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<span>Algorithms</span> </div> </a> <button aria-controls="toc-Algorithms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Algorithms subsection</span> </button> <ul id="toc-Algorithms-sublist" class="vector-toc-list"> <li id="toc-Connectivity-based_clustering_(hierarchical_clustering)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connectivity-based_clustering_(hierarchical_clustering)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Connectivity-based clustering (hierarchical clustering)</span> </div> </a> <ul id="toc-Connectivity-based_clustering_(hierarchical_clustering)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Centroid-based_clustering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Centroid-based_clustering"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Centroid-based clustering</span> </div> </a> <ul id="toc-Centroid-based_clustering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Model-based_clustering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Model-based_clustering"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Model-based clustering</span> </div> </a> <ul id="toc-Model-based_clustering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Density-based_clustering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Density-based_clustering"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Density-based clustering</span> </div> </a> <ul id="toc-Density-based_clustering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Grid-based_clustering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grid-based_clustering"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Grid-based clustering</span> </div> </a> <ul id="toc-Grid-based_clustering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Recent_developments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Recent_developments"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Recent developments</span> </div> </a> <ul id="toc-Recent_developments-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Evaluation_and_assessment" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Evaluation_and_assessment"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Evaluation and assessment</span> </div> </a> <button aria-controls="toc-Evaluation_and_assessment-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Evaluation and assessment subsection</span> </button> <ul id="toc-Evaluation_and_assessment-sublist" class="vector-toc-list"> <li id="toc-Internal_evaluation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Internal_evaluation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Internal evaluation</span> </div> </a> <ul id="toc-Internal_evaluation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_evaluation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#External_evaluation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>External evaluation</span> </div> </a> <ul id="toc-External_evaluation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cluster_tendency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cluster_tendency"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Cluster tendency</span> </div> </a> <ul id="toc-Cluster_tendency-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Biology,_computational_biology_and_bioinformatics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biology,_computational_biology_and_bioinformatics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Biology, computational biology and bioinformatics</span> </div> </a> <ul id="toc-Biology,_computational_biology_and_bioinformatics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Medicine" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Medicine"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Medicine</span> </div> </a> <ul id="toc-Medicine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Business_and_marketing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Business_and_marketing"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Business and marketing</span> </div> </a> <ul id="toc-Business_and_marketing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-World_Wide_Web" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#World_Wide_Web"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>World Wide Web</span> </div> </a> <ul id="toc-World_Wide_Web-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computer_science" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computer_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Computer science</span> </div> </a> <ul id="toc-Computer_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Social_science" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Social_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Social science</span> </div> </a> <ul id="toc-Social_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Others" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Others"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Others</span> </div> </a> <ul id="toc-Others-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle See also subsection</span> </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Specialized_types_of_cluster_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specialized_types_of_cluster_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Specialized types of cluster analysis</span> </div> </a> <ul id="toc-Specialized_types_of_cluster_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Techniques_used_in_cluster_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Techniques_used_in_cluster_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Techniques used in cluster analysis</span> </div> </a> <ul id="toc-Techniques_used_in_cluster_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Data_projection_and_preprocessing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Data_projection_and_preprocessing"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Data projection and preprocessing</span> </div> </a> <ul id="toc-Data_projection_and_preprocessing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Other</span> </div> </a> <ul id="toc-Other-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Cluster analysis</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 40 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-40" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">40 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%B9%D9%86%D9%82%D9%88%D8%AF%D9%8A" title="تحليل عنقودي – Arabic" lang="ar" hreflang="ar" data-title="تحليل عنقودي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%B2%E0%A6%BE%E0%A6%B8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%B0_%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%8D%E0%A6%B2%E0%A7%87%E0%A6%B7%E0%A6%A3" title="ক্লাস্টার বিশ্লেষণ – Bangla" lang="bn" hreflang="bn" data-title="ক্লাস্টার বিশ্লেষণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BB%D1%8A%D1%81%D1%82%D0%B5%D1%80%D0%B5%D0%BD_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Клъстерен анализ – Bulgarian" lang="bg" hreflang="bg" data-title="Клъстерен анализ" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Clusteritzaci%C3%B3_de_dades" title="Clusterització de dades – Catalan" lang="ca" hreflang="ca" data-title="Clusterització de dades" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Shlukov%C3%A1_anal%C3%BDza" title="Shluková analýza – Czech" lang="cs" hreflang="cs" data-title="Shluková analýza" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Segmentanalyse" title="Segmentanalyse – Danish" lang="da" hreflang="da" data-title="Segmentanalyse" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Clusteranalyse" title="Clusteranalyse – German" lang="de" hreflang="de" data-title="Clusteranalyse" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Klasteranal%C3%BC%C3%BCs" title="Klasteranalüüs – Estonian" lang="et" hreflang="et" data-title="Klasteranalüüs" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CF%83%CF%84%CE%B1%CE%B4%CE%BF%CF%80%CE%BF%CE%AF%CE%B7%CF%83%CE%B7" title="Συσταδοποίηση – Greek" lang="el" hreflang="el" data-title="Συσταδοποίηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_de_grupos" title="Análisis de grupos – Spanish" lang="es" hreflang="es" data-title="Análisis de grupos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Multzokatze_(estatistika)" title="Multzokatze (estatistika) – Basque" lang="eu" hreflang="eu" data-title="Multzokatze (estatistika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AE%D9%88%D8%B4%D9%87%E2%80%8C%D8%A8%D9%86%D8%AF%DB%8C" title="خوشه‌بندی – Persian" lang="fa" hreflang="fa" data-title="خوشه‌بندی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Partitionnement_de_donn%C3%A9es" title="Partitionnement de données – French" lang="fr" hreflang="fr" data-title="Partitionnement de données" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%81%B4%EB%9F%AC%EC%8A%A4%ED%84%B0_%EB%B6%84%EC%84%9D" title="클러스터 분석 – Korean" lang="ko" hreflang="ko" data-title="클러스터 분석" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%AC%D5%A1%D5%BD%D5%BF%D5%A5%D6%80-%D5%A1%D5%B6%D5%A1%D5%AC%D5%AB%D5%A6" title="Կլաստեր-անալիզ – Armenian" lang="hy" hreflang="hy" data-title="Կլաստեր-անալիզ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%81%E0%A4%82%E0%A4%9C_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="पुंज विश्लेषण – Hindi" lang="hi" hreflang="hi" data-title="पुंज विश्लेषण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Grupiranje" title="Grupiranje – Croatian" lang="hr" hreflang="hr" data-title="Grupiranje" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Analisis_kelompok" title="Analisis kelompok – Indonesian" lang="id" hreflang="id" data-title="Analisis kelompok" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Clustering" title="Clustering – Italian" lang="it" hreflang="it" data-title="Clustering" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%99%D7%AA%D7%95%D7%97_%D7%90%D7%A9%D7%9B%D7%95%D7%9C%D7%95%D7%AA" title="ניתוח אשכולות – Hebrew" lang="he" hreflang="he" data-title="ניתוח אשכולות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Klasteru_anal%C4%ABze" title="Klasteru analīze – Latvian" lang="lv" hreflang="lv" data-title="Klasteru analīze" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Klaszteranal%C3%ADzis" title="Klaszteranalízis – Hungarian" lang="hu" hreflang="hu" data-title="Klaszteranalízis" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Clusteranalyse" title="Clusteranalyse – Dutch" lang="nl" hreflang="nl" data-title="Clusteranalyse" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%87%E3%83%BC%E3%82%BF%E3%83%BB%E3%82%AF%E3%83%A9%E3%82%B9%E3%82%BF%E3%83%AA%E3%83%B3%E3%82%B0" title="データ・クラスタリング – Japanese" lang="ja" hreflang="ja" data-title="データ・クラスタリング" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Klyngeanalyse" title="Klyngeanalyse – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Klyngeanalyse" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_skupie%C5%84" title="Analiza skupień – Polish" lang="pl" hreflang="pl" data-title="Analiza skupień" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Clustering" title="Clustering – Portuguese" lang="pt" hreflang="pt" data-title="Clustering" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%82%D0%B5%D1%80%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Кластерный анализ – Russian" lang="ru" hreflang="ru" data-title="Кластерный анализ" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a 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a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a><br />and <a href="/wiki/Data_mining" title="Data mining">data mining</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Paradigms</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Semi-supervised_learning" class="mw-redirect" title="Semi-supervised learning">Semi-supervised learning</a></li> <li><a href="/wiki/Self-supervised_learning" title="Self-supervised learning">Self-supervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Meta-learning_(computer_science)" title="Meta-learning (computer science)">Meta-learning</a></li> <li><a href="/wiki/Online_machine_learning" title="Online machine learning">Online learning</a></li> <li><a href="/wiki/Batch_learning" class="mw-redirect" title="Batch learning">Batch learning</a></li> <li><a href="/wiki/Curriculum_learning" title="Curriculum learning">Curriculum learning</a></li> <li><a href="/wiki/Rule-based_machine_learning" title="Rule-based machine learning">Rule-based learning</a></li> <li><a href="/wiki/Neuro-symbolic_AI" title="Neuro-symbolic AI">Neuro-symbolic AI</a></li> <li><a href="/wiki/Neuromorphic_engineering" class="mw-redirect" title="Neuromorphic engineering">Neuromorphic engineering</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Problems</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Generative_model" title="Generative model">Generative modeling</a></li> <li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression</a></li> <li><a class="mw-selflink selflink">Clustering</a></li> <li><a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></li> <li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li> <li><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></li> <li><a href="/wiki/Data_cleaning" class="mw-redirect" title="Data cleaning">Data cleaning</a></li> <li><a href="/wiki/Automated_machine_learning" title="Automated machine learning">AutoML</a></li> <li><a href="/wiki/Association_rule_learning" title="Association rule learning">Association rules</a></li> <li><a href="/wiki/Semantic_analysis_(machine_learning)" title="Semantic analysis (machine learning)">Semantic analysis</a></li> <li><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></li> <li><a href="/wiki/Feature_engineering" title="Feature engineering">Feature engineering</a></li> <li><a href="/wiki/Feature_learning" title="Feature learning">Feature learning</a></li> <li><a href="/wiki/Learning_to_rank" title="Learning to rank">Learning to rank</a></li> <li><a href="/wiki/Grammar_induction" title="Grammar induction">Grammar induction</a></li> <li><a href="/wiki/Ontology_learning" title="Ontology learning">Ontology learning</a></li> <li><a href="/wiki/Multimodal_learning" title="Multimodal learning">Multimodal learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a><br /><span class="nobold"><span style="font-size:85%;">(<b><a href="/wiki/Statistical_classification" title="Statistical classification">classification</a></b> • <b><a href="/wiki/Regression_analysis" title="Regression analysis">regression</a></b>)</span></span> </div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Apprenticeship_learning" title="Apprenticeship learning">Apprenticeship learning</a></li> <li><a href="/wiki/Decision_tree_learning" title="Decision tree learning">Decision trees</a></li> <li><a href="/wiki/Ensemble_learning" title="Ensemble learning">Ensembles</a> <ul><li><a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">Bagging</a></li> <li><a href="/wiki/Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting</a></li> <li><a href="/wiki/Random_forest" title="Random forest">Random forest</a></li></ul></li> <li><a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-NN</a></li> <li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Naive_Bayes_classifier" title="Naive Bayes classifier">Naive Bayes</a></li> <li><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural networks</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a></li> <li><a href="/wiki/Perceptron" title="Perceptron">Perceptron</a></li> <li><a href="/wiki/Relevance_vector_machine" title="Relevance vector machine">Relevance vector machine (RVM)</a></li> <li><a href="/wiki/Support_vector_machine" title="Support vector machine">Support vector machine (SVM)</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a class="mw-selflink selflink">Clustering</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/BIRCH" title="BIRCH">BIRCH</a></li> <li><a href="/wiki/CURE_algorithm" title="CURE algorithm">CURE</a></li> <li><a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">Hierarchical</a></li> <li><a href="/wiki/K-means_clustering" title="K-means clustering"><i>k</i>-means</a></li> <li><a href="/wiki/Fuzzy_clustering" title="Fuzzy clustering">Fuzzy</a></li> <li><a href="/wiki/Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">Expectation–maximization (EM)</a></li> <li><br /><a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a></li> <li><a href="/wiki/OPTICS_algorithm" title="OPTICS algorithm">OPTICS</a></li> <li><a href="/wiki/Mean_shift" title="Mean shift">Mean shift</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">CCA</a></li> <li><a href="/wiki/Independent_component_analysis" title="Independent component analysis">ICA</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">LDA</a></li> <li><a href="/wiki/Non-negative_matrix_factorization" title="Non-negative matrix factorization">NMF</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">PCA</a></li> <li><a href="/wiki/Proper_generalized_decomposition" title="Proper generalized decomposition">PGD</a></li> <li><a href="/wiki/T-distributed_stochastic_neighbor_embedding" title="T-distributed stochastic neighbor embedding">t-SNE</a></li> <li><a href="/wiki/Sparse_dictionary_learning" title="Sparse dictionary learning">SDL</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Graphical_model" title="Graphical model">Graphical models</a> <ul><li><a href="/wiki/Bayesian_network" title="Bayesian network">Bayes net</a></li> <li><a href="/wiki/Conditional_random_field" title="Conditional random field">Conditional random field</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Random_sample_consensus" title="Random sample consensus">RANSAC</a></li> <li><a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-NN</a></li> <li><a href="/wiki/Local_outlier_factor" title="Local outlier factor">Local outlier factor</a></li> <li><a href="/wiki/Isolation_forest" title="Isolation forest">Isolation forest</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural network</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Autoencoder" title="Autoencoder">Autoencoder</a></li> <li><a href="/wiki/Deep_learning" title="Deep learning">Deep learning</a></li> <li><a href="/wiki/Feedforward_neural_network" title="Feedforward neural network">Feedforward neural network</a></li> <li><a href="/wiki/Recurrent_neural_network" title="Recurrent neural network">Recurrent neural network</a> <ul><li><a href="/wiki/Long_short-term_memory" title="Long short-term memory">LSTM</a></li> <li><a href="/wiki/Gated_recurrent_unit" title="Gated recurrent unit">GRU</a></li> <li><a href="/wiki/Echo_state_network" title="Echo state network">ESN</a></li> <li><a href="/wiki/Reservoir_computing" title="Reservoir computing">reservoir computing</a></li></ul></li> <li><a href="/wiki/Boltzmann_machine" title="Boltzmann machine">Boltzmann machine</a> <ul><li><a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">Restricted</a></li></ul></li> <li><a href="/wiki/Generative_adversarial_network" title="Generative adversarial network">GAN</a></li> <li><a href="/wiki/Diffusion_model" title="Diffusion model">Diffusion model</a></li> <li><a href="/wiki/Self-organizing_map" title="Self-organizing map">SOM</a></li> <li><a href="/wiki/Convolutional_neural_network" title="Convolutional neural network">Convolutional neural network</a> <ul><li><a href="/wiki/U-Net" title="U-Net">U-Net</a></li> <li><a href="/wiki/LeNet" title="LeNet">LeNet</a></li> <li><a href="/wiki/AlexNet" title="AlexNet">AlexNet</a></li> <li><a href="/wiki/DeepDream" title="DeepDream">DeepDream</a></li></ul></li> <li><a href="/wiki/Neural_radiance_field" title="Neural radiance field">Neural radiance field</a></li> <li><a href="/wiki/Transformer_(machine_learning_model)" class="mw-redirect" title="Transformer (machine learning model)">Transformer</a> <ul><li><a href="/wiki/Vision_transformer" title="Vision transformer">Vision</a></li></ul></li> <li><a href="/wiki/Mamba_(deep_learning_architecture)" title="Mamba (deep learning architecture)">Mamba</a></li> <li><a href="/wiki/Spiking_neural_network" title="Spiking neural network">Spiking neural network</a></li> <li><a href="/wiki/Memtransistor" title="Memtransistor">Memtransistor</a></li> <li><a href="/wiki/Electrochemical_RAM" title="Electrochemical RAM">Electrochemical RAM</a> (ECRAM)</li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Q-learning" title="Q-learning">Q-learning</a></li> <li><a href="/wiki/State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action" title="State–action–reward–state–action">SARSA</a></li> <li><a href="/wiki/Temporal_difference_learning" title="Temporal difference learning">Temporal difference (TD)</a></li> <li><a href="/wiki/Multi-agent_reinforcement_learning" title="Multi-agent reinforcement learning">Multi-agent</a> <ul><li><a href="/wiki/Self-play_(reinforcement_learning_technique)" class="mw-redirect" title="Self-play (reinforcement learning technique)">Self-play</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Learning with humans</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Active_learning_(machine_learning)" title="Active learning (machine learning)">Active learning</a></li> <li><a href="/wiki/Crowdsourcing" title="Crowdsourcing">Crowdsourcing</a></li> <li><a href="/wiki/Human-in-the-loop" title="Human-in-the-loop">Human-in-the-loop</a></li> <li><a href="/wiki/Reinforcement_learning_from_human_feedback" title="Reinforcement learning from human feedback">RLHF</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Model diagnostics</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li> <li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li> <li><a href="/wiki/Learning_curve_(machine_learning)" title="Learning curve (machine learning)">Learning curve</a></li> <li><a href="/wiki/Receiver_operating_characteristic" title="Receiver operating characteristic">ROC curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Mathematical foundations</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Kernel_machines" class="mw-redirect" title="Kernel machines">Kernel machines</a></li> <li><a href="/wiki/Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></li> <li><a href="/wiki/Computational_learning_theory" title="Computational learning theory">Computational learning theory</a></li> <li><a href="/wiki/Empirical_risk_minimization" title="Empirical risk minimization">Empirical risk minimization</a></li> <li><a href="/wiki/Occam_learning" title="Occam learning">Occam learning</a></li> <li><a href="/wiki/Probably_approximately_correct_learning" title="Probably approximately correct learning">PAC learning</a></li> <li><a href="/wiki/Statistical_learning_theory" title="Statistical learning theory">Statistical learning</a></li> <li><a href="/wiki/Vapnik%E2%80%93Chervonenkis_theory" title="Vapnik–Chervonenkis theory">VC theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Journals and conferences</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/ECML_PKDD" title="ECML PKDD">ECML PKDD</a></li> <li><a href="/wiki/Conference_on_Neural_Information_Processing_Systems" title="Conference on Neural Information Processing Systems">NeurIPS</a></li> <li><a href="/wiki/International_Conference_on_Machine_Learning" title="International Conference on Machine Learning">ICML</a></li> <li><a href="/wiki/International_Conference_on_Learning_Representations" title="International Conference on Learning Representations">ICLR</a></li> <li><a href="/wiki/International_Joint_Conference_on_Artificial_Intelligence" title="International Joint Conference on Artificial Intelligence">IJCAI</a></li> <li><a href="/wiki/Machine_Learning_(journal)" title="Machine Learning (journal)">ML</a></li> <li><a href="/wiki/Journal_of_Machine_Learning_Research" title="Journal of Machine Learning Research">JMLR</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Related articles</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Glossary_of_artificial_intelligence" title="Glossary of artificial intelligence">Glossary of artificial intelligence</a></li> <li><a href="/wiki/List_of_datasets_for_machine-learning_research" title="List of datasets for machine-learning research">List of datasets for machine-learning research</a> <ul><li><a href="/wiki/List_of_datasets_in_computer_vision_and_image_processing" title="List of datasets in computer vision and image processing">List of datasets in computer vision and image processing</a></li></ul></li> <li><a href="/wiki/Outline_of_machine_learning" title="Outline of machine learning">Outline of machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Machine_learning" title="Template:Machine learning"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Machine_learning" title="Template talk:Machine learning"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Machine_learning" title="Special:EditPage/Template:Machine learning"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Cluster analysis</b> or <b>clustering</b> is the task of grouping a set of objects in such a way that objects in the same group (called a <b>cluster</b>) are more <a href="/wiki/Similarity_measure" title="Similarity measure">similar</a> (in some specific sense defined by the analyst) to each other than to those in other groups (clusters). It is a main task of <a href="/wiki/Exploratory_data_analysis" title="Exploratory data analysis">exploratory data analysis</a>, and a common technique for <a href="/wiki/Statistics" title="Statistics">statistical</a> <a href="/wiki/Data_analysis" title="Data analysis">data analysis</a>, used in many fields, including <a href="/wiki/Pattern_recognition" title="Pattern recognition">pattern recognition</a>, <a href="/wiki/Image_analysis" title="Image analysis">image analysis</a>, <a href="/wiki/Information_retrieval" title="Information retrieval">information retrieval</a>, <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a>, <a href="/wiki/Data_compression" title="Data compression">data compression</a>, <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a> and <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>. </p><p>Cluster analysis refers to a family of algorithms and tasks rather than one specific <a href="/wiki/Algorithm" title="Algorithm">algorithm</a>. It can be achieved by various algorithms that differ significantly in their understanding of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with small <a href="/wiki/Distance_function" class="mw-redirect" title="Distance function">distances</a> between cluster members, dense areas of the data space, intervals or particular <a href="/wiki/Statistical_distribution" class="mw-redirect" title="Statistical distribution">statistical distributions</a>. Clustering can therefore be formulated as a <a href="/wiki/Multi-objective_optimization" title="Multi-objective optimization">multi-objective optimization</a> problem. The appropriate clustering algorithm and parameter settings (including parameters such as the <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">distance function</a> to use, a density threshold or the number of expected clusters) depend on the individual <a href="/wiki/Data_set" title="Data set">data set</a> and intended use of the results. Cluster analysis as such is not an automatic task, but an iterative process of <a href="/wiki/Knowledge_discovery" class="mw-redirect" title="Knowledge discovery">knowledge discovery</a> or interactive multi-objective optimization that involves trial and failure. It is often necessary to modify <a href="/wiki/Data_preprocessing" title="Data preprocessing">data preprocessing</a> and model parameters until the result achieves the desired properties. </p><p>Besides the term <i>clustering</i>, there is a number of terms with similar meanings, including <i>automatic <a href="/wiki/Statistical_classification" title="Statistical classification">classification</a></i>, <i><a href="/wiki/Numerical_taxonomy" title="Numerical taxonomy">numerical taxonomy</a></i>, <i>botryology</i> (from <a href="/wiki/Greek_language" title="Greek language">Greek</a>: <span lang="el">βότρυς</span> <span class="gloss-quot">'</span><span class="gloss-text">grape</span><span class="gloss-quot">'</span>), <i>typological analysis</i>, and <i><a href="/wiki/Community_structure" title="Community structure">community detection</a></i>. The subtle differences are often in the use of the results: while in data mining, the resulting groups are the matter of interest, in automatic classification the resulting discriminative power is of interest. </p><p>Cluster analysis originated in anthropology by Driver and Kroeber in 1932<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and introduced to psychology by <a href="/wiki/Joseph_Zubin" title="Joseph Zubin">Joseph Zubin</a> in 1938<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Robert_Tryon" title="Robert Tryon">Robert Tryon</a> in 1939<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and famously used by <a href="/wiki/Raymond_Cattell" title="Raymond Cattell">Cattell</a> beginning in 1943<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> for trait theory classification in <a href="/wiki/Personality_psychology" title="Personality psychology">personality psychology</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The notion of a "cluster" cannot be precisely defined, which is one of the reasons why there are so many clustering algorithms.<sup id="cite_ref-estivill_5-0" class="reference"><a href="#cite_note-estivill-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> There is a common denominator: a group of data objects. However, different researchers employ different cluster models, and for each of these cluster models again different algorithms can be given. The notion of a cluster, as found by different algorithms, varies significantly in its properties. Understanding these "cluster models" is key to understanding the differences between the various algorithms. Typical cluster models include: </p> <ul><li><i><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="Connectivity_model"></span><span class="vanchor-text">Connectivity model</span></span>s</i>: for example, <a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">hierarchical clustering</a> builds models based on distance connectivity.</li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Centroid_model"></span><span class="vanchor-text">Centroid model</span></span>s</i>: for example, the <a href="/wiki/K-means_algorithm" class="mw-redirect" title="K-means algorithm">k-means algorithm</a> represents each cluster by a single mean vector.</li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Distribution_model"></span><span class="vanchor-text">Distribution model</span></span>s</i>: clusters are modeled using statistical distributions, such as <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distributions</a> used by the <a href="/wiki/Expectation-maximization_algorithm" class="mw-redirect" title="Expectation-maximization algorithm">expectation-maximization algorithm</a>.</li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Density_model"></span><span class="vanchor-text">Density model</span></span>s</i>: for example, <a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a> and <a href="/wiki/OPTICS" class="mw-redirect" title="OPTICS">OPTICS</a> defines clusters as connected dense regions in the data space.</li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Subspace_model"></span><span class="vanchor-text">Subspace model</span></span>s</i>: in <a href="/wiki/Biclustering" title="Biclustering">biclustering</a> (also known as co-clustering or two-mode-clustering), clusters are modeled with both cluster members and relevant attributes.</li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Group_model"></span><span class="vanchor-text">Group model</span></span>s</i>: some algorithms do not provide a refined model for their results and just provide the grouping information.</li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Graph-based_model"></span><span class="vanchor-text">Graph-based model</span></span>s</i>: a <a href="/wiki/Clique_(graph_theory)" title="Clique (graph theory)">clique</a>, that is, a subset of nodes in a <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> such that every two nodes in the subset are connected by an edge can be considered as a prototypical form of cluster. Relaxations of the complete connectivity requirement (a fraction of the edges can be missing) are known as quasi-cliques, as in the <a href="/wiki/HCS_clustering_algorithm" title="HCS clustering algorithm">HCS clustering algorithm</a>.</li> <li><i>Signed graph models</i>: Every <a href="/wiki/Path_(graph_theory)" title="Path (graph theory)">path</a> in a <a href="/wiki/Signed_graph" title="Signed graph">signed graph</a> has a <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">sign</a> from the product of the signs on the edges. Under the assumptions of <a href="/wiki/Balance_theory" title="Balance theory">balance theory</a>, edges may change sign and result in a bifurcated graph. The weaker "clusterability axiom" (no <a href="/wiki/Cycle_(graph_theory)" title="Cycle (graph theory)">cycle</a> has exactly one negative edge) yields results with more than two clusters, or subgraphs with only positive edges.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Neural_model"></span><span class="vanchor-text">Neural model</span></span>s</i>: the most well-known <a href="/wiki/Unsupervised_learning" title="Unsupervised learning">unsupervised</a> <a href="/wiki/Neural_network" title="Neural network">neural network</a> is the <a href="/wiki/Self-organizing_map" title="Self-organizing map">self-organizing map</a> and these models can usually be characterized as similar to one or more of the above models, and including subspace models when neural networks implement a form of <a href="/wiki/Principal_Component_Analysis" class="mw-redirect" title="Principal Component Analysis">Principal Component Analysis</a> or <a href="/wiki/Independent_Component_Analysis" class="mw-redirect" title="Independent Component Analysis">Independent Component Analysis</a>.</li></ul> <p>A "clustering" is essentially a set of such clusters, usually containing all objects in the data set. Additionally, it may specify the relationship of the clusters to each other, for example, a hierarchy of clusters embedded in each other. Clusterings can be roughly distinguished as: </p> <ul><li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Hard_clustering"></span><span class="vanchor-text">Hard clustering</span></span></i>: each object belongs to a cluster or not</li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Soft_clustering"></span><span class="vanchor-text">Soft clustering</span></span></i> (also: <i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="[[fuzzy_clustering]]"></span><span class="vanchor-text"><a href="/wiki/Fuzzy_clustering" title="Fuzzy clustering">fuzzy clustering</a></span></span></i>): each object belongs to each cluster to a certain degree (for example, a likelihood of belonging to the cluster)</li></ul> <p>There are also finer distinctions possible, for example: </p> <ul><li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Strict_partitioning_clustering"></span><span class="vanchor-text">Strict partitioning clustering</span></span></i>: each object belongs to exactly one cluster</li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Strict_partitioning_clustering_with_outliers"></span><span class="vanchor-text">Strict partitioning clustering with outliers</span></span></i>: objects can also belong to no cluster; in which case they are considered <a href="/wiki/Anomaly_detection" title="Anomaly detection">outliers</a></li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Overlapping_clustering"></span><span class="vanchor-text">Overlapping clustering</span></span></i> (also: <i>alternative clustering</i>, <i>multi-view clustering</i>): objects may belong to more than one cluster; usually involving hard clusters</li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="Hierarchical_clustering"></span><span class="vanchor-text">Hierarchical clustering</span></span></i>: objects that belong to a child cluster also belong to the parent cluster</li> <li><i><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="[[Subspace_clustering]]"></span><span class="vanchor-text"><a href="/wiki/Subspace_clustering" class="mw-redirect" title="Subspace clustering">Subspace clustering</a></span></span></i>: while an overlapping clustering, within a uniquely defined subspace, clusters are not expected to overlap</li></ul> <div class="mw-heading mw-heading2"><h2 id="Algorithms">Algorithms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=2" title="Edit section: Algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main category: <a href="/wiki/Category:Cluster_analysis_algorithms" title="Category:Cluster analysis algorithms">Cluster analysis algorithms</a></div> <p>As listed above, clustering algorithms can be categorized based on their cluster model. The following overview will only list the most prominent examples of clustering algorithms, as there are possibly over 100 published clustering algorithms. Not all provide models for their clusters and can thus not easily be categorized. An overview of algorithms explained in Wikipedia can be found in the <a href="/wiki/List_of_algorithms#Statistics" title="List of algorithms">list of statistics algorithms</a>. </p><p>There is no objectively "correct" clustering algorithm, but as it was noted, "clustering is in the eye of the beholder."<sup id="cite_ref-estivill_5-1" class="reference"><a href="#cite_note-estivill-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The most appropriate clustering algorithm for a particular problem often needs to be chosen experimentally, unless there is a mathematical reason to prefer one cluster model over another. An algorithm that is designed for one kind of model will generally fail on a data set that contains a radically different kind of model.<sup id="cite_ref-estivill_5-2" class="reference"><a href="#cite_note-estivill-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> For example, k-means cannot find non-convex clusters.<sup id="cite_ref-estivill_5-3" class="reference"><a href="#cite_note-estivill-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Most traditional clustering methods assume the clusters exhibit a spherical, elliptical or convex shape.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Connectivity-based_clustering_(hierarchical_clustering)"><span id="Connectivity-based_clustering_.28hierarchical_clustering.29"></span>Connectivity-based clustering (hierarchical clustering)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=3" title="Edit section: Connectivity-based clustering (hierarchical clustering)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">Hierarchical clustering</a></div> <p>Connectivity-based clustering, also known as <i><a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">hierarchical clustering</a></i>, is based on the core idea of objects being more related to nearby objects than to objects farther away. These algorithms connect "objects" to form "clusters" based on their distance. A cluster can be described largely by the maximum distance needed to connect parts of the cluster. At different distances, different clusters will form, which can be represented using a <a href="/wiki/Dendrogram" title="Dendrogram">dendrogram</a>, which explains where the common name "<a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">hierarchical clustering</a>" comes from: these algorithms do not provide a single partitioning of the data set, but instead provide an extensive hierarchy of clusters that merge with each other at certain distances. In a dendrogram, the y-axis marks the distance at which the clusters merge, while the objects are placed along the x-axis such that the clusters don't mix. </p><p>Connectivity-based clustering is a whole family of methods that differ by the way distances are computed. Apart from the usual choice of <a href="/wiki/Distance_function" class="mw-redirect" title="Distance function">distance functions</a>, the user also needs to decide on the linkage criterion (since a cluster consists of multiple objects, there are multiple candidates to compute the distance) to use. Popular choices are known as <a href="/wiki/Single-linkage_clustering" title="Single-linkage clustering">single-linkage clustering</a> (the minimum of object distances), <a href="/wiki/Complete_linkage_clustering" class="mw-redirect" title="Complete linkage clustering">complete linkage clustering</a> (the maximum of object distances), and <a href="/wiki/UPGMA" title="UPGMA">UPGMA</a> or <a href="/wiki/WPGMA" title="WPGMA">WPGMA</a> ("Unweighted or Weighted Pair Group Method with Arithmetic Mean", also known as average linkage clustering). Furthermore, hierarchical clustering can be agglomerative (starting with single elements and aggregating them into clusters) or divisive (starting with the complete data set and dividing it into partitions). </p><p>These methods will not produce a unique partitioning of the data set, but a hierarchy from which the user still needs to choose appropriate clusters. They are not very robust towards outliers, which will either show up as additional clusters or even cause other clusters to merge (known as "chaining phenomenon", in particular with <a href="/wiki/Single-linkage_clustering" title="Single-linkage clustering">single-linkage clustering</a>). In the general case, the complexity is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(n^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(n^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff78e74de3bf7a5246829c66bc5acf0c2a94b67c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.108ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}(n^{3})}"></span> for agglomerative clustering and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(2^{n-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(2^{n-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/309e446c28aed3412bfeb2108eb3ae9ed8245d6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.141ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}(2^{n-1})}"></span> for <a href="/wiki/Divisive_clustering" class="mw-redirect" title="Divisive clustering">divisive clustering</a>,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> which makes them too slow for large data sets. For some special cases, optimal efficient methods (of complexity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(n^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(n^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4441d9689c0e6b2c47994e2f587ac5378faeefba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.108ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}(n^{2})}"></span>) are known: SLINK<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> for single-linkage and CLINK<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> for complete-linkage clustering. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerycaption">Linkage clustering examples</li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 230px;"><span typeof="mw:File"><a href="/wiki/File:SLINK-Gaussian-data.svg" class="mw-file-description" title="Single-linkage on Gaussian data. At 35 clusters, the biggest cluster starts fragmenting into smaller parts, while before it was still connected to the second largest due to the single-link effect."><img alt="Single-linkage on Gaussian data. At 35 clusters, the biggest cluster starts fragmenting into smaller parts, while before it was still connected to the second largest due to the single-link effect." src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/SLINK-Gaussian-data.svg/186px-SLINK-Gaussian-data.svg.png" decoding="async" width="186" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/SLINK-Gaussian-data.svg/279px-SLINK-Gaussian-data.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/SLINK-Gaussian-data.svg/372px-SLINK-Gaussian-data.svg.png 2x" data-file-width="434" data-file-height="467" /></a></span></div> <div class="gallerytext">Single-linkage on Gaussian data. At 35 clusters, the biggest cluster starts fragmenting into smaller parts, while before it was still connected to the second largest due to the single-link effect.</div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 230px;"><span typeof="mw:File"><a href="/wiki/File:SLINK-density-data.svg" class="mw-file-description" title="Single-linkage on density-based clusters. 20 clusters extracted, most of which contain single elements, since linkage clustering does not have a notion of "noise"."><img alt="Single-linkage on density-based clusters. 20 clusters extracted, most of which contain single elements, since linkage clustering does not have a notion of "noise"." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/SLINK-density-data.svg/186px-SLINK-density-data.svg.png" decoding="async" width="186" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/SLINK-density-data.svg/279px-SLINK-density-data.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/SLINK-density-data.svg/372px-SLINK-density-data.svg.png 2x" data-file-width="434" data-file-height="467" /></a></span></div> <div class="gallerytext">Single-linkage on density-based clusters. 20 clusters extracted, most of which contain single elements, since linkage clustering does not have a notion of "noise".</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Centroid-based_clustering">Centroid-based clustering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=4" title="Edit section: Centroid-based clustering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/K-means_clustering" title="K-means clustering">k-means clustering</a></div> <p>In centroid-based clustering, each cluster is represented by a central vector, which is not necessarily a member of the data set. When the number of clusters is fixed to <i>k</i>, <a href="/wiki/K-means_clustering" title="K-means clustering"><i>k</i>-means clustering</a> gives a formal definition as an optimization problem: find the <i>k</i> cluster centers and assign the objects to the nearest cluster center, such that the squared distances from the cluster are minimized. </p><p>The optimization problem itself is known to be <a href="/wiki/NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a>, and thus the common approach is to search only for approximate solutions. A particularly well-known approximate method is <a href="/wiki/Lloyd%27s_algorithm" title="Lloyd's algorithm">Lloyd's algorithm</a>,<sup id="cite_ref-lloyd_11-0" class="reference"><a href="#cite_note-lloyd-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> often just referred to as "<i>k-means algorithm</i>" (although <a href="/wiki/K-means_clustering#History" title="K-means clustering">another algorithm introduced this name</a>). It does however only find a <a href="/wiki/Local_optimum" class="mw-redirect" title="Local optimum">local optimum</a>, and is commonly run multiple times with different random initializations. Variations of <i>k</i>-means often include such optimizations as choosing the best of multiple runs, but also restricting the centroids to members of the data set (<a href="/wiki/K-medoids" title="K-medoids"><i>k</i>-medoids</a>), choosing <a href="/wiki/Median" title="Median">medians</a> (<a href="/wiki/K-medians_clustering" title="K-medians clustering"><i>k</i>-medians clustering</a>), choosing the initial centers less randomly (<a href="/wiki/K-means%2B%2B" title="K-means++"><i>k</i>-means++</a>) or allowing a fuzzy cluster assignment (<a href="/wiki/Fuzzy_clustering" title="Fuzzy clustering">fuzzy c-means</a>). </p><p>Most <i>k</i>-means-type algorithms require the <a href="/wiki/Determining_the_number_of_clusters_in_a_data_set" title="Determining the number of clusters in a data set">number of clusters</a> – <i>k</i> – to be specified in advance, which is considered to be one of the biggest drawbacks of these algorithms. Furthermore, the algorithms prefer clusters of approximately similar size, as they will always assign an object to the nearest centroid. This often leads to incorrectly cut borders of clusters (which is not surprising since the algorithm optimizes cluster centers, not cluster borders). </p><p>K-means has a number of interesting theoretical properties. First, it partitions the data space into a structure known as a <a href="/wiki/Voronoi_diagram" title="Voronoi diagram">Voronoi diagram</a>. Second, it is conceptually close to nearest neighbor classification, and as such is popular in <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>. Third, it can be seen as a variation of model-based clustering, and Lloyd's algorithm as a variation of the <a href="/wiki/Expectation-maximization_algorithm" class="mw-redirect" title="Expectation-maximization algorithm">Expectation-maximization algorithm</a> for this model discussed below. </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerycaption"><i>k</i>-means clustering examples</li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 230px;"><span typeof="mw:File"><a href="/wiki/File:KMeans-Gaussian-data.svg" class="mw-file-description" title="k-means separates data into Voronoi cells, which assumes equal-sized clusters (not adequate here)."><img alt="k-means separates data into Voronoi cells, which assumes equal-sized clusters (not adequate here)." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/KMeans-Gaussian-data.svg/186px-KMeans-Gaussian-data.svg.png" decoding="async" width="186" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/KMeans-Gaussian-data.svg/279px-KMeans-Gaussian-data.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/KMeans-Gaussian-data.svg/372px-KMeans-Gaussian-data.svg.png 2x" data-file-width="434" data-file-height="467" /></a></span></div> <div class="gallerytext"><i>k</i>-means separates data into Voronoi cells, which assumes equal-sized clusters (not adequate here).</div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 230px;"><span typeof="mw:File"><a href="/wiki/File:KMeans-density-data.svg" class="mw-file-description" title="k-means cannot represent density-based clusters."><img alt="k-means cannot represent density-based clusters." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/KMeans-density-data.svg/199px-KMeans-density-data.svg.png" decoding="async" width="199" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/KMeans-density-data.svg/298px-KMeans-density-data.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/KMeans-density-data.svg/398px-KMeans-density-data.svg.png 2x" data-file-width="345" data-file-height="347" /></a></span></div> <div class="gallerytext"><i>k</i>-means cannot represent density-based clusters.</div> </li> </ul> <p>Centroid-based clustering problems such as <i>k</i>-means and <i>k</i>-medoids are special cases of the uncapacitated, metric <a href="/wiki/Optimal_facility_location" title="Optimal facility location">facility location problem</a>, a canonical problem in the operations research and computational geometry communities. In a basic facility location problem (of which there are numerous variants that model more elaborate settings), the task is to find the best warehouse locations to optimally service a given set of consumers. One may view "warehouses" as cluster centroids and "consumer locations" as the data to be clustered. This makes it possible to apply the well-developed algorithmic solutions from the facility location literature to the presently considered centroid-based clustering problem. </p> <div class="mw-heading mw-heading3"><h3 id="Model-based_clustering">Model-based clustering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=5" title="Edit section: Model-based clustering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The clustering framework most closely related to statistics is <a href="/wiki/Model-based_clustering" title="Model-based clustering">model-based clustering</a>, which is based on <a href="/wiki/Probability_distribution" title="Probability distribution">distribution models</a>. This approach models the data as arising from a mixture of probability distributions. It has the advantages of providing principled statistical answers to questions such as how many clusters there are, what clustering method or model to use, and how to detect and deal with outliers. </p><p>While the theoretical foundation of these methods is excellent, they suffer from <a href="/wiki/Overfitting" title="Overfitting">overfitting</a> unless constraints are put on the model complexity. A more complex model will usually be able to explain the data better, which makes choosing the appropriate model complexity inherently difficult. Standard <a href="/wiki/Model-based_clustering" title="Model-based clustering">model-based clustering</a> methods include more parsimonious models based on the <a href="/wiki/Eigenvalue_decomposition" class="mw-redirect" title="Eigenvalue decomposition">eigenvalue decomposition</a> of the covariance matrices, that provide a balance between overfitting and fidelity to the data. </p><p>One prominent method is known as Gaussian mixture models (using the <a href="/wiki/Expectation-maximization_algorithm" class="mw-redirect" title="Expectation-maximization algorithm">expectation-maximization algorithm</a>). Here, the data set is usually modeled with a fixed (to avoid overfitting) number of <a href="/wiki/Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian distributions</a> that are initialized randomly and whose parameters are iteratively optimized to better fit the data set. This will converge to a <a href="/wiki/Local_optimum" class="mw-redirect" title="Local optimum">local optimum</a>, so multiple runs may produce different results. In order to obtain a hard clustering, objects are often then assigned to the Gaussian distribution they most likely belong to; for soft clusterings, this is not necessary. </p><p>Distribution-based clustering produces complex models for clusters that can capture <a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">correlation and dependence</a> between attributes. However, these algorithms put an extra burden on the user: for many real data sets, there may be no concisely defined mathematical model (e.g. assuming Gaussian distributions is a rather strong assumption on the data). </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerycaption">Gaussian mixture model clustering examples</li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 230px;"><span typeof="mw:File"><a href="/wiki/File:EM-Gaussian-data.svg" class="mw-file-description" title="On Gaussian-distributed data, EM works well, since it uses Gaussians for modelling clusters."><img alt="On Gaussian-distributed data, EM works well, since it uses Gaussians for modelling clusters." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/EM-Gaussian-data.svg/186px-EM-Gaussian-data.svg.png" decoding="async" width="186" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/EM-Gaussian-data.svg/279px-EM-Gaussian-data.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/EM-Gaussian-data.svg/372px-EM-Gaussian-data.svg.png 2x" data-file-width="434" data-file-height="467" /></a></span></div> <div class="gallerytext">On Gaussian-distributed data, <abbr title="expectation–maximization">EM</abbr> works well, since it uses Gaussians for modelling clusters.</div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 230px;"><span typeof="mw:File"><a href="/wiki/File:EM-density-data.svg" class="mw-file-description" title="Density-based clusters cannot be modeled using Gaussian distributions."><img alt="Density-based clusters cannot be modeled using Gaussian distributions." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/EM-density-data.svg/186px-EM-density-data.svg.png" decoding="async" width="186" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/EM-density-data.svg/279px-EM-density-data.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/EM-density-data.svg/372px-EM-density-data.svg.png 2x" data-file-width="434" data-file-height="467" /></a></span></div> <div class="gallerytext">Density-based clusters cannot be modeled using Gaussian distributions.</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Density-based_clustering">Density-based clustering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=6" title="Edit section: Density-based clustering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In density-based clustering,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> clusters are defined as areas of higher density than the remainder of the data set. Objects in sparse areas – that are required to separate clusters – are usually considered to be noise and border points. </p><p>The most popular<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> density-based clustering method is <a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In contrast to many newer methods, it features a well-defined cluster model called "density-reachability". Similar to linkage-based clustering, it is based on connecting points within certain distance thresholds. However, it only connects points that satisfy a density criterion, in the original variant defined as a minimum number of other objects within this radius. A cluster consists of all density-connected objects (which can form a cluster of an arbitrary shape, in contrast to many other methods) plus all objects that are within these objects' range. Another interesting property of DBSCAN is that its complexity is fairly low – it requires a linear number of range queries on the database – and that it will discover essentially the same results (it is <a href="/wiki/Deterministic_algorithm" title="Deterministic algorithm">deterministic</a> for core and noise points, but not for border points) in each run, therefore there is no need to run it multiple times. <a href="/wiki/OPTICS_algorithm" title="OPTICS algorithm">OPTICS</a><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> is a generalization of DBSCAN that removes the need to choose an appropriate value for the range parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span>, and produces a hierarchical result related to that of <a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">linkage clustering</a>. DeLi-Clu,<sup id="cite_ref-ReferenceA_16-0" class="reference"><a href="#cite_note-ReferenceA-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Density-Link-Clustering combines ideas from <a href="/wiki/Single-linkage_clustering" title="Single-linkage clustering">single-linkage clustering</a> and OPTICS, eliminating the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"></span> parameter entirely and offering performance improvements over OPTICS by using an <a href="/wiki/R-tree" title="R-tree">R-tree</a> index. </p><p>The key drawback of <a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a> and <a href="/wiki/OPTICS" class="mw-redirect" title="OPTICS">OPTICS</a> is that they expect some kind of density drop to detect cluster borders. On data sets with, for example, overlapping Gaussian distributions – a common use case in artificial data – the cluster borders produced by these algorithms will often look arbitrary, because the cluster density decreases continuously. On a data set consisting of mixtures of Gaussians, these algorithms are nearly always outperformed by methods such as <a href="/wiki/Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">EM clustering</a> that are able to precisely model this kind of data. </p><p><a href="/wiki/Mean-shift" class="mw-redirect" title="Mean-shift">Mean-shift</a> is a clustering approach where each object is moved to the densest area in its vicinity, based on <a href="/wiki/Kernel_density_estimation" title="Kernel density estimation">kernel density estimation</a>. Eventually, objects converge to local maxima of density. Similar to k-means clustering, these "density attractors" can serve as representatives for the data set, but mean-shift can detect arbitrary-shaped clusters similar to DBSCAN. Due to the expensive iterative procedure and density estimation, mean-shift is usually slower than DBSCAN or k-Means. Besides that, the applicability of the mean-shift algorithm to multidimensional data is hindered by the unsmooth behaviour of the kernel density estimate, which results in over-fragmentation of cluster tails.<sup id="cite_ref-ReferenceA_16-1" class="reference"><a href="#cite_note-ReferenceA-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerycaption">Density-based clustering examples</li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 230px;"><span typeof="mw:File"><a href="/wiki/File:DBSCAN-density-data.svg" class="mw-file-description" title="Density-based clustering with DBSCAN"><img alt="Density-based clustering with DBSCAN" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/DBSCAN-density-data.svg/199px-DBSCAN-density-data.svg.png" decoding="async" width="199" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/DBSCAN-density-data.svg/298px-DBSCAN-density-data.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/DBSCAN-density-data.svg/398px-DBSCAN-density-data.svg.png 2x" data-file-width="320" data-file-height="322" /></a></span></div> <div class="gallerytext">Density-based clustering with <a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 230px;"><span typeof="mw:File"><a href="/wiki/File:DBSCAN-Gaussian-data.svg" class="mw-file-description" title="DBSCAN assumes clusters of similar density, and may have problems separating nearby clusters."><img alt="DBSCAN assumes clusters of similar density, and may have problems separating nearby clusters." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/DBSCAN-Gaussian-data.svg/186px-DBSCAN-Gaussian-data.svg.png" decoding="async" width="186" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/DBSCAN-Gaussian-data.svg/279px-DBSCAN-Gaussian-data.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/DBSCAN-Gaussian-data.svg/372px-DBSCAN-Gaussian-data.svg.png 2x" data-file-width="434" data-file-height="467" /></a></span></div> <div class="gallerytext"><a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a> assumes clusters of similar density, and may have problems separating nearby clusters.</div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 230px;"><span typeof="mw:File"><a href="/wiki/File:OPTICS-Gaussian-data.svg" class="mw-file-description" title="OPTICS is a DBSCAN variant, improving handling of different densities clusters."><img alt="OPTICS is a DBSCAN variant, improving handling of different densities clusters." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OPTICS-Gaussian-data.svg/186px-OPTICS-Gaussian-data.svg.png" decoding="async" width="186" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OPTICS-Gaussian-data.svg/279px-OPTICS-Gaussian-data.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OPTICS-Gaussian-data.svg/372px-OPTICS-Gaussian-data.svg.png 2x" data-file-width="434" data-file-height="467" /></a></span></div> <div class="gallerytext"><a href="/wiki/OPTICS_algorithm" title="OPTICS algorithm">OPTICS</a> is a DBSCAN variant, improving handling of different densities clusters.</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Grid-based_clustering">Grid-based clustering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=7" title="Edit section: Grid-based clustering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The grid-based technique is used for a <a href="/wiki/Multidimensional_scaling" title="Multidimensional scaling">multi-dimensional</a> data set.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> In this technique, we create a grid structure, and the comparison is performed on grids (also known as cells). The grid-based technique is fast and has low computational complexity. There are two types of grid-based clustering methods: STING and CLIQUE. Steps involved in grid-based clustering <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> are: </p> <ol><li>Divide data space into a finite number of cells.</li> <li>Randomly select a cell ‘c’, where c should not be traversed beforehand.</li> <li>Calculate the density of ‘c’</li> <li>If the density of ‘c’ greater than threshold density <ol><li>Mark cell ‘c’ as a new cluster</li> <li>Calculate the density of all the neighbors of ‘c’</li> <li>If the density of a neighboring cell is greater than threshold density then, add the cell in the cluster and repeat steps 4.2 and 4.3 till there is no neighbor with a density greater than threshold density.</li></ol></li> <li>Repeat steps 2,3 and 4 till all the cells are traversed.</li> <li>Stop.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Recent_developments">Recent developments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=8" title="Edit section: Recent developments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In recent years, considerable effort has been put into improving the performance of existing algorithms.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Among them are <i>CLARANS</i>,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> and <i><a href="/wiki/Birch_(data_clustering)" class="mw-redirect" title="Birch (data clustering)">BIRCH</a></i>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> With the recent need to process larger and larger data sets (also known as <a href="/wiki/Big_data" title="Big data">big data</a>), the willingness to trade semantic meaning of the generated clusters for performance has been increasing. This led to the development of pre-clustering methods such as <a href="/wiki/Canopy_clustering_algorithm" title="Canopy clustering algorithm">canopy clustering</a>, which can process huge data sets efficiently, but the resulting "clusters" are merely a rough pre-partitioning of the data set to then analyze the partitions with existing slower methods such as <a href="/wiki/K-means_clustering" title="K-means clustering">k-means clustering</a>. </p><p>For <a href="/wiki/High-dimensional_data" class="mw-redirect" title="High-dimensional data">high-dimensional data</a>, many of the existing methods fail due to the <a href="/wiki/Curse_of_dimensionality" title="Curse of dimensionality">curse of dimensionality</a>, which renders particular distance functions problematic in high-dimensional spaces. This led to new <a href="/wiki/Clustering_high-dimensional_data" title="Clustering high-dimensional data">clustering algorithms for high-dimensional data</a> that focus on <a href="/wiki/Subspace_clustering" class="mw-redirect" title="Subspace clustering">subspace clustering</a> (where only some attributes are used, and cluster models include the relevant attributes for the cluster) and <a href="/wiki/Correlation_clustering" title="Correlation clustering">correlation clustering</a> that also looks for arbitrary rotated ("correlated") subspace clusters that can be modeled by giving a <a href="/wiki/Correlation" title="Correlation">correlation</a> of their attributes.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Examples for such clustering algorithms are CLIQUE<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/SUBCLU" title="SUBCLU">SUBCLU</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>Ideas from density-based clustering methods (in particular the <a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a>/<a href="/wiki/OPTICS" class="mw-redirect" title="OPTICS">OPTICS</a> family of algorithms) have been adapted to subspace clustering (HiSC,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> hierarchical subspace clustering and DiSH<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>) and correlation clustering (HiCO,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> hierarchical correlation clustering, 4C<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> using "correlation connectivity" and ERiC<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> exploring hierarchical density-based correlation clusters). </p><p>Several different clustering systems based on <a href="/wiki/Mutual_information" title="Mutual information">mutual information</a> have been proposed. One is Marina Meilă's <i><a href="/wiki/Variation_of_information" title="Variation of information">variation of information</a></i> metric;<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> another provides hierarchical clustering.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Using genetic algorithms, a wide range of different fit-functions can be optimized, including mutual information.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Also <a href="/wiki/Belief_propagation" title="Belief propagation">belief propagation</a>, a recent development in <a href="/wiki/Computer_science" title="Computer science">computer science</a> and <a href="/wiki/Statistical_physics" class="mw-redirect" title="Statistical physics">statistical physics</a>, has led to the creation of new types of clustering algorithms.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Evaluation_and_assessment">Evaluation and assessment</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=9" title="Edit section: Evaluation and assessment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Evaluation (or "validation") of clustering results is as difficult as the clustering itself.<sup id="cite_ref-pfitzner_34-0" class="reference"><a href="#cite_note-pfitzner-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Popular approaches involve "<i>internal</i>" evaluation, where the clustering is summarized to a single quality score, "<i>external</i>" evaluation, where the clustering is compared to an existing "ground truth" classification, "<i>manual</i>" evaluation by a human expert, and "<i>indirect</i>" evaluation by evaluating the utility of the clustering in its intended application.<sup id="cite_ref-:0_35-0" class="reference"><a href="#cite_note-:0-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>Internal evaluation measures suffer from the problem that they represent functions that themselves can be seen as a clustering objective. For example, one could cluster the data set by the Silhouette coefficient; except that there is no known efficient algorithm for this. By using such an internal measure for evaluation, one rather compares the similarity of the optimization problems,<sup id="cite_ref-:0_35-1" class="reference"><a href="#cite_note-:0-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> and not necessarily how useful the clustering is. </p><p>External evaluation has similar problems: if we have such "ground truth" labels, then we would not need to cluster; and in practical applications we usually do not have such labels. On the other hand, the labels only reflect one possible partitioning of the data set, which does not imply that there does not exist a different, and maybe even better, clustering. </p><p>Neither of these approaches can therefore ultimately judge the actual quality of a clustering, but this needs human evaluation,<sup id="cite_ref-:0_35-2" class="reference"><a href="#cite_note-:0-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> which is highly subjective. Nevertheless, such statistics can be quite informative in identifying bad clusterings,<sup id="cite_ref-:1_36-0" class="reference"><a href="#cite_note-:1-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> but one should not dismiss subjective human evaluation.<sup id="cite_ref-:1_36-1" class="reference"><a href="#cite_note-:1-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Internal_evaluation">Internal evaluation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=10" title="Edit section: Internal evaluation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Determining_the_number_of_clusters_in_a_data_set" title="Determining the number of clusters in a data set">Determining the number of clusters in a data set</a></div> <p>When a clustering result is evaluated based on the data that was clustered itself, this is called internal evaluation. These methods usually assign the best score to the algorithm that produces clusters with high similarity within a cluster and low similarity between clusters. One drawback of using internal criteria in cluster evaluation is that high scores on an internal measure do not necessarily result in effective information retrieval applications.<sup id="cite_ref-Christopher_D._Manning,_Prabhakar_Raghavan_&_Hinrich_Schutze_37-0" class="reference"><a href="#cite_note-Christopher_D._Manning,_Prabhakar_Raghavan_&_Hinrich_Schutze-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> Additionally, this evaluation is biased towards algorithms that use the same cluster model. For example, k-means clustering naturally optimizes object distances, and a distance-based internal criterion will likely overrate the resulting clustering. </p><p>Therefore, the internal evaluation measures are best suited to get some insight into situations where one algorithm performs better than another, but this shall not imply that one algorithm produces more valid results than another.<sup id="cite_ref-estivill_5-4" class="reference"><a href="#cite_note-estivill-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Validity as measured by such an index depends on the claim that this kind of structure exists in the data set. An algorithm designed for some kind of models has no chance if the data set contains a radically different set of models, or if the evaluation measures a radically different criterion.<sup id="cite_ref-estivill_5-5" class="reference"><a href="#cite_note-estivill-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> For example, k-means clustering can only find convex clusters, and many evaluation indexes assume convex clusters. On a data set with non-convex clusters neither the use of <i>k</i>-means, nor of an evaluation criterion that assumes convexity, is sound. </p><p>More than a dozen of internal evaluation measures exist, usually based on the intuition that items in the same cluster should be more similar than items in different clusters.<sup id="cite_ref-:2_38-0" class="reference"><a href="#cite_note-:2-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 115–121">: 115–121 </span></sup> For example, the following methods can be used to assess the quality of clustering algorithms based on internal criterion: </p> <ul><li><b><a href="/wiki/Davies%E2%80%93Bouldin_index" title="Davies–Bouldin index">Davies–Bouldin index</a></b></li></ul> <dl><dd>The <a href="/wiki/Davies%E2%80%93Bouldin_index" title="Davies–Bouldin index">Davies–Bouldin index</a> can be calculated by the following formula: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DB={\frac {1}{n}}\sum _{i=1}^{n}\max _{j\neq i}\left({\frac {\sigma _{i}+\sigma _{j}}{d(c_{i},c_{j})}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DB={\frac {1}{n}}\sum _{i=1}^{n}\max _{j\neq i}\left({\frac {\sigma _{i}+\sigma _{j}}{d(c_{i},c_{j})}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088c4609673b8a85f7901d9027215bca7f3c71be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.899ex; height:6.843ex;" alt="{\displaystyle DB={\frac {1}{n}}\sum _{i=1}^{n}\max _{j\neq i}\left({\frac {\sigma _{i}+\sigma _{j}}{d(c_{i},c_{j})}}\right)}"></span></dd></dl></dd> <dd>where <i>n</i> is the number of clusters, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}"></span> is the <a href="/wiki/Centroid" title="Centroid">centroid</a> of cluster <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab3208a7d0c634ef720e03ff5a9949e8310edc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.009ex;" alt="{\displaystyle \sigma _{i}}"></span> is the average distance of all elements in cluster <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> to centroid <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(c_{i},c_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(c_{i},c_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed8aadd6df55d30f58f08354478a71aec4f40d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.782ex; height:3.009ex;" alt="{\displaystyle d(c_{i},c_{j})}"></span> is the distance between centroids <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a844d180d176af828d1636d4e85aa534d0b77baa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.917ex; height:2.343ex;" alt="{\displaystyle c_{j}}"></span>. Since algorithms that produce clusters with low intra-cluster distances (high intra-cluster similarity) and high inter-cluster distances (low inter-cluster similarity) will have a low Davies–Bouldin index, the clustering algorithm that produces a collection of clusters with the smallest <a href="/wiki/Davies%E2%80%93Bouldin_index" title="Davies–Bouldin index">Davies–Bouldin index</a> is considered the best algorithm based on this criterion.</dd></dl> <ul><li><b><a href="/wiki/Dunn_index" title="Dunn index">Dunn index</a></b></li></ul> <dl><dd>The Dunn index aims to identify dense and well-separated clusters. It is defined as the ratio between the minimal inter-cluster distance to maximal intra-cluster distance. For each cluster partition, the Dunn index can be calculated by the following formula:<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\frac {\min _{1\leq i<j\leq n}d(i,j)}{\max _{1\leq k\leq n}d^{\prime }(k)}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D={\frac {\min _{1\leq i<j\leq n}d(i,j)}{\max _{1\leq k\leq n}d^{\prime }(k)}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/420730949ae2ca35dc53316964f9c054030026ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.096ex; height:6.509ex;" alt="{\displaystyle D={\frac {\min _{1\leq i<j\leq n}d(i,j)}{\max _{1\leq k\leq n}d^{\prime }(k)}}\,,}"></span></dd></dl></dd> <dd>where <i>d</i>(<i>i</i>,<i>j</i>) represents the distance between clusters <i>i</i> and <i>j</i>, and <i>d</i> '(<i>k</i>) measures the intra-cluster distance of cluster <i>k</i>. The inter-cluster distance <i>d</i>(<i>i</i>,<i>j</i>) between two clusters may be any number of distance measures, such as the distance between the <a href="/wiki/Centroids" class="mw-redirect" title="Centroids">centroids</a> of the clusters. Similarly, the intra-cluster distance <i>d</i> '(<i>k</i>) may be measured in a variety of ways, such as the maximal distance between any pair of elements in cluster <i>k</i>. Since internal criterion seek clusters with high intra-cluster similarity and low inter-cluster similarity, algorithms that produce clusters with high Dunn index are more desirable.</dd></dl> <ul><li><b><a href="/wiki/Silhouette_(clustering)" title="Silhouette (clustering)">Silhouette coefficient</a></b></li></ul> <dl><dd>The silhouette coefficient contrasts the average distance to elements in the same cluster with the average distance to elements in other clusters. Objects with a high silhouette value are considered well clustered, objects with a low value may be outliers. This index works well with <i>k</i>-means clustering, and is also used to determine the optimal number of clusters.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="External_evaluation">External evaluation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=11" title="Edit section: External evaluation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In external evaluation, clustering results are evaluated based on data that was not used for clustering, such as known class labels and external benchmarks. Such benchmarks consist of a set of pre-classified items, and these sets are often created by (expert) humans. Thus, the benchmark sets can be thought of as a <a href="/wiki/Gold_standard_(test)" title="Gold standard (test)">gold standard</a> for evaluation.<sup id="cite_ref-pfitzner_34-1" class="reference"><a href="#cite_note-pfitzner-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> These types of evaluation methods measure how close the clustering is to the predetermined benchmark classes. However, it has recently been discussed whether this is adequate for real data, or only on synthetic data sets with a factual ground truth, since classes can contain internal structure, the attributes present may not allow separation of clusters or the classes may contain <a href="/wiki/Anomaly_detection" title="Anomaly detection">anomalies</a>.<sup id="cite_ref-Faerberetal2010_41-0" class="reference"><a href="#cite_note-Faerberetal2010-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> Additionally, from a <a href="/wiki/Knowledge_discovery" class="mw-redirect" title="Knowledge discovery">knowledge discovery</a> point of view, the reproduction of known knowledge may not necessarily be the intended result.<sup id="cite_ref-Faerberetal2010_41-1" class="reference"><a href="#cite_note-Faerberetal2010-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> In the special scenario of <a href="/wiki/Constrained_clustering" title="Constrained clustering">constrained clustering</a>, where meta information (such as class labels) is used already in the clustering process, the hold-out of information for evaluation purposes is non-trivial.<sup id="cite_ref-pourrajabi_42-0" class="reference"><a href="#cite_note-pourrajabi-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p><p>A number of measures are adapted from variants used to evaluate classification tasks. In place of counting the number of times a class was correctly assigned to a single data point (known as <a href="/wiki/True_positive" class="mw-redirect" title="True positive">true positives</a>), such <i>pair counting</i> metrics assess whether each pair of data points that is truly in the same cluster is predicted to be in the same cluster.<sup id="cite_ref-pfitzner_34-2" class="reference"><a href="#cite_note-pfitzner-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>As with internal evaluation, several external evaluation measures exist,<sup id="cite_ref-:2_38-1" class="reference"><a href="#cite_note-:2-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 125–129">: 125–129 </span></sup> for example: </p> <ul><li><b>Purity</b>: Purity is a measure of the extent to which clusters contain a single class.<sup id="cite_ref-Christopher_D._Manning,_Prabhakar_Raghavan_&_Hinrich_Schutze_37-1" class="reference"><a href="#cite_note-Christopher_D._Manning,_Prabhakar_Raghavan_&_Hinrich_Schutze-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> Its calculation can be thought of as follows: For each cluster, count the number of data points from the most common class in said cluster. Now take the sum over all clusters and divide by the total number of data points. Formally, given some set of clusters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> and some set of classes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>, both partitioning <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> data points, purity can be defined as:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{N}}\sum _{m\in M}\max _{d\in D}{|m\cap d|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>∈</mo> <mi>M</mi> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>∈</mo> <mi>D</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mo>∩</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{N}}\sum _{m\in M}\max _{d\in D}{|m\cap d|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ce692b2f952dca7400b95d70699a6896adeeab8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.785ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{N}}\sum _{m\in M}\max _{d\in D}{|m\cap d|}}"></span></dd></dl></dd> <dd>This measure doesn't penalize having many clusters, and more clusters will make it easier to produce a high purity. A purity score of 1 is always possible by putting each data point in its own cluster. Also, purity doesn't work well for imbalanced data, where even poorly performing clustering algorithms will give a high purity value. For example, if a size 1000 dataset consists of two classes, one containing 999 points and the other containing 1 point, then every possible partition will have a purity of at least 99.9%.</dd></dl> <ul><li><b><a href="/wiki/Rand_index" title="Rand index">Rand index</a></b><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup></li></ul> <dl><dd>The Rand index computes how similar the clusters (returned by the clustering algorithm) are to the benchmark classifications. It can be computed using the following formula: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RI={\frac {TP+TN}{TP+FP+FN+TN}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>I</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>T</mi> <mi>N</mi> </mrow> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>N</mi> <mo>+</mo> <mi>T</mi> <mi>N</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RI={\frac {TP+TN}{TP+FP+FN+TN}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ec80af7aba3d18700e0c06940c3f277990f95c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.764ex; height:5.343ex;" alt="{\displaystyle RI={\frac {TP+TN}{TP+FP+FN+TN}}}"></span></dd></dl></dd> <dd>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f2291e501a1c99921208d8d34b5a175731781f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.382ex; height:2.176ex;" alt="{\displaystyle TP}"></span> is the number of true positives, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TN}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd09d1b81f8d232dcf7bc9dfe5f0d08810aa4b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.7ex; height:2.176ex;" alt="{\displaystyle TN}"></span> is the number of <a href="/wiki/True_negative" class="mw-redirect" title="True negative">true negatives</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79879ec9236e6d6a087517c11e2638827c09595a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.486ex; height:2.176ex;" alt="{\displaystyle FP}"></span> is the number of <a href="/wiki/False_positives" class="mw-redirect" title="False positives">false positives</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FN}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1c5b779dac08ca6c5a0bb2cc3d64e8375b5941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.804ex; height:2.176ex;" alt="{\displaystyle FN}"></span> is the number of <a href="/wiki/False_negatives" class="mw-redirect" title="False negatives">false negatives</a>. The instances being counted here are the number of correct <i>pairwise</i> assignments. That is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f2291e501a1c99921208d8d34b5a175731781f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.382ex; height:2.176ex;" alt="{\displaystyle TP}"></span> is the number of pairs of points that are clustered together in the predicted partition and in the ground truth partition, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79879ec9236e6d6a087517c11e2638827c09595a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.486ex; height:2.176ex;" alt="{\displaystyle FP}"></span> is the number of pairs of points that are clustered together in the predicted partition but not in the ground truth partition etc. If the dataset is of size N, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TP+TN+FP+FN={\binom {N}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>T</mi> <mi>N</mi> <mo>+</mo> <mi>F</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>N</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>N</mi> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TP+TN+FP+FN={\binom {N}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f94a2667918c49c47e98a5c5fb70acc348fbc576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.477ex; height:6.176ex;" alt="{\displaystyle TP+TN+FP+FN={\binom {N}{2}}}"></span>.</dd></dl> <p>One issue with the <a href="/wiki/Rand_index" title="Rand index">Rand index</a> is that <a href="/wiki/False_positive" class="mw-redirect" title="False positive">false positives</a> and <a href="/wiki/False_negative" class="mw-redirect" title="False negative">false negatives</a> are equally weighted. This may be an undesirable characteristic for some clustering applications. The F-measure addresses this concern,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="it does not achieve the same for chance correction as ARI (May 2018)">citation needed</span></a></i>]</sup> as does the chance-corrected <a href="/wiki/Adjusted_Rand_index" class="mw-redirect" title="Adjusted Rand index">adjusted Rand index</a>. </p> <ul><li><b><a href="/wiki/F-measure" class="mw-redirect" title="F-measure">F-measure</a></b></li></ul> <dl><dd>The F-measure can be used to balance the contribution of <a href="/wiki/False_negative" class="mw-redirect" title="False negative">false negatives</a> by weighting <a href="/wiki/Recall_(information_retrieval)" class="mw-redirect" title="Recall (information retrieval)">recall</a> through a parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78f6fcdda6c164f2e0c8df177d9917a5d0c94214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta \geq 0}"></span>. Let <b><a href="/wiki/Precision_(information_retrieval)" class="mw-redirect" title="Precision (information retrieval)">precision</a></b> and <b><a href="/wiki/Recall_(information_retrieval)" class="mw-redirect" title="Recall (information retrieval)">recall</a></b> (both external evaluation measures in themselves) be defined as follows: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\frac {TP}{TP+FP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>P</mi> </mrow> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>P</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\frac {TP}{TP+FP}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/374965799df4b54302f62e443e26c4f28bed0d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.388ex; height:5.343ex;" alt="{\displaystyle P={\frac {TP}{TP+FP}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\frac {TP}{TP+FN}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>P</mi> </mrow> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>N</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\frac {TP}{TP+FN}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc8bbe6cd9485c02eb84737170cbb2d0a3fcbd88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.725ex; height:5.343ex;" alt="{\displaystyle R={\frac {TP}{TP+FN}}}"></span></dd></dl></dd> <dd>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is the <a href="/wiki/Precision_(information_retrieval)" class="mw-redirect" title="Precision (information retrieval)">precision</a> rate and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the <a href="/wiki/Recall_(information_retrieval)" class="mw-redirect" title="Recall (information retrieval)">recall</a> rate. We can calculate the F-measure by using the following formula:<sup id="cite_ref-Christopher_D._Manning,_Prabhakar_Raghavan_&_Hinrich_Schutze_37-2" class="reference"><a href="#cite_note-Christopher_D._Manning,_Prabhakar_Raghavan_&_Hinrich_Schutze-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\beta }={\frac {(\beta ^{2}+1)\cdot P\cdot R}{\beta ^{2}\cdot P+R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>P</mi> <mo>⋅</mo> <mi>R</mi> </mrow> <mrow> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>P</mi> <mo>+</mo> <mi>R</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\beta }={\frac {(\beta ^{2}+1)\cdot P\cdot R}{\beta ^{2}\cdot P+R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c73e2dc1bd386bcab8e59dac8b850cbf7d345ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.674ex; height:6.509ex;" alt="{\displaystyle F_{\beta }={\frac {(\beta ^{2}+1)\cdot P\cdot R}{\beta ^{2}\cdot P+R}}}"></span></dd></dl></dd> <dd>When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b5e78663eba7ba08e0dd4915251e6261f4f35c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta =0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{0}=P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{0}=P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80cde00007f6bf8aa22dc038074468dc29dc8aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.393ex; height:2.509ex;" alt="{\displaystyle F_{0}=P}"></span>. In other words, <a href="/wiki/Recall_(information_retrieval)" class="mw-redirect" title="Recall (information retrieval)">recall</a> has no impact on the F-measure when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b5e78663eba7ba08e0dd4915251e6261f4f35c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta =0}"></span>, and increasing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> allocates an increasing amount of weight to recall in the final F-measure.</dd> <dd>Also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TN}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd09d1b81f8d232dcf7bc9dfe5f0d08810aa4b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.7ex; height:2.176ex;" alt="{\displaystyle TN}"></span> is not taken into account and can vary from 0 upward without bound.</dd></dl> <ul><li><b><a href="/wiki/Jaccard_coefficient" class="mw-redirect" title="Jaccard coefficient">Jaccard index</a></b></li></ul> <dl><dd>The Jaccard index is used to quantify the similarity between two datasets. The <a href="/wiki/Jaccard_coefficient" class="mw-redirect" title="Jaccard coefficient">Jaccard index</a> takes on a value between 0 and 1. An index of 1 means that the two dataset are identical, and an index of 0 indicates that the datasets have no common elements. The Jaccard index is defined by the following formula: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(A,B)={\frac {|A\cap B|}{|A\cup B|}}={\frac {TP}{TP+FP+FN}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>P</mi> </mrow> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>N</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J(A,B)={\frac {|A\cap B|}{|A\cup B|}}={\frac {TP}{TP+FP+FN}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c70373ac9a71a49940a816f591bbe35c2aa918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.427ex; height:6.509ex;" alt="{\displaystyle J(A,B)={\frac {|A\cap B|}{|A\cup B|}}={\frac {TP}{TP+FP+FN}}}"></span></dd></dl></dd> <dd>This is simply the number of unique elements common to both sets divided by the total number of unique elements in both sets.</dd> <dd>Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TN}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd09d1b81f8d232dcf7bc9dfe5f0d08810aa4b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.7ex; height:2.176ex;" alt="{\displaystyle TN}"></span> is not taken into account.</dd></dl> <ul><li><b><a href="/wiki/S%C3%B8rensen%E2%80%93Dice_coefficient" class="mw-redirect" title="Sørensen–Dice coefficient">Dice index</a></b></li></ul> <dl><dd>The Dice symmetric measure doubles the weight on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f2291e501a1c99921208d8d34b5a175731781f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.382ex; height:2.176ex;" alt="{\displaystyle TP}"></span> while still ignoring <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TN}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd09d1b81f8d232dcf7bc9dfe5f0d08810aa4b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.7ex; height:2.176ex;" alt="{\displaystyle TN}"></span>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DSC={\frac {2TP}{2TP+FP+FN}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>S</mi> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>T</mi> <mi>P</mi> </mrow> <mrow> <mn>2</mn> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>N</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DSC={\frac {2TP}{2TP+FP+FN}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/174f40f295f784c6fc6f78d359503821b757a353" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.64ex; height:5.343ex;" alt="{\displaystyle DSC={\frac {2TP}{2TP+FP+FN}}}"></span></dd></dl></dd></dl> <ul><li><b><a href="/wiki/Fowlkes%E2%80%93Mallows_Index" class="mw-redirect" title="Fowlkes–Mallows Index">Fowlkes–Mallows index</a></b><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup></li></ul> <dl><dd>The Fowlkes–Mallows index computes the similarity between the clusters returned by the clustering algorithm and the benchmark classifications. The higher the value of the Fowlkes–Mallows index the more similar the clusters and the benchmark classifications are. It can be computed using the following formula: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FM={\sqrt {{\frac {TP}{TP+FP}}\cdot {\frac {TP}{TP+FN}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>P</mi> </mrow> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>P</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>P</mi> </mrow> <mrow> <mi>T</mi> <mi>P</mi> <mo>+</mo> <mi>F</mi> <mi>N</mi> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FM={\sqrt {{\frac {TP}{TP+FP}}\cdot {\frac {TP}{TP+FN}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0397c64b73657216f7df54272cd7ef1f73a1c864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.691ex; height:6.343ex;" alt="{\displaystyle FM={\sqrt {{\frac {TP}{TP+FP}}\cdot {\frac {TP}{TP+FN}}}}}"></span></dd></dl></dd> <dd>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle TP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f2291e501a1c99921208d8d34b5a175731781f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.382ex; height:2.176ex;" alt="{\displaystyle TP}"></span> is the number of <a href="/wiki/True_positive" class="mw-redirect" title="True positive">true positives</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FP}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79879ec9236e6d6a087517c11e2638827c09595a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.486ex; height:2.176ex;" alt="{\displaystyle FP}"></span> is the number of <a href="/wiki/False_positives" class="mw-redirect" title="False positives">false positives</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FN}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1c5b779dac08ca6c5a0bb2cc3d64e8375b5941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.804ex; height:2.176ex;" alt="{\displaystyle FN}"></span> is the number of <a href="/wiki/False_negatives" class="mw-redirect" title="False negatives">false negatives</a>. The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be621ec797984559fcca151cbf33fed409f4ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.183ex; height:2.176ex;" alt="{\displaystyle FM}"></span> index is the geometric mean of the <a href="/wiki/Precision_(information_retrieval)" class="mw-redirect" title="Precision (information retrieval)">precision</a> and <a href="/wiki/Recall_(information_retrieval)" class="mw-redirect" title="Recall (information retrieval)">recall</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>, and is thus also known as the <a href="/wiki/G-measure" title="G-measure">G-measure</a>, while the F-measure is their harmonic mean.<sup id="cite_ref-powers_45-0" class="reference"><a href="#cite_note-powers-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> Moreover, <a href="/wiki/Precision_(information_retrieval)" class="mw-redirect" title="Precision (information retrieval)">precision</a> and <a href="/wiki/Recall_(information_retrieval)" class="mw-redirect" title="Recall (information retrieval)">recall</a> are also known as Wallace's indices <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B^{I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c615f6c6d649f468aa611867b86f4eeed49c7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.825ex; height:2.676ex;" alt="{\displaystyle B^{I}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{II}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>I</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B^{II}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e40a5d5e8ef2c20a5802ced82194dff36ee98ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.653ex; height:2.676ex;" alt="{\displaystyle B^{II}}"></span>.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> Chance normalized versions of recall, precision and G-measure correspond to <a href="/wiki/Informedness" class="mw-redirect" title="Informedness">Informedness</a>, <a href="/wiki/Markedness" title="Markedness">Markedness</a> and <a href="/wiki/Matthews_correlation_coefficient" class="mw-redirect" title="Matthews correlation coefficient">Matthews Correlation</a> and relate strongly to <a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Kappa</a>.<sup id="cite_ref-kappa_48-0" class="reference"><a href="#cite_note-kappa-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup></dd></dl> <ul><li><b>Chi Index</b><sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> is an external validation index that measure the clustering results by applying the <a href="/wiki/Chi-squared_test" title="Chi-squared test">chi-squared statistic</a>. This index scores positively the fact that the labels are as sparse as possible across the clusters, i.e., that each cluster has as few different labels as possible. The higher the value of the Chi Index the greater the relationship between the resulting clusters and the label used.</li> <li>The <b><a href="/wiki/Mutual_information" title="Mutual information">mutual information</a></b> is an <a href="/wiki/Information_theory" title="Information theory">information theoretic</a> measure of how much information is shared between a clustering and a ground-truth classification that can detect a non-linear similarity between two clusterings. <a href="/wiki/Adjusted_mutual_information" title="Adjusted mutual information">Normalized mutual information</a> is a family of corrected-for-chance variants of this that has a reduced bias for varying cluster numbers.<sup id="cite_ref-pfitzner_34-3" class="reference"><a href="#cite_note-pfitzner-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup></li> <li><b><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a></b></li></ul> <dl><dd>A confusion matrix can be used to quickly visualize the results of a classification (or clustering) algorithm. It shows how different a cluster is from the gold standard cluster.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Cluster_tendency">Cluster tendency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=12" title="Edit section: Cluster tendency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To measure cluster tendency is to measure to what degree clusters exist in the data to be clustered, and may be performed as an initial test, before attempting clustering. One way to do this is to compare the data against random data. On average, random data should not have clusters. </p> <ul><li><b><a href="/wiki/Hopkins_statistic" title="Hopkins statistic">Hopkins statistic</a></b></li></ul> <dl><dd>There are multiple formulations of the <a href="/wiki/Hopkins_statistic" title="Hopkins statistic">Hopkins statistic</a>.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> A typical one is as follows.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> be the set of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> data points in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> dimensional space. Consider a random sample (without replacement) of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\ll n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≪</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\ll n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691a26b06c3d6d03b77e3f1507fe9fd8ef145924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.049ex; height:1.843ex;" alt="{\displaystyle m\ll n}"></span> data points with members <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span>. Also generate a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> uniformly randomly distributed data points. Now define two distance measures, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f13cb025ff2e136dcbd2fc81ddf965b728e3d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}"></span> to be the distance of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}\in Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}\in Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0717b773e813620a14036f6db3e96a589fb301c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.553ex; height:2.509ex;" alt="{\displaystyle y_{i}\in Y}"></span> from its nearest neighbor in X and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe22f0329d3ecb2e1880d44d191aba0e5475db68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle w_{i}}"></span> to be the distance of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ea6d0515864dd80d66a0bdc4cc074088a7647f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.95ex; height:2.509ex;" alt="{\displaystyle x_{i}\in X}"></span> from its nearest neighbor in X. We then define the Hopkins statistic as: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H={\frac {\sum _{i=1}^{m}{u_{i}^{d}}}{\sum _{i=1}^{m}{u_{i}^{d}}+\sum _{i=1}^{m}{w_{i}^{d}}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> </mrow> </mrow> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msubsup> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H={\frac {\sum _{i=1}^{m}{u_{i}^{d}}}{\sum _{i=1}^{m}{u_{i}^{d}}+\sum _{i=1}^{m}{w_{i}^{d}}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/465b3eb8d1577be4c32a8314ab2d572098f8e167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.533ex; height:7.176ex;" alt="{\displaystyle H={\frac {\sum _{i=1}^{m}{u_{i}^{d}}}{\sum _{i=1}^{m}{u_{i}^{d}}+\sum _{i=1}^{m}{w_{i}^{d}}}}\,,}"></span></dd></dl></dd> <dd>With this definition, uniform random data should tend to have values near to 0.5, and clustered data should tend to have values nearer to 1.</dd> <dd>However, data containing just a single Gaussian will also score close to 1, as this statistic measures deviation from a <i>uniform</i> distribution, not <a href="/wiki/Multimodal_distribution" title="Multimodal distribution">multimodality</a>, making this statistic largely useless in application (as real data never is remotely uniform).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=13" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Cluster_analysis" title="Special:EditPage/Cluster analysis">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">November 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Biology,_computational_biology_and_bioinformatics"><span id="Biology.2C_computational_biology_and_bioinformatics"></span>Biology, computational biology and bioinformatics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=14" title="Edit section: Biology, computational biology and bioinformatics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Distance_matrices_in_phylogeny" title="Distance matrices in phylogeny">Distance matrices in phylogeny</a></div> <dl><dt><a href="/wiki/Plant" title="Plant">Plant</a> and <a href="/wiki/Animal" title="Animal">animal</a> <a href="/wiki/Ecology" title="Ecology">ecology</a></dt> <dd>Cluster analysis is used to describe and to make spatial and temporal comparisons of communities (assemblages) of organisms in heterogeneous environments. It is also used in <a href="/wiki/Systematics" title="Systematics">plant systematics</a> to generate artificial <a href="/wiki/Phylogeny" class="mw-redirect" title="Phylogeny">phylogenies</a> or clusters of organisms (individuals) at the species, genus or higher level that share a number of attributes.</dd> <dt><a href="/wiki/Transcriptomics" class="mw-redirect" title="Transcriptomics">Transcriptomics</a></dt> <dd>Clustering is used to build groups of <a href="/wiki/Genes" class="mw-redirect" title="Genes">genes</a> with related expression patterns (also known as coexpressed genes) as in <a href="/wiki/HCS_clustering_algorithm" title="HCS clustering algorithm">HCS clustering algorithm</a>.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> Often such groups contain functionally related proteins, such as <a href="/wiki/Enzyme" title="Enzyme">enzymes</a> for a specific <a href="/wiki/Metabolic_pathway" title="Metabolic pathway">pathway</a>, or genes that are co-regulated. High throughput experiments using <a href="/wiki/Expressed_sequence_tag" title="Expressed sequence tag">expressed sequence tags</a> (ESTs) or <a href="/wiki/DNA_microarray" title="DNA microarray">DNA microarrays</a> can be a powerful tool for <a href="/wiki/Genome_annotation" class="mw-redirect" title="Genome annotation">genome annotation</a> – a general aspect of <a href="/wiki/Genomics" title="Genomics">genomics</a>.</dd> <dt><a href="/wiki/Sequence_analysis" title="Sequence analysis">Sequence analysis</a></dt> <dd><a href="/wiki/Sequence_clustering" title="Sequence clustering">Sequence clustering</a> is used to group homologous sequences into <a href="/wiki/List_of_gene_families" title="List of gene families">gene families</a>.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> This is a very important concept in <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a>, and <a href="/wiki/Evolutionary_biology" title="Evolutionary biology">evolutionary biology</a> in general. See evolution by <a href="/wiki/Gene_duplication" title="Gene duplication">gene duplication</a>.</dd> <dt>High-throughput <a href="/wiki/Genotype" title="Genotype">genotyping</a> platforms</dt> <dd>Clustering algorithms are used to automatically assign genotypes.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup></dd> <dt><a href="/wiki/Human_genetic_clustering" title="Human genetic clustering">Human genetic clustering</a></dt> <dd>The similarity of genetic data is used in clustering to infer population structures.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Medicine"><a href="/wiki/Medicine" title="Medicine">Medicine</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=15" title="Edit section: Medicine"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt><a href="/wiki/Medical_imaging" title="Medical imaging">Medical imaging</a></dt> <dd>On <a href="/wiki/PET_scan" class="mw-redirect" title="PET scan">PET scans</a>, cluster analysis can be used to differentiate between different types of <a href="/wiki/Tissue_(biology)" title="Tissue (biology)">tissue</a> in a three-dimensional image for many different purposes.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup></dd> <dt>Analysis of antimicrobial activity</dt> <dd>Cluster analysis can be used to analyse patterns of antibiotic resistance, to classify antimicrobial compounds according to their mechanism of action, to classify antibiotics according to their antibacterial activity.</dd> <dt>IMRT segmentation</dt> <dd>Clustering can be used to divide a fluence map into distinct regions for conversion into deliverable fields in MLC-based Radiation Therapy.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Business_and_marketing">Business and marketing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=16" title="Edit section: Business and marketing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt><a href="/wiki/Market_research" title="Market research">Market research</a></dt> <dd>Cluster analysis is widely used in market research when working with multivariate data from <a href="/wiki/Statistical_survey" class="mw-redirect" title="Statistical survey">surveys</a> and test panels. Market researchers use cluster analysis to partition the general <a href="/wiki/Population" title="Population">population</a> of <a href="/wiki/Consumer" title="Consumer">consumers</a> into market segments and to better understand the relationships between different groups of consumers/potential <a href="/wiki/Customers" class="mw-redirect" title="Customers">customers</a>, and for use in <a href="/wiki/Market_segmentation" title="Market segmentation">market segmentation</a>, <a href="/wiki/Positioning_(marketing)" title="Positioning (marketing)">product positioning</a>, <a href="/wiki/New_product_development" title="New product development">new product development</a> and selecting test markets.</dd> <dt>Grouping of shopping items</dt> <dd>Clustering can be used to group all the shopping items available on the web into a set of unique products. For example, all the items on eBay can be grouped into unique products (eBay does not have the concept of a <a href="/wiki/Stock-keeping_unit" class="mw-redirect" title="Stock-keeping unit">SKU</a>).</dd></dl> <div class="mw-heading mw-heading3"><h3 id="World_Wide_Web"><a href="/wiki/World_Wide_Web" title="World Wide Web">World Wide Web</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=17" title="Edit section: World Wide Web"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Social network analysis</dt> <dd>In the study of <a href="/wiki/Social_network" title="Social network">social networks</a>, clustering may be used to recognize <a href="/wiki/Communities" class="mw-redirect" title="Communities">communities</a> within large groups of people.</dd> <dt>Search result grouping</dt> <dd>In the process of intelligent grouping of the files and websites, clustering may be used to create a more relevant set of search results compared to normal search engines like <a href="/wiki/Google" title="Google">Google</a><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2018)">citation needed</span></a></i>]</sup>. There are currently a number of web-based clustering tools such as <a href="/wiki/Clusty" class="mw-redirect" title="Clusty">Clusty</a>. It also may be used to return a more comprehensive set of results in cases where a search term could refer to vastly different things. Each distinct use of the term corresponds to a unique cluster of results, allowing a ranking algorithm to return comprehensive results by picking the top result from each cluster.<sup id="cite_ref-mitpressjournals.org_57-0" class="reference"><a href="#cite_note-mitpressjournals.org-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup></dd> <dt>Slippy map optimization</dt> <dd><a href="/wiki/Flickr" title="Flickr">Flickr</a>'s map of photos and other map sites use clustering to reduce the number of markers on a map.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2023)">citation needed</span></a></i>]</sup> This makes it both faster and reduces the amount of visual clutter.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Computer_science"><a href="/wiki/Computer_science" title="Computer science">Computer science</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=18" title="Edit section: Computer science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt><a href="/wiki/Software_evolution" title="Software evolution">Software evolution</a></dt> <dd>Clustering is useful in software evolution as it helps to reduce legacy properties in code by reforming functionality that has become dispersed. It is a form of restructuring and hence is a way of direct preventative maintenance.</dd> <dt><a href="/wiki/Image_segmentation" title="Image segmentation">Image segmentation</a></dt> <dd>Clustering can be used to divide a <a href="/wiki/Digital_data" title="Digital data">digital</a> <a href="/wiki/Image" title="Image">image</a> into distinct regions for <a href="/wiki/Border_detection" class="mw-redirect" title="Border detection">border detection</a> or <a href="/wiki/Object_recognition" class="mw-redirect" title="Object recognition">object recognition</a>.<sup id="cite_ref-panSearch_58-0" class="reference"><a href="#cite_note-panSearch-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup></dd> <dt><a href="/wiki/Evolutionary_algorithms" class="mw-redirect" title="Evolutionary algorithms">Evolutionary algorithms</a></dt> <dd>Clustering may be used to identify different niches within the population of an evolutionary algorithm so that reproductive opportunity can be distributed more evenly amongst the evolving species or subspecies.</dd> <dt><a href="/wiki/Recommender_systems" class="mw-redirect" title="Recommender systems">Recommender systems</a></dt> <dd>Recommender systems are designed to recommend new items based on a user's tastes. They sometimes use clustering algorithms to predict a user's preferences based on the preferences of other users in the user's cluster.</dd> <dt><a href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo methods</a></dt> <dd>Clustering is often utilized to locate and characterize extrema in the target distribution.</dd> <dt><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></dt> <dd>Anomalies/outliers are typically – be it explicitly or implicitly – defined with respect to clustering structure in data.</dd> <dt><a href="/wiki/Natural_language_processing" title="Natural language processing">Natural language processing</a></dt> <dd>Clustering can be used to resolve <a href="/wiki/Lexical_ambiguity" class="mw-redirect" title="Lexical ambiguity">lexical ambiguity</a>.<sup id="cite_ref-mitpressjournals.org_57-1" class="reference"><a href="#cite_note-mitpressjournals.org-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup></dd> <dt><a href="/wiki/DevOps" title="DevOps">DevOps</a></dt> <dd>Clustering has been used to analyse the effectiveness of DevOps teams.<sup id="cite_ref-stateofdevopsreport_59-0" class="reference"><a href="#cite_note-stateofdevopsreport-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Social_science">Social science</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=19" title="Edit section: Social science"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt><a href="/wiki/Sequence_analysis_in_social_sciences" title="Sequence analysis in social sciences">Sequence analysis in social sciences</a></dt> <dd>Cluster analysis is used to identify patterns of family life trajectories, professional careers, and daily or weekly time use for example.</dd> <dt><a href="/wiki/Crime_analysis" title="Crime analysis">Crime analysis</a></dt> <dd>Cluster analysis can be used to identify areas where there are greater incidences of particular types of crime. By identifying these distinct areas or "hot spots" where a similar crime has happened over a period of time, it is possible to manage law enforcement resources more effectively.</dd> <dt><a href="/wiki/Educational_data_mining" title="Educational data mining">Educational data mining</a></dt> <dd>Cluster analysis is for example used to identify groups of schools or students with similar properties.</dd> <dt>Typologies</dt> <dd>From poll data, projects such as those undertaken by the Pew Research Center use cluster analysis to discern typologies of opinions, habits, and demographics that may be useful in politics and marketing.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Others">Others</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=20" title="Edit section: Others"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Field robotics</dt> <dd>Clustering algorithms are used for robotic situational awareness to track objects and detect outliers in sensor data.<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dt><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></dt> <dd>To find structural similarity, etc., for example, 3000 chemical compounds were clustered in the space of 90 <a href="/wiki/Topological_index" title="Topological index">topological indices</a>.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dt><a href="/wiki/Climatology" title="Climatology">Climatology</a></dt> <dd>To find weather regimes or preferred sea level pressure atmospheric patterns.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dt>Finance</dt> <dd>Cluster analysis has been used to cluster stocks into sectors.<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup></dd></dl> <dl><dt>Petroleum geology</dt> <dd>Cluster analysis is used to reconstruct missing bottom hole core data or missing log curves in order to evaluate reservoir properties.</dd></dl> <dl><dt>Geochemistry</dt> <dd>The clustering of chemical properties in different sample locations.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Cluster_analysis" class="extiw" title="commons:Category:Cluster analysis">Cluster analysis</a></span>.</div></div> </div> <div class="mw-heading mw-heading3"><h3 id="Specialized_types_of_cluster_analysis">Specialized types of cluster analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=22" title="Edit section: Specialized types of cluster analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Automatic_clustering_algorithms" title="Automatic clustering algorithms">Automatic clustering algorithms</a></li> <li><a href="/wiki/Balanced_clustering" title="Balanced clustering">Balanced clustering</a></li> <li><a href="/wiki/Clustering_high-dimensional_data" title="Clustering high-dimensional data">Clustering high-dimensional data</a></li> <li><a href="/wiki/Conceptual_clustering" title="Conceptual clustering">Conceptual clustering</a></li> <li><a href="/wiki/Consensus_clustering" title="Consensus clustering">Consensus clustering</a></li> <li><a href="/wiki/Constrained_clustering" title="Constrained clustering">Constrained clustering</a></li> <li><a href="/wiki/Community_structure#Algorithms_for_finding_communities" title="Community structure">Community detection</a></li> <li><a href="/wiki/Data_stream_clustering" title="Data stream clustering">Data stream clustering</a></li> <li><a href="/wiki/HCS_clustering_algorithm" title="HCS clustering algorithm">HCS clustering</a></li> <li><a href="/wiki/Sequence_clustering" title="Sequence clustering">Sequence clustering</a></li> <li><a href="/wiki/Spectral_clustering" title="Spectral clustering">Spectral clustering</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Techniques_used_in_cluster_analysis">Techniques used in cluster analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=23" title="Edit section: Techniques used in cluster analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural network</a> (ANN)</li> <li><a href="/wiki/Nearest_neighbor_search" title="Nearest neighbor search">Nearest neighbor search</a></li> <li><a href="/wiki/Neighbourhood_components_analysis" title="Neighbourhood components analysis">Neighbourhood components analysis</a></li> <li><a href="/wiki/Latent_class_model" title="Latent class model">Latent class analysis</a></li> <li><a href="/wiki/Affinity_propagation" title="Affinity propagation">Affinity propagation</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Data_projection_and_preprocessing">Data projection and preprocessing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=24" title="Edit section: Data projection and preprocessing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Dimension_reduction" class="mw-redirect" title="Dimension reduction">Dimension reduction</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal component analysis</a></li> <li><a href="/wiki/Multidimensional_scaling" title="Multidimensional scaling">Multidimensional scaling</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Other">Other</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=25" title="Edit section: Other"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cluster-weighted_modeling" title="Cluster-weighted modeling">Cluster-weighted modeling</a></li> <li><a href="/wiki/Curse_of_dimensionality" title="Curse of dimensionality">Curse of dimensionality</a></li> <li><a href="/wiki/Determining_the_number_of_clusters_in_a_data_set" title="Determining the number of clusters in a data set">Determining the number of clusters in a data set</a></li> <li><a href="/wiki/Parallel_coordinates" title="Parallel coordinates">Parallel coordinates</a></li> <li><a href="/wiki/Structured_data_analysis_(statistics)" title="Structured data analysis (statistics)">Structured data analysis</a></li> <li><a href="/wiki/Linear_separability" title="Linear separability">Linear separability</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cluster_analysis&action=edit&section=26" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output 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href="/wiki/Loss_functions_for_classification" title="Loss functions for classification">Loss functions</a></li> <li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression</a> <ul><li><a href="/wiki/Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></li> <li><a href="/wiki/Double_descent" title="Double descent">Double descent</a></li> <li><a href="/wiki/Overfitting" title="Overfitting">Overfitting</a></li></ul></li> <li><a class="mw-selflink selflink">Clustering</a></li> <li><a href="/wiki/Gradient_descent" title="Gradient descent">Gradient descent</a> <ul><li><a href="/wiki/Stochastic_gradient_descent" title="Stochastic gradient descent">SGD</a></li> <li><a href="/wiki/Quasi-Newton_method" title="Quasi-Newton method">Quasi-Newton method</a></li> <li><a href="/wiki/Conjugate_gradient_method" title="Conjugate gradient method">Conjugate gradient method</a></li></ul></li> <li><a href="/wiki/Backpropagation" title="Backpropagation">Backpropagation</a></li> <li><a href="/wiki/Attention_(machine_learning)" title="Attention (machine learning)">Attention</a></li> <li><a href="/wiki/Convolution" title="Convolution">Convolution</a></li> <li><a href="/wiki/Normalization_(machine_learning)" title="Normalization (machine learning)">Normalization</a> <ul><li><a href="/wiki/Batch_normalization" title="Batch normalization">Batchnorm</a></li></ul></li> <li><a href="/wiki/Activation_function" title="Activation function">Activation</a> <ul><li><a href="/wiki/Softmax_function" title="Softmax function">Softmax</a></li> <li><a href="/wiki/Sigmoid_function" title="Sigmoid function">Sigmoid</a></li> <li><a href="/wiki/Rectifier_(neural_networks)" title="Rectifier (neural networks)">Rectifier</a></li></ul></li> <li><a href="/wiki/Gating_mechanism" title="Gating mechanism">Gating</a></li> <li><a href="/wiki/Weight_initialization" title="Weight initialization">Weight initialization</a></li> <li><a href="/wiki/Regularization_(mathematics)" title="Regularization (mathematics)">Regularization</a></li> <li><a href="/wiki/Training,_validation,_and_test_data_sets" title="Training, validation, and test data sets">Datasets</a> <ul><li><a href="/wiki/Data_augmentation" title="Data augmentation">Augmentation</a></li></ul></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a> <ul><li><a href="/wiki/Q-learning" title="Q-learning">Q-learning</a></li> <li><a href="/wiki/State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action" title="State–action–reward–state–action">SARSA</a></li> <li><a href="/wiki/Imitation_learning" title="Imitation learning">Imitation</a></li></ul></li> <li><a href="/wiki/Diffusion_process" title="Diffusion process">Diffusion</a></li> <li><a href="/wiki/Latent_diffusion_model" title="Latent diffusion model">Latent diffusion model</a></li> <li><a href="/wiki/Autoregressive_model" title="Autoregressive model">Autoregression</a></li> <li><a href="/wiki/Adversarial_machine_learning" title="Adversarial machine learning">Adversary</a></li> <li><a href="/wiki/Retrieval-augmented_generation" title="Retrieval-augmented generation">RAG</a></li> <li><a href="/wiki/Reinforcement_learning_from_human_feedback" title="Reinforcement learning from human feedback">RLHF</a></li> <li><a href="/wiki/Self-supervised_learning" title="Self-supervised learning">Self-supervised learning</a></li> <li><a href="/wiki/Prompt_engineering" title="Prompt engineering">Prompt engineering</a></li> <li><a href="/wiki/Word_embedding" title="Word embedding">Word embedding</a></li> <li><a href="/wiki/Hallucination_(artificial_intelligence)" title="Hallucination (artificial intelligence)">Hallucination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a> <ul><li><a href="/wiki/Prompt_engineering#In-context_learning" title="Prompt engineering">In-context learning</a></li></ul></li> <li><a href="/wiki/Neural_network_(machine_learning)" title="Neural network (machine learning)">Artificial neural network</a> <ul><li><a href="/wiki/Deep_learning" title="Deep learning">Deep learning</a></li></ul></li> <li><a href="/wiki/Language_model" title="Language model">Language model</a> <ul><li><a href="/wiki/Large_language_model" title="Large language model">Large language model</a></li> <li><a href="/wiki/Neural_machine_translation" title="Neural machine translation">NMT</a></li></ul></li> <li><a href="/wiki/Artificial_general_intelligence" title="Artificial general intelligence">Artificial general intelligence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Implementations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Audio–visual</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/AlexNet" title="AlexNet">AlexNet</a></li> <li><a href="/wiki/WaveNet" title="WaveNet">WaveNet</a></li> <li><a href="/wiki/Human_image_synthesis" title="Human image synthesis">Human image synthesis</a></li> <li><a href="/wiki/Handwriting_recognition" title="Handwriting recognition">HWR</a></li> <li><a href="/wiki/Optical_character_recognition" title="Optical character recognition">OCR</a></li> <li><a href="/wiki/Deep_learning_speech_synthesis" title="Deep learning speech synthesis">Speech synthesis</a> <ul><li><a href="/wiki/ElevenLabs" title="ElevenLabs">ElevenLabs</a></li></ul></li> <li><a href="/wiki/Speech_recognition" title="Speech recognition">Speech recognition</a> <ul><li><a href="/wiki/Whisper_(speech_recognition_system)" title="Whisper (speech recognition system)">Whisper</a></li></ul></li> <li><a href="/wiki/Facial_recognition_system" title="Facial recognition system">Facial recognition</a></li> <li><a href="/wiki/AlphaFold" title="AlphaFold">AlphaFold</a></li> <li><a href="/wiki/Text-to-image_model" title="Text-to-image model">Text-to-image models</a> <ul><li><a href="/wiki/DALL-E" title="DALL-E">DALL-E</a></li> <li><a href="/wiki/Flux_(text-to-image_model)" title="Flux (text-to-image model)">Flux</a></li> <li><a href="/wiki/Ideogram_(text-to-image_model)" title="Ideogram (text-to-image model)">Ideogram</a></li> <li><a href="/wiki/Midjourney" title="Midjourney">Midjourney</a></li> <li><a href="/wiki/Stable_Diffusion" title="Stable Diffusion">Stable Diffusion</a></li></ul></li> <li><a href="/wiki/Text-to-video_model" title="Text-to-video model">Text-to-video models</a> <ul><li><a href="/wiki/Sora_(text-to-video_model)" title="Sora (text-to-video model)">Sora</a></li> <li><a href="/wiki/Dream_Machine_(text-to-video_model)" title="Dream Machine (text-to-video model)">Dream Machine</a></li> <li><a href="/wiki/VideoPoet" title="VideoPoet">VideoPoet</a></li></ul></li> <li><a href="/wiki/Music_and_artificial_intelligence" title="Music and artificial intelligence">Music generation</a> <ul><li><a href="/wiki/Suno_AI" title="Suno AI">Suno AI</a></li> <li><a href="/wiki/Udio" title="Udio">Udio</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Text</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Word2vec" title="Word2vec">Word2vec</a></li> <li><a href="/wiki/Seq2seq" title="Seq2seq">Seq2seq</a></li> <li><a href="/wiki/GloVe" title="GloVe">GloVe</a></li> <li><a href="/wiki/BERT_(language_model)" title="BERT (language model)">BERT</a></li> <li><a href="/wiki/T5_(language_model)" title="T5 (language model)">T5</a></li> <li><a href="/wiki/Llama_(language_model)" title="Llama (language model)">Llama</a></li> <li><a href="/wiki/Chinchilla_(language_model)" title="Chinchilla (language model)">Chinchilla AI</a></li> <li><a href="/wiki/PaLM" title="PaLM">PaLM</a></li> <li><a href="/wiki/Generative_pre-trained_transformer" title="Generative pre-trained transformer">GPT</a> <ul><li><a href="/wiki/GPT-1" title="GPT-1">1</a></li> <li><a href="/wiki/GPT-2" title="GPT-2">2</a></li> <li><a href="/wiki/GPT-3" title="GPT-3">3</a></li> <li><a href="/wiki/GPT-J" title="GPT-J">J</a></li> <li><a href="/wiki/ChatGPT" title="ChatGPT">ChatGPT</a></li> <li><a href="/wiki/GPT-4" title="GPT-4">4</a></li> <li><a href="/wiki/GPT-4o" title="GPT-4o">4o</a></li> <li><a href="/wiki/OpenAI_o1" title="OpenAI o1">o1</a></li></ul></li> <li><a href="/wiki/Claude_(language_model)" title="Claude (language model)">Claude</a></li> <li><a href="/wiki/Gemini_(language_model)" title="Gemini (language model)">Gemini</a></li> <li><a href="/wiki/Grok_(chatbot)" title="Grok (chatbot)">Grok</a></li> <li><a href="/wiki/LaMDA" title="LaMDA">LaMDA</a></li> <li><a href="/wiki/BLOOM_(language_model)" title="BLOOM (language model)">BLOOM</a></li> <li><a href="/wiki/Project_Debater" title="Project Debater">Project Debater</a></li> <li><a href="/wiki/IBM_Watson" title="IBM Watson">IBM Watson</a></li> <li><a href="/wiki/IBM_Watsonx" title="IBM Watsonx">IBM Watsonx</a></li> <li><a href="/wiki/IBM_Granite" title="IBM Granite">Granite</a></li> <li><a href="/wiki/Huawei_PanGu" title="Huawei PanGu">PanGu-Σ</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decisional</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/AlphaGo" title="AlphaGo">AlphaGo</a></li> <li><a href="/wiki/AlphaZero" title="AlphaZero">AlphaZero</a></li> <li><a href="/wiki/OpenAI_Five" title="OpenAI Five">OpenAI Five</a></li> <li><a href="/wiki/Self-driving_car" title="Self-driving car">Self-driving car</a></li> <li><a href="/wiki/MuZero" title="MuZero">MuZero</a></li> <li><a href="/wiki/Action_selection" title="Action selection">Action selection</a> <ul><li><a href="/wiki/AutoGPT" title="AutoGPT">AutoGPT</a></li></ul></li> <li><a href="/wiki/Robot_control" title="Robot control">Robot control</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a></li> <li><a href="/wiki/Warren_Sturgis_McCulloch" title="Warren Sturgis McCulloch">Warren Sturgis McCulloch</a></li> <li><a href="/wiki/Walter_Pitts" title="Walter Pitts">Walter Pitts</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a></li> <li><a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a></li> <li><a href="/wiki/Marvin_Minsky" title="Marvin Minsky">Marvin Minsky</a></li> <li><a href="/wiki/John_McCarthy_(computer_scientist)" title="John McCarthy (computer scientist)">John McCarthy</a></li> <li><a href="/wiki/Nathaniel_Rochester_(computer_scientist)" title="Nathaniel Rochester (computer scientist)">Nathaniel Rochester</a></li> <li><a href="/wiki/Allen_Newell" title="Allen Newell">Allen Newell</a></li> <li><a href="/wiki/Cliff_Shaw" title="Cliff Shaw">Cliff Shaw</a></li> <li><a href="/wiki/Herbert_A._Simon" title="Herbert A. Simon">Herbert A. Simon</a></li> <li><a href="/wiki/Oliver_Selfridge" title="Oliver Selfridge">Oliver Selfridge</a></li> <li><a href="/wiki/Frank_Rosenblatt" title="Frank Rosenblatt">Frank Rosenblatt</a></li> <li><a href="/wiki/Bernard_Widrow" title="Bernard Widrow">Bernard Widrow</a></li> <li><a href="/wiki/Joseph_Weizenbaum" title="Joseph Weizenbaum">Joseph Weizenbaum</a></li> <li><a href="/wiki/Seymour_Papert" title="Seymour Papert">Seymour Papert</a></li> <li><a href="/wiki/Seppo_Linnainmaa" title="Seppo Linnainmaa">Seppo Linnainmaa</a></li> <li><a href="/wiki/Paul_Werbos" title="Paul Werbos">Paul Werbos</a></li> <li><a href="/wiki/J%C3%BCrgen_Schmidhuber" title="Jürgen Schmidhuber">Jürgen Schmidhuber</a></li> <li><a href="/wiki/Yann_LeCun" title="Yann LeCun">Yann LeCun</a></li> <li><a href="/wiki/Geoffrey_Hinton" title="Geoffrey Hinton">Geoffrey Hinton</a></li> <li><a href="/wiki/John_Hopfield" title="John Hopfield">John Hopfield</a></li> <li><a href="/wiki/Yoshua_Bengio" title="Yoshua Bengio">Yoshua Bengio</a></li> <li><a href="/wiki/Lotfi_A._Zadeh" title="Lotfi A. Zadeh">Lotfi A. Zadeh</a></li> <li><a href="/wiki/Stephen_Grossberg" title="Stephen Grossberg">Stephen Grossberg</a></li> <li><a href="/wiki/Alex_Graves_(computer_scientist)" title="Alex Graves (computer scientist)">Alex Graves</a></li> <li><a href="/wiki/Andrew_Ng" title="Andrew Ng">Andrew Ng</a></li> <li><a href="/wiki/Fei-Fei_Li" title="Fei-Fei Li">Fei-Fei Li</a></li> <li><a href="/wiki/Alex_Krizhevsky" title="Alex Krizhevsky">Alex Krizhevsky</a></li> <li><a href="/wiki/Ilya_Sutskever" title="Ilya Sutskever">Ilya Sutskever</a></li> <li><a href="/wiki/Demis_Hassabis" title="Demis Hassabis">Demis Hassabis</a></li> <li><a href="/wiki/David_Silver_(computer_scientist)" title="David Silver (computer scientist)">David Silver</a></li> <li><a href="/wiki/Ian_Goodfellow" title="Ian Goodfellow">Ian Goodfellow</a></li> <li><a href="/wiki/Andrej_Karpathy" title="Andrej Karpathy">Andrej Karpathy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anthropic" title="Anthropic">Anthropic</a></li> <li><a href="/wiki/EleutherAI" title="EleutherAI">EleutherAI</a></li> <li><a href="/wiki/Google_DeepMind" title="Google DeepMind">Google DeepMind</a></li> <li><a href="/wiki/Hugging_Face" title="Hugging Face">Hugging Face</a></li> <li><a href="/wiki/Kuaishou" title="Kuaishou">Kuaishou</a></li> <li><a href="/wiki/Meta_AI" title="Meta AI">Meta AI</a></li> <li><a href="/wiki/Mila_(research_institute)" title="Mila (research institute)">Mila</a></li> <li><a href="/wiki/MiniMax_(company)" title="MiniMax (company)">MiniMax</a></li> <li><a href="/wiki/Mistral_AI" title="Mistral AI">Mistral AI</a></li> <li><a href="/wiki/MIT_Computer_Science_and_Artificial_Intelligence_Laboratory" title="MIT Computer Science and Artificial Intelligence Laboratory">MIT CSAIL</a></li> <li><a href="/wiki/OpenAI" title="OpenAI">OpenAI</a></li> <li><a href="/wiki/Runway_(company)" title="Runway (company)">Runway</a></li> <li><a href="/wiki/Stability_AI" title="Stability AI">Stability AI</a></li> <li><a href="/wiki/XAI_(company)" title="XAI (company)">xAI</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Architectures</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Neural_Turing_machine" title="Neural Turing machine">Neural Turing machine</a></li> <li><a href="/wiki/Differentiable_neural_computer" title="Differentiable neural computer">Differentiable neural computer</a></li> <li><a href="/wiki/Transformer_(deep_learning_architecture)" title="Transformer (deep learning architecture)">Transformer</a> <ul><li><a href="/wiki/Vision_transformer" title="Vision transformer">Vision transformer (ViT)</a></li></ul></li> <li><a href="/wiki/Recurrent_neural_network" title="Recurrent neural network">Recurrent neural network (RNN)</a></li> <li><a href="/wiki/Long_short-term_memory" title="Long short-term memory">Long short-term memory (LSTM)</a></li> <li><a href="/wiki/Gated_recurrent_unit" title="Gated recurrent unit">Gated recurrent unit (GRU)</a></li> <li><a href="/wiki/Echo_state_network" title="Echo state network">Echo state network</a></li> <li><a href="/wiki/Multilayer_perceptron" title="Multilayer perceptron">Multilayer perceptron (MLP)</a></li> <li><a href="/wiki/Convolutional_neural_network" title="Convolutional neural network">Convolutional neural network (CNN)</a></li> <li><a href="/wiki/Residual_neural_network" title="Residual neural network">Residual neural network (RNN)</a></li> <li><a href="/wiki/Highway_network" title="Highway network">Highway network</a></li> <li><a href="/wiki/Mamba_(deep_learning_architecture)" title="Mamba (deep learning architecture)">Mamba</a></li> <li><a href="/wiki/Autoencoder" title="Autoencoder">Autoencoder</a></li> <li><a href="/wiki/Variational_autoencoder" title="Variational autoencoder">Variational autoencoder (VAE)</a></li> <li><a href="/wiki/Generative_adversarial_network" title="Generative adversarial network">Generative adversarial network (GAN)</a></li> <li><a href="/wiki/Graph_neural_network" title="Graph neural network">Graph neural network (GNN)</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span> Portals <ul><li><a href="/wiki/Portal:Technology" title="Portal:Technology">Technology</a></li></ul></li> <li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> Categories <ul><li><a href="/wiki/Category:Artificial_neural_networks" title="Category:Artificial neural networks">Artificial neural networks</a></li> <li><a href="/wiki/Category:Machine_learning" title="Category:Machine learning">Machine learning</a></li></ul></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Statistics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Statistics" title="Template:Statistics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Statistics" title="Template talk:Statistics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Statistics" title="Special:EditPage/Template:Statistics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Outline_of_statistics" title="Outline of statistics">Outline</a></li> <li><a href="/wiki/List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics" style="font-size:114%;margin:0 4em"><a href="/wiki/Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mean" title="Mean">Mean</a> <ul><li><a href="/wiki/Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li> <li><a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li> <li><a href="/wiki/Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li> <li><a href="/wiki/Cubic_mean" title="Cubic mean">Cubic</a></li> <li><a href="/wiki/Generalized_mean" title="Generalized mean">Generalized/power</a></li> <li><a href="/wiki/Geometric_mean" title="Geometric mean">Geometric</a></li> <li><a href="/wiki/Harmonic_mean" title="Harmonic mean">Harmonic</a></li> <li><a href="/wiki/Heronian_mean" title="Heronian mean">Heronian</a></li> <li><a href="/wiki/Heinz_mean" title="Heinz mean">Heinz</a></li> <li><a href="/wiki/Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li> <li><a href="/wiki/Median" title="Median">Median</a></li> <li><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li> <li><a href="/wiki/Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li> <li><a href="/wiki/Interquartile_range" title="Interquartile range">Interquartile range</a></li> <li><a href="/wiki/Percentile" title="Percentile">Percentile</a></li> <li><a href="/wiki/Range_(statistics)" title="Range (statistics)">Range</a></li> <li><a href="/wiki/Standard_deviation" title="Standard deviation">Standard deviation</a></li> <li><a href="/wiki/Variance#Sample_variance" title="Variance">Variance</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">Moments</a> <ul><li><a href="/wiki/Kurtosis" title="Kurtosis">Kurtosis</a></li> <li><a href="/wiki/L-moment" title="L-moment">L-moments</a></li> <li><a href="/wiki/Skewness" title="Skewness">Skewness</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li> <li><a href="/wiki/Grouped_data" title="Grouped data">Grouped data</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li> <li><a href="/wiki/Rank_correlation" title="Rank correlation">Rank correlation</a> <ul><li><a href="/wiki/Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li> <li><a href="/wiki/Spearman%27s_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bar_chart" title="Bar chart">Bar chart</a></li> <li><a href="/wiki/Biplot" title="Biplot">Biplot</a></li> <li><a href="/wiki/Box_plot" title="Box plot">Box plot</a></li> <li><a href="/wiki/Control_chart" title="Control chart">Control chart</a></li> <li><a href="/wiki/Correlogram" title="Correlogram">Correlogram</a></li> <li><a href="/wiki/Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li> <li><a href="/wiki/Forest_plot" title="Forest plot">Forest plot</a></li> <li><a href="/wiki/Histogram" title="Histogram">Histogram</a></li> <li><a href="/wiki/Pie_chart" title="Pie chart">Pie chart</a></li> <li><a href="/wiki/Q%E2%80%93Q_plot" title="Q–Q plot">Q–Q plot</a></li> <li><a href="/wiki/Radar_chart" title="Radar chart">Radar chart</a></li> <li><a href="/wiki/Run_chart" title="Run chart">Run chart</a></li> <li><a href="/wiki/Scatter_plot" title="Scatter plot">Scatter plot</a></li> <li><a href="/wiki/Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li> <li><a href="/wiki/Violin_plot" title="Violin plot">Violin plot</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effect_size" title="Effect size">Effect size</a></li> <li><a href="/wiki/Missing_data" title="Missing data">Missing data</a></li> <li><a href="/wiki/Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li> <li><a href="/wiki/Statistical_population" title="Statistical population">Population</a></li> <li><a href="/wiki/Replication_(statistics)" title="Replication (statistics)">Replication</a></li> <li><a href="/wiki/Sample_size_determination" title="Sample size determination">Sample size determination</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">Sampling</a> <ul><li><a href="/wiki/Cluster_sampling" title="Cluster sampling">Cluster</a></li> <li><a href="/wiki/Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li> <li><a href="/wiki/Opinion_poll" title="Opinion poll">Opinion poll</a></li> <li><a href="/wiki/Questionnaire" title="Questionnaire">Questionnaire</a></li> <li><a href="/wiki/Standard_error" title="Standard error">Standard error</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li> <li><a href="/wiki/Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li> <li><a href="/wiki/Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li> <li><a href="/wiki/Random_assignment" title="Random assignment">Random assignment</a></li> <li><a href="/wiki/Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li> <li><a href="/wiki/Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li> <li><a href="/wiki/Scientific_control" title="Scientific control">Scientific control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li> <li><a href="/wiki/Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li> <li><a href="/wiki/Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohort_study" title="Cohort study">Cohort study</a></li> <li><a href="/wiki/Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li> <li><a href="/wiki/Natural_experiment" title="Natural experiment">Natural experiment</a></li> <li><a href="/wiki/Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference" style="font-size:114%;margin:0 4em"><a href="/wiki/Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li> <li><a href="/wiki/Statistic" title="Statistic">Statistic</a></li> <li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a></li> <li><a href="/wiki/Sampling_distribution" title="Sampling distribution">Sampling distribution</a> <ul><li><a href="/wiki/Order_statistic" title="Order statistic">Order statistic</a></li></ul></li> <li><a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a> <ul><li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li></ul></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a> <ul><li><a href="/wiki/Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li> <li><a href="/wiki/Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li> <li><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameter</a> <ul><li><a href="/wiki/Location_parameter" title="Location parameter">location</a></li> <li><a href="/wiki/Scale_parameter" title="Scale parameter">scale</a></li> <li><a href="/wiki/Shape_parameter" title="Shape parameter">shape</a></li></ul></li> <li><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric family</a> <ul><li><a href="/wiki/Likelihood_function" title="Likelihood function">Likelihood</a> <a href="/wiki/Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size:85%;">(monotone)</span></a></li> <li><a href="/wiki/Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li> <li><a href="/wiki/Exponential_family" title="Exponential family">Exponential family</a></li></ul></li> <li><a href="/wiki/Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li> <li><a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/U-statistic" title="U-statistic">U</a></li> <li><a href="/wiki/V-statistic" title="V-statistic">V</a></li></ul></li> <li><a href="/wiki/Optimal_decision" title="Optimal decision">Optimal decision</a> <ul><li><a href="/wiki/Loss_function" title="Loss function">loss function</a></li></ul></li> <li><a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li> <li><a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a> <ul><li><a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li> <li><a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li> <li><a href="/wiki/Robust_statistics" title="Robust statistics">Robustness</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estimating_equations" title="Estimating equations">Estimating equations</a> <ul><li><a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li> <li><a href="/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li> <li><a href="/wiki/M-estimator" title="M-estimator">M-estimator</a></li> <li><a href="/wiki/Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li> <li><a href="/wiki/Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a> <ul><li><a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a> <ul><li><a href="/wiki/Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li> <li><a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li> <li><a href="/wiki/Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li> <li><a href="/wiki/Plug-in_principle" class="mw-redirect" title="Plug-in principle">Plug-in</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Confidence_interval" title="Confidence interval">Confidence interval</a></li> <li><a href="/wiki/Pivotal_quantity" title="Pivotal quantity">Pivot</a></li> <li><a href="/wiki/Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li> <li><a href="/wiki/Prediction_interval" title="Prediction interval">Prediction interval</a></li> <li><a href="/wiki/Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a> <ul><li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li> <li><a href="/wiki/Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li> <li><a href="/wiki/Power_(statistics)" title="Power (statistics)">Power</a> <ul><li><a href="/wiki/Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li> <li><a href="/wiki/Permutation_test" title="Permutation test">Permutation test</a> <ul><li><a href="/wiki/Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li> <li><a href="/wiki/Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li> <li><a href="/wiki/Score_test" title="Score test">Score/Lagrange multiplier</a></li> <li><a href="/wiki/Wald_test" title="Wald test">Wald</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Z-test" title="Z-test"><i>Z</i>-test <span style="font-size:85%;">(normal)</span></a></li> <li><a href="/wiki/Student%27s_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li> <li><a href="/wiki/F-test" title="F-test"><i>F</i>-test</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chi-squared_test" title="Chi-squared test">Chi-squared</a></li> <li><a href="/wiki/G-test" title="G-test"><i>G</i>-test</a></li> <li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li> <li><a href="/wiki/Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li> <li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors</a></li> <li><a href="/wiki/Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li> <li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size:85%;">(Shapiro–Wilk)</span></a></li> <li><a href="/wiki/Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li> <li><a href="/wiki/Model_selection" title="Model selection">Model selection</a> <ul><li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li> <li><a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">AIC</a></li> <li><a href="/wiki/Bayesian_information_criterion" title="Bayesian information criterion">BIC</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sign_test" title="Sign test">Sign</a> <ul><li><a href="/wiki/Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li> <li><a href="/wiki/Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size:85%;">(Wilcoxon)</span></a> <ul><li><a href="/wiki/Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li> <li><a href="/wiki/Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size:85%;">(Mann–Whitney)</span></a></li> <li><a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="/wiki/Analysis_of_variance" title="Analysis of variance">anova</a> <ul><li><a href="/wiki/Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size:85%;">(Kruskal–Wallis)</span></a></li> <li><a href="/wiki/Friedman_test" title="Friedman test">2-way <span style="font-size:85%;">(Friedman)</span></a></li> <li><a href="/wiki/Jonckheere%27s_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size:85%;">(Jonckheere–Terpstra)</span></a></li></ul></li> <li><a href="/wiki/Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a> <ul><li><a href="/wiki/Prior_probability" title="Prior probability">prior</a></li> <li><a href="/wiki/Posterior_probability" title="Posterior probability">posterior</a></li></ul></li> <li><a href="/wiki/Credible_interval" title="Credible interval">Credible interval</a></li> <li><a href="/wiki/Bayes_factor" title="Bayes factor">Bayes factor</a></li> <li><a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayesian estimator</a> <ul><li><a href="/wiki/Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li> <li><a href="/wiki/Partial_correlation" title="Partial correlation">Partial correlation</a></li> <li><a href="/wiki/Confounding" title="Confounding">Confounding variable</a></li> <li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li> <li><a href="/wiki/Regression_validation" title="Regression validation">Regression validation</a></li> <li><a href="/wiki/Mixed_model" title="Mixed model">Mixed effects models</a></li> <li><a href="/wiki/Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li> <li><a href="/wiki/Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li> <li><a href="/wiki/Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li> <li><a href="/wiki/General_linear_model" title="General linear model">General linear model</a></li> <li><a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li> <li><a href="/wiki/Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li> <li><a href="/wiki/Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li> <li><a href="/wiki/Isotonic_regression" title="Isotonic regression">Isotonic</a></li> <li><a href="/wiki/Robust_regression" title="Robust regression">Robust</a></li> <li><a href="/wiki/Heteroscedasticity" class="mw-redirect" title="Heteroscedasticity">Heteroscedasticity</a></li> <li><a href="/wiki/Homoscedasticity" class="mw-redirect" title="Homoscedasticity">Homoscedasticity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exponential_family" title="Exponential family">Exponential families</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic <span style="font-size:85%;">(Bernoulli)</span></a> / <a href="/wiki/Binomial_regression" title="Binomial regression">Binomial</a> / <a href="/wiki/Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li> <li><a href="/wiki/Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li> <li><a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_Multivariate_/_Time-series_/_Survival_analysis" style="font-size:114%;margin:0 4em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a> / <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a> / <a href="/wiki/Time_series" title="Time series">Time-series</a> / <a href="/wiki/Survival_analysis" title="Survival analysis">Survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cohen%27s_kappa" title="Cohen's kappa">Cohen's kappa</a></li> <li><a href="/wiki/Contingency_table" title="Contingency table">Contingency table</a></li> <li><a href="/wiki/Graphical_model" title="Graphical model">Graphical model</a></li> <li><a href="/wiki/Poisson_regression" title="Poisson regression">Log-linear model</a></li> <li><a href="/wiki/McNemar%27s_test" title="McNemar's test">McNemar's test</a></li> <li><a href="/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_model" title="General linear model">Regression</a></li> <li><a href="/wiki/Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal components</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li> <li><a class="mw-selflink selflink">Cluster analysis</a></li> <li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li> <li><a href="/wiki/Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a> <ul><li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a> <ul><li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li> <li><a href="/wiki/Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationarity</a></li> <li><a href="/wiki/Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li> <li><a href="/wiki/Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li> <li><a href="/wiki/Cointegration" title="Cointegration">Cointegration</a></li> <li><a href="/wiki/Structural_break" title="Structural break">Structural break</a></li> <li><a href="/wiki/Granger_causality" title="Granger causality">Granger causality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li> <li><a href="/wiki/Johansen_test" title="Johansen test">Johansen</a></li> <li><a href="/wiki/Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size:85%;">(Ljung–Box)</span></a></li> <li><a href="/wiki/Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li> <li><a href="/wiki/Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a> <ul><li><a href="/wiki/Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li> <li><a href="/wiki/Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li> <li><a href="/wiki/Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size:85%;">(Box–Jenkins)</span></a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li> <li><a href="/wiki/Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Wavelet" title="Wavelet">Wavelet</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Survival_function" title="Survival function">Survival function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li> <li><a href="/wiki/Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li> <li><a href="/wiki/Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li> <li><a href="/wiki/First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Log-rank_test" class="mw-redirect" title="Log-rank test">Log-rank test</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications" style="font-size:114%;margin:0 4em"><a href="/wiki/List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bioinformatics" title="Bioinformatics">Bioinformatics</a></li> <li><a href="/wiki/Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="/wiki/Clinical_study_design" title="Clinical study design">studies</a></li> <li><a href="/wiki/Epidemiology" title="Epidemiology">Epidemiology</a></li> <li><a href="/wiki/Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chemometrics" title="Chemometrics">Chemometrics</a></li> <li><a href="/wiki/Methods_engineering" title="Methods engineering">Methods engineering</a></li> <li><a href="/wiki/Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li> <li><a href="/wiki/Statistical_process_control" title="Statistical process control">Process</a> / <a href="/wiki/Quality_control" title="Quality control">quality control</a></li> <li><a href="/wiki/Reliability_engineering" title="Reliability engineering">Reliability</a></li> <li><a href="/wiki/System_identification" title="System identification">System identification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="/wiki/Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_science" title="Actuarial science">Actuarial science</a></li> <li><a href="/wiki/Census" title="Census">Census</a></li> <li><a href="/wiki/Crime_statistics" title="Crime statistics">Crime statistics</a></li> <li><a href="/wiki/Demographic_statistics" title="Demographic statistics">Demography</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Jurimetrics" title="Jurimetrics">Jurimetrics</a></li> <li><a href="/wiki/National_accounts" title="National accounts">National accounts</a></li> <li><a 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title="Kriging">Kriging</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span><b><a href="/wiki/Category:Statistics" title="Category:Statistics">Category</a></b></li> <li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" 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