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cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Tecniche</span> </button> <ul id="toc-Tecniche-sublist" class="vector-toc-list"> <li id="toc-Clustering_partizionale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Clustering_partizionale"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Clustering partizionale</span> </div> </a> <ul id="toc-Clustering_partizionale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Clustering_gerarchico" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Clustering_gerarchico"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Clustering gerarchico</span> </div> </a> <ul id="toc-Clustering_gerarchico-sublist" class="vector-toc-list"> <li id="toc-Misure_utilizzate_nel_clustering_gerarchico" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Misure_utilizzate_nel_clustering_gerarchico"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Misure utilizzate nel clustering gerarchico</span> </div> </a> <ul id="toc-Misure_utilizzate_nel_clustering_gerarchico-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Clustering_density-based" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Clustering_density-based"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Clustering density-based</span> </div> </a> <ul id="toc-Clustering_density-based-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algoritmi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algoritmi"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Algoritmi</span> </div> </a> <button aria-controls="toc-Algoritmi-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>Attiva/disattiva la sottosezione Algoritmi</span> </button> <ul id="toc-Algoritmi-sublist" class="vector-toc-list"> <li id="toc-QT_clustering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#QT_clustering"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>QT clustering</span> </div> </a> <ul id="toc-QT_clustering-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Clustering</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="Vai a una voce in un'altra lingua. Disponibile in 40 lingue" > <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 lingue</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 - arabo" lang="ar" hreflang="ar" data-title="تحليل عنقودي" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</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 - bulgaro" lang="bg" hreflang="bg" data-title="Клъстерен анализ" data-language-autonym="Български" data-language-local-name="bulgaro" class="interlanguage-link-target"><span>Български</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 - bengalese" lang="bn" hreflang="bn" data-title="ক্লাস্টার বিশ্লেষণ" data-language-autonym="বাংলা" data-language-local-name="bengalese" class="interlanguage-link-target"><span>বাংলা</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 - catalano" lang="ca" hreflang="ca" data-title="Clusterització de dades" data-language-autonym="Català" data-language-local-name="catalano" 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 - ceco" lang="cs" hreflang="cs" data-title="Shluková analýza" data-language-autonym="Čeština" data-language-local-name="ceco" 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 - danese" lang="da" hreflang="da" data-title="Segmentanalyse" data-language-autonym="Dansk" data-language-local-name="danese" 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 - tedesco" lang="de" hreflang="de" data-title="Clusteranalyse" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</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 - greco" lang="el" hreflang="el" data-title="Συσταδοποίηση" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Cluster_analysis" title="Cluster analysis - inglese" lang="en" hreflang="en" data-title="Cluster analysis" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</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 - spagnolo" lang="es" hreflang="es" data-title="Análisis de grupos" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</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 - estone" lang="et" hreflang="et" data-title="Klasteranalüüs" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Multzokatze_(estatistika)" title="Multzokatze (estatistika) - basco" lang="eu" hreflang="eu" data-title="Multzokatze (estatistika)" data-language-autonym="Euskara" data-language-local-name="basco" 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 - persiano" lang="fa" hreflang="fa" data-title="خوشه‌بندی" data-language-autonym="فارسی" data-language-local-name="persiano" 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 - francese" lang="fr" hreflang="fr" data-title="Partitionnement de données" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</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 - ebraico" lang="he" hreflang="he" data-title="ניתוח אשכולות" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</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 - croato" lang="hr" hreflang="hr" data-title="Grupiranje" data-language-autonym="Hrvatski" data-language-local-name="croato" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Klaszteranal%C3%ADzis" title="Klaszteranalízis - ungherese" lang="hu" hreflang="hu" data-title="Klaszteranalízis" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</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 - armeno" lang="hy" hreflang="hy" data-title="Կլաստեր-անալիզ" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Analisis_kelompok" title="Analisis kelompok - indonesiano" lang="id" hreflang="id" data-title="Analisis kelompok" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</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 - giapponese" 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="giapponese" class="interlanguage-link-target"><span>日本語</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 - coreano" lang="ko" hreflang="ko" data-title="클러스터 분석" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</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 - lettone" lang="lv" hreflang="lv" data-title="Klasteru analīze" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Clusteranalyse" title="Clusteranalyse - olandese" lang="nl" hreflang="nl" data-title="Clusteranalyse" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Klyngeanalyse" title="Klyngeanalyse - norvegese bokmål" lang="nb" hreflang="nb" data-title="Klyngeanalyse" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese 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 - polacco" lang="pl" hreflang="pl" data-title="Analiza skupień" data-language-autonym="Polski" data-language-local-name="polacco" 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 - portoghese" lang="pt" hreflang="pt" data-title="Clustering" data-language-autonym="Português" data-language-local-name="portoghese" 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 - russo" lang="ru" hreflang="ru" data-title="Кластерный анализ" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Grupiranje" title="Grupiranje - serbo-croato" lang="sh" hreflang="sh" data-title="Grupiranje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croato" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Cluster_analysis" title="Cluster analysis - Simple English" lang="en-simple" hreflang="en-simple" data-title="Cluster analysis" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Zhlukov%C3%A1_anal%C3%BDza" title="Zhluková analýza - slovacco" lang="sk" hreflang="sk" data-title="Zhluková analýza" data-language-autonym="Slovenčina" data-language-local-name="slovacco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Grupiranje" title="Grupiranje - sloveno" lang="sl" hreflang="sl" data-title="Grupiranje" data-language-autonym="Slovenščina" data-language-local-name="sloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Klasterska_analiza" title="Klasterska analiza - serbo" lang="sr" hreflang="sr" data-title="Klasterska analiza" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Klusteranalys_(datavetenskap)" title="Klusteranalys (datavetenskap) - svedese" lang="sv" hreflang="sv" data-title="Klusteranalys (datavetenskap)" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%88%E0%B8%B1%E0%B8%9A%E0%B8%81%E0%B8%A5%E0%B8%B8%E0%B9%88%E0%B8%A1%E0%B8%82%E0%B9%89%E0%B8%AD%E0%B8%A1%E0%B8%B9%E0%B8%A5" title="การจับกลุ่มข้อมูล - thailandese" lang="th" hreflang="th" data-title="การจับกลุ่มข้อมูล" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C3%BCmeleme_analizi" title="Kümeleme analizi - turco" lang="tr" hreflang="tr" data-title="Kümeleme analizi" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%82%D0%B5%D1%80%D0%BD%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" 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class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">nascondi</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Da Wikipedia, l'enciclopedia libera.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><p>In <a href="/wiki/Statistica" title="Statistica">statistica</a>, il <b>clustering</b> o <b>analisi dei gruppi</b> (dal termine <a href="/wiki/Lingua_inglese" title="Lingua inglese">inglese</a> <i>cluster analysis</i>, introdotto da <a href="/wiki/Robert_Tryon" title="Robert Tryon">Robert Tryon</a> nel <a href="/wiki/1939" title="1939">1939</a>) è un insieme di tecniche di <a href="/wiki/Statistica_multivariata" title="Statistica multivariata">analisi multivariata dei dati</a> volte alla selezione e raggruppamento di elementi omogenei in un insieme di dati. </p><p>Le tecniche di <i>clustering</i> si basano su misure relative alla somiglianza tra gli elementi. In molti approcci questa similarità, o meglio, dissimilarità, è concepita in termini di distanza in uno spazio multidimensionale. La bontà delle analisi ottenute dagli algoritmi di <i>clustering</i> dipende molto dalla scelta della <a href="/wiki/Distanza_(matematica)" title="Distanza (matematica)">metrica</a>, e quindi da come è calcolata la distanza. Gli <a href="/wiki/Algoritmo" title="Algoritmo">algoritmi</a> di <i>clustering</i> raggruppano gli elementi sulla base della loro distanza reciproca, e quindi l'appartenenza o meno a un <a href="/wiki/Insieme" title="Insieme">insieme</a> dipende da quanto l'elemento preso in esame è distante dall'insieme stesso. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Tecniche">Tecniche</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=1" title="Modifica la sezione Tecniche" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=1" title="Edit section's source code: Tecniche"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le tecniche di <i>clustering</i> si possono basare principalmente su due "filosofie": </p> <ul><li>Dal basso verso l'alto (<i>metodi aggregativi o bottom-up</i>):</li></ul> <dl><dd>Questa filosofia prevede che inizialmente tutti gli elementi siano considerati <i>cluster</i> a sé, e poi l'algoritmo provvede ad unire i <i>cluster</i> più vicini. L'algoritmo continua ad unire elementi al <i>cluster</i> fino ad ottenere un numero prefissato di <i>cluster</i>, oppure fino a che la distanza minima tra i <i>cluster</i> non supera un certo valore, o ancora in relazione ad un determinato criterio statistico prefissato.</dd></dl> <ul><li>Dall'alto verso il basso (<i>metodi divisivi o top-down</i>):</li></ul> <dl><dd>All'inizio tutti gli elementi sono un unico <i>cluster</i>, e poi l'algoritmo inizia a dividere il <i>cluster</i> in tanti <i>cluster</i> di dimensioni inferiori. Il criterio che guida la divisione è naturalmente quello di ottenere gruppi sempre più omogenei. L'algoritmo procede fino a che non viene soddisfatta una regola di arresto generalmente legata al raggiungimento di un numero prefissato di <i>cluster</i>.</dd></dl> <p>Esistono varie classificazioni delle tecniche di clustering comunemente utilizzate. Una prima categorizzazione dipende dalla possibilità che un elemento possa o meno essere assegnato a più cluster: </p> <ul><li><i>clustering esclusivo</i>: ogni elemento può essere assegnato ad uno e ad un solo gruppo. Quindi i cluster risultanti non possono avere elementi in comune. Questo approccio è detto anche <i>hard clustering</i>.</li> <li><i>clustering non-esclusivo</i>, in cui un elemento può appartenere a più cluster con gradi di appartenenza diversi. Questo approccio è noto anche con il nome di <i>soft clustering</i> o <i>fuzzy clustering,</i> dal termine usato per indicare la <a href="/wiki/Logica_fuzzy" title="Logica fuzzy">logica fuzzy</a>.</li></ul> <p>Un'altra suddivisione delle tecniche di clustering tiene conto del tipo di algoritmo utilizzato per dividere lo spazio: </p> <ul><li><i>clustering partizionale (detto anche non gerarchico, o k-clustering)</i>, in cui per definire l'appartenenza ad un gruppo viene utilizzata una distanza da un punto rappresentativo del cluster (centroide, medioide, ecc.), avendo prefissato il numero di gruppi della partizione risultato. Si tratta di derivazioni del più noto algoritmo di clustering, quello detto delle <a href="/wiki/K-means" title="K-means">k-means</a>, introdotto da MacQueen nel 1967.</li> <li><i>Clustering gerarchico</i>, in cui viene costruita una gerarchia di partizioni caratterizzate da un numero (de)crescente di gruppi, visualizzabile mediante una rappresentazione ad albero (dendrogramma), in cui sono rappresentati i passi di accorpamento/divisione dei gruppi.</li></ul> <p>Queste due suddivisioni sono del tutto trasversali, e molti algoritmi nati come "esclusivi" sono stati in seguito adattati nel caso "non-esclusivo" e viceversa. </p> <div class="mw-heading mw-heading3"><h3 id="Clustering_partizionale">Clustering partizionale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=2" title="Modifica la sezione Clustering partizionale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=2" title="Edit section's source code: Clustering partizionale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gli algoritmi di clustering di questa famiglia creano una <a href="/wiki/Partizione_(teoria_degli_insiemi)" title="Partizione (teoria degli insiemi)">partizione</a> delle osservazioni minimizzando una certa funzione di costo: </p> <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 \sum _{j=1}^{k}E(C_{j}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mi>E</mi> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=1}^{k}E(C_{j}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f9295fccb49688cacc17ec434003acb08a5baa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:10.545ex; height:7.676ex;" alt="{\displaystyle \sum _{j=1}^{k}E(C_{j}),}"></span></dd></dl> <p>dove <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> è il numero dei 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 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/4598eb9b4e1c079e0973de79383fc6f898744e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.572ex; height:2.843ex;" alt="{\displaystyle C_{j}}"></span> è il <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>-esimo cluster e <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 E\colon C\rightarrow \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>:</mo> <mi>C</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\colon C\rightarrow \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17e69880e3f7cd1901ac9bedf565ad40f6278cc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.379ex; height:2.509ex;" alt="{\displaystyle E\colon C\rightarrow \mathbb {R} ^{+}}"></span> è la funzione di costo associata al singolo cluster. L'algoritmo più famoso appartenente a questa famiglia è il <a href="/wiki/K-means" title="K-means">k-means</a>, proposto da MacQueen nel <a href="/wiki/1967" title="1967">1967</a>. Un altro algoritmo abbastanza conosciuto appartenente a questa classe è il <a href="/w/index.php?title=Partitioning_Around_Medioid&action=edit&redlink=1" class="new" title="Partitioning Around Medioid (la pagina non esiste)">Partitioning Around Medioid</a> (PAM). </p> <div class="mw-heading mw-heading3"><h3 id="Clustering_gerarchico">Clustering gerarchico</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=3" title="Modifica la sezione Clustering gerarchico" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=3" title="Edit section's source code: Clustering gerarchico"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le tecniche di <a href="/wiki/Clustering_gerarchico" title="Clustering gerarchico">clustering gerarchico</a> non producono un partizionamento <i>flat</i> dei punti, ma una rappresentazione gerarchica ad albero. <span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Hierarchical1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/02/Hierarchical1.jpg" decoding="async" width="410" height="209" class="mw-file-element" data-file-width="410" data-file-height="209" /></a></span> </p><p>Questi algoritmi sono a loro volta suddivisi in due classi: </p> <ul><li><i>Agglomerativo</i> - Questi algoritmi assumono che inizialmente ogni cluster (foglia) contenga un singolo punto; ad ogni passo, poi, vengono fusi i cluster più "vicini" fino ad ottenere un singolo grande cluster. Questi algoritmi necessitano di misure per valutare la similarità tra cluster, per scegliere la coppia di cluster da fondere ad ogni passo.</li> <li><i>Divisivo</i> - Questi algoritmi, invece, partono considerando lo spazio organizzato in un singolo grande cluster contenente tutti i punti, e via via lo dividono in due. Ad ogni passo viene selezionato un cluster in base ad una misura, ed esso viene suddiviso in due cluster più piccoli. Normalmente viene fissato un numero minimo di punti sotto il quale il cluster non viene ulteriormente suddiviso (nel caso estremo questo valore è 1). Questi tipi di algoritmi necessitano di definire una funzione per scegliere il cluster da suddividere.</li></ul> <p>Una rappresentazione grafica del processo di clustering è fornita dal <a href="/wiki/Dendrogramma" title="Dendrogramma">dendrogramma</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Misure_utilizzate_nel_clustering_gerarchico">Misure utilizzate nel clustering gerarchico</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=4" title="Modifica la sezione Misure utilizzate nel clustering gerarchico" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=4" title="Edit section's source code: Misure utilizzate nel clustering gerarchico"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In entrambi i tipi di clustering gerarchico sono necessarie funzioni per selezionare la coppia di cluster da fondere ("agglomerativo"), oppure il cluster da dividere ("divisivo"). </p><p>Nel primo caso, sono necessarie funzioni che misurino la <i>similarità</i> (o, indistintamente, la <i>distanza</i>) tra due cluster, in modo da fondere quelli più simili. Le funzioni utilizzate nel caso agglomerativo sono: </p> <ul><li><i>Single-link proximity</i></li></ul> <dl><dd>Calcola la distanza tra i due cluster come la distanza minima tra elementi appartenenti a cluster diversi:</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 D(C_{i},C_{j})=\min _{x\in C_{i},y\in C_{j}}d(x,y)}"> <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> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(C_{i},C_{j})=\min _{x\in C_{i},y\in C_{j}}d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237d7d0a32073e74cd94d5e396205a287258d911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.927ex; height:4.676ex;" alt="{\displaystyle D(C_{i},C_{j})=\min _{x\in C_{i},y\in C_{j}}d(x,y)}"></span></dd></dl> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/File:Singlelink.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/f3/Singlelink.jpg" decoding="async" width="437" height="172" class="mw-file-element" data-file-width="437" data-file-height="172" /></a><figcaption></figcaption></figure> <ul><li><i>Average-link proximity</i></li></ul> <dl><dd>Questa funzione calcola la distanza tra i due cluster come la media delle distanze tra i singoli elementi: <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})={\frac {1}{|C_{i}||C_{j}|}}\sum _{x\in C_{i},y\in C_{j}}d(x,y)}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(C_{i},C_{j})={\frac {1}{|C_{i}||C_{j}|}}\sum _{x\in C_{i},y\in C_{j}}d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/980cacfb9f8be52e66dd7a2ca4f4e3ec6c2ad0f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:36.77ex; height:7.009ex;" alt="{\displaystyle D(C_{i},C_{j})={\frac {1}{|C_{i}||C_{j}|}}\sum _{x\in C_{i},y\in C_{j}}d(x,y)}"></span></dd></dl> <ul><li><i>Complete-link proximity</i></li></ul> <dl><dd>Questa funzione calcola la distanza tra i due cluster come la distanza massima tra elementi appartenenti ai due 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 D(C_{i},C_{j})=\max _{x\in C_{i},y\in C_{j}}d(x,y)}"> <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> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(C_{i},C_{j})=\max _{x\in C_{i},y\in C_{j}}d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91bc409c3fa22339a4f7c99431457eb7f463668c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.927ex; height:4.676ex;" alt="{\displaystyle D(C_{i},C_{j})=\max _{x\in C_{i},y\in C_{j}}d(x,y)}"></span></dd></dl> <ul><li><i>Distanza tra centroidi</i></li></ul> <dl><dd>La distanza tra i due cluster coincide con la distanza calcolata tra i centroidi (o medioidi):</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 D(C_{i},C_{j})=d({\hat {c_{i}}},{\hat {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> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(C_{i},C_{j})=d({\hat {c_{i}}},{\hat {c_{j}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c64f5a8c6bc1a4fecdc79da38a72b09a5127b5c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.681ex; height:3.009ex;" alt="{\displaystyle D(C_{i},C_{j})=d({\hat {c_{i}}},{\hat {c_{j}}})}"></span>.</dd></dl> <ul><li><i>Dunn Index</i></li></ul> <dl><dd>L'indice di Dunn mira a identificare cluster densi e ben separati. È definito come il rapporto tra la minima distanza inter-cluster e la massima distanza intra-cluster. Per ogni partizione del cluster, l'indice di Dunn può essere calcolato con la seguente formula:<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></dd></dl> <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 D(C_{i},C_{j})={\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 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> <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(C_{i},C_{j})={\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/148ada727bbfa4dc0e894d85dddc27febc5cc1ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.972ex; height:6.509ex;" alt="{\displaystyle D(C_{i},C_{j})={\frac {\min _{1\leq i<j\leq n}d(i,j)}{\max _{1\leq k\leq n}d^{\prime }(k)}}\,,}"></span></dd></dl></dd></dl> <dl><dd>dove <i>d</i>(<i>i</i>,<i>j</i>) rappresenta la distanza tra i cluster <i>i</i> e <i>j</i> e <i>d</i> '(<i>k</i>) misura la distanza intra-cluster del cluster <i>k</i>. La distanza inter-cluster <i>d</i>(<i>i</i>,<i>j</i>) tra due cluster può essere una qualsiasi misura di distanza, come la distanza tra i centroidi dei cluster. Allo stesso modo, la distanza intra-cluster 'd<i> '(</i>k<i>) può essere misurata in vari modi, come la distanza massima tra qualsiasi coppia di elementi nel cluster </i>k<i>. Poiché il criterio interno cerca cluster con un'alta somiglianza intra-cluster e una bassa somiglianza inter-cluster, gli algoritmi che producono cluster con un alto indice di Dunn sono più desiderabili<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>.</i></dd></dl> <p>Nei casi precedenti, <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(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3772957879a8bbf7946bddf5743c508a1d5072c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.544ex; height:2.843ex;" alt="{\displaystyle d(x,y)}"></span> indica una qualsiasi funzione distanza su uno spazio metrico. </p><p>Invece nel clustering divisivo è necessario individuare il cluster da suddividere in due sottogruppi. Per questa ragione sono necessarie funzioni che misurino la compattezza del cluster, la densità o la sparsità dei punti assegnati ad un cluster. Le funzioni normalmente utilizzate nel caso divisivo sono: </p> <ul><li><i>Average internal similarity</i></li></ul> <dl><dd>Questa funzione valuta la similarità media tra i documenti interni ad un cluster: più sono tra loro dissimili (valori bassi di similarità), più il cluster è da suddividere in sottogruppi:</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 D(C_{i})={\frac {1}{|C_{i}|(|C_{i}|-1)}}\sum _{x,y\in C_{i},x\neq y}d(x,y)}"> <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 stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>x</mi> <mo>≠</mo> <mi>y</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(C_{i})={\frac {1}{|C_{i}|(|C_{i}|-1)}}\sum _{x,y\in C_{i},x\neq y}d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b6322e3c0f2a9d88988b9ef04ae6a2b67f05e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:39.375ex; height:6.843ex;" alt="{\displaystyle D(C_{i})={\frac {1}{|C_{i}|(|C_{i}|-1)}}\sum _{x,y\in C_{i},x\neq y}d(x,y)}"></span></dd></dl> <ul><li><i>Maximum internal distance</i></li></ul> <dl><dd>Questa funzione valuta la distanza massima tra due punti interni ad un cluster. Tale valore è noto anche come 'diametro del cluster': più tale valore è basso, più il cluster è compatto:</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 D(C_{i})=\max _{x,y\in C_{i}}d(x,y)}"> <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 stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(C_{i})=\max _{x,y\in C_{i}}d(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf42547e611d7ba70dfc4da7543062bd3e6770c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.336ex; height:4.343ex;" alt="{\displaystyle D(C_{i})=\max _{x,y\in C_{i}}d(x,y)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Clustering_density-based">Clustering density-based</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=5" title="Modifica la sezione Clustering density-based" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=5" title="Edit section's source code: Clustering density-based"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nel <i>Clustering density-based</i>, il raggruppamento avviene analizzando l'<a href="/wiki/Intorno" title="Intorno">intorno</a> di ogni punto dello spazio. In particolare, viene considerata la densità di punti in un intorno di raggio fissato. </p><p>Un esempio è il metodo di clustering <a href="/wiki/Dbscan" class="mw-redirect" title="Dbscan">Dbscan</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Algoritmi">Algoritmi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=6" title="Modifica la sezione Algoritmi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=6" title="Edit section's source code: Algoritmi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Algoritmi di clustering molto usati sono: </p> <ul><li><a href="/wiki/K-means" title="K-means">K-means</a></li> <li><a href="/wiki/K-medoids" title="K-medoids">K-medoids</a></li> <li><a href="/w/index.php?title=Fuzzy_c-means&action=edit&redlink=1" class="new" title="Fuzzy c-means (la pagina non esiste)">fuzzy c-means</a></li> <li><a href="/w/index.php?title=Qt_clustering&action=edit&redlink=1" class="new" title="Qt clustering (la pagina non esiste)">qt clustering</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="QT_clustering">QT clustering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=7" title="Modifica la sezione QT clustering" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=7" title="Edit section's source code: QT clustering"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Il <b>QT</b> (<i>Quality Threshold</i>) <b>Clustering</b> (Heyer et al., 1999) è un metodo alternativo di partizionare i dati, inventato per il clustering dei <a href="/wiki/Gene" title="Gene">geni</a>. Richiede più potenza di calcolo rispetto al <i>K</i>-Means, ma non richiede di specificare il numero di cluster <i>a priori</i>, e restituisce sempre lo stesso risultato quando si ripete diverse volte. </p><p>L'algoritmo è: </p> <ul><li>L'utente sceglie un diametro massimo per i cluster;</li> <li>Costruzione di un cluster candidato per ogni punto, includendo il punto più vicino, il prossimo più vicino, e così via, fino a che il diametro del cluster non supera la soglia;</li> <li>Salvataggio del cluster candidato con la maggior parte dei punti come primo vero cluster, e rimozione di tutti i punti nel cluster da ulteriori considerazioni;</li> <li><a href="/wiki/Ricorsione" class="mw-redirect" title="Ricorsione">Ricorsione</a> col ridotto insieme di cluster.</li></ul> <p>La distanza tra un punto ed un gruppo di punti è calcolata usando il concatenamento completo, cioè come la massima distanza dal punto di ciascun membro del gruppo (vedi il "Clustering gerarchico agglomerativo" sulla distanza tra i cluster nella sezione <a href="#Clustering_gerarchico">clustering gerarchico</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=8" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=8" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation pubblicazione" style="font-style:normal"> J. Dunn, <a rel="nofollow" class="external text" href="https://oadoi.org/10.1080/01969727408546059"><span style="font-style:italic;">Well separated clusters and optimal fuzzy partitions</span></a>, in <span style="font-style:italic;">Journal of Cybernetics</span>, vol. 4, 1974, pp. 95–104, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080%2F01969727408546059">10.1080/01969727408546059</a>.</cite></span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://python.engineering/dunn-index-and-db-index-cluster-validity-indices-set/"><span style="font-style:italic;">Dunn index in Python</span></a>, su <span style="font-style:italic;">Python.Engineering</span>, 13 dicembre 2022.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=9" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=9" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation libro" style="font-style:normal"> Roberto Todeschini, <span style="font-style:italic;">Introduzione alla chemiometria</span>, 1ª ed., Napoli, EdiSES, 2003, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-7959-146-0" title="Speciale:RicercaISBN/88-7959-146-0">88-7959-146-0</a>.</cite></li> <li>G.Susinno, M.A.Miceli, <i>Ultrametricity in Fund of Funds Diversification</i>, Physica A 344 (1-2) (2004) 95-99</li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=10" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=10" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Mean_shift" title="Mean shift">Mean shift</a></li> <li><a href="/wiki/Region_growing" title="Region growing">Region growing</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=11" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=11" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a href="https://it.wikibooks.org/wiki/Intelligenza_artificiale/Clustering" class="extiw" title="b:Intelligenza artificiale/Clustering">Wikibooks</a></li> <li class="" title=""><a href="https://it.wiktionary.org/wiki/clustering" class="extiw" title="wikt:clustering">Wikizionario</a></li> <li class="" title=""><span class="plainlinks" title="commons:Category:Cluster analysis"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Cluster_analysis?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://it.wikibooks.org/wiki/" title="Collabora a Wikibooks"><img alt="Collabora a Wikibooks" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/18px-Wikibooks-logo.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/27px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/36px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></a></span> <a href="https://it.wikibooks.org/wiki/" class="extiw" title="b:">Wikibooks</a> contiene testi o manuali sul <b><a href="https://it.wikibooks.org/wiki/Intelligenza_artificiale/Clustering" class="extiw" title="b:Intelligenza artificiale/Clustering">clustering</a></b></li> <li><span typeof="mw:File"><a href="https://it.wiktionary.org/wiki/" title="Collabora a Wikizionario"><img alt="Collabora a Wikizionario" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/18px-Wiktionary_small.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/27px-Wiktionary_small.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/36px-Wiktionary_small.svg.png 2x" data-file-width="350" data-file-height="350" /></a></span> <a href="https://it.wiktionary.org/wiki/" class="extiw" title="wikt:">Wikizionario</a> contiene il lemma di dizionario «<b><a href="https://it.wiktionary.org/wiki/clustering" class="extiw" title="wikt:clustering">clustering</a></b>»</li> <li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file sul <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Cluster_analysis?uselang=it">clustering</a></span></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Clustering&veaction=edit&section=12" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Clustering&action=edit&section=12" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFSapere.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.sapere.it/enciclopedia/cluster+analysis.html"><span style="font-style:italic;">cluster analysis</span></a>, su <span style="font-style:italic;">sapere.it</span>, <a href="/wiki/De_Agostini" title="De Agostini">De Agostini</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q622825#P6706" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFBritannica.com" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Jorge Bacallao Gallestey, <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/cluster-analysis"><span style="font-style:italic;">cluster analysis</span></a>, su <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a></span>, Encyclopædia Britannica, Inc.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q622825#P1417" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFMathWorld" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Eric W. Weisstein, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ClusterAnalysis.html"><span style="font-style:italic;">Clustering</span></a>, su <span style="font-style:italic;"><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></span>, Wolfram Research.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q622825#P2812" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><a rel="nofollow" class="external text" href="http://www.matematicamente.it/rivista-il-magazine/numero-9-aprile-2009/112-data-mining-esplorando-le-miniere-alla-ricerca-della-conoscenza-nascosta-clustering/">(IT) Articolo divulgativo su Clustering e Data Mining.</a></li> <li>Gaetano Zazzaro, Angelo Martone, "ECF-means – Ensemble Clustering Fuzzification Means. A novel algorithm for clustering aggregation, fuzzification, and optimization" <a rel="nofollow" class="external autonumber" href="https://www.thinkmind.org/index.php?view=article&articleid=immm_2018_2_10_50010">[1]</a>, Eighth International Conferenceon Advances in Information Mining and Management, IMMM 2018, Barcelona, Spain, July 22 - 26, 2018.</li></ul> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output .navbox_abovebelow{background:#ddf;font-size:90%;font-weight:normal}.mw-parser-output .navbox_group{background:#ddf;font-size:90%;padding:0 10px;white-space:nowrap}.mw-parser-output .navbox_list{font-size:90%;width:100%}.mw-parser-output .navbox_list a{white-space:nowrap}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_odd{background:#fdfdfd;color:var(--color-base,black)}html:not(.vector-feature-night-mode-enabled) .mw-parser-output .navbox_even{background:#f7f7f7;color:var(--color-base,black)}.mw-parser-output .navbox a.mw-selflink{color:var(--color-base,black)}.mw-parser-output .navbox_center{text-align:center}.mw-parser-output .navbox .navbox_image{padding-left:7px;vertical-align:middle;width:0}.mw-parser-output .navbox+.navbox{margin-top:-1px}.mw-parser-output .navbox .mw-collapsible-toggle{font-weight:normal;text-align:right;width:7em}body.skin--responsive .mw-parser-output .navbox_image img{max-width:none!important}.mw-parser-output .subnavbox{margin:-3px;width:100%}.mw-parser-output .subnavbox_group{background:#e6e6ff;padding:0 10px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-night .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-night .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-night .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbox>tbody>tr:first-child>th{background:var(--background-color-interactive)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox th{color:var(--color-base)!important}html.skin-theme-clientpref-os .mw-parser-output .navbox_abovebelow,html.skin-theme-clientpref-os .mw-parser-output .navbox_group{background:var(--background-color-interactive-subtle)!important}html.skin-theme-clientpref-os .mw-parser-output .subnavbox_group{background:var(--background-color-neutral-subtle)!important}}</style><table class="navbox mw-collapsible mw-collapsed noprint metadata" id="navbox-Apprendimento_automatico"><tbody><tr><th colspan="3"><div class="navbox_navbar"><div class="noprint plainlinks" style="background-color:transparent; padding:0; font-size:xx-small; color:var(--color-base, #000000); white-space:nowrap;"><a href="/wiki/Template:Apprendimento_automatico" title="Template:Apprendimento automatico"><span title="Vai alla pagina del template">V</span></a> · <a href="/w/index.php?title=Discussioni_template:Apprendimento_automatico&action=edit&redlink=1" class="new" title="Discussioni template:Apprendimento automatico (la pagina non esiste)"><span title="Discuti del template">D</span></a> · <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Template:Apprendimento_automatico&action=edit"><span title="Modifica il template. Usa l'anteprima prima di salvare">M</span></a></div></div><span class="navbox_title"><a href="/wiki/Apprendimento_automatico" title="Apprendimento automatico">Apprendimento automatico</a></span></th></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;">Problemi</th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Teoria_dell%27apprendimento_statistico" title="Teoria dell'apprendimento statistico">Teoria dell'apprendimento statistico</a><b> ·</b> <a href="/wiki/Classificazione" title="Classificazione">Classificazione</a><b> ·</b> <a href="/wiki/Analisi_della_regressione" title="Analisi della regressione">Regressione</a><b> ·</b> <a href="/wiki/Regole_di_associazione" title="Regole di associazione">Regole di associazione</a><b> ·</b> <a href="/wiki/Apprendimento_non_supervisionato" title="Apprendimento non supervisionato">Apprendimento non supervisionato</a><b> ·</b> <a href="/wiki/Apprendimento_supervisionato" title="Apprendimento supervisionato">Apprendimento supervisionato</a><b> ·</b> <a href="/wiki/Apprendimento_per_rinforzo" title="Apprendimento per rinforzo">Apprendimento per rinforzo</a><b> ·</b> <a href="/wiki/Apprendimento_profondo" title="Apprendimento profondo">Apprendimento profondo</a></td><td rowspan="8" class="navbox_image"><span typeof="mw:File"><a href="/wiki/File:Kernel_Machine.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Kernel_Machine.svg/150px-Kernel_Machine.svg.png" decoding="async" width="150" height="68" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Kernel_Machine.svg/225px-Kernel_Machine.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Kernel_Machine.svg/300px-Kernel_Machine.svg.png 2x" data-file-width="512" data-file-height="233" /></a></span><br /><br /><span typeof="mw:File"><a href="/wiki/File:Random_forest_model_space.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Random_forest_model_space.png/70px-Random_forest_model_space.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Random_forest_model_space.png/105px-Random_forest_model_space.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Random_forest_model_space.png/140px-Random_forest_model_space.png 2x" data-file-width="512" data-file-height="512" /></a></span></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;"><a href="/wiki/Apprendimento_non_supervisionato" title="Apprendimento non supervisionato">Apprendimento non supervisionato</a></th><td colspan="1" class="navbox_list navbox_even"><a class="mw-selflink selflink">Clustering</a><b> ·</b> <a href="/wiki/Clustering_gerarchico" title="Clustering gerarchico">Clustering gerarchico</a><b> ·</b> <a href="/wiki/K-means" title="K-means">K-means</a><b> ·</b> <a href="/wiki/Algoritmo_EM" title="Algoritmo EM">Algoritmo EM</a><b> ·</b> <a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a><b> ·</b> <a href="/wiki/Mean_shift" title="Mean shift">Mean shift</a><b> ·</b> <a href="/wiki/Rete_generativa_avversaria" title="Rete generativa avversaria">Rete generativa avversaria</a> (cGAN<b> ·</b> VAE-GAN<b> ·</b> cycleGAN)</td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;"><a href="/wiki/Apprendimento_supervisionato" title="Apprendimento supervisionato">Apprendimento supervisionato</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Albero_di_decisione" title="Albero di decisione">Albero di decisione</a><b> ·</b> <a href="/wiki/Foresta_casuale" title="Foresta casuale">Foresta casuale</a><b> ·</b> <a href="/wiki/Conditional_random_field" title="Conditional random field">Conditional random fields</a> CRF<b> ·</b> <a href="/wiki/Modello_di_Markov_nascosto" title="Modello di Markov nascosto">Modello di Markov nascosto</a><b> ·</b> <a href="/wiki/K-nearest_neighbors" title="K-nearest neighbors">K-nearest neighbors</a><b> ·</b> <a href="/wiki/Classificatore_bayesiano" title="Classificatore bayesiano">Classificatore bayesiano</a><b> ·</b> <a href="/wiki/Rete_neurale_artificiale" title="Rete neurale artificiale">Rete neurale artificiale</a><b> ·</b> <a href="/wiki/Regressione_lineare" title="Regressione lineare">Regressione lineare</a><b> ·</b> <a href="/wiki/Modello_logit" title="Modello logit">Regressione logistica</a><b> ·</b> <a href="/wiki/Modello_grafico" title="Modello grafico">Modelli grafici</a><b> ·</b> <a href="/wiki/Macchine_a_vettori_di_supporto" title="Macchine a vettori di supporto">Macchine a vettori di supporto</a></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;"><a href="/wiki/Apprendimento_per_rinforzo" title="Apprendimento per rinforzo">Apprendimento per rinforzo</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Q-learning" title="Q-learning">Q-learning</a><b> ·</b> <a href="/wiki/SARSA" title="SARSA">SARSA</a><b> ·</b> <a href="/wiki/Temporal_difference_learning" title="Temporal difference learning">TD</a></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;"><a href="/wiki/Riduzione_della_dimensionalit%C3%A0" title="Riduzione della dimensionalità">Riduzione della dimensionalità</a></th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Analisi_fattoriale" title="Analisi fattoriale">Analisi fattoriale</a><b> ·</b> <a href="/wiki/Analisi_della_correlazione_canonica" title="Analisi della correlazione canonica">Analisi della correlazione canonica</a> (CCA)<b> ·</b> <a href="/wiki/Analisi_delle_componenti_indipendenti" title="Analisi delle componenti indipendenti">Analisi delle componenti indipendenti</a> (ICA)<b> ·</b> <a href="/wiki/Analisi_discriminante_lineare" title="Analisi discriminante lineare">Analisi discriminante lineare</a> (LDA)<b> ·</b> <a href="/wiki/Analisi_delle_componenti_principali" title="Analisi delle componenti principali">Analisi delle componenti principali</a> (PCA)<b> ·</b> <a href="/wiki/Selezione_delle_caratteristiche" title="Selezione delle caratteristiche">Selezione delle caratteristiche</a><b> ·</b> <a href="/wiki/Estrazione_di_caratteristiche" title="Estrazione di caratteristiche">Estrazione di caratteristiche</a><b> ·</b> <a href="/wiki/T-distributed_stochastic_neighbor_embedding" title="T-distributed stochastic neighbor embedding">t-distributed stochastic neighbor embedding</a> (t-SNE)</td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;"><a href="/wiki/Rete_neurale_artificiale" title="Rete neurale artificiale">Reti neurali artificiali</a></th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Percettrone" title="Percettrone">Percettrone</a><b> ·</b> <a href="/wiki/Rete_neurale_a_base_radiale" title="Rete neurale a base radiale">Rete neurale a base radiale</a><b> ·</b> <a href="/wiki/Rete_bayesiana" title="Rete bayesiana">Rete bayesiana</a><b> ·</b> <a href="/wiki/Rete_neurale_feed-forward" title="Rete neurale feed-forward">Rete neurale feed-forward</a><b> ·</b> <a href="/wiki/Rete_di_Hopfield" title="Rete di Hopfield">Rete di Hopfield</a><b> ·</b> <a href="/wiki/Percettrone_multistrato" title="Percettrone multistrato">Percettrone multistrato</a><b> ·</b> <a href="/wiki/Rete_neurale_ricorrente" title="Rete neurale ricorrente">Rete neurale ricorrente</a> (<a href="/w/index.php?title=Long_short-term_memory&action=edit&redlink=1" class="new" title="Long short-term memory (la pagina non esiste)">LSTM</a>)<b> ·</b> <a href="/wiki/Macchina_di_Boltzmann_ristretta" title="Macchina di Boltzmann ristretta">Macchina di Boltzmann ristretta</a><b> ·</b> <a href="/wiki/Mappa_auto-organizzata" title="Mappa auto-organizzata">Mappa auto-organizzata</a><b> ·</b> <a href="/wiki/Rete_neurale_convoluzionale" title="Rete neurale convoluzionale">Rete neurale convoluzionale</a><b> ·</b> <a href="/wiki/Rete_neurale_a_ritardo" title="Rete neurale a ritardo">Rete neurale a ritardo</a><b> ·</b> <a href="/wiki/Rete_neurale_spiking" title="Rete neurale spiking">Rete neurale spiking</a><b> ·</b> <a href="/wiki/Trasformatore_(informatica)" title="Trasformatore (informatica)">Trasformatore</a></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;">Software</th><td colspan="1" class="navbox_list navbox_odd"><a href="/wiki/Keras" title="Keras">Keras</a><b> ·</b> <a href="/wiki/Microsoft_Cognitive_Toolkit" title="Microsoft Cognitive Toolkit">Microsoft Cognitive Toolkit</a><b> ·</b> <a href="/wiki/Scikit-learn" title="Scikit-learn">Scikit-learn</a><b> ·</b> <a href="/wiki/TensorFlow" title="TensorFlow">TensorFlow</a><b> ·</b> <a href="/wiki/Theano" title="Theano">Theano</a><b> ·</b> <a href="/w/index.php?title=Torch_(libreria_software)&action=edit&redlink=1" class="new" title="Torch (libreria software) (la pagina non 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