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class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Storia</span> </div> </a> <ul id="toc-Storia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Presupposti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Presupposti"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Presupposti</span> </div> </a> <ul id="toc-Presupposti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regressione_lineare" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Regressione_lineare"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Regressione lineare</span> </div> </a> <button aria-controls="toc-Regressione_lineare-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 Regressione lineare</span> </button> <ul id="toc-Regressione_lineare-sublist" class="vector-toc-list"> <li id="toc-La_regressione_multipla" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#La_regressione_multipla"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>La regressione multipla</span> </div> </a> <ul id="toc-La_regressione_multipla-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analisi_di_bontà_del_modello" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analisi_di_bontà_del_modello"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Analisi di bontà del modello</span> </div> </a> <ul id="toc-Analisi_di_bontà_del_modello-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regressione_per_variabili_discrete:_i_modelli_lineari_generalizzati" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Regressione_per_variabili_discrete:_i_modelli_lineari_generalizzati"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Regressione per variabili discrete: i modelli lineari generalizzati</span> </div> </a> <ul id="toc-Regressione_per_variabili_discrete:_i_modelli_lineari_generalizzati-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpolazione_e_estrapolazione" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpolazione_e_estrapolazione"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Interpolazione e estrapolazione</span> </div> </a> <ul id="toc-Interpolazione_e_estrapolazione-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Regressione_non_lineare" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Regressione_non_lineare"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Regressione non lineare</span> </div> </a> <ul id="toc-Regressione_non_lineare-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_metodi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_metodi"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Altri metodi</span> </div> </a> <ul id="toc-Altri_metodi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Software" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Software"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Software</span> </div> </a> <ul id="toc-Software-sublist" class="vector-toc-list"> </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">7</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">8</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">9</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">10</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">11</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">Analisi della regressione</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 52 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-52" 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">52 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%A7%D9%84%D8%A7%D9%86%D8%AD%D8%AF%D8%A7%D8%B1" title="تحليل الانحدار - arabo" lang="ar" hreflang="ar" data-title="تحليل الانحدار" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Anal%C3%ADs_de_la_regresi%C3%B3n" title="Analís de la regresión - asturiano" lang="ast" hreflang="ast" data-title="Analís de la regresión" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Reqressiya_analizi" title="Reqressiya analizi - azerbaigiano" lang="az" hreflang="az" data-title="Reqressiya analizi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaigiano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B3%D1%80%D0%B5%D1%81%D0%B8%D0%BE%D0%BD%D0%B5%D0%BD_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Регресионен анализ - bulgaro" lang="bg" hreflang="bg" data-title="Регресионен 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%A8%E0%A6%BF%E0%A6%B0%E0%A7%8D%E0%A6%AD%E0%A6%B0%E0%A6%A3_%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="নির্ভরণ বিশ্লেষণ - bengalese" lang="bn" hreflang="bn" data-title="নির্ভরণ 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/An%C3%A0lisi_de_la_regressi%C3%B3" title="Anàlisi de la regressió - catalano" lang="ca" hreflang="ca" data-title="Anàlisi de la regressió" 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/Regresn%C3%AD_anal%C3%BDza" title="Regresní analýza - ceco" lang="cs" hreflang="cs" data-title="Regresní 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/Regressionsanalyse" title="Regressionsanalyse - danese" lang="da" hreflang="da" data-title="Regressionsanalyse" 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/Regressionsanalyse" title="Regressionsanalyse - tedesco" lang="de" hreflang="de" data-title="Regressionsanalyse" 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%A0%CE%B1%CE%BB%CE%B9%CE%BD%CE%B4%CF%81%CF%8C%CE%BC%CE%B7%CF%83%CE%B7_(%CF%83%CF%84%CE%B1%CF%84%CE%B9%CF%83%CF%84%CE%B9%CE%BA%CE%AE)" title="Παλινδρόμηση (στατιστική) - greco" lang="el" hreflang="el" data-title="Παλινδρόμηση (στατιστική)" 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/Regression_analysis" title="Regression analysis - inglese" lang="en" hreflang="en" data-title="Regression analysis" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Regreso_(statistiko)" title="Regreso (statistiko) - esperanto" lang="eo" hreflang="eo" data-title="Regreso (statistiko)" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_de_la_regresi%C3%B3n" title="Análisis de la regresión - spagnolo" lang="es" hreflang="es" data-title="Análisis de la regresión" 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/Regressioonanal%C3%BC%C3%BCs" title="Regressioonanalüüs - estone" lang="et" hreflang="et" data-title="Regressioonanalüüs" 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/Erregresio-analisi" title="Erregresio-analisi - basco" lang="eu" hreflang="eu" data-title="Erregresio-analisi" 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%AA%D8%AD%D9%84%DB%8C%D9%84_%D8%B1%DA%AF%D8%B1%D8%B3%DB%8C%D9%88%D9%86" title="تحلیل رگرسیون - persiano" lang="fa" hreflang="fa" data-title="تحلیل رگرسیون" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Regressioanalyysi" title="Regressioanalyysi - finlandese" lang="fi" hreflang="fi" data-title="Regressioanalyysi" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/R%C3%A9gression_(statistiques)" title="Régression (statistiques) - francese" lang="fr" hreflang="fr" data-title="Régression (statistiques)" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Anail%C3%ADs_ar_ch%C3%BAlch%C3%A9imni%C3%BA" title="Anailís ar chúlchéimniú - irlandese" lang="ga" hreflang="ga" data-title="Anailís ar chúlchéimniú" data-language-autonym="Gaeilge" data-language-local-name="irlandese" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/An%C3%A1lise_da_regresi%C3%B3n" title="Análise da regresión - galiziano" lang="gl" hreflang="gl" data-title="Análise da regresión" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A8%D7%92%D7%A8%D7%A1%D7%99%D7%94_(%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94)" title="רגרסיה (אנליזה) - ebraico" lang="he" hreflang="he" data-title="רגרסיה (אנליזה)" 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%B8%E0%A4%AE%E0%A4%BE%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A4%AF%E0%A4%A3_%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="समाश्रयण विश्लेषण - hindi" lang="hi" hreflang="hi" data-title="समाश्रयण विश्लेषण" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Regresszi%C3%B3sz%C3%A1m%C3%ADt%C3%A1s" title="Regressziószámítás - ungherese" lang="hu" hreflang="hu" data-title="Regressziószámítás" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Analisis_regresi" title="Analisis regresi - indonesiano" lang="id" hreflang="id" data-title="Analisis regresi" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/A%C3%B0hvarfsgreining" title="Aðhvarfsgreining - islandese" lang="is" hreflang="is" data-title="Aðhvarfsgreining" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9B%9E%E5%B8%B0%E5%88%86%E6%9E%90" title="回帰分析 - giapponese" lang="ja" hreflang="ja" data-title="回帰分析" 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-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Analisis_r%C3%A9gr%C3%A8si" title="Analisis régrèsi - giavanese" lang="jv" hreflang="jv" data-title="Analisis régrèsi" data-language-autonym="Jawa" data-language-local-name="giavanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%9A%8C%EA%B7%80_%EB%B6%84%EC%84%9D" title="회귀 분석 - coreano" lang="ko" hreflang="ko" data-title="회귀 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/Line%C4%81r%C4%81s_regresijas_anal%C4%ABze" title="Lineārās regresijas analīze - lettone" lang="lv" hreflang="lv" data-title="Lineārās regresijas analīze" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B3%D1%80%D0%B5%D1%81%D1%81%D0%B8%D0%B9%D0%BD_%D1%88%D0%B8%D0%BD%D0%B6%D0%B8%D0%BB%D0%B3%D1%8D%D1%8D" title="Регрессийн шинжилгээ - mongolo" lang="mn" hreflang="mn" data-title="Регрессийн шинжилгээ" data-language-autonym="Монгол" data-language-local-name="mongolo" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Regressieanalyse" title="Regressieanalyse - olandese" lang="nl" hreflang="nl" data-title="Regressieanalyse" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Regresjonsanalyse" title="Regresjonsanalyse - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Regresjonsanalyse" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Regresjonsanalyse" title="Regresjonsanalyse - norvegese bokmål" lang="nb" hreflang="nb" data-title="Regresjonsanalyse" 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/Regresja_(statystyka)" title="Regresja (statystyka) - polacco" lang="pl" hreflang="pl" data-title="Regresja (statystyka)" 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/Regress%C3%A3o_(estat%C3%ADstica)" title="Regressão (estatística) - portoghese" lang="pt" hreflang="pt" data-title="Regressão (estatística)" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Analiza_de_regresie" title="Analiza de regresie - rumeno" lang="ro" hreflang="ro" data-title="Analiza de regresie" data-language-autonym="Română" data-language-local-name="rumeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B3%D1%80%D0%B5%D1%81%D1%81%D0%B8%D0%BE%D0%BD%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Регрессионный анализ - russo" lang="ru" hreflang="ru" data-title="Регрессионный анализ" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Regression_analysis" title="Regression analysis - Simple English" lang="en-simple" hreflang="en-simple" data-title="Regression 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Regresija" title="Regresija - sloveno" lang="sl" hreflang="sl" data-title="Regresija" 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/%D0%A0%D0%B5%D0%B3%D1%80%D0%B5%D1%81%D0%B8%D0%BE%D0%BD%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0" title="Регресиона анализа - serbo" lang="sr" hreflang="sr" data-title="Регресиона анализа" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Analisis_r%C3%A9gr%C3%A9si" title="Analisis régrési - sundanese" lang="su" hreflang="su" data-title="Analisis régrési" data-language-autonym="Sunda" data-language-local-name="sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Regressionsanalys" title="Regressionsanalys - svedese" lang="sv" hreflang="sv" data-title="Regressionsanalys" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A2%D0%B0%D2%B3%D0%BB%D0%B8%D0%BB%D0%B8_%D1%80%D0%B5%D0%B3%D1%80%D0%B5%D1%81%D1%81%D0%B8%D1%8F" title="Таҳлили регрессия - tagico" lang="tg" hreflang="tg" data-title="Таҳлили регрессия" data-language-autonym="Тоҷикӣ" data-language-local-name="tagico" class="interlanguage-link-target"><span>Тоҷикӣ</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%A7%E0%B8%B4%E0%B9%80%E0%B8%84%E0%B8%A3%E0%B8%B2%E0%B8%B0%E0%B8%AB%E0%B9%8C%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%96%E0%B8%94%E0%B8%96%E0%B8%AD%E0%B8%A2" title="การวิเคราะห์การถดถอย - thailandese" lang="th" hreflang="th" data-title="การวิเคราะห์การถดถอย" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Regresyong_analisis" title="Regresyong analisis - tagalog" lang="tl" hreflang="tl" data-title="Regresyong analisis" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Regresyon_analizi" title="Regresyon analizi - turco" lang="tr" hreflang="tr" data-title="Regresyon 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%A0%D0%B5%D0%B3%D1%80%D0%B5%D1%81%D1%96%D0%B9%D0%BD%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Регресійний аналіз - ucraino" lang="uk" hreflang="uk" data-title="Регресійний аналіз" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Regressiya_(matematika)" title="Regressiya (matematika) - uzbeco" lang="uz" hreflang="uz" data-title="Regressiya (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a 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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"><style data-mw-deduplicate="TemplateStyles:r130658281">body:not(.skin-minerva) .mw-parser-output .hatnote.nota-disambigua{clear:both;margin-top:0;padding:.05em .5em}</style> <style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint nota-disambigua"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/18px-Nota_disambigua.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/27px-Nota_disambigua.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/36px-Nota_disambigua.svg.png 2x" data-file-width="200" data-file-height="200" /></span></span> <span class="hatnote-text"><a href="/wiki/Aiuto:Disambiguazione" title="Aiuto:Disambiguazione">Disambiguazione</a> – "Regressione" rimanda qui. Se stai cercando altri significati, vedi <b><a href="/wiki/Regressione_(disambigua)" class="mw-disambig" title="Regressione (disambigua)">Regressione (disambigua)</a></b>.</span></div> </div> <p>L'<b>analisi della regressione</b> è una tecnica usata per analizzare una serie di dati che consistono in una <a href="/wiki/Variabile_dipendente" class="mw-redirect" title="Variabile dipendente">variabile dipendente</a> e una o più <a href="/wiki/Variabili_indipendenti" class="mw-redirect" title="Variabili indipendenti">variabili indipendenti</a>. Lo scopo è stimare un'eventuale relazione funzionale esistente tra la variabile dipendente e le variabili indipendenti. La variabile dipendente nell'<i>equazione di regressione</i> è una funzione delle variabili indipendenti più un <i><a href="/wiki/Errore_statistico" title="Errore statistico">termine d'errore</a></i>. Quest'ultimo è una <a href="/wiki/Variabile_casuale" title="Variabile casuale">variabile casuale</a> e rappresenta una variazione non controllabile e imprevedibile nella variabile dipendente. I parametri sono stimati in modo da descrivere al meglio i dati. Il metodo più comunemente utilizzato per ottenere le migliori stime è il metodo dei <a href="/wiki/Minimi_quadrati" class="mw-redirect" title="Minimi quadrati">"minimi quadrati" (OLS)</a>, ma sono utilizzati anche altri metodi. </p><p>Il <i>data modeling</i> può essere usato senza alcuna conoscenza dei processi sottostanti che hanno generato i dati;<sup id="cite_ref-Berk_1-0" class="reference"><a href="#cite_note-Berk-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> in questo caso il modello è un modello empirico. Nella modellizzazione, inoltre, non è richiesta la conoscenza della <a href="/wiki/Distribuzione_di_probabilit%C3%A0" class="mw-redirect" title="Distribuzione di probabilità">distribuzione di probabilità</a> degli errori. L'analisi della regressione richiede ipotesi riguardanti la <a href="/wiki/Distribuzione_di_probabilit%C3%A0" class="mw-redirect" title="Distribuzione di probabilità">distribuzione di probabilità</a> degli errori. Test statistici vengono effettuati sulla base di tali ipotesi. Nell'analisi della regressione il termine "modello" comprende sia la funzione usata per modellare i dati che le assunzioni concernenti la distribuzione di probabilità. </p><p>L'analisi della regressione può essere usata per effettuare previsioni (ad esempio per prevedere dati futuri di una serie temporale), <a href="/wiki/Inferenza_statistica" title="Inferenza statistica">inferenza statistica</a>, per testare ipotesi o per modellare delle relazioni di dipendenza. Questi usi della regressione dipendono fortemente dal fatto che le assunzioni di partenza siano verificate. L'uso dell'analisi della regressione è stato criticato in diversi casi in cui le ipotesi di partenza non possono essere verificate.<sup id="cite_ref-Berk_1-1" class="reference"><a href="#cite_note-Berk-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> Un fattore che contribuisce all'uso improprio della regressione è che richiede più competenze per criticare un modello che per adattarlo.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Storia">Storia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_della_regressione&veaction=edit&section=1" title="Modifica la sezione Storia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_della_regressione&action=edit&section=1" title="Edit section's source code: Storia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La prima forma di regressione fu il <a href="/wiki/Metodo_dei_minimi_quadrati" title="Metodo dei minimi quadrati">metodo dei minimi quadrati</a>, pubblicato da <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Legendre</a> nel 1805,<sup id="cite_ref-Legendre_4-0" class="reference"><a href="#cite_note-Legendre-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> e da <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> nel 1809.<sup id="cite_ref-Gauss_5-0" class="reference"><a href="#cite_note-Gauss-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Il termine “minimi quadrati” deriva dall'espressione francese usata da Legendre, <i>moindres carrés</i>. Tuttavia, Gauss affermò di essere a conoscenza di questo metodo fin dal 1795. </p><p>Legendre e Gauss applicarono entrambi il metodo al problema di determinare, a partire da osservazioni astronomiche, l'orbita dei pianeti attorno al Sole. Già <a href="/wiki/Eulero" title="Eulero">Eulero</a> aveva lavorato sullo stesso problema, intorno al 1748, ma senza successo.<style data-mw-deduplicate="TemplateStyles:r140554517">.mw-parser-output .chiarimento{background:#ffeaea;color:#444444}.mw-parser-output .chiarimento-apice{color:#EE0700}@media screen{html.skin-theme-clientpref-night .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-night .mw-parser-output .chiarimento-apice{color:#b32424}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .chiarimento{background:rgba(179,36,36,0.21);color:inherit}html.skin-theme-clientpref-os .mw-parser-output .chiarimento-apice{color:#b32424}}</style><span class="chiarimento" title="Queste informazioni non sono comprovate da fonti attendibili."></span><sup class="noprint chiarimento-apice" title="Queste informazioni non sono comprovate da fonti attendibili.">[<i><a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">senza fonte</a></i>]</sup> Gauss pubblicò un ulteriore sviluppo della teoria dei minimi quadrati nel 1821,<sup id="cite_ref-Gauss2_6-0" class="reference"><a href="#cite_note-Gauss2-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> includendo una versione del <a href="/wiki/Teorema_di_Gauss-Markov" title="Teorema di Gauss-Markov">teorema di Gauss-Markov</a>. </p><p>Il termine "regressione" venne coniato nel <a href="/wiki/XIX_secolo" title="XIX secolo">XIX secolo</a> per descrivere il fenomeno biologico per il quale la progenie di individui eccezionali tende in genere ad essere meno eccezionale dei propri genitori e più simile ai loro avi più distanti. <a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a>, un cugino di <a href="/wiki/Charles_Darwin" title="Charles Darwin">Charles Darwin</a>, studiò questo fenomeno e usò i termini, vagamente fuorvianti, di "regressione verso il centro" e "regressione verso la media". Per Galton la regressione aveva solo questo significato biologico, ma il suo lavoro<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> venne in seguito esteso da <a href="/w/index.php?title=Udny_Yule&action=edit&redlink=1" class="new" title="Udny Yule (la pagina non esiste)">Udny Yule</a> e <a href="/wiki/Karl_Pearson" title="Karl Pearson">Karl Pearson</a> in un contesto statistico più generale.<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> Oggi il termine "regressione" viene spesso usato come sinonimo di "curva intercetta dei minimi quadrati". </p> <div class="mw-heading mw-heading2"><h2 id="Presupposti">Presupposti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_della_regressione&veaction=edit&section=2" title="Modifica la sezione Presupposti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_della_regressione&action=edit&section=2" title="Edit section's source code: Presupposti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Il campione deve essere rappresentativo della popolazione per la quale si vuole effettuare la previsione.</li> <li>La variabile dipendente è soggetta ad errore. Tale errore si assume sia una <a href="/wiki/Variabile_casuale" title="Variabile casuale">variabile casuale</a>, con <a href="/wiki/Valore_atteso" title="Valore atteso">media</a> zero. L'<a href="/wiki/Errore_sistematico" title="Errore sistematico">errore sistematico</a> può essere presente ma il suo trattamento esula dallo scopo dell'analisi della regressione.</li> <li>Le variabili indipendenti non hanno errore. Se così non fosse, la modellizzazione dovrebbe essere fatta usando le tecniche <a href="/w/index.php?title=Errors-in-variables&action=edit&redlink=1" class="new" title="Errors-in-variables (la pagina non esiste)">errors-in-variables</a>.</li> <li>Le variabili predittive devono essere <a href="/wiki/Linearmente_indipendenti" class="mw-redirect" title="Linearmente indipendenti">linearmente indipendenti</a>, ossia non deve essere possibile esprimere un qualunque predittore come combinazione lineare degli altri. Vedi <a href="/w/index.php?title=Multicollinearit%C3%A0&action=edit&redlink=1" class="new" title="Multicollinearità (la pagina non esiste)">multicollinearità</a>.</li> <li>Gli errori sono <a href="/wiki/Correlazione_(statistica)" title="Correlazione (statistica)">incorrelati</a>, ossia, la <a href="/wiki/Matrice_delle_covarianze" title="Matrice delle covarianze">matrice di varianza e covarianza</a> degli errori è diagonale e ogni elemento non-nullo è la varianza dell'errore.</li> <li>La varianza dell'errore è costante (<a href="/wiki/Omoschedasticit%C3%A0" title="Omoschedasticità">omoschedasticità</a>). In caso contrario, si deve utilizzare il metodo dei minimi quadrati pesati, o altri metodi.</li> <li>Gli errori seguono una <a href="/wiki/Distribuzione_normale" title="Distribuzione normale">distribuzione normale</a>. Altrimenti, dovrebbe essere usato il <a href="/wiki/Modello_lineare_generalizzato" title="Modello lineare generalizzato">modello lineare generalizzato</a>.</li></ul> <p>Queste condizioni sono sufficienti (ma non tutte necessarie) perché lo <a href="/wiki/Stimatore" title="Stimatore">stimatore</a> dei minimi quadrati goda di buone proprietà. In particolare queste assunzioni implicano che lo stimatore sia <a href="/wiki/Bias_(statistica)" title="Bias (statistica)">non distorto</a>, <a href="/wiki/Consistenza_(statistica)" title="Consistenza (statistica)">consistente</a> ed <a href="/wiki/Efficienza_(statistica)" title="Efficienza (statistica)">efficiente</a> nella classe degli stimatori lineari non distorti. Molte di queste assunzioni possono essere rilassate in analisi più avanzate. </p> <div class="mw-heading mw-heading2"><h2 id="Regressione_lineare">Regressione lineare</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_della_regressione&veaction=edit&section=3" title="Modifica la sezione Regressione lineare" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_della_regressione&action=edit&section=3" title="Edit section's source code: Regressione lineare"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Regressione_lineare" title="Regressione lineare">Regressione lineare</a></b>.</span></div> </div> <p>Nella regressione lineare, il modello assume che la variabile dipendente, <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d30d30b6c2dbe4d6f150d699de040937ecc95f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}"></span> sia una <a href="/wiki/Combinazione_lineare" title="Combinazione lineare">combinazione lineare</a> dei <i>parametri</i> (ma non è necessario che sia lineare nella <i>variabile indipendente</i>). Ad esempio, nella regressione lineare semplice con <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> osservazioni ci sono una variabile indipendente: <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>, e due parametri, <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"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </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/40b42f71f244103a8fca3c76885c7580a92831c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{0}}"></span> 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 \beta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeeccd8b585b819e38f9c1fe5e9816a3ea01804c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{1}}"></span>: </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 y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i},\quad i=1,\ldots ,N.}"> <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> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i},\quad i=1,\ldots ,N.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88d9edb8d3e166a2821080cc08dace01d5914fa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.779ex; height:2.509ex;" alt="{\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i},\quad i=1,\ldots ,N.}"></span></dd></dl> <p>Nella regressione lineare multipla, ci sono più variabili indipendenti o funzioni di variabili indipendenti. Ad esempio, aggiungendo un termine 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="{\textstyle x_{i}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x_{i}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec71c47bd2f4c314eb9ef381ee0e7768bab6a2d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:3.176ex;" alt="{\textstyle x_{i}^{2}}"></span> alla regressione precedente si ottiene: </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 y_{i}=\beta _{0}+\beta _{1}x_{i}+\beta _{2}x_{i}^{2}+\varepsilon _{i},\ i=1,\ldots ,N.}"> <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> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\beta _{2}x_{i}^{2}+\varepsilon _{i},\ i=1,\ldots ,N.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db8012906ccc78b31e4a3efb5a06838e18319cd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.632ex; height:3.176ex;" alt="{\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\beta _{2}x_{i}^{2}+\varepsilon _{i},\ i=1,\ldots ,N.}"></span></dd></dl> <p>Si tratta ancora di una regressione lineare: sebbene l'espressione sulla destra sia quadratica nella variabile indipendente <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>, è comunque lineare nei parametri <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"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </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/40b42f71f244103a8fca3c76885c7580a92831c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; 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 \beta _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeeccd8b585b819e38f9c1fe5e9816a3ea01804c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{1}}"></span> 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 \beta _{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b33e51caf90d7efa53cc46d78fb5e35bbf87101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.017ex; height:2.509ex;" alt="{\displaystyle \beta _{2}.}"></span> </p><p>In entrambi i casi, <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 _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e1b6ad3cbad4af49bf21a3ad2dc379ff045079" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{i}}"></span> è un termine di errore e l'indice <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> identifica una particolare osservazione. Dato un campione casuale della popolazione, stimiamo i parametri della popolazione e otteniamo il modello di regressione lineare semplice: </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 y_{i}={\widehat {\beta }}_{0}+{\widehat {\beta }}_{1}X_{i}+e_{i}.}"> <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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}={\widehat {\beta }}_{0}+{\widehat {\beta }}_{1}X_{i}+e_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76df50f633af13579ba76d8cd1156aead021f182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.113ex; height:3.509ex;" alt="{\displaystyle y_{i}={\widehat {\beta }}_{0}+{\widehat {\beta }}_{1}X_{i}+e_{i}.}"></span></dd></dl> <p>Il termine <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}"></span> è il residuo, <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_{i}=y_{i}-{\widehat {y}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}=y_{i}-{\widehat {y}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f8cd75171211a603cef88560273253f8afb7f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.863ex; height:2.676ex;" alt="{\displaystyle e_{i}=y_{i}-{\widehat {y}}_{i}}"></span>. Un metodo di stima è quello dei <a href="/wiki/Metodo_dei_minimi_quadrati" title="Metodo dei minimi quadrati">minimi quadrati ordinari</a>. Questo metodo ottiene le stime dei parametri che minimizzano la somma dei quadrati dei <a href="/wiki/Residuo_(analisi_complessa)" title="Residuo (analisi complessa)">residui</a>, SSE: </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 SSE=\sum _{i=1}^{N}e_{i}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>S</mi> <mi>E</mi> <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>N</mi> </mrow> </munderover> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SSE=\sum _{i=1}^{N}e_{i}^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca250c7099946fb0d96e98b3a28e1665b6bb73c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.399ex; height:7.343ex;" alt="{\displaystyle SSE=\sum _{i=1}^{N}e_{i}^{2}.}"></span></dd></dl> <p>La minimizzazione di questa funzione risulta essere un sistema di <a href="/w/index.php?title=Minimi_quadrati_lineari&action=edit&redlink=1" class="new" title="Minimi quadrati lineari (la pagina non esiste)">equazioni normali</a>, un insieme di equazioni lineari simultanee nei parametri, che vengono risolte per trovare le stime dei parametri, <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 {\widehat {\beta }}_{0},{\widehat {\beta }}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta }}_{0},{\widehat {\beta }}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c651a093fe15ffd6d21104c29f6b05704ea9ecc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.175ex; height:3.509ex;" alt="{\displaystyle {\widehat {\beta }}_{0},{\widehat {\beta }}_{1}}"></span>. Vedi <a href="/w/index.php?title=Stima_dei_minimi_quadrati_dei_coefficienti_di_regressione_lineare&action=edit&redlink=1" class="new" title="Stima dei minimi quadrati dei coefficienti di regressione lineare (la pagina non esiste)">coefficienti di regressione</a> per informazioni sulle proprietà statistiche di tali stimatori. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:LinearRegression.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/LinearRegression.svg/220px-LinearRegression.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/LinearRegression.svg/330px-LinearRegression.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/LinearRegression.svg/440px-LinearRegression.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>Illustrazione della regressione lineare su un insieme di dati (punti rossi).</figcaption></figure> <p>Nel caso della regressione semplice, le formule per le stime dei minimi quadrati sono </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 {\widehat {\beta _{1}}}={\frac {\sum (x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sum (x_{i}-{\bar {x}})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>∑</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\beta _{1}}}={\frac {\sum (x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sum (x_{i}-{\bar {x}})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ea2adcf26d298036a5eba675c40816a0df02be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.758ex; height:6.509ex;" alt="{\displaystyle {\widehat {\beta _{1}}}={\frac {\sum (x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sum (x_{i}-{\bar {x}})^{2}}}}"></span> 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 {\hat {\beta _{0}}}={\bar {y}}-{\widehat {\beta _{1}}}{\bar {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\beta _{0}}}={\bar {y}}-{\widehat {\beta _{1}}}{\bar {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b12579b2373ece4bf75196929fb51a58bbde97b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.311ex; height:3.343ex;" alt="{\displaystyle {\hat {\beta _{0}}}={\bar {y}}-{\widehat {\beta _{1}}}{\bar {x}}}"></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 {\bar {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}"></span> è la <a href="/wiki/Media_aritmetica" class="mw-redirect" title="Media aritmetica">media</a> (media) dei valori <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/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 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 {\bar {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b298744237368f34e61ff7dc90b34016a7037af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.343ex;" alt="{\displaystyle {\bar {y}}}"></span> è la media dei valori <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/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. Sotto l'ipotesi che il termine di errore della popolazione abbia una varianza costante, la stima di quella varianza è data da: <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 {\hat {\sigma _{\varepsilon }}}={\sqrt {\frac {SSE}{N-2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>S</mi> <mi>S</mi> <mi>E</mi> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma _{\varepsilon }}}={\sqrt {\frac {SSE}{N-2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e52c79a2551b4896805dab42976deab4fdb4684" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.651ex; height:6.176ex;" alt="{\displaystyle {\hat {\sigma _{\varepsilon }}}={\sqrt {\frac {SSE}{N-2}}}}"></span> Questo è la radice dell'<a href="/wiki/Errore_quadratico_medio" title="Errore quadratico medio">errore quadratico medio</a> (RMSE) della regressione. Gli <a href="/wiki/Errore_standard" title="Errore standard">errori standard</a> delle stime dei parametri sono dati da </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 {\hat {\sigma }}_{\beta _{0}}={\hat {\sigma }}_{\varepsilon }{\sqrt {{\frac {1}{N}}+{\frac {{\bar {x}}^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>∑</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}_{\beta _{0}}={\hat {\sigma }}_{\varepsilon }{\sqrt {{\frac {1}{N}}+{\frac {{\bar {x}}^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c7b8b3499a5f364e6d0731eedd701f7c9478ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:29.267ex; height:7.676ex;" alt="{\displaystyle {\hat {\sigma }}_{\beta _{0}}={\hat {\sigma }}_{\varepsilon }{\sqrt {{\frac {1}{N}}+{\frac {{\bar {x}}^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}}}}"></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 {\hat {\sigma }}_{\beta _{1}}={\hat {\sigma }}_{\varepsilon }{\sqrt {\frac {1}{\sum (x_{i}-{\bar {x}})^{2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mrow> <mo>∑</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}_{\beta _{1}}={\hat {\sigma }}_{\varepsilon }{\sqrt {\frac {1}{\sum (x_{i}-{\bar {x}})^{2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95fb4c65a2ced0ecb8bfa509b760189fca855e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:24.174ex; height:7.676ex;" alt="{\displaystyle {\hat {\sigma }}_{\beta _{1}}={\hat {\sigma }}_{\varepsilon }{\sqrt {\frac {1}{\sum (x_{i}-{\bar {x}})^{2}}}}.}"></span></dd></dl> <p>Sotto l'ulteriore ipotesi che il termine di errore della popolazione abbia distribuzione normale, il ricercatore può usare questi errori standard stimati per creare intervalli di confidenza e condurre test d'ipotesi sui parametri della popolazione. </p> <div class="mw-heading mw-heading3"><h3 id="La_regressione_multipla">La regressione multipla</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_della_regressione&veaction=edit&section=4" title="Modifica la sezione La regressione multipla" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_della_regressione&action=edit&section=4" title="Edit section's source code: La regressione multipla"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nel più generale modello di regressione multipla, ci sono <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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> variabili indipendenti: </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 y_{i}=\beta _{0}+\beta _{1}x_{1i}+\cdots +\beta _{p}x_{pi}+\varepsilon _{i}.}"> <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> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{1i}+\cdots +\beta _{p}x_{pi}+\varepsilon _{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1725959c90d924ccb64f09ccdef3bb1b384ad55f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.675ex; height:2.843ex;" alt="{\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{1i}+\cdots +\beta _{p}x_{pi}+\varepsilon _{i}.}"></span></dd></dl> <p>Le stime dei parametri dei minimi quadrati sono ottenute da <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/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> equazioni normali. Il residuo può essere scritto come </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 e_{i}=y_{i}-{\hat {\beta }}_{0}-{\hat {\beta }}_{1}x_{1}-\cdots -{\hat {\beta }}_{p}x_{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mo>⋯</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}=y_{i}-{\hat {\beta }}_{0}-{\hat {\beta }}_{1}x_{1}-\cdots -{\hat {\beta }}_{p}x_{p}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af58bb67893038863b54f50899b147716da0fe03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.947ex; height:3.676ex;" alt="{\displaystyle e_{i}=y_{i}-{\hat {\beta }}_{0}-{\hat {\beta }}_{1}x_{1}-\cdots -{\hat {\beta }}_{p}x_{p}.}"></span></dd></dl> <p>Le <a href="/w/index.php?title=Equazioni_normali&action=edit&redlink=1" class="new" title="Equazioni normali (la pagina non esiste)">equazioni normali</a> sono </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 _{i=1}^{n}\sum _{k=1}^{p}X_{ij}X_{ik}{\hat {\beta }}_{k}=\sum _{i=1}^{n}X_{ij}y_{i},\ j=1,...,p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <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>n</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}\sum _{k=1}^{p}X_{ij}X_{ik}{\hat {\beta }}_{k}=\sum _{i=1}^{n}X_{ij}y_{i},\ j=1,...,p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1002ed4651bf0dbe5ac7f4b5fe59cb54eafcdb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.36ex; height:7.009ex;" alt="{\displaystyle \sum _{i=1}^{n}\sum _{k=1}^{p}X_{ij}X_{ik}{\hat {\beta }}_{k}=\sum _{i=1}^{n}X_{ij}y_{i},\ j=1,...,p}"></span></dd></dl> <p>In notazione matriciale, le equazioni normali sono scritte come </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 \mathbf {\left(X^{T}X\right){\hat {\boldsymbol {\beta }}}=X^{T}y} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </msup> <mi mathvariant="bold">X</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">β</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo mathvariant="bold">=</mo> <msup> <mi mathvariant="bold">X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </msup> <mi mathvariant="bold">y</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\left(X^{T}X\right){\hat {\boldsymbol {\beta }}}=X^{T}y} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bfe8f761a223508a97783aa29838a61227f8179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.753ex; height:3.509ex;" alt="{\displaystyle \mathbf {\left(X^{T}X\right){\hat {\boldsymbol {\beta }}}=X^{T}y} .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Analisi_di_bontà_del_modello"><span id="Analisi_di_bont.C3.A0_del_modello"></span>Analisi di bontà del modello</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_della_regressione&veaction=edit&section=5" title="Modifica la sezione Analisi di bontà del modello" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_della_regressione&action=edit&section=5" title="Edit section's source code: Analisi di bontà del modello"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una volta costruito un modello di regressione, è importante confermare la <a href="/w/index.php?title=Bont%C3%A0_di_adattamento&action=edit&redlink=1" class="new" title="Bontà di adattamento (la pagina non esiste)">bontà di adattamento</a> del modello e la <a href="/wiki/Significativit%C3%A0_statistica" class="mw-redirect" title="Significatività statistica">significatività statistica</a> dei parametri stimati. I controlli della bontà di adattamento comunemente usati includono l'indice <a href="/wiki/Coefficiente_di_determinazione" title="Coefficiente di determinazione">R-quadro</a>, analisi dei <a href="/w/index.php?title=Errori_e_residui&action=edit&redlink=1" class="new" title="Errori e residui (la pagina non esiste)">residui</a> e <a href="/wiki/Test_di_verifica_d%27ipotesi" title="Test di verifica d'ipotesi">test di ipotesi</a>. La significatività statistica è verificata con un <a href="/wiki/Test_F" title="Test F">test F</a> dell'adattamento globale, seguito da <a href="/wiki/T-test" class="mw-redirect" title="T-test">t-test</a> per ogni singolo parametro. </p><p>L'interpretazione di questi test dipende fortemente dalle assunzioni sul modello. Nonostante l'analisi dei residui sia usata per determinare la bontà di un modello, i risultati dei <a href="/w/index.php?title=Test-T&action=edit&redlink=1" class="new" title="Test-T (la pagina non esiste)">test-T</a> e dei <a href="/w/index.php?title=Test-F&action=edit&redlink=1" class="new" title="Test-F (la pagina non esiste)">test-F</a> sono difficili da interpretare nel caso in cui le assunzioni di partenza non siano soddisfatte. Ad esempio, se la distribuzione degli errori non è normale, può accadere che in campioni di numerosità ridotta le stime dei parametri non seguano una distribuzione normale, cosa che complica l'inferenza. Per grandi campioni, il <a href="/wiki/Teorema_del_limite_centrale" class="mw-redirect" title="Teorema del limite centrale">teorema del limite centrale</a> permette di effettuare i test usando un'approssimazione asintotica delle distribuzioni. </p> <div class="mw-heading mw-heading3"><h3 id="Regressione_per_variabili_discrete:_i_modelli_lineari_generalizzati">Regressione per variabili discrete: i modelli lineari generalizzati</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_della_regressione&veaction=edit&section=6" title="Modifica la sezione Regressione per variabili discrete: i modelli lineari generalizzati" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_della_regressione&action=edit&section=6" title="Edit section's source code: Regressione per variabili discrete: i modelli lineari generalizzati"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La variabile risposta può essere non continua. Per le variabili binarie (zero/uno), si può procedere con un particolare tipo di modello lineare <a href="/w/index.php?title=Linear_probability_model&action=edit&redlink=1" class="new" title="Linear probability model (la pagina non esiste)">linear probability model</a>. Se si usa un modello non-lineare i modelli più utilizzati sono il <a href="/wiki/Modello_probit" title="Modello probit">probit</a> e il <a href="/wiki/Regressione_logistica" class="mw-redirect" title="Regressione logistica">modello logit</a>. Il modello <a href="/w/index.php?title=Probit_multivariato&action=edit&redlink=1" class="new" title="Probit multivariato (la pagina non esiste)">probit multivariato</a> rende possibile stimare congiuntamente la relazione tra più variabili binarie dipendenti e alcune variabili indipendenti. Per <a href="/w/index.php?title=Variabili_categoriche&action=edit&redlink=1" class="new" title="Variabili categoriche (la pagina non esiste)">variabili categoriche</a> con più di due valori si utilizza il modello <a href="/w/index.php?title=Logit_multinomiale&action=edit&redlink=1" class="new" title="Logit multinomiale (la pagina non esiste)">logit multinomiale</a>. Per <a href="/w/index.php?title=Variabili_ordinali&action=edit&redlink=1" class="new" title="Variabili ordinali (la pagina non esiste)">variabili ordinali</a> con più di due valori, si utilizzano i modelli <a href="/w/index.php?title=Logit_cumulativo&action=edit&redlink=1" class="new" title="Logit cumulativo (la pagina non esiste)">logit cumulativo</a> e <a href="/w/index.php?title=Probit_cumulativo&action=edit&redlink=1" class="new" title="Probit cumulativo (la pagina non esiste)">probit cumulativo</a>. Un'alternativa a tali procedure è la regressione lineare basata su polychoric o polyserial correlazioni tra le variabili categoriche. Tali procedure differiscono nelle ipotesi fatte sulla distribuzione delle variabili nella popolazione. Se la variabile rappresenta una ripetizione di un evento nel tempo, è positiva e con poche realizzazioni ("eventi rari"), si possono utilizzare modelli di <a href="/wiki/Distribuzione_di_Poisson" title="Distribuzione di Poisson">Poisson</a> o <a href="/w/index.php?title=Binomiale_negativa&action=edit&redlink=1" class="new" title="Binomiale negativa (la pagina non esiste)">binomiale negativa</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Interpolazione_e_estrapolazione">Interpolazione e estrapolazione</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_della_regressione&veaction=edit&section=7" title="Modifica la sezione Interpolazione e estrapolazione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_della_regressione&action=edit&section=7" title="Edit section's source code: Interpolazione e estrapolazione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I modelli di regressione predicono una variabile <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/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> partendo dai valori di altre variabili <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/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. Se i valori della previsione sono compresi nell'intervallo dei valori delle variabili <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/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> utilizzate per la costruzione del modello si parla di <a href="/wiki/Interpolazione" title="Interpolazione">interpolazione</a>. Se i valori escono dal range delle variabili esplicative si parla di <a href="/wiki/Estrapolazione" title="Estrapolazione">estrapolazione</a>. In questo caso la previsione diventa più rischiosa. </p> <div class="mw-heading mw-heading2"><h2 id="Regressione_non_lineare">Regressione non lineare</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_della_regressione&veaction=edit&section=8" title="Modifica la sezione Regressione non lineare" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_della_regressione&action=edit&section=8" title="Edit section's source code: Regressione non lineare"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Regressione_nonlineare" title="Regressione nonlineare">Regressione nonlineare</a></b>.</span></div> </div> <p>Quando la funzione del modello non è lineare nei parametri la somma dei quadrati deve essere minimizzata da una procedura iterativa. </p> <div class="mw-heading mw-heading2"><h2 id="Altri_metodi">Altri metodi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_della_regressione&veaction=edit&section=9" title="Modifica la sezione Altri metodi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_della_regressione&action=edit&section=9" title="Edit section's source code: Altri metodi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sebbene i parametri di un modello di regressione siano di solito stimati usando il metodo dei minimi quadrati, altri metodi includono: </p> <ul><li>metodi <a href="/w/index.php?title=Bayesiani&action=edit&redlink=1" class="new" title="Bayesiani (la pagina non esiste)">Bayesiani</a>, per esempio la <a href="/w/index.php?title=Regressione_lineare_Bayesiana&action=edit&redlink=1" class="new" title="Regressione lineare Bayesiana (la pagina non esiste)">regressione lineare Bayesiana</a>;</li> <li>minimizzazione delle deviazioni assolute, che porta alla <a href="/wiki/Regressione_dei_quantili" title="Regressione dei quantili">regressione dei quantili</a>;</li> <li><a href="/w/index.php?title=Regressione_non_parametrica&action=edit&redlink=1" class="new" title="Regressione non parametrica (la pagina non esiste)">regressione non parametrica</a>, tale approccio richiede un ampio numero di osservazioni, poiché i dati sono usati sia per costruire la struttura del modello che per stimare i parametri del modello. Normalmente richiedono un elevato sforzo computazionale.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Software">Software</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_della_regressione&veaction=edit&section=10" title="Modifica la sezione Software" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_della_regressione&action=edit&section=10" title="Edit section's source code: Software"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tutti i principali <a href="/w/index.php?title=Elenco_dei_pacchetti_statistici&action=edit&redlink=1" class="new" title="Elenco dei pacchetti statistici (la pagina non esiste)">pacchetti statistici</a> eseguono i tipi comuni di analisi di regressione correttamente e in modo semplice. La <a href="/w/index.php?title=Regressione_lineare_semplice&action=edit&redlink=1" class="new" title="Regressione lineare semplice (la pagina non esiste)">regressione lineare semplice</a> può essere fatta in alcuni <a href="/wiki/Fogli_elettronici" class="mw-redirect" title="Fogli elettronici">fogli elettronici</a>. C'è una quantità di programmi che esegue forme specializzate di regressione, e gli esperti possono scegliere di scrivere il loro proprio codice per usare <a href="/w/index.php?title=Categoria:Linguaggi_di_programmazione_statistica&action=edit&redlink=1" class="new" title="Categoria:Linguaggi di programmazione statistica (la pagina non esiste)">linguaggi di programmazione statistica</a> o <a href="/w/index.php?title=Elenco_di_software_per_analisi_numerica&action=edit&redlink=1" class="new" title="Elenco di software per analisi numerica (la pagina non esiste)">software per analisi numerica</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=Analisi_della_regressione&veaction=edit&section=11" 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=Analisi_della_regressione&action=edit&section=11" 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-Berk-1"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-Berk_1-0">a</a></i></sup> <sup><i><a href="#cite_ref-Berk_1-1">b</a></i></sup></span> <span class="reference-text">Richard A. Berk, <i>Regression Analysis: A Constructive Critique</i>, Sage Publications (2004)</span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text">David A. Freedman, <i>Statistical Models: Theory and Practice</i>, Cambridge University Press (2005)</span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"><a rel="nofollow" class="external autonumber" href="https://links.jstor.org/sici?sici=0081-1750%281982%2913%3C313%3ACAIAIR%3E2.0.CO%3B2-3">[1]</a> R. Dennis Cook; Sanford Weisberg "Criticism and Influence Analysis in Regression", <i>Sociological Methodology</i>, Vol. 13. (1982), pp. 313-361.</span> </li> <li id="cite_note-Legendre-4"><a href="#cite_ref-Legendre_4-0"><b>^</b></a> <span class="reference-text"><a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">A.M. Legendre</a>. <i>Nouvelles méthodes pour la détermination des orbites des comètes</i> (1805). “Sur la Méthode des moindres quarrés” appears as an appendix.</span> </li> <li id="cite_note-Gauss-5"><a href="#cite_ref-Gauss_5-0"><b>^</b></a> <span class="reference-text"><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">C.F. Gauss</a>. <i>Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientum</i>. (1809)</span> </li> <li id="cite_note-Gauss2-6"><a href="#cite_ref-Gauss2_6-0"><b>^</b></a> <span class="reference-text">C.F. Gauss. <i>Theoria combinationis observationum erroribus minimis obnoxiae</i>. (1821/1823)</span> </li> <li id="cite_note-7"><a href="#cite_ref-7"><b>^</b></a> <span class="reference-text"><a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a>. "Typical laws of heredity", Nature 15 (1877), 492-495, 512-514, 532-533. <i>(Galton usa il termine "reversion" in questo articolo, che tratta della grandezza dei piselli.)</i>; Francis Galton. Presidential address, Section H, Anthropology. (1885) <i>(in questo documento,che tratta dell'altezza degli esseri umani, Galton utilizza il termine "regressione".)</i></span> </li> <li id="cite_note-8"><a href="#cite_ref-8"><b>^</b></a> <span class="reference-text"><a href="/w/index.php?title=G._Udny_Yule&action=edit&redlink=1" class="new" title="G. Udny Yule (la pagina non esiste)">G. Udny Yule</a>. "On the Theory of Correlation", J. Royal Statist. Soc., 1897, p. 812-54. <a href="/wiki/Karl_Pearson" title="Karl Pearson">Karl Pearson</a>, G. U. Yule, Norman Blanchard, e Alice Lee. "The Law of Ancestral Heredity", <i><a href="/wiki/Biometrika" title="Biometrika">Biometrika</a></i> (1903). Nel lavoro di Yule e Pearson, la distribuzione congiunta della variabile risposta e delle variabili esplicative è ipotizzata essere una <a href="/wiki/Distribuzione_normale" title="Distribuzione normale">distribuzione normale</a>. Questa ipotesi fu notevolmente indebolita da <a href="/w/index.php?title=Ronald_A._Fisher&action=edit&redlink=1" class="new" title="Ronald A. Fisher (la pagina non esiste)">R.A. Fisher</a> nei suoi lavori del 1922 e del 1925 (R.A. Fisher, "The goodness of fit of regression formulae, and the distribution of regression coefficients", J. Royal Statist. Soc., 85, 597-612 del 1922 e <i><a href="/w/index.php?title=Statistical_Methods_for_Research_Workers&action=edit&redlink=1" class="new" title="Statistical Methods for Research Workers (la pagina non esiste)">Statistical Methods for Research Workers</a></i> del 1925). Fisher ipotizzava che la distribuzione condizionata della variabile risposta fosse normale, ma non poneva condizioni sulla distribuzione congiunta. Sotto questo aspetto, l'ipotesi di Fisher è più vicina alla formulazione di Gauss del 1821.</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=Analisi_della_regressione&veaction=edit&section=12" 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=Analisi_della_regressione&action=edit&section=12" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Audi, R., Ed. (1996). "curve fitting problem," <i>The Cambridge Dictionary of Philosophy</i>. Cambridge, Cambridge University Press. pp. 172–173.</li> <li><a href="/wiki/William_Kruskal" title="William Kruskal">William H. Kruskal</a> e Judith M. Tanur, ed. (1978), "Linear Hypotheses," <i>International Encyclopedia of Statistics</i>. Free Press, v. 1,</li></ul> <dl><dd>Evan J. Williams, "I. Regression," pp. 523-41.</dd> <dd><a href="/w/index.php?title=Julian_C._Stanley&action=edit&redlink=1" class="new" title="Julian C. Stanley (la pagina non esiste)">Julian C. Stanley</a>, "II. Analysis of Variance," pp. 541-554.</dd></dl> <ul><li><a href="/w/index.php?title=D.V._Lindley&action=edit&redlink=1" class="new" title="D.V. Lindley (la pagina non esiste)">Lindley, D.V.</a> (1987). "Regression and correlation analysis," <a href="/w/index.php?title=New_Palgrave:_A_Dictionary_of_Economics&action=edit&redlink=1" class="new" title="New Palgrave: A Dictionary of Economics (la pagina non esiste)">New Palgrave: A Dictionary of Economics</a>, v. 4, pp. 120–23.</li> <li>Birkes, David e Yadolah Dodge, <i>Alternative Methods of Regression</i>. <a href="/wiki/Speciale:RicercaISBN/0471568813" class="internal mw-magiclink-isbn">ISBN 0-471-56881-3</a></li> <li>Chatfield, C. (1993) "Calculating Interval Forecasts," <i>Journal of Business and Economic Statistics,</i> <b>11</b>. pp. 121–135.</li> <li>Draper, N.R. e Smith, H. (1998).<i>Applied Regression Analysis</i> Wiley Series in Probability and Statistics</li> <li>Fox, J. (1997). <i>Applied Regression Analysis, Linear Models and Related Methods.</i> Sage</li> <li>Hardle, W., <i>Applied Nonparametric Regression</i> (1990), <a href="/wiki/Speciale:RicercaISBN/0521429501" class="internal mw-magiclink-isbn">ISBN 0-521-42950-1</a></li> <li>Meade, N. e T. Islam (1995) "Prediction Intervals for Growth Curve Forecasts," <i>Journal of Forecasting,</i> <b>14</b>, pp. 413–430.</li> <li>Munro, Barbara Hazard (2005) "Statistical Methods for Health Care Research" Lippincott Williams & Wilkins, 5th ed.</li> <li>Gujarati, Basic Econometrics, 4ª edizione</li> <li>Sykes, A.O. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080920190058/http://www.law.uchicago.edu/Lawecon/WkngPprs_01-25/20.Sykes.Regression.pdf">"An Introduction to Regression Analysis"</a> (Inaugural Coase Lecture)</li> <li>S. Kotsiantis, D. Kanellopoulos, P. Pintelas, Local Additive Regression of Decision Stumps, Lecture Notes in Artificial Intelligence, Springer-Verlag, Vol. 3955, SETN 2006, pp. 148 – 157, 2006</li> <li>S. Kotsiantis, P. Pintelas, Selective Averaging of Regression Models, Annals of Mathematics, Computing & TeleInformatics, Vol 1, No 3, 2005, pp. 66–75</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=Analisi_della_regressione&veaction=edit&section=13" 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=Analisi_della_regressione&action=edit&section=13" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="-moz-column-count:2; column-count:2;"> <ul><li><a href="/w/index.php?title=Confidence_region&action=edit&redlink=1" class="new" title="Confidence region (la pagina non esiste)">Confidence region</a></li> <li><a href="/wiki/Distanza_di_Cook" title="Distanza di Cook">Distanza di Cook</a></li> <li><a href="/wiki/Distribuzione_normale_multivariata" title="Distribuzione normale multivariata">Distribuzione normale multivariata</a></li> <li><a href="/wiki/Estrapolazione" title="Estrapolazione">Estrapolazione</a></li> <li><a href="/w/index.php?title=Forecasting&action=edit&redlink=1" class="new" title="Forecasting (la pagina non esiste)">Forecasting</a></li> <li><a href="/wiki/Funzione_di_Huber" title="Funzione di Huber">Funzione di Huber</a></li> <li><a href="/wiki/Intervallo_di_confidenza" title="Intervallo di confidenza">Intervallo di confidenza</a></li> <li><a href="/wiki/Intervallo_di_previsione" title="Intervallo di previsione">Intervallo di previsione</a></li> <li><a href="/wiki/Kriging" title="Kriging">Kriging</a> (un algoritmo di stima dei minimi quadrati lineari)</li> <li><a href="/w/index.php?title=Robust_regression&action=edit&redlink=1" class="new" title="Robust regression (la pagina non esiste)">Robust regression</a></li> <li><a href="/w/index.php?title=Segmented_regression&action=edit&redlink=1" class="new" title="Segmented regression (la pagina non esiste)">Segmented regression</a></li> <li><a href="/wiki/Statistica" title="Statistica">Statistica</a></li> <li><a href="/w/index.php?title=Stima_del_trend&action=edit&redlink=1" class="new" title="Stima del trend (la pagina non esiste)">Stima del trend</a></li></ul> </div> <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=Analisi_della_regressione&veaction=edit&section=14" 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=Analisi_della_regressione&action=edit&section=14" 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=""><span class="plainlinks" title="commons:Category:Regression analysis"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Regression_analysis?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><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 sull'<b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Regression_analysis?uselang=it">analisi della regressione</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=Analisi_della_regressione&veaction=edit&section=15" 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=Analisi_della_regressione&action=edit&section=15" 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="CITEREFBritannica.com" 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://www.britannica.com/science/regression-analysis"><span style="font-style:italic;">regression 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/Q208042#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 class="citation web" style="font-style:normal">(<span style="font-weight:bolder; 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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 esiste)">Torch</a><b> ·</b> <a href="/wiki/Weka_(software)" title="Weka (software)">Weka</a></td></tr><tr><th colspan="1" class="navbox_group" style="text-align:center;">Altro</th><td colspan="1" class="navbox_list navbox_even"><a href="/wiki/Algoritmo_genetico" title="Algoritmo genetico">Algoritmo genetico</a><b> ·</b> <a href="/wiki/Particle_Swarm_Optimization" title="Particle Swarm Optimization">Particle Swarm Optimization</a><b> ·</b> <a href="/wiki/Caratteristica_(apprendimento_automatico)" title="Caratteristica (apprendimento automatico)">Caratteristica</a><b> ·</b> <a href="/wiki/Compromesso_bias-varianza" title="Compromesso bias-varianza">Compromesso bias-varianza</a><b> ·</b> <a href="/wiki/Minimizzazione_del_rischio_empirico" title="Minimizzazione del rischio empirico">Minimizzazione del rischio empirico</a></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid 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href="http://id.loc.gov/authorities/subjects/sh85112392">sh85112392</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4129903-6">4129903-6</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Francia" title="Biblioteca nazionale di Francia">BNF</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="francese">FR</abbr></span>) <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb119445648">cb119445648</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb119445648">(data)</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr></span>) <a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007529518905171">987007529518905171</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_della_Dieta_nazionale_del_Giappone" title="Biblioteca della Dieta nazionale del Giappone">NDL</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="giapponese">JA</abbr></span>) <a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00564579">00564579</a></span></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-basis: calc( 100% / 2 - 8px / 2 );"><style 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