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.window-content {\n background-color: #fafafa;\n}\n\n.ty-preferences .nav-group-item.active {\n color: white;\n background: #999;\n}\n\n :root {--mermaid-font-zoom:1em ;} .typora-export li, .typora-export p, .typora-export, .footnote-line {white-space: normal;} \n<\/style>\n<\/head>\n<body class='typora-export'>\n<div id='write' class=''><h1><a name=\"data-mining-cheat-sheet\" class=\"md-header-anchor\"><\/a><span>Data Mining Cheat Sheet<\/span><\/h1><h2><a name=\"introduction\" class=\"md-header-anchor\"><\/a><span>Introduction<\/span><\/h2><p><strong><span>Definition 1:<\/span><\/strong><span>\t<\/span><span>The nontrivial extraction of implicit, previous unknown, and potentially useful information from data.<\/span><\/p><p><strong><span>Definition 2:<\/span><\/strong><span>\t<\/span><span>Data mining is the process of automatically discovering useful information in large data repositories.<\/span><\/p><p><strong><span>Directed Knowledge Discovery:<\/span><\/strong><span>\t<\/span><span>Goal-oriented. Example: classification<\/span><\/p><p><strong><span>Undirected Knowledge Discovery:<\/span><\/strong><span>\t<\/span><span>Bottom-up approach, no target field. Example: clustering<\/span><\/p><p><strong><span>Data Mining Tasks:<\/span><\/strong><\/p><p><span>Classifcation, Regression, Clustering, Association Rule Discovery, Anomaly Detection<\/span><\/p><h2><a name=\"data-preprocessing\" class=\"md-header-anchor\"><\/a><span>Data Preprocessing<\/span><\/h2><p><strong><span>4 types of attributes:<\/span><\/strong><\/p><p><span>Nominal, Ordinal, Interval and Ratio<\/span><\/p><p><strong><span>Data quality examples:<\/span><\/strong><\/p><p><span>Missing values, duplication and inconsistent data, noise and outliers<\/span><\/p><p><strong><span>Methods for improving data quality:<\/span><\/strong><\/p><p><span>Sampling, aggregation, dimensionality reduction, feature subset selection, feature creation, discretization and binarization, attribute transformation<\/span><\/p><h2><a name=\"clustering\" class=\"md-header-anchor\"><\/a><span>Clustering<\/span><\/h2><p><strong><span>Similarity:<\/span><\/strong><span> often falls in the range [0,1] or [-1,1]<\/span><\/p><p><strong><span>Dissimilarity:<\/span><\/strong><span>\t<\/span><span>upper bound can be 1 or <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.323ex\" height=\"1.41ex\" viewBox=\"0 -504.6 1000 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E1-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E1-MJMAIN-221E\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\infty<\/script><\/p><h3><a name=\"different-types-of-similarities\" class=\"md-header-anchor\"><\/a><span>Different types of similarities<\/span><\/h3><p><strong><span>Euclidean Distance:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.635ex\" height=\"2.227ex\" viewBox=\"0 -755.9 1134.6 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E2-MJMATHI-4C\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path stroke-width=\"0\" id=\"E2-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E2-MJMATHI-4C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E2-MJMAIN-32\" x=\"963\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">L_2<\/script><span> Norm<\/span><\/p><p><strong><span>City Block:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"28.047ex\" height=\"2.928ex\" viewBox=\"0 -856.4 12075.6 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E3-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E3-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-69\" x=\"523\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-73\" x=\"868\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-74\" x=\"1337\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-28\" x=\"1698\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-78\" x=\"2087\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-2C\" x=\"2659\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-79\" x=\"3103\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-29\" x=\"3600\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-3D\" x=\"4267\" y=\"0\"><\/use><g transform=\"translate(5323,0)\"><use xlink:href=\"#E3-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMATHI-70\" x=\"1493\" y=\"681\"><\/use><g transform=\"translate(1056,-307)\"><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E3-MJMAIN-7C\" x=\"7793\" y=\"0\"><\/use><g transform=\"translate(8071,0)\"><use xlink:href=\"#E3-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMATHI-6D\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E3-MJMAIN-2212\" x=\"9586\" y=\"0\"><\/use><g transform=\"translate(10586,0)\"><use xlink:href=\"#E3-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMATHI-6D\" x=\"692\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E3-MJMAIN-7C\" x=\"11797\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">dist(x,y) = \\sum ^p _{i=1} |x_m - y_m|<\/script><\/p><p><strong><span>Minkowski Distance:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"31.038ex\" height=\"3.628ex\" viewBox=\"0 -1158 13363.6 1562\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E4-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-3BB\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E4-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMATHI-69\" x=\"523\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMATHI-73\" x=\"868\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMATHI-74\" x=\"1337\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMAIN-28\" x=\"1698\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMATHI-78\" x=\"2087\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMAIN-2C\" x=\"2659\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMATHI-79\" x=\"3103\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMAIN-29\" x=\"3600\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMAIN-3D\" x=\"4267\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMAIN-28\" x=\"5323\" y=\"0\"><\/use><g transform=\"translate(5712,0)\"><use xlink:href=\"#E4-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMATHI-70\" x=\"1493\" y=\"681\"><\/use><g transform=\"translate(1056,-307)\"><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E4-MJMAIN-7C\" x=\"8182\" y=\"0\"><\/use><g transform=\"translate(8460,0)\"><use xlink:href=\"#E4-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E4-MJMAIN-2212\" x=\"9598\" y=\"0\"><\/use><g transform=\"translate(10598,0)\"><use xlink:href=\"#E4-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><g transform=\"translate(11432,0)\"><use xlink:href=\"#E4-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMATHI-3BB\" x=\"393\" y=\"674\"><\/use><\/g><g transform=\"translate(12223,0)\"><use xlink:href=\"#E4-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,438)\"><g transform=\"translate(120,0)\"><rect stroke=\"none\" width=\"411\" height=\"60\" x=\"0\" y=\"146\"><\/rect><use transform=\"scale(0.5)\" xlink:href=\"#E4-MJMAIN-31\" x=\"161\" y=\"662\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E4-MJMATHI-3BB\" x=\"119\" y=\"-649\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">dist(x,y) = (\\sum ^p _{i = 1}|x_i - y_i|^ \\lambda ) ^{\\frac{1}{\\lambda}}<\/script><\/p><p><strong><span>Supremum Distance:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"25.954ex\" height=\"2.577ex\" viewBox=\"0 -806.1 11174.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E5-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E5-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-69\" x=\"523\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-73\" x=\"868\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-74\" x=\"1337\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMAIN-28\" x=\"1698\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-78\" x=\"2087\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMAIN-2C\" x=\"2659\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-79\" x=\"3103\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMAIN-29\" x=\"3600\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMAIN-3D\" x=\"4267\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-6D\" x=\"5323\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-61\" x=\"6201\" y=\"0\"><\/use><g transform=\"translate(6730,0)\"><use xlink:href=\"#E5-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E5-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E5-MJMAIN-7C\" x=\"7646\" y=\"0\"><\/use><g transform=\"translate(7924,0)\"><use xlink:href=\"#E5-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E5-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E5-MJMAIN-2212\" x=\"9062\" y=\"0\"><\/use><g transform=\"translate(10062,0)\"><use xlink:href=\"#E5-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E5-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E5-MJMAIN-7C\" x=\"10896\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">dist(x,y) = max_i |x_i - y_i|<\/script><\/p><p><strong><span>Cosine Similarity:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"27.79ex\" height=\"5.846ex\" viewBox=\"0 -1258.5 11965.3 2517\" role=\"img\" focusable=\"false\" style=\"vertical-align: -2.923ex;\"><defs><path stroke-width=\"0\" id=\"E6-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 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y=\"-429\"><\/use><\/g><g transform=\"translate(2884,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E6-MJSZ1-2211\" x=\"0\" y=\"-1\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMATHI-70\" x=\"1493\" y=\"681\"><\/use><g transform=\"translate(746,-217)\"><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(4679,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E6-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMAIN-32\" x=\"808\" y=\"487\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMATHI-69\" x=\"808\" y=\"-429\"><\/use><\/g><g transform=\"translate(5404,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E6-MJMAIN-5D\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(196,438)\"><g 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y=\"681\"><\/use><g transform=\"translate(746,-217)\"><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-28\" x=\"2581\" y=\"0\"><\/use><g transform=\"translate(2100,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-2212\" x=\"3886\" y=\"0\"><\/use><g transform=\"translate(3298,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-AF\" x=\"75\" y=\"4\"><\/use><\/g><g transform=\"translate(3705,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-32\" x=\"550\" y=\"674\"><\/use><\/g><g transform=\"translate(4467,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJSZ1-2211\" x=\"0\" y=\"-1\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMATHI-70\" x=\"1493\" y=\"681\"><\/use><g transform=\"translate(746,-217)\"><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-28\" x=\"8621\" y=\"0\"><\/use><g transform=\"translate(6371,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMATHI-69\" x=\"692\" y=\"-340\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-2212\" x=\"9844\" y=\"0\"><\/use><g transform=\"translate(7511,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-AF\" x=\"83\" y=\"5\"><\/use><\/g><g transform=\"translate(7923,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-32\" x=\"550\" y=\"674\"><\/use><\/g><g transform=\"translate(8519,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-5D\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(196,438)\"><g transform=\"translate(120,0)\"><rect stroke=\"none\" width=\"370\" height=\"60\" x=\"0\" y=\"95\"><\/rect><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-31\" x=\"119\" y=\"558\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-32\" x=\"119\" y=\"-725\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">corr(x,y) = \\frac{\\sum ^p _{i=1} (x_i - \\bar{x})(y_i - \\bar{y})}{[\\sum ^p _{i=1} (x_i - \\bar{x})^2 \\sum^p _{i=1} (y_i - \\bar{y})^2]^{\\frac{1}{2}}}<\/script><\/p><p><strong><span>Mahalanobis Distance:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"31.862ex\" height=\"2.811ex\" viewBox=\"0 -906.7 13718.2 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E8-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 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xlink:href=\"#E8-MJMATHI-78\" x=\"11037\" y=\"0\"><\/use><use xlink:href=\"#E8-MJMAIN-2212\" x=\"11832\" y=\"0\"><\/use><use xlink:href=\"#E8-MJMATHI-79\" x=\"12832\" y=\"0\"><\/use><use xlink:href=\"#E8-MJMAIN-29\" x=\"13329\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">dist(x,y) = (x - y)^TS ^{-1}(x - y)<\/script><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"38.167ex\" height=\"3.395ex\" viewBox=\"0 -956.9 16432.8 1461.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E9-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 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278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E9-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(613,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6B\" x=\"412\" y=\"0\"><\/use><\/g><use xlink:href=\"#E9-MJMAIN-3D\" x=\"1650\" y=\"0\"><\/use><g transform=\"translate(2428,0)\"><g transform=\"translate(397,0)\"><rect stroke=\"none\" width=\"1447\" height=\"60\" x=\"0\" y=\"220\"><\/rect><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMAIN-31\" x=\"773\" y=\"571\"><\/use><g transform=\"translate(60,-376)\"><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMAIN-2212\" x=\"600\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMAIN-31\" x=\"1378\" y=\"0\"><\/use><\/g><\/g><\/g><g transform=\"translate(4560,0)\"><use 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xlink:href=\"#E9-MJMAIN-AF\" x=\"274\" y=\"214\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6A\" x=\"1204\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E9-MJMAIN-29\" x=\"11182\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMAIN-28\" x=\"11571\" y=\"0\"><\/use><g transform=\"translate(11960,0)\"><use xlink:href=\"#E9-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(828,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6B\" x=\"345\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E9-MJMAIN-2212\" x=\"13723\" y=\"0\"><\/use><g transform=\"translate(14723,0)\"><use xlink:href=\"#E9-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMAIN-AF\" x=\"274\" y=\"214\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6B\" x=\"1204\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E9-MJMAIN-29\" x=\"16043\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S_{jk} = \\frac{1}{n-1}\\sum ^n _{i=1} (X_{ij} - \\bar{X}_j)(X_{ik} - \\bar{X}_k)<\/script><\/p><h3><a name=\"sequential-k-means\" class=\"md-header-anchor\"><\/a><span>Sequential K-Means<\/span><\/h3><ol start='' ><li><p><span>make the initial guess for the means <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.384ex\" height=\"1.76ex\" viewBox=\"0 -504.6 4901.3 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E29-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E29-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E29-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E29-MJMAIN-2C\" x=\"1331\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"1776\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"2220\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"2665\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2C\" x=\"3110\" y=\"0\"><\/use><g transform=\"translate(3554,0)\"><use xlink:href=\"#E29-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E29-MJMATHI-6B\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">m_1,...,m_k<\/script><\/p><\/li><li><p><span>set the counts <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.092ex\" height=\"1.76ex\" viewBox=\"0 -504.6 4345.3 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E18-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E18-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E18-MJMAIN-31\" x=\"848\" y=\"-213\"><\/use><use xlink:href=\"#E18-MJMAIN-2C\" x=\"1053\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"1498\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"1942\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"2387\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2C\" x=\"2832\" y=\"0\"><\/use><g transform=\"translate(3276,0)\"><use xlink:href=\"#E18-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E18-MJMATHI-6B\" x=\"848\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">n_1,...,n_k<\/script><span> to 0<\/span><\/p><\/li><li><p><span>until Interrupted<\/span><\/p><ol start='' ><li><span>acquire the next example <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><\/li><li><span>if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span> is closest to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E36-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i<\/script><span>, then increase <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.192ex\" height=\"1.76ex\" viewBox=\"0 -504.6 944 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E15-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E15-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E15-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E15-MJMATHI-69\" x=\"848\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">n_i<\/script><span> and update <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E36-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i<\/script><span> by <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"16.877ex\" height=\"3.511ex\" viewBox=\"0 -956.9 7266.3 1511.8\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.289ex;\"><defs><path stroke-width=\"0\" id=\"E17-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E17-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E17-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E17-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E17-MJMAIN-2B\" x=\"1444\" y=\"0\"><\/use><g transform=\"translate(2222,0)\"><g transform=\"translate(342,0)\"><rect stroke=\"none\" width=\"787\" height=\"60\" x=\"0\" y=\"220\"><\/rect><use transform=\"scale(0.707)\" xlink:href=\"#E17-MJMAIN-31\" x=\"306\" y=\"571\"><\/use><g transform=\"translate(60,-345)\"><use transform=\"scale(0.707)\" xlink:href=\"#E17-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E17-MJMATHI-69\" x=\"848\" y=\"-213\"><\/use><\/g><\/g><\/g><use xlink:href=\"#E17-MJMAIN-28\" x=\"3471\" y=\"0\"><\/use><use xlink:href=\"#E17-MJMATHI-78\" x=\"3860\" y=\"0\"><\/use><use xlink:href=\"#E17-MJMAIN-2212\" x=\"4655\" y=\"0\"><\/use><g transform=\"translate(5655,0)\"><use xlink:href=\"#E17-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E17-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E17-MJMAIN-29\" x=\"6877\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i + \\frac{1}{n_i}(x - m_i)<\/script><\/li><\/ol><\/li><li><p><span>end<\/span><\/p><\/li><\/ol><h3><a name=\"forgetful-sequence-k-means\" class=\"md-header-anchor\"><\/a><span>Forgetful Sequence K-Means<\/span><\/h3><p><span>Replace the counts <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.092ex\" height=\"1.76ex\" viewBox=\"0 -504.6 4345.3 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E18-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E18-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E18-MJMAIN-31\" x=\"848\" y=\"-213\"><\/use><use xlink:href=\"#E18-MJMAIN-2C\" x=\"1053\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"1498\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"1942\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"2387\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2C\" x=\"2832\" y=\"0\"><\/use><g transform=\"translate(3276,0)\"><use xlink:href=\"#E18-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E18-MJMATHI-6B\" x=\"848\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">n_1,...,n_k<\/script><span> with a constant <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.715ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3752.2 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E19-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E19-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-2208\" x=\"806\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-5B\" x=\"1751\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-30\" x=\"2029\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-2C\" x=\"2529\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-31\" x=\"2974\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-5D\" x=\"3474\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">a \\in [0,1]<\/script><span>, update <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E36-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i<\/script><span> by <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"15.719ex\" height=\"2.577ex\" viewBox=\"0 -806.1 6767.8 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E21-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 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103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E21-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E21-MJMAIN-2B\" x=\"1444\" y=\"0\"><\/use><use xlink:href=\"#E21-MJMATHI-61\" x=\"2444\" y=\"0\"><\/use><use xlink:href=\"#E21-MJMAIN-28\" x=\"2973\" y=\"0\"><\/use><use xlink:href=\"#E21-MJMATHI-78\" x=\"3362\" y=\"0\"><\/use><use xlink:href=\"#E21-MJMAIN-2212\" x=\"4156\" y=\"0\"><\/use><g transform=\"translate(5156,0)\"><use xlink:href=\"#E21-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E21-MJMAIN-29\" x=\"6378\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i + a (x - m_i)<\/script><\/p><h3><a name=\"expection-maximization-algorithm-gaussian-mixture-methods\" class=\"md-header-anchor\"><\/a><span>Expection-Maximization Algorithm (Gaussian Mixture Methods)<\/span><\/h3><p><span>Initialize the model parameters<\/span><\/p><p><span>Until there is no change in any parameters<\/span><\/p><p><span>Expectation Step : calculate the probability that each object belongs to each cluster<\/span><\/p><p><span>Maximization Step : find the new estimates of the parameters that maximize the expected likelihood<\/span><\/p><p><span>end until<\/span><\/p><h3><a name=\"kernel-k-means\" class=\"md-header-anchor\"><\/a><span>Kernel K-Means<\/span><\/h3><p><span>Transform datapoint <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.127ex\" height=\"1.76ex\" viewBox=\"0 -504.6 916 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E22-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E22-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E22-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x_i<\/script><span> into a feature space <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.319ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2290 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E23-MJMATHI-3D5\" d=\"M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 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726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E23-MJMATHI-3D5\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E23-MJMAIN-28\" x=\"596\" y=\"0\"><\/use><g transform=\"translate(985,0)\"><use xlink:href=\"#E23-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E23-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E23-MJMAIN-29\" x=\"1900\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\phi(x_i)<\/script><span> and then apply the basic K-means<\/span><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"24.502ex\" 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xlink:href=\"#E24-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E24-MJMAIN-2C\" x=\"2193\" y=\"0\"><\/use><g transform=\"translate(2638,0)\"><use xlink:href=\"#E24-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E24-MJMATHI-6A\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E24-MJMAIN-29\" x=\"3601\" y=\"0\"><\/use><use xlink:href=\"#E24-MJMAIN-3D\" x=\"4268\" y=\"0\"><\/use><use xlink:href=\"#E24-MJMATHI-3D5\" x=\"5324\" y=\"0\"><\/use><use xlink:href=\"#E24-MJMAIN-28\" x=\"5920\" y=\"0\"><\/use><g transform=\"translate(6309,0)\"><use xlink:href=\"#E24-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E24-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><g transform=\"translate(7225,0)\"><use xlink:href=\"#E24-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E24-MJMATHI-54\" x=\"550\" y=\"513\"><\/use><\/g><use xlink:href=\"#E24-MJMATHI-3D5\" x=\"8212\" y=\"0\"><\/use><use xlink:href=\"#E24-MJMAIN-28\" x=\"8808\" y=\"0\"><\/use><g transform=\"translate(9197,0)\"><use xlink:href=\"#E24-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E24-MJMATHI-6A\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E24-MJMAIN-29\" x=\"10160\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K(x_i,x_j) = \\phi(x_i)^T \\phi(x_j)<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"54.968ex\" height=\"3.161ex\" viewBox=\"0 -1007.2 23666.6 1361\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E25-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMATHI-3D5\" d=\"M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 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78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 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y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-2C\" x=\"16156\" y=\"0\"><\/use><g transform=\"translate(16600,0)\"><use xlink:href=\"#E25-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMATHI-6A\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-29\" x=\"17564\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-2212\" x=\"18175\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-32\" x=\"19175\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMATHI-4B\" x=\"19675\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-28\" x=\"20564\" y=\"0\"><\/use><g transform=\"translate(20953,0)\"><use xlink:href=\"#E25-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-2C\" x=\"21869\" y=\"0\"><\/use><g transform=\"translate(22314,0)\"><use xlink:href=\"#E25-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMATHI-6A\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-29\" x=\"23277\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">||\\phi(x_i) -\\phi(x_j)||^2 = K(x_i,x_i) + K(x_j,x_j) - 2K(x_i,x_j)<\/script><\/p><p><strong><span>Homogeneous polynomial kernel:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.046ex\" height=\"2.928ex\" viewBox=\"0 -906.7 7769.6 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E26-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 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440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 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inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"21.89ex\" height=\"2.928ex\" viewBox=\"0 -906.7 9425 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E27-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 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x=\"572\" y=\"288\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">K(x,y) = e^{-\\frac{||x-y||^2}{2\\sigma ^2}}<\/script><\/p><h3><a name=\"self-organizing-feature-maps\" class=\"md-header-anchor\"><\/a><span>Self-Organizing Feature Maps<\/span><\/h3><p><span>Unlike sequential K-Means, the centres (called <\/span><strong><span>units or neurons)<\/span><\/strong><span> have a pre-determined topographic ordering relationship<\/span><\/p><ol start='' ><li><p><span>make initial guesses for centroids <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.384ex\" height=\"1.76ex\" viewBox=\"0 -504.6 4901.3 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E29-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E29-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E29-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E29-MJMAIN-2C\" x=\"1331\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"1776\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"2220\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"2665\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2C\" x=\"3110\" y=\"0\"><\/use><g transform=\"translate(3554,0)\"><use xlink:href=\"#E29-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E29-MJMATHI-6B\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">m_1,...,m_k<\/script><\/p><\/li><li><p><span>repeat<\/span><\/p><ol start='' ><li><span>acquire the next example <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><\/li><li><span>if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E36-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i<\/script><span> is closest to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span>, then update <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E36-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i<\/script><span> by <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"15.719ex\" height=\"2.577ex\" viewBox=\"0 -806.1 6767.8 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E34-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E34-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E34-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E34-MJMAIN-2B\" x=\"1444\" y=\"0\"><\/use><use xlink:href=\"#E34-MJMATHI-61\" x=\"2444\" y=\"0\"><\/use><use xlink:href=\"#E34-MJMAIN-28\" x=\"2973\" y=\"0\"><\/use><use xlink:href=\"#E34-MJMATHI-78\" x=\"3362\" y=\"0\"><\/use><use xlink:href=\"#E34-MJMAIN-2212\" x=\"4156\" y=\"0\"><\/use><g transform=\"translate(5156,0)\"><use xlink:href=\"#E34-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E34-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E34-MJMAIN-29\" x=\"6378\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i + a(x - m_i)<\/script><span>; update <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.948ex\" height=\"1.994ex\" viewBox=\"0 -504.6 1269.3 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E35-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E35-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E35-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E35-MJMATHI-6A\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_j<\/script><span> (neighbor of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g 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y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_j + a_n (x - m_j)<\/script><\/li><\/ol><\/li><li><p><span>end until centroids do not change or a threshold is exceeded<\/span><\/p><\/li><\/ol><h3><a name=\"agglomerative-hierarchical-clustering\" class=\"md-header-anchor\"><\/a><span>Agglomerative Hierarchical Clustering<\/span><\/h3><p><span>Start out with each data point forming its own cluster and gradually merge clusters until all points have been gathered together in one big cluster<\/span><\/p><ol start='' ><li><p><span>compute the proximity matrix<\/span><\/p><\/li><li><p><span>repeat<\/span><\/p><ol start='' ><li><span>merge the two closest clusters<\/span><\/li><li><span>update the proximity matrix to reflect the proximity between the new cluster and the original clusters<\/span><\/li><\/ol><\/li><li><p><span>until only one cluster remains<\/span><\/p><\/li><\/ol><p><strong><span>Different methods to define the distances between clusters:<\/span><\/strong><\/p><p><strong><span>Single Linkage Clustering:<\/span><\/strong><span>\t<\/span><span>distance between groups is defined as that of the <\/span><strong><span>closest<\/span><\/strong><span> pair of data<\/span><\/p><p><span>Pros: can handle non-elliptical shapes Cons: sensitive to noise and outliers <\/span><\/p><p><strong><span>Complete Linkage Clustering:<\/span><\/strong><span>\t<\/span><span>distance between two clusters is given by the distance between their most <\/span><strong><span>distant<\/span><\/strong><span> members<\/span><\/p><p><span>Pros: less susceptible to noise and outliers Cons: tend to break large clusters; biased towards globular clusters<\/span><\/p><p><strong><span>Group Average Clustering:<\/span><\/strong><span>\t<\/span><span>distance between two clusters is defined as the <\/span><strong><span>average<\/span><\/strong><span> of the distances between <\/span><strong><span>all<\/span><\/strong><span> pairs of records<\/span><\/p><p><span>Pros: less susceptible to noise and outliers Cons: biased towards globular clusters<\/span><\/p><p><strong><span>Centroid Clustering:<\/span><\/strong><span>\t<\/span><span>distance between two clusters is defined as the distance between the <\/span><strong><span>mean vectors<\/span><\/strong><span> of the two clusters<\/span><\/p><p><strong><span>Median Clustering:<\/span><\/strong><span>\t<\/span><span>distance between two clusters is defined as the distance between the <\/span><strong><span>prototypes<\/span><\/strong><span> of the two clusters<\/span><\/p><p><span>the prototype is the data point when there is only one point in the clusterl; When we merge two clusters, the new prototype is the mid point of the two prototypes<\/span><\/p><p><strong><span>Ward's Method:<\/span><\/strong><span>\t<\/span><span>Similarity of two clusters is based on the <\/span><strong><span>increase in squared error<\/span><\/strong><span> when two clusters are 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y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E38-MJMATHI-6A\" x=\"852\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E38-MJMAIN-7C\" x=\"13606\" y=\"0\"><\/use><g transform=\"translate(13884,0)\"><use xlink:href=\"#E38-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E38-MJMAIN-32\" x=\"393\" y=\"674\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\Delta SSE(C_i,C_j) = \\frac{n_in_j}{n_i+n_j}||\\mu _i - \\mu _j||^2<\/script><\/p><h3><a name=\"dbscan\" class=\"md-header-anchor\"><\/a><span>DBSCAN<\/span><\/h3><p><strong><span>Core point:<\/span><\/strong><span>\t<\/span><span>has more than a specified number of points (MinPts) within Eps<\/span><\/p><p><strong><span>Border point:<\/span><\/strong><span>\t<\/span><span>has fewer than MinPts within Eps, but is in the neighborhood of a core point<\/span><\/p><p><strong><span>Noise point:<\/span><\/strong><span>\t<\/span><span>neither is core point nor border point<\/span><\/p><ol start='' ><li><span>Label all points as core, border, or noise points<\/span><\/li><li><span>Eliminate noise points<\/span><\/li><li><span>Put an edge between all core points that are within Eps of each other<\/span><\/li><li><span>Make each group of connected core points into a separated cluster<\/span><\/li><li><span>Assign each border point to one of the clusters containing a core point from which the border point is directly density- reachable<\/span><\/li><\/ol><p><span>Pros: resistant to noise, can handle clusters of different shapes and sizes <\/span><\/p><p><span>Cons: sensitive to denses, can not handle high dimensional data well <\/span><\/p><h3><a name=\"cluster-evaluation\" class=\"md-header-anchor\"><\/a><span>Cluster Evaluation<\/span><\/h3><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"26.798ex\" height=\"3.161ex\" viewBox=\"0 -906.7 11538 1361\" role=\"img\" 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xlink:href=\"#E40-MJMAIN-3D\" x=\"2326\" y=\"0\"><\/use><g transform=\"translate(3382,0)\"><use xlink:href=\"#E40-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E40-MJMATHI-69\" x=\"1493\" y=\"-404\"><\/use><\/g><use xlink:href=\"#E40-MJMAIN-7C\" x=\"4949\" y=\"0\"><\/use><g transform=\"translate(5227,0)\"><use xlink:href=\"#E40-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E40-MJMATHI-69\" x=\"1011\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E40-MJMAIN-7C\" x=\"6286\" y=\"0\"><\/use><use xlink:href=\"#E40-MJMAIN-28\" x=\"6564\" y=\"0\"><\/use><use xlink:href=\"#E40-MJMATHI-63\" x=\"6953\" y=\"0\"><\/use><use xlink:href=\"#E40-MJMAIN-2212\" x=\"7608\" y=\"0\"><\/use><g transform=\"translate(8608,0)\"><use xlink:href=\"#E40-MJMATHI-63\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E40-MJMATHI-69\" x=\"612\" y=\"-213\"><\/use><\/g><g transform=\"translate(9385,0)\"><use 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40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" 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637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E41-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMATHI-63\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E41-MJMATHI-54\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMATHI-53\" x=\"704\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMATHI-53\" x=\"1349\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMAIN-3D\" x=\"2271\" y=\"0\"><\/use><g transform=\"translate(3327,0)\"><use xlink:href=\"#E41-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMATHI-4B\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(5797,0)\"><use xlink:href=\"#E41-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMAIN-2208\" x=\"572\" y=\"0\"><\/use><g transform=\"translate(876,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E41-MJMATHI-69\" x=\"1011\" y=\"-213\"><\/use><\/g><\/g><\/g><use xlink:href=\"#E41-MJMAIN-28\" x=\"8578\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMATHI-78\" x=\"8967\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMAIN-2212\" x=\"9761\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMATHI-63\" x=\"10762\" y=\"0\"><\/use><g transform=\"translate(11195,0)\"><use xlink:href=\"#E41-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMAIN-32\" x=\"550\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">TSS = \\sum _{i=1} ^K \\sum _{x \\in C_i} (x-c)^2<\/script><\/p><p><strong><span>Silhouette Coefficient:<\/span><\/strong><span>\t<\/span><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.318ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1428.7 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E42-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E42-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E42-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E42-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E42-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><use xlink:href=\"#E42-MJMAIN-3A\" x=\"1150\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">a_i:<\/script><span>\t<\/span><span>average distance of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"0.801ex\" height=\"1.994ex\" viewBox=\"0 -755.9 345 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E56-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E56-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i<\/script><span> to the points in its cluster<\/span><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.795ex\" height=\"2.344ex\" viewBox=\"0 -755.9 773 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E44-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E44-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E44-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E44-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">b_i<\/script><span>:<\/span><span>\t<\/span><span>min (average distance of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"0.801ex\" height=\"1.994ex\" viewBox=\"0 -755.9 345 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E56-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E56-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i<\/script><span> to points in another cluster)<\/span><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.71ex\" height=\"4.212ex\" viewBox=\"0 -1107.7 5472.5 1813.3\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.639ex;\"><defs><path stroke-width=\"0\" id=\"E46-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E46-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E46-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 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307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E46-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" 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transform=\"scale(0.707)\" xlink:href=\"#E46-MJMATHI-78\" x=\"1406\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E46-MJMAIN-28\" x=\"1979\" y=\"0\"><\/use><g transform=\"translate(1674,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E46-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E46-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E46-MJMAIN-2C\" x=\"3240\" y=\"0\"><\/use><g transform=\"translate(2488,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E46-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E46-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E46-MJMAIN-29\" x=\"4291\" y=\"0\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">s = \\frac{b_i - a_i}{max(a_i,b_i)}<\/script><\/p><h2><a name=\"classification\" class=\"md-header-anchor\"><\/a><span>Classification<\/span><\/h2><h3><a name=\"minimum-distance-classifier\" class=\"md-header-anchor\"><\/a><span>Minimum Distance Classifier<\/span><\/h3><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.482ex\" height=\"3.745ex\" viewBox=\"0 -956.9 7527.1 1612.3\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.522ex;\"><defs><path stroke-width=\"0\" id=\"E47-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJMAIN-3D\" d=\"M56 347Q56 360 70 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2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E47-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMATHI-6A\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E47-MJMAIN-3D\" x=\"1547\" y=\"0\"><\/use><g transform=\"translate(2325,0)\"><g transform=\"translate(397,0)\"><rect stroke=\"none\" width=\"964\" height=\"60\" x=\"0\" y=\"220\"><\/rect><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMAIN-31\" x=\"432\" y=\"571\"><\/use><g transform=\"translate(60,-388)\"><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMATHI-4E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E47-MJMATHI-6A\" x=\"1135\" y=\"-213\"><\/use><\/g><\/g><\/g><g transform=\"translate(3974,0)\"><use xlink:href=\"#E47-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMAIN-2208\" x=\"572\" y=\"0\"><\/use><g transform=\"translate(876,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E47-MJMATHI-6A\" x=\"1011\" y=\"-213\"><\/use><\/g><\/g><\/g><use xlink:href=\"#E47-MJMATHI-78\" x=\"6955\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_j = \\frac{1}{N_j} \\sum _{x \\in C_j} x<\/script><\/p><p><span>The goal is to minimize <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.763ex\" height=\"2.694ex\" viewBox=\"0 -806.1 8078.7 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E48-MJMATHI-44\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 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140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E48-MJMATHI-44\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E48-MJMATHI-6A\" x=\"1170\" y=\"-213\"><\/use><use xlink:href=\"#E48-MJMAIN-28\" x=\"1219\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMATHI-78\" x=\"1608\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-29\" x=\"2180\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-3D\" x=\"2847\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-7C\" x=\"3902\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-7C\" x=\"4180\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMATHI-78\" x=\"4458\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-2212\" x=\"5253\" y=\"0\"><\/use><g transform=\"translate(6253,0)\"><use xlink:href=\"#E48-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E48-MJMATHI-6A\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E48-MJMAIN-7C\" x=\"7522\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-7C\" x=\"7800\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">D_j(x) = ||x - m_j||<\/script><\/p><p><span>Which is equivalent with maximizing <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"23.978ex\" height=\"3.395ex\" viewBox=\"0 -956.9 10323.8 1461.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E49-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E49-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E49-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E49-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" 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193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E49-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 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429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E49-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-6A\" x=\"735\" y=\"-213\"><\/use><use xlink:href=\"#E49-MJMAIN-28\" x=\"911\" y=\"0\"><\/use><use xlink:href=\"#E49-MJMATHI-78\" x=\"1300\" y=\"0\"><\/use><use xlink:href=\"#E49-MJMAIN-29\" x=\"1872\" y=\"0\"><\/use><use xlink:href=\"#E49-MJMAIN-3D\" x=\"2539\" y=\"0\"><\/use><g transform=\"translate(3594,0)\"><use xlink:href=\"#E49-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-54\" x=\"1241\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-6A\" x=\"1241\" y=\"-429\"><\/use><\/g><use xlink:href=\"#E49-MJMATHI-78\" x=\"5070\" y=\"0\"><\/use><use xlink:href=\"#E49-MJMAIN-2212\" x=\"5864\" y=\"0\"><\/use><g transform=\"translate(6642,0)\"><g transform=\"translate(342,0)\"><rect stroke=\"none\" width=\"473\" height=\"60\" x=\"0\" y=\"220\"><\/rect><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMAIN-31\" x=\"84\" y=\"571\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMAIN-32\" x=\"84\" y=\"-531\"><\/use><\/g><\/g><g transform=\"translate(7578,0)\"><use xlink:href=\"#E49-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-54\" x=\"1241\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-6A\" x=\"1241\" y=\"-429\"><\/use><\/g><g transform=\"translate(9054,0)\"><use xlink:href=\"#E49-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-6A\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">d_j(x) = m_j^Tx - \\frac{1}{2} m_j^Tm_j<\/script><span>\t<\/span><span>the decision function for class j<\/span><\/p><h3><a name=\"linear-discriminant-classifier\" class=\"md-header-anchor\"><\/a><span>Linear Discriminant Classifier<\/span><\/h3><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.632ex\" height=\"2.811ex\" viewBox=\"0 -906.7 8022 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E50-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 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1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 674T327 656Q322 641 318 637H297Q236 634 232 620Q262 160 266 136L501 550L499 587Q496 629 489 632Q483 636 447 637Q428 637 422 639T416 648Q416 650 418 660Q419 664 420 669T421 676T424 680T428 682T436 683Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMATHI-6F\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E50-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMAIN-28\" x=\"480\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMATHI-78\" x=\"869\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMAIN-29\" x=\"1441\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMAIN-3D\" x=\"2107\" y=\"0\"><\/use><g transform=\"translate(3163,0)\"><use xlink:href=\"#E50-MJMATHI-57\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E50-MJMATHI-54\" x=\"1526\" y=\"513\"><\/use><\/g><use xlink:href=\"#E50-MJMATHI-78\" x=\"4840\" y=\"0\"><\/use><use xlink:href=\"#E50-MJMAIN-2B\" x=\"5634\" y=\"0\"><\/use><g transform=\"translate(6635,0)\"><use xlink:href=\"#E50-MJMATHI-57\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E50-MJMATHI-6F\" x=\"1335\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">g(x) = W^Tx + W_o<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span> is assigned to one class if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.509ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3663.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E52-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 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139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E52-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E52-MJMAIN-3E\" d=\"M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z\"><\/path><path stroke-width=\"0\" id=\"E52-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E52-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-28\" x=\"480\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMATHI-78\" x=\"869\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-29\" x=\"1441\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-3E\" x=\"2107\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-30\" x=\"3163\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">g(x) \\gt 0<\/script><span> and another class if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.509ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3663.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E53-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E53-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E53-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E53-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E53-MJMAIN-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path stroke-width=\"0\" id=\"E53-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E53-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E53-MJMAIN-28\" x=\"480\" y=\"0\"><\/use><use xlink:href=\"#E53-MJMATHI-78\" x=\"869\" y=\"0\"><\/use><use xlink:href=\"#E53-MJMAIN-29\" x=\"1441\" y=\"0\"><\/use><use xlink:href=\"#E53-MJMAIN-3C\" x=\"2107\" y=\"0\"><\/use><use xlink:href=\"#E53-MJMAIN-30\" x=\"3163\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">g(x) \\lt 0<\/script><\/p><h3><a name=\"k-nearest-neighbor\" class=\"md-header-anchor\"><\/a><span>k Nearest Neighbor<\/span><\/h3><p><span>lazy learners, does not build model explicitly<\/span><\/p><p><span>can have arbitary decision boundries<\/span><\/p><h3><a name=\"bayes-classifier\" class=\"md-header-anchor\"><\/a><span>Bayes Classifier<\/span><\/h3><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"30.276ex\" height=\"2.577ex\" viewBox=\"0 -806.1 13035.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E54-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E54-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-28\" x=\"751\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-41\" x=\"1140\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-7C\" x=\"1890\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-43\" x=\"2168\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-29\" x=\"2928\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-3D\" x=\"3594\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-50\" x=\"4650\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-28\" x=\"5401\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-41\" x=\"5790\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-29\" x=\"6540\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-50\" x=\"6929\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-28\" x=\"7680\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-43\" x=\"8069\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-7C\" x=\"8829\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-41\" x=\"9107\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-29\" x=\"9857\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-2F\" x=\"10246\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-50\" x=\"10746\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-28\" x=\"11497\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-43\" x=\"11886\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-29\" x=\"12646\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P(A|C) = P(A)P(C|A) \/ P(C)<\/script><\/p><p><span>A point <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span> belongs to class <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"0.801ex\" height=\"1.994ex\" viewBox=\"0 -755.9 345 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E56-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E56-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i<\/script><span> if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.364ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3601 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E57-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E57-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMAIN-28\" x=\"751\" y=\"0\"><\/use><g transform=\"translate(1140,0)\"><use xlink:href=\"#E57-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E57-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E57-MJMAIN-7C\" x=\"2361\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMATHI-78\" x=\"2639\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMAIN-29\" x=\"3211\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P(m_i | x) <\/script><span> is the largest<\/span><\/p><p><span>Normally we are given <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.364ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3601 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E58-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E58-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMAIN-28\" x=\"751\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMATHI-78\" x=\"1140\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMAIN-7C\" x=\"1712\" y=\"0\"><\/use><g transform=\"translate(1990,0)\"><use xlink:href=\"#E58-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E58-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E58-MJMAIN-29\" x=\"3211\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P(x|m_i)<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.389ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2751 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E59-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E59-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E59-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E59-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E59-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E59-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E59-MJMAIN-28\" x=\"751\" y=\"0\"><\/use><g transform=\"translate(1140,0)\"><use xlink:href=\"#E59-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E59-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E59-MJMAIN-29\" x=\"2361\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P(m_i)<\/script><span> to compute <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.364ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3601 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E60-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E60-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMAIN-28\" x=\"751\" y=\"0\"><\/use><g transform=\"translate(1140,0)\"><use xlink:href=\"#E60-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E60-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E60-MJMAIN-7C\" x=\"2361\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMATHI-78\" x=\"2639\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMAIN-29\" x=\"3211\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P(m_i|x)<\/script><\/p><p><span>the decision function is <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"21.418ex\" height=\"2.694ex\" viewBox=\"0 -806.1 9221.5 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E61-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E61-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E61-MJMATHI-6A\" x=\"735\" y=\"-213\"><\/use><use xlink:href=\"#E61-MJMAIN-28\" x=\"911\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMATHI-78\" x=\"1300\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMAIN-29\" x=\"1872\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMAIN-3D\" x=\"2539\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMATHI-70\" x=\"3594\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMAIN-28\" x=\"4097\" y=\"0\"><\/use><g transform=\"translate(4486,0)\"><use xlink:href=\"#E61-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E61-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E61-MJMAIN-29\" x=\"5594\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMATHI-70\" x=\"5983\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMAIN-28\" x=\"6486\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMATHI-78\" x=\"6875\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMAIN-7C\" x=\"7447\" y=\"0\"><\/use><g transform=\"translate(7725,0)\"><use xlink:href=\"#E61-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E61-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E61-MJMAIN-29\" x=\"8832\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">d_j(x) = p(w_j)p(x|w_j)<\/script><\/p><p><span>How to estimate <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.612ex\" height=\"2.694ex\" viewBox=\"-39 -806.1 3277.3 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex; margin-left: -0.091ex;\"><defs><path stroke-width=\"0\" id=\"E62-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 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transform=\"scale(0.707)\" xlink:href=\"#E63-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E63-MJMAIN-29\" x=\"8122\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-2E\" x=\"8511\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-2E\" x=\"8956\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-2E\" x=\"9401\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMATHI-50\" x=\"9845\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-28\" x=\"10596\" y=\"0\"><\/use><g transform=\"translate(10985,0)\"><use xlink:href=\"#E63-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E63-MJMATHI-6E\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E63-MJMAIN-7C\" x=\"12082\" y=\"0\"><\/use><g transform=\"translate(12360,0)\"><use xlink:href=\"#E63-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E63-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E63-MJMAIN-29\" x=\"13467\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">p(x|w_j) = P(x_1|w_j)...P(x_n|w_j)<\/script><\/p><p> <\/p><\/div>\n<\/body>\n<\/html>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Data Mining Cheat Sheet Data Mining Cheat Sheet Introduction Definition 1: The nontrivial extraction of implicit, previous unknown, and potentially… <\/p>\n<div class=\"art-el-more\"><a href=\"https:\/\/yunpeng.site\/index.php\/2021\/04\/16\/dmsum\/\" class=\"art-link art-color-link art-w-chevron\">Read more<\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"spay_email":"","footnotes":""},"categories":[3],"tags":[5],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts\/122"}],"collection":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/comments?post=122"}],"version-history":[{"count":4,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts\/122\/revisions"}],"predecessor-version":[{"id":227,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts\/122\/revisions\/227"}],"wp:attachment":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/media?parent=122"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/categories?post=122"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/tags?post=122"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

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\n\t\t\t\n\n\n\nData Mining Cheat Sheet<\/title><link href='https:\/\/fonts.loli.net\/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text\/css' \/><style type='text\/css'>html {overflow-x: initial !important;}:root { --bg-color: #ffffff; --text-color: #333333; --select-text-bg-color: #B5D6FC; --select-text-font-color: auto; --monospace: \"Lucida Console\",Consolas,\"Courier\",monospace; --title-bar-height: 20px; }\n.mac-os-11 { --title-bar-height: 28px; }\nhtml { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: \"Helvetica Neue\", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; }\nbody { margin: 0px; padding: 0px; height: auto; bottom: 0px; top: 0px; left: 0px; right: 0px; font-size: 1rem; line-height: 1.42857143; overflow-x: hidden; background-image: inherit; background-size: inherit; background-attachment: inherit; background-origin: inherit; background-clip: inherit; 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margin-top:20px;\n padding-bottom:20px;\n}\n\n.sidebar-tabs {\n border-bottom: none;\n}\n\n#typora-quick-open {\n border: 1px solid #ddd;\n background-color: #f8f8f8;\n}\n\n#typora-quick-open-item {\n background-color: #FAFAFA;\n border-color: #FEFEFE #e5e5e5 #e5e5e5 #eee;\n border-style: solid;\n border-width: 1px;\n}\n\n\/** focus mode *\/\n.on-focus-mode blockquote {\n border-left-color: rgba(85, 85, 85, 0.12);\n}\n\nheader, .context-menu, .megamenu-content, footer{\n font-family: \"Segoe UI\", \"Arial\", sans-serif;\n}\n\n.file-node-content:hover .file-node-icon,\n.file-node-content:hover .file-node-open-state{\n visibility: visible;\n}\n\n.mac-seamless-mode #typora-sidebar {\n background-color: #fafafa;\n background-color: var(--side-bar-bg-color);\n}\n\n.md-lang {\n color: #b4654d;\n}\n\n.html-for-mac .context-menu {\n --item-hover-bg-color: #E6F0FE;\n}\n\n#md-notification .btn {\n border: 0;\n}\n\n.dropdown-menu .divider {\n border-color: #e5e5e5;\n}\n\n.ty-preferences .window-content {\n background-color: #fafafa;\n}\n\n.ty-preferences .nav-group-item.active {\n color: white;\n background: #999;\n}\n\n :root {--mermaid-font-zoom:1em ;} .typora-export li, .typora-export p, .typora-export, .footnote-line {white-space: normal;} \n<\/style>\n<\/head>\n<body class='typora-export'>\n<div id='write' class=''><h1><a name=\"data-mining-cheat-sheet\" class=\"md-header-anchor\"><\/a><span>Data Mining Cheat Sheet<\/span><\/h1><h2><a name=\"introduction\" class=\"md-header-anchor\"><\/a><span>Introduction<\/span><\/h2><p><strong><span>Definition 1:<\/span><\/strong><span>\t<\/span><span>The nontrivial extraction of implicit, previous unknown, and potentially useful information from data.<\/span><\/p><p><strong><span>Definition 2:<\/span><\/strong><span>\t<\/span><span>Data mining is the process of automatically discovering useful information in large data repositories.<\/span><\/p><p><strong><span>Directed Knowledge Discovery:<\/span><\/strong><span>\t<\/span><span>Goal-oriented. Example: classification<\/span><\/p><p><strong><span>Undirected Knowledge Discovery:<\/span><\/strong><span>\t<\/span><span>Bottom-up approach, no target field. Example: clustering<\/span><\/p><p><strong><span>Data Mining Tasks:<\/span><\/strong><\/p><p><span>Classifcation, Regression, Clustering, Association Rule Discovery, Anomaly Detection<\/span><\/p><h2><a name=\"data-preprocessing\" class=\"md-header-anchor\"><\/a><span>Data Preprocessing<\/span><\/h2><p><strong><span>4 types of attributes:<\/span><\/strong><\/p><p><span>Nominal, Ordinal, Interval and Ratio<\/span><\/p><p><strong><span>Data quality examples:<\/span><\/strong><\/p><p><span>Missing values, duplication and inconsistent data, noise and outliers<\/span><\/p><p><strong><span>Methods for improving data quality:<\/span><\/strong><\/p><p><span>Sampling, aggregation, dimensionality reduction, feature subset selection, feature creation, discretization and binarization, attribute transformation<\/span><\/p><h2><a name=\"clustering\" class=\"md-header-anchor\"><\/a><span>Clustering<\/span><\/h2><p><strong><span>Similarity:<\/span><\/strong><span> often falls in the range [0,1] or [-1,1]<\/span><\/p><p><strong><span>Dissimilarity:<\/span><\/strong><span>\t<\/span><span>upper bound can be 1 or <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.323ex\" height=\"1.41ex\" viewBox=\"0 -504.6 1000 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E1-MJMAIN-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E1-MJMAIN-221E\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\infty<\/script><\/p><h3><a name=\"different-types-of-similarities\" class=\"md-header-anchor\"><\/a><span>Different types of similarities<\/span><\/h3><p><strong><span>Euclidean Distance:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.635ex\" height=\"2.227ex\" viewBox=\"0 -755.9 1134.6 958.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.472ex;\"><defs><path stroke-width=\"0\" id=\"E2-MJMATHI-4C\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path stroke-width=\"0\" id=\"E2-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E2-MJMATHI-4C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E2-MJMAIN-32\" x=\"963\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">L_2<\/script><span> Norm<\/span><\/p><p><strong><span>City Block:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"28.047ex\" height=\"2.928ex\" viewBox=\"0 -856.4 12075.6 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E3-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E3-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E3-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-69\" x=\"523\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-73\" x=\"868\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-74\" x=\"1337\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-28\" x=\"1698\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-78\" x=\"2087\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-2C\" x=\"2659\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMATHI-79\" x=\"3103\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-29\" x=\"3600\" y=\"0\"><\/use><use xlink:href=\"#E3-MJMAIN-3D\" x=\"4267\" y=\"0\"><\/use><g transform=\"translate(5323,0)\"><use xlink:href=\"#E3-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMATHI-70\" x=\"1493\" y=\"681\"><\/use><g transform=\"translate(1056,-307)\"><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E3-MJMAIN-7C\" x=\"7793\" y=\"0\"><\/use><g transform=\"translate(8071,0)\"><use xlink:href=\"#E3-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMATHI-6D\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E3-MJMAIN-2212\" x=\"9586\" y=\"0\"><\/use><g transform=\"translate(10586,0)\"><use xlink:href=\"#E3-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E3-MJMATHI-6D\" x=\"692\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E3-MJMAIN-7C\" x=\"11797\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">dist(x,y) = \\sum ^p _{i=1} |x_m - y_m|<\/script><\/p><p><strong><span>Minkowski Distance:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"31.038ex\" height=\"3.628ex\" viewBox=\"0 -1158 13363.6 1562\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.938ex;\"><defs><path stroke-width=\"0\" id=\"E4-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E4-MJMATHI-3BB\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E4-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMATHI-69\" x=\"523\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMATHI-73\" x=\"868\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMATHI-74\" x=\"1337\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMAIN-28\" x=\"1698\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMATHI-78\" x=\"2087\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMAIN-2C\" x=\"2659\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMATHI-79\" x=\"3103\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMAIN-29\" x=\"3600\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMAIN-3D\" x=\"4267\" y=\"0\"><\/use><use xlink:href=\"#E4-MJMAIN-28\" x=\"5323\" y=\"0\"><\/use><g transform=\"translate(5712,0)\"><use xlink:href=\"#E4-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMATHI-70\" x=\"1493\" y=\"681\"><\/use><g transform=\"translate(1056,-307)\"><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E4-MJMAIN-7C\" x=\"8182\" y=\"0\"><\/use><g transform=\"translate(8460,0)\"><use xlink:href=\"#E4-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E4-MJMAIN-2212\" x=\"9598\" y=\"0\"><\/use><g transform=\"translate(10598,0)\"><use xlink:href=\"#E4-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><g transform=\"translate(11432,0)\"><use xlink:href=\"#E4-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E4-MJMATHI-3BB\" x=\"393\" y=\"674\"><\/use><\/g><g transform=\"translate(12223,0)\"><use xlink:href=\"#E4-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(389,438)\"><g transform=\"translate(120,0)\"><rect stroke=\"none\" width=\"411\" height=\"60\" x=\"0\" y=\"146\"><\/rect><use transform=\"scale(0.5)\" xlink:href=\"#E4-MJMAIN-31\" x=\"161\" y=\"662\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E4-MJMATHI-3BB\" x=\"119\" y=\"-649\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">dist(x,y) = (\\sum ^p _{i = 1}|x_i - y_i|^ \\lambda ) ^{\\frac{1}{\\lambda}}<\/script><\/p><p><strong><span>Supremum Distance:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"25.954ex\" height=\"2.577ex\" viewBox=\"0 -806.1 11174.5 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E5-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E5-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E5-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-69\" x=\"523\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-73\" x=\"868\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-74\" x=\"1337\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMAIN-28\" x=\"1698\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-78\" x=\"2087\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMAIN-2C\" x=\"2659\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-79\" x=\"3103\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMAIN-29\" x=\"3600\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMAIN-3D\" x=\"4267\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-6D\" x=\"5323\" y=\"0\"><\/use><use xlink:href=\"#E5-MJMATHI-61\" x=\"6201\" y=\"0\"><\/use><g transform=\"translate(6730,0)\"><use xlink:href=\"#E5-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E5-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E5-MJMAIN-7C\" x=\"7646\" y=\"0\"><\/use><g transform=\"translate(7924,0)\"><use xlink:href=\"#E5-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E5-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E5-MJMAIN-2212\" x=\"9062\" y=\"0\"><\/use><g transform=\"translate(10062,0)\"><use xlink:href=\"#E5-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E5-MJMATHI-69\" x=\"692\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E5-MJMAIN-7C\" x=\"10896\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">dist(x,y) = max_i |x_i - y_i|<\/script><\/p><p><strong><span>Cosine Similarity:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"27.79ex\" height=\"5.846ex\" viewBox=\"0 -1258.5 11965.3 2517\" role=\"img\" focusable=\"false\" style=\"vertical-align: -2.923ex;\"><defs><path stroke-width=\"0\" id=\"E6-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E6-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 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y=\"-429\"><\/use><\/g><g transform=\"translate(2884,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E6-MJSZ1-2211\" x=\"0\" y=\"-1\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMATHI-70\" x=\"1493\" y=\"681\"><\/use><g transform=\"translate(746,-217)\"><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(4679,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E6-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMAIN-32\" x=\"808\" y=\"487\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E6-MJMATHI-69\" x=\"808\" y=\"-429\"><\/use><\/g><g transform=\"translate(5404,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E6-MJMAIN-5D\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(196,438)\"><g 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y=\"681\"><\/use><g transform=\"translate(746,-217)\"><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-28\" x=\"2581\" y=\"0\"><\/use><g transform=\"translate(2100,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-2212\" x=\"3886\" y=\"0\"><\/use><g transform=\"translate(3298,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-AF\" x=\"75\" y=\"4\"><\/use><\/g><g transform=\"translate(3705,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-32\" x=\"550\" y=\"674\"><\/use><\/g><g transform=\"translate(4467,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJSZ1-2211\" x=\"0\" y=\"-1\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMATHI-70\" x=\"1493\" y=\"681\"><\/use><g transform=\"translate(746,-217)\"><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-28\" x=\"8621\" y=\"0\"><\/use><g transform=\"translate(6371,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMATHI-79\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMATHI-69\" x=\"692\" y=\"-340\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-2212\" x=\"9844\" y=\"0\"><\/use><g transform=\"translate(7511,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMATHI-79\" x=\"1\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-AF\" x=\"83\" y=\"5\"><\/use><\/g><g transform=\"translate(7923,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-32\" x=\"550\" y=\"674\"><\/use><\/g><g transform=\"translate(8519,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E7-MJMAIN-5D\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(196,438)\"><g transform=\"translate(120,0)\"><rect stroke=\"none\" width=\"370\" height=\"60\" x=\"0\" y=\"95\"><\/rect><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-31\" x=\"119\" y=\"558\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E7-MJMAIN-32\" x=\"119\" y=\"-725\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">corr(x,y) = \\frac{\\sum ^p _{i=1} (x_i - \\bar{x})(y_i - \\bar{y})}{[\\sum ^p _{i=1} (x_i - \\bar{x})^2 \\sum^p _{i=1} (y_i - \\bar{y})^2]^{\\frac{1}{2}}}<\/script><\/p><p><strong><span>Mahalanobis Distance:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"31.862ex\" height=\"2.811ex\" viewBox=\"0 -906.7 13718.2 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E8-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-74\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E8-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 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xlink:href=\"#E8-MJMATHI-78\" x=\"11037\" y=\"0\"><\/use><use xlink:href=\"#E8-MJMAIN-2212\" x=\"11832\" y=\"0\"><\/use><use xlink:href=\"#E8-MJMATHI-79\" x=\"12832\" y=\"0\"><\/use><use xlink:href=\"#E8-MJMAIN-29\" x=\"13329\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">dist(x,y) = (x - y)^TS ^{-1}(x - y)<\/script><span>\t\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"38.167ex\" height=\"3.395ex\" viewBox=\"0 -956.9 16432.8 1461.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E9-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 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265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMATHI-58\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMAIN-AF\" d=\"M69 544V590H430V544H69Z\"><\/path><path stroke-width=\"0\" id=\"E9-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E9-MJMATHI-53\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(613,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6A\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6B\" x=\"412\" y=\"0\"><\/use><\/g><use xlink:href=\"#E9-MJMAIN-3D\" x=\"1650\" y=\"0\"><\/use><g transform=\"translate(2428,0)\"><g transform=\"translate(397,0)\"><rect stroke=\"none\" width=\"1447\" height=\"60\" x=\"0\" y=\"220\"><\/rect><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMAIN-31\" x=\"773\" y=\"571\"><\/use><g transform=\"translate(60,-376)\"><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMAIN-2212\" x=\"600\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMAIN-31\" x=\"1378\" y=\"0\"><\/use><\/g><\/g><\/g><g transform=\"translate(4560,0)\"><use 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xlink:href=\"#E9-MJMAIN-AF\" x=\"274\" y=\"214\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6A\" x=\"1204\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E9-MJMAIN-29\" x=\"11182\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMAIN-28\" x=\"11571\" y=\"0\"><\/use><g transform=\"translate(11960,0)\"><use xlink:href=\"#E9-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(828,-150)\"><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6B\" x=\"345\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E9-MJMAIN-2212\" x=\"13723\" y=\"0\"><\/use><g transform=\"translate(14723,0)\"><use xlink:href=\"#E9-MJMATHI-58\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E9-MJMAIN-AF\" x=\"274\" y=\"214\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E9-MJMATHI-6B\" x=\"1204\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E9-MJMAIN-29\" x=\"16043\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">S_{jk} = \\frac{1}{n-1}\\sum ^n _{i=1} (X_{ij} - \\bar{X}_j)(X_{ik} - \\bar{X}_k)<\/script><\/p><h3><a name=\"sequential-k-means\" class=\"md-header-anchor\"><\/a><span>Sequential K-Means<\/span><\/h3><ol start='' ><li><p><span>make the initial guess for the means <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.384ex\" height=\"1.76ex\" viewBox=\"0 -504.6 4901.3 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E29-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E29-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E29-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E29-MJMAIN-2C\" x=\"1331\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"1776\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"2220\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"2665\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2C\" x=\"3110\" y=\"0\"><\/use><g transform=\"translate(3554,0)\"><use xlink:href=\"#E29-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E29-MJMATHI-6B\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">m_1,...,m_k<\/script><\/p><\/li><li><p><span>set the counts <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.092ex\" height=\"1.76ex\" viewBox=\"0 -504.6 4345.3 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E18-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E18-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E18-MJMAIN-31\" x=\"848\" y=\"-213\"><\/use><use xlink:href=\"#E18-MJMAIN-2C\" x=\"1053\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"1498\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"1942\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"2387\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2C\" x=\"2832\" y=\"0\"><\/use><g transform=\"translate(3276,0)\"><use xlink:href=\"#E18-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E18-MJMATHI-6B\" x=\"848\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">n_1,...,n_k<\/script><span> to 0<\/span><\/p><\/li><li><p><span>until Interrupted<\/span><\/p><ol start='' ><li><span>acquire the next example <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><\/li><li><span>if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span> is closest to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E36-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i<\/script><span>, then increase <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.192ex\" height=\"1.76ex\" viewBox=\"0 -504.6 944 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E15-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E15-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E15-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E15-MJMATHI-69\" x=\"848\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">n_i<\/script><span> and update <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E36-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i<\/script><span> by <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"16.877ex\" height=\"3.511ex\" viewBox=\"0 -956.9 7266.3 1511.8\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.289ex;\"><defs><path stroke-width=\"0\" id=\"E17-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E17-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E17-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E17-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E17-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E17-MJMAIN-2B\" x=\"1444\" y=\"0\"><\/use><g transform=\"translate(2222,0)\"><g transform=\"translate(342,0)\"><rect stroke=\"none\" width=\"787\" height=\"60\" x=\"0\" y=\"220\"><\/rect><use transform=\"scale(0.707)\" xlink:href=\"#E17-MJMAIN-31\" x=\"306\" y=\"571\"><\/use><g transform=\"translate(60,-345)\"><use transform=\"scale(0.707)\" xlink:href=\"#E17-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E17-MJMATHI-69\" x=\"848\" y=\"-213\"><\/use><\/g><\/g><\/g><use xlink:href=\"#E17-MJMAIN-28\" x=\"3471\" y=\"0\"><\/use><use xlink:href=\"#E17-MJMATHI-78\" x=\"3860\" y=\"0\"><\/use><use xlink:href=\"#E17-MJMAIN-2212\" x=\"4655\" y=\"0\"><\/use><g transform=\"translate(5655,0)\"><use xlink:href=\"#E17-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E17-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E17-MJMAIN-29\" x=\"6877\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i + \\frac{1}{n_i}(x - m_i)<\/script><\/li><\/ol><\/li><li><p><span>end<\/span><\/p><\/li><\/ol><h3><a name=\"forgetful-sequence-k-means\" class=\"md-header-anchor\"><\/a><span>Forgetful Sequence K-Means<\/span><\/h3><p><span>Replace the counts <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"10.092ex\" height=\"1.76ex\" viewBox=\"0 -504.6 4345.3 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E18-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E18-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E18-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E18-MJMAIN-31\" x=\"848\" y=\"-213\"><\/use><use xlink:href=\"#E18-MJMAIN-2C\" x=\"1053\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"1498\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"1942\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2E\" x=\"2387\" y=\"0\"><\/use><use xlink:href=\"#E18-MJMAIN-2C\" x=\"2832\" y=\"0\"><\/use><g transform=\"translate(3276,0)\"><use xlink:href=\"#E18-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E18-MJMATHI-6B\" x=\"848\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">n_1,...,n_k<\/script><span> with a constant <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.715ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3752.2 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E19-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E19-MJMAIN-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E19-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-2208\" x=\"806\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-5B\" x=\"1751\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-30\" x=\"2029\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-2C\" x=\"2529\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-31\" x=\"2974\" y=\"0\"><\/use><use xlink:href=\"#E19-MJMAIN-5D\" x=\"3474\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">a \\in [0,1]<\/script><span>, update <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E36-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i<\/script><span> by <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"15.719ex\" height=\"2.577ex\" viewBox=\"0 -806.1 6767.8 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E21-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 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103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E21-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E21-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E21-MJMAIN-2B\" x=\"1444\" y=\"0\"><\/use><use xlink:href=\"#E21-MJMATHI-61\" x=\"2444\" y=\"0\"><\/use><use xlink:href=\"#E21-MJMAIN-28\" x=\"2973\" y=\"0\"><\/use><use xlink:href=\"#E21-MJMATHI-78\" x=\"3362\" y=\"0\"><\/use><use xlink:href=\"#E21-MJMAIN-2212\" x=\"4156\" y=\"0\"><\/use><g transform=\"translate(5156,0)\"><use xlink:href=\"#E21-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E21-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E21-MJMAIN-29\" x=\"6378\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i + a (x - m_i)<\/script><\/p><h3><a name=\"expection-maximization-algorithm-gaussian-mixture-methods\" class=\"md-header-anchor\"><\/a><span>Expection-Maximization Algorithm (Gaussian Mixture Methods)<\/span><\/h3><p><span>Initialize the model parameters<\/span><\/p><p><span>Until there is no change in any parameters<\/span><\/p><p><span>Expectation Step : calculate the probability that each object belongs to each cluster<\/span><\/p><p><span>Maximization Step : find the new estimates of the parameters that maximize the expected likelihood<\/span><\/p><p><span>end until<\/span><\/p><h3><a name=\"kernel-k-means\" class=\"md-header-anchor\"><\/a><span>Kernel K-Means<\/span><\/h3><p><span>Transform datapoint <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.127ex\" height=\"1.76ex\" viewBox=\"0 -504.6 916 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E22-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E22-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E22-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E22-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x_i<\/script><span> into a feature space <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"5.319ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2290 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E23-MJMATHI-3D5\" d=\"M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 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726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E23-MJMATHI-3D5\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E23-MJMAIN-28\" x=\"596\" y=\"0\"><\/use><g transform=\"translate(985,0)\"><use xlink:href=\"#E23-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E23-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E23-MJMAIN-29\" x=\"1900\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">\\phi(x_i)<\/script><span> and then apply the basic K-means<\/span><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"24.502ex\" 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y=\"0\"><\/use><use xlink:href=\"#E24-MJMAIN-28\" x=\"8808\" y=\"0\"><\/use><g transform=\"translate(9197,0)\"><use xlink:href=\"#E24-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E24-MJMATHI-6A\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E24-MJMAIN-29\" x=\"10160\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">K(x_i,x_j) = \\phi(x_i)^T \\phi(x_j)<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"54.968ex\" height=\"3.161ex\" viewBox=\"0 -1007.2 23666.6 1361\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E25-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMATHI-3D5\" d=\"M409 688Q413 694 421 694H429H442Q448 688 448 686Q448 679 418 563Q411 535 404 504T392 458L388 442Q388 441 397 441T429 435T477 418Q521 397 550 357T579 260T548 151T471 65T374 11T279 -10H275L251 -105Q245 -128 238 -160Q230 -192 227 -198T215 -205H209Q189 -205 189 -198Q189 -193 211 -103L234 -11Q234 -10 226 -10Q221 -10 206 -8T161 6T107 36T62 89T43 171Q43 231 76 284T157 370T254 422T342 441Q347 441 348 445L378 567Q409 686 409 688ZM122 150Q122 116 134 91T167 53T203 35T237 27H244L337 404Q333 404 326 403T297 395T255 379T211 350T170 304Q152 276 137 237Q122 191 122 150ZM500 282Q500 320 484 347T444 385T405 400T381 404H378L332 217L284 29Q284 27 285 27Q293 27 317 33T357 47Q400 66 431 100T475 170T494 234T500 282Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E25-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E25-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-7C\" x=\"278\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMATHI-3D5\" x=\"556\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-28\" x=\"1152\" y=\"0\"><\/use><g transform=\"translate(1541,0)\"><use xlink:href=\"#E25-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-29\" x=\"2456\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-2212\" x=\"3068\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMATHI-3D5\" x=\"4068\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-28\" x=\"4664\" y=\"0\"><\/use><g transform=\"translate(5053,0)\"><use xlink:href=\"#E25-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMATHI-6A\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-29\" x=\"6016\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-7C\" x=\"6405\" y=\"0\"><\/use><g transform=\"translate(6683,0)\"><use xlink:href=\"#E25-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMAIN-32\" x=\"393\" y=\"674\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-3D\" x=\"7693\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMATHI-4B\" x=\"8748\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-28\" x=\"9637\" y=\"0\"><\/use><g transform=\"translate(10026,0)\"><use xlink:href=\"#E25-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-2C\" x=\"10942\" y=\"0\"><\/use><g transform=\"translate(11387,0)\"><use xlink:href=\"#E25-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-29\" x=\"12303\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-2B\" x=\"12914\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMATHI-4B\" x=\"13914\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-28\" x=\"14803\" y=\"0\"><\/use><g transform=\"translate(15192,0)\"><use xlink:href=\"#E25-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMATHI-6A\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-2C\" x=\"16156\" y=\"0\"><\/use><g transform=\"translate(16600,0)\"><use xlink:href=\"#E25-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMATHI-6A\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-29\" x=\"17564\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-2212\" x=\"18175\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-32\" x=\"19175\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMATHI-4B\" x=\"19675\" y=\"0\"><\/use><use xlink:href=\"#E25-MJMAIN-28\" x=\"20564\" y=\"0\"><\/use><g transform=\"translate(20953,0)\"><use xlink:href=\"#E25-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMATHI-69\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-2C\" x=\"21869\" y=\"0\"><\/use><g transform=\"translate(22314,0)\"><use xlink:href=\"#E25-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E25-MJMATHI-6A\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E25-MJMAIN-29\" x=\"23277\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">||\\phi(x_i) -\\phi(x_j)||^2 = K(x_i,x_i) + K(x_j,x_j) - 2K(x_i,x_j)<\/script><\/p><p><strong><span>Homogeneous polynomial kernel:<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.046ex\" height=\"2.928ex\" viewBox=\"0 -906.7 7769.6 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E26-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMATHI-71\" d=\"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMATHI-79\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E26-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E26-MJMATHI-4B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E26-MJMATHI-71\" x=\"1200\" y=\"-213\"><\/use><use xlink:href=\"#E26-MJMAIN-28\" x=\"1274\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMATHI-78\" x=\"1663\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMAIN-2C\" x=\"2235\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMATHI-79\" x=\"2679\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMAIN-29\" x=\"3176\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMAIN-3D\" x=\"3843\" y=\"0\"><\/use><use xlink:href=\"#E26-MJMAIN-28\" x=\"4899\" y=\"0\"><\/use><g transform=\"translate(5288,0)\"><use xlink:href=\"#E26-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E26-MJMATHI-54\" x=\"808\" y=\"513\"><\/use><\/g><use xlink:href=\"#E26-MJMATHI-79\" x=\"6458\" y=\"0\"><\/use><g transform=\"translate(6955,0)\"><use xlink:href=\"#E26-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E26-MJMATHI-71\" x=\"550\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">K_q(x,y) = (x^Ty)^q<\/script><\/p><p><strong><span>Inhomogeneous polynomial kernel (c >= 0):<\/span><\/strong><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"21.89ex\" height=\"2.928ex\" viewBox=\"0 -906.7 9425 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E27-MJMATHI-4B\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 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x=\"572\" y=\"288\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">K(x,y) = e^{-\\frac{||x-y||^2}{2\\sigma ^2}}<\/script><\/p><h3><a name=\"self-organizing-feature-maps\" class=\"md-header-anchor\"><\/a><span>Self-Organizing Feature Maps<\/span><\/h3><p><span>Unlike sequential K-Means, the centres (called <\/span><strong><span>units or neurons)<\/span><\/strong><span> have a pre-determined topographic ordering relationship<\/span><\/p><ol start='' ><li><p><span>make initial guesses for centroids <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"11.384ex\" height=\"1.76ex\" viewBox=\"0 -504.6 4901.3 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E29-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E29-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E29-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E29-MJMAIN-31\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E29-MJMAIN-2C\" x=\"1331\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"1776\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"2220\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2E\" x=\"2665\" y=\"0\"><\/use><use xlink:href=\"#E29-MJMAIN-2C\" x=\"3110\" y=\"0\"><\/use><g transform=\"translate(3554,0)\"><use xlink:href=\"#E29-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E29-MJMATHI-6B\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">m_1,...,m_k<\/script><\/p><\/li><li><p><span>repeat<\/span><\/p><ol start='' ><li><span>acquire the next example <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><\/li><li><span>if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E36-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i<\/script><span> is closest to <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span>, then update <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E36-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i<\/script><span> by <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"15.719ex\" height=\"2.577ex\" viewBox=\"0 -806.1 6767.8 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E34-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E34-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E34-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E34-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E34-MJMAIN-2B\" x=\"1444\" y=\"0\"><\/use><use xlink:href=\"#E34-MJMATHI-61\" x=\"2444\" y=\"0\"><\/use><use xlink:href=\"#E34-MJMAIN-28\" x=\"2973\" y=\"0\"><\/use><use xlink:href=\"#E34-MJMATHI-78\" x=\"3362\" y=\"0\"><\/use><use xlink:href=\"#E34-MJMAIN-2212\" x=\"4156\" y=\"0\"><\/use><g transform=\"translate(5156,0)\"><use xlink:href=\"#E34-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E34-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E34-MJMAIN-29\" x=\"6378\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i + a(x - m_i)<\/script><span>; update <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.948ex\" height=\"1.994ex\" viewBox=\"0 -504.6 1269.3 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E35-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E35-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E35-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E35-MJMATHI-6A\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_j<\/script><span> (neighbor of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"2.838ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1222 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E36-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E36-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E36-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E36-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_i<\/script><span>) by <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.157ex\" height=\"2.694ex\" viewBox=\"0 -806.1 7386.8 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E37-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 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y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_j + a_n (x - m_j)<\/script><\/li><\/ol><\/li><li><p><span>end until centroids do not change or a threshold is exceeded<\/span><\/p><\/li><\/ol><h3><a name=\"agglomerative-hierarchical-clustering\" class=\"md-header-anchor\"><\/a><span>Agglomerative Hierarchical Clustering<\/span><\/h3><p><span>Start out with each data point forming its own cluster and gradually merge clusters until all points have been gathered together in one big cluster<\/span><\/p><ol start='' ><li><p><span>compute the proximity matrix<\/span><\/p><\/li><li><p><span>repeat<\/span><\/p><ol start='' ><li><span>merge the two closest clusters<\/span><\/li><li><span>update the proximity matrix to reflect the proximity between the new cluster and the original clusters<\/span><\/li><\/ol><\/li><li><p><span>until only one cluster remains<\/span><\/p><\/li><\/ol><p><strong><span>Different methods to define the distances between clusters:<\/span><\/strong><\/p><p><strong><span>Single Linkage Clustering:<\/span><\/strong><span>\t<\/span><span>distance between groups is defined as that of the <\/span><strong><span>closest<\/span><\/strong><span> pair of data<\/span><\/p><p><span>Pros: can handle non-elliptical shapes Cons: sensitive to noise and outliers <\/span><\/p><p><strong><span>Complete Linkage Clustering:<\/span><\/strong><span>\t<\/span><span>distance between two clusters is given by the distance between their most <\/span><strong><span>distant<\/span><\/strong><span> members<\/span><\/p><p><span>Pros: less susceptible to noise and outliers Cons: tend to break large clusters; biased towards globular clusters<\/span><\/p><p><strong><span>Group Average Clustering:<\/span><\/strong><span>\t<\/span><span>distance between two clusters is defined as the <\/span><strong><span>average<\/span><\/strong><span> of the distances between <\/span><strong><span>all<\/span><\/strong><span> pairs of records<\/span><\/p><p><span>Pros: less susceptible to noise and outliers Cons: biased towards globular clusters<\/span><\/p><p><strong><span>Centroid Clustering:<\/span><\/strong><span>\t<\/span><span>distance between two clusters is defined as the distance between the <\/span><strong><span>mean vectors<\/span><\/strong><span> of the two clusters<\/span><\/p><p><strong><span>Median Clustering:<\/span><\/strong><span>\t<\/span><span>distance between two clusters is defined as the distance between the <\/span><strong><span>prototypes<\/span><\/strong><span> of the two clusters<\/span><\/p><p><span>the prototype is the data point when there is only one point in the clusterl; When we merge two clusters, the new prototype is the mid point of the two prototypes<\/span><\/p><p><strong><span>Ward's Method:<\/span><\/strong><span>\t<\/span><span>Similarity of two clusters is based on the <\/span><strong><span>increase in squared error<\/span><\/strong><span> when two clusters are 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transform=\"translate(60,-345)\"><use transform=\"scale(0.707)\" xlink:href=\"#E38-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E38-MJMATHI-69\" x=\"848\" y=\"-213\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E38-MJMAIN-2B\" x=\"943\" y=\"0\"><\/use><g transform=\"translate(1217,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E38-MJMATHI-6E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E38-MJMATHI-6A\" x=\"848\" y=\"-213\"><\/use><\/g><\/g><\/g><\/g><use xlink:href=\"#E38-MJMAIN-7C\" x=\"9887\" y=\"0\"><\/use><use xlink:href=\"#E38-MJMAIN-7C\" x=\"10165\" y=\"0\"><\/use><g transform=\"translate(10443,0)\"><use xlink:href=\"#E38-MJMATHI-3BC\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E38-MJMATHI-69\" x=\"852\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E38-MJMAIN-2212\" x=\"11612\" y=\"0\"><\/use><g transform=\"translate(12612,0)\"><use xlink:href=\"#E38-MJMATHI-3BC\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E38-MJMATHI-6A\" x=\"852\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E38-MJMAIN-7C\" x=\"13606\" y=\"0\"><\/use><g transform=\"translate(13884,0)\"><use xlink:href=\"#E38-MJMAIN-7C\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E38-MJMAIN-32\" x=\"393\" y=\"674\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">\\Delta SSE(C_i,C_j) = \\frac{n_in_j}{n_i+n_j}||\\mu _i - \\mu _j||^2<\/script><\/p><h3><a name=\"dbscan\" class=\"md-header-anchor\"><\/a><span>DBSCAN<\/span><\/h3><p><strong><span>Core point:<\/span><\/strong><span>\t<\/span><span>has more than a specified number of points (MinPts) within Eps<\/span><\/p><p><strong><span>Border point:<\/span><\/strong><span>\t<\/span><span>has fewer than MinPts within Eps, but is in the neighborhood of a core point<\/span><\/p><p><strong><span>Noise point:<\/span><\/strong><span>\t<\/span><span>neither is core point nor border point<\/span><\/p><ol start='' ><li><span>Label all points as core, border, or noise points<\/span><\/li><li><span>Eliminate noise points<\/span><\/li><li><span>Put an edge between all core points that are within Eps of each other<\/span><\/li><li><span>Make each group of connected core points into a separated cluster<\/span><\/li><li><span>Assign each border point to one of the clusters containing a core point from which the border point is directly density- reachable<\/span><\/li><\/ol><p><span>Pros: resistant to noise, can handle clusters of different shapes and sizes <\/span><\/p><p><span>Cons: sensitive to denses, can not handle high dimensional data well <\/span><\/p><h3><a name=\"cluster-evaluation\" class=\"md-header-anchor\"><\/a><span>Cluster Evaluation<\/span><\/h3><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"26.798ex\" height=\"3.161ex\" viewBox=\"0 -906.7 11538 1361\" role=\"img\" 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xlink:href=\"#E39-MJMAIN-32\" x=\"550\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">SSE = \\sum _i \\sum _{x \\in C_i}(x - c_i)^2<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"23.756ex\" height=\"2.928ex\" viewBox=\"0 -906.7 10228.1 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E40-MJMATHI-53\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 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xlink:href=\"#E40-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E40-MJMAIN-32\" x=\"550\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">SSB = \\sum_i |C_i| (c - c_i)^2<\/script><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"27.959ex\" height=\"3.511ex\" viewBox=\"0 -1057.4 12037.8 1511.8\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.055ex;\"><defs><path stroke-width=\"0\" id=\"E41-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 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637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E41-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMATHI-63\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E41-MJMAIN-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E41-MJMATHI-54\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMATHI-53\" x=\"704\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMATHI-53\" x=\"1349\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMAIN-3D\" x=\"2271\" y=\"0\"><\/use><g transform=\"translate(3327,0)\"><use xlink:href=\"#E41-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMATHI-4B\" x=\"1493\" y=\"674\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMAIN-3D\" x=\"345\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMAIN-31\" x=\"1123\" y=\"0\"><\/use><\/g><\/g><g transform=\"translate(5797,0)\"><use xlink:href=\"#E41-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMAIN-2208\" x=\"572\" y=\"0\"><\/use><g transform=\"translate(876,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E41-MJMATHI-69\" x=\"1011\" y=\"-213\"><\/use><\/g><\/g><\/g><use xlink:href=\"#E41-MJMAIN-28\" x=\"8578\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMATHI-78\" x=\"8967\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMAIN-2212\" x=\"9761\" y=\"0\"><\/use><use xlink:href=\"#E41-MJMATHI-63\" x=\"10762\" y=\"0\"><\/use><g transform=\"translate(11195,0)\"><use xlink:href=\"#E41-MJMAIN-29\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E41-MJMAIN-32\" x=\"550\" y=\"513\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">TSS = \\sum _{i=1} ^K \\sum _{x \\in C_i} (x-c)^2<\/script><\/p><p><strong><span>Silhouette Coefficient:<\/span><\/strong><span>\t<\/span><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"3.318ex\" height=\"1.76ex\" viewBox=\"0 -504.6 1428.7 757.9\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E42-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E42-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E42-MJMAIN-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E42-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E42-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><use xlink:href=\"#E42-MJMAIN-3A\" x=\"1150\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">a_i:<\/script><span>\t<\/span><span>average distance of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"0.801ex\" height=\"1.994ex\" viewBox=\"0 -755.9 345 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E56-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E56-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i<\/script><span> to the points in its cluster<\/span><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.795ex\" height=\"2.344ex\" viewBox=\"0 -755.9 773 1009.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.588ex;\"><defs><path stroke-width=\"0\" id=\"E44-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path stroke-width=\"0\" id=\"E44-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E44-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E44-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">b_i<\/script><span>:<\/span><span>\t<\/span><span>min (average distance of <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"0.801ex\" height=\"1.994ex\" viewBox=\"0 -755.9 345 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E56-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E56-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i<\/script><span> to points in another cluster)<\/span><\/p><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"12.71ex\" height=\"4.212ex\" viewBox=\"0 -1107.7 5472.5 1813.3\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.639ex;\"><defs><path stroke-width=\"0\" id=\"E46-MJMATHI-73\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path stroke-width=\"0\" id=\"E46-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E46-MJMATHI-62\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 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153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E46-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E46-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" 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transform=\"scale(0.707)\" xlink:href=\"#E46-MJMATHI-78\" x=\"1406\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E46-MJMAIN-28\" x=\"1979\" y=\"0\"><\/use><g transform=\"translate(1674,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E46-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E46-MJMATHI-69\" x=\"748\" y=\"-213\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E46-MJMAIN-2C\" x=\"3240\" y=\"0\"><\/use><g transform=\"translate(2488,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E46-MJMATHI-62\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E46-MJMATHI-69\" x=\"606\" y=\"-213\"><\/use><\/g><use transform=\"scale(0.707)\" xlink:href=\"#E46-MJMAIN-29\" x=\"4291\" y=\"0\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">s = \\frac{b_i - a_i}{max(a_i,b_i)}<\/script><\/p><h2><a name=\"classification\" class=\"md-header-anchor\"><\/a><span>Classification<\/span><\/h2><h3><a name=\"minimum-distance-classifier\" class=\"md-header-anchor\"><\/a><span>Minimum Distance Classifier<\/span><\/h3><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"17.482ex\" height=\"3.745ex\" viewBox=\"0 -956.9 7527.1 1612.3\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.522ex;\"><defs><path stroke-width=\"0\" id=\"E47-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJMAIN-3D\" d=\"M56 347Q56 360 70 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2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJSZ1-2211\" d=\"M61 748Q64 750 489 750H913L954 640Q965 609 976 579T993 533T999 516H979L959 517Q936 579 886 621T777 682Q724 700 655 705T436 710H319Q183 710 183 709Q186 706 348 484T511 259Q517 250 513 244L490 216Q466 188 420 134T330 27L149 -187Q149 -188 362 -188Q388 -188 436 -188T506 -189Q679 -189 778 -162T936 -43Q946 -27 959 6H999L913 -249L489 -250Q65 -250 62 -248Q56 -246 56 -239Q56 -234 118 -161Q186 -81 245 -11L428 206Q428 207 242 462L57 717L56 728Q56 744 61 748Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJMAIN-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E47-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E47-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMATHI-6A\" x=\"1241\" y=\"-213\"><\/use><use xlink:href=\"#E47-MJMAIN-3D\" x=\"1547\" y=\"0\"><\/use><g transform=\"translate(2325,0)\"><g transform=\"translate(397,0)\"><rect stroke=\"none\" width=\"964\" height=\"60\" x=\"0\" y=\"220\"><\/rect><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMAIN-31\" x=\"432\" y=\"571\"><\/use><g transform=\"translate(60,-388)\"><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMATHI-4E\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E47-MJMATHI-6A\" x=\"1135\" y=\"-213\"><\/use><\/g><\/g><\/g><g transform=\"translate(3974,0)\"><use xlink:href=\"#E47-MJSZ1-2211\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(1056,-286)\"><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMAIN-2208\" x=\"572\" y=\"0\"><\/use><g transform=\"translate(876,0)\"><use transform=\"scale(0.707)\" xlink:href=\"#E47-MJMATHI-43\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.5)\" xlink:href=\"#E47-MJMATHI-6A\" x=\"1011\" y=\"-213\"><\/use><\/g><\/g><\/g><use xlink:href=\"#E47-MJMATHI-78\" x=\"6955\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">m_j = \\frac{1}{N_j} \\sum _{x \\in C_j} x<\/script><\/p><p><span>The goal is to minimize <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.763ex\" height=\"2.694ex\" viewBox=\"0 -806.1 8078.7 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E48-MJMATHI-44\" d=\"M287 628Q287 635 230 637Q207 637 200 638T193 647Q193 655 197 667T204 682Q206 683 403 683Q570 682 590 682T630 676Q702 659 752 597T803 431Q803 275 696 151T444 3L430 1L236 0H125H72Q48 0 41 2T33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM703 469Q703 507 692 537T666 584T629 613T590 629T555 636Q553 636 541 636T512 636T479 637H436Q392 637 386 627Q384 623 313 339T242 52Q242 48 253 48T330 47Q335 47 349 47T373 46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 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140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E48-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E48-MJMATHI-44\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E48-MJMATHI-6A\" x=\"1170\" y=\"-213\"><\/use><use xlink:href=\"#E48-MJMAIN-28\" x=\"1219\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMATHI-78\" x=\"1608\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-29\" x=\"2180\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-3D\" x=\"2847\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-7C\" x=\"3902\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-7C\" x=\"4180\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMATHI-78\" x=\"4458\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-2212\" x=\"5253\" y=\"0\"><\/use><g transform=\"translate(6253,0)\"><use xlink:href=\"#E48-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E48-MJMATHI-6A\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E48-MJMAIN-7C\" x=\"7522\" y=\"0\"><\/use><use xlink:href=\"#E48-MJMAIN-7C\" x=\"7800\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">D_j(x) = ||x - m_j||<\/script><\/p><p><span>Which is equivalent with maximizing <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"23.978ex\" height=\"3.395ex\" viewBox=\"0 -956.9 10323.8 1461.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -1.172ex;\"><defs><path stroke-width=\"0\" id=\"E49-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 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319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E49-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" 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193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E49-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 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429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E49-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-6A\" x=\"735\" y=\"-213\"><\/use><use xlink:href=\"#E49-MJMAIN-28\" x=\"911\" y=\"0\"><\/use><use xlink:href=\"#E49-MJMATHI-78\" x=\"1300\" y=\"0\"><\/use><use xlink:href=\"#E49-MJMAIN-29\" x=\"1872\" y=\"0\"><\/use><use xlink:href=\"#E49-MJMAIN-3D\" x=\"2539\" y=\"0\"><\/use><g transform=\"translate(3594,0)\"><use xlink:href=\"#E49-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-54\" x=\"1241\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-6A\" x=\"1241\" y=\"-429\"><\/use><\/g><use xlink:href=\"#E49-MJMATHI-78\" x=\"5070\" y=\"0\"><\/use><use xlink:href=\"#E49-MJMAIN-2212\" x=\"5864\" y=\"0\"><\/use><g transform=\"translate(6642,0)\"><g transform=\"translate(342,0)\"><rect stroke=\"none\" width=\"473\" height=\"60\" x=\"0\" y=\"220\"><\/rect><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMAIN-31\" x=\"84\" y=\"571\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMAIN-32\" x=\"84\" y=\"-531\"><\/use><\/g><\/g><g transform=\"translate(7578,0)\"><use xlink:href=\"#E49-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-54\" x=\"1241\" y=\"487\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-6A\" x=\"1241\" y=\"-429\"><\/use><\/g><g transform=\"translate(9054,0)\"><use xlink:href=\"#E49-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E49-MJMATHI-6A\" x=\"1241\" y=\"-213\"><\/use><\/g><\/g><\/svg><\/span><script type=\"math\/tex\">d_j(x) = m_j^Tx - \\frac{1}{2} m_j^Tm_j<\/script><span>\t<\/span><span>the decision function for class j<\/span><\/p><h3><a name=\"linear-discriminant-classifier\" class=\"md-header-anchor\"><\/a><span>Linear Discriminant Classifier<\/span><\/h3><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.632ex\" height=\"2.811ex\" viewBox=\"0 -906.7 8022 1210.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E50-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 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1034 683Q1048 683 1048 672Q1048 666 1045 655T1038 640T1028 637Q1006 637 988 631T958 617T939 600T927 584L923 578L754 282Q586 -14 585 -15Q579 -22 561 -22Q546 -22 542 -17Q539 -14 523 229T506 480L494 462Q472 425 366 239Q222 -13 220 -15T215 -19Q210 -22 197 -22Q178 -22 176 -15Q176 -12 154 304T131 622Q129 631 121 633T82 637H58Q51 644 51 648Q52 671 64 683H76Q118 680 176 680Q301 680 313 683H323Q329 677 329 674T327 656Q322 641 318 637H297Q236 634 232 620Q262 160 266 136L501 550L499 587Q496 629 489 632Q483 636 447 637Q428 637 422 639T416 648Q416 650 418 660Q419 664 420 669T421 676T424 680T428 682T436 683Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMATHI-54\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMAIN-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path stroke-width=\"0\" id=\"E50-MJMATHI-6F\" d=\"M201 -11Q126 -11 80 38T34 156Q34 221 64 279T146 380Q222 441 301 441Q333 441 341 440Q354 437 367 433T402 417T438 387T464 338T476 268Q476 161 390 75T201 -11ZM121 120Q121 70 147 48T206 26Q250 26 289 58T351 142Q360 163 374 216T388 308Q388 352 370 375Q346 405 306 405Q243 405 195 347Q158 303 140 230T121 120Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use 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xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span> is assigned to one class if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.509ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3663.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E52-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 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145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E52-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-28\" x=\"480\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMATHI-78\" x=\"869\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-29\" x=\"1441\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-3E\" x=\"2107\" y=\"0\"><\/use><use xlink:href=\"#E52-MJMAIN-30\" x=\"3163\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">g(x) \\gt 0<\/script><span> and another class if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.509ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3663.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E53-MJMATHI-67\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path stroke-width=\"0\" id=\"E53-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E53-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E53-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E53-MJMAIN-3C\" d=\"M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z\"><\/path><path stroke-width=\"0\" id=\"E53-MJMAIN-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E53-MJMATHI-67\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E53-MJMAIN-28\" x=\"480\" y=\"0\"><\/use><use xlink:href=\"#E53-MJMATHI-78\" x=\"869\" y=\"0\"><\/use><use xlink:href=\"#E53-MJMAIN-29\" x=\"1441\" y=\"0\"><\/use><use xlink:href=\"#E53-MJMAIN-3C\" x=\"2107\" y=\"0\"><\/use><use xlink:href=\"#E53-MJMAIN-30\" x=\"3163\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">g(x) \\lt 0<\/script><\/p><h3><a name=\"k-nearest-neighbor\" class=\"md-header-anchor\"><\/a><span>k Nearest Neighbor<\/span><\/h3><p><span>lazy learners, does not build model explicitly<\/span><\/p><p><span>can have arbitary decision boundries<\/span><\/p><h3><a name=\"bayes-classifier\" class=\"md-header-anchor\"><\/a><span>Bayes Classifier<\/span><\/h3><p><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"30.276ex\" height=\"2.577ex\" viewBox=\"0 -806.1 13035.6 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E54-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-41\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMATHI-43\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E54-MJMAIN-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E54-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-28\" x=\"751\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-41\" x=\"1140\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-7C\" x=\"1890\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-43\" x=\"2168\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-29\" x=\"2928\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-3D\" x=\"3594\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-50\" x=\"4650\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-28\" x=\"5401\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-41\" x=\"5790\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-29\" x=\"6540\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-50\" x=\"6929\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-28\" x=\"7680\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-43\" x=\"8069\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-7C\" x=\"8829\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-41\" x=\"9107\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-29\" x=\"9857\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-2F\" x=\"10246\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-50\" x=\"10746\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-28\" x=\"11497\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMATHI-43\" x=\"11886\" y=\"0\"><\/use><use xlink:href=\"#E54-MJMAIN-29\" x=\"12646\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P(A|C) = P(A)P(C|A) \/ P(C)<\/script><\/p><p><span>A point <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"1.329ex\" height=\"1.41ex\" viewBox=\"0 -504.6 572 607.1\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E55-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E55-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">x<\/script><span> belongs to class <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"0.801ex\" height=\"1.994ex\" viewBox=\"0 -755.9 345 858.4\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.238ex;\"><defs><path stroke-width=\"0\" id=\"E56-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E56-MJMATHI-69\" x=\"0\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">i<\/script><span> if <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.364ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3601 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E57-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E57-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E57-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMAIN-28\" x=\"751\" y=\"0\"><\/use><g transform=\"translate(1140,0)\"><use xlink:href=\"#E57-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E57-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E57-MJMAIN-7C\" x=\"2361\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMATHI-78\" x=\"2639\" y=\"0\"><\/use><use xlink:href=\"#E57-MJMAIN-29\" x=\"3211\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P(m_i | x) <\/script><span> is the largest<\/span><\/p><p><span>Normally we are given <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.364ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3601 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E58-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E58-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E58-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMAIN-28\" x=\"751\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMATHI-78\" x=\"1140\" y=\"0\"><\/use><use xlink:href=\"#E58-MJMAIN-7C\" x=\"1712\" y=\"0\"><\/use><g transform=\"translate(1990,0)\"><use xlink:href=\"#E58-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E58-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E58-MJMAIN-29\" x=\"3211\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P(x|m_i)<\/script><span> and <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"6.389ex\" height=\"2.577ex\" viewBox=\"0 -806.1 2751 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E59-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E59-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E59-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E59-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E59-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E59-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E59-MJMAIN-28\" x=\"751\" y=\"0\"><\/use><g transform=\"translate(1140,0)\"><use xlink:href=\"#E59-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E59-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E59-MJMAIN-29\" x=\"2361\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P(m_i)<\/script><span> to compute <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"8.364ex\" height=\"2.577ex\" viewBox=\"0 -806.1 3601 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex;\"><defs><path stroke-width=\"0\" id=\"E60-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMATHI-6D\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMATHI-69\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E60-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E60-MJMATHI-50\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMAIN-28\" x=\"751\" y=\"0\"><\/use><g transform=\"translate(1140,0)\"><use xlink:href=\"#E60-MJMATHI-6D\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E60-MJMATHI-69\" x=\"1241\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E60-MJMAIN-7C\" x=\"2361\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMATHI-78\" x=\"2639\" y=\"0\"><\/use><use xlink:href=\"#E60-MJMAIN-29\" x=\"3211\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">P(m_i|x)<\/script><\/p><p><span>the decision function is <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"21.418ex\" height=\"2.694ex\" viewBox=\"0 -806.1 9221.5 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex;\"><defs><path stroke-width=\"0\" id=\"E61-MJMATHI-64\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E61-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E61-MJMATHI-64\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E61-MJMATHI-6A\" x=\"735\" y=\"-213\"><\/use><use xlink:href=\"#E61-MJMAIN-28\" x=\"911\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMATHI-78\" x=\"1300\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMAIN-29\" x=\"1872\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMAIN-3D\" x=\"2539\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMATHI-70\" x=\"3594\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMAIN-28\" x=\"4097\" y=\"0\"><\/use><g transform=\"translate(4486,0)\"><use xlink:href=\"#E61-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E61-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E61-MJMAIN-29\" x=\"5594\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMATHI-70\" x=\"5983\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMAIN-28\" x=\"6486\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMATHI-78\" x=\"6875\" y=\"0\"><\/use><use xlink:href=\"#E61-MJMAIN-7C\" x=\"7447\" y=\"0\"><\/use><g transform=\"translate(7725,0)\"><use xlink:href=\"#E61-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E61-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E61-MJMAIN-29\" x=\"8832\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">d_j(x) = p(w_j)p(x|w_j)<\/script><\/p><p><span>How to estimate <\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"7.612ex\" height=\"2.694ex\" viewBox=\"-39 -806.1 3277.3 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex; margin-left: -0.091ex;\"><defs><path stroke-width=\"0\" id=\"E62-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E62-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E62-MJMATHI-70\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E62-MJMAIN-28\" x=\"503\" y=\"0\"><\/use><use xlink:href=\"#E62-MJMATHI-78\" x=\"892\" y=\"0\"><\/use><use xlink:href=\"#E62-MJMAIN-7C\" x=\"1464\" y=\"0\"><\/use><g transform=\"translate(1742,0)\"><use xlink:href=\"#E62-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E62-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E62-MJMAIN-29\" x=\"2849\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">p(x|w_j)<\/script><span>:<\/span><span>\t<\/span><span class=\"MathJax_SVG\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"32.273ex\" height=\"2.694ex\" viewBox=\"-39 -806.1 13895.4 1160\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.822ex; margin-left: -0.091ex;\"><defs><path stroke-width=\"0\" id=\"E63-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E63-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E63-MJMATHI-78\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path stroke-width=\"0\" id=\"E63-MJMAIN-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path stroke-width=\"0\" id=\"E63-MJMATHI-77\" d=\"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z\"><\/path><path stroke-width=\"0\" id=\"E63-MJMATHI-6A\" d=\"M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z\"><\/path><path stroke-width=\"0\" id=\"E63-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E63-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E63-MJMATHI-50\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path stroke-width=\"0\" id=\"E63-MJMAIN-31\" d=\"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\"><\/path><path stroke-width=\"0\" id=\"E63-MJMAIN-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path stroke-width=\"0\" id=\"E63-MJMATHI-6E\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E63-MJMATHI-70\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-28\" x=\"503\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMATHI-78\" x=\"892\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-7C\" x=\"1464\" y=\"0\"><\/use><g transform=\"translate(1742,0)\"><use xlink:href=\"#E63-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E63-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E63-MJMAIN-29\" x=\"2849\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-3D\" x=\"3516\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMATHI-50\" x=\"4571\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-28\" x=\"5322\" y=\"0\"><\/use><g transform=\"translate(5711,0)\"><use xlink:href=\"#E63-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E63-MJMAIN-31\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E63-MJMAIN-7C\" x=\"6737\" y=\"0\"><\/use><g transform=\"translate(7015,0)\"><use xlink:href=\"#E63-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E63-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E63-MJMAIN-29\" x=\"8122\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-2E\" x=\"8511\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-2E\" x=\"8956\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-2E\" x=\"9401\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMATHI-50\" x=\"9845\" y=\"0\"><\/use><use xlink:href=\"#E63-MJMAIN-28\" x=\"10596\" y=\"0\"><\/use><g transform=\"translate(10985,0)\"><use xlink:href=\"#E63-MJMATHI-78\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E63-MJMATHI-6E\" x=\"808\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E63-MJMAIN-7C\" x=\"12082\" y=\"0\"><\/use><g transform=\"translate(12360,0)\"><use xlink:href=\"#E63-MJMATHI-77\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E63-MJMATHI-6A\" x=\"1012\" y=\"-213\"><\/use><\/g><use xlink:href=\"#E63-MJMAIN-29\" x=\"13467\" y=\"0\"><\/use><\/g><\/svg><\/span><script type=\"math\/tex\">p(x|w_j) = P(x_1|w_j)...P(x_n|w_j)<\/script><\/p><p> <\/p><\/div>\n<\/body>\n<\/html>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Data Mining Cheat Sheet Data Mining Cheat Sheet Introduction Definition 1: The nontrivial extraction of implicit, previous unknown, and potentially… <\/p>\n<div class=\"art-el-more\"><a href=\"https:\/\/yunpeng.site\/index.php\/2021\/04\/16\/dmsum\/\" class=\"art-link art-color-link art-w-chevron\">Read more<\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"spay_email":"","footnotes":""},"categories":[3],"tags":[5],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts\/122"}],"collection":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/comments?post=122"}],"version-history":[{"count":4,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts\/122\/revisions"}],"predecessor-version":[{"id":227,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/posts\/122\/revisions\/227"}],"wp:attachment":[{"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/media?parent=122"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/categories?post=122"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yunpeng.site\/index.php\/wp-json\/wp\/v2\/tags?post=122"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}