{"id":8393,"date":"2020-03-27T15:49:01","date_gmt":"2020-03-27T14:49:01","guid":{"rendered":"https:\/\/complex-systems-ai.com\/?page_id=8393"},"modified":"2024-02-25T13:30:26","modified_gmt":"2024-02-25T12:30:26","slug":"criteres-de-qualite-internes","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/","title":{"rendered":"Internal quality criteria"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"8393\" class=\"elementor elementor-8393\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-28a9395 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"28a9395\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3a62fc3\" data-id=\"3a62fc3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-42d4b9f elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"42d4b9f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Data partitioning<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\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<div 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class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Wiki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-41cb066 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"41cb066\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7b4fa9a\" data-id=\"7b4fa9a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fdb584a elementor-widget elementor-widget-heading\" data-id=\"fdb584a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_88 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewbox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewbox=\"0 0 24 24\" version=\"1.2\" baseprofile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Criteres-de-qualite-internes\" >Internal quality criteria<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Liste\" >List<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Comment-lire-les-mesures-de-qualite-interne\" >How to Read Internal Quality Metrics<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Somme-de-lerreur-quadratique\" >Sum of squared error<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Criteres-de-dispersion\" >Dispersion criteria<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Uutilitaire-de-categorie\" >Category utility<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Mesure-de-Coupe\" >Cutting Measurement<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Ball-Hall\" >Ball Hall<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Banfeld-Raftery\" >Banfeld-Raftery<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Concordet\" >Concordet<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Critere-C\" >Criterion C<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Calinski-Harabasz\" >Calinski-Harabasz<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Davies-Bouldin\" >Davies-Bouldin<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Det-Ratio\" >Det_Ratio<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Dunn\" >Dunn<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#GDImn\" >GDImn<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Baker-Hubert-Gamma\" >Baker-Hubert Gamma<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#G\" >G+<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Ksq-DetW\" >Ksq_DetW<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Log-Det-Ratio\" >Log_Det_Ratio<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Log-SS-Ratio\" >Log_SS_Ratio<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#McClain-Rao\" >McClain-Rao<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#PBM\" >PBM<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Point-Biserial\" >Point-Biserial<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Ratkowsky-Lance\" >Ratkowsky-Lance<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Ray-Turi\" >Ray-Turi<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Scott-Symons\" >Scott Symons<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#SD-Scat-et-SD-Dis\" >SD_Scat and SD_Dis<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#S-Dbw\" >S_Dbw<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Silhouette\" >Silhouette<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Tau\" >tau<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Trace-W\" >Trace_W<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Trace-WiB\" >Trace_WiB<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-34\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Wemmert-Gancarski\" >Wemmert-Gan\u00e7arski<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-35\" href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/internal-quality-criteria\/#Xie-Beni\" >Xie-Beni<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Criteres-de-qualite-internes\"><\/span>Internal quality criteria<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5949388a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5949388a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-73abcf85\" data-id=\"73abcf85\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5d1ad416 elementor-widget elementor-widget-text-editor\" data-id=\"5d1ad416\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><\/p>\n<p class=\"wp-block-paragraph\">Internal quality criteria typically measure the compactness of clusters using a similarity measure. It typically measures intra-cluster homogeneity, inter-cluster separability, or a combination of these two. It does not use external information alongside the data itself.&nbsp;<\/p>\n<p><img decoding=\"async\" class=\"aligncenter wp-image-11096 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/09\/cropped-Capture.png\" alt=\"internal quality criteria\" width=\"97\" height=\"97\" title=\"\"><\/p>\n<p><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7b2fc8f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7b2fc8f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a2552c4\" data-id=\"a2552c4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8118be8 elementor-widget elementor-widget-heading\" data-id=\"8118be8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Liste\"><\/span>List<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ecbaee7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ecbaee7\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f938b08\" data-id=\"f938b08\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-78b43e0 elementor-widget elementor-widget-text-editor\" data-id=\"78b43e0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ul>\n<li>Sum of the quadratic error<\/li>\n<li>Dispersion criteria<\/li>\n<li><span style=\"color: var( --e-global-color-text ); font-family: var( --e-global-typography-text-font-family ), Sans-serif; font-weight: var( --e-global-typography-text-font-weight ); font-size: 1.125rem;\">Category utility metric<\/span><\/li>\n<li>Cutting measures<\/li>\n<li>Ball Hall<\/li>\n<li>Banfeld-Raftery<\/li>\n<li>Condorcet criterion<\/li>\n<li>Criterion C<\/li>\n<li>Calinski-Harabasz<\/li>\n<li>Davies-Bouldin<\/li>\n<li>Det_Ratio<\/li>\n<li>Dunn<\/li>\n<li>GDImn<\/li>\n<li>Gamma<\/li>\n<li>G+<\/li>\n<li>Ksq_DetW<\/li>\n<li>Log_Det_Ratio<\/li>\n<li>Log_SS_Ratio<\/li>\n<li>McClain-Rao<\/li>\n<li>PBM<\/li>\n<li>Biserial point<\/li>\n<li>Ratkawsky-Lance<\/li>\n<li>Ray-Turi<\/li>\n<li>Scott Symons<\/li>\n<li>SD_Scat<\/li>\n<li>SD_Dis<\/li>\n<li>S_Dbw<\/li>\n<li>Silhouette<\/li>\n<li>Trace W<\/li>\n<li>WiB trace<\/li>\n<li>Wemmert-Gan\u00e7arski<\/li>\n<li>Xie-Beni<\/li>\n<\/ul>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e2152c2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e2152c2\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f66ada4\" data-id=\"f66ada4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-78c3f1f elementor-widget elementor-widget-heading\" data-id=\"78c3f1f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Comment-lire-les-mesures-de-qualite-interne\"><\/span>How to Read Internal Quality Metrics<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e186eca elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e186eca\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f3bf07b\" data-id=\"f3bf07b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-327af23 elementor-widget elementor-widget-text-editor\" data-id=\"327af23\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>In order to find the best partition of the data, we generally run a <a href=\"https:\/\/complex-systems-ai.com\/en\/algorithmic\/\">algorithm<\/a> of <a href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/\">clustering<\/a> with different values of the expected number of clusters K: let&#039;s say that Km \u2264 K \u2264 KM. The applied clustering algorithm could be Ascending Hierarchical Clustering (AHC) or k-means algorithm or any other technique. We then calculate a QK quality index for each value of K and select the partition that led to the \u201cbest\u201d value for QK.<\/p>\n<p>This section explains what is considered the \u201cbest\u201d value for the different quality indices.<\/p>\n<p>The table summarizes, for each index, which rule should be applied in order to determine the best index value. For example, in the case of the Calinski-Harabasz index, if the quality index has been calculated for different partitions of the data, the best partition is the one corresponding to the largest value of the index.<\/p>\n<p><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-21108 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_interne1.png\" alt=\"internal quality\" width=\"226\" height=\"525\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_interne1.png 226w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_interne1-129x300.png 129w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_interne1-5x12.png 5w\" sizes=\"(max-width: 226px) 100vw, 226px\" \/><\/p>\n<p>The decision rules called max and min in the table mean selecting the largest or smallest index value, respectively. The decision rule called max diff means that the best value of K is that corresponding to the greatest difference between two successive slopes. On a chart representing index values versus <a href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/quality-over-number-of-clusters\/\">number of clusters<\/a> selected, this corresponds to an elbow.<\/p>\n<p><img decoding=\"async\" class=\"alignnone wp-image-21109 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_interne2.png\" alt=\"internal quality\" width=\"417\" height=\"207\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_interne2.png 417w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_interne2-300x149.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_interne2-18x9.png 18w\" sizes=\"(max-width: 417px) 100vw, 417px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-259f237 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"259f237\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-efb29a5\" data-id=\"efb29a5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8967d82 elementor-widget elementor-widget-heading\" data-id=\"8967d82\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Somme-de-lerreur-quadratique\"><\/span>Sum of squared error\n<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e16f51a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e16f51a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-77f77c5\" data-id=\"77f77c5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e9b3d9f elementor-widget elementor-widget-text-editor\" data-id=\"e9b3d9f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Internal quality metrics typically measure cluster compactness using a measure of similarity (such as Sum of Squared Error). It typically measures intra-cluster homogeneity, inter-cluster separability, or a combination of these two. It does not use external information alongside the data itself.<\/p>\n<p>Sum of squared error is the simplest and most widely used criterion measure for clustering. It is calculated as:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Sum of Quadratic Error Sum of Quadratic Error\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval1.png\" alt=\"Sum of the quadratic error\" width=\"249\" height=\"73\" \/><\/figure>\n<p>where C_k is the set of instances of cluster k; \u03bc_k is the vector mean of cluster k. The components of \u03bc_k are calculated as:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Sum of Quadratic Error Sum of Quadratic Error\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval2.png\" alt=\"Sum of the quadratic error\" width=\"184\" height=\"59\" \/><\/figure>\n<p>where N_k = | C_k | is the number of instances belonging to cluster k.<\/p>\n<p>The methods of\u00a0<a href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/\">partitioning<\/a>\u00a0which minimize the SSE criterion are often called minimum variance partitions, because by simple algebraic manipulation the SSE criterion can be written:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Sum of Quadratic Error Sum of Quadratic Error\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval3.png\" alt=\"Sum of the quadratic error\" width=\"320\" height=\"186\" \/><\/figure>\n<p>The SSE criterion function is suitable for cases where the clusters form compact clouds well separated from each other.<\/p>\n<p>Additional minimum criteria for SSE can be produced by replacing the value of S_k with expressions such as:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Sum of Quadratic Error Sum of Quadratic Error\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval4.png\" alt=\"Sum of the quadratic error\" width=\"238\" height=\"133\" \/><\/figure>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b91cd09 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b91cd09\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5eb3547\" data-id=\"5eb3547\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c705845 elementor-widget elementor-widget-heading\" data-id=\"c705845\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Criteres-de-dispersion\"><\/span>Dispersion criteria<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-211bf88 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"211bf88\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e63ecac\" data-id=\"e63ecac\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a0c3525 elementor-widget elementor-widget-text-editor\" data-id=\"a0c3525\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Internal quality metrics generally measure the compactness of clusters using a measure of similarity (such as Dispersion criteria: trace, determinant, invariance). It typically measures intra-cluster homogeneity, inter-cluster separability, or a combination of these two. It does not use external information alongside the data itself.<\/p>\n<p>The scalar diffusion criteria are derived from the diffusion matrices, reflecting the intra-cluster diffusion, the inter-cluster diffusion and their summation \u2013 the total diffusion matrix. For the k-th cluster, the diffusion matrix can be calculated as follows:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Dispersal Criteria Dispersal Criteria\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval5.png\" alt=\"Dispersion criteria\" width=\"251\" height=\"62\" \/><\/figure>\n<p>The intra-cluster dispersion matrix is calculated as the sum of the last definition over all W clusters:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Dispersal Criteria Dispersal Criteria\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval6.png\" alt=\"Dispersion criteria\" width=\"132\" height=\"68\" \/><\/figure>\n<p>The diffusion matrix between clusters can be calculated as follows:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Dispersal Criteria Dispersal Criteria\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval7.png\" alt=\"Dispersion criteria\" width=\"288\" height=\"75\" \/><\/figure>\n<p>where \u03bc is the total mean vector and is defined as:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Dispersal Criteria Dispersal Criteria\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval8.png\" alt=\"Dispersion criteria\" width=\"160\" height=\"72\" \/><\/figure>\n<p>The total diffusion matrix should be calculated as follows:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Dispersal Criteria Dispersal Criteria\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval9.png\" alt=\"Dispersion criteria\" width=\"287\" height=\"64\" \/><\/figure>\n<p>Three scalar criteria can be derived from S_W, S_B and S_T.<span id=\"La-trace\" class=\"ez-toc-section\"><\/span><\/p>\n<p>The trace is the sum of the diagonal elements of a matrix. Minimizing the trace of S_W is similar to minimizing\u00a0<a href=\"https:\/\/complex-systems-ai.com\/en\/partitionnement-de-donnees\/somme-de-lerreur-quadratique\/\">HSE<\/a>\u00a0and is therefore commonly used. This criterion, representing the intra-cluster dispersion, is calculated as follows:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Dispersal Criteria Dispersal Criteria\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval10.png\" alt=\"Trace dispersion criteria\" width=\"279\" height=\"71\" \/><\/figure>\n<p>Another criterion, which can be maximized, is the criterion between clusters:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Dispersal Criteria Dispersal Criteria\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval11.png\" alt=\"Trace dispersion criteria\" width=\"227\" height=\"65\" \/><span id=\"Le-determinant\" class=\"ez-toc-section\"><\/span><\/figure>\n<p>The determinant of a scattering matrix measures approximately the square of the scattering volume. Since S_B will be singular if the\u00a0<a href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/quality-over-number-of-clusters\/\">number of clusters<\/a>\u00a0is less than or equal to dimensionality, or if mc is less than dimensionality, its determinant is not an appropriate criterion. If we assume that S_W is not singular, the function of the determinant criterion is:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Dispersal Criteria Dispersal Criteria\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval12.png\" alt=\"Decisive dispersion criteria\" width=\"180\" height=\"70\" \/><span id=\"Linvariance\" class=\"ez-toc-section\"><\/span><\/figure>\n<p>The eigenvalues \u03bb_1, \u03bb_2 ,. . . , \u03bb_d of S_W * S_B are the basic linear invariants of the diffusion matrices. The good partitions are those for which the non-zero eigenvalues are large. As a result, several criteria can be derived, including eigenvalues. Three of these criteria are:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Dispersal Criteria Dispersal Criteria\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval13.png\" alt=\"Dispersion invariance criteria\" width=\"262\" height=\"163\" \/><\/figure>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-60f0862 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"60f0862\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-18fb3f5\" data-id=\"18fb3f5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d3630c2 elementor-widget elementor-widget-heading\" data-id=\"d3630c2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Uutilitaire-de-categorie\"><\/span>Category utility<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-39418bd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"39418bd\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-27f25a3\" data-id=\"27f25a3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f299b12 elementor-widget elementor-widget-text-editor\" data-id=\"f299b12\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Category utility is defined as increasing the expected number of entity values that can be correctly predicted given a certain grouping. This metric is useful for problems that contain a relatively small number of nominal features each having a small cardinality.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-59aabb1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"59aabb1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c19d0d7\" data-id=\"c19d0d7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-11082dc elementor-widget elementor-widget-heading\" data-id=\"11082dc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Mesure-de-Coupe\"><\/span>Cutting Measurement<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ea23454 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ea23454\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-caedc2f\" data-id=\"caedc2f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-09af387 elementor-widget elementor-widget-text-editor\" data-id=\"09af387\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>In some cases, it is useful to represent the clustering problem as a minimal cutting problem. In such cases, the quality is measured as the ratio of the remaining weights to the total cut weights. If there is no restriction on the size of the clusters, it is easy to find the optimal value. Thus, the min-cut measurement is revised to penalize unbalanced structures.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-79822b2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"79822b2\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-12b2cd1\" data-id=\"12b2cd1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-928e803 elementor-widget elementor-widget-heading\" data-id=\"928e803\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Ball-Hall\"><\/span>Ball Hall<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-da5b85d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"da5b85d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-093a76e\" data-id=\"093a76e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b3c6735 elementor-widget elementor-widget-text-editor\" data-id=\"b3c6735\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The average dispersion of a cluster is the average of the squares of the distances of the points in the cluster from their barycenter. The Ball-Hall index is the average, across all clusters, of their average dispersion:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21050\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/ball-hall.png\" alt=\"ball hall\" width=\"239\" height=\"57\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/ball-hall.png 239w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/ball-hall-18x4.png 18w\" sizes=\"(max-width: 239px) 100vw, 239px\" \/><\/p>\n<p>In the particular case where all the clusters have the same size N\/K, this sum is reduced to WGSS\/N.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6c760ce elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6c760ce\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6be47e5\" data-id=\"6be47e5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0568a1a elementor-widget elementor-widget-heading\" data-id=\"0568a1a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Banfeld-Raftery\"><\/span>Banfeld-Raftery <span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f7c1874 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f7c1874\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a3b1c40\" data-id=\"a3b1c40\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-218d47f elementor-widget elementor-widget-text-editor\" data-id=\"218d47f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>This index is the weighted sum of the logarithms of the traces of the variance-covariance matrix of each cluster.<\/p>\n<p>The index can be written as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21051\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Banfeld-Raftery.png\" alt=\"Banfeld-Raftery\" width=\"201\" height=\"52\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Banfeld-Raftery.png 201w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Banfeld-Raftery-18x5.png 18w\" sizes=\"(max-width: 201px) 100vw, 201px\" \/><\/p>\n<p>The quantity Tr(WG{k})\/nk can be interpreted as the average of the squares of the distances between the points of the cluster Ck and their barycenter G{k}. If a cluster contains a single point, this trace is equal to 0 and the logarithm is undefined.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ac4d565 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ac4d565\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9a9a7bd\" data-id=\"9a9a7bd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dceb6d5 elementor-widget elementor-widget-heading\" data-id=\"dceb6d5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Concordet\"><\/span>Concordet<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9592941 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9592941\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6cefd17\" data-id=\"6cefd17\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fcc9541 elementor-widget elementor-widget-text-editor\" data-id=\"fcc9541\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Another suitable approach is to apply the Condorcet solution to the clustering problem. In this case, the criterion is calculated as follows:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval14.png\" alt=\"internal quality criteria condorcet criterion\" width=\"507\" height=\"86\" title=\"\"><\/figure>\n<p>where s (x_j, x_k) and d (x_j, x_k) measure the similarity and distance of vectors x_j and x_k.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-88e6ed3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"88e6ed3\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ec991f1\" data-id=\"ec991f1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e8fa1f1 elementor-widget elementor-widget-heading\" data-id=\"e8fa1f1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Critere-C\"><\/span>Criterion C<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-97b904f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"97b904f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b809285\" data-id=\"b809285\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-50c8aee elementor-widget elementor-widget-text-editor\" data-id=\"50c8aee\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Criterion C is an extension of Condorcet&#039;s criterion and is defined as (where \u03b3 is a threshold value):<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval15.png\" alt=\"internal quality criteria criterion C\" width=\"581\" height=\"91\" title=\"\"><\/figure>\n<p>If we consider the NT distances between pairs of points as a series of values sorted in ascending order, the index C uses the smallest NW values and the largest NW values to calculate the sums Smin and Smax: the sum S implies the NW distances in this sequence which correspond to pairs present in a cluster (i.e. pairs whose two points are in the same cluster). At most, 3NW distances are effectively retained in the calculation of this index.<\/p>\n<p>Criterion C is then written as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21052\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/C-index.png\" alt=\"C-index\" width=\"122\" height=\"40\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/C-index.png 122w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/C-index-18x6.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/C-index-120x40.png 120w\" sizes=\"(max-width: 122px) 100vw, 122px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b8bad90 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b8bad90\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2af6886\" data-id=\"2af6886\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ef01086 elementor-widget elementor-widget-heading\" data-id=\"ef01086\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Calinski-Harabasz\"><\/span>Calinski-Harabasz<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8daff3b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8daff3b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5004fac\" data-id=\"5004fac\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e6dbf82 elementor-widget elementor-widget-text-editor\" data-id=\"e6dbf82\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Performance based on\u00a0<a href=\"https:\/\/complex-systems-ai.com\/en\/partitionnement-de-donnees\/somme-de-lerreur-quadratique\/\">HSE<\/a>\u00a0average intra and inter-cluster (Tr):<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette Calinski-Harabasz\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval20.png\" alt=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette\" width=\"291\" height=\"75\" \/><\/figure>\n<p>where B_k is the matrix of\u00a0<a href=\"https:\/\/complex-systems-ai.com\/en\/partitionnement-de-donnees\/criteres-de-dispersion\/\">dispersion<\/a>\u00a0between clusters and W_k is the intra-cluster scatter matrix defined by:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette Calinski-Harabasz\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval21.png\" alt=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette\" width=\"382\" height=\"177\" \/><\/figure>\n<p>with N the number of points in our data, C_q the set of points in cluster q, c_q the center of cluster q, c the center of E, n_q the number of points in cluster q.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-efd8400 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"efd8400\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c6235a4\" data-id=\"c6235a4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6687e37 elementor-widget elementor-widget-heading\" data-id=\"6687e37\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Davies-Bouldin\"><\/span>Davies-Bouldin<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2a3e98d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2a3e98d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1dac4a2\" data-id=\"1dac4a2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cd6340a elementor-widget elementor-widget-text-editor\" data-id=\"cd6340a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>This index treats each cluster individually and seeks to measure how similar it is to the cluster closest to it. The DB index is formulated as follows:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette Calinski-Harabasz\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval25.png\" alt=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette\" width=\"263\" height=\"57\" \/><\/figure>\n<p>I (c_i) represents the average of the distances between the objects belonging to the cluster C_i and its center. And I (c_i, c_j) represents the distance between the centers of the two clusters C_i and C_j.<\/p>\n<p>For each cluster i of the partition, we seek the cluster j which maximizes the index described as follows:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette Calinski-Harabasz\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval26.png\" alt=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette\" width=\"131\" height=\"45\" \/><\/figure>\n<p>The best partition is therefore the one that minimizes the average of the value calculated for each cluster. In other words, the best partition is the one that minimizes the similarity between clusters.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1f6a47b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1f6a47b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-80b6592\" data-id=\"80b6592\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7fcc0d5 elementor-widget elementor-widget-heading\" data-id=\"7fcc0d5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Det-Ratio\"><\/span>Det_Ratio<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d2e7242 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d2e7242\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5d23446\" data-id=\"5d23446\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cdb4309 elementor-widget elementor-widget-text-editor\" data-id=\"cdb4309\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Det_Ratio index is defined like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21053\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Det_Ratio.png\" alt=\"Det_Ratio\" width=\"94\" height=\"42\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Det_Ratio.png 94w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Det_Ratio-18x8.png 18w\" sizes=\"(max-width: 94px) 100vw, 94px\" \/><\/p>\n<p>T denotes the total diffusion matrix. It is the sum of the BG and WG matrices.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f315920 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f315920\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c5f63e3\" data-id=\"c5f63e3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f7391f8 elementor-widget elementor-widget-heading\" data-id=\"f7391f8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Dunn\"><\/span>Dunn<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-afe7606 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"afe7606\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-fb8dfe7\" data-id=\"fb8dfe7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c101781 elementor-widget elementor-widget-text-editor\" data-id=\"c101781\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Dunn&#039;s Index is another internal cluster validation metric that can be calculated as follows:<\/p>\n<ol>\n<li>For each cluster, calculate the distance between each of the objects of the cluster and the objects of the other clusters<\/li>\n<li>Use the minimum of this distance per pair as inter-cluster separation (min.separation)<\/li>\n<li>For each cluster, calculate the distance between objects in the same cluster.<\/li>\n<li>Use maximum intra-cluster distance (i.e. maximum diameter) as intra-cluster compactness<\/li>\n<li>Calculate Dunn&#039;s Index (D) as follows:<\/li>\n<\/ol>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette Calinski-Harabasz\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval27.png\" alt=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette\" width=\"188\" height=\"53\" \/><\/figure>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d182825 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d182825\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-64fbf91\" data-id=\"64fbf91\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-78a30d7 elementor-widget elementor-widget-heading\" data-id=\"78a30d7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"GDImn\"><\/span>GDImn<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-086b7e3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"086b7e3\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-45dd064\" data-id=\"45dd064\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2966835 elementor-widget elementor-widget-text-editor\" data-id=\"2966835\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Generalized Dunn Index (GDI) gives a value based on the \u201cgood behavior\u201d of clusters and their members, measured based on distances between clusters and within clusters.<\/p>\n<p>Let us denote by the letter \u03b4 a measure of the inter-cluster distance and by \u0394 a measure of the intra-cluster distance (which is also called cluster diameter). The GDI index, relating to these distances, is defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21061\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/GDI1.png\" alt=\"GDI\" width=\"155\" height=\"40\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/GDI1.png 155w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/GDI1-150x40.png 150w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/GDI1-18x5.png 18w\" sizes=\"(max-width: 155px) 100vw, 155px\" \/><\/p>\n<p>With k and k&#039; between 1 and K.<\/p>\n<p>Six different definitions of \u03b4 (denoted \u03b41 to \u03b46) and three definitions of \u0394 (denoted \u03941 to \u03943) have been suggested. This leads to 18 different indices denoted Cuv: here u is an integer designating the distance between the clusters (1 \u2264 u \u2264 6) and v an integer designating the distance within the groups (1 \u2264 v \u2264 3). The definitions of the distances \u0394 within the cluster are:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21062\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/GDI2.png\" alt=\"GDI\" width=\"294\" height=\"144\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/GDI2.png 294w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/GDI2-18x9.png 18w\" sizes=\"(max-width: 294px) 100vw, 294px\" \/><\/p>\n<p>Here d is the Euclidean distance. The factor 2 in the definition of \u03943 allows the value to be interpreted as a diameter rather than a radius. The definitions of the distances between clusters \u03b4 are:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21063 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/GDI3.png\" alt=\"GDI\" width=\"453\" height=\"250\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/GDI3.png 453w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/GDI3-300x166.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/GDI3-18x10.png 18w\" sizes=\"(max-width: 453px) 100vw, 453px\" \/><\/p>\n<p>The first four distances (\u03b41 to \u03b44) appear in bottom-up clustering algorithms and are called single linkage, full linkage, average linkage, and centroid linkage, respectively. The measure \u03b45 is the weighted average (with the weights nk and nk\u2032) of the average distances between the points of the clusters Ck and Ck\u2032 and their respective barycenter. The measure \u03b46 is the Hausdorff distance DH.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0fb66c7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0fb66c7\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a9a8f9b\" data-id=\"a9a8f9b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ab35f1f elementor-widget elementor-widget-heading\" data-id=\"ab35f1f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Baker-Hubert-Gamma\"><\/span>Baker-Hubert Gamma <span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c0e8be7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c0e8be7\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ea3d5c8\" data-id=\"ea3d5c8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-23ddf12 elementor-widget elementor-widget-text-editor\" data-id=\"23ddf12\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Baker-Hubert Gamma index is an adaptation, within the framework of clustering, of the \u0393 index of <a href=\"https:\/\/complex-systems-ai.com\/en\/correlation-and-regressions\/\">correlation<\/a> between two data vectors A and B of the same size.<\/p>\n<p>Generally, for two indices i and j such that ai &lt; aj , we say that the two vectors are concordant if bi &lt; bj , in other words if the values are classified in the same order in the two vectors. We calculate the number s+ of concordant pairs {i, j} and the number s\u2212 of discordant pairs. Note that inequalities are strict, meaning ties are removed. In this context, the index \u0393 is classically defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21059\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Gamma1.png\" alt=\"Gamma\" width=\"120\" height=\"39\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Gamma1.png 120w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Gamma1-18x6.png 18w\" sizes=\"(max-width: 120px) 100vw, 120px\" \/><\/p>\n<p>The value is between -1 and 1.<\/p>\n<p>In the context of a partition, the first vector A is chosen to be the set of distances dij between pairs of points {Mi,Mj} (with i &lt; j). The second vector B is a binary vector: in this vector, the coordinate corresponding to a pair {Mi,Mj} is worth 0 if the two points are in the same cluster and 1 otherwise. These two vectors have a length NT = N(N \u2212 1)\/2.<\/p>\n<p>The number s+ represents the number of times that a distance between two points belonging to the same cluster (i.e. a couple for which the value of vector B is 0) is strictly less than the distance between two points n not belonging to the same cluster. cluster (i.e. a couple for which the value of vector B is 1).<\/p>\n<p>The number s\u2212 represents the number of times where the opposite situation occurs, that is to say that a distance between two points belonging to the same cluster (value 0 in B) is strictly greater than a distance between two points not belonging to the same cluster. (value 1 in B). Cases where there is a tie (tie or ex-aequos) are not taken into account.<\/p>\n<p>There are NB inter-cluster distances and, for each of them, we compare with the NW intra-cluster distances: we finally carry out NB \u00d7 NW comparisons. We can write the numbers s+ and s\u2212 in the following form:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21060 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Gamma2.png\" alt=\"Gamma\" width=\"395\" height=\"178\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Gamma2.png 395w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Gamma2-300x135.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Gamma2-18x8.png 18w\" sizes=\"(max-width: 395px) 100vw, 395px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2863f02 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2863f02\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0b9bd11\" data-id=\"0b9bd11\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0ed4003 elementor-widget elementor-widget-heading\" data-id=\"0ed4003\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"G\"><\/span>G+<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-de18e53 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"de18e53\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-11dc790\" data-id=\"11dc790\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c3009fb elementor-widget elementor-widget-text-editor\" data-id=\"c3009fb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Using the same notations as for the Baker-Hubert \u0393 index, the G+ index is defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21064\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/G.png\" alt=\"G+\" width=\"238\" height=\"42\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/G.png 238w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/G-18x3.png 18w\" sizes=\"(max-width: 238px) 100vw, 238px\" \/><\/p>\n<p>It is the proportion of discordant pairs among all pairs of distinct points.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ac3fd45 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ac3fd45\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4bec7cd\" data-id=\"4bec7cd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-25032a7 elementor-widget elementor-widget-heading\" data-id=\"25032a7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Ksq-DetW\"><\/span>Ksq_DetW<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9e96afc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9e96afc\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3b7c85f\" data-id=\"3b7c85f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1034b2d elementor-widget elementor-widget-text-editor\" data-id=\"1034b2d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>As its name suggests, its formula is K\u00b2|WG|.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-bc96c18 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bc96c18\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c242462\" data-id=\"c242462\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e73a419 elementor-widget elementor-widget-heading\" data-id=\"e73a419\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Log-Det-Ratio\"><\/span>Log_Det_Ratio<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fe9cff0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fe9cff0\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a962b59\" data-id=\"a962b59\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-96ea979 elementor-widget elementor-widget-text-editor\" data-id=\"96ea979\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Log_Det_Ratio index is defined like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21065\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Log_Det_Ratio.png\" alt=\"Log_Det_Ratio\" width=\"161\" height=\"39\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Log_Det_Ratio.png 161w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Log_Det_Ratio-18x4.png 18w\" sizes=\"(max-width: 161px) 100vw, 161px\" \/><\/p>\n<p>where T is the diffusion matrix and WG. This is a logarithmic variant of the Det_Ratio index.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-58e0e89 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"58e0e89\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1753697\" data-id=\"1753697\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b117979 elementor-widget elementor-widget-heading\" data-id=\"b117979\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Log-SS-Ratio\"><\/span>Log_SS_Ratio<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4746e87 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4746e87\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0623ff1\" data-id=\"0623ff1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d9458bd elementor-widget elementor-widget-text-editor\" data-id=\"d9458bd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Log_SS_Ratio index is defined like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21066\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Log_SS_Ratio.png\" alt=\"Log_SS_Ratio\" width=\"128\" height=\"41\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Log_SS_Ratio.png 128w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Log_SS_Ratio-18x6.png 18w\" sizes=\"(max-width: 128px) 100vw, 128px\" \/><\/p>\n<p>where BGSS and WGSS are the traces of the BG and WG matrices respectively.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-670bef1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"670bef1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5e7e0e4\" data-id=\"5e7e0e4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-02e59b8 elementor-widget elementor-widget-heading\" data-id=\"02e59b8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"McClain-Rao\"><\/span>McClain-Rao<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-356350c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"356350c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5c4b167\" data-id=\"5c4b167\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e96f30f elementor-widget elementor-widget-text-editor\" data-id=\"e96f30f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>As for the index C, let SW denote the sum of the intra-cluster distances:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21067 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McClain-Rao1.png\" alt=\"McClain-Rao\" width=\"325\" height=\"60\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McClain-Rao1.png 325w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McClain-Rao1-300x55.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McClain-Rao1-18x3.png 18w\" sizes=\"(max-width: 325px) 100vw, 325px\" \/><\/p>\n<p>Recall that the total number of distances between pairs of points belonging to the same cluster is NW. Let us denote SB the sum of the distances between clusters:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21068 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McClain-Rao2.png\" alt=\"McClain-Rao\" width=\"351\" height=\"56\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McClain-Rao2.png 351w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McClain-Rao2-300x48.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McClain-Rao2-18x3.png 18w\" sizes=\"(max-width: 351px) 100vw, 351px\" \/><\/p>\n<p>The total number of distances between pairs of points that do not belong to the same cluster is NB = N(N \u2212 1)\/2 \u2212 NW. The McClain-Rao index is defined as the quotient between the average distances within a cluster and between clusters:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21069\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McClain-Rao3.png\" alt=\"McClain-Rao\" width=\"169\" height=\"37\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McClain-Rao3.png 169w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McClain-Rao3-18x4.png 18w\" sizes=\"(max-width: 169px) 100vw, 169px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-196a0a7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"196a0a7\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-541f1ef\" data-id=\"541f1ef\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-65335bd elementor-widget elementor-widget-heading\" data-id=\"65335bd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"PBM\"><\/span>PBM<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7557679 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7557679\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-29c5270\" data-id=\"29c5270\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-00ab12b elementor-widget elementor-widget-text-editor\" data-id=\"00ab12b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The PBM index (acronym consisting of the initials of the names of its authors, Pakhira, Bandyopadhyay and Maulik) is calculated from the distances between the points and their barycenters and the distances between the barycenters themselves.<\/p>\n<p>Let us denote by DB the greatest distance between two cluster barycenters:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21070\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/PBM.png\" alt=\"PBM\" width=\"191\" height=\"50\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/PBM.png 191w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/PBM-18x5.png 18w\" sizes=\"(max-width: 191px) 100vw, 191px\" \/><\/p>\n<p>On the other hand, let us denote EW the sum of the distances of the points of each cluster to their barycenter and ET the sum of the distances of all the points to the barycenter G of the whole data:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21071\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/PBM2.png\" alt=\"PBM\" width=\"216\" height=\"108\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/PBM2.png 216w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/PBM2-18x9.png 18w\" sizes=\"(max-width: 216px) 100vw, 216px\" \/><\/p>\n<p>The PBM index is defined like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21072\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/PBM3.png\" alt=\"PBM\" width=\"172\" height=\"45\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/PBM3.png 172w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/PBM3-18x5.png 18w\" sizes=\"(max-width: 172px) 100vw, 172px\" \/><\/p>\n<p>ET is a constant that does not depend on the partition or the number of clusters.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f6e7e33 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f6e7e33\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3388b20\" data-id=\"3388b20\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ca94ff7 elementor-widget elementor-widget-heading\" data-id=\"ca94ff7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Point-Biserial\"><\/span>Point-Biserial<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-75ab1c8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"75ab1c8\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-899c1c5\" data-id=\"899c1c5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a6818be elementor-widget elementor-widget-text-editor\" data-id=\"a6818be\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Generally speaking, in statistics, the point-biserial coefficient is a measure of correlation between a continuous variable A and a binary variable B (i.e. a variable whose values are 0 or 1). A and B are sets of the same length n.<\/p>\n<p>The values of A are divided into two groups A0 and A1 depending on whether the corresponding value in B is 0 or 1. Let MA0 and MA1 denote the averages in A0 and A1, and nA0 and nA1 the number of elements in each group. The point-biserial correlation coefficient is defined as the quantity:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21076\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/point-biserial1.png\" alt=\"point-biserial\" width=\"249\" height=\"50\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/point-biserial1.png 249w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/point-biserial1-18x4.png 18w\" sizes=\"(max-width: 249px) 100vw, 249px\" \/><\/p>\n<p>where sn is the standard deviation of A.<\/p>\n<p>In the context of a comparison between different clusterings, the term sn can be omitted because it does not depend on the partitions but only on the entire data set.<\/p>\n<p>As in the case of the index \u0393, we adapt this definition by choosing A as the set of NT distances between pairs of points Mi and Mj. The corresponding value in B is 1 if the two points are in the same cluster and 0 otherwise:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21077\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/point-biserial2.png\" alt=\"point-biserial\" width=\"198\" height=\"79\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/point-biserial2.png 198w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/point-biserial2-18x7.png 18w\" sizes=\"(max-width: 198px) 100vw, 198px\" \/><\/p>\n<p>MA1 is the average of all distances within the cluster and MA0 is the average of all distances between clusters. Thus, the definition of the point-biserial index is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21078 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/point-biserial3.png\" alt=\"point-biserial\" width=\"350\" height=\"42\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/point-biserial3.png 350w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/point-biserial3-300x36.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/point-biserial3-18x2.png 18w\" sizes=\"(max-width: 350px) 100vw, 350px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b22f56d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b22f56d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6e5b837\" data-id=\"6e5b837\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-231339d elementor-widget elementor-widget-heading\" data-id=\"231339d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Ratkowsky-Lance\"><\/span>Ratkowsky-Lance<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3895b7e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3895b7e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2e7289f\" data-id=\"2e7289f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1e73917 elementor-widget elementor-widget-text-editor\" data-id=\"1e73917\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>We calculate the average \u00afR of the quotients between BGSS and TSS for each dimension of the data, that is to say for each column of the matrix A. Note:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21079\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ratkowsky-Lance1.png\" alt=\"Ratkowsky-Lance\" width=\"251\" height=\"181\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ratkowsky-Lance1.png 251w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ratkowsky-Lance1-18x12.png 18w\" sizes=\"(max-width: 251px) 100vw, 251px\" \/><\/p>\n<p>BGSSj is in fact the j-th diagonal term of the BG matrix. The Ratkowsky-Lance index (\u00afc\/\u221aK) is defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21080\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ratkowsky-Lance2.png\" alt=\"Ratkowsky-Lance\" width=\"115\" height=\"45\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ratkowsky-Lance2.png 115w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ratkowsky-Lance2-18x7.png 18w\" sizes=\"(max-width: 115px) 100vw, 115px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9218dd5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9218dd5\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f50f3fc\" data-id=\"f50f3fc\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a137a00 elementor-widget elementor-widget-heading\" data-id=\"a137a00\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Ray-Turi\"><\/span>Ray-Turi<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-54dd638 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"54dd638\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e8526db\" data-id=\"e8526db\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-89edfad elementor-widget elementor-widget-text-editor\" data-id=\"89edfad\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Ray-Turi index is defined as a quotient:<\/p>\n<p>\u2013 the numerator is the average of the squares of the distances of all the points relative to the barycenter of the cluster to which they belong:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21082 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ray-Turi1.png\" alt=\"Ray-Turi\" width=\"418\" height=\"65\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ray-Turi1.png 418w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ray-Turi1-300x47.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ray-Turi1-18x3.png 18w\" sizes=\"(max-width: 418px) 100vw, 418px\" \/><\/p>\n<p>\u2013 the denominator is the minimum of the squares of the distances \u0394kk\u2032 between all the barycenters of the cluster:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21083 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ray-Turi2.png\" alt=\"Ray-Turi\" width=\"389\" height=\"40\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ray-Turi2.png 389w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ray-Turi2-300x31.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ray-Turi2-18x2.png 18w\" sizes=\"(max-width: 389px) 100vw, 389px\" \/><\/p>\n<p>The Ray-Turi index can therefore be written as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21084\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ray-Turi3.png\" alt=\"Ray-Turi\" width=\"112\" height=\"51\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ray-Turi3.png 112w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Ray-Turi3-18x8.png 18w\" sizes=\"(max-width: 112px) 100vw, 112px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-733ac7d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"733ac7d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0ecab9d\" data-id=\"0ecab9d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-238db92 elementor-widget elementor-widget-heading\" data-id=\"238db92\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Scott-Symons\"><\/span>Scott Symons<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fb3045d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fb3045d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2217bdd\" data-id=\"2217bdd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-75cdf6b elementor-widget elementor-widget-text-editor\" data-id=\"75cdf6b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>This index is the weighted sum of the logarithms of the determinants of the variance-covariance matrix of each cluster. This can be written as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21085\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Scott-Symons.png\" alt=\"Scott Symons\" width=\"193\" height=\"52\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Scott-Symons.png 193w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Scott-Symons-18x5.png 18w\" sizes=\"(max-width: 193px) 100vw, 193px\" \/><\/p>\n<p>The determinants of WG{k} matrices are greater than or equal to 0 because these matrices are positive semi-definite. If either of them is 0, the index is undefined.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b578942 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b578942\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-47e1f41\" data-id=\"47e1f41\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-06ccfbe elementor-widget elementor-widget-heading\" data-id=\"06ccfbe\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"SD-Scat-et-SD-Dis\"><\/span>SD_Scat and SD_Dis<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e29c493 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e29c493\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1659231\" data-id=\"1659231\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a72a87d elementor-widget elementor-widget-text-editor\" data-id=\"a72a87d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>We define two quantities S and D called respectively average diffusion of clusters and total separation between clusters.<\/p>\n<p>The average cluster diffusion, denoted S, is defined as follows. Consider the vector of variances for each variable in the dataset. It is a vector V of size p defined by:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21086\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD1.png\" alt=\"SD\" width=\"186\" height=\"30\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD1.png 186w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD1-18x3.png 18w\" sizes=\"(max-width: 186px) 100vw, 186px\" \/><\/p>\n<p>Similarly, we define variance vectors V{k} for each cluster Ck:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21087\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD2.png\" alt=\"SD\" width=\"240\" height=\"40\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD2.png 240w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD2-18x3.png 18w\" sizes=\"(max-width: 240px) 100vw, 240px\" \/><\/p>\n<p>The quantity S is the average of the norms of the vectors V{k} divided by the norm of the vector V:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21088\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD3.png\" alt=\"SD\" width=\"137\" height=\"75\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD3.png 137w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD3-18x10.png 18w\" sizes=\"(max-width: 137px) 100vw, 137px\" \/><\/p>\n<p>On the other hand, the total separation between clusters, denoted D, is defined as follows. Let us denote Dmax and Dmin respectively as the largest and smallest distance between the barycenters of the clusters:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21089\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD4.png\" alt=\"SD\" width=\"261\" height=\"185\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD4.png 261w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD4-18x12.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD4-120x85.png 120w\" sizes=\"(max-width: 261px) 100vw, 261px\" \/><\/p>\n<p>The SD index is finally defined like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21090\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD5.png\" alt=\"SD\" width=\"84\" height=\"19\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD5.png 84w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/SD5-18x4.png 18w\" sizes=\"(max-width: 84px) 100vw, 84px\" \/><\/p>\n<p>where \u03b1 is a weight equal to the value of D obtained for the partition with the greatest number of clusters. In order to compare several partitions of the data, it is first necessary to calculate the value of D corresponding to the greatest number of clusters in order to find the value of the coefficient \u03b1 then calculate the other indices from this coefficient.<\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-482bd2f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"482bd2f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d4ab056\" data-id=\"d4ab056\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a86cb8b elementor-widget elementor-widget-heading\" data-id=\"a86cb8b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"S-Dbw\"><\/span>S_Dbw<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a9a0234 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a9a0234\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-adeb914\" data-id=\"adeb914\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-36c2356 elementor-widget elementor-widget-text-editor\" data-id=\"36c2356\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>This index is based on the notion of density of points belonging to two clusters. We first define a limit value \u03c3 equal to the square root of the sum of the norms of the variance vectors V{k} divided by the number of clusters:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21091\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw1.png\" alt=\"S_Dbw\" width=\"156\" height=\"65\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw1.png 156w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw1-18x8.png 18w\" sizes=\"(max-width: 156px) 100vw, 156px\" \/><\/p>\n<p>The density \u03b3kk\u2032 for a given point, relating to two clusters Ck and Ck\u2032, is equal to the number of points in these two clusters whose distance from this point is less than \u03c3. Geometrically, this amounts to considering the ball of radius \u03c3 centered at a given point and counting the number of points of Ck \u222a Ck\u2032 located in this ball.<\/p>\n<p>For each pair of clusters, let us evaluate the densities for the barycenters G{k} and G{k\u2032} of the clusters and for their midpoint Hkk\u2032. We form the quotient Rkk&#039; between the density in the middle and the greatest density at the two barycenters:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21092 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw2.png\" alt=\"S_Dbw\" width=\"270\" height=\"51\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw2.png 270w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw2-18x3.png 18w\" sizes=\"(max-width: 270px) 100vw, 270px\" \/><\/p>\n<p>On the other hand, we define an inter-cluster density G as the average of the quotients Rkk\u2032:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21093\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw3.png\" alt=\"S_Dbw\" width=\"173\" height=\"50\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw3.png 173w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw3-18x5.png 18w\" sizes=\"(max-width: 173px) 100vw, 173px\" \/><\/p>\n<p>The S-Dbw index is defined as the sum of the average dispersion in clusters S and the inter-cluster density G:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21094\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw4.png\" alt=\"S_Dbw\" width=\"71\" height=\"15\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw4.png 71w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/S_Dbw4-18x4.png 18w\" sizes=\"(max-width: 71px) 100vw, 71px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c2b7719 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c2b7719\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-aaadc71\" data-id=\"aaadc71\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-eb345d9 elementor-widget elementor-widget-heading\" data-id=\"eb345d9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Silhouette\"><\/span>Silhouette<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-71115ac elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"71115ac\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a49fcc5\" data-id=\"a49fcc5\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6c0e6f7 elementor-widget elementor-widget-text-editor\" data-id=\"6c0e6f7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Validates performance based on intra and inter-cluster distances:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette Calinski-Harabasz\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval22.png\" alt=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette\" width=\"217\" height=\"73\" \/><\/figure>\n<p>with a (i) the average dissimilarity with the other data of the cluster and b (i) the weakest dissimilarity with any non-member cluster for each x_i and center of the cluster y:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette Calinski-Harabasz\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval23.png\" alt=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette\" width=\"350\" height=\"167\" \/><\/figure>\n<p>The silhouette coefficient varies between -1 (worst ranking) and 1 (best ranking). Silhouette&#039;s overall average is often calculated.<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette Calinski-Harabasz\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval24.png\" alt=\"Calinski-Harabasz, Davies-Bouldin, Dunn and Silhouette\" width=\"638\" height=\"300\" \/><\/figure>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-03af559 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"03af559\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c972bcd\" data-id=\"c972bcd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ed97439 elementor-widget elementor-widget-heading\" data-id=\"ed97439\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Tau\"><\/span>tau<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-62a2c12 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"62a2c12\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-06cb92f\" data-id=\"06cb92f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b9dabc0 elementor-widget elementor-widget-text-editor\" data-id=\"b9dabc0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Using the same notations as for the Gamma index, the Kendall \u03c4 index between two data vectors of length NT is classically defined in statistics as the quantity:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21095\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau1.png\" alt=\"tau\" width=\"132\" height=\"61\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau1.png 132w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau1-18x8.png 18w\" sizes=\"(max-width: 132px) 100vw, 132px\" \/><\/p>\n<p>The numbers s+ and s\u2212 do not count links, so if an inter-cluster distance and an intra-cluster distance are equal, they do not enter the numerator. In order to take equalities into account, we modify the denominator and define the corrected index \u03c4c like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21096\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau2.png\" alt=\"tau\" width=\"178\" height=\"48\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau2.png 178w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau2-18x5.png 18w\" sizes=\"(max-width: 178px) 100vw, 178px\" \/><\/p>\n<p>with<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21097\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau3.png\" alt=\"tau\" width=\"173\" height=\"136\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau3.png 173w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau3-15x12.png 15w\" sizes=\"(max-width: 173px) 100vw, 173px\" \/><\/p>\n<p>where ti is the number of values in the i-th group of links for vector A and uj is the number of values in the j-th group of links for vector B. Here vector B consists only of values 0 and 1 (corresponding respectively to the inter-cluster and intra-cluster pairs) and we thus have:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21098\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau4.png\" alt=\"tau\" width=\"267\" height=\"26\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau4.png 267w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau4-18x2.png 18w\" sizes=\"(max-width: 267px) 100vw, 267px\" \/><\/p>\n<p>A simple calculation shows that \u03bd0 \u2212 \u03bd2 = NBNW. If we make the reasonable assumption that the vector A contains few identical values, we can estimate that \u03bd2 is negligible compared to \u03bd0. This justifies the following definition of the Tau clustering index:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21099\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau5.png\" alt=\"tau\" width=\"211\" height=\"73\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau5.png 211w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Tau5-18x6.png 18w\" sizes=\"(max-width: 211px) 100vw, 211px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2a87e5e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2a87e5e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c78aba0\" data-id=\"c78aba0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6b07908 elementor-widget elementor-widget-heading\" data-id=\"6b07908\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Trace-W\"><\/span>Trace_W<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-af26b9f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"af26b9f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-16906d0\" data-id=\"16906d0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ceb1398 elementor-widget elementor-widget-text-editor\" data-id=\"ceb1398\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Trace_W index is defined like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21100\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Trace_W.png\" alt=\"Trace_W\" width=\"157\" height=\"18\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Trace_W.png 157w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Trace_W-150x18.png 150w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Trace_W-18x2.png 18w\" sizes=\"(max-width: 157px) 100vw, 157px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-45a60ed elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"45a60ed\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-defb9ad\" data-id=\"defb9ad\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9645602 elementor-widget elementor-widget-heading\" data-id=\"9645602\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Trace-WiB\"><\/span>Trace_WiB<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-61a417f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"61a417f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e986e70\" data-id=\"e986e70\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fdf042e elementor-widget elementor-widget-text-editor\" data-id=\"fdf042e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Trace_WiB index is defined like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21101\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Trace_WiB.png\" alt=\"Trace_WiB\" width=\"139\" height=\"22\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Trace_WiB.png 139w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Trace_WiB-18x3.png 18w\" sizes=\"(max-width: 139px) 100vw, 139px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d20c853 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d20c853\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b209e29\" data-id=\"b209e29\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b242fa9 elementor-widget elementor-widget-heading\" data-id=\"b242fa9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Wemmert-Gancarski\"><\/span>Wemmert-Gan\u00e7arski<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-26f43b6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"26f43b6\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-24107a4\" data-id=\"24107a4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a9a35b6 elementor-widget elementor-widget-text-editor\" data-id=\"a9a35b6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Wemmert-Gan\u00e7arski index is constructed from the quotients of distances between the points and the barycenters of all the clusters.<\/p>\n<p>For a point M belonging to cluster Ck, we form the quotient R(M) between the distance of this point from the barycenter of the cluster to which it belongs and the smallest distance from this point to the barycenters of all the other clusters:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21102\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski1.png\" alt=\"Wemmert-Gan\u00e7arski\" width=\"199\" height=\"60\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski1.png 199w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski1-18x5.png 18w\" sizes=\"(max-width: 199px) 100vw, 199px\" \/><\/p>\n<p>We then average these quotients in each cluster. If this average is greater than 1, we ignore it, otherwise we take its complement to 1. Specifically, let&#039;s define:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21103\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski2.png\" alt=\"Wemmert-Gan\u00e7arski\" width=\"234\" height=\"45\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski2.png 234w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski2-18x3.png 18w\" sizes=\"(max-width: 234px) 100vw, 234px\" \/><\/p>\n<p>The Wemmert-Gan\u00e7arski index is defined as the weighted average, for all clusters, of the quantities Jk like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21104\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski3.png\" alt=\"Wemmert-Gan\u00e7arski\" width=\"111\" height=\"51\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski3.png 111w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski3-18x8.png 18w\" sizes=\"(max-width: 111px) 100vw, 111px\" \/><\/p>\n<p>Which can be rewritten as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21105\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski4.png\" alt=\"Wemmert-Gan\u00e7arski\" width=\"257\" height=\"64\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski4.png 257w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Wemmert-Gancarski4-18x4.png 18w\" sizes=\"(max-width: 257px) 100vw, 257px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d61bcbb elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d61bcbb\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2af5b65\" data-id=\"2af5b65\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-31b2660 elementor-widget elementor-widget-heading\" data-id=\"31b2660\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Xie-Beni\"><\/span>Xie-Beni<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-33bef5d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"33bef5d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2ad1fbe\" data-id=\"2ad1fbe\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-89cb38a elementor-widget elementor-widget-text-editor\" data-id=\"89cb38a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Xie-Beni index is a fuzzy clustering index, but it also applies to sharp clustering.<\/p>\n<p>It is defined as the quotient between the mean square error and the minimum of the minimum square distances between the points of the clusters. The mean square error, in the case of net clustering, is simply the quantity 1\/N*WGSS, in other words the average of the squares of the distances of all points relative to the barycenter of the cluster to which they belong.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21106\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Xie-Beni1.png\" alt=\"Xie-Beni\" width=\"207\" height=\"49\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Xie-Beni1.png 207w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Xie-Beni1-18x4.png 18w\" sizes=\"(max-width: 207px) 100vw, 207px\" \/><\/p>\n<p>and the Xie-Beni index can be written as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21107\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Xie-Beni2.png\" alt=\"Xie-Beni\" width=\"161\" height=\"51\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Xie-Beni2.png 161w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Xie-Beni2-18x6.png 18w\" sizes=\"(max-width: 161px) 100vw, 161px\" \/><\/p>\t\t\t\t\t\t\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<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Data Partitioning Wiki Home Internal Quality Criteria Internal quality criteria typically measure the compactness of clusters using a metric... <\/p>","protected":false},"author":1,"featured_media":0,"parent":8271,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-8393","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/8393","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/comments?post=8393"}],"version-history":[{"count":15,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/8393\/revisions"}],"predecessor-version":[{"id":21112,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/8393\/revisions\/21112"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/8271"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=8393"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}