,4835,4838,4841,4844,4847,4850,4853,4856,4859,4863,4866,4869,4872,4875,4878,4881,4884,4887,4890,4893,4896,4899,4902],{},[119,4818],{"id":4819,"d":3010},"MJX-47-TEX-I-1D715",[119,4821],{"id":4822,"d":1706},"MJX-47-TEX-I-1D438",[119,4824],{"id":4825,"d":774},"MJX-47-TEX-I-1D464",[119,4827],{"id":4828,"d":778},"MJX-47-TEX-N-28",[119,4830],{"id":4831,"d":122},"MJX-47-TEX-N-31",[119,4833],{"id":4834,"d":359},"MJX-47-TEX-N-2C",[119,4836],{"id":4837,"d":126},"MJX-47-TEX-N-30",[119,4839],{"id":4840,"d":791},"MJX-47-TEX-N-29",[119,4842],{"id":4843,"d":870},"MJX-47-TEX-I-1D457",[119,4845],{"id":4846,"d":874},"MJX-47-TEX-I-1D456",[119,4848],{"id":4849,"d":1122},"MJX-47-TEX-N-3D",[119,4851],{"id":4852,"d":1713},"MJX-47-TEX-SO-2211",[119,4854],{"id":4855,"d":1680},"MJX-47-TEX-I-1D43E",[119,4857],{"id":4858,"d":1729},"MJX-47-TEX-I-1D458",[119,4860],{"id":4861,"d":4862},"MJX-47-TEX-LO-7B","M547 -643L541 -649H528Q515 -649 503 -645Q324 -582 293 -466Q289 -449 289 -428T287 -200L286 42L284 53Q274 98 248 135T196 190T146 222L121 235Q119 239 119 250Q119 262 121 266T133 273Q262 336 284 449L286 460L287 701Q287 737 287 794Q288 949 292 963Q293 966 293 967Q325 1080 508 1148Q516 1150 527 1150H541L547 1144V1130Q547 1117 546 1115T536 1109Q480 1086 437 1046T381 950L379 940L378 699Q378 657 378 594Q377 452 374 438Q373 437 373 436Q350 348 243 282Q192 257 186 254L176 251L188 245Q211 236 234 223T287 189T340 135T373 65Q373 64 374 63Q377 49 378 -93Q378 -156 378 -198L379 -438L381 -449Q393 -504 436 -544T536 -608Q544 -611 545 -613T547 -629V-643Z",[119,4864],{"id":4865,"d":1548},"MJX-47-TEX-N-2212",[119,4867],{"id":4868,"d":363},"MJX-47-TEX-N-32",[119,4870],{"id":4871,"d":1111},"MJX-47-TEX-I-1D451",[119,4873],{"id":4874,"d":1390},"MJX-47-TEX-I-1D45C",[119,4876],{"id":4877,"d":3430},"MJX-47-TEX-N-22C5",[119,4879],{"id":4880,"d":1957},"MJX-47-TEX-I-1D446",[119,4882],{"id":4883,"d":2669},"MJX-47-TEX-V-2032",[119,4885],{"id":4886,"d":2004},"MJX-47-TEX-LO-28",[119,4888],{"id":4889,"d":2008},"MJX-47-TEX-I-1D45B",[119,4891],{"id":4892,"d":2012},"MJX-47-TEX-I-1D452",[119,4894],{"id":4895,"d":2016},"MJX-47-TEX-I-1D461",[119,4897],{"id":4898,"d":2020},"MJX-47-TEX-LO-29",[119,4900],{"id":4901,"d":1100},"MJX-47-TEX-I-1D465",[119,4903],{"id":4904,"d":4905},"MJX-47-TEX-LO-7D","M119 1130Q119 1144 121 1147T135 1150H139Q151 1150 182 1138T252 1105T326 1046T373 964Q378 942 378 702Q378 469 379 462Q386 394 439 339Q482 296 535 272Q544 268 545 266T547 251Q547 241 547 238T542 231T531 227T510 217T477 194Q390 129 379 39Q378 32 378 -201Q378 -441 373 -463Q342 -580 165 -644Q152 -649 139 -649Q125 -649 122 -646T119 -629Q119 -622 119 -619T121 -614T124 -610T132 -607T143 -602Q195 -579 235 -539T285 -447Q286 -435 287 -199T289 51Q294 74 300 91T329 138T390 197Q412 213 436 226T475 244L489 250L472 258Q455 265 430 279T377 313T327 366T293 434Q289 451 289 472T287 699Q286 941 285 948Q279 978 262 1005T227 1048T184 1080T151 1100T129 1109L127 1110Q119 1113 119 1130Z",[128,4907,4908],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,4909,4910,4982,4987,5016],{"dataMmlNode":135},[128,4911,4912,4924,4980],{"dataMmlNode":2600},[128,4913,4914,4919],{"dataMmlNode":2085,"transform":3093},[128,4915,4916],{"dataMmlNode":381},[143,4917],{"dataC":3098,"xLinkHref":4918},"#MJX-47-TEX-I-1D715",[128,4920,4921],{"dataMmlNode":381,"transform":3102},[143,4922],{"dataC":1759,"xLinkHref":4923},"#MJX-47-TEX-I-1D438",[128,4925,4926,4930],{"dataMmlNode":2085,"transform":3108},[128,4927,4928],{"dataMmlNode":381},[143,4929],{"dataC":3098,"xLinkHref":4918},[128,4931,4932,4937,4964],{"dataMmlNode":881,"transform":3102},[128,4933,4934],{"dataMmlNode":381},[143,4935],{"dataC":802,"xLinkHref":4936},"#MJX-47-TEX-I-1D464",[128,4938,4939,4944,4949,4954,4959],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,4940,4941],{"dataMmlNode":394},[143,4942],{"dataC":811,"xLinkHref":4943},"#MJX-47-TEX-N-28",[128,4945,4946],{"dataMmlNode":141,"transform":815},[143,4947],{"dataC":145,"xLinkHref":4948},"#MJX-47-TEX-N-31",[128,4950,4951],{"dataMmlNode":394,"transform":821},[143,4952],{"dataC":398,"xLinkHref":4953},"#MJX-47-TEX-N-2C",[128,4955,4956],{"dataMmlNode":141,"transform":827},[143,4957],{"dataC":149,"xLinkHref":4958},"#MJX-47-TEX-N-30",[128,4960,4961],{"dataMmlNode":394,"transform":833},[143,4962],{"dataC":836,"xLinkHref":4963},"#MJX-47-TEX-N-29",[128,4965,4966,4971,4975],{"dataMmlNode":154,"transform":917,"dataMjxTexclass":156},[128,4967,4968],{"dataMmlNode":381},[143,4969],{"dataC":922,"xLinkHref":4970},"#MJX-47-TEX-I-1D457",[128,4972,4973],{"dataMmlNode":394,"transform":926},[143,4974],{"dataC":398,"xLinkHref":4953},[128,4976,4977],{"dataMmlNode":381,"transform":931},[143,4978],{"dataC":934,"xLinkHref":4979},"#MJX-47-TEX-I-1D456",[2643,4981],{"width":3157,"height":2646,"x":2647,"y":2648},[128,4983,4984],{"dataMmlNode":394,"transform":3518},[143,4985],{"dataC":1209,"xLinkHref":4986},"#MJX-47-TEX-N-3D",[128,4988,4989,4994,5001],{"dataMmlNode":1769,"transform":3524},[128,4990,4991],{"dataMmlNode":394},[143,4992],{"dataC":1775,"xLinkHref":4993},"#MJX-47-TEX-SO-2211",[128,4995,4996],{"dataMmlNode":154,"transform":1779,"dataMjxTexclass":156},[128,4997,4998],{"dataMmlNode":381},[143,4999],{"dataC":1689,"xLinkHref":5000},"#MJX-47-TEX-I-1D43E",[128,5002,5003,5008,5012],{"dataMmlNode":154,"transform":1787,"dataMjxTexclass":156},[128,5004,5005],{"dataMmlNode":381},[143,5006],{"dataC":1823,"xLinkHref":5007},"#MJX-47-TEX-I-1D458",[128,5009,5010],{"dataMmlNode":394,"transform":1827},[143,5011],{"dataC":1209,"xLinkHref":4986},[128,5013,5014],{"dataMmlNode":141,"transform":1832},[143,5015],{"dataC":145,"xLinkHref":4948},[128,5017,5018,5023,5029,5035,5040,5052,5057,5069,5074,5080,5093,5141,5146,5189,5194,5205,5248,5253,5265],{"dataMmlNode":2085,"transform":4434},[128,5019,5020],{"dataMmlNode":394,"transform":2089},[143,5021],{"dataC":1144,"xLinkHref":5022},"#MJX-47-TEX-LO-7B",[128,5024,5026],{"dataMmlNode":394,"transform":5025},"translate(667,0)",[143,5027],{"dataC":1613,"xLinkHref":5028},"#MJX-47-TEX-N-2212",[128,5030,5032],{"dataMmlNode":141,"transform":5031},"translate(1445,0)",[143,5033],{"dataC":411,"xLinkHref":5034},"#MJX-47-TEX-N-32",[128,5036,5038],{"dataMmlNode":394,"transform":5037},"translate(1945,0)",[143,5039],{"dataC":811,"xLinkHref":4943},[128,5041,5043,5048],{"dataMmlNode":378,"transform":5042},"translate(2334,0)",[128,5044,5045],{"dataMmlNode":381},[143,5046],{"dataC":1179,"xLinkHref":5047},"#MJX-47-TEX-I-1D451",[128,5049,5050],{"dataMmlNode":381,"transform":1183},[143,5051],{"dataC":1823,"xLinkHref":5007},[128,5053,5055],{"dataMmlNode":394,"transform":5054},"translate(3527.6,0)",[143,5056],{"dataC":1613,"xLinkHref":5028},[128,5058,5060,5065],{"dataMmlNode":378,"transform":5059},"translate(4527.8,0)",[128,5061,5062],{"dataMmlNode":381},[143,5063],{"dataC":1404,"xLinkHref":5064},"#MJX-47-TEX-I-1D45C",[128,5066,5067],{"dataMmlNode":381,"transform":1408},[143,5068],{"dataC":1823,"xLinkHref":5007},[128,5070,5072],{"dataMmlNode":394,"transform":5071},"translate(5464.2,0)",[143,5073],{"dataC":836,"xLinkHref":4963},[128,5075,5077],{"dataMmlNode":394,"transform":5076},"translate(6075.5,0)",[143,5078],{"dataC":3562,"xLinkHref":5079},"#MJX-47-TEX-N-22C5",[128,5081,5083,5088],{"dataMmlNode":138,"transform":5082},"translate(6575.7,0)",[128,5084,5085],{"dataMmlNode":381},[143,5086],{"dataC":1966,"xLinkHref":5087},"#MJX-47-TEX-I-1D446",[128,5089,5090],{"dataMmlNode":394,"transform":2701},[143,5091],{"dataC":2704,"xLinkHref":5092},"#MJX-47-TEX-V-2032",[128,5094,5096,5101,5106,5111,5136],{"dataMmlNode":2085,"transform":5095},"translate(7716.4,0)",[128,5097,5098],{"dataMmlNode":394,"transform":2089},[143,5099],{"dataC":811,"xLinkHref":5100},"#MJX-47-TEX-LO-28",[128,5102,5103],{"dataMmlNode":381,"transform":2095},[143,5104],{"dataC":2098,"xLinkHref":5105},"#MJX-47-TEX-I-1D45B",[128,5107,5108],{"dataMmlNode":381,"transform":2102},[143,5109],{"dataC":2105,"xLinkHref":5110},"#MJX-47-TEX-I-1D452",[128,5112,5113,5118,5132],{"dataMmlNode":881,"transform":2109},[128,5114,5115],{"dataMmlNode":381},[143,5116],{"dataC":2114,"xLinkHref":5117},"#MJX-47-TEX-I-1D461",[128,5119,5120,5124,5128],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},[128,5121,5122],{"dataMmlNode":394},[143,5123],{"dataC":811,"xLinkHref":4943},[128,5125,5126],{"dataMmlNode":141,"transform":815},[143,5127],{"dataC":411,"xLinkHref":5034},[128,5129,5130],{"dataMmlNode":394,"transform":821},[143,5131],{"dataC":836,"xLinkHref":4963},[128,5133,5134],{"dataMmlNode":381,"transform":2397},[143,5135],{"dataC":1823,"xLinkHref":5007},[128,5137,5138],{"dataMmlNode":394,"transform":2138},[143,5139],{"dataC":836,"xLinkHref":5140},"#MJX-47-TEX-LO-29",[128,5142,5144],{"dataMmlNode":394,"transform":5143},"translate(11546.3,0)",[143,5145],{"dataC":3562,"xLinkHref":5079},[128,5147,5149,5153,5175],{"dataMmlNode":881,"transform":5148},"translate(12046.5,0)",[128,5150,5151],{"dataMmlNode":381},[143,5152],{"dataC":802,"xLinkHref":4936},[128,5154,5155,5159,5163,5167,5171],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,5156,5157],{"dataMmlNode":394},[143,5158],{"dataC":811,"xLinkHref":4943},[128,5160,5161],{"dataMmlNode":141,"transform":815},[143,5162],{"dataC":411,"xLinkHref":5034},[128,5164,5165],{"dataMmlNode":394,"transform":821},[143,5166],{"dataC":398,"xLinkHref":4953},[128,5168,5169],{"dataMmlNode":141,"transform":827},[143,5170],{"dataC":145,"xLinkHref":4948},[128,5172,5173],{"dataMmlNode":394,"transform":833},[143,5174],{"dataC":836,"xLinkHref":4963},[128,5176,5177,5181,5185],{"dataMmlNode":154,"transform":2470,"dataMjxTexclass":156},[128,5178,5179],{"dataMmlNode":381},[143,5180],{"dataC":1823,"xLinkHref":5007},[128,5182,5183],{"dataMmlNode":394,"transform":1827},[143,5184],{"dataC":398,"xLinkHref":4953},[128,5186,5187],{"dataMmlNode":381,"transform":2481},[143,5188],{"dataC":922,"xLinkHref":4970},[128,5190,5192],{"dataMmlNode":394,"transform":5191},"translate(14521.6,0)",[143,5193],{"dataC":3562,"xLinkHref":5079},[128,5195,5197,5201],{"dataMmlNode":138,"transform":5196},"translate(15021.8,0)",[128,5198,5199],{"dataMmlNode":381},[143,5200],{"dataC":1966,"xLinkHref":5087},[128,5202,5203],{"dataMmlNode":394,"transform":2701},[143,5204],{"dataC":2704,"xLinkHref":5092},[128,5206,5208,5212,5216,5220,5244],{"dataMmlNode":2085,"transform":5207},"translate(16162.5,0)",[128,5209,5210],{"dataMmlNode":394,"transform":2089},[143,5211],{"dataC":811,"xLinkHref":5100},[128,5213,5214],{"dataMmlNode":381,"transform":2095},[143,5215],{"dataC":2098,"xLinkHref":5105},[128,5217,5218],{"dataMmlNode":381,"transform":2102},[143,5219],{"dataC":2105,"xLinkHref":5110},[128,5221,5222,5226,5240],{"dataMmlNode":881,"transform":2109},[128,5223,5224],{"dataMmlNode":381},[143,5225],{"dataC":2114,"xLinkHref":5117},[128,5227,5228,5232,5236],{"dataMmlNode":154,"transform":2118,"dataMjxTexclass":156},[128,5229,5230],{"dataMmlNode":394},[143,5231],{"dataC":811,"xLinkHref":4943},[128,5233,5234],{"dataMmlNode":141,"transform":815},[143,5235],{"dataC":145,"xLinkHref":4948},[128,5237,5238],{"dataMmlNode":394,"transform":821},[143,5239],{"dataC":836,"xLinkHref":4963},[128,5241,5242],{"dataMmlNode":381,"transform":2133},[143,5243],{"dataC":922,"xLinkHref":4970},[128,5245,5246],{"dataMmlNode":394,"transform":2138},[143,5247],{"dataC":836,"xLinkHref":5140},[128,5249,5251],{"dataMmlNode":394,"transform":5250},"translate(19992.4,0)",[143,5252],{"dataC":3562,"xLinkHref":5079},[128,5254,5256,5261],{"dataMmlNode":378,"transform":5255},"translate(20492.6,0)",[128,5257,5258],{"dataMmlNode":381},[143,5259],{"dataC":1157,"xLinkHref":5260},"#MJX-47-TEX-I-1D465",[128,5262,5263],{"dataMmlNode":381,"transform":1161},[143,5264],{"dataC":934,"xLinkHref":4979},[128,5266,5268],{"dataMmlNode":394,"transform":5267},"translate(21391.6,0) translate(0 -0.5)",[143,5269],{"dataC":1254,"xLinkHref":5270},"#MJX-47-TEX-LO-7D",[16,5272,5273,5274,5347,5348,5433,5434,52],{},"This formula elegantly demonstrates how the error ",[99,5275,5277],{"className":5276,"jax":103},[102],[105,5278,5282,5302],{"style":2657,"xmlns":108,"width":5279,"height":5280,"role":111,"focusable":112,"viewBox":5281,"xmlnsXLink":114},"8.842ex","2.262ex","0 -750 3908.2 1000",[116,5283,5284,5287,5290,5293,5296,5299],{},[119,5285],{"id":5286,"d":778},"MJX-48-TEX-N-28",[119,5288],{"id":5289,"d":1111},"MJX-48-TEX-I-1D451",[119,5291],{"id":5292,"d":1729},"MJX-48-TEX-I-1D458",[119,5294],{"id":5295,"d":1548},"MJX-48-TEX-N-2212",[119,5297],{"id":5298,"d":1390},"MJX-48-TEX-I-1D45C",[119,5300],{"id":5301,"d":791},"MJX-48-TEX-N-29",[128,5303,5304],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,5305,5306,5311,5323,5329,5341],{"dataMmlNode":135},[128,5307,5308],{"dataMmlNode":394},[143,5309],{"dataC":811,"xLinkHref":5310},"#MJX-48-TEX-N-28",[128,5312,5313,5318],{"dataMmlNode":378,"transform":815},[128,5314,5315],{"dataMmlNode":381},[143,5316],{"dataC":1179,"xLinkHref":5317},"#MJX-48-TEX-I-1D451",[128,5319,5320],{"dataMmlNode":381,"transform":1183},[143,5321],{"dataC":1823,"xLinkHref":5322},"#MJX-48-TEX-I-1D458",[128,5324,5326],{"dataMmlNode":394,"transform":5325},"translate(1582.6,0)",[143,5327],{"dataC":1613,"xLinkHref":5328},"#MJX-48-TEX-N-2212",[128,5330,5332,5337],{"dataMmlNode":378,"transform":5331},"translate(2582.8,0)",[128,5333,5334],{"dataMmlNode":381},[143,5335],{"dataC":1404,"xLinkHref":5336},"#MJX-48-TEX-I-1D45C",[128,5338,5339],{"dataMmlNode":381,"transform":1408},[143,5340],{"dataC":1823,"xLinkHref":5322},[128,5342,5344],{"dataMmlNode":394,"transform":5343},"translate(3519.2,0)",[143,5345],{"dataC":836,"xLinkHref":5346},"#MJX-48-TEX-N-29"," is \"back-propagated\" through the weights ",[99,5349,5351],{"className":5350,"jax":103},[102],[105,5352,5353,5379],{"style":3298,"xmlns":108,"width":766,"height":3299,"role":111,"focusable":112,"viewBox":3300,"xmlnsXLink":114},[116,5354,5355,5358,5361,5364,5367,5370,5373,5376],{},[119,5356],{"id":5357,"d":774},"MJX-49-TEX-I-1D464",[119,5359],{"id":5360,"d":778},"MJX-49-TEX-N-28",[119,5362],{"id":5363,"d":363},"MJX-49-TEX-N-32",[119,5365],{"id":5366,"d":359},"MJX-49-TEX-N-2C",[119,5368],{"id":5369,"d":122},"MJX-49-TEX-N-31",[119,5371],{"id":5372,"d":791},"MJX-49-TEX-N-29",[119,5374],{"id":5375,"d":1729},"MJX-49-TEX-I-1D458",[119,5377],{"id":5378,"d":870},"MJX-49-TEX-I-1D457",[128,5380,5381],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,5382,5383],{"dataMmlNode":135},[128,5384,5385,5390,5417],{"dataMmlNode":881},[128,5386,5387],{"dataMmlNode":381},[143,5388],{"dataC":802,"xLinkHref":5389},"#MJX-49-TEX-I-1D464",[128,5391,5392,5397,5402,5407,5412],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,5393,5394],{"dataMmlNode":394},[143,5395],{"dataC":811,"xLinkHref":5396},"#MJX-49-TEX-N-28",[128,5398,5399],{"dataMmlNode":141,"transform":815},[143,5400],{"dataC":411,"xLinkHref":5401},"#MJX-49-TEX-N-32",[128,5403,5404],{"dataMmlNode":394,"transform":821},[143,5405],{"dataC":398,"xLinkHref":5406},"#MJX-49-TEX-N-2C",[128,5408,5409],{"dataMmlNode":141,"transform":827},[143,5410],{"dataC":145,"xLinkHref":5411},"#MJX-49-TEX-N-31",[128,5413,5414],{"dataMmlNode":394,"transform":833},[143,5415],{"dataC":836,"xLinkHref":5416},"#MJX-49-TEX-N-29",[128,5418,5419,5424,5428],{"dataMmlNode":154,"transform":2470,"dataMjxTexclass":156},[128,5420,5421],{"dataMmlNode":381},[143,5422],{"dataC":1823,"xLinkHref":5423},"#MJX-49-TEX-I-1D458",[128,5425,5426],{"dataMmlNode":394,"transform":1827},[143,5427],{"dataC":398,"xLinkHref":5406},[128,5429,5430],{"dataMmlNode":381,"transform":2481},[143,5431],{"dataC":922,"xLinkHref":5432},"#MJX-49-TEX-I-1D457"," to adjust the deeper hidden weights ",[99,5435,5437],{"className":5436,"jax":103},[102],[105,5438,5439,5465],{"style":844,"xmlns":108,"width":766,"height":845,"role":111,"focusable":112,"viewBox":846,"xmlnsXLink":114},[116,5440,5441,5444,5447,5450,5453,5456,5459,5462],{},[119,5442],{"id":5443,"d":774},"MJX-50-TEX-I-1D464",[119,5445],{"id":5446,"d":778},"MJX-50-TEX-N-28",[119,5448],{"id":5449,"d":122},"MJX-50-TEX-N-31",[119,5451],{"id":5452,"d":359},"MJX-50-TEX-N-2C",[119,5454],{"id":5455,"d":126},"MJX-50-TEX-N-30",[119,5457],{"id":5458,"d":791},"MJX-50-TEX-N-29",[119,5460],{"id":5461,"d":870},"MJX-50-TEX-I-1D457",[119,5463],{"id":5464,"d":874},"MJX-50-TEX-I-1D456",[128,5466,5467],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,5468,5469],{"dataMmlNode":135},[128,5470,5471,5476,5503],{"dataMmlNode":881},[128,5472,5473],{"dataMmlNode":381},[143,5474],{"dataC":802,"xLinkHref":5475},"#MJX-50-TEX-I-1D464",[128,5477,5478,5483,5488,5493,5498],{"dataMmlNode":154,"transform":889,"dataMjxTexclass":156},[128,5479,5480],{"dataMmlNode":394},[143,5481],{"dataC":811,"xLinkHref":5482},"#MJX-50-TEX-N-28",[128,5484,5485],{"dataMmlNode":141,"transform":815},[143,5486],{"dataC":145,"xLinkHref":5487},"#MJX-50-TEX-N-31",[128,5489,5490],{"dataMmlNode":394,"transform":821},[143,5491],{"dataC":398,"xLinkHref":5492},"#MJX-50-TEX-N-2C",[128,5494,5495],{"dataMmlNode":141,"transform":827},[143,5496],{"dataC":149,"xLinkHref":5497},"#MJX-50-TEX-N-30",[128,5499,5500],{"dataMmlNode":394,"transform":833},[143,5501],{"dataC":836,"xLinkHref":5502},"#MJX-50-TEX-N-29",[128,5504,5505,5510,5514],{"dataMmlNode":154,"transform":917,"dataMjxTexclass":156},[128,5506,5507],{"dataMmlNode":381},[143,5508],{"dataC":922,"xLinkHref":5509},"#MJX-50-TEX-I-1D457",[128,5511,5512],{"dataMmlNode":394,"transform":926},[143,5513],{"dataC":398,"xLinkHref":5492},[128,5515,5516],{"dataMmlNode":381,"transform":931},[143,5517],{"dataC":934,"xLinkHref":5518},"#MJX-50-TEX-I-1D456",[54,5520,5522],{"id":5521},"enhancing-backpropagation","Enhancing Backpropagation",[16,5524,5525],{},"Standard Backpropagation can be slow. Several techniques are utilized to speed up the learning process:",[25,5527,5528,5534,5540],{},[28,5529,5530,5533],{},[20,5531,5532],{},"Momentum Terms:"," Adds a fraction of the previous weight update to the current one, preventing the algorithm from getting stuck in shallow local minima and smoothing out oscillations in the gradient.",[28,5535,5536,5539],{},[20,5537,5538],{},"Adaptive Learning Rates (Delta-Bar-Delta):"," Adjusts the step size dynamically. If the error is decreasing consistently, the learning rate increases; if the error oscillates, the rate decreases.",[28,5541,5542,5545],{},[20,5543,5544],{},"Quickprop:"," A second-order optimization method that uses a mathematical trick (assuming the error surface is a parabola) to jump directly to the minimum.",[11,5547,5549],{"id":5548},"_6-applications-of-neural-networks","6. Applications of Neural Networks",[16,5551,5552],{},"Neural networks excel in environments where data is noisy, relationships are non-linear, and traditional algorithmic logic fails.",[54,5554,5556],{"id":5555},"general-functional-categories","General Functional Categories",[286,5558,5559,5565,5571,5626],{},[28,5560,5561,5564],{},[20,5562,5563],{},"Clustering (Unsupervised):"," Exploring similarities between data patterns and grouping them. Use cases include data compression and data mining.",[28,5566,5567,5570],{},[20,5568,5569],{},"Classification \u002F Pattern Recognition:"," Assigning an input pattern (e.g., a handwritten symbol or an image) to one of many predefined classes.",[28,5572,5573,5576,5577,5625],{},[20,5574,5575],{},"Function Approximation:"," Finding an estimate of an unknown mathematical function ",[99,5578,5580],{"className":5579,"jax":103},[102],[105,5581,5584,5598],{"style":2657,"xmlns":108,"width":5582,"height":5280,"role":111,"focusable":112,"viewBox":5583,"xmlnsXLink":114},"4.299ex","0 -750 1900 1000",[116,5585,5586,5589,5592,5595],{},[119,5587],{"id":5588,"d":593},"MJX-51-TEX-I-1D453",[119,5590],{"id":5591,"d":778},"MJX-51-TEX-N-28",[119,5593],{"id":5594,"d":1100},"MJX-51-TEX-I-1D465",[119,5596],{"id":5597,"d":791},"MJX-51-TEX-N-29",[128,5599,5600],{"stroke":130,"fill":130,"stroke-width":131,"transform":132},[128,5601,5602,5607,5613,5619],{"dataMmlNode":135},[128,5603,5604],{"dataMmlNode":381},[143,5605],{"dataC":602,"xLinkHref":5606},"#MJX-51-TEX-I-1D453",[128,5608,5610],{"dataMmlNode":394,"transform":5609},"translate(550,0)",[143,5611],{"dataC":811,"xLinkHref":5612},"#MJX-51-TEX-N-28",[128,5614,5616],{"dataMmlNode":381,"transform":5615},"translate(939,0)",[143,5617],{"dataC":1157,"xLinkHref":5618},"#MJX-51-TEX-I-1D465",[128,5620,5622],{"dataMmlNode":394,"transform":5621},"translate(1511,0)",[143,5623],{"dataC":836,"xLinkHref":5624},"#MJX-51-TEX-N-29"," subject to noise. This is heavily used in scientific and engineering disciplines.",[28,5627,5628,5631],{},[20,5629,5630],{},"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.",[54,5633,5635],{"id":5634},"specific-real-world-applications","Specific Real-World Applications",[25,5637,5638,5663,5685,5702],{},[28,5639,5640,5643],{},[20,5641,5642],{},"Medical Diagnosis:",[25,5644,5645,5651,5657],{},[28,5646,5647,5650],{},[738,5648,5649],{},"Input:"," Patient manifestations (symptoms, lab results, blood tests).",[28,5652,5653,5656],{},[738,5654,5655],{},"Output:"," Predicted disease states (e.g., probability of prostate cancer or Hepatitis B).",[28,5658,5659,5662],{},[738,5660,5661],{},"Advantage:"," Circumvents the need for explicit causal rules, which are often impossible to define in complex human biology.",[28,5664,5665,5668],{},[20,5666,5667],{},"Process Control:",[25,5669,5670,5675,5680],{},[28,5671,5672,5674],{},[738,5673,5649],{}," Environmental parameters and sensor readings.",[28,5676,5677,5679],{},[738,5678,5655],{}," Automated control parameters (e.g., adjusting valves, regulating temperature).",[28,5681,5682,5684],{},[738,5683,5661],{}," Learns ill-structured control functions that are too mathematically complex for standard PID controllers.",[28,5686,5687,5690],{},[20,5688,5689],{},"Financial Forecasting:",[25,5691,5692,5697],{},[28,5693,5694,5696],{},[738,5695,5649],{}," Macroeconomic factors (CPI, interest rates) and historical stock quotes.",[28,5698,5699,5701],{},[738,5700,5655],{}," Forecasts of future stock prices or major indices (like the S&P 500).",[28,5703,5704,5707],{},[20,5705,5706],{},"Consumer Credit Evaluation:",[25,5708,5709,5714],{},[28,5710,5711,5713],{},[738,5712,5649],{}," Personal financial metrics (income, debt-to-income ratio, payment history).",[28,5715,5716,5718],{},[738,5717,5655],{}," Risk assessment or credit rating scores.",[11,5720,5722],{"id":5721},"_7-evaluation-of-backpropagation-networks","7. Evaluation of Backpropagation Networks",[16,5724,5725],{},"While Multi-Layer Perceptrons trained with Backpropagation are exceptionally powerful, the architecture has both distinct advantages and inherent limitations.",[54,5727,5729],{"id":5728},"strengths","Strengths",[25,5731,5732,5738,5744,5750],{},[28,5733,5734,5737],{},[20,5735,5736],{},"Great Representation Power:"," The inclusion of non-linear hidden layers allows the network to approximate virtually any continuous function.",[28,5739,5740,5743],{},[20,5741,5742],{},"Wide Practical Applicability:"," Easily adapted to classification, regression, and prediction tasks across countless domains.",[28,5745,5746,5749],{},[20,5747,5748],{},"Easy to Implement:"," The calculus behind backpropagation is complex, but the algorithmic implementation is straightforward matrix multiplication.",[28,5751,5752,5755],{},[20,5753,5754],{},"Good Generalization:"," When trained correctly (often utilizing cross-validation testing), ANNs can make highly accurate predictions on entirely unseen data.",[54,5757,5759],{"id":5758},"limitations-and-problems","Limitations and Problems",[25,5761,5762,5768,5778,5792,5798,5804,5810,5816,5822],{},[28,5763,5764,5767],{},[20,5765,5766],{},"Slow Convergence:"," Gradient descent can require tens of thousands of epochs (passes through the data) to reach an acceptable error rate.",[28,5769,5770,5773,5774,5777],{},[20,5771,5772],{},"The \"Black Box\" Problem:"," It is nearly impossible to inspect the thousands of weights to understand ",[738,5775,5776],{},"why"," the network made a specific decision.",[28,5779,5780,5783,5784,5787,5788,5791],{},[20,5781,5782],{},"Local Minima:"," Gradient descent only guarantees finding a ",[738,5785,5786],{},"local"," minimum error, not necessarily the ",[738,5789,5790],{},"global"," minimum.",[28,5793,5794,5797],{},[20,5795,5796],{},"Representational Limits:"," Not absolutely every conceivable function can be learned easily within standard operational timeframes.",[28,5799,5800,5803],{},[20,5801,5802],{},"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.",[28,5805,5806,5809],{},[20,5807,5808],{},"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.",[28,5811,5812,5815],{},[20,5813,5814],{},"Network Paralysis:"," If weights become too large, activation functions are driven into deep saturation regions. The derivatives become near zero, freezing the network.",[28,5817,5818,5821],{},[20,5819,5820],{},"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.",[28,5823,5824,5827],{},[20,5825,5826],{},"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.",[5829,5830,5831],"style",{},"\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":5833,"searchDepth":5834,"depth":5834,"links":5835},"",2,[5836,5840,5844,5848,5849,5857,5861],{"id":13,"depth":5834,"text":14,"children":5837},[5838],{"id":56,"depth":5839,"text":57},3,{"id":273,"depth":5834,"text":274,"children":5841},[5842,5843],{"id":280,"depth":5839,"text":281},{"id":314,"depth":5839,"text":315},{"id":631,"depth":5834,"text":632,"children":5845},[5846,5847],{"id":641,"depth":5839,"text":642},{"id":659,"depth":5839,"text":660},{"id":687,"depth":5834,"text":688},{"id":714,"depth":5834,"text":715,"children":5850},[5851,5852,5853,5854,5855,5856],{"id":732,"depth":5839,"text":733},{"id":744,"depth":5839,"text":745},{"id":1908,"depth":5839,"text":1909},{"id":2517,"depth":5839,"text":2518},{"id":2857,"depth":5839,"text":2858},{"id":5521,"depth":5839,"text":5522},{"id":5548,"depth":5834,"text":5549,"children":5858},[5859,5860],{"id":5555,"depth":5839,"text":5556},{"id":5634,"depth":5839,"text":5635},{"id":5721,"depth":5834,"text":5722,"children":5862},[5863,5864],{"id":5728,"depth":5839,"text":5729},{"id":5758,"depth":5839,"text":5759},"2026-04-22",false,"md",null,{},true,"\u002Fblog\u002FArtificial-Neural-Networks-and-Backpropagation",{"title":5,"description":5833},{"loc":5871},"blog\u002FArtificial-Neural-Networks-and-Backpropagation","iBQnHDQ-AM8VA9UOWJY-gOrwTJHTXqC-ZjOzB9wODu0",102,{"id":5878,"extension":5879,"meta":5880,"series":5881,"stem":5985,"__hash__":5986},"series\u002Fseries.json","json",{},{"微積分教學":5882,"生活紀錄":5885,"Motor Control":5887,"生活隨筆":5901,"Motor learning":5905,"小兒物治":5923,"中風":5938,"平衡":5949,"Network Communication":5959,"CSA":5966,"機器學習":5972,"小腦":5976},[5883,5884],"微積分隨筆-未完成版","2025數學回顧",[5886],"一個漂流到地球的故事",[5888,5889,5890,5891,5892,5893,5894,5895,5896,5897,5898,5899,5900],"控制自己-Be-water-my-friend","控制自己-Be-water-my-friend（二）","控制自己-Be-water-my-friend（三）","控制自己-Be-water-my-friend（四）","控制自己-Be-water-my-friend（五）","進階控制制制制","周圍理論學派（一）反射理論","周圍理論學派（二）階層理論","中樞理論學派（一）CPG","中樞理論學派（二）Motor-Program","模組理論","系統理論","動態模組理論",[5902,5903,5904],"你好，世界。","根本沒人在乎你的部落格","早安-午安-晚安",[5906,5907,5908,5909,5910,5911,5912,5913,5914,5915,5916,5917,5918,5919,5920,5921,5922],"動作學習（一）介紹","動作學習（二）form-of-learning","動作學習（三）Measurement-of-learning","動作學習（四）理論","動作學習（五）理論-2","動作學習（六）理論-3","動作學習（七）練習方式-1","動作學習（八）練習方式-2","動作學習（九）回饋-1","動作學習（十）回饋-2-擴增性(KR)","動作學習（十一）回饋-3-擴增性(KP)","動作學習（十一）回饋-4-(間隔+物理引導)","動作學習（十二）神經可塑性","動作學習（十二）神經可塑性2","動作學習（十三）臨床應用","動作學習（十四）記憶","動作學習（十五）影響表現的因素",[5924,5925,5926,5927,5928,5929,5930,5931,5932,5933,5934,5935,5936,5937],"腦性痲痺-CP","CP補充（一）","CP—Rood-&-Bobath","CP—Rood-&-Bobath（二）","Motor-Learning","Motor-Learning小兒（二）","Gait-analysis小兒（一）","Gait-analysis小兒（二）","小兒發展（一）","小兒發展（二）","小兒發展（三）","小兒發展（四）","小兒發展（五）","GMFCS",[5939,5940,5941,5942,5943,5944,5945,5946,5947,5948],"腦血管病變（CVA）（中風）(一)","CVA（二）","CVA（三）血管症候群-i","CVA（四）血管症候群-(ii)","CVA（四）","CVA（六）","CVA（七）評估-(i)","CVA（八）評估-(ii)","CVA（九）復健—手部-(i)","CVA跑台（一）",[5950,5951,5952,5953,5954,5955,5956,5957,5958],"平衡與前庭失調（一）","Balance（二）前庭覺-(i)","Balance（三）","Balance（四）評估","Balance（五）復健","Balance（六）功能恢復","Balance（七）前庭障礙","Balance（八）檢查","Balance（九）干預",[5960,5961,5962,5963,5964,5965],"Network-Communication,-Chapter-1","Network-Communication,-Chapter-2","Network-Communication,-Chapter-3","Network-Communication-Chapter-4","Network-Communications,-Chapter-5","Network-Communication,-Chapter-6",[5967,5968,5969,5970,5971],"Week-1-—-Introduction-to-Computer-Systems","Computer-Systems-Architecture-Understanding-Performance","A-Top-Level-View-of-Computer-Function-and-Interconnection","The-Memory-Hierarchy-Understanding-Cache-Memory","Internal-Memory-How-Your-Computer-Remembers-Things",[5973,5974,5975],"機器學習導論","資料前處理與迴歸分析","決策樹",[5977,5978,5979,5980,5981,5982,5983,5984],"小腦（一）","小腦（二）","小腦（三）功能","小腦（四）損傷","小腦（五）各功能障礙","小腦（六）評估","小腦（七）評估(ii)","小腦（八）治療","series","sJucyaG7gC0dv353KrH133RVxHX0BiFnZMLor9gTQ3o",[5988,6007,6027,6046,6063,6080,6095,6112,6130,6152],{"id":5989,"title":5990,"avatar":5991,"banner":5868,"bio":5992,"body":5993,"description":5833,"extension":5867,"meta":5997,"name":5990,"navigation":5870,"path":5998,"seo":5999,"sitemap":6000,"social":6001,"stem":6005,"__hash__":6006},"authors\u002Fauthors\u002Fautomata.md","Automata","\u002Fimages\u002Fuploads\u002Fnier-automata-2b.jpg","一隻吐司天喵，漂浮在銀河星辰中",{"type":8,"value":5994,"toc":5995},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":5996},[],{},"\u002Fauthors\u002Fautomata",{"description":5833},{"loc":5998},{"website":6002,"twitter":6003,"github":6004},"https:\u002F\u002Freurl.cc\u002FWOeM29","https:\u002F\u002Freurl.cc\u002FLnvLEy","https:\u002F\u002Fgithub.com\u002FAutomata-0","authors\u002Fautomata","IkVbO2zA7revgYq624iVWpSZQUyMmWa82tw_EbWXViE",{"id":6008,"title":6009,"avatar":6010,"banner":6011,"bio":6012,"body":6013,"description":5833,"extension":5867,"meta":6017,"name":6018,"navigation":5870,"path":6019,"seo":6020,"sitemap":6021,"social":6022,"stem":6025,"__hash__":6026},"authors\u002Fauthors\u002Fchinono.md","Chinono","\u002Fimages\u002Fuploads\u002F103467998_p0 copy.png","\u002Fimages\u002Fbackground_light.jpg","我不是女生！",{"type":8,"value":6014,"toc":6015},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6016},[],{},"七糯糯","\u002Fauthors\u002Fchinono",{"description":5833},{"loc":6019},{"github":6023,"twitter":5833,"website":6024},"https:\u002F\u002Fgithub.com\u002FChinHongTan","https:\u002F\u002Fchinono.dev","authors\u002Fchinono","jj1J9mFh3InZFL6XtCzGBQ5jPip0EwBDE3mjGvnN6jE",{"id":6028,"title":6029,"avatar":6030,"banner":6031,"bio":6032,"body":6033,"description":5833,"extension":5867,"meta":6037,"name":6038,"navigation":5870,"path":6039,"seo":6040,"sitemap":6041,"social":6042,"stem":6044,"__hash__":6045},"authors\u002Fauthors\u002Fhibiki12141132.md","Hibiki12141132","https:\u002F\u002Favatars.githubusercontent.com\u002Fu\u002F265822020?v=4","\u002Fimages\u002Fuploads\u002F1773978423557-___.jpg","享受著知識強姦大腦的過程 (內文含個人發癲 不要再意)",{"type":8,"value":6034,"toc":6035},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6036},[],{},"HiBiKi","\u002Fauthors\u002Fhibiki12141132",{"description":5833},{"loc":6039},{"github":6043,"twitter":5833},"https:\u002F\u002Fgithub.com\u002FHiBiKi12141132","authors\u002Fhibiki12141132","dbRnKEcYeCH_faD8R7AUmPPcwgc26s_fR4Q_lu4qtA4",{"id":6047,"title":6048,"avatar":6049,"banner":5868,"bio":6050,"body":6051,"description":5833,"extension":5867,"meta":6055,"name":6048,"navigation":5870,"path":6056,"seo":6057,"sitemap":6058,"social":6059,"stem":6061,"__hash__":6062},"authors\u002Fauthors\u002Fmahiro.md","Mahiro","https:\u002F\u002Ftruth.bahamut.com.tw\u002Fs01\u002F202601\u002F2a29b047d341f840b2ce89f7d65b2ba3.JPG","一個致力於逃離新竹的電機系小雜魚",{"type":8,"value":6052,"toc":6053},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6054},[],{},"\u002Fauthors\u002Fmahiro",{"description":5833},{"loc":6056},{"github":6060},"https:\u002F\u002Fgithub.com\u002Fwifekurumi","authors\u002Fmahiro","b435tdWu9eXUf06WroCge0I405cqA0FhLlUUhoPk14k",{"id":6064,"title":6065,"avatar":6066,"banner":5868,"bio":6067,"body":6068,"description":5833,"extension":5867,"meta":6072,"name":6065,"navigation":5870,"path":6073,"seo":6074,"sitemap":6075,"social":6076,"stem":6078,"__hash__":6079},"authors\u002Fauthors\u002Fosborrrrn.md","Osborrrrn","\u002Fimages\u002Fuploads\u002Frectangle_large_type_2_c516437ed713e5de1f7d2dca8a20cd81.jpg","別人笑我太瘋癲，我笑他人看不穿。\n不見五陵豪傑墓，無花無酒鋤就田",{"type":8,"value":6069,"toc":6070},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6071},[],{},"\u002Fauthors\u002Fosborrrrn",{"description":5833},{"loc":6073},{"github":6077},"https:\u002F\u002Fgithub.com\u002FOsborrrrn","authors\u002Fosborrrrn","w6VWZKPUwvXn5i7MKXOpU2Jeqr3BrdTKVCeDOF2jZlU",{"id":6081,"title":6082,"avatar":5868,"banner":5868,"bio":6083,"body":6084,"description":5833,"extension":5867,"meta":6088,"name":6082,"navigation":5870,"path":6089,"seo":6090,"sitemap":6091,"social":6092,"stem":6093,"__hash__":6094},"authors\u002Fauthors\u002F法法.md","法法","123",{"type":8,"value":6085,"toc":6086},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6087},[],{},"\u002Fauthors\u002F法法",{"description":5833},{"loc":6089},{"github":5833},"authors\u002F法法","o5pdVuPCfTmhkDCpvgy4YmAP0CGdvFluPvjhgvQVbsI",{"id":6096,"title":6097,"avatar":6098,"banner":5868,"bio":6099,"body":6100,"description":5833,"extension":5867,"meta":6104,"name":6097,"navigation":5870,"path":6105,"seo":6106,"sitemap":6107,"social":6108,"stem":6110,"__hash__":6111},"authors\u002Fauthors\u002F灰海獅.md","灰海獅","\u002Fimages\u002Fuploads\u002Fimg_3279.jpeg","守夜人",{"type":8,"value":6101,"toc":6102},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6103},[],{},"\u002Fauthors\u002F灰海獅",{"description":5833},{"loc":6105},{"github":6109},"https:\u002F\u002Fgithub.com\u002Fyuiri333","authors\u002F灰海獅","iZoSIFbQdS-6v3LiK1txgxnIMKy-d2CyZXQk9CMua_s",{"id":6113,"title":6114,"avatar":6115,"banner":6116,"bio":6117,"body":6118,"description":5833,"extension":5867,"meta":6122,"name":6114,"navigation":5870,"path":6123,"seo":6124,"sitemap":6125,"social":6126,"stem":6128,"__hash__":6129},"authors\u002Fauthors\u002F花夜.md","花夜","\u002Fimages\u002Fuploads\u002F1772719470518-791_20260218161129.png","\u002Fimages\u002Fuploads\u002Fimg_2446.png","無論你身在何處，我都會在這裡等你",{"type":8,"value":6119,"toc":6120},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6121},[],{},"\u002Fauthors\u002F花夜",{"description":5833},{"loc":6123},{"github":6127,"twitter":5833},"https:\u002F\u002Fgithub.com\u002Fflowernight0709","authors\u002F花夜","a7jeQiF_JkawgYIR-aYSGceJdDP6Z-OWydsICvgSIzs",{"id":6131,"title":6132,"avatar":6133,"banner":6134,"bio":6135,"body":6136,"description":6140,"extension":5867,"meta":6143,"name":6132,"navigation":5870,"path":6144,"seo":6145,"sitemap":6146,"social":6147,"stem":6150,"__hash__":6151},"authors\u002Fauthors\u002F輝月.md","輝月","\u002Fimages\u002Fuploads\u002Ffb_img_1771085634823.jpg","\u002Fimages\u002Fuploads\u002Fimg_1751.jpg","天下布魔好好玩",{"type":8,"value":6137,"toc":6141},[6138],[16,6139,6140],{},"準大學生，目前正在製作TFR模組",{"title":5833,"searchDepth":5834,"depth":5834,"links":6142},[],{},"\u002Fauthors\u002F輝月",{"description":6140},{"loc":6144},{"twitter":6148,"github":6149},"https:\u002F\u002Fx.com\u002Fhuiyue945","https:\u002F\u002Fgithub.com\u002Fhuiyueyea","authors\u002F輝月","koUocBihphDy3453-nAcolM7JJYwI7UMBpVkf1JQrMQ",{"id":6153,"title":6154,"avatar":6155,"banner":5868,"bio":6156,"body":6157,"description":6161,"extension":5867,"meta":6174,"name":6154,"navigation":5870,"path":6175,"seo":6176,"sitemap":6177,"social":6178,"stem":6180,"__hash__":6181},"authors\u002Fauthors\u002F阿西狄亞.md","阿西狄亞","\u002Fimages\u002Fuploads\u002Fimg_20251215_121849_589.jpg","君は実に馬鹿だな",{"type":8,"value":6158,"toc":6172},[6159,6162],[16,6160,6161],{},"我是阿西狄亞，阿西狄亞的阿，阿西狄亞的西，阿西狄亞的狄，阿西狄亞的亞，你可以叫我阿西。",[16,6163,6164,6167,6168,6171],{},[20,6165,6166],{},"我說的所有事情都抱有極度主觀的看法以及意見","，如果你有其他想法，",[20,6169,6170],{},"你是對的","。",{"title":5833,"searchDepth":5834,"depth":5834,"links":6173},[],{},"\u002Fauthors\u002F阿西狄亞",{"description":6161},{"loc":6175},{"github":6179},"https:\u002F\u002Fgithub.com\u002FAcedia0130","authors\u002F阿西狄亞","q5ECEDl8-0Y33tPck0lYZnzPjFdJkrOnBN7HkAO3pls",[],[6184,6193,6202,6211,6220,6229,6238,6247,6256,6267],{"id":5989,"title":5990,"avatar":5991,"banner":5868,"bio":5992,"body":6185,"description":5833,"extension":5867,"meta":6189,"name":5990,"navigation":5870,"path":5998,"seo":6190,"sitemap":6191,"social":6192,"stem":6005,"__hash__":6006},{"type":8,"value":6186,"toc":6187},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6188},[],{},{"description":5833},{"loc":5998},{"website":6002,"twitter":6003,"github":6004},{"id":6008,"title":6009,"avatar":6010,"banner":6011,"bio":6012,"body":6194,"description":5833,"extension":5867,"meta":6198,"name":6018,"navigation":5870,"path":6019,"seo":6199,"sitemap":6200,"social":6201,"stem":6025,"__hash__":6026},{"type":8,"value":6195,"toc":6196},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6197},[],{},{"description":5833},{"loc":6019},{"github":6023,"twitter":5833,"website":6024},{"id":6028,"title":6029,"avatar":6030,"banner":6031,"bio":6032,"body":6203,"description":5833,"extension":5867,"meta":6207,"name":6038,"navigation":5870,"path":6039,"seo":6208,"sitemap":6209,"social":6210,"stem":6044,"__hash__":6045},{"type":8,"value":6204,"toc":6205},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6206},[],{},{"description":5833},{"loc":6039},{"github":6043,"twitter":5833},{"id":6047,"title":6048,"avatar":6049,"banner":5868,"bio":6050,"body":6212,"description":5833,"extension":5867,"meta":6216,"name":6048,"navigation":5870,"path":6056,"seo":6217,"sitemap":6218,"social":6219,"stem":6061,"__hash__":6062},{"type":8,"value":6213,"toc":6214},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6215},[],{},{"description":5833},{"loc":6056},{"github":6060},{"id":6064,"title":6065,"avatar":6066,"banner":5868,"bio":6067,"body":6221,"description":5833,"extension":5867,"meta":6225,"name":6065,"navigation":5870,"path":6073,"seo":6226,"sitemap":6227,"social":6228,"stem":6078,"__hash__":6079},{"type":8,"value":6222,"toc":6223},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6224},[],{},{"description":5833},{"loc":6073},{"github":6077},{"id":6081,"title":6082,"avatar":5868,"banner":5868,"bio":6083,"body":6230,"description":5833,"extension":5867,"meta":6234,"name":6082,"navigation":5870,"path":6089,"seo":6235,"sitemap":6236,"social":6237,"stem":6093,"__hash__":6094},{"type":8,"value":6231,"toc":6232},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6233},[],{},{"description":5833},{"loc":6089},{"github":5833},{"id":6096,"title":6097,"avatar":6098,"banner":5868,"bio":6099,"body":6239,"description":5833,"extension":5867,"meta":6243,"name":6097,"navigation":5870,"path":6105,"seo":6244,"sitemap":6245,"social":6246,"stem":6110,"__hash__":6111},{"type":8,"value":6240,"toc":6241},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6242},[],{},{"description":5833},{"loc":6105},{"github":6109},{"id":6113,"title":6114,"avatar":6115,"banner":6116,"bio":6117,"body":6248,"description":5833,"extension":5867,"meta":6252,"name":6114,"navigation":5870,"path":6123,"seo":6253,"sitemap":6254,"social":6255,"stem":6128,"__hash__":6129},{"type":8,"value":6249,"toc":6250},[],{"title":5833,"searchDepth":5834,"depth":5834,"links":6251},[],{},{"description":5833},{"loc":6123},{"github":6127,"twitter":5833},{"id":6131,"title":6132,"avatar":6133,"banner":6134,"bio":6135,"body":6257,"description":6140,"extension":5867,"meta":6263,"name":6132,"navigation":5870,"path":6144,"seo":6264,"sitemap":6265,"social":6266,"stem":6150,"__hash__":6151},{"type":8,"value":6258,"toc":6261},[6259],[16,6260,6140],{},{"title":5833,"searchDepth":5834,"depth":5834,"links":6262},[],{},{"description":6140},{"loc":6144},{"twitter":6148,"github":6149},{"id":6153,"title":6154,"avatar":6155,"banner":5868,"bio":6156,"body":6268,"description":6161,"extension":5867,"meta":6280,"name":6154,"navigation":5870,"path":6175,"seo":6281,"sitemap":6282,"social":6283,"stem":6180,"__hash__":6181},{"type":8,"value":6269,"toc":6278},[6270,6272],[16,6271,6161],{},[16,6273,6274,6167,6276,6171],{},[20,6275,6166],{},[20,6277,6170],{},{"title":5833,"searchDepth":5834,"depth":5834,"links":6279},[],{},{"description":6161},{"loc":6175},{"github":6179},1776857454591]