[{"data":1,"prerenderedAt":7583},["ShallowReactive",2],{"\u002Fblog\u002FDecision-Trees":3,"post-count":7175,"series-global-data":7176,"authors-all":7286,"series-sidebar-none":7481,"sidebar-authors":7482},{"id":4,"title":5,"author":6,"body":7,"date":7163,"description":756,"draft":7164,"edited_at":7165,"extension":7166,"featured_image":7167,"meta":7168,"navigation":7169,"path":7170,"pinned":7164,"seo":7171,"sitemap":7172,"stem":7173,"tags":7167,"__hash__":7174},"blog\u002Fblog\u002FDecision-Trees.md","Decision Trees","chinono",{"type":8,"value":9,"toc":7136},"minimark",[10,15,19,22,29,32,40,687,690,696,700,703,739,743,746,757,763,979,983,986,991,996,1286,1353,1356,1371,2175,2203,2440,2443,2446,2452,2884,2990,2997,5099,5103,5106,5192,5316,5321,5596,5601,5911,5916,6208,6211,6215,6222,6259,6263,6266,6429,6605,6608,6628,6635,6638,6644,6647,6670,6674,6677,6753,6812,6815,6819,6822,6826,6829,6835,6841,6845,6848,6870,6874,6888,6892,7010,7052,7056,7060,7077,7081,7098,7102,7105,7125,7127,7132],[11,12,14],"h2",{"id":13},"what-is-a-decision-tree","What Is a Decision Tree?",[16,17,18],"p",{},"A decision tree is exactly what it sounds like — a tree-shaped model that makes decisions. 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is one of the best things about decision trees, you can always convert them into human-readable if-then rules.",[16,691,692],{},[25,693],{"alt":694,"src":695},"0.98","https:\u002F\u002Fraw.githubusercontent.com\u002FChinHongTan\u002Fblog\u002Fmain\u002Fpublic\u002Fimages\u002Fuploads\u002F1777359225311-a-describes-the-components-of-a-decision-tree-the-Nodes-represent-the-possible.webp",[11,697,699],{"id":698},"why-decision-trees","Why Decision Trees?",[16,701,702],{},"Decision trees have several properties that make them a solid starting point for classification problems:",[41,704,705,711,717,723,729],{},[44,706,707,710],{},[36,708,709],{},"They handle both discrete and continuous features."," Discrete features like color (red, blue, green) work naturally. For continuous features like temperature, the tree can split on a threshold (e.g., temperature \u003C 30°C vs. ≥ 30°C).",[44,712,713,716],{},[36,714,715],{},"They naturally represent disjunctive (OR) concepts."," Some classification boundaries are complex unions of conditions, but decision trees handle this natively.",[44,718,719,722],{},[36,720,721],{},"They're robust to noise."," Both mislabeled examples and noisy feature values can be tolerated.",[44,724,725,728],{},[36,726,727],{},"They handle missing values."," If a data point doesn't have a value for some feature, the algorithm can still work with it.",[44,730,731,734,735,738],{},[36,732,733],{},"Classification trees"," produce discrete class labels at the leaves, while ",[36,736,737],{},"regression trees"," output continuous real-valued predictions.",[11,740,742],{"id":741},"how-do-we-build-a-decision-tree","How Do We Build a Decision Tree?",[16,744,745],{},"The core algorithm is a top-down, recursive, divide-and-conquer approach. Here's the pseudocode:",[747,748,753],"pre",{"className":749,"code":751,"language":752},[750],"language-text","DTree(examples, features):\n    If all examples belong to one class:\n        return a leaf with that class label\n \n    If no features are left:\n        return a leaf with the majority class\n \n    Pick the \"best\" feature F\n    Create a node for F\n    For each value v of F:\n        Let examples_v = subset of examples where F = v\n        If examples_v is empty:\n            attach a leaf with the majority class of examples\n        Else:\n            attach DTree(examples_v, features - {F})\n \n    Return the tree rooted at this node\n","text",[754,755,751],"code",{"__ignoreMap":756},"",[16,758,759,760],{},"The key question is: ",[36,761,762],{},"how do we pick the \"best\" feature?",[764,765,769,773],"details",{"className":766},[767,768],"info-box","info-box-info",[770,771,772],"summary",{},"Pseudocode Explaination",[774,775,778,783,786,829,834,837,872,877,880,939,944],"div",{"className":776},[777],"info-box-content",[16,779,780],{},[36,781,782],{},"1. Check the Stopping Rules (Base Cases)",[16,784,785],{},"Before trying to split the data, the algorithm checks if it should stop growing the tree:",[41,787,788,802],{},[44,789,790,317,793,796,797],{},[36,791,792],{},"Pure Data:",[754,794,795],{},"If all examples belong to one class:"," If every item in the current dataset belongs to the exact same category (e.g., they are all \"Spam\"), it stops and creates a leaf node predicting that exact category.\n",[41,798,799],{},[44,800,801],{},"> Are all the fruits in the basket apples? No, it’s a mix.",[44,803,804,317,807,810,811,814,815],{},[36,805,806],{},"Out of Features:",[754,808,809],{},"If no features are left:"," If the algorithm has used up all the available features to split on, but there are still mixed classes in the dataset, it stops. It creates a leaf node predicting the ",[36,812,813],{},"majority class"," (whichever category is most common in that remaining subset).\n",[41,816,817],{},[44,818,819,820,824,825,828],{},"Have we run out of questions to ask? No, we still have ",[821,822,823],"em",{},"Size"," and ",[821,826,827],{},"Color",".",[16,830,831],{},[36,832,833],{},"2. Choose the Split (The Divide Step)",[16,835,836],{},"If the algorithm hasn't stopped, it needs to figure out how to divide the data:",[41,838,839,858],{},[44,840,841,844,845],{},[754,842,843],{},"Pick the \"best\" feature F:"," It evaluates all available features and chooses the one that best separates the classes.\n",[41,846,847],{},[44,848,849,850,824,852,854,855,857],{},"The algorithm evaluates ",[821,851,823],{},[821,853,827],{}," to see which question separates the fruits best. Let's say it determines that asking about ",[36,856,823],{}," is the best first move, because it easily separates the grapes from the larger fruits.",[44,859,860,863,864],{},[754,861,862],{},"Create a node for F:"," It makes this chosen feature a decision point (a node) in the tree.\n",[41,865,866],{},[44,867,868,869],{},"It creates the very first decision point (the root node) in our tree: ",[36,870,871],{},"\"What is the Size?\"",[16,873,874],{},[36,875,876],{},"3. Split the Data and Repeat (The Conquer Step)",[16,878,879],{},"Once the best feature is chosen, the algorithm branches out:",[41,881,882,888,899,908],{},[44,883,884,887],{},[754,885,886],{},"For each value v of F:"," It creates a branch for every possible answer to the feature's question (e.g., if the feature is \"Weather\", the branches might be \"Sunny\", \"Rainy\", and \"Cloudy\").",[44,889,890,893,894],{},[754,891,892],{},"Let examples_v = subset...:"," It divides the current dataset, sending the relevant examples down their respective branches.\n",[41,895,896],{},[44,897,898],{},"It gathers all the \"Small\" fruits into a new subset.",[44,900,901,317,904,907],{},[36,902,903],{},"Handle Empty Branches:",[754,905,906],{},"If examples_v is empty:"," If a branch ends up with absolutely no data, the algorithm doesn't crash; it simply caps that branch with a leaf node predicting the majority class of the parent data.",[44,909,910,317,913,916,917,920,921,924,925],{},[36,911,912],{},"Recursion:",[754,914,915],{},"Else: attach DTree(...)"," If the branch ",[821,918,919],{},"does"," have data, the algorithm calls itself to repeat the exact same process on this new, smaller subset of data, using only the features that haven't been used yet (",[754,922,923],{},"features - {F}",").\n",[41,926,927,930],{},[44,928,929],{},"It runs the algorithm again on just the small fruits.",[44,931,932,935,936,828],{},[821,933,934],{},"Recursion kicks in:"," When looking at this \"Small\" subset, it realizes every single fruit here is a Grape! It hits the first stopping rule (If all examples belong to one class) and instantly creates a ",[36,937,938],{},"Leaf: Grape",[16,940,941],{},[36,942,943],{},"4. Return the Result",[41,945,946],{},[44,947,948,951,952],{},[754,949,950],{},"Return the tree rooted at this node:"," Finally, it passes the fully assembled sub-tree back up the chain until the entire decision tree is built.\n",[41,953,954],{},[44,955,956,957],{},"The algorithm stitches all these sub-trees and leaves together. It passes the finished flowchart back to you:\n",[41,958,959,964,969,974],{},[44,960,961],{},[821,962,963],{},"First, check Size.",[44,965,966],{},[821,967,968],{},"If Small ➔ It's a Grape.",[44,970,971],{},[821,972,973],{},"If Big ➔ Check Color.",[44,975,976],{},[821,977,978],{},"If Red\u002FGreen ➔ It's an Apple. If Yellow ➔ It's a Lemon.",[11,980,982],{"id":981},"entropy-and-information-gain","Entropy and Information Gain",[16,984,985],{},"This is where information theory comes in. We want to choose the feature that best separates the data — the one that brings us closest to pure (single-class) subsets.",[987,988,990],"h3",{"id":989},"entropy","Entropy",[16,992,993,995],{},[36,994,990],{}," measures the impurity or disorder of a set. For a binary classification:",[16,997,998],{},[47,999,1001],{"className":1000,"jax":51},[50],[53,1002,1005,1077],{"style":55,"xmlns":56,"width":1003,"height":58,"role":25,"focusable":59,"viewBox":1004,"xmlnsXLink":61},"39.565ex","0 -750 17487.7 1000",[63,1006,1007,1011,1015,1019,1023,1027,1031,1035,1038,1042,1045,1048,1052,1056,1059,1062,1066,1070,1073],{},[66,1008],{"id":1009,"d":1010},"MJX-6-TEX-I-1D438","M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 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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",[66,1043],{"id":1044,"d":113},"MJX-6-TEX-N-29",[66,1046],{"id":1047,"d":97},"MJX-6-TEX-N-3D",[66,1049],{"id":1050,"d":1051},"MJX-6-TEX-N-2212","M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z",[66,1053],{"id":1054,"d":1055},"MJX-6-TEX-N-31","M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",[66,1057],{"id":1058,"d":85},"MJX-6-TEX-N-6C",[66,1060],{"id":1061,"d":89},"MJX-6-TEX-N-6F",[66,1063],{"id":1064,"d":1065},"MJX-6-TEX-N-67","M329 409Q373 453 429 453Q459 453 472 434T485 396Q485 382 476 371T449 360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 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16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z",[147,1078,1079],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1080,1081,1088,1095,1102,1109,1116,1123,1130,1136,1143,1149,1155,1162,1178,1201,1208,1212,1223,1228,1233,1246,1261,1266,1270,1281],{"dataMmlNode":154},[147,1082,1084],{"dataMmlNode":1083},"mi",[159,1085],{"dataC":1086,"xLinkHref":1087},"1D438","#MJX-6-TEX-I-1D438",[147,1089,1091],{"dataMmlNode":1083,"transform":1090},"translate(764,0)",[159,1092],{"dataC":1093,"xLinkHref":1094},"1D45B","#MJX-6-TEX-I-1D45B",[147,1096,1098],{"dataMmlNode":1083,"transform":1097},"translate(1364,0)",[159,1099],{"dataC":1100,"xLinkHref":1101},"1D461","#MJX-6-TEX-I-1D461",[147,1103,1105],{"dataMmlNode":1083,"transform":1104},"translate(1725,0)",[159,1106],{"dataC":1107,"xLinkHref":1108},"1D45F","#MJX-6-TEX-I-1D45F",[147,1110,1112],{"dataMmlNode":1083,"transform":1111},"translate(2176,0)",[159,1113],{"dataC":1114,"xLinkHref":1115},"1D45C","#MJX-6-TEX-I-1D45C",[147,1117,1119],{"dataMmlNode":1083,"transform":1118},"translate(2661,0)",[159,1120],{"dataC":1121,"xLinkHref":1122},"1D45D","#MJX-6-TEX-I-1D45D",[147,1124,1126],{"dataMmlNode":1083,"transform":1125},"translate(3164,0)",[159,1127],{"dataC":1128,"xLinkHref":1129},"1D466","#MJX-6-TEX-I-1D466",[147,1131,1133],{"dataMmlNode":157,"transform":1132},"translate(3654,0)",[159,1134],{"dataC":161,"xLinkHref":1135},"#MJX-6-TEX-N-28",[147,1137,1139],{"dataMmlNode":1083,"transform":1138},"translate(4043,0)",[159,1140],{"dataC":1141,"xLinkHref":1142},"1D446","#MJX-6-TEX-I-1D446",[147,1144,1146],{"dataMmlNode":157,"transform":1145},"translate(4688,0)",[159,1147],{"dataC":234,"xLinkHref":1148},"#MJX-6-TEX-N-29",[147,1150,1152],{"dataMmlNode":157,"transform":1151},"translate(5354.8,0)",[159,1153],{"dataC":204,"xLinkHref":1154},"#MJX-6-TEX-N-3D",[147,1156,1158],{"dataMmlNode":157,"transform":1157},"translate(6410.6,0)",[159,1159],{"dataC":1160,"xLinkHref":1161},"2212","#MJX-6-TEX-N-2212",[147,1163,1166,1170],{"dataMmlNode":1164,"transform":1165},"msub","translate(7188.6,0)",[147,1167,1168],{"dataMmlNode":1083},[159,1169],{"dataC":1121,"xLinkHref":1122},[147,1171,1174],{"dataMmlNode":1172,"transform":1173},"mn","translate(536,-150) scale(0.707)",[159,1175],{"dataC":1176,"xLinkHref":1177},"31","#MJX-6-TEX-N-31",[147,1179,1181,1194],{"dataMmlNode":1164,"transform":1180},"translate(8294.8,0)",[147,1182,1183,1186,1190],{"dataMmlNode":1083},[159,1184],{"dataC":183,"xLinkHref":1185},"#MJX-6-TEX-N-6C",[159,1187],{"dataC":188,"xLinkHref":1188,"transform":1189},"#MJX-6-TEX-N-6F","translate(278,0)",[159,1191],{"dataC":1192,"xLinkHref":1193,"transform":175},"67","#MJX-6-TEX-N-67",[147,1195,1197],{"dataMmlNode":1172,"transform":1196},"translate(1311,-241.4) scale(0.707)",[159,1198],{"dataC":1199,"xLinkHref":1200},"32","#MJX-6-TEX-N-32",[147,1202,1204],{"dataMmlNode":157,"transform":1203},"translate(10009.3,0)",[159,1205],{"dataC":1206,"xLinkHref":1207},"2061","#MJX-6-TEX-N-2061",[147,1209,1210],{"dataMmlNode":157,"transform":1203},[159,1211],{"dataC":161,"xLinkHref":1135},[147,1213,1215,1219],{"dataMmlNode":1164,"transform":1214},"translate(10398.3,0)",[147,1216,1217],{"dataMmlNode":1083},[159,1218],{"dataC":1121,"xLinkHref":1122},[147,1220,1221],{"dataMmlNode":1172,"transform":1173},[159,1222],{"dataC":1176,"xLinkHref":1177},[147,1224,1226],{"dataMmlNode":157,"transform":1225},"translate(11337.9,0)",[159,1227],{"dataC":234,"xLinkHref":1148},[147,1229,1231],{"dataMmlNode":157,"transform":1230},"translate(11949.1,0)",[159,1232],{"dataC":1160,"xLinkHref":1161},[147,1234,1236,1240],{"dataMmlNode":1164,"transform":1235},"translate(12949.3,0)",[147,1237,1238],{"dataMmlNode":1083},[159,1239],{"dataC":1121,"xLinkHref":1122},[147,1241,1242],{"dataMmlNode":1172,"transform":1173},[159,1243],{"dataC":1244,"xLinkHref":1245},"30","#MJX-6-TEX-N-30",[147,1247,1249,1257],{"dataMmlNode":1164,"transform":1248},"translate(14055.5,0)",[147,1250,1251,1253,1255],{"dataMmlNode":1083},[159,1252],{"dataC":183,"xLinkHref":1185},[159,1254],{"dataC":188,"xLinkHref":1188,"transform":1189},[159,1256],{"dataC":1192,"xLinkHref":1193,"transform":175},[147,1258,1259],{"dataMmlNode":1172,"transform":1196},[159,1260],{"dataC":1199,"xLinkHref":1200},[147,1262,1264],{"dataMmlNode":157,"transform":1263},"translate(15770.1,0)",[159,1265],{"dataC":1206,"xLinkHref":1207},[147,1267,1268],{"dataMmlNode":157,"transform":1263},[159,1269],{"dataC":161,"xLinkHref":1135},[147,1271,1273,1277],{"dataMmlNode":1164,"transform":1272},"translate(16159.1,0)",[147,1274,1275],{"dataMmlNode":1083},[159,1276],{"dataC":1121,"xLinkHref":1122},[147,1278,1279],{"dataMmlNode":1172,"transform":1173},[159,1280],{"dataC":1244,"xLinkHref":1245},[147,1282,1284],{"dataMmlNode":157,"transform":1283},"translate(17098.7,0)",[159,1285],{"dataC":234,"xLinkHref":1148},[16,1287,1288,1289,1322,1323,1352],{},"where ",[47,1290,1292],{"className":1291,"jax":51},[50],[53,1293,1298,1306],{"style":1294,"xmlns":56,"width":1295,"height":1296,"role":25,"focusable":59,"viewBox":1297,"xmlnsXLink":61},"vertical-align: -0.439ex;","2.126ex","1.439ex","0 -442 939.6 636",[63,1299,1300,1303],{},[66,1301],{"id":1302,"d":1030},"MJX-7-TEX-I-1D45D",[66,1304],{"id":1305,"d":1055},"MJX-7-TEX-N-31",[147,1307,1308],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1309,1310],{"dataMmlNode":154},[147,1311,1312,1317],{"dataMmlNode":1164},[147,1313,1314],{"dataMmlNode":1083},[159,1315],{"dataC":1121,"xLinkHref":1316},"#MJX-7-TEX-I-1D45D",[147,1318,1319],{"dataMmlNode":1172,"transform":1173},[159,1320],{"dataC":1176,"xLinkHref":1321},"#MJX-7-TEX-N-31"," is the fraction of positive examples and ",[47,1324,1326],{"className":1325,"jax":51},[50],[53,1327,1328,1336],{"style":1294,"xmlns":56,"width":1295,"height":1296,"role":25,"focusable":59,"viewBox":1297,"xmlnsXLink":61},[63,1329,1330,1333],{},[66,1331],{"id":1332,"d":1030},"MJX-8-TEX-I-1D45D",[66,1334],{"id":1335,"d":1076},"MJX-8-TEX-N-30",[147,1337,1338],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1339,1340],{"dataMmlNode":154},[147,1341,1342,1347],{"dataMmlNode":1164},[147,1343,1344],{"dataMmlNode":1083},[159,1345],{"dataC":1121,"xLinkHref":1346},"#MJX-8-TEX-I-1D45D",[147,1348,1349],{"dataMmlNode":1172,"transform":1173},[159,1350],{"dataC":1244,"xLinkHref":1351},"#MJX-8-TEX-N-30"," is the fraction of negatives.",[16,1354,1355],{},"Some intuition:",[41,1357,1358,1364],{},[44,1359,1360,1361,1363],{},"If all examples are the same class → entropy = ",[36,1362,150],{}," (perfectly pure, no uncertainty)",[44,1365,1366,1367,1370],{},"If examples are evenly split (50\u002F50) → entropy = ",[36,1368,1369],{},"1"," (maximum uncertainty)",[774,1372,1374,1377,1441,1703,1752,1758,1763,2002,2029,2143,2146,2150,2161],{"className":1373},[767,768],[16,1375,1376],{},"Let's break down this equation piece by piece.",[1378,1379,1381,1382,824,1411,1440],"h4",{"id":1380},"_1-the-fractions-and","1. The Fractions (",[47,1383,1385],{"className":1384,"jax":51},[50],[53,1386,1387,1395],{"style":1294,"xmlns":56,"width":1295,"height":1296,"role":25,"focusable":59,"viewBox":1297,"xmlnsXLink":61},[63,1388,1389,1392],{},[66,1390],{"id":1391,"d":1030},"MJX-9-TEX-I-1D45D",[66,1393],{"id":1394,"d":1055},"MJX-9-TEX-N-31",[147,1396,1397],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1398,1399],{"dataMmlNode":154},[147,1400,1401,1406],{"dataMmlNode":1164},[147,1402,1403],{"dataMmlNode":1083},[159,1404],{"dataC":1121,"xLinkHref":1405},"#MJX-9-TEX-I-1D45D",[147,1407,1408],{"dataMmlNode":1172,"transform":1173},[159,1409],{"dataC":1176,"xLinkHref":1410},"#MJX-9-TEX-N-31",[47,1412,1414],{"className":1413,"jax":51},[50],[53,1415,1416,1424],{"style":1294,"xmlns":56,"width":1295,"height":1296,"role":25,"focusable":59,"viewBox":1297,"xmlnsXLink":61},[63,1417,1418,1421],{},[66,1419],{"id":1420,"d":1030},"MJX-10-TEX-I-1D45D",[66,1422],{"id":1423,"d":1076},"MJX-10-TEX-N-30",[147,1425,1426],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1427,1428],{"dataMmlNode":154},[147,1429,1430,1435],{"dataMmlNode":1164},[147,1431,1432],{"dataMmlNode":1083},[159,1433],{"dataC":1121,"xLinkHref":1434},"#MJX-10-TEX-I-1D45D",[147,1436,1437],{"dataMmlNode":1172,"transform":1173},[159,1438],{"dataC":1244,"xLinkHref":1439},"#MJX-10-TEX-N-30",")",[41,1442,1443,1542],{},[44,1444,1445,1474,1475,828],{},[47,1446,1448],{"className":1447,"jax":51},[50],[53,1449,1450,1458],{"style":1294,"xmlns":56,"width":1295,"height":1296,"role":25,"focusable":59,"viewBox":1297,"xmlnsXLink":61},[63,1451,1452,1455],{},[66,1453],{"id":1454,"d":1030},"MJX-11-TEX-I-1D45D",[66,1456],{"id":1457,"d":1055},"MJX-11-TEX-N-31",[147,1459,1460],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1461,1462],{"dataMmlNode":154},[147,1463,1464,1469],{"dataMmlNode":1164},[147,1465,1466],{"dataMmlNode":1083},[159,1467],{"dataC":1121,"xLinkHref":1468},"#MJX-11-TEX-I-1D45D",[147,1470,1471],{"dataMmlNode":1172,"transform":1173},[159,1472],{"dataC":1176,"xLinkHref":1473},"#MJX-11-TEX-N-31"," is the fraction of the basket that is Apples. If you have 10 fruits and 7 are Apples, ",[47,1476,1478],{"className":1477,"jax":51},[50],[53,1479,1483,1505],{"style":1294,"xmlns":56,"width":1480,"height":1481,"role":25,"focusable":59,"viewBox":1482,"xmlnsXLink":61},"8.034ex","1.968ex","0 -676 3551.1 870",[63,1484,1485,1488,1491,1494,1497,1501],{},[66,1486],{"id":1487,"d":1030},"MJX-12-TEX-I-1D45D",[66,1489],{"id":1490,"d":1055},"MJX-12-TEX-N-31",[66,1492],{"id":1493,"d":97},"MJX-12-TEX-N-3D",[66,1495],{"id":1496,"d":1076},"MJX-12-TEX-N-30",[66,1498],{"id":1499,"d":1500},"MJX-12-TEX-N-2E","M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z",[66,1502],{"id":1503,"d":1504},"MJX-12-TEX-N-37","M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z",[147,1506,1507],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1508,1509,1521,1527],{"dataMmlNode":154},[147,1510,1511,1516],{"dataMmlNode":1164},[147,1512,1513],{"dataMmlNode":1083},[159,1514],{"dataC":1121,"xLinkHref":1515},"#MJX-12-TEX-I-1D45D",[147,1517,1518],{"dataMmlNode":1172,"transform":1173},[159,1519],{"dataC":1176,"xLinkHref":1520},"#MJX-12-TEX-N-31",[147,1522,1524],{"dataMmlNode":157,"transform":1523},"translate(1217.3,0)",[159,1525],{"dataC":204,"xLinkHref":1526},"#MJX-12-TEX-N-3D",[147,1528,1530,1533,1538],{"dataMmlNode":1172,"transform":1529},"translate(2273.1,0)",[159,1531],{"dataC":1244,"xLinkHref":1532},"#MJX-12-TEX-N-30",[159,1534],{"dataC":1535,"xLinkHref":1536,"transform":1537},"2E","#MJX-12-TEX-N-2E","translate(500,0)",[159,1539],{"dataC":1540,"xLinkHref":1541,"transform":175},"37","#MJX-12-TEX-N-37",[44,1543,1544,1573,1574,1631,1632,317,1635,317,1666,317,1669,317,1700],{},[47,1545,1547],{"className":1546,"jax":51},[50],[53,1548,1549,1557],{"style":1294,"xmlns":56,"width":1295,"height":1296,"role":25,"focusable":59,"viewBox":1297,"xmlnsXLink":61},[63,1550,1551,1554],{},[66,1552],{"id":1553,"d":1030},"MJX-13-TEX-I-1D45D",[66,1555],{"id":1556,"d":1076},"MJX-13-TEX-N-30",[147,1558,1559],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1560,1561],{"dataMmlNode":154},[147,1562,1563,1568],{"dataMmlNode":1164},[147,1564,1565],{"dataMmlNode":1083},[159,1566],{"dataC":1121,"xLinkHref":1567},"#MJX-13-TEX-I-1D45D",[147,1569,1570],{"dataMmlNode":1172,"transform":1173},[159,1571],{"dataC":1244,"xLinkHref":1572},"#MJX-13-TEX-N-30"," is the fraction that is Lemons. In that same basket, 3 are Lemons, so ",[47,1575,1577],{"className":1576,"jax":51},[50],[53,1578,1581,1599],{"style":1294,"xmlns":56,"width":1480,"height":1579,"role":25,"focusable":59,"viewBox":1580,"xmlnsXLink":61},"1.946ex","0 -666 3551.1 860",[63,1582,1583,1586,1589,1592,1595],{},[66,1584],{"id":1585,"d":1030},"MJX-14-TEX-I-1D45D",[66,1587],{"id":1588,"d":1076},"MJX-14-TEX-N-30",[66,1590],{"id":1591,"d":97},"MJX-14-TEX-N-3D",[66,1593],{"id":1594,"d":1500},"MJX-14-TEX-N-2E",[66,1596],{"id":1597,"d":1598},"MJX-14-TEX-N-33","M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z",[147,1600,1601],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1602,1603,1615,1620],{"dataMmlNode":154},[147,1604,1605,1610],{"dataMmlNode":1164},[147,1606,1607],{"dataMmlNode":1083},[159,1608],{"dataC":1121,"xLinkHref":1609},"#MJX-14-TEX-I-1D45D",[147,1611,1612],{"dataMmlNode":1172,"transform":1173},[159,1613],{"dataC":1244,"xLinkHref":1614},"#MJX-14-TEX-N-30",[147,1616,1617],{"dataMmlNode":157,"transform":1523},[159,1618],{"dataC":204,"xLinkHref":1619},"#MJX-14-TEX-N-3D",[147,1621,1622,1624,1627],{"dataMmlNode":1172,"transform":1529},[159,1623],{"dataC":1244,"xLinkHref":1614},[159,1625],{"dataC":1535,"xLinkHref":1626,"transform":1537},"#MJX-14-TEX-N-2E",[159,1628],{"dataC":1629,"xLinkHref":1630,"transform":175},"33","#MJX-14-TEX-N-33",".\n",[821,1633,1634],{},"(Note:",[821,1636,1637],{},[47,1638,1640],{"className":1639,"jax":51},[50],[53,1641,1642,1650],{"style":1294,"xmlns":56,"width":1295,"height":1296,"role":25,"focusable":59,"viewBox":1297,"xmlnsXLink":61},[63,1643,1644,1647],{},[66,1645],{"id":1646,"d":1030},"MJX-15-TEX-I-1D45D",[66,1648],{"id":1649,"d":1055},"MJX-15-TEX-N-31",[147,1651,1652],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1653,1654],{"dataMmlNode":154},[147,1655,1656,1661],{"dataMmlNode":1164},[147,1657,1658],{"dataMmlNode":1083},[159,1659],{"dataC":1121,"xLinkHref":1660},"#MJX-15-TEX-I-1D45D",[147,1662,1663],{"dataMmlNode":1172,"transform":1173},[159,1664],{"dataC":1176,"xLinkHref":1665},"#MJX-15-TEX-N-31",[821,1667,1668],{},"and",[821,1670,1671],{},[47,1672,1674],{"className":1673,"jax":51},[50],[53,1675,1676,1684],{"style":1294,"xmlns":56,"width":1295,"height":1296,"role":25,"focusable":59,"viewBox":1297,"xmlnsXLink":61},[63,1677,1678,1681],{},[66,1679],{"id":1680,"d":1030},"MJX-16-TEX-I-1D45D",[66,1682],{"id":1683,"d":1076},"MJX-16-TEX-N-30",[147,1685,1686],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1687,1688],{"dataMmlNode":154},[147,1689,1690,1695],{"dataMmlNode":1164},[147,1691,1692],{"dataMmlNode":1083},[159,1693],{"dataC":1121,"xLinkHref":1694},"#MJX-16-TEX-I-1D45D",[147,1696,1697],{"dataMmlNode":1172,"transform":1173},[159,1698],{"dataC":1244,"xLinkHref":1699},"#MJX-16-TEX-N-30",[821,1701,1702],{},"must always add up to 1, representing 100% of the basket).",[1378,1704,1706,1707,1440],{"id":1705},"_2-the-logarithm","2. The Logarithm (",[47,1708,1710],{"className":1709,"jax":51},[50],[53,1711,1716,1730],{"style":1712,"xmlns":56,"width":1713,"height":1714,"role":25,"focusable":59,"viewBox":1715,"xmlnsXLink":61},"vertical-align: -0.546ex;","3.879ex","2.116ex","0 -694 1714.6 935.4",[63,1717,1718,1721,1724,1727],{},[66,1719],{"id":1720,"d":85},"MJX-17-TEX-N-6C",[66,1722],{"id":1723,"d":89},"MJX-17-TEX-N-6F",[66,1725],{"id":1726,"d":1065},"MJX-17-TEX-N-67",[66,1728],{"id":1729,"d":1069},"MJX-17-TEX-N-32",[147,1731,1732],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1733,1734],{"dataMmlNode":154},[147,1735,1736,1747],{"dataMmlNode":1164},[147,1737,1738,1741,1744],{"dataMmlNode":1083},[159,1739],{"dataC":183,"xLinkHref":1740},"#MJX-17-TEX-N-6C",[159,1742],{"dataC":188,"xLinkHref":1743,"transform":1189},"#MJX-17-TEX-N-6F",[159,1745],{"dataC":1192,"xLinkHref":1746,"transform":175},"#MJX-17-TEX-N-67",[147,1748,1749],{"dataMmlNode":1172,"transform":1196},[159,1750],{"dataC":1199,"xLinkHref":1751},"#MJX-17-TEX-N-32",[16,1753,1754,1755],{},"Why use a base-2 logarithm? It measures \"bits\" of information. Think of it as asking: ",[821,1756,1757],{},"\"How many yes\u002Fno questions (bits) would I need to ask to figure out the identity of a randomly picked fruit?\"",[16,1759,1760],{},[36,1761,1762],{},"The log of a fraction (any number between 0 and 1) is always a negative number.",[41,1764,1765,1886],{},[44,1766,1767],{},[47,1768,1770],{"className":1769,"jax":51},[50],[53,1771,1774,1816],{"style":55,"xmlns":56,"width":1772,"height":58,"role":25,"focusable":59,"viewBox":1773,"xmlnsXLink":61},"14.439ex","0 -750 6382.1 1000",[63,1775,1776,1779,1782,1785,1788,1791,1794,1797,1800,1804,1807,1810,1813],{},[66,1777],{"id":1778,"d":85},"MJX-18-TEX-N-6C",[66,1780],{"id":1781,"d":89},"MJX-18-TEX-N-6F",[66,1783],{"id":1784,"d":1065},"MJX-18-TEX-N-67",[66,1786],{"id":1787,"d":1069},"MJX-18-TEX-N-32",[66,1789],{"id":1790,"d":756},"MJX-18-TEX-N-2061",[66,1792],{"id":1793,"d":69},"MJX-18-TEX-N-28",[66,1795],{"id":1796,"d":1076},"MJX-18-TEX-N-30",[66,1798],{"id":1799,"d":1500},"MJX-18-TEX-N-2E",[66,1801],{"id":1802,"d":1803},"MJX-18-TEX-N-35","M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z",[66,1805],{"id":1806,"d":113},"MJX-18-TEX-N-29",[66,1808],{"id":1809,"d":97},"MJX-18-TEX-N-3D",[66,1811],{"id":1812,"d":1051},"MJX-18-TEX-N-2212",[66,1814],{"id":1815,"d":1055},"MJX-18-TEX-N-31",[147,1817,1818],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,1819,1820,1838,1844,1849,1862,1868,1874,1880],{"dataMmlNode":154},[147,1821,1822,1833],{"dataMmlNode":1164},[147,1823,1824,1827,1830],{"dataMmlNode":1083},[159,1825],{"dataC":183,"xLinkHref":1826},"#MJX-18-TEX-N-6C",[159,1828],{"dataC":188,"xLinkHref":1829,"transform":1189},"#MJX-18-TEX-N-6F",[159,1831],{"dataC":1192,"xLinkHref":1832,"transform":175},"#MJX-18-TEX-N-67",[147,1834,1835],{"dataMmlNode":1172,"transform":1196},[159,1836],{"dataC":1199,"xLinkHref":1837},"#MJX-18-TEX-N-32",[147,1839,1841],{"dataMmlNode":157,"transform":1840},"translate(1714.6,0)",[159,1842],{"dataC":1206,"xLinkHref":1843},"#MJX-18-TEX-N-2061",[147,1845,1846],{"dataMmlNode":157,"transform":1840},[159,1847],{"dataC":161,"xLinkHref":1848},"#MJX-18-TEX-N-28",[147,1850,1852,1855,1858],{"dataMmlNode":1172,"transform":1851},"translate(2103.6,0)",[159,1853],{"dataC":1244,"xLinkHref":1854},"#MJX-18-TEX-N-30",[159,1856],{"dataC":1535,"xLinkHref":1857,"transform":1537},"#MJX-18-TEX-N-2E",[159,1859],{"dataC":1860,"xLinkHref":1861,"transform":175},"35","#MJX-18-TEX-N-35",[147,1863,1865],{"dataMmlNode":157,"transform":1864},"translate(3381.6,0)",[159,1866],{"dataC":234,"xLinkHref":1867},"#MJX-18-TEX-N-29",[147,1869,1871],{"dataMmlNode":157,"transform":1870},"translate(4048.3,0)",[159,1872],{"dataC":204,"xLinkHref":1873},"#MJX-18-TEX-N-3D",[147,1875,1877],{"dataMmlNode":157,"transform":1876},"translate(5104.1,0)",[159,1878],{"dataC":1160,"xLinkHref":1879},"#MJX-18-TEX-N-2212",[147,1881,1883],{"dataMmlNode":1172,"transform":1882},"translate(5882.1,0)",[159,1884],{"dataC":1176,"xLinkHref":1885},"#MJX-18-TEX-N-31",[44,1887,1888],{},[47,1889,1891],{"className":1890,"jax":51},[50],[53,1892,1895,1933],{"style":55,"xmlns":56,"width":1893,"height":58,"role":25,"focusable":59,"viewBox":1894,"xmlnsXLink":61},"15.57ex","0 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The Negative Sign (",[47,2007,2009],{"className":2008,"jax":51},[50],[53,2010,2015,2020],{"style":2011,"xmlns":56,"width":2012,"height":2013,"role":25,"focusable":59,"viewBox":2014,"xmlnsXLink":61},"vertical-align: -0.186ex;","1.76ex","1.505ex","0 -583 778 665",[63,2016,2017],{},[66,2018],{"id":2019,"d":1051},"MJX-20-TEX-N-2212",[147,2021,2022],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,2023,2024],{"dataMmlNode":154},[147,2025,2026],{"dataMmlNode":157},[159,2027],{"dataC":1160,"xLinkHref":2028},"#MJX-20-TEX-N-2212",[16,2030,2031,2032,2142],{},"Because the logarithm of a fraction gives us a negative number, calculating ",[47,2033,2035],{"className":2034,"jax":51},[50],[53,2036,2039,2072],{"style":55,"xmlns":56,"width":2037,"height":58,"role":25,"focusable":59,"viewBox":2038,"xmlnsXLink":61},"12.656ex","0 -750 5594.1 1000",[63,2040,2041,2044,2047,2051,2054,2057,2060,2063,2066,2069],{},[66,2042],{"id":2043,"d":1030},"MJX-21-TEX-I-1D45D",[66,2045],{"id":2046,"d":1055},"MJX-21-TEX-N-31",[66,2048],{"id":2049,"d":2050},"MJX-21-TEX-N-D7","M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z",[66,2052],{"id":2053,"d":85},"MJX-21-TEX-N-6C",[66,2055],{"id":2056,"d":89},"MJX-21-TEX-N-6F",[66,2058],{"id":2059,"d":1065},"MJX-21-TEX-N-67",[66,2061],{"id":2062,"d":1069},"MJX-21-TEX-N-32",[66,2064],{"id":2065,"d":756},"MJX-21-TEX-N-2061",[66,2067],{"id":2068,"d":69},"MJX-21-TEX-N-28",[66,2070],{"id":2071,"d":113},"MJX-21-TEX-N-29",[147,2073,2074],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,2075,2076,2088,2095,2114,2120,2125,2136],{"dataMmlNode":154},[147,2077,2078,2083],{"dataMmlNode":1164},[147,2079,2080],{"dataMmlNode":1083},[159,2081],{"dataC":1121,"xLinkHref":2082},"#MJX-21-TEX-I-1D45D",[147,2084,2085],{"dataMmlNode":1172,"transform":1173},[159,2086],{"dataC":1176,"xLinkHref":2087},"#MJX-21-TEX-N-31",[147,2089,2091],{"dataMmlNode":157,"transform":2090},"translate(1161.8,0)",[159,2092],{"dataC":2093,"xLinkHref":2094},"D7","#MJX-21-TEX-N-D7",[147,2096,2098,2109],{"dataMmlNode":1164,"transform":2097},"translate(2162,0)",[147,2099,2100,2103,2106],{"dataMmlNode":1083},[159,2101],{"dataC":183,"xLinkHref":2102},"#MJX-21-TEX-N-6C",[159,2104],{"dataC":188,"xLinkHref":2105,"transform":1189},"#MJX-21-TEX-N-6F",[159,2107],{"dataC":1192,"xLinkHref":2108,"transform":175},"#MJX-21-TEX-N-67",[147,2110,2111],{"dataMmlNode":1172,"transform":1196},[159,2112],{"dataC":1199,"xLinkHref":2113},"#MJX-21-TEX-N-32",[147,2115,2117],{"dataMmlNode":157,"transform":2116},"translate(3876.6,0)",[159,2118],{"dataC":1206,"xLinkHref":2119},"#MJX-21-TEX-N-2061",[147,2121,2122],{"dataMmlNode":157,"transform":2116},[159,2123],{"dataC":161,"xLinkHref":2124},"#MJX-21-TEX-N-28",[147,2126,2128,2132],{"dataMmlNode":1164,"transform":2127},"translate(4265.6,0)",[147,2129,2130],{"dataMmlNode":1083},[159,2131],{"dataC":1121,"xLinkHref":2082},[147,2133,2134],{"dataMmlNode":1172,"transform":1173},[159,2135],{"dataC":1176,"xLinkHref":2087},[147,2137,2139],{"dataMmlNode":157,"transform":2138},"translate(5205.1,0)",[159,2140],{"dataC":234,"xLinkHref":2141},"#MJX-21-TEX-N-29"," would result in a negative score. But we want our \"messiness\" score to be a positive number (between 0 and 1). So, we put a negative sign in front of the whole thing to cancel out the negative log and flip the final result to a positive number.",[2144,2145],"hr",{},[1378,2147,2149],{"id":2148},"why-is-this-useful-for-decision-trees","Why is this useful for Decision Trees?",[16,2151,2152,2153,2156,2157,2160],{},"When the algorithm is picking the \"best\" feature to split the data (from step 2 of our pseudocode), it calculates the entropy ",[821,2154,2155],{},"before"," the split, and compares it to the entropy ",[821,2158,2159],{},"after"," the split.",[16,2162,2163,2164,2167,2168,2171,2172,828],{},"If asking \"Is the fruit Big?\" drops the entropy from ",[36,2165,2166],{},"1.0"," (a total mess) down to ",[36,2169,2170],{},"0.2"," (mostly organized), that means it's a fantastic question to ask! This drop in entropy is exactly what we call ",[36,2173,2174],{},"Information Gain",[16,2176,2177,2178,2202],{},"For multi-class problems with ",[47,2179,2181],{"className":2180,"jax":51},[50],[53,2182,2186,2192],{"style":323,"xmlns":56,"width":2183,"height":2184,"role":25,"focusable":59,"viewBox":2185,"xmlnsXLink":61},"0.98ex","1.025ex","0 -442 433 453",[63,2187,2188],{},[66,2189],{"id":2190,"d":2191},"MJX-22-TEX-I-1D450","M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z",[147,2193,2194],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,2195,2196],{"dataMmlNode":154},[147,2197,2198],{"dataMmlNode":1083},[159,2199],{"dataC":2200,"xLinkHref":2201},"1D450","#MJX-22-TEX-I-1D450"," classes, it generalizes to:",[47,2204,2207],{"className":2205,"jax":51,"display":2206},[50],"true",[53,2208,2213,2280],{"style":2209,"xmlns":56,"width":2210,"height":2211,"role":25,"focusable":59,"viewBox":2212,"xmlnsXLink":61},"vertical-align: -2.819ex;","29.68ex","6.354ex","0 -1562.5 13118.3 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scale(0.707)",[147,2374,2375],{"dataMmlNode":1083},[159,2376],{"dataC":2200,"xLinkHref":2377},"#MJX-23-TEX-I-1D450",[147,2379,2381],{"dataMmlNode":157,"transform":2380},"translate(8021.2,0)",[159,2382],{"dataC":1160,"xLinkHref":2383},"#MJX-23-TEX-N-2212",[147,2385,2387,2391],{"dataMmlNode":1164,"transform":2386},"translate(8799.2,0)",[147,2388,2389],{"dataMmlNode":1083},[159,2390],{"dataC":1121,"xLinkHref":2313},[147,2392,2393],{"dataMmlNode":1083,"transform":1173},[159,2394],{"dataC":2357,"xLinkHref":2358},[147,2396,2398,2409],{"dataMmlNode":1164,"transform":2397},"translate(9795.8,0)",[147,2399,2400,2403,2406],{"dataMmlNode":1083},[159,2401],{"dataC":183,"xLinkHref":2402},"#MJX-23-TEX-N-6C",[159,2404],{"dataC":188,"xLinkHref":2405,"transform":1189},"#MJX-23-TEX-N-6F",[159,2407],{"dataC":1192,"xLinkHref":2408,"transform":175},"#MJX-23-TEX-N-67",[147,2410,2411],{"dataMmlNode":1172,"transform":1196},[159,2412],{"dataC":1199,"xLinkHref":2413},"#MJX-23-TEX-N-32",[147,2415,2417],{"dataMmlNode":157,"transform":2416},"translate(11510.4,0)",[159,2418],{"dataC":1206,"xLinkHref":2419},"#MJX-23-TEX-N-2061",[147,2421,2422],{"dataMmlNode":157,"transform":2416},[159,2423],{"dataC":161,"xLinkHref":2323},[147,2425,2427,2431],{"dataMmlNode":1164,"transform":2426},"translate(11899.4,0)",[147,2428,2429],{"dataMmlNode":1083},[159,2430],{"dataC":1121,"xLinkHref":2313},[147,2432,2433],{"dataMmlNode":1083,"transform":1173},[159,2434],{"dataC":2357,"xLinkHref":2358},[147,2436,2438],{"dataMmlNode":157,"transform":2437},"translate(12729.3,0)",[159,2439],{"dataC":234,"xLinkHref":2333},[16,2441,2442],{},"You 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scale(0.707)",[147,3156,3157],{"dataMmlNode":1083},[159,3158],{"dataC":2200,"xLinkHref":3159},"#MJX-29-TEX-I-1D450",[147,3161,3163,3168,3172],{"dataMmlNode":2350,"transform":3162,"dataMjxTexclass":2352},"translate(1089,-285.4) 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Explanation:","\nThis is the exact same equation as the binary one you saw earlier, just generalized so it works for ",[821,3246,3247],{},"any"," number of categories, not just two.",[41,3250,3251,3281,3345],{},[44,3252,3253,3280],{},[36,3254,3255,3279],{},[47,3256,3258],{"className":3257,"jax":51},[50],[53,3259,3263,3269],{"style":2949,"xmlns":56,"width":3260,"height":3261,"role":25,"focusable":59,"viewBox":3262,"xmlnsXLink":61},"1.633ex","1.545ex","0 -683 722 683",[63,3264,3265],{},[66,3266],{"id":3267,"d":3268},"MJX-30-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",[147,3270,3271],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,3272,3273],{"dataMmlNode":154},[147,3274,3275],{"dataMmlNode":1083},[159,3276],{"dataC":3277,"xLinkHref":3278},"3A3","#MJX-30-TEX-N-3A3",":"," This just means \"add everything together.\"",[44,3282,3283,3304,3305,828],{},[36,3284,3285,3279],{},[47,3286,3288],{"className":3287,"jax":51},[50],[53,3289,3290,3295],{"style":323,"xmlns":56,"width":2183,"height":2184,"role":25,"focusable":59,"viewBox":2185,"xmlnsXLink":61},[63,3291,3292],{},[66,3293],{"id":3294,"d":2191},"MJX-31-TEX-I-1D450",[147,3296,3297],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,3298,3299],{"dataMmlNode":154},[147,3300,3301],{"dataMmlNode":1083},[159,3302],{"dataC":2200,"xLinkHref":3303},"#MJX-31-TEX-I-1D450"," The total number of classes. In our previous example (Apples, Lemons, Grapes), ",[47,3306,3308],{"className":3307,"jax":51},[50],[53,3309,3313,3324],{"style":2011,"xmlns":56,"width":3310,"height":3311,"role":25,"focusable":59,"viewBox":3312,"xmlnsXLink":61},"5.128ex","1.69ex","0 -665 2266.6 747",[63,3314,3315,3318,3321],{},[66,3316],{"id":3317,"d":2191},"MJX-32-TEX-I-1D450",[66,3319],{"id":3320,"d":97},"MJX-32-TEX-N-3D",[66,3322],{"id":3323,"d":1598},"MJX-32-TEX-N-33",[147,3325,3326],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,3327,3328,3333,3339],{"dataMmlNode":154},[147,3329,3330],{"dataMmlNode":1083},[159,3331],{"dataC":2200,"xLinkHref":3332},"#MJX-32-TEX-I-1D450",[147,3334,3336],{"dataMmlNode":157,"transform":3335},"translate(710.8,0)",[159,3337],{"dataC":204,"xLinkHref":3338},"#MJX-32-TEX-N-3D",[147,3340,3342],{"dataMmlNode":1172,"transform":3341},"translate(1766.6,0)",[159,3343],{"dataC":1629,"xLinkHref":3344},"#MJX-32-TEX-N-33",[44,3346,3347,3380,3381,828],{},[36,3348,3349,3279],{},[47,3350,3352],{"className":3351,"jax":51},[50],[53,3353,3356,3364],{"style":1294,"xmlns":56,"width":3354,"height":1296,"role":25,"focusable":59,"viewBox":3355,"xmlnsXLink":61},"1.878ex","0 -442 830 636",[63,3357,3358,3361],{},[66,3359],{"id":3360,"d":1030},"MJX-33-TEX-I-1D45D",[66,3362],{"id":3363,"d":2255},"MJX-33-TEX-I-1D456",[147,3365,3366],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,3367,3368],{"dataMmlNode":154},[147,3369,3370,3375],{"dataMmlNode":1164},[147,3371,3372],{"dataMmlNode":1083},[159,3373],{"dataC":1121,"xLinkHref":3374},"#MJX-33-TEX-I-1D45D",[147,3376,3377],{"dataMmlNode":1083,"transform":1173},[159,3378],{"dataC":2357,"xLinkHref":3379},"#MJX-33-TEX-I-1D456"," The fraction of the dataset that belongs to class ",[47,3382,3384],{"className":3383,"jax":51},[50],[53,3385,3389,3394],{"style":323,"xmlns":56,"width":3386,"height":3387,"role":25,"focusable":59,"viewBox":3388,"xmlnsXLink":61},"0.781ex","1.52ex","0 -661 345 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It calculates exactly ",[821,4092,4093],{},"how good"," a specific feature (",[47,4096,4098],{"className":4097,"jax":51},[50],[53,4099,4100,4105],{"style":2949,"xmlns":56,"width":2950,"height":2951,"role":25,"focusable":59,"viewBox":2952,"xmlnsXLink":61},[63,4101,4102],{},[66,4103],{"id":4104,"d":2490},"MJX-37-TEX-I-1D439",[147,4106,4107],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4108,4109],{"dataMmlNode":154},[147,4110,4111],{"dataMmlNode":1083},[159,4112],{"dataC":2610,"xLinkHref":4113},"#MJX-37-TEX-I-1D439",") is at organizing the data.",[41,4116,4117,4238,4269,4425,4534],{},[44,4118,4119,4214,4215,4217,4218,4237],{},[36,4120,4121,3279],{},[47,4122,4124],{"className":4123,"jax":51},[50],[53,4125,4128,4160],{"style":55,"xmlns":56,"width":4126,"height":58,"role":25,"focusable":59,"viewBox":4127,"xmlnsXLink":61},"11.486ex","0 -750 5077 1000",[63,4129,4130,4133,4136,4139,4142,4145,4148,4151,4154,4157],{},[66,4131],{"id":4132,"d":1010},"MJX-38-TEX-I-1D438",[66,4134],{"id":4135,"d":1014},"MJX-38-TEX-I-1D45B",[66,4137],{"id":4138,"d":1018},"MJX-38-TEX-I-1D461",[66,4140],{"id":4141,"d":1022},"MJX-38-TEX-I-1D45F",[66,4143],{"id":4144,"d":1026},"MJX-38-TEX-I-1D45C",[66,4146],{"id":4147,"d":1030},"MJX-38-TEX-I-1D45D",[66,4149],{"id":4150,"d":1034},"MJX-38-TEX-I-1D466",[66,4152],{"id":4153,"d":69},"MJX-38-TEX-N-28",[66,4155],{"id":4156,"d":1041},"MJX-38-TEX-I-1D446",[66,4158],{"id":4159,"d":113},"MJX-38-TEX-N-29",[147,4161,4162],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4163,4164,4169,4174,4179,4184,4189,4194,4199,4204,4209],{"dataMmlNode":154},[147,4165,4166],{"dataMmlNode":1083},[159,4167],{"dataC":1086,"xLinkHref":4168},"#MJX-38-TEX-I-1D438",[147,4170,4171],{"dataMmlNode":1083,"transform":1090},[159,4172],{"dataC":1093,"xLinkHref":4173},"#MJX-38-TEX-I-1D45B",[147,4175,4176],{"dataMmlNode":1083,"transform":1097},[159,4177],{"dataC":1100,"xLinkHref":4178},"#MJX-38-TEX-I-1D461",[147,4180,4181],{"dataMmlNode":1083,"transform":1104},[159,4182],{"dataC":1107,"xLinkHref":4183},"#MJX-38-TEX-I-1D45F",[147,4185,4186],{"dataMmlNode":1083,"transform":1111},[159,4187],{"dataC":1114,"xLinkHref":4188},"#MJX-38-TEX-I-1D45C",[147,4190,4191],{"dataMmlNode":1083,"transform":1118},[159,4192],{"dataC":1121,"xLinkHref":4193},"#MJX-38-TEX-I-1D45D",[147,4195,4196],{"dataMmlNode":1083,"transform":1125},[159,4197],{"dataC":1128,"xLinkHref":4198},"#MJX-38-TEX-I-1D466",[147,4200,4201],{"dataMmlNode":157,"transform":1132},[159,4202],{"dataC":161,"xLinkHref":4203},"#MJX-38-TEX-N-28",[147,4205,4206],{"dataMmlNode":1083,"transform":1138},[159,4207],{"dataC":1141,"xLinkHref":4208},"#MJX-38-TEX-I-1D446",[147,4210,4211],{"dataMmlNode":157,"transform":1145},[159,4212],{"dataC":234,"xLinkHref":4213},"#MJX-38-TEX-N-29"," This is the messiness of your data ",[821,4216,2155],{}," you split it. (",[47,4219,4221],{"className":4220,"jax":51},[50],[53,4222,4223,4228],{"style":2925,"xmlns":56,"width":2926,"height":2927,"role":25,"focusable":59,"viewBox":2928,"xmlnsXLink":61},[63,4224,4225],{},[66,4226],{"id":4227,"d":1041},"MJX-39-TEX-I-1D446",[147,4229,4230],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4231,4232],{"dataMmlNode":154},[147,4233,4234],{"dataMmlNode":1083},[159,4235],{"dataC":1141,"xLinkHref":4236},"#MJX-39-TEX-I-1D446"," stands for the original Set of data).",[44,4239,4240,4262,4263,4265,4266,4268],{},[36,4241,4242,4243,1440],{},"The Subtraction (",[47,4244,4246],{"className":4245,"jax":51},[50],[53,4247,4248,4253],{"style":2011,"xmlns":56,"width":2012,"height":2013,"role":25,"focusable":59,"viewBox":2014,"xmlnsXLink":61},[63,4249,4250],{},[66,4251],{"id":4252,"d":1051},"MJX-40-TEX-N-2212",[147,4254,4255],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4256,4257],{"dataMmlNode":154},[147,4258,4259],{"dataMmlNode":157},[159,4260],{"dataC":1160,"xLinkHref":4261},"#MJX-40-TEX-N-2212",": We are taking the messiness ",[821,4264,2155],{}," the split, and subtracting the messiness ",[821,4267,2159],{}," the split. If the number goes down a lot, our \"Gain\" is high.",[44,4270,4271,4384,4385,4404,4405,4424],{},[36,4272,4273,3279],{},[47,4274,4276],{"className":4275,"jax":51},[50],[53,4277,4280,4315],{"style":55,"xmlns":56,"width":4278,"height":58,"role":25,"focusable":59,"viewBox":4279,"xmlnsXLink":61},"14.338ex","0 -750 6337.6 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This just means \"for every branch we create.\" If the feature ",[47,4386,4388],{"className":4387,"jax":51},[50],[53,4389,4390,4395],{"style":2949,"xmlns":56,"width":2950,"height":2951,"role":25,"focusable":59,"viewBox":2952,"xmlnsXLink":61},[63,4391,4392],{},[66,4393],{"id":4394,"d":2490},"MJX-42-TEX-I-1D439",[147,4396,4397],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4398,4399],{"dataMmlNode":154},[147,4400,4401],{"dataMmlNode":1083},[159,4402],{"dataC":2610,"xLinkHref":4403},"#MJX-42-TEX-I-1D439"," is \"Color\", then 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Weight):"," The straight bars ",[47,4606,4608],{"className":4607,"jax":51},[50],[53,4609,4614,4619],{"style":4610,"xmlns":56,"width":4611,"height":4612,"role":25,"focusable":59,"viewBox":4613,"xmlnsXLink":61},"vertical-align: -0.564ex;","1.258ex","2.26ex","0 -749.5 556 999",[63,4615,4616],{},[66,4617],{"id":4618,"d":2552},"MJX-46-TEX-N-7C",[147,4620,4621],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4622,4623,4628],{"dataMmlNode":154},[147,4624,4625],{"dataMmlNode":157,"transform":2777},[159,4626],{"dataC":2780,"xLinkHref":4627},"#MJX-46-TEX-N-7C",[147,4629,4631],{"dataMmlNode":157,"transform":4630},"translate(278,0) translate(0 -0.5)",[159,4632],{"dataC":2780,"xLinkHref":4627}," mean \"count\". ",[47,4635,4637],{"className":4636,"jax":51},[50],[53,4638,4641,4652],{"style":4610,"xmlns":56,"width":4639,"height":4612,"role":25,"focusable":59,"viewBox":4640,"xmlnsXLink":61},"3.608ex","0 -749.5 1594.9 999",[63,4642,4643,4646,4649],{},[66,4644],{"id":4645,"d":2552},"MJX-47-TEX-N-7C",[66,4647],{"id":4648,"d":1041},"MJX-47-TEX-I-1D446",[66,4650],{"id":4651,"d":2524},"MJX-47-TEX-I-1D463",[147,4653,4654],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4655,4656,4661,4673],{"dataMmlNode":154},[147,4657,4658],{"dataMmlNode":157,"transform":2777},[159,4659],{"dataC":2780,"xLinkHref":4660},"#MJX-47-TEX-N-7C",[147,4662,4663,4668],{"dataMmlNode":1164,"transform":1189},[147,4664,4665],{"dataMmlNode":1083},[159,4666],{"dataC":1141,"xLinkHref":4667},"#MJX-47-TEX-I-1D446",[147,4669,4670],{"dataMmlNode":1083,"transform":2790},[159,4671],{"dataC":2703,"xLinkHref":4672},"#MJX-47-TEX-I-1D463",[147,4674,4675],{"dataMmlNode":157,"transform":2795},[159,4676],{"dataC":2780,"xLinkHref":4660}," is the number of items in this specific branch, and ",[47,4679,4681],{"className":4680,"jax":51},[50],[53,4682,4685,4693],{"style":4610,"xmlns":56,"width":4683,"height":4612,"role":25,"focusable":59,"viewBox":4684,"xmlnsXLink":61},"2.717ex","0 -749.5 1201 999",[63,4686,4687,4690],{},[66,4688],{"id":4689,"d":2552},"MJX-48-TEX-N-7C",[66,4691],{"id":4692,"d":1041},"MJX-48-TEX-I-1D446",[147,4694,4695],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4696,4697,4702,4707],{"dataMmlNode":154},[147,4698,4699],{"dataMmlNode":157,"transform":2777},[159,4700],{"dataC":2780,"xLinkHref":4701},"#MJX-48-TEX-N-7C",[147,4703,4704],{"dataMmlNode":1083,"transform":1189},[159,4705],{"dataC":1141,"xLinkHref":4706},"#MJX-48-TEX-I-1D446",[147,4708,4709],{"dataMmlNode":157,"transform":2811},[159,4710],{"dataC":2780,"xLinkHref":4701}," is the total number of items overall. This creates a fraction (e.g., 4 items out of 10 = ",[47,4713,4715],{"className":4714,"jax":51},[50],[53,4716,4720,4732],{"style":2925,"xmlns":56,"width":4717,"height":4718,"role":25,"focusable":59,"viewBox":4719,"xmlnsXLink":61},"2.891ex","1.581ex","0 -677 1278 699",[63,4721,4722,4725,4728],{},[66,4723],{"id":4724,"d":1076},"MJX-49-TEX-N-30",[66,4726],{"id":4727,"d":1500},"MJX-49-TEX-N-2E",[66,4729],{"id":4730,"d":4731},"MJX-49-TEX-N-34","M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z",[147,4733,4734],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4735,4736],{"dataMmlNode":154},[147,4737,4738,4741,4744],{"dataMmlNode":1172},[159,4739],{"dataC":1244,"xLinkHref":4740},"#MJX-49-TEX-N-30",[159,4742],{"dataC":1535,"xLinkHref":4743,"transform":1537},"#MJX-49-TEX-N-2E",[159,4745],{"dataC":4746,"xLinkHref":4747,"transform":175},"34","#MJX-49-TEX-N-34",").",[16,4750,4751,4754,4755],{},[36,4752,4753],{},"Why do we need the weight?:","\nImagine you have 100 fruits. You ask a terrible question: ",[821,4756,4757],{},"\"Does the fruit have exactly 341 tiny dimples on its skin?\"",[41,4759,4760,4766],{},[44,4761,4762,4765],{},[36,4763,4764],{},"Branch 1 (Yes):"," Only 1 fruit goes here. It happens to be an Apple. The entropy of this branch is 0 (perfectly pure!).",[44,4767,4768,4771],{},[36,4769,4770],{},"Branch 2 (No):"," 99 fruits go here. It's a complete, jumbled mess of Apples, Lemons, and Grapes. The entropy is very high.",[16,4773,4774,4775,4836],{},"If we just averaged the two entropies together, the 1-fruit branch would artificially make the split look really good. By multiplying by ",[47,4776,4778],{"className":4777,"jax":51},[50],[53,4779,4780,4791],{"style":3729,"xmlns":56,"width":4543,"height":3731,"role":25,"focusable":59,"viewBox":4544,"xmlnsXLink":61},[63,4781,4782,4785,4788],{},[66,4783],{"id":4784,"d":2552},"MJX-50-TEX-N-7C",[66,4786],{"id":4787,"d":1041},"MJX-50-TEX-I-1D446",[66,4789],{"id":4790,"d":2524},"MJX-50-TEX-I-1D463",[147,4792,4793],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4794,4795],{"dataMmlNode":154},[147,4796,4797,4820,4834],{"dataMmlNode":2769},[147,4798,4799,4804,4816],{"dataMmlNode":2773,"transform":3986},[147,4800,4801],{"dataMmlNode":157,"transform":2777},[159,4802],{"dataC":2780,"xLinkHref":4803},"#MJX-50-TEX-N-7C",[147,4805,4806,4811],{"dataMmlNode":1164,"transform":1189},[147,4807,4808],{"dataMmlNode":1083},[159,4809],{"dataC":1141,"xLinkHref":4810},"#MJX-50-TEX-I-1D446",[147,4812,4813],{"dataMmlNode":1083,"transform":2790},[159,4814],{"dataC":2703,"xLinkHref":4815},"#MJX-50-TEX-I-1D463",[147,4817,4818],{"dataMmlNode":157,"transform":2795},[159,4819],{"dataC":2780,"xLinkHref":4803},[147,4821,4822,4826,4830],{"dataMmlNode":2773,"transform":4008},[147,4823,4824],{"dataMmlNode":157,"transform":2777},[159,4825],{"dataC":2780,"xLinkHref":4803},[147,4827,4828],{"dataMmlNode":1083,"transform":1189},[159,4829],{"dataC":1141,"xLinkHref":4810},[147,4831,4832],{"dataMmlNode":157,"transform":2811},[159,4833],{"dataC":2780,"xLinkHref":4803},[2815,4835],{"width":4023,"height":2818,"x":2819,"y":2820},", we \"weight\" the branches by their size.",[41,4838,4839,4889],{},[44,4840,4841,4842,4888],{},"Branch 1 gets a weight of ",[47,4843,4845],{"className":4844,"jax":51},[50],[53,4846,4849,4861],{"style":55,"xmlns":56,"width":4847,"height":58,"role":25,"focusable":59,"viewBox":4848,"xmlnsXLink":61},"5.656ex","0 -750 2500 1000",[63,4850,4851,4854,4858],{},[66,4852],{"id":4853,"d":1055},"MJX-51-TEX-N-31",[66,4855],{"id":4856,"d":4857},"MJX-51-TEX-N-2F","M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z",[66,4859],{"id":4860,"d":1076},"MJX-51-TEX-N-30",[147,4862,4863],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4864,4865,4870,4878],{"dataMmlNode":154},[147,4866,4867],{"dataMmlNode":1172},[159,4868],{"dataC":1176,"xLinkHref":4869},"#MJX-51-TEX-N-31",[147,4871,4872],{"dataMmlNode":2350,"dataMjxTexclass":2352,"transform":1537},[147,4873,4874],{"dataMmlNode":157},[159,4875],{"dataC":4876,"xLinkHref":4877},"2F","#MJX-51-TEX-N-2F",[147,4879,4881,4883,4886],{"dataMmlNode":1172,"transform":4880},"translate(1000,0)",[159,4882],{"dataC":1176,"xLinkHref":4869},[159,4884],{"dataC":1244,"xLinkHref":4885,"transform":1537},"#MJX-51-TEX-N-30",[159,4887],{"dataC":1244,"xLinkHref":4885,"transform":4880}," (barely counts).",[44,4890,4891,4892,4944],{},"Branch 2 gets a weight of ",[47,4893,4895],{"className":4894,"jax":51},[50],[53,4896,4899,4914],{"style":55,"xmlns":56,"width":4897,"height":58,"role":25,"focusable":59,"viewBox":4898,"xmlnsXLink":61},"6.787ex","0 -750 3000 1000",[63,4900,4901,4905,4908,4911],{},[66,4902],{"id":4903,"d":4904},"MJX-52-TEX-N-39","M352 287Q304 211 232 211Q154 211 104 270T44 396Q42 412 42 436V444Q42 537 111 606Q171 666 243 666Q245 666 249 666T257 665H261Q273 665 286 663T323 651T370 619T413 560Q456 472 456 334Q456 194 396 97Q361 41 312 10T208 -22Q147 -22 108 7T68 93T121 149Q143 149 158 135T173 96Q173 78 164 65T148 49T135 44L131 43Q131 41 138 37T164 27T206 22H212Q272 22 313 86Q352 142 352 280V287ZM244 248Q292 248 321 297T351 430Q351 508 343 542Q341 552 337 562T323 588T293 615T246 625Q208 625 181 598Q160 576 154 546T147 441Q147 358 152 329T172 282Q197 248 244 248Z",[66,4906],{"id":4907,"d":4857},"MJX-52-TEX-N-2F",[66,4909],{"id":4910,"d":1055},"MJX-52-TEX-N-31",[66,4912],{"id":4913,"d":1076},"MJX-52-TEX-N-30",[147,4915,4916],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4917,4918,4926,4933],{"dataMmlNode":154},[147,4919,4920,4924],{"dataMmlNode":1172},[159,4921],{"dataC":4922,"xLinkHref":4923},"39","#MJX-52-TEX-N-39",[159,4925],{"dataC":4922,"xLinkHref":4923,"transform":1537},[147,4927,4928],{"dataMmlNode":2350,"dataMjxTexclass":2352,"transform":4880},[147,4929,4930],{"dataMmlNode":157},[159,4931],{"dataC":4876,"xLinkHref":4932},"#MJX-52-TEX-N-2F",[147,4934,4936,4939,4942],{"dataMmlNode":1172,"transform":4935},"translate(1500,0)",[159,4937],{"dataC":1176,"xLinkHref":4938},"#MJX-52-TEX-N-31",[159,4940],{"dataC":1244,"xLinkHref":4941,"transform":1537},"#MJX-52-TEX-N-30",[159,4943],{"dataC":1244,"xLinkHref":4941,"transform":4880}," (counts heavily).",[16,4946,4947,4950,5082,5083,5086,5087,828],{},[36,4948,4949],{},"Bringing it all together:",[47,4951,4953],{"className":4952,"jax":51},[50],[53,4954,4959,4993],{"style":4955,"xmlns":56,"width":4956,"height":4957,"role":25,"focusable":59,"viewBox":4958,"xmlnsXLink":61},"vertical-align: -0.464ex;","18.208ex","2.059ex","0 -705 8048 910",[63,4960,4961,4965,4968,4972,4975,4978,4981,4984,4987,4990],{},[66,4962],{"id":4963,"d":4964},"MJX-53-TEX-I-1D43C","M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z",[66,4966],{"id":4967,"d":1014},"MJX-53-TEX-I-1D45B",[66,4969],{"id":4970,"d":4971},"MJX-53-TEX-I-1D453","M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z",[66,4973],{"id":4974,"d":1026},"MJX-53-TEX-I-1D45C",[66,4976],{"id":4977,"d":1022},"MJX-53-TEX-I-1D45F",[66,4979],{"id":4980,"d":3459},"MJX-53-TEX-I-1D45A",[66,4982],{"id":4983,"d":2470},"MJX-53-TEX-I-1D44E",[66,4985],{"id":4986,"d":1018},"MJX-53-TEX-I-1D461",[66,4988],{"id":4989,"d":2255},"MJX-53-TEX-I-1D456",[66,4991],{"id":4992,"d":2466},"MJX-53-TEX-I-1D43A",[147,4994,4995],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,4996,4997,5003,5009,5016,5022,5027,5033,5039,5045,5051,5056,5061,5067,5072,5077],{"dataMmlNode":154},[147,4998,4999],{"dataMmlNode":1083},[159,5000],{"dataC":5001,"xLinkHref":5002},"1D43C","#MJX-53-TEX-I-1D43C",[147,5004,5006],{"dataMmlNode":1083,"transform":5005},"translate(504,0)",[159,5007],{"dataC":1093,"xLinkHref":5008},"#MJX-53-TEX-I-1D45B",[147,5010,5012],{"dataMmlNode":1083,"transform":5011},"translate(1104,0)",[159,5013],{"dataC":5014,"xLinkHref":5015},"1D453","#MJX-53-TEX-I-1D453",[147,5017,5019],{"dataMmlNode":1083,"transform":5018},"translate(1654,0)",[159,5020],{"dataC":1114,"xLinkHref":5021},"#MJX-53-TEX-I-1D45C",[147,5023,5024],{"dataMmlNode":1083,"transform":267},[159,5025],{"dataC":1107,"xLinkHref":5026},"#MJX-53-TEX-I-1D45F",[147,5028,5030],{"dataMmlNode":1083,"transform":5029},"translate(2590,0)",[159,5031],{"dataC":3628,"xLinkHref":5032},"#MJX-53-TEX-I-1D45A",[147,5034,5036],{"dataMmlNode":1083,"transform":5035},"translate(3468,0)",[159,5037],{"dataC":2572,"xLinkHref":5038},"#MJX-53-TEX-I-1D44E",[147,5040,5042],{"dataMmlNode":1083,"transform":5041},"translate(3997,0)",[159,5043],{"dataC":1100,"xLinkHref":5044},"#MJX-53-TEX-I-1D461",[147,5046,5048],{"dataMmlNode":1083,"transform":5047},"translate(4358,0)",[159,5049],{"dataC":2357,"xLinkHref":5050},"#MJX-53-TEX-I-1D456",[147,5052,5054],{"dataMmlNode":1083,"transform":5053},"translate(4703,0)",[159,5055],{"dataC":1114,"xLinkHref":5021},[147,5057,5059],{"dataMmlNode":1083,"transform":5058},"translate(5188,0)",[159,5060],{"dataC":1093,"xLinkHref":5008},[147,5062,5064],{"dataMmlNode":1083,"transform":5063},"translate(5788,0)",[159,5065],{"dataC":2565,"xLinkHref":5066},"#MJX-53-TEX-I-1D43A",[147,5068,5070],{"dataMmlNode":1083,"transform":5069},"translate(6574,0)",[159,5071],{"dataC":2572,"xLinkHref":5038},[147,5073,5075],{"dataMmlNode":1083,"transform":5074},"translate(7103,0)",[159,5076],{"dataC":2357,"xLinkHref":5050},[147,5078,5080],{"dataMmlNode":1083,"transform":5079},"translate(7448,0)",[159,5081],{"dataC":1093,"xLinkHref":5008}," = ",[754,5084,5085],{},"(Messiness of Parent)"," MINUS ",[754,5088,5089],{},"(Weighted Average Messiness of the Children)",[16,5091,5092,5093,5095,5096,5098],{},"When the decision tree is sitting at Step 2 of the pseudocode (\"Pick the best feature\"), it calculates this exact Information Gain equation for ",[821,5094,823],{},", and then for ",[821,5097,827],{},". Whichever feature gives the highest number (meaning it removed the most messiness) is the winner, and becomes the next node in the tree!",[987,5100,5102],{"id":5101},"a-worked-example","A Worked Example",[16,5104,5105],{},"Consider four examples with features (size, color, shape) and a binary class 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1000",[63,5720,5721,5724,5727,5730,5733,5736,5739,5742,5745,5748,5751,5754,5757,5760,5763,5766,5769,5772,5775],{},[66,5722],{"id":5723,"d":2466},"MJX-60-TEX-I-1D43A",[66,5725],{"id":5726,"d":2470},"MJX-60-TEX-I-1D44E",[66,5728],{"id":5729,"d":2255},"MJX-60-TEX-I-1D456",[66,5731],{"id":5732,"d":1014},"MJX-60-TEX-I-1D45B",[66,5734],{"id":5735,"d":97},"MJX-60-TEX-N-3D",[66,5737],{"id":5738,"d":1055},"MJX-60-TEX-N-31",[66,5740],{"id":5741,"d":1051},"MJX-60-TEX-N-2212",[66,5743],{"id":5744,"d":69},"MJX-60-TEX-N-28",[66,5746],{"id":5747,"d":1076},"MJX-60-TEX-N-30",[66,5749],{"id":5750,"d":1500},"MJX-60-TEX-N-2E",[66,5752],{"id":5753,"d":1504},"MJX-60-TEX-N-37",[66,5755],{"id":5756,"d":1803},"MJX-60-TEX-N-35",[66,5758],{"id":5759,"d":2050},"MJX-60-TEX-N-D7",[66,5761],{"id":5762,"d":4904},"MJX-60-TEX-N-39",[66,5764],{"id":5765,"d":5636},"MJX-60-TEX-N-38",[66,5767],{"id":5768,"d":5478},"MJX-60-TEX-N-2B",[66,5770],{"id":5771,"d":1069},"MJX-60-TEX-N-32",[66,5773],{"id":5774,"d":113},"MJX-60-TEX-N-29",[66,5776],{"id":5777,"d":1598},"MJX-60-TEX-N-33",[147,5779,5780],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,5781,5782,5787,5792,5797,5802,5807,5812,5817,5822,5836,5842,5857,5863,5875,5880,5885,5891,5896],{"dataMmlNode":154},[147,5783,5784],{"dataMmlNode":1083},[159,5785],{"dataC":2565,"xLinkHref":5786},"#MJX-60-TEX-I-1D43A",[147,5788,5789],{"dataMmlNode":1083,"transform":2569},[159,5790],{"dataC":2572,"xLinkHref":5791},"#MJX-60-TEX-I-1D44E",[147,5793,5794],{"dataMmlNode":1083,"transform":2576},[159,5795],{"dataC":2357,"xLinkHref":5796},"#MJX-60-TEX-I-1D456",[147,5798,5799],{"dataMmlNode":1083,"transform":2582},[159,5800],{"dataC":1093,"xLinkHref":5801},"#MJX-60-TEX-I-1D45B",[147,5803,5804],{"dataMmlNode":157,"transform":5508},[159,5805],{"dataC":204,"xLinkHref":5806},"#MJX-60-TEX-N-3D",[147,5808,5809],{"dataMmlNode":1172,"transform":5514},[159,5810],{"dataC":1176,"xLinkHref":5811},"#MJX-60-TEX-N-31",[147,5813,5814],{"dataMmlNode":157,"transform":5520},[159,5815],{"dataC":1160,"xLinkHref":5816},"#MJX-60-TEX-N-2212",[147,5818,5819],{"dataMmlNode":157,"transform":5526},[159,5820],{"dataC":161,"xLinkHref":5821},"#MJX-60-TEX-N-28",[147,5823,5824,5827,5830,5833],{"dataMmlNode":1172,"transform":5532},[159,5825],{"dataC":1244,"xLinkHref":5826},"#MJX-60-TEX-N-30",[159,5828],{"dataC":1535,"xLinkHref":5829,"transform":1537},"#MJX-60-TEX-N-2E",[159,5831],{"dataC":1540,"xLinkHref":5832,"transform":175},"#MJX-60-TEX-N-37",[159,5834],{"dataC":1860,"xLinkHref":5835,"transform":1978},"#MJX-60-TEX-N-35",[147,5837,5839],{"dataMmlNode":157,"transform":5838},"translate(7705.2,0)",[159,5840],{"dataC":2093,"xLinkHref":5841},"#MJX-60-TEX-N-D7",[147,5843,5845,5847,5849,5852,5854],{"dataMmlNode":1172,"transform":5844},"translate(8705.4,0)",[159,5846],{"dataC":1244,"xLinkHref":5826},[159,5848],{"dataC":1535,"xLinkHref":5829,"transform":1537},[159,5850],{"dataC":4922,"xLinkHref":5851,"transform":175},"#MJX-60-TEX-N-39",[159,5853],{"dataC":1176,"xLinkHref":5811,"transform":1978},[159,5855],{"dataC":5667,"xLinkHref":5856,"transform":5669},"#MJX-60-TEX-N-38",[147,5858,5860],{"dataMmlNode":157,"transform":5859},"translate(11205.7,0)",[159,5861],{"dataC":5558,"xLinkHref":5862},"#MJX-60-TEX-N-2B",[147,5864,5866,5868,5870,5873],{"dataMmlNode":1172,"transform":5865},"translate(12205.9,0)",[159,5867],{"dataC":1244,"xLinkHref":5826},[159,5869],{"dataC":1535,"xLinkHref":5829,"transform":1537},[159,5871],{"dataC":1199,"xLinkHref":5872,"transform":175},"#MJX-60-TEX-N-32",[159,5874],{"dataC":1860,"xLinkHref":5835,"transform":1978},[147,5876,5878],{"dataMmlNode":157,"transform":5877},"translate(14206.1,0)",[159,5879],{"dataC":2093,"xLinkHref":5841},[147,5881,5883],{"dataMmlNode":1172,"transform":5882},"translate(15206.3,0)",[159,5884],{"dataC":1244,"xLinkHref":5826},[147,5886,5888],{"dataMmlNode":157,"transform":5887},"translate(15706.3,0)",[159,5889],{"dataC":234,"xLinkHref":5890},"#MJX-60-TEX-N-29",[147,5892,5894],{"dataMmlNode":157,"transform":5893},"translate(16373.1,0)",[159,5895],{"dataC":204,"xLinkHref":5806},[147,5897,5899,5901,5903,5906,5908],{"dataMmlNode":1172,"transform":5898},"translate(17428.9,0)",[159,5900],{"dataC":1244,"xLinkHref":5826},[159,5902],{"dataC":1535,"xLinkHref":5829,"transform":1537},[159,5904],{"dataC":1629,"xLinkHref":5905,"transform":175},"#MJX-60-TEX-N-33",[159,5907],{"dataC":1176,"xLinkHref":5811,"transform":1978},[159,5909],{"dataC":1176,"xLinkHref":5811,"transform":5669}," — much better.",[16,5912,5913],{},[36,5914,5915],{},"Splitting on Shape:",[41,5917,5918,5980,6018],{},[44,5919,5920,5921],{},"circle → 2+, 1− → ",[47,5922,5924],{"className":5923,"jax":51},[50],[53,5925,5926,5949],{"style":2011,"xmlns":56,"width":5611,"height":5332,"role":25,"focusable":59,"viewBox":5612,"xmlnsXLink":61},[63,5927,5928,5931,5934,5937,5940,5943,5946],{},[66,5929],{"id":5930,"d":1010},"MJX-61-TEX-I-1D438",[66,5932],{"id":5933,"d":97},"MJX-61-TEX-N-3D",[66,5935],{"id":5936,"d":1076},"MJX-61-TEX-N-30",[66,5938],{"id":5939,"d":1500},"MJX-61-TEX-N-2E",[66,5941],{"id":5942,"d":4904},"MJX-61-TEX-N-39",[66,5944],{"id":5945,"d":1055},"MJX-61-TEX-N-31",[66,5947],{"id":5948,"d":5636},"MJX-61-TEX-N-38",[147,5950,5951],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,5952,5953,5958,5963],{"dataMmlNode":154},[147,5954,5955],{"dataMmlNode":1083},[159,5956],{"dataC":1086,"xLinkHref":5957},"#MJX-61-TEX-I-1D438",[147,5959,5960],{"dataMmlNode":157,"transform":5362},[159,5961],{"dataC":204,"xLinkHref":5962},"#MJX-61-TEX-N-3D",[147,5964,5965,5968,5971,5974,5977],{"dataMmlNode":1172,"transform":5368},[159,5966],{"dataC":1244,"xLinkHref":5967},"#MJX-61-TEX-N-30",[159,5969],{"dataC":1535,"xLinkHref":5970,"transform":1537},"#MJX-61-TEX-N-2E",[159,5972],{"dataC":4922,"xLinkHref":5973,"transform":175},"#MJX-61-TEX-N-39",[159,5975],{"dataC":1176,"xLinkHref":5976,"transform":1978},"#MJX-61-TEX-N-31",[159,5978],{"dataC":5667,"xLinkHref":5979,"transform":5669},"#MJX-61-TEX-N-38",[44,5981,5982,5983],{},"square → 0+, 1− → ",[47,5984,5986],{"className":5985,"jax":51},[50],[53,5987,5988,5999],{"style":2011,"xmlns":56,"width":5678,"height":5332,"role":25,"focusable":59,"viewBox":5679,"xmlnsXLink":61},[63,5989,5990,5993,5996],{},[66,5991],{"id":5992,"d":1010},"MJX-62-TEX-I-1D438",[66,5994],{"id":5995,"d":97},"MJX-62-TEX-N-3D",[66,5997],{"id":5998,"d":1076},"MJX-62-TEX-N-30",[147,6000,6001],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,6002,6003,6008,6013],{"dataMmlNode":154},[147,6004,6005],{"dataMmlNode":1083},[159,6006],{"dataC":1086,"xLinkHref":6007},"#MJX-62-TEX-I-1D438",[147,6009,6010],{"dataMmlNode":157,"transform":5362},[159,6011],{"dataC":204,"xLinkHref":6012},"#MJX-62-TEX-N-3D",[147,6014,6015],{"dataMmlNode":1172,"transform":5368},[159,6016],{"dataC":1244,"xLinkHref":6017},"#MJX-62-TEX-N-30",[44,6019,6020,6207],{},[47,6021,6023],{"className":6022,"jax":51},[50],[53,6024,6025,6084],{"style":55,"xmlns":56,"width":5717,"height":58,"role":25,"focusable":59,"viewBox":5718,"xmlnsXLink":61},[63,6026,6027,6030,6033,6036,6039,6042,6045,6048,6051,6054,6057,6060,6063,6066,6069,6072,6075,6078,6081],{},[66,6028],{"id":6029,"d":2466},"MJX-63-TEX-I-1D43A",[66,6031],{"id":6032,"d":2470},"MJX-63-TEX-I-1D44E",[66,6034],{"id":6035,"d":2255},"MJX-63-TEX-I-1D456",[66,6037],{"id":6038,"d":1014},"MJX-63-TEX-I-1D45B",[66,6040],{"id":6041,"d":97},"MJX-63-TEX-N-3D",[66,6043],{"id":6044,"d":1055},"MJX-63-TEX-N-31",[66,6046],{"id":6047,"d":1051},"MJX-63-TEX-N-2212",[66,6049],{"id":6050,"d":69},"MJX-63-TEX-N-28",[66,6052],{"id":6053,"d":1076},"MJX-63-TEX-N-30",[66,6055],{"id":6056,"d":1500},"MJX-63-TEX-N-2E",[66,6058],{"id":6059,"d":1504},"MJX-63-TEX-N-37",[66,6061],{"id":6062,"d":1803},"MJX-63-TEX-N-35",[66,6064],{"id":6065,"d":2050},"MJX-63-TEX-N-D7",[66,6067],{"id":6068,"d":4904},"MJX-63-TEX-N-39",[66,6070],{"id":6071,"d":5636},"MJX-63-TEX-N-38",[66,6073],{"id":6074,"d":5478},"MJX-63-TEX-N-2B",[66,6076],{"id":6077,"d":1069},"MJX-63-TEX-N-32",[66,6079],{"id":6080,"d":113},"MJX-63-TEX-N-29",[66,6082],{"id":6083,"d":1598},"MJX-63-TEX-N-33",[147,6085,6086],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,6087,6088,6093,6098,6103,6108,6113,6118,6123,6128,6142,6147,6161,6166,6177,6181,6185,6190,6194],{"dataMmlNode":154},[147,6089,6090],{"dataMmlNode":1083},[159,6091],{"dataC":2565,"xLinkHref":6092},"#MJX-63-TEX-I-1D43A",[147,6094,6095],{"dataMmlNode":1083,"transform":2569},[159,6096],{"dataC":2572,"xLinkHref":6097},"#MJX-63-TEX-I-1D44E",[147,6099,6100],{"dataMmlNode":1083,"transform":2576},[159,6101],{"dataC":2357,"xLinkHref":6102},"#MJX-63-TEX-I-1D456",[147,6104,6105],{"dataMmlNode":1083,"transform":2582},[159,6106],{"dataC":1093,"xLinkHref":6107},"#MJX-63-TEX-I-1D45B",[147,6109,6110],{"dataMmlNode":157,"transform":5508},[159,6111],{"dataC":204,"xLinkHref":6112},"#MJX-63-TEX-N-3D",[147,6114,6115],{"dataMmlNode":1172,"transform":5514},[159,6116],{"dataC":1176,"xLinkHref":6117},"#MJX-63-TEX-N-31",[147,6119,6120],{"dataMmlNode":157,"transform":5520},[159,6121],{"dataC":1160,"xLinkHref":6122},"#MJX-63-TEX-N-2212",[147,6124,6125],{"dataMmlNode":157,"transform":5526},[159,6126],{"dataC":161,"xLinkHref":6127},"#MJX-63-TEX-N-28",[147,6129,6130,6133,6136,6139],{"dataMmlNode":1172,"transform":5532},[159,6131],{"dataC":1244,"xLinkHref":6132},"#MJX-63-TEX-N-30",[159,6134],{"dataC":1535,"xLinkHref":6135,"transform":1537},"#MJX-63-TEX-N-2E",[159,6137],{"dataC":1540,"xLinkHref":6138,"transform":175},"#MJX-63-TEX-N-37",[159,6140],{"dataC":1860,"xLinkHref":6141,"transform":1978},"#MJX-63-TEX-N-35",[147,6143,6144],{"dataMmlNode":157,"transform":5838},[159,6145],{"dataC":2093,"xLinkHref":6146},"#MJX-63-TEX-N-D7",[147,6148,6149,6151,6153,6156,6158],{"dataMmlNode":1172,"transform":5844},[159,6150],{"dataC":1244,"xLinkHref":6132},[159,6152],{"dataC":1535,"xLinkHref":6135,"transform":1537},[159,6154],{"dataC":4922,"xLinkHref":6155,"transform":175},"#MJX-63-TEX-N-39",[159,6157],{"dataC":1176,"xLinkHref":6117,"transform":1978},[159,6159],{"dataC":5667,"xLinkHref":6160,"transform":5669},"#MJX-63-TEX-N-38",[147,6162,6163],{"dataMmlNode":157,"transform":5859},[159,6164],{"dataC":5558,"xLinkHref":6165},"#MJX-63-TEX-N-2B",[147,6167,6168,6170,6172,6175],{"dataMmlNode":1172,"transform":5865},[159,6169],{"dataC":1244,"xLinkHref":6132},[159,6171],{"dataC":1535,"xLinkHref":6135,"transform":1537},[159,6173],{"dataC":1199,"xLinkHref":6174,"transform":175},"#MJX-63-TEX-N-32",[159,6176],{"dataC":1860,"xLinkHref":6141,"transform":1978},[147,6178,6179],{"dataMmlNode":157,"transform":5877},[159,6180],{"dataC":2093,"xLinkHref":6146},[147,6182,6183],{"dataMmlNode":1172,"transform":5882},[159,6184],{"dataC":1244,"xLinkHref":6132},[147,6186,6187],{"dataMmlNode":157,"transform":5887},[159,6188],{"dataC":234,"xLinkHref":6189},"#MJX-63-TEX-N-29",[147,6191,6192],{"dataMmlNode":157,"transform":5893},[159,6193],{"dataC":204,"xLinkHref":6112},[147,6195,6196,6198,6200,6203,6205],{"dataMmlNode":1172,"transform":5898},[159,6197],{"dataC":1244,"xLinkHref":6132},[159,619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— tied with color.",[16,6209,6210],{},"So we'd pick either color or shape first (both have gain = 0.311), and definitely not size (gain = 0).",[11,6212,6214],{"id":6213},"the-id3-algorithm","The ID3 Algorithm",[16,6216,6217,6218,6221],{},"The algorithm we've been describing is called ",[36,6219,6220],{},"ID3",", invented by J. Ross Quinlan in 1979. It uses Shannon's information theory (1948) to select features via information gain. Key characteristics:",[41,6223,6224,6231,6237,6244],{},[44,6225,6226,6227,6230],{},"Builds the tree ",[36,6228,6229],{},"top-down"," with no backtracking (the algorithm is stubbornly committed to its choices, greedy algorithm)",[44,6232,6233,6234,6236],{},"Selects the feature with the ",[36,6235,2995],{}," at each step",[44,6238,6239,6240,6243],{},"Runs ",[36,6241,6242],{},"recursively"," on non-leaf branches until all data is classified",[44,6245,6246,6247,6250,6251,6254,6255,6258],{},"Searches the ",[36,6248,6249],{},"entire dataset"," to construct the tree\nLater improvements led to ",[36,6252,6253],{},"C4.5"," (Quinlan, 1993), which handles continuous attributes, missing values, and pruning. Another parallel development was ",[36,6256,6257],{},"CART"," (Classification and Regression Trees) by Breiman and Friedman.",[11,6260,6262],{"id":6261},"a-fun-example-classifying-simpsons-characters","A Fun Example: Classifying Simpsons Characters",[16,6264,6265],{},"Let's say we want to classify Simpsons characters as Male or Female using three features: hair length, weight, and age.",[5107,6267,6268,6286],{},[5110,6269,6270],{},[5113,6271,6272,6275,6278,6281,6284],{},[5116,6273,6274],{},"Person",[5116,6276,6277],{},"Hair Length",[5116,6279,6280],{},"Weight",[5116,6282,6283],{},"Age",[5116,6285,5128],{},[5130,6287,6288,6305,6321,6337,6353,6368,6384,6400,6414],{},[5113,6289,6290,6293,6296,6299,6302],{},[5135,6291,6292],{},"Homer",[5135,6294,6295],{},"0\"",[5135,6297,6298],{},"250",[5135,6300,6301],{},"36",[5135,6303,6304],{},"M",[5113,6306,6307,6310,6313,6316,6318],{},[5135,6308,6309],{},"Marge",[5135,6311,6312],{},"10\"",[5135,6314,6315],{},"150",[5135,6317,4746],{},[5135,6319,6320],{},"F",[5113,6322,6323,6326,6329,6332,6335],{},[5135,6324,6325],{},"Bart",[5135,6327,6328],{},"2\"",[5135,6330,6331],{},"90",[5135,6333,6334],{},"10",[5135,6336,6304],{},[5113,6338,6339,6342,6345,6348,6351],{},[5135,6340,6341],{},"Lisa",[5135,6343,6344],{},"6\"",[5135,6346,6347],{},"78",[5135,6349,6350],{},"8",[5135,6352,6320],{},[5113,6354,6355,6358,6361,6364,6366],{},[5135,6356,6357],{},"Maggie",[5135,6359,6360],{},"4\"",[5135,6362,6363],{},"20",[5135,6365,1369],{},[5135,6367,6320],{},[5113,6369,6370,6373,6376,6379,6382],{},[5135,6371,6372],{},"Abe",[5135,6374,6375],{},"1\"",[5135,6377,6378],{},"170",[5135,6380,6381],{},"70",[5135,6383,6304],{},[5113,6385,6386,6389,6392,6395,6398],{},[5135,6387,6388],{},"Selma",[5135,6390,6391],{},"8\"",[5135,6393,6394],{},"160",[5135,6396,6397],{},"41",[5135,6399,6320],{},[5113,6401,6402,6405,6407,6410,6412],{},[5135,6403,6404],{},"Otto",[5135,6406,6312],{},[5135,6408,6409],{},"180",[5135,6411,5667],{},[5135,6413,6304],{},[5113,6415,6416,6419,6421,6424,6427],{},[5135,6417,6418],{},"Krusty",[5135,6420,6344],{},[5135,6422,6423],{},"200",[5135,6425,6426],{},"45",[5135,6428,6304],{},[16,6430,6431,6432,828],{},"The dataset has 4 females and 5 males, with ",[47,6433,6435],{"className":6434,"jax":51},[50],[53,6436,6439,6499],{"style":55,"xmlns":56,"width":6437,"height":58,"role":25,"focusable":59,"viewBox":6438,"xmlnsXLink":61},"26.67ex","0 -750 11788.2 1000",[63,6440,6441,6444,6447,6450,6453,6456,6459,6462,6465,6468,6471,6474,6477,6481,6484,6487,6490,6493,6496],{},[66,6442],{"id":6443,"d":1010},"MJX-64-TEX-I-1D438",[66,6445],{"id":6446,"d":1014},"MJX-64-TEX-I-1D45B",[66,6448],{"id":6449,"d":1018},"MJX-64-TEX-I-1D461",[66,6451],{"id":6452,"d":1022},"MJX-64-TEX-I-1D45F",[66,6454],{"id":6455,"d":1026},"MJX-64-TEX-I-1D45C",[66,6457],{"id":6458,"d":1030},"MJX-64-TEX-I-1D45D",[66,6460],{"id":6461,"d":1034},"MJX-64-TEX-I-1D466",[66,6463],{"id":6464,"d":69},"MJX-64-TEX-N-28",[66,6466],{"id":6467,"d":4731},"MJX-64-TEX-N-34",[66,6469],{"id":6470,"d":2490},"MJX-64-TEX-I-1D439",[66,6472],{"id":6473,"d":2486},"MJX-64-TEX-N-2C",[66,6475],{"id":6476,"d":1803},"MJX-64-TEX-N-35",[66,6478],{"id":6479,"d":6480},"MJX-64-TEX-I-1D440","M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",[66,6482],{"id":6483,"d":113},"MJX-64-TEX-N-29",[66,6485],{"id":6486,"d":97},"MJX-64-TEX-N-3D",[66,6488],{"id":6489,"d":1076},"MJX-64-TEX-N-30",[66,6491],{"id":6492,"d":1500},"MJX-64-TEX-N-2E",[66,6494],{"id":6495,"d":4904},"MJX-64-TEX-N-39",[66,6497],{"id":6498,"d":1055},"MJX-64-TEX-N-31",[147,6500,6501],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,6502,6503,6508,6513,6518,6523,6528,6533,6538,6543,6548,6554,6560,6566,6573,6579,6585],{"dataMmlNode":154},[147,6504,6505],{"dataMmlNode":1083},[159,6506],{"dataC":1086,"xLinkHref":6507},"#MJX-64-TEX-I-1D438",[147,6509,6510],{"dataMmlNode":1083,"transform":1090},[159,6511],{"dataC":1093,"xLinkHref":6512},"#MJX-64-TEX-I-1D45B",[147,6514,6515],{"dataMmlNode":1083,"transform":1097},[159,6516],{"dataC":1100,"xLinkHref":6517},"#MJX-64-TEX-I-1D461",[147,6519,6520],{"dataMmlNode":1083,"transform":1104},[159,6521],{"dataC":1107,"xLinkHref":6522},"#MJX-64-TEX-I-1D45F",[147,6524,6525],{"dataMmlNode":1083,"transform":1111},[159,6526],{"dataC":1114,"xLinkHref":6527},"#MJX-64-TEX-I-1D45C",[147,6529,6530],{"dataMmlNode":1083,"transform":1118},[159,6531],{"dataC":1121,"xLinkHref":6532},"#MJX-64-TEX-I-1D45D",[147,6534,6535],{"dataMmlNode":1083,"transform":1125},[159,6536],{"dataC":1128,"xLinkHref":6537},"#MJX-64-TEX-I-1D466",[147,6539,6540],{"dataMmlNode":157,"transform":1132},[159,6541],{"dataC":161,"xLinkHref":6542},"#MJX-64-TEX-N-28",[147,6544,6545],{"dataMmlNode":1172,"transform":1138},[159,6546],{"dataC":4746,"xLinkHref":6547},"#MJX-64-TEX-N-34",[147,6549,6551],{"dataMmlNode":1083,"transform":6550},"translate(4543,0)",[159,6552],{"dataC":2610,"xLinkHref":6553},"#MJX-64-TEX-I-1D439",[147,6555,6557],{"dataMmlNode":157,"transform":6556},"translate(5292,0)",[159,6558],{"dataC":2603,"xLinkHref":6559},"#MJX-64-TEX-N-2C",[147,6561,6563],{"dataMmlNode":1172,"transform":6562},"translate(5736.7,0)",[159,6564],{"dataC":1860,"xLinkHref":6565},"#MJX-64-TEX-N-35",[147,6567,6569],{"dataMmlNode":1083,"transform":6568},"translate(6236.7,0)",[159,6570],{"dataC":6571,"xLinkHref":6572},"1D440","#MJX-64-TEX-I-1D440",[147,6574,6576],{"dataMmlNode":157,"transform":6575},"translate(7287.7,0)",[159,6577],{"dataC":234,"xLinkHref":6578},"#MJX-64-TEX-N-29",[147,6580,6582],{"dataMmlNode":157,"transform":6581},"translate(7954.4,0)",[159,6583],{"dataC":204,"xLinkHref":6584},"#MJX-64-TEX-N-3D",[147,6586,6588,6591,6594,6597,6599,6602],{"dataMmlNode":1172,"transform":6587},"translate(9010.2,0)",[159,6589],{"dataC":1244,"xLinkHref":6590},"#MJX-64-TEX-N-30",[159,6592],{"dataC":1535,"xLinkHref":6593,"transform":1537},"#MJX-64-TEX-N-2E",[159,6595],{"dataC":4922,"xLinkHref":6596,"transform":175},"#MJX-64-TEX-N-39",[159,6598],{"dataC":4922,"xLinkHref":6596,"transform":1978},[159,6600],{"dataC":1176,"xLinkHref":6601,"transform":5669},"#MJX-64-TEX-N-31",[159,6603],{"dataC":1176,"xLinkHref":6601,"transform":6604},"translate(2278,0)",[16,6606,6607],{},"We compute the information gain for splitting on each feature:",[41,6609,6610,6616,6622],{},[44,6611,6612,6615],{},[36,6613,6614],{},"Hair Length ≤ 5?"," → Gain = 0.0911",[44,6617,6618,6621],{},[36,6619,6620],{},"Weight ≤ 160?"," → Gain = 0.5900 ← Winner!",[44,6623,6624,6627],{},[36,6625,6626],{},"Age ≤ 40?"," → Gain = 0.0183\nWeight gives us the best split. Everyone over 160 lbs turns out to be male (entropy = 0 on that branch — done!). But the ≤160 group still has a mix (4F, 1M), so we recurse.",[16,6629,6630,6631,6634],{},"On the second split, we find that ",[36,6632,6633],{},"Hair Length ≤ 2"," perfectly separates the remaining group: Bart (short hair, male) vs. the females (longer hair).",[16,6636,6637],{},"The final tree:",[747,6639,6642],{"className":6640,"code":6641,"language":752},[750],"            Weight ≤ 160?\n            \u002F           \\\n          yes            no\n          \u002F               \\\n   Hair Length ≤ 2?      Male\n      \u002F        \\\n    yes         no\n    \u002F             \\\n  Male          Female\n",[754,6643,6641],{"__ignoreMap":756},[16,6645,6646],{},"And as rules:",[41,6648,6649,6655,6660],{},[44,6650,6651,6652],{},"If Weight > 160 → ",[36,6653,6654],{},"Male",[44,6656,6657,6658],{},"Else if Hair Length ≤ 2 → ",[36,6659,6654],{},[44,6661,6662,6663,6666,6667,6669],{},"Else → ",[36,6664,6665],{},"Female","\nNow if Comic Book Guy shows up (Hair: 8\", Weight: 290, Age: 38), the tree says: Weight 290 > 160 → ",[36,6668,6654],{},". Correct!",[11,6671,6673],{"id":6672},"the-overfitting-problem","The Overfitting Problem",[16,6675,6676],{},"Here's the catch: a tree that perfectly classifies every training example isn't necessarily a good tree.",[16,6678,6679,6682,6683,6707,6708,6744,6745,6748,6749,6752],{},[36,6680,6681],{},"Overfitting"," happens when a model fits the training data too well — including its noise and quirks — and performs poorly on new, unseen data. Formally, hypothesis ",[47,6684,6686],{"className":6685,"jax":51},[50],[53,6687,6691,6697],{"style":323,"xmlns":56,"width":6688,"height":6689,"role":25,"focusable":59,"viewBox":6690,"xmlnsXLink":61},"1.303ex","1.595ex","0 -694 576 705",[63,6692,6693],{},[66,6694],{"id":6695,"d":6696},"MJX-65-TEX-I-210E","M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z",[147,6698,6699],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,6700,6701],{"dataMmlNode":154},[147,6702,6703],{"dataMmlNode":1083},[159,6704],{"dataC":6705,"xLinkHref":6706},"210E","#MJX-65-TEX-I-210E"," overfits if there exists another hypothesis ",[47,6709,6711],{"className":6710,"jax":51},[50],[53,6712,6716,6725],{"style":323,"xmlns":56,"width":6713,"height":6714,"role":25,"focusable":59,"viewBox":6715,"xmlnsXLink":61},"1.931ex","1.742ex","0 -759 853.5 770",[63,6717,6718,6721],{},[66,6719],{"id":6720,"d":6696},"MJX-66-TEX-I-210E",[66,6722],{"id":6723,"d":6724},"MJX-66-TEX-V-2032","M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z",[147,6726,6727],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,6728,6729],{"dataMmlNode":154},[147,6730,6732,6737],{"dataMmlNode":6731},"msup",[147,6733,6734],{"dataMmlNode":1083},[159,6735],{"dataC":6705,"xLinkHref":6736},"#MJX-66-TEX-I-210E",[147,6738,6740],{"dataMmlNode":157,"transform":6739},"translate(609,363) scale(0.707)",[159,6741],{"dataC":6742,"xLinkHref":6743},"2032","#MJX-66-TEX-V-2032"," that has ",[821,6746,6747],{},"higher"," training error but ",[821,6750,6751],{},"lower"," test error.",[16,6754,6755,6756,6807,6808,6811],{},"Think of it this way: if you measure 10 data points for Ohm's Law (",[47,6757,6759],{"className":6758,"jax":51},[50],[53,6760,6764,6779],{"style":2011,"xmlns":56,"width":6761,"height":6762,"role":25,"focusable":59,"viewBox":6763,"xmlnsXLink":61},"7.614ex","1.731ex","0 -683 3365.6 765",[63,6765,6766,6769,6772,6775],{},[66,6767],{"id":6768,"d":2532},"MJX-67-TEX-I-1D449",[66,6770],{"id":6771,"d":97},"MJX-67-TEX-N-3D",[66,6773],{"id":6774,"d":4964},"MJX-67-TEX-I-1D43C",[66,6776],{"id":6777,"d":6778},"MJX-67-TEX-I-1D445","M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z",[147,6780,6781],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,6782,6783,6788,6794,6800],{"dataMmlNode":154},[147,6784,6785],{"dataMmlNode":1083},[159,6786],{"dataC":2717,"xLinkHref":6787},"#MJX-67-TEX-I-1D449",[147,6789,6791],{"dataMmlNode":157,"transform":6790},"translate(1046.8,0)",[159,6792],{"dataC":204,"xLinkHref":6793},"#MJX-67-TEX-N-3D",[147,6795,6797],{"dataMmlNode":1083,"transform":6796},"translate(2102.6,0)",[159,6798],{"dataC":5001,"xLinkHref":6799},"#MJX-67-TEX-I-1D43C",[147,6801,6803],{"dataMmlNode":1083,"transform":6802},"translate(2606.6,0)",[159,6804],{"dataC":6805,"xLinkHref":6806},"1D445","#MJX-67-TEX-I-1D445",") and fit a 9th-degree polynomial, you'll pass through every point perfectly. But that wiggly curve is ",[821,6809,6810],{},"way"," worse at predicting new measurements than a simple straight line that misses a few points.",[16,6813,6814],{},"The same thing happens with decision trees — noise in the data can cause the algorithm to create extra branches that capture randomness rather than real patterns.",[987,6816,6818],{"id":6817},"how-noise-causes-overfitting","How Noise Causes Overfitting",[16,6820,6821],{},"Imagine our color\u002Fshape example, and one training instance gets mislabeled. The tree might need to add extra splits on irrelevant features (like size) just to account for that one noisy data point. The result is a bigger, more complex tree that generalizes poorly.",[11,6823,6825],{"id":6824},"pruning-the-cure-for-overfitting","Pruning: The Cure for Overfitting",[16,6827,6828],{},"There are two main strategies:",[16,6830,6831,6834],{},[36,6832,6833],{},"Prepruning"," — Stop growing the tree early, before it perfectly fits the training data. For example, stop when a node has too few examples to split reliably.",[16,6836,6837,6840],{},[36,6838,6839],{},"Postpruning"," — Grow the full tree first, then cut back branches that don't help generalization. This is generally more effective since it's hard to know when to stop growing.",[987,6842,6844],{"id":6843},"reduced-error-pruning","Reduced Error Pruning",[16,6846,6847],{},"A common postpruning method:",[6849,6850,6851,6854,6857,6860,6863],"ol",{},[44,6852,6853],{},"Split your data into a \"grow\" set and a \"validation\" set",[44,6855,6856],{},"Build a complete tree from the grow set",[44,6858,6859],{},"For each internal node, try replacing its subtree with a leaf (majority class)",[44,6861,6862],{},"If the replacement improves accuracy on the validation set, make it permanent",[44,6864,6865,6866,6869],{},"Repeat until no more pruning helps\nThe downside is that you \"waste\" some training data on validation. An alternative is to run multiple trials with different random splits, average the resulting tree complexity, then grow a final tree from ",[821,6867,6868],{},"all"," the data up to that average complexity.",[987,6871,6873],{"id":6872},"other-pruning-approaches","Other Pruning Approaches",[41,6875,6876,6882],{},[44,6877,6878,6881],{},[36,6879,6880],{},"Statistical tests"," — Check whether the pattern at a node is statistically significant or just random chance",[44,6883,6884,6887],{},[36,6885,6886],{},"Minimum Description Length (MDL)"," — Ask whether the extra tree complexity is worth it compared to just memorizing the exceptions",[11,6889,6891],{"id":6890},"computational-complexity","Computational Complexity",[16,6893,6894,6895,6965,6966,6987,6988,7009],{},"In the worst case (a complete tree testing every feature on every path), building a decision tree takes ",[47,6896,6898],{"className":6897,"jax":51},[50],[53,6899,6903,6924],{"style":55,"xmlns":56,"width":6900,"height":6901,"role":25,"focusable":59,"viewBox":6902,"xmlnsXLink":61},"7.818ex","2.452ex","0 -833.9 3455.6 1083.9",[63,6904,6905,6909,6912,6915,6918,6921],{},[66,6906],{"id":6907,"d":6908},"MJX-68-TEX-I-1D442","M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435ZM637 476Q637 565 591 615T476 665Q396 665 322 605Q242 542 200 428T157 216Q157 126 200 73T314 19Q404 19 485 98T608 313Q637 408 637 476Z",[66,6910],{"id":6911,"d":69},"MJX-68-TEX-N-28",[66,6913],{"id":6914,"d":1014},"MJX-68-TEX-I-1D45B",[66,6916],{"id":6917,"d":3459},"MJX-68-TEX-I-1D45A",[66,6919],{"id":6920,"d":1069},"MJX-68-TEX-N-32",[66,6922],{"id":6923,"d":113},"MJX-68-TEX-N-29",[147,6925,6926],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,6927,6928,6934,6940,6945,6959],{"dataMmlNode":154},[147,6929,6930],{"dataMmlNode":1083},[159,6931],{"dataC":6932,"xLinkHref":6933},"1D442","#MJX-68-TEX-I-1D442",[147,6935,6937],{"dataMmlNode":157,"transform":6936},"translate(763,0)",[159,6938],{"dataC":161,"xLinkHref":6939},"#MJX-68-TEX-N-28",[147,6941,6942],{"dataMmlNode":1083,"transform":2714},[159,6943],{"dataC":1093,"xLinkHref":6944},"#MJX-68-TEX-I-1D45B",[147,6946,6948,6953],{"dataMmlNode":6731,"transform":6947},"translate(1752,0)",[147,6949,6950],{"dataMmlNode":1083},[159,6951],{"dataC":3628,"xLinkHref":6952},"#MJX-68-TEX-I-1D45A",[147,6954,6956],{"dataMmlNode":1172,"transform":6955},"translate(911,363) scale(0.707)",[159,6957],{"dataC":1199,"xLinkHref":6958},"#MJX-68-TEX-N-32",[147,6960,6962],{"dataMmlNode":157,"transform":6961},"translate(3066.6,0)",[159,6963],{"dataC":234,"xLinkHref":6964},"#MJX-68-TEX-N-29"," time, where ",[47,6967,6969],{"className":6968,"jax":51},[50],[53,6970,6973,6978],{"style":323,"xmlns":56,"width":6971,"height":2184,"role":25,"focusable":59,"viewBox":6972,"xmlnsXLink":61},"1.357ex","0 -442 600 453",[63,6974,6975],{},[66,6976],{"id":6977,"d":1014},"MJX-69-TEX-I-1D45B",[147,6979,6980],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,6981,6982],{"dataMmlNode":154},[147,6983,6984],{"dataMmlNode":1083},[159,6985],{"dataC":1093,"xLinkHref":6986},"#MJX-69-TEX-I-1D45B"," is the number of examples and ",[47,6989,6991],{"className":6990,"jax":51},[50],[53,6992,6995,7000],{"style":323,"xmlns":56,"width":6993,"height":2184,"role":25,"focusable":59,"viewBox":6994,"xmlnsXLink":61},"1.986ex","0 -442 878 453",[63,6996,6997],{},[66,6998],{"id":6999,"d":3459},"MJX-70-TEX-I-1D45A",[147,7001,7002],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,7003,7004],{"dataMmlNode":154},[147,7005,7006],{"dataMmlNode":1083},[159,7007],{"dataC":3628,"xLinkHref":7008},"#MJX-70-TEX-I-1D45A"," is the number of features.",[16,7011,7012,7013,824,7032,7051],{},"In practice, the tree is rarely complete, and the complexity is roughly linear in both ",[47,7014,7016],{"className":7015,"jax":51},[50],[53,7017,7018,7023],{"style":323,"xmlns":56,"width":6971,"height":2184,"role":25,"focusable":59,"viewBox":6972,"xmlnsXLink":61},[63,7019,7020],{},[66,7021],{"id":7022,"d":1014},"MJX-71-TEX-I-1D45B",[147,7024,7025],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,7026,7027],{"dataMmlNode":154},[147,7028,7029],{"dataMmlNode":1083},[159,7030],{"dataC":1093,"xLinkHref":7031},"#MJX-71-TEX-I-1D45B",[47,7033,7035],{"className":7034,"jax":51},[50],[53,7036,7037,7042],{"style":323,"xmlns":56,"width":6993,"height":2184,"role":25,"focusable":59,"viewBox":6994,"xmlnsXLink":61},[63,7038,7039],{},[66,7040],{"id":7041,"d":3459},"MJX-72-TEX-I-1D45A",[147,7043,7044],{"stroke":149,"fill":149,"stroke-width":150,"transform":151},[147,7045,7046],{"dataMmlNode":154},[147,7047,7048],{"dataMmlNode":1083},[159,7049],{"dataC":3628,"xLinkHref":7050},"#MJX-72-TEX-I-1D45A"," — which is why decision trees scale well to large datasets.",[11,7053,7055],{"id":7054},"strengths-and-limitations","Strengths and Limitations",[987,7057,7059],{"id":7058},"strengths","Strengths",[41,7061,7062,7065,7068,7071,7074],{},[44,7063,7064],{},"Produces interpretable, human-readable rules",[44,7066,7067],{},"Fast to build and fast to classify new instances",[44,7069,7070],{},"Handles both categorical and numerical features",[44,7072,7073],{},"Robust to noise and missing data",[44,7075,7076],{},"No need for feature scaling or normalization",[987,7078,7080],{"id":7079},"limitations","Limitations",[41,7082,7083,7086,7089,7092,7095],{},[44,7084,7085],{},"Prone to overfitting, especially with small datasets",[44,7087,7088],{},"Tests only one feature at a time (axis-aligned splits) — can't easily capture diagonal decision boundaries",[44,7090,7091],{},"Information gain has a bias toward features with many values",[44,7093,7094],{},"Greedy construction doesn't guarantee the globally optimal tree (finding the smallest consistent tree is NP-hard)",[44,7096,7097],{},"Produces a single deterministic hypothesis with no confidence estimates",[11,7099,7101],{"id":7100},"what-comes-next","What Comes Next?",[16,7103,7104],{},"Decision trees are a foundation for more powerful methods:",[41,7106,7107,7113,7119],{},[44,7108,7109,7112],{},[36,7110,7111],{},"Random Forests"," — build many decision trees on random subsets of features and data, then vote",[44,7114,7115,7118],{},[36,7116,7117],{},"Gradient Boosted Trees"," (XGBoost, LightGBM) — build trees sequentially, each one correcting the errors of the previous",[44,7120,7121,7124],{},[36,7122,7123],{},"C4.5 \u002F C5.0"," — Quinlan's improvements to ID3 with better handling of continuous features, missing values, and pruning\nThese ensemble methods are among the most successful algorithms in practice, especially for structured\u002Ftabular data — and they all build on the decision tree concepts we've covered here.",[2144,7126],{},[16,7128,7129],{},[821,7130,7131],{},"These notes are based on Week 4 of WIA1006\u002FWID3006 Machine Learning at Universiti Malaya (Semester 2, 2025\u002F2026). If you found this helpful, stay tuned for more ML concepts explained from scratch!",[7133,7134,7135],"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  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3;\n}\n",{"title":756,"searchDepth":7137,"depth":7137,"links":7138},2,[7139,7140,7141,7142,7148,7149,7150,7153,7157,7158,7162],{"id":13,"depth":7137,"text":14},{"id":698,"depth":7137,"text":699},{"id":741,"depth":7137,"text":742},{"id":981,"depth":7137,"text":982,"children":7143},[7144,7146,7147],{"id":989,"depth":7145,"text":990},3,{"id":2445,"depth":7145,"text":2174},{"id":5101,"depth":7145,"text":5102},{"id":6213,"depth":7137,"text":6214},{"id":6261,"depth":7137,"text":6262},{"id":6672,"depth":7137,"text":6673,"children":7151},[7152],{"id":6817,"depth":7145,"text":6818},{"id":6824,"depth":7137,"text":6825,"children":7154},[7155,7156],{"id":6843,"depth":7145,"text":6844},{"id":6872,"depth":7145,"text":6873},{"id":6890,"depth":7137,"text":6891},{"id":7054,"depth":7137,"text":7055,"children":7159},[7160,7161],{"id":7058,"depth":7145,"text":7059},{"id":7079,"depth":7145,"text":7080},{"id":7100,"depth":7137,"text":7101},"2026-04-28",false,"2026-04-30","md",null,{},true,"\u002Fblog\u002FDecision-Trees",{"title":5,"description":756},{"loc":7170},"blog\u002FDecision-Trees","scI7o6eDpOq-nKdwjALM6YkGpkrtOMrm7xqoLV0lAS4",105,{"id":7177,"extension":7178,"meta":7179,"series":7180,"stem":7284,"__hash__":7285},"series\u002Fseries.json","json",{},{"微積分教學":7181,"生活紀錄":7184,"Motor 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