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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</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">6</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">7</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-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 delle componenti principali</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 34 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-34" 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">34 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%B9%D9%86%D8%B5%D8%B1_%D8%A7%D9%84%D8%B1%D8%A6%D9%8A%D8%B3%D9%8A" title="تحليل العنصر الرئيسي - arabo" lang="ar" hreflang="ar" data-title="تحليل العنصر الرئيسي" 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_componentes_principales" title="Analís de componentes principales - asturiano" lang="ast" hreflang="ast" data-title="Analís de componentes principales" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_de_components_principals" title="Anàlisi de components principals - catalano" lang="ca" hreflang="ca" data-title="Anàlisi de components principals" 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/Anal%C3%BDza_hlavn%C3%ADch_komponent" title="Analýza hlavních komponent - ceco" lang="cs" hreflang="cs" data-title="Analýza hlavních komponent" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hauptkomponentenanalyse" title="Hauptkomponentenanalyse - tedesco" lang="de" hreflang="de" data-title="Hauptkomponentenanalyse" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Principal_component_analysis" title="Principal component analysis - inglese" lang="en" hreflang="en" data-title="Principal component 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/Analizo_al_precipaj_konsisteroj" title="Analizo al precipaj konsisteroj - esperanto" lang="eo" hreflang="eo" data-title="Analizo al precipaj konsisteroj" 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_componentes_principales" title="Análisis de componentes principales - spagnolo" lang="es" hreflang="es" data-title="Análisis de componentes principales" 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/Peakomponentide_anal%C3%BC%C3%BCs" title="Peakomponentide analüüs - estone" lang="et" hreflang="et" data-title="Peakomponentide analüüs" 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/Osagai_nagusien_analisi" title="Osagai nagusien analisi - basco" lang="eu" hreflang="eu" data-title="Osagai nagusien 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_%D9%85%D8%A4%D9%84%D9%81%D9%87%E2%80%8C%D9%87%D8%A7%DB%8C_%D8%A7%D8%B5%D9%84%DB%8C" title="تحلیل مؤلفه‌های اصلی - persiano" lang="fa" hreflang="fa" data-title="تحلیل مؤلفه‌های اصلی" 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/P%C3%A4%C3%A4komponenttianalyysi" title="Pääkomponenttianalyysi - finlandese" lang="fi" hreflang="fi" data-title="Pääkomponenttianalyysi" 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/Analyse_en_composantes_principales" title="Analyse en composantes principales - francese" lang="fr" hreflang="fr" data-title="Analyse en composantes principales" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/An%C3%A1lise_de_compo%C3%B1entes_principais" title="Análise de compoñentes principais - galiziano" lang="gl" hreflang="gl" data-title="Análise de compoñentes principais" 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%A0%D7%99%D7%AA%D7%95%D7%97_%D7%A8%D7%9B%D7%99%D7%91%D7%99%D7%9D_%D7%A2%D7%99%D7%A7%D7%A8%D7%99%D7%99%D7%9D" title="ניתוח רכיבים עיקריים - ebraico" lang="he" hreflang="he" data-title="ניתוח רכיבים עיקריים" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/F%C5%91komponens-anal%C3%ADzis" title="Főkomponens-analízis - ungherese" lang="hu" hreflang="hu" data-title="Főkomponens-analízis" 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_komponen_utama" title="Analisis komponen utama - indonesiano" lang="id" hreflang="id" data-title="Analisis komponen utama" 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/Meginhlutagreining" title="Meginhlutagreining - islandese" lang="is" hreflang="is" data-title="Meginhlutagreining" 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/%E4%B8%BB%E6%88%90%E5%88%86%E5%88%86%E6%9E%90" title="主成分分析 - giapponese" lang="ja" hreflang="ja" data-title="主成分分析" 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A3%BC%EC%84%B1%EB%B6%84_%EB%B6%84%EC%84%9D" title="주성분 분석 - coreano" lang="ko" hreflang="ko" data-title="주성분 분석" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hoofdcomponentenanalyse" title="Hoofdcomponentenanalyse - olandese" lang="nl" hreflang="nl" data-title="Hoofdcomponentenanalyse" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_g%C5%82%C3%B3wnych_sk%C5%82adowych" title="Analiza głównych składowych - polacco" lang="pl" hreflang="pl" data-title="Analiza głównych składowych" 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/An%C3%A1lise_de_componentes_principais" title="Análise de componentes principais - portoghese" lang="pt" hreflang="pt" data-title="Análise de componentes principais" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%B3%D0%BB%D0%B0%D0%B2%D0%BD%D1%8B%D1%85_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%82" title="Метод главных компонент - russo" lang="ru" hreflang="ru" data-title="Метод главных компонент" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Anal%C3%BDza_hlavn%C3%BDch_komponentov" title="Analýza hlavných komponentov - slovacco" lang="sk" hreflang="sk" data-title="Analýza hlavných komponentov" data-language-autonym="Slovenčina" data-language-local-name="slovacco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Analiza_glavnih_komponenti" title="Analiza glavnih komponenti - serbo" lang="sr" hreflang="sr" data-title="Analiza glavnih komponenti" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Principalkomponentanalys" title="Principalkomponentanalys - svedese" lang="sv" hreflang="sv" data-title="Principalkomponentanalys" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Analisis_ng_pangunahing_bahagi" title="Analisis ng pangunahing bahagi - tagalog" lang="tl" hreflang="tl" data-title="Analisis ng pangunahing bahagi" 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/Temel_bile%C5%9Fen_analizi" title="Temel bileşen analizi - turco" lang="tr" hreflang="tr" data-title="Temel bileşen 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%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%B3%D0%BE%D0%BB%D0%BE%D0%B2%D0%BD%D0%B8%D1%85_%D0%BA%D0%BE%D0%BC%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%82" title="Метод головних компонент - ucraino" lang="uk" hreflang="uk" data-title="Метод головних компонент" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_ph%C3%A2n_t%C3%ADch_th%C3%A0nh_ph%E1%BA%A7n_ch%C3%ADnh" title="Phép phân tích thành phần chính - vietnamita" lang="vi" hreflang="vi" data-title="Phép phân tích thành phần chính" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%BB%E6%88%90%E5%88%86%E5%88%86%E6%9E%90" title="主成分分析 - cinese" lang="zh" hreflang="zh" data-title="主成分分析" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C3%BA-s%C3%AAng-h%C5%ABn_hun-sek" title="Chú-sêng-hūn hun-sek - min nan" lang="nan" hreflang="nan" data-title="Chú-sêng-hūn hun-sek" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%B8%BB%E6%88%90%E4%BB%BD%E5%88%86%E6%9E%90" title="主成份分析 - cantonese" lang="yue" hreflang="yue" data-title="主成份分析" data-language-autonym="粵語" data-language-local-name="cantonese" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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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:r133964453">.mw-parser-output .avviso .mbox-text-div>div,.mw-parser-output .avviso .mbox-text-full-div>div{font-size:90%}.mw-parser-output .avviso .mbox-image{flex-basis:52px;flex-grow:0;flex-shrink:0}.mw-parser-output .avviso .mbox-text-full-div .hide-when-compact{display:block}</style><div style="" class="ambox metadata noprint plainlinks avviso avviso-contenuto"> <div class="avviso-immagine mbox-image noprint"><span typeof="mw:File"><a href="/wiki/File:Question_book_magnify.svg" class="mw-file-description" title="Nessuna nota a piè di pagina"><img alt="Nessuna nota a piè di pagina" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Question_book_magnify.svg/45px-Question_book_magnify.svg.png" decoding="async" 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esterni">collegamenti esterni</a>, manca la contestualizzazione delle fonti con <a href="/wiki/Aiuto:Note" title="Aiuto:Note">note a piè di pagina</a> o altri riferimenti precisi che indichino puntualmente la provenienza delle informazioni. Puoi <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Analisi_delle_componenti_principali&action=edit">migliorare questa voce</a> <a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">citando le fonti</a> più precisamente. Segui i suggerimenti del <a href="/wiki/Progetto:Statistica" title="Progetto:Statistica">progetto di riferimento</a>.</div> </div> </div> </div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:GaussianScatterPCA.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/GaussianScatterPCA.svg/220px-GaussianScatterPCA.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/GaussianScatterPCA.svg/330px-GaussianScatterPCA.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/GaussianScatterPCA.svg/440px-GaussianScatterPCA.svg.png 2x" data-file-width="720" data-file-height="720" /></a><figcaption>Esempio di PCA. I vettori indicati sono gli <a href="/wiki/Autovettore_e_autovalore" title="Autovettore e autovalore">autovettori</a> della <a href="/wiki/Matrice_di_covarianza" class="mw-redirect" title="Matrice di covarianza">matrice di covarianza</a>.</figcaption></figure> <p>L'<b>analisi delle componenti principali</b> (in <a href="/wiki/Lingua_inglese" title="Lingua inglese">inglese</a> <span dir="ltr" lang="en"><i>principal component analysis</i></span> o abbreviata <i>PCA</i>), anche nota come <i>trasformata di Karhunen-Loève</i><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>, è una tecnica per la semplificazione dei dati di <a href="/wiki/Riduzione_della_dimensionalit%C3%A0" title="Riduzione della dimensionalità">riduzione della dimensionalità</a> lineare utilizzata nell'ambito della <a href="/wiki/Statistica_multivariata" title="Statistica multivariata">statistica multivariata</a>.<sup id="cite_ref-Tuffery_definizione_2-0" class="reference"><a href="#cite_note-Tuffery_definizione-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Questo metodo fu proposto per la prima volta nel 1901 da <a href="/wiki/Karl_Pearson" title="Karl Pearson">Karl Pearson</a> e sviluppato poi da <a href="/wiki/Harold_Hotelling" title="Harold Hotelling">Harold Hotelling</a> nel 1933, e fa parte dell'<a href="/wiki/Analisi_fattoriale" title="Analisi fattoriale">analisi fattoriale</a>. La tecnica ha lo scopo principale di ridurre il numero più o meno elevato di variabili che descrivono un insieme di dati a un numero minore di variabili latenti, limitando il più possibile la perdita di informazioni.<sup id="cite_ref-Tuffery_definizione_2-1" class="reference"><a href="#cite_note-Tuffery_definizione-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Risoluzione_lineare">Risoluzione lineare</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_delle_componenti_principali&veaction=edit&section=1" title="Modifica la sezione Risoluzione lineare" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_delle_componenti_principali&action=edit&section=1" title="Edit section's source code: Risoluzione lineare"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La risoluzione lineare avviene tramite una <a href="/wiki/Trasformazione_lineare" title="Trasformazione lineare">trasformazione lineare</a> delle variabili che proietta quelle originarie in un nuovo <a href="/wiki/Sistema_cartesiano" class="mw-redirect" title="Sistema cartesiano">sistema cartesiano</a> in cui la nuova variabile con la maggiore <a href="/wiki/Varianza" title="Varianza">varianza</a> viene proiettata sul primo asse, la variabile nuova, seconda per dimensione della varianza, sul secondo asse e così via. </p><p>La riduzione della complessità avviene limitandosi ad analizzare le principali, per varianza, tra le nuove variabili. </p><p>Diversamente da altre trasformazioni lineari di variabili praticate nell'ambito della statistica, in questa tecnica sono gli stessi dati che determinano i vettori di trasformazione. </p><p>Assumendo che a ciascuna delle variabili originarie venga sottratta la loro media e pertanto la nuova 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 (X_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </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/67f488841364dd170c2f46faae8e2c3010f4cdb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.533ex; height:2.843ex;" alt="{\displaystyle (X_{i})}"></span> abbia media nulla, </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 {w} _{1}={\underset {\Vert \mathbf {w} \Vert =1}{\operatorname {\arg \,max} }}\,E\left\{\left(\mathbf {w} ^{T}\mathbf {x} \right)^{2}\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>arg</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> </mrow> <mspace width="thinmathspace" /> <mi>E</mi> <mrow> <mo>{</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{1}={\underset {\Vert \mathbf {w} \Vert =1}{\operatorname {\arg \,max} }}\,E\left\{\left(\mathbf {w} ^{T}\mathbf {x} \right)^{2}\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b08cb4c60843570e8538e326b3eb63409b42ae40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.02ex; height:6.009ex;" alt="{\displaystyle \mathbf {w} _{1}={\underset {\Vert \mathbf {w} \Vert =1}{\operatorname {\arg \,max} }}\,E\left\{\left(\mathbf {w} ^{T}\mathbf {x} \right)^{2}\right\},}"></span></dd></dl> <p>dove <a href="/wiki/Arg_max" title="Arg max">arg max</a> indica l'insieme degli argomenti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b1e0c8e1be5ebe69d18a8010676fa42d7961e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}"></span> in cui è raggiunto il massimo. Con le prime <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21363ebd7038c93aae93127e7d910fc1b2e2c745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k-1}"></span> componenti, la <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>-esima componente può essere trovata sottraendo le prime <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21363ebd7038c93aae93127e7d910fc1b2e2c745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k-1}"></span> componenti principali a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>: </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 {\hat {x}} _{k}=\mathbf {x} -\sum _{i=1}^{k-1}\mathbf {w} _{i}\mathbf {w} _{i}^{T}\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">x</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {x}} _{k}=\mathbf {x} -\sum _{i=1}^{k-1}\mathbf {w} _{i}\mathbf {w} _{i}^{T}\mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95a3cb1deeb83600b81a7c5738f086147a92b433" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.054ex; height:7.343ex;" alt="{\displaystyle \mathbf {\hat {x}} _{k}=\mathbf {x} -\sum _{i=1}^{k-1}\mathbf {w} _{i}\mathbf {w} _{i}^{T}\mathbf {x} }"></span></dd></dl> <p>e sostituendo questo </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 {w} _{k}={\underset {\Vert \mathbf {w} \Vert =1}{\operatorname {arg\,max} }}\,E\left\{\left(\mathbf {w} ^{T}\mathbf {\hat {x}} _{k-1}\right)^{2}\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> </munder> </mrow> <mspace width="thinmathspace" /> <mi>E</mi> <mrow> <mo>{</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">x</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{k}={\underset {\Vert \mathbf {w} \Vert =1}{\operatorname {arg\,max} }}\,E\left\{\left(\mathbf {w} ^{T}\mathbf {\hat {x}} _{k-1}\right)^{2}\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19fa8885b2c5714371e92eec694a4338b17e5a93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.857ex; height:6.009ex;" alt="{\displaystyle \mathbf {w} _{k}={\underset {\Vert \mathbf {w} \Vert =1}{\operatorname {arg\,max} }}\,E\left\{\left(\mathbf {w} ^{T}\mathbf {\hat {x}} _{k-1}\right)^{2}\right\}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Risoluzione_matriciale">Risoluzione matriciale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_delle_componenti_principali&veaction=edit&section=2" title="Modifica la sezione Risoluzione matriciale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_delle_componenti_principali&action=edit&section=2" title="Edit section's source code: Risoluzione matriciale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un metodo più semplice per calcolare la componente <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 {w} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738edcf12148b7bd3c07ba5459be825cb6089aa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.731ex; height:2.009ex;" alt="{\displaystyle \mathbf {w} _{i}}"></span> utilizza la <a href="/wiki/Matrice_delle_covarianze" title="Matrice delle covarianze">matrice delle covarianze</a> di <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 {x} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d2ef3df60acdb53bdf90535264041fea7231cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{i}}"></span>. La stessa operazione può essere eseguita partendo dalla matrice dei coefficienti di correlazione anziché dalla matrice delle covarianze.<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><p>Innanzitutto si devono trovare gli <a href="/wiki/Autovalori" class="mw-redirect" title="Autovalori">autovalori</a> della matrice di covarianza o della matrice dei coefficienti di correlazione. Si ottengono tanti autovalori quante sono le 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 \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span>. Se viene utilizzata la matrice di correlazione, l'autovalore relativo alla prima componente principale, ossia quella con varianza massima, sarà pari a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>. In ogni caso l'autovalore con il maggiore valore corrisponde alla dimensione <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 {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20795664b5b048744a2fd88977851104cc5816f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }"></span> che ha la maggiore varianza: esso sarà dunque la varianza della componente principale 1. In ordine decrescente, il secondo autovalore sarà la varianza della componente principale 2, e così via per gli n autovalori. Per ciascun autovalore viene calcolato il corrispondente <a href="/wiki/Autovettore" class="mw-redirect" title="Autovettore">autovettore</a>, ossia la matrice (riga vettore) dei coefficienti che moltiplicano le vecchie 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 \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span> nella <a href="/wiki/Combinazione_lineare" title="Combinazione lineare">combinazione lineare</a> per l'ottenimento delle nuove 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 \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20795664b5b048744a2fd88977851104cc5816f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }"></span>. Questi coefficienti sono anche definiti <a href="/w/index.php?title=Loadings&action=edit&redlink=1" class="new" title="Loadings (la pagina non esiste)">loading</a>. La matrice degli autovettori, ossia la matrice che ha per riga ciascun autovettore prima calcolato, è la cosiddetta <i>matrice di rotazione</i> <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>. Eseguendo l'operazione matriciale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=V\cdot X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>V</mi> <mo>⋅</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=V\cdot X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbe0faa8e11edb5b8893e4a39b4824c060914b1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.98ex; height:2.176ex;" alt="{\displaystyle W=V\cdot X}"></span>, 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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> è il vettore colonna avente come elementi le nuove 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 w_{1},\,w_{2},\,\ldots ,\,w_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{1},\,w_{2},\,\ldots ,\,w_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72358566a593793c770f57ac34bff181991fba1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.693ex; height:2.009ex;" alt="{\displaystyle w_{1},\,w_{2},\,\ldots ,\,w_{n}}"></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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> è il vettore colonna avente come elementi le "vecchie 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_{1},\,x_{2},\,\ldots ,\,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},\,x_{2},\,\ldots ,\,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0db44dc23fd4c08d9adfe63d60de8e59f67e92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.689ex; height:2.009ex;" alt="{\displaystyle x_{1},\,x_{2},\,\ldots ,\,x_{n}}"></span>, si possono trovare le coordinate di ciascun punto nel nuovo <a href="/wiki/Spazio_vettoriale" title="Spazio vettoriale">spazio vettoriale</a>. Utilizzando le coordinate per ciascun punto relative alle componenti principali si costruisce il grafico denominato <i>score plot</i>. Se le componenti principali sono 3 si avrà un grafico tridimensionale, se sono 2 sarà bidimensionale, se invece si è scelta una sola componente principale lo score plot sarà allora monodimensionale. Mediante lo score plot è possibile verificare quali dati sono simili tra di loro e quindi si può ad esempio dedurre quali campioni presentano la medesima composizione. </p><p>In PCA esiste anche un altro tipo di grafico, definito loading plot, in cui sono le 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> ad essere riportate nel nuovo sistema avente per assi le componenti principali. Con questo tipo di grafico è possibile osservare se due variabili sono simili, e pertanto forniscono lo stesso tipo di informazione, oppure se sono distanti (e quindi non sono simili). </p><p>Quindi gli elementi dell'autovettore colonna corrispondente a un autovalore esprimono il legame tra le variabili di partenza e la componente considerata attraverso dei pesi. Il numero di variabili latenti da considerare come componenti principali si fonda sulla grandezza relativa di un autovalore rispetto agli altri. Invece nel caso in cui sia l'operatore a scegliere le componenti principali senza considerare la relativa varianza espressa dai rispettivi autovalori, si ha un <i>supervised pattern recognition</i>. </p><p>Si può costruire la <a href="/wiki/Matrice" title="Matrice">matrice</a> dei fattori, in pratica una matrice modale, che elenca per riga le variabili originarie e per colonna le variabili latenti: ogni valore, compreso tra 0 e 1, dice quanto le seconde incidano sulle prime. </p><p>Invece la matrice del punteggio fattoriale ha la stessa struttura della precedente, ma dice quanto le singole variabili originarie abbiano pesato sulla determinazione della grandezza di quelle latenti. </p> <div class="mw-heading mw-heading2"><h2 id="Esempio">Esempio</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_delle_componenti_principali&veaction=edit&section=3" title="Modifica la sezione Esempio" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analisi_delle_componenti_principali&action=edit&section=3" title="Edit section's source code: Esempio"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si supponga di disporre di un'indagine che riporta per 10 soggetti: voto medio (da 0 a 33), intelligenza (da 0 a 10), media ore studiate in un giorno e zona d'origine, che varia da 1 a 3. Si standardizzino i valori con la formula: </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 z_{i}={\frac {x_{i}-\mu _{X}}{SD}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mrow> <mrow> <mi>S</mi> <mi>D</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z_{i}={\frac {x_{i}-\mu _{X}}{SD}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d9a12e72c6b25f1191fe3829fc146444e2a691" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.466ex; height:5.343ex;" alt="{\displaystyle z_{i}={\frac {x_{i}-\mu _{X}}{SD}},}"></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 \mu _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfe6d3f115b8d6cb595119ea9bc7962a11db65a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.034ex; height:2.176ex;" alt="{\displaystyle \mu _{X}}"></span> è la <a href="/wiki/Media_(statistica)#Media_aritmetica" title="Media (statistica)">media aritmetica</a> di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> 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 SD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af326440837e67870d4bee5bf63f49c37c9d89c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.423ex; height:2.176ex;" alt="{\displaystyle SD}"></span> è la <a href="/wiki/Deviazione_standard" class="mw-redirect" title="Deviazione standard">deviazione standard</a>. </p><p>La matrice dei <a href="/wiki/Coefficiente_di_correlazione" class="mw-redirect" title="Coefficiente di correlazione">coefficienti di correlazione</a> è: </p> <table class="wikitable"> <tbody><tr> <th> </th> <th>Z<sub>score</sub> (Voto medio) </th> <th>Z<sub>score</sub> (Intelligenza) </th> <th>Z<sub>score</sub> (Provenienza) </th> <th>Z<sub>score</sub> (Media ore di studio) </th></tr> <tr> <td>Z<sub>score</sub> (Voto medio)</td> <td>1,000</td> <td>0,600</td> <td>-0,838</td> <td>0,788 </td></tr> <tr> <td>Z<sub>score</sub> (Intelligenza)</td> <td>0,600</td> <td>1,000</td> <td>-0,222</td> <td>0,022 </td></tr> <tr> <td>Z<sub>score</sub> (Provenienza)</td> <td>-0,838</td> <td>-0,222</td> <td>1,000</td> <td>-0,918 </td></tr> <tr> <td>Z<sub>score</sub> (Media ore di studio)</td> <td>0,788</td> <td>0,022</td> <td>-0,918</td> <td>1,000 </td></tr> </tbody></table> <p>La <a href="/wiki/Diagonale_principale" title="Diagonale principale">diagonale principale</a> è composta da valori uguali a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> perché è il coefficiente di correlazione di una variabile con sé stessa. La matrice è <a href="/wiki/Matrice_simmetrica" title="Matrice simmetrica">simmetrica</a> perché il coefficiente di correlazione tra la 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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> e la 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/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> è uguale a quello tra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> 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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>. Si vede come ci sia un forte legame tra voto, media ore studio e provenienza. </p><p>Dall'analisi degli autovalori si possono trarre conclusioni: </p> <table class="wikitable"> <tbody><tr> <th rowspan="2">Componente</th> <th colspan="3">Autovalori iniziali</th> <th colspan="3">Extraction sums of squared loadings </th></tr> <tr> <td>Totale</td> <td>Varianza %</td> <td>% Cumulative</td> <td>Totale</td> <td>Varianza %</td> <td>% Cumulative </td></tr> <tr> <td>1</td> <td>2,828</td> <td>70,708</td> <td>70,708</td> <td>2,828</td> <td>70,708</td> <td>70,708 </td></tr> <tr> <td>2</td> <td>1,070</td> <td>26,755</td> <td>97,463</td> <td>1,070</td> <td>26,755</td> <td>97,496 </td></tr> <tr> <td>3</td> <td>0,084</td> <td>2,088</td> <td>99,551 </td></tr> <tr> <td>4</td> <td>0,018</td> <td>0,449</td> <td>100,000 </td></tr></tbody></table> <p>Gli autovalori sono in ordine decrescente e il loro rapporto con la somma degli autovalori dà la percentuale di varianza che rappresentano. Sono stati selezionati arbitrariamente solo quelli che hanno valore maggiore di <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> in quanto più significativi, che spiegano il 70,708% e il 26,755% rispettivamente. </p><p>Si osservi la matrice delle componenti principali: </p> <table class="wikitable"> <tbody><tr> <th rowspan="2"></th> <th colspan="2">Componente </th></tr> <tr> <th>1</th> <th>2 </th></tr> <tr> <td>Z<sub>score</sub> (Voto medio)</td> <td>0,966</td> <td>0,204 </td></tr> <tr> <td>Z<sub>score</sub> (Intelligenza)</td> <td>0,442</td> <td>0,894 </td></tr> <tr> <td>Z<sub>score</sub> (Provenienza)</td> <td>-0,947</td> <td>0,228 </td></tr> <tr> <td>Z<sub>score</sub> (Media ore di studio)</td> <td>0,897</td> <td>-0,420 </td></tr></tbody></table> <p>Il fattore 1 pesa fortemente sul voto medio e, in maniera negativa, sulla variabile della zona di origine. </p><p>Si calcoli quindi la matrice di punteggio fattoriale: </p> <table class="wikitable"> <tbody><tr> <th rowspan="2"></th> <th colspan="2">Componente </th></tr> <tr> <th>1</th> <th>2 </th></tr> <tr> <td>Z<sub>score</sub> (Voto medio)</td> <td>0,341</td> <td>0,191 </td></tr> <tr> <td>Z<sub>score</sub> (Intelligenza)</td> <td>0,156</td> <td>0,836 </td></tr> <tr> <td>Z<sub>score</sub> (Provenienza)</td> <td>-0,335</td> <td>0,213 </td></tr> <tr> <td>Z<sub>score</sub> (Media ore di studio)</td> <td>0,317</td> <td>-0,392 </td></tr></tbody></table> <p>Come si vede la variabile provenienza continua ad avere un influsso di segno negativo sull'autovalore principale. Le altre variabili invece hanno peso positivo. </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_delle_componenti_principali&veaction=edit&section=4" 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_delle_componenti_principali&action=edit&section=4" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text">viene chiamata anche <i>trasformata di Hotelling</i> oppure <i>decomposizione ortogonale propria</i></span> </li> <li id="cite_note-Tuffery_definizione-2"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-Tuffery_definizione_2-0">a</a></i></sup> <sup><i><a href="#cite_ref-Tuffery_definizione_2-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Stéphane Tufféry, <a rel="nofollow" class="external text" href="https://archive.org/details/dataminingstatis00tuff"><span style="font-style:italic;">Factor analysis</span></a>, in <span style="font-style:italic;">Data mining and statistics for decision making</span>, Wiley, 2011, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/dataminingstatis00tuff/page/175">175</a>-180, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-470-68829-8" title="Speciale:RicercaISBN/978-0-470-68829-8">978-0-470-68829-8</a>.</cite></span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal"> Paola Pozzolo, <a rel="nofollow" class="external text" href="https://paolapozzolo.it/analisi-delle-componenti-principali-criteri/"><span style="font-style:italic;">Analisi delle componenti principali: da dove partire</span></a>, su <span style="font-style:italic;">Paola Pozzolo</span>, 5 dicembre 2020. <small>URL consultato il 28 luglio 2024</small>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_delle_componenti_principali&veaction=edit&section=5" 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_delle_componenti_principali&action=edit&section=5" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation libro" style="font-style:normal"> Sergio Bolasco, <span style="font-style:italic;">Analisi multidimensionale dei dati. Metodi, strategie e criteri d'interpretazione</span>, 6ª ed., Roma, Carocci, 2014 <abbr title="Data di edizione originale">[1999]</abbr>, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-430-1401-3" title="Speciale:RicercaISBN/88-430-1401-3">88-430-1401-3</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal"> Roberto Todeschini, <span style="font-style:italic;">Introduzione alla chemiometria</span>, 1ª ed., Napoli, EdiSES, 2003, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-7959-146-0" title="Speciale:RicercaISBN/88-7959-146-0">88-7959-146-0</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Stéphane Tufféry, <span style="font-style:italic;">Data mining and statistics for decision making</span>, Wiley, 2011, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-470-68829-8" title="Speciale:RicercaISBN/978-0-470-68829-8">978-0-470-68829-8</a>.</cite></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_delle_componenti_principali&veaction=edit&section=6" 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_delle_componenti_principali&action=edit&section=6" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Analisi_della_correlazione_canonica" title="Analisi della correlazione canonica">Analisi della correlazione canonica</a></li> <li><a href="/wiki/Analisi_fattoriale" title="Analisi fattoriale">Analisi fattoriale</a></li> <li><a href="/wiki/Riduzione_della_dimensionalit%C3%A0" title="Riduzione della dimensionalità">Riduzione della dimensionalità</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analisi_delle_componenti_principali&veaction=edit&section=7" 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_delle_componenti_principali&action=edit&section=7" 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; 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selflink">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 #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .CdA th{background-color:#202122}}</style><table class="CdA"><tbody><tr><th><a href="/wiki/Aiuto:Controllo_di_autorit%C3%A0" title="Aiuto:Controllo di autorità">Controllo di autorità</a></th><td><a href="/wiki/Nuovo_soggettario" title="Nuovo soggettario">Thesaurus BNCF</a> <span class="uid"><a rel="nofollow" class="external text" href="https://thes.bncf.firenze.sbn.it/termine.php?id=52507">52507</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://id.loc.gov/authorities/subjects/sh85106729">sh85106729</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/4129174-8">4129174-8</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/cb11942895w">cb11942895w</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11942895w">(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=987007536366205171">987007536366205171</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% / 3 - 8px * 2/3 );"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night 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