{"id":8410,"date":"2020-03-27T17:24:33","date_gmt":"2020-03-27T16:24:33","guid":{"rendered":"https:\/\/complex-systems-ai.com\/?page_id=8410"},"modified":"2024-02-25T18:31:59","modified_gmt":"2024-02-25T17:31:59","slug":"criteres-de-qualite-externes","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/external-quality-criteria\/","title":{"rendered":"External quality criteria"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"8410\" class=\"elementor elementor-8410\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1a27ac6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1a27ac6\" 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-6d0e1cd\" data-id=\"6d0e1cd\" 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-da704fb elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"da704fb\" 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-783e2e4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"783e2e4\" 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-0bc361c\" data-id=\"0bc361c\" 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-636ba42 elementor-widget elementor-widget-heading\" data-id=\"636ba42\" 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_82_2 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\/external-quality-criteria\/#Criteres-de-qualite-externes\" >External 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\/external-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\/external-quality-criteria\/#Notation\" >Rating<\/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\/external-quality-criteria\/#Mesure-de-rappel-de-precision-et-F-mesure\" >Precision recall measurement and F-measure<\/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\/external-quality-criteria\/#Variables-indicatrices\" >Indicator variables<\/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\/external-quality-criteria\/#Mesure-fondee-sur-linformation-mutuelle\" >Measure based on mutual information<\/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\/external-quality-criteria\/#Entropie-purete-et-V-mesure\" >Entropy, purity and V-measure<\/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\/external-quality-criteria\/#Czekanowski-Dice\" >Czekanowski-Dice<\/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\/external-quality-criteria\/#Folkes-Mallows\" >Folkes-Mallows<\/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\/external-quality-criteria\/#Hubert-%CE%93\" >Hubert \u0393<\/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\/external-quality-criteria\/#Jaccard\" >Jaccard<\/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\/external-quality-criteria\/#Kulczynski\" >Kulczynski<\/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\/external-quality-criteria\/#McNemar\" >McNemar<\/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\/external-quality-criteria\/#Phi\" >Phi<\/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\/external-quality-criteria\/#Rand\" >Rand<\/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\/external-quality-criteria\/#Rogers-Tanimoto\" >Rogers-Tanimoto<\/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\/external-quality-criteria\/#Russel-Rao\" >Russell Rao<\/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\/external-quality-criteria\/#Sokal-Sneath\" >Sokal-Sneath<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Criteres-de-qualite-externes\"><\/span>External 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-c0ca37d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c0ca37d\" 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-30e8c37\" data-id=\"30e8c37\" 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-2720cf2 elementor-widget elementor-widget-text-editor\" data-id=\"2720cf2\" 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>External quality indices are indices intended to measure the similarity between two partitions. They only take into account the distribution of points in the different clusters and do not make it possible to measure the quality of this distribution.<\/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=\"external quality criteria\" width=\"97\" height=\"97\" title=\"\"><\/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-1d82c86 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1d82c86\" 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-013fc7f\" data-id=\"013fc7f\" 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-236e726 elementor-widget elementor-widget-heading\" data-id=\"236e726\" 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-c8d47da elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c8d47da\" 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-b75e1e8\" data-id=\"b75e1e8\" 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-6c8adb7 elementor-widget elementor-widget-text-editor\" data-id=\"6c8adb7\" 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>Precision recall measurement<\/li>\n<li>Indicator variables<\/li>\n<li>Measure based on mutual information<\/li>\n<li>Entropy, purity and V-measure<\/li>\n<li>Czekanowski-Dice<\/li>\n<li>Folkes-Mallows<\/li>\n<li>Hubert \u0393<\/li>\n<li><a href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/similarity-function\/\">Jaccard<\/a><\/li>\n<li>Kulczynski<\/li>\n<li>McNemar<\/li>\n<li>Phi<\/li>\n<li>Rand<\/li>\n<li>Rogers-Tanimoto<\/li>\n<li>Russell Rao<\/li>\n<li>Sokal-Sneath<\/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-0806166 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0806166\" 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-f3d6ea9\" data-id=\"f3d6ea9\" 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-714ed91 elementor-widget elementor-widget-heading\" data-id=\"714ed91\" 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=\"Notation\"><\/span>Rating<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-798ca46 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"798ca46\" 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-2d7ebed\" data-id=\"2d7ebed\" 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-9406cbf elementor-widget elementor-widget-text-editor\" data-id=\"9406cbf\" 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>All the proposed indices are based on a confusion matrix representing the counting of pairs of points depending on whether or not they are considered to belong to the same cluster according to partition P1 or partition P2. There are therefore four possibilities:<\/p>\n<p>\u2022 the two points belong to the same cluster, according to P1 and P2<\/p>\n<p>\u2022 the two points belong to the same cluster according to P1 but not to P2<\/p>\n<p>\u2022 the two points belong to the same cluster according to P2 but not to P1<\/p>\n<p>\u2022 the two points do not belong to the same cluster, according to P1 and P2.<\/p>\n<p>Let us note yy, yn, ny, nn (y means yes, and n means no) the number of points belonging respectively to these four categories. NT being the total number of pairs of points, we have:<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-21118\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe1.png\" alt=\"external quality\" width=\"272\" height=\"44\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe1.png 272w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe1-18x3.png 18w\" sizes=\"(max-width: 272px) 100vw, 272px\" \/><\/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-0e5dc61 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0e5dc61\" 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-fa5a643\" data-id=\"fa5a643\" 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-afd84a3 elementor-widget elementor-widget-heading\" data-id=\"afd84a3\" 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-rappel-de-precision-et-F-mesure\"><\/span>Precision recall measurement and F-measure<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-2507304 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2507304\" 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-0a81bad\" data-id=\"0a81bad\" 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-32cf5be elementor-widget elementor-widget-text-editor\" data-id=\"32cf5be\" 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>If the partition P1 is used as a reference, we define the precision coefficient as the proportion of points precisely grouped in P2, that is to say which are also grouped according to the reference partition P1. Among the points yy + ny grouped according to P2, yy are rightly grouped. So we have :<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-21119\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/precision_recall1.png\" alt=\"precision recall\" width=\"97\" height=\"42\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/precision_recall1.png 97w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/precision_recall1-18x8.png 18w\" sizes=\"(max-width: 97px) 100vw, 97px\" \/><\/p>\n<p>Likewise, we define the recall coefficient as the proportion of points grouped in P1 which are also grouped in the partition P2. This is the proportion of points which are supposed to be grouped according to the reference partition P1 and which are actually identified as such by the partition P2. Among the points yy+yn grouped in P1, yy are also grouped in P2. So we have :<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21120\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/precision_recall2.png\" alt=\"precision recall\" width=\"110\" height=\"44\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/precision_recall2.png 110w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/precision_recall2-18x7.png 18w\" sizes=\"(max-width: 110px) 100vw, 110px\" \/><\/p>\n<p>In terms of conditional probabilities, we can write<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21121\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/precision_recall3.png\" alt=\"precision recall\" width=\"283\" height=\"38\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/precision_recall3.png 283w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/precision_recall3-18x2.png 18w\" sizes=\"(max-width: 283px) 100vw, 283px\" \/><\/p>\n<p>where the events gp1 and gp2 mean that two points are grouped into P1 and P2 respectively.<\/p>\n<p>The F measure is the harmonic average of the precision and recall coefficients:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-21122\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/F-mesure1-300x66.png\" alt=\"F-measure\" width=\"300\" height=\"66\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/F-mesure1-300x66.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/F-mesure1-18x4.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/F-mesure1.png 307w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>There is also a weighted version of this measure, called the F\u03b1 measure, defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21123\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/F-mesure2.png\" alt=\"F-measure\" width=\"225\" height=\"51\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/F-mesure2.png 225w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/F-mesure2-18x4.png 18w\" sizes=\"(max-width: 225px) 100vw, 225px\" \/><\/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-bbb39b3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bbb39b3\" 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-eff761b\" data-id=\"eff761b\" 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-7b17aee elementor-widget elementor-widget-heading\" data-id=\"7b17aee\" 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=\"Variables-indicatrices\"><\/span>Indicator variables<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-6c2e23a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6c2e23a\" 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-4070894\" data-id=\"4070894\" 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-8853e15 elementor-widget elementor-widget-text-editor\" data-id=\"8853e15\" 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>Let us associate with each partition Pa (a = 1, 2) the binary random variable Xa defined on the set of indices i and j such that i &lt; j as follows: its value is 1 if the points Mi and Mj are classified in the same cluster only in partition Pa and 0 otherwise. The variable Xa functions as an indicator variable.<\/p>\n<p>There exist NT pairs of points and we are only interested in the indices i and j such that i &lt; j. Consider the mean and standard deviation of Xa:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21128\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe2.png\" alt=\"External quality\" width=\"251\" height=\"98\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe2.png 251w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe2-18x7.png 18w\" sizes=\"(max-width: 251px) 100vw, 251px\" \/><\/p>\n<p>The following formulas relate these random variables to the matching and discordant count variables:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21129\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe3.png\" alt=\"External quality\" width=\"240\" height=\"136\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe3.png 240w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe3-18x10.png 18w\" sizes=\"(max-width: 240px) 100vw, 240px\" \/><\/p>\n<p>From here we get:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21130 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe4.png\" alt=\"External quality\" width=\"400\" height=\"82\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe4.png 400w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe4-300x62.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Qualite_externe4-18x4.png 18w\" sizes=\"(max-width: 400px) 100vw, 400px\" \/><\/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-ff81926 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ff81926\" 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-7d83f46\" data-id=\"7d83f46\" 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-b3c4490 elementor-widget elementor-widget-heading\" data-id=\"b3c4490\" 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-fondee-sur-linformation-mutuelle\"><\/span>Measure based on mutual information<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-43c26d7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"43c26d7\" 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-ca01066\" data-id=\"ca01066\" 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-d855bc2 elementor-widget elementor-widget-text-editor\" data-id=\"d855bc2\" 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 mutual information criterion can be used as an external measure for clustering. The measure for m instances grouped together using C = {C_1 ,. . . , C_g} and referring to the target attribute y whose domain is dom (y) = {c_1 ,. . . , c_k} is defined as follows:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"sizeSlug\":\"large\"} --><\/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\/eval16.png\" alt=\"external quality criteria (measurement based on mutual information, precision recall measurement, RAND index)\" width=\"325\" height=\"68\" title=\"\"><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>where m_l, h indicates the number of instances that are in cluster C_l and also in class c_h. m., h indicates the total number of instances in class c_h. Likewise, m_l ,. indicates the number of instances of the C_l cluster.<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:paragraph --><\/p>\n<p>MI is combined with entropy in the NMI:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"sizeSlug\":\"large\"} --><\/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\/eval33.png\" alt=\"external quality criteria (measurement based on mutual information, precision recall measurement, RAND index)\" width=\"292\" height=\"69\" title=\"\"><\/figure>\n<p><!-- \/wp:image --><!-- wp:paragraph --><\/p>\n<p>MI is combined with entropy in AMI:<\/p>\n<p><!-- \/wp:paragraph --><!-- wp:image {\"sizeSlug\":\"large\"} --><\/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\/eval32.png\" alt=\"external quality criteria (measurement based on mutual information, precision recall measurement, RAND index)\" width=\"441\" height=\"68\" title=\"\"><\/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-a093d54 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a093d54\" 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-643b842\" data-id=\"643b842\" 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-6184715 elementor-widget elementor-widget-heading\" data-id=\"6184715\" 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=\"Entropie-purete-et-V-mesure\"><\/span>Entropy, purity and V-measure<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-b816b87 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b816b87\" 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-f47693b\" data-id=\"f47693b\" 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-0e131ae elementor-widget elementor-widget-text-editor\" data-id=\"0e131ae\" 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>Since the complete cluster (all objects of a same class are assigned to a single cluster) and the homogeneous cluster (each cluster contains only objects of a same class) are rarely achieved, we aim to achieve an equilibrium satisfactory between these two approaches. Therefore, we generally apply five well-known clustering criteria in order to evaluate partition performance, which are purity, H-entropy, V-metric, RAND index, and F-metric. This page exposes the three first. The others are exposed in another page.<\/p>\n<p>The entropy measure is used to show how sentence clusters are partitioned within each cluster, and it is known as the average of the weighted values in each cluster entropy over all clusters C = {c_1, \u2026, c_n} :<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Entropy, Purity And V-Measure Entropy\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval28-1.png\" alt=\"Entropy purity and V-measure\" width=\"452\" height=\"90\" \/><\/figure>\n<p>The purity of a cluster is the fraction of the size of the cluster represented by the largest class of sentences assigned to this cluster, namely:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Entropy, Purity And V-Measure Entropy\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval29.png\" alt=\"Entropy purity and V-measure\" width=\"210\" height=\"65\" \/><\/figure>\n<p>The overall purity is the weighted sum of the purities of the individual clusters given by:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Entropy, Purity And V-Measure Entropy\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval30.png\" alt=\"Entropy purity and V-measure\" width=\"218\" height=\"72\" \/><\/figure>\n<p>Although purity and entropy are useful for comparing\u00a0<a href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/\">partitioning<\/a>\u00a0with the same\u00a0<a href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/quality-over-number-of-clusters\/\">number of clusters<\/a>, they are not reliable when comparing <a href=\"https:\/\/complex-systems-ai.com\/en\/data-partitioning\/\">partitioning<\/a> with different numbers of clusters. This is because entropy and purity operate on how sets of sentences are partitioned within each cluster, and this will lead to a case of homogeneity. The highest purity scores and lowest entropy scores are usually obtained when the total number of clusters is too large, where this step will lead to being the lowest in completeness. The next measure considers both the completeness and homogeneity approaches.<\/p>\n<p>The V measure is known as the harmonic mean of homogeneity and completeness; that is, V = homogeneity * completeness \/ (homogeneity + completeness), where homogeneity and completeness are defined as homogeneity = 1-H (C | L) \/ H (C) and completeness = 1-H (L | C) \/ H (L) where:<\/p>\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" title=\"Entropy, Purity And V-Measure Entropy\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/03\/eval31.png\" alt=\"Entropy purity and V-measure\" width=\"690\" height=\"152\" \/><\/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-9dd4f5e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9dd4f5e\" 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-bd8bb45\" data-id=\"bd8bb45\" 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-f32762d elementor-widget elementor-widget-heading\" data-id=\"f32762d\" 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=\"Czekanowski-Dice\"><\/span>Czekanowski-Dice<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-3efeb3b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3efeb3b\" 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-9537368\" data-id=\"9537368\" 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-4f52771 elementor-widget elementor-widget-text-editor\" data-id=\"4f52771\" 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 Czekanowski-Dice index (aka the Ochiai index) is defined like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21131\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Czekanowski-Dice1.png\" alt=\"Czekanowski-Dice\" width=\"132\" height=\"39\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Czekanowski-Dice1.png 132w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Czekanowski-Dice1-18x5.png 18w\" sizes=\"(max-width: 132px) 100vw, 132px\" \/><\/p>\n<p>This index is the harmonic average of the precision and recall coefficients, that is to say it is identical to the F-measure:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21132\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Czekanowski-Dice2.png\" alt=\"Czekanowski-Dice\" width=\"105\" height=\"41\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Czekanowski-Dice2.png 105w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Czekanowski-Dice2-18x7.png 18w\" sizes=\"(max-width: 105px) 100vw, 105px\" \/><\/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-cf60f1a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cf60f1a\" 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-4fbee6d\" data-id=\"4fbee6d\" 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-1e424b2 elementor-widget elementor-widget-heading\" data-id=\"1e424b2\" 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=\"Folkes-Mallows\"><\/span>Folkes-Mallows<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-e190e05 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e190e05\" 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-87d8efc\" data-id=\"87d8efc\" 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-57dbe89 elementor-widget elementor-widget-text-editor\" data-id=\"57dbe89\" 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 Folkes-Mallows index is defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21133\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Folkes-Mallows.png\" alt=\"Folkes-Mallows\" width=\"196\" height=\"38\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Folkes-Mallows.png 196w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Folkes-Mallows-18x3.png 18w\" sizes=\"(max-width: 196px) 100vw, 196px\" \/><\/p>\n<p>This index is the geometric mean (square root of the multiplication) of the precision and recall coefficients.<\/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-94da3b6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"94da3b6\" 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-df91ace\" data-id=\"df91ace\" 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-01835a6 elementor-widget elementor-widget-heading\" data-id=\"01835a6\" 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=\"Hubert-%CE%93\"><\/span>Hubert \u0393<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-2c575e6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2c575e6\" 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-0529dec\" data-id=\"0529dec\" 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-34bd208 elementor-widget elementor-widget-text-editor\" data-id=\"34bd208\" 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 Hubert index \u02c6\u0393 is the coefficient of <a href=\"https:\/\/complex-systems-ai.com\/en\/correlation-and-regressions\/\">correlation<\/a> indicator variables. It is defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21134 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Hubert1.png\" alt=\"Hubert Gamma\" width=\"386\" height=\"60\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Hubert1.png 386w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Hubert1-300x47.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Hubert1-18x3.png 18w\" sizes=\"(max-width: 386px) 100vw, 386px\" \/><\/p>\n<p>The Hubert index \u02c6\u0393 appears as a standardized variant (centered and reduced) of the Russell-Rao index. Its value is between -1 and 1. We can write the index \u02c6\u0393 as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-21135\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Hubert2-300x55.png\" alt=\"Hubert Gamma\" width=\"300\" height=\"55\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Hubert2-300x55.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Hubert2-18x3.png 18w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Hubert2.png 328w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/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-fb9d6c9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fb9d6c9\" 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-7a936e6\" data-id=\"7a936e6\" 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-9587eb8 elementor-widget elementor-widget-heading\" data-id=\"9587eb8\" 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=\"Jaccard\"><\/span>Jaccard<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-3437e5a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3437e5a\" 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-d5fc1e6\" data-id=\"d5fc1e6\" 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-55a0800 elementor-widget elementor-widget-text-editor\" data-id=\"55a0800\" 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 Jaccard index is defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21136\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Jaccard.png\" alt=\"Jaccard\" width=\"139\" height=\"35\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Jaccard.png 139w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Jaccard-18x5.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-3934b78 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3934b78\" 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-c534c01\" data-id=\"c534c01\" 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-e279b70 elementor-widget elementor-widget-heading\" data-id=\"e279b70\" 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=\"Kulczynski\"><\/span>Kulczynski<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-32354b9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"32354b9\" 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-9340501\" data-id=\"9340501\" 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-ab9b493 elementor-widget elementor-widget-text-editor\" data-id=\"ab9b493\" 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 Kulczynski index is defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21137\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Kulczynski.png\" alt=\"Kulczynski\" width=\"192\" height=\"41\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Kulczynski.png 192w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Kulczynski-18x4.png 18w\" sizes=\"(max-width: 192px) 100vw, 192px\" \/><\/p>\n<p>This index is the arithmetic average of the precision and recall coefficients.<\/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-c8121ba elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c8121ba\" 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-8cc549e\" data-id=\"8cc549e\" 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-90ceaa7 elementor-widget elementor-widget-heading\" data-id=\"90ceaa7\" 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=\"McNemar\"><\/span>McNemar<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-f088d4a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f088d4a\" 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-f8cbc20\" data-id=\"f8cbc20\" 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-f91fe22 elementor-widget elementor-widget-text-editor\" data-id=\"f91fe22\" 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 McNemar index is defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21138\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McNemar1.png\" alt=\"McNemar\" width=\"100\" height=\"39\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McNemar1.png 100w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McNemar1-18x7.png 18w\" sizes=\"(max-width: 100px) 100vw, 100px\" \/><\/p>\n<p>Under the null hypothesis H0 that the mismatches between partitions P1 and P2 are random, the index C approximately follows a normal distribution. This is an adaptation of the non-parametric McNemar test for comparing frequencies between two paired samples: the McNemar test statistic (called \u03c72 distance) is the square of the index:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21139\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McNemar2.png\" alt=\"McNemar\" width=\"128\" height=\"46\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McNemar2.png 128w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/McNemar2-18x6.png 18w\" sizes=\"(max-width: 128px) 100vw, 128px\" \/><\/p>\n<p>and follows, under the null hypothesis of marginal homogeneity of the contingency table, a Chi-square distribution with 1 degree of freedom.<\/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-bf4ac82 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bf4ac82\" 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-a121d9d\" data-id=\"a121d9d\" 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-ab5a9c4 elementor-widget elementor-widget-heading\" data-id=\"ab5a9c4\" 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=\"Phi\"><\/span>Phi<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-bba9fdd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bba9fdd\" 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-fe849fd\" data-id=\"fe849fd\" 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-686b4fd elementor-widget elementor-widget-text-editor\" data-id=\"686b4fd\" 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 Phi index is a classic measure of the correlation between two dichotomous variables. It is defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21140\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Phi.png\" alt=\"Phi\" width=\"293\" height=\"36\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Phi.png 293w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Phi-18x2.png 18w\" sizes=\"(max-width: 293px) 100vw, 293px\" \/><\/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-2f809ec elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2f809ec\" 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-26e71d7\" data-id=\"26e71d7\" 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-1519600 elementor-widget elementor-widget-heading\" data-id=\"1519600\" 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=\"Rand\"><\/span>Rand<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-67d92a4a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"67d92a4a\" 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-6519ef83\" data-id=\"6519ef83\" 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-1f15dfbf elementor-widget elementor-widget-text-editor\" data-id=\"1f15dfbf\" 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<!-- wp:paragraph --><!-- \/wp:heading --><!-- wp:paragraph -->\n<figure class=\"wp-block-image size-large\"><\/figure>\n<!-- \/wp:image --><!-- wp:heading --><!-- \/wp:heading --><!-- wp:paragraph --><!-- \/wp:heading --><!-- wp:paragraph -->\n<p>The Rand index is a simple criterion used to compare an induced aggregation structure (C1) with a given aggregation structure (C2). Let a be the number of pairs of instances assigned to the same cluster in C1 and in the same cluster in C2; let b be the number of pairs of instances which are in the same cluster C1, but not in the same cluster C2; let c be the number of pairs of instances which are in the same cluster C2, but not in the same cluster C1; and d the number of pairs of instances allocated to different clusters than C1 and C2.<\/p>\n<p>The quantities a and d can be interpreted as agreements, and b and c as disagreements. The Rand index is defined as:<\/p>\n<!-- \/wp:paragraph --><!-- wp:image {\"sizeSlug\":\"large\"} -->\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\/eval17.png\" alt=\"external quality criteria (measurement based on mutual information, precision recall measurement, RAND index)\" width=\"222\" height=\"49\" title=\"\"><\/figure>\n<p>What comes down with the rating system is<\/p>\n<p>(yy+nn)\/NT<\/p>\n<!-- \/wp:image --><!-- wp:paragraph -->\n<p>The Rand index is between 0 and 1. When the two partitions match perfectly, the Rand index is 1.<\/p>\n<!-- \/wp:paragraph --><!-- wp:paragraph -->\n<p>One problem with the Rand index is that its expected value of two random groupings does not take a constant value (such as zero). Hubert and Arabia in 1985 suggest an adjusted Rand index which overcomes this drawback.<\/p>\n<!-- \/wp:paragraph -->\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-874a20c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"874a20c\" 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-8f88ea6\" data-id=\"8f88ea6\" 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-0049040 elementor-widget elementor-widget-heading\" data-id=\"0049040\" 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=\"Rogers-Tanimoto\"><\/span>Rogers-Tanimoto<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-597d039 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"597d039\" 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-29dea5c\" data-id=\"29dea5c\" 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-159f296 elementor-widget elementor-widget-text-editor\" data-id=\"159f296\" 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 Rogers-Tanimoto index is defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21141\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Rogers-Tanimoto.png\" alt=\"Rogers-Tanimoto\" width=\"178\" height=\"39\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Rogers-Tanimoto.png 178w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Rogers-Tanimoto-18x4.png 18w\" sizes=\"(max-width: 178px) 100vw, 178px\" \/><\/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-a3215ce elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a3215ce\" 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-001a62c\" data-id=\"001a62c\" 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-29f457b elementor-widget elementor-widget-heading\" data-id=\"29f457b\" 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=\"Russel-Rao\"><\/span>Russell 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-b157227 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b157227\" 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-b4102f4\" data-id=\"b4102f4\" 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-51c28c6 elementor-widget elementor-widget-text-editor\" data-id=\"51c28c6\" 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 Russell-Rao index measures the proportion of matches between the two partitions. It is defined as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21142\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Russel-Rao1.png\" alt=\"Russell Rao\" width=\"57\" height=\"34\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Russel-Rao1.png 57w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Russel-Rao1-18x12.png 18w\" sizes=\"(max-width: 57px) 100vw, 57px\" \/><\/p>\n<p>This index can also be written:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21143\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Russel-Rao2.png\" alt=\"Russell Rao\" width=\"207\" height=\"64\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Russel-Rao2.png 207w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Russel-Rao2-18x6.png 18w\" sizes=\"(max-width: 207px) 100vw, 207px\" \/><\/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-da101dd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"da101dd\" 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-e33cbeb\" data-id=\"e33cbeb\" 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-3946e81 elementor-widget elementor-widget-heading\" data-id=\"3946e81\" 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=\"Sokal-Sneath\"><\/span>Sokal-Sneath<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-fde1b86 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fde1b86\" 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-dcb657a\" data-id=\"dcb657a\" 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-9d75654 elementor-widget elementor-widget-text-editor\" data-id=\"9d75654\" 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>There are two versions of the Sokal-Sneath index. They are defined respectively as follows:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-21144\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Sokal-Sneath.png\" alt=\"Sokal-Sneath\" width=\"230\" height=\"89\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Sokal-Sneath.png 230w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/Sokal-Sneath-18x7.png 18w\" sizes=\"(max-width: 230px) 100vw, 230px\" \/><\/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 External Quality Criteria External quality indices are indices intended to measure the similarity between two\u2026 <\/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-8410","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/8410","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=8410"}],"version-history":[{"count":14,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/8410\/revisions"}],"predecessor-version":[{"id":21147,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/8410\/revisions\/21147"}],"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=8410"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}