典型相关 - 维基百科，自由的百科全书

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href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-计算" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#计算"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>计算</span> </div> </a> <button aria-controls="toc-计算-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>开关计算子章节</span> </button> <ul id="toc-计算-sublist" class="vector-toc-list"> <li id="toc-推导" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#推导"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>推导</span> </div> </a> <ul id="toc-推导-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-解法" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#解法"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>解法</span> </div> </a> <ul id="toc-解法-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-实现" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#实现"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>实现</span> </div> </a> <ul id="toc-实现-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-假设检验" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#假设检验"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>假设检验</span> </div> </a> <ul id="toc-假设检验-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-实际运用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#实际运用"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>实际运用</span> </div> </a> <ul id="toc-实际运用-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-例子" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#例子"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>例子</span> </div> </a> <ul id="toc-例子-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-与principal_angles的连接" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#与principal_angles的连接"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>与principal angles的连接</span> </div> </a> <ul id="toc-与principal_angles的连接-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>参见</span> </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考文献" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>参考文献</span> </div> </a> <ul id="toc-参考文献-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部链接" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部链接"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>外部链接</span> </div> </a> <ul id="toc-外部链接-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="目录" 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="开关目录" > <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">开关目录</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">典型相关</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="前往另一种语言写成的文章。14种语言可用" > <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-14" 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">14种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Correlaci%C3%B3_can%C3%B2nica" title="Correlació canònica – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Correlació canònica" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kanonische_Korrelation" title="Kanonische Korrelation – 德语" lang="de" hreflang="de" data-title="Kanonische Korrelation" data-language-autonym="Deutsch" data-language-local-name="德语" 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/Canonical_correlation" title="Canonical correlation – 英语" lang="en" hreflang="en" data-title="Canonical correlation" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_de_la_correlaci%C3%B3n_can%C3%B3nica" title="Análisis de la correlación canónica – 西班牙语" lang="es" hreflang="es" data-title="Análisis de la correlación canónica" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Korrelazio_kanoniko" title="Korrelazio kanoniko – 巴斯克语" lang="eu" hreflang="eu" data-title="Korrelazio kanoniko" data-language-autonym="Euskara" data-language-local-name="巴斯克语" 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/%D9%87%D9%85%D8%A8%D8%B3%D8%AA%DA%AF%DB%8C_%DA%A9%D8%A7%D9%86%D9%88%D9%86%DB%8C" title="همبستگی کانونی – 波斯语" lang="fa" hreflang="fa" data-title="همبستگی کانونی" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Analyse_canonique_des_corr%C3%A9lations" title="Analyse canonique des corrélations – 法语" lang="fr" hreflang="fr" data-title="Analyse canonique des corrélations" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Analisi_della_correlazione_canonica" title="Analisi della correlazione canonica – 意大利语" lang="it" hreflang="it" data-title="Analisi della correlazione canonica" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AB%E3%83%8E%E3%83%8B%E3%82%AB%E3%83%AB%E7%9B%B8%E9%96%A2" title="カノニカル相関 – 日语" lang="ja" hreflang="ja" data-title="カノニカル相関" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Korelacja_kanoniczna" title="Korelacja kanoniczna – 波兰语" lang="pl" hreflang="pl" data-title="Korelacja kanoniczna" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BD%D0%BE%D0%BD%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%BA%D0%BE%D1%80%D1%80%D0%B5%D0%BB%D1%8F%D1%86%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="Канонический корреляционный анализ – 俄语" lang="ru" hreflang="ru" data-title="Канонический корреляционный анализ" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Kor%C3%A9lasi_kanonik" title="Korélasi kanonik – 巽他语" lang="su" hreflang="su" data-title="Korélasi kanonik" data-language-autonym="Sunda" data-language-local-name="巽他语" 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/Kanonisk_korrelationsanalys" title="Kanonisk korrelationsanalys – 瑞典语" lang="sv" hreflang="sv" data-title="Kanonisk korrelationsanalys" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BD%D0%BE%D0%BD%D1%96%D1%87%D0%BD%D0%B0_%D0%BA%D0%BE%D1%80%D0%B5%D0%BB%D1%8F%D1%86%D1%96%D1%8F" title="Канонічна кореляція – 乌克兰语" lang="uk" hreflang="uk" data-title="Канонічна кореляція" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q115542#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3" rel="discussion" title="关于此页面的讨论[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="更改语言变体" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">不转换</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="selected ca-variants-zh mw-list-item"><a href="/zh/%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3" lang="zh-Hans" 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</div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-noteTA-307af9e9" class="mw-indicator"><div class="mw-parser-output"><span class="skin-invert" typeof="mw:File"><span title="本页使用了标题或全文手工转换"><img alt="本页使用了标题或全文手工转换" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/35px-Zh_conversion_icon_m.svg.png" decoding="async" width="35" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/53px-Zh_conversion_icon_m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/70px-Zh_conversion_icon_m.svg.png 2x" data-file-width="32" data-file-height="20" /></span></span></div></div> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</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="zh" dir="ltr"><div id="noteTA-307af9e9" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Math"></div></div></div> <p>在<a href="/wiki/%E7%BB%9F%E8%AE%A1%E5%AD%A6" title="统计学">统计学</a>中，<b>典型相关分析</b>（英語：<span lang="en">Canonical Correlation Analysis</span>）是对<a href="/wiki/%E4%BA%92%E5%8D%8F%E6%96%B9%E5%B7%AE" title="互协方差">互协方差</a>矩阵的一种理解。如果我们有两个<a href="/wiki/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F" title="随机变量">随机变量</a>向量 <i>X</i> = (<i>X</i><sub>1</sub>, ..., <i>X</i><sub><i>n</i></sub>) 和 <i>Y</i> = (<i>Y</i><sub>1</sub>, ..., <i>Y</i><sub><i>m</i></sub>) 并且它们是<a href="/wiki/%E7%9B%B8%E5%85%B3_(%E6%A6%82%E7%8E%87%E8%AE%BA)" title="相关 (概率论)">相關</a>的，那么典型相关分析会找出 <i>X</i><sub><i>i</i></sub> 和 <i>Y</i><sub><i>j</i></sub> 的相互相关最大的线性组合。<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>T·R·Knapp指出“几乎所有常见的<a href="/w/index.php?title=%E5%8F%82%E6%95%B0%E6%B5%8B%E8%AF%95&action=edit&redlink=1" class="new" title="参数测试（页面不存在）">参数测试</a>的意义可视为特殊情况的典型相关分析，这是研究两组变量之间关系的一般步骤。”<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> 这个方法在1936年由<a href="/wiki/%E5%93%88%E7%BE%85%E5%BE%B7%C2%B7%E9%9C%8D%E7%89%B9%E6%9E%97" title="哈羅德·霍特林">哈罗德·霍特林</a>首次引入。<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>给定两个随机向量<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=(x_{1},\dots ,x_{n})'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=(x_{1},\dots ,x_{n})'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7cbf9b188ec8afc2098dc179c9187601a555a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.683ex; height:3.009ex;" alt="{\displaystyle X=(x_{1},\dots ,x_{n})'}"></span>和<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=(y_{1},\dots ,y_{m})'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=(y_{1},\dots ,y_{m})'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/384bcfa6d635d14e396e93977fec508768f455e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.552ex; height:3.009ex;" alt="{\displaystyle Y=(y_{1},\dots ,y_{m})'}"></span>，我们可以定义<a href="/wiki/%E4%BA%92%E5%8D%8F%E6%96%B9%E5%B7%AE" title="互协方差">互协方差</a>矩阵 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{XY}=\operatorname {cov} (X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{XY}=\operatorname {cov} (X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe34aee6af5acb5960284d0c79f627fe1048b566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.682ex; height:2.843ex;" alt="{\displaystyle \Sigma _{XY}=\operatorname {cov} (X,Y)}"></span> 为 <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\times m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82325a2a02ad79bc7c347ba9702ad46eb0de824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle n\times m}"></span> 的<a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a>，其中 <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,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}"></span> 是<a href="/wiki/%E5%8D%8F%E6%96%B9%E5%B7%AE" title="协方差">协方差</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 \operatorname {cov} (x_{i},y_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (x_{i},y_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3f82dad7b28c28c140c59dfcf37afbada0516c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.444ex; height:3.009ex;" alt="{\displaystyle \operatorname {cov} (x_{i},y_{j})}"></span>。实际上，我们可以基于 <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> 和 <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> 的采样数据来估计协方差矩阵。(如从一对数据矩阵)。 </p><p>典型相关分析求出向量 <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> 和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> 使得随机变量 <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 a'X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a'X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21ceeb84d7f3fb8392c3fe96d01e92e7ee004cd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.895ex; height:2.509ex;" alt="{\displaystyle a'X}"></span> 和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b'Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b'Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13426384a5a0ba5295422f66993026227de8f439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.456ex; height:2.509ex;" alt="{\displaystyle b'Y}"></span> 的<a href="/wiki/%E7%9B%B8%E5%85%B3_(%E6%A6%82%E7%8E%87%E8%AE%BA)" title="相关 (概率论)">相關</a>性 <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 \rho =\operatorname {corr} (a'X,b'Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mi>corr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>X</mi> <mo>,</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\operatorname {corr} (a'X,b'Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/402a6ff39bf181076a33e6a5c33da8ca57f510ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.512ex; height:3.009ex;" alt="{\displaystyle \rho =\operatorname {corr} (a'X,b'Y)}"></span> 最大。随机变量 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=a'X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=a'X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2088998ac0c8e2cb4d31a6816a1fe5709b256eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.776ex; height:2.509ex;" alt="{\displaystyle U=a'X}"></span> 和 <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=b'Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=b'Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab58835b2205050344881348e1500f9f824d49b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.341ex; height:2.509ex;" alt="{\displaystyle V=b'Y}"></span> 是 <i><b>第一对典型变量</b></i>。然后寻求一个依然最大化相关但与第一对典型变量不相关的向量；这样就得到了 <i><b>第二对典型变量</b></i>。这个步骤会进行 <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 \min\{m,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min\{m,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a50e7d8d8a2fdf94f29f0ed7e8f94a8a7200806b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.669ex; height:2.843ex;" alt="{\displaystyle \min\{m,n\}}"></span> 次。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="计算"><span id=".E8.AE.A1.E7.AE.97"></span>计算</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&action=edit&section=1" title="编辑章节：计算"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="推导"><span id=".E6.8E.A8.E5.AF.BC"></span>推导</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&action=edit&section=2" title="编辑章节：推导"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>设 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{XX}=\operatorname {cov} (X,X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> </msub> <mo>=</mo> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{XX}=\operatorname {cov} (X,X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13e451b2d418294eef9e5788350c9c4ce339e1a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.035ex; height:2.843ex;" alt="{\displaystyle \Sigma _{XX}=\operatorname {cov} (X,X)}"></span> 和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{YY}=\operatorname {cov} (Y,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{YY}=\operatorname {cov} (Y,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f78e782c41004c9a3883e4aca839385c568a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.329ex; height:2.843ex;" alt="{\displaystyle \Sigma _{YY}=\operatorname {cov} (Y,Y)}"></span>。需要最大化的参数为 </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 \rho ={\frac {a'\Sigma _{XY}b}{{\sqrt {a'\Sigma _{XX}a}}{\sqrt {b'\Sigma _{YY}b}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <mi>b</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mo>′</mo> </msup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> </msub> <mi>a</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mo>′</mo> </msup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> </msub> <mi>b</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {a'\Sigma _{XY}b}{{\sqrt {a'\Sigma _{XX}a}}{\sqrt {b'\Sigma _{YY}b}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d60e42ccbeb70058417e2bf31103cc354574c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.996ex; height:6.843ex;" alt="{\displaystyle \rho ={\frac {a'\Sigma _{XY}b}{{\sqrt {a'\Sigma _{XX}a}}{\sqrt {b'\Sigma _{YY}b}}}}.}"></span></dd></dl> <p>第一步是定义一个<a href="/wiki/%E5%9F%BA%E5%8F%98%E6%9B%B4" title="基变更">基变更</a>以及 </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 c=\Sigma _{XX}^{1/2}a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=\Sigma _{XX}^{1/2}a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a33c4208e8f26f34c33c8670da992462973db871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.692ex; height:3.676ex;" alt="{\displaystyle c=\Sigma _{XX}^{1/2}a,}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=\Sigma _{YY}^{1/2}b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=\Sigma _{YY}^{1/2}b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc260f82a84df13e653d7cc4fb53ee3b69e50e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.377ex; height:3.676ex;" alt="{\displaystyle d=\Sigma _{YY}^{1/2}b.}"></span></dd></dl> <p>因此我们有 </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 \rho ={\frac {c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}d}{{\sqrt {c'c}}{\sqrt {d'd}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mo>′</mo> </msup> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>c</mi> <mo>′</mo> </msup> <mi>c</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>d</mi> <mo>′</mo> </msup> <mi>d</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}d}{{\sqrt {c'c}}{\sqrt {d'd}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c775dfeb9534f99dc77680bc0ad609b8fa7cb9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.565ex; height:7.509ex;" alt="{\displaystyle \rho ={\frac {c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}d}{{\sqrt {c'c}}{\sqrt {d'd}}}}.}"></span></dd></dl> <p>根据<a href="/wiki/%E6%9F%AF%E8%A5%BF-%E6%96%BD%E7%93%A6%E8%8C%A8%E4%B8%8D%E7%AD%89%E5%BC%8F" title="柯西-施瓦茨不等式">柯西-施瓦茨不等式</a>，我们有 </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 \left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}\right)d\leq \left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}\Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c\right)^{1/2}\left(d'd\right)^{1/2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mo>′</mo> </msup> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mo>≤</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mo>′</mo> </msup> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>X</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>d</mi> <mo>′</mo> </msup> <mi>d</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}\right)d\leq \left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}\Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c\right)^{1/2}\left(d'd\right)^{1/2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d3d3d4982ae06e64d2e624fc2db107c1110d92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:73.235ex; height:5.343ex;" alt="{\displaystyle \left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}\right)d\leq \left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}\Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c\right)^{1/2}\left(d'd\right)^{1/2},}"></span></dd></dl> <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 \rho \leq {\frac {\left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}c\right)^{1/2}}{\left(c'c\right)^{1/2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mo>′</mo> </msup> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>X</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mo>′</mo> </msup> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \leq {\frac {\left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}c\right)^{1/2}}{\left(c'c\right)^{1/2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ceec321cf159e36d44f6efe37b5fc91d5c941e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.812ex; height:9.343ex;" alt="{\displaystyle \rho \leq {\frac {\left(c'\Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}c\right)^{1/2}}{\left(c'c\right)^{1/2}}}.}"></span></dd></dl> <p>如果向量 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> 和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>X</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47322ff0814d5d690e92e238c7e6a436515061c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.881ex; height:3.676ex;" alt="{\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c}"></span> 共线，那么上式相等。此外，如果 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> 是矩阵 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>X</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc01c692383c5305fce38eadefe22df171525ed0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.856ex; height:3.676ex;" alt="{\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}}"></span> (见<a href="/w/index.php?title=Rayleigh_quotient&action=edit&redlink=1" class="new" title="Rayleigh quotient（页面不存在）">Rayleigh quotient</a>) 最大特征值对应的<a href="/wiki/%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F" class="mw-redirect" title="特征向量">特征向量</a>，那么就可以得到相关的最大值。随后的典型变量对可以通过减少<a href="/wiki/%E7%89%B9%E5%BE%81%E5%80%BC" class="mw-redirect" title="特征值">特征值</a>的量级来得到。正交性保证了相关矩阵的对称性。 </p> <div class="mw-heading mw-heading3"><h3 id="解法"><span id=".E8.A7.A3.E6.B3.95"></span>解法</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&action=edit&section=3" title="编辑章节：解法"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>因此解法是： </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> 是 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>X</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc01c692383c5305fce38eadefe22df171525ed0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.856ex; height:3.676ex;" alt="{\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1/2}}"></span> 的一个特征向量。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> 是 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>X</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47322ff0814d5d690e92e238c7e6a436515061c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.881ex; height:3.676ex;" alt="{\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1/2}c}"></span> 的比例项。</li></ul> <p>相反地，也有： </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> 是 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1}\Sigma _{XY}\Sigma _{YY}^{-1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>X</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1}\Sigma _{XY}\Sigma _{YY}^{-1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e9f003d93cb883819447090a4096e6b63afd9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.149ex; height:3.676ex;" alt="{\displaystyle \Sigma _{YY}^{-1/2}\Sigma _{YX}\Sigma _{XX}^{-1}\Sigma _{XY}\Sigma _{YY}^{-1/2}}"></span> 的一个特征向量。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> 是 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b45ce8c6e6bab5e9637b273fc3d6028f10b1527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.09ex; height:3.676ex;" alt="{\displaystyle \Sigma _{XX}^{-1/2}\Sigma _{XY}\Sigma _{YY}^{-1/2}d}"></span> 的比例项。</li></ul> <p>把坐标反过来，我们有 </p> <ul><li><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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> 是 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{XX}^{-1}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{XX}^{-1}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82849653c09d68b492d67c0a0ddbd03dab6216ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.257ex; height:3.343ex;" alt="{\displaystyle \Sigma _{XX}^{-1}\Sigma _{XY}\Sigma _{YY}^{-1}\Sigma _{YX}}"></span> 的一个特征向量。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> 是 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1}\Sigma _{XY}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>X</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1}\Sigma _{XY}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dba23cfb16bfc2f20ea30d074208b7e9e0b5fb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.257ex; height:3.343ex;" alt="{\displaystyle \Sigma _{YY}^{-1}\Sigma _{YX}\Sigma _{XX}^{-1}\Sigma _{XY}}"></span> 的一个特征向量。</li> <li><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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> 是 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{XX}^{-1}\Sigma _{XY}b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>Y</mi> </mrow> </msub> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{XX}^{-1}\Sigma _{XY}b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48166b247061493414ae98be671b04bde0dcacce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.272ex; height:3.343ex;" alt="{\displaystyle \Sigma _{XX}^{-1}\Sigma _{XY}b}"></span> 的比例项。</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> 是 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{YY}^{-1}\Sigma _{YX}a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>X</mi> </mrow> </msub> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{YY}^{-1}\Sigma _{YX}a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3af11da5c7e0b8f9e792cb9bc78e097fad00b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.212ex; height:3.343ex;" alt="{\displaystyle \Sigma _{YY}^{-1}\Sigma _{YX}a}"></span> 的比例项。</li></ul> <p>那么相关变量定义为： </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 U=c'\Sigma _{XX}^{-1/2}X=a'X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mi>X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>X</mi> <mo>=</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=c'\Sigma _{XX}^{-1/2}X=a'X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7339d19e2c2b4a438d7695f3ac3d3f839899df70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.2ex; height:3.676ex;" alt="{\displaystyle U=c'\Sigma _{XX}^{-1/2}X=a'X}"></span></dd></dl> <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 V=d'\Sigma _{YY}^{-1/2}Y=b'Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mi>Y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mi>Y</mi> <mo>=</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=d'\Sigma _{YY}^{-1/2}Y=b'Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b8f271854e28ad60cd66c88b40220231531b6f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.771ex; height:3.676ex;" alt="{\displaystyle V=d'\Sigma _{YY}^{-1/2}Y=b'Y}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="实现"><span id=".E5.AE.9E.E7.8E.B0"></span>实现</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&action=edit&section=4" title="编辑章节：实现"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>典型相关分析可以用一个相关矩阵的<a href="/wiki/%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3" title="奇异值分解">奇异值分解</a>来解决。<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> 以下是它在一些语言中的函数 <sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/MATLAB" title="MATLAB">MATLAB</a> as <a rel="nofollow" class="external text" href="http://www.mathworks.co.uk/help/stats/canoncorr.html">canoncorr</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20140830030857/http://www.mathworks.co.uk/help/stats/canoncorr.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a href="/w/index.php?title=R_(programming_language)&action=edit&redlink=1" class="new" title="R (programming language)（页面不存在）">R</a> as <a rel="nofollow" class="external text" href="http://stat.ethz.ch/R-manual/R-devel/library/stats/html/cancor.html">cancor</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20200917212423/http://stat.ethz.ch/R-manual/R-devel/library/stats/html/cancor.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） or in <a rel="nofollow" class="external text" href="http://factominer.free.fr/">FactoMineR</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20201109000911/http://factominer.free.fr/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a href="/w/index.php?title=SAS_language&action=edit&redlink=1" class="new" title="SAS language（页面不存在）">SAS</a> as <a rel="nofollow" class="external text" href="https://go.documentation.sas.com/?cdcId=pgmsascdc&cdcVersion=9.4_3.5&docsetId=statug&docsetTarget=statug_cancorr_example.htm&locale=en">The CANCORR Procedure</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20200915021737/https://go.documentation.sas.com/?cdcId=pgmsascdc&cdcVersion=9.4_3.5&docsetId=statug&docsetTarget=statug_cancorr_example.htm&locale=en">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a href="/w/index.php?title=Scikit-Learn&action=edit&redlink=1" class="new" title="Scikit-Learn（页面不存在）">Scikit-Learn</a>, <a href="/wiki/Python" title="Python">Python</a> as <a rel="nofollow" class="external text" href="http://scikit-learn.org/stable/modules/cross_decomposition.html">Cross decomposition</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20200918133514/http://scikit-learn.org/stable/modules/cross_decomposition.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li></ul> <div class="mw-heading mw-heading2"><h2 id="假设检验"><span id=".E5.81.87.E8.AE.BE.E6.A3.80.E9.AA.8C"></span>假设检验</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&action=edit&section=5" title="编辑章节：假设检验"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>每一行可以用下面的方法检测其重要性。由于相关是排好序的，也就是说行 <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> 为 0 意味着所有后续的相关都为 0。如果我们在一个样本中有 <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> 个独立观测，对 <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=1,\dots ,\min\{m,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1,\dots ,\min\{m,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb31b83d0640a3ff10ea0fee351de8d28d8baac7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.911ex; height:2.843ex;" alt="{\displaystyle i=1,\dots ,\min\{m,n\}}"></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 {\widehat {\rho }}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\rho }}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592f706e62b776a40db2f748cfe41f7c00ab888e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.175ex; height:2.676ex;" alt="{\displaystyle {\widehat {\rho }}_{i}}"></span> 是其估计相关。对第 <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> 行，测试统计为： </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 \chi ^{2}=-\left(p-1-{\frac {1}{2}}(m+n+1)\right)\ln \prod _{j=i}^{\min\{m,n\}}(1-{\widehat {\rho }}_{j}^{2}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mi>ln</mi> <mo>⁡</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ρ</mi> <mo>^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ^{2}=-\left(p-1-{\frac {1}{2}}(m+n+1)\right)\ln \prod _{j=i}^{\min\{m,n\}}(1-{\widehat {\rho }}_{j}^{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3feb748522990b15a4d0e22bca2f2342bfc252c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:52.196ex; height:8.009ex;" alt="{\displaystyle \chi ^{2}=-\left(p-1-{\frac {1}{2}}(m+n+1)\right)\ln \prod _{j=i}^{\min\{m,n\}}(1-{\widehat {\rho }}_{j}^{2}),}"></span></dd></dl> <p>上面渐近为一个对大 <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> 有 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m-i+1)(n-i+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m-i+1)(n-i+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adeef2e90e37ee755fb28111e544a8b837c049ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.345ex; height:2.843ex;" alt="{\displaystyle (m-i+1)(n-i+1)}"></span> 个<a href="/wiki/%E8%87%AA%E7%94%B1%E5%BA%A6" class="mw-disambig" title="自由度">自由度</a>的<a href="/wiki/%E5%8D%A1%E6%96%B9%E5%88%86%E5%B8%83" class="mw-redirect" title="卡方分布">卡方分布</a>。<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> 由于所有从 <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 \min\{m,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min\{m,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a50e7d8d8a2fdf94f29f0ed7e8f94a8a7200806b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.669ex; height:2.843ex;" alt="{\displaystyle \min\{m,n\}}"></span> 到 <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> 的相关从逻辑上来说都是 0，所以在这一点之后的乘积都是不相关的。 </p> <div class="mw-heading mw-heading2"><h2 id="实际运用"><span id=".E5.AE.9E.E9.99.85.E8.BF.90.E7.94.A8"></span>实际运用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&action=edit&section=6" title="编辑章节：实际运用"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading2"><h2 id="例子"><span id=".E4.BE.8B.E5.AD.90"></span>例子</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&action=edit&section=7" title="编辑章节：例子"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading2"><h2 id="与principal_angles的连接"><span id=".E4.B8.8Eprincipal_angles.E7.9A.84.E8.BF.9E.E6.8E.A5"></span>与principal angles的连接</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&action=edit&section=8" title="编辑章节：与principal angles的连接"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading2"><h2 id="参见"><span id=".E5.8F.82.E8.A7.81"></span>参见</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&action=edit&section=9" title="编辑章节：参见"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Generalized_Canonical_Correlation&action=edit&redlink=1" class="new" title="Generalized Canonical Correlation（页面不存在）">Generalized Canonical Correlation</a></li> <li><a href="/w/index.php?title=Multilinear_subspace_learning&action=edit&redlink=1" class="new" title="Multilinear subspace learning（页面不存在）">Multilinear subspace learning</a></li> <li><a href="/w/index.php?title=RV_coefficient&action=edit&redlink=1" class="new" title="RV coefficient（页面不存在）">RV coefficient</a></li> <li><a href="/w/index.php?title=Principal_angles&action=edit&redlink=1" class="new" title="Principal angles（页面不存在）">Principal angles</a></li> <li><a href="/wiki/%E4%B8%BB%E6%88%90%E5%88%86%E5%88%86%E6%9E%90" title="主成分分析">主成分分析</a></li> <li><a href="/w/index.php?title=Regularized_canonical_correlation_analysis&action=edit&redlink=1" class="new" title="Regularized canonical correlation analysis（页面不存在）">Regularized canonical correlation analysis</a></li> <li><a href="/wiki/%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3" title="奇异值分解">奇异值分解</a></li> <li><a href="/w/index.php?title=Partial_least_squares_regression&action=edit&redlink=1" class="new" title="Partial least squares regression（页面不存在）">Partial least squares regression</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="参考文献"><span id=".E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE"></span>参考文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&action=edit&section=10" title="编辑章节：参考文献"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation book">Härdle, Wolfgang; Simar, Léopold. Canonical Correlation Analysis. Applied Multivariate Statistical Analysis. 2007: 321–330. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-3-540-72243-4" title="Special:网络书源/978-3-540-72243-4"><span title="国际标准书号">ISBN</span> 978-3-540-72243-4</a>. <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-72244-1_14"><span title="數位物件識別號">doi:10.1007/978-3-540-72244-1_14</span></a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&rft.atitle=Canonical+Correlation+Analysis&rft.au=Simar%2C+L%C3%A9opold&rft.aufirst=Wolfgang&rft.aulast=H%C3%A4rdle&rft.btitle=Applied+Multivariate+Statistical+Analysis&rft.date=2007&rft.genre=bookitem&rft.isbn=978-3-540-72243-4&rft.pages=321-330&rft_id=info%3Adoi%2F10.1007%2F978-3-540-72244-1_14&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation journal">Knapp, T. R. <a rel="nofollow" class="external text" href="https://archive.org/details/sim_psychological-bulletin_1978-03_85_2/page/410">Canonical correlation analysis: A general parametric significance-testing system</a>. Psychological Bulletin. 1978, <b>85</b> (2): 410–416. <a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F0033-2909.85.2.410"><span title="數位物件識別號">doi:10.1037/0033-2909.85.2.410</span></a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&rft.atitle=Canonical+correlation+analysis%3A+A+general+parametric+significance-testing+system&rft.aufirst=T.+R.&rft.aulast=Knapp&rft.date=1978&rft.genre=article&rft.issue=2&rft.jtitle=Psychological+Bulletin&rft.pages=410-416&rft.volume=85&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsim_psychological-bulletin_1978-03_85_2%2Fpage%2F410&rft_id=info%3Adoi%2F10.1037%2F0033-2909.85.2.410&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation journal"><a href="/wiki/Harold_Hotelling" class="mw-redirect" title="Harold Hotelling">Hotelling, H.</a> Relations Between Two Sets of Variates. Biometrika. 1936, <b>28</b> (3–4): 321–377. <a rel="nofollow" class="external text" href="//www.jstor.org/stable/2333955"><span title="JSTOR">JSTOR 2333955</span></a>. <a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbiomet%2F28.3-4.321"><span title="數位物件識別號">doi:10.1093/biomet/28.3-4.321</span></a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&rft.atitle=Relations+Between+Two+Sets+of+Variates&rft.aufirst=H.&rft.aulast=Hotelling&rft.date=1936&rft.genre=article&rft.issue=3%E2%80%934&rft.jtitle=Biometrika&rft.pages=321-377&rft.volume=28&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2333955&rft_id=info%3Adoi%2F10.1093%2Fbiomet%2F28.3-4.321&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation journal">Hsu, D.; Kakade, S. 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Journal of Computer and System Sciences. 2012, <b>78</b> (5): 1460 <span class="reference-accessdate"> [<span class="nowrap">2015-09-10</span>]</span>. <span class="plainlinks"><a rel="nofollow" class="external text" href="//arxiv.org/abs/0811.4413"><span title="arXiv">arXiv:0811.4413</span></a> <span typeof="mw:File"><span title="可免费查阅"><img alt="可免费查阅" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span>. <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jcss.2011.12.025"><span title="數位物件識別號">doi:10.1016/j.jcss.2011.12.025</span></a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20201001073457/https://www.cs.mcgill.ca/~colt2009/papers/011.pdf">存档</a> <span style="font-size:85%;">(PDF)</span>于2020-10-01）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&rft.atitle=A+spectral+algorithm+for+learning+Hidden+Markov+Models&rft.au=Kakade%2C+S.+M.&rft.au=Zhang%2C+T.&rft.aufirst=D.&rft.aulast=Hsu&rft.date=2012&rft.genre=article&rft.issue=5&rft.jtitle=Journal+of+Computer+and+System+Sciences&rft.pages=1460&rft.volume=78&rft_id=http%3A%2F%2Fwww.cs.mcgill.ca%2F~colt2009%2Fpapers%2F011.pdf&rft_id=info%3Aarxiv%2F0811.4413&rft_id=info%3Adoi%2F10.1016%2Fj.jcss.2011.12.025&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation journal">Huang, S. Y.; Lee, M. H.; Hsiao, C. K. <a rel="nofollow" class="external text" href="http://www.stat.sinica.edu.tw/syhuang/papersdownload/KCCA-080906.pdf">Nonlinear measures of association with kernel canonical correlation analysis and applications</a> <span style="font-size:85%;">(PDF)</span>. Journal of Statistical Planning and Inference. 2009, <b>139</b> (7): 2162 <span class="reference-accessdate"> [<span class="nowrap">2015-09-10</span>]</span>. <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jspi.2008.10.011"><span title="數位物件識別號">doi:10.1016/j.jspi.2008.10.011</span></a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20170313203427/http://www.stat.sinica.edu.tw/syhuang/papersdownload/KCCA-080906.pdf">存档</a> <span style="font-size:85%;">(PDF)</span>于2017-03-13）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&rft.atitle=Nonlinear+measures+of+association+with+kernel+canonical+correlation+analysis+and+applications&rft.au=Hsiao%2C+C.+K.&rft.au=Lee%2C+M.+H.&rft.aufirst=S.+Y.&rft.aulast=Huang&rft.date=2009&rft.genre=article&rft.issue=7&rft.jtitle=Journal+of+Statistical+Planning+and+Inference&rft.pages=2162&rft.volume=139&rft_id=http%3A%2F%2Fwww.stat.sinica.edu.tw%2Fsyhuang%2Fpapersdownload%2FKCCA-080906.pdf&rft_id=info%3Adoi%2F10.1016%2Fj.jspi.2008.10.011&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/w/index.php?title=Kanti_V._Mardia&action=edit&redlink=1" class="new" title="Kanti V. Mardia（页面不存在）">Kanti V. Mardia</a>, J. T. Kent and J. M. Bibby. <a rel="nofollow" class="external text" href="https://archive.org/details/multivariateanal0000mard">Multivariate Analysis</a>. <a href="/w/index.php?title=Academic_Press&action=edit&redlink=1" class="new" title="Academic Press（页面不存在）">Academic Press</a>. 1979.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&rft.au=Kanti+V.+Mardia%2C+J.+T.+Kent+and+J.+M.+Bibby&rft.btitle=Multivariate+Analysis&rft.date=1979&rft.genre=book&rft.pub=Academic+Press&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmultivariateanal0000mard&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="外部链接"><span id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5"></span>外部链接</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&action=edit&section=11" title="编辑章节：外部链接"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation journal">Hardoon, D. R.; Szedmak, S.; Shawe-Taylor, J. Canonical Correlation Analysis: An Overview with Application to Learning Methods. Neural Computation. 2004, <b>16</b> (12): 2639–2664. <a rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pubmed/15516276"><span title="公共医学识别码">PMID 15516276</span></a>. <a rel="nofollow" class="external text" href="https://doi.org/10.1162%2F0899766042321814"><span title="數位物件識別號">doi:10.1162/0899766042321814</span></a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%85%B8%E5%9E%8B%E7%9B%B8%E5%85%B3&rft.atitle=Canonical+Correlation+Analysis%3A+An+Overview+with+Application+to+Learning+Methods&rft.au=Shawe-Taylor%2C+J.&rft.au=Szedmak%2C+S.&rft.aufirst=D.+R.&rft.aulast=Hardoon&rft.date=2004&rft.genre=article&rft.issue=12&rft.jtitle=Neural+Computation&rft.pages=2639-2664&rft.volume=16&rft_id=info%3Adoi%2F10.1162%2F0899766042321814&rft_id=info%3Apmid%2F15516276&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://mpra.ub.uni-muenchen.de/12796/">A note on the ordinal canonical-correlation analysis of two sets of ranking scores</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20200918003744/http://mpra.ub.uni-muenchen.de/12796/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） (Also provides a FORTRAN program)- in J. of Quantitative Economics 7(2), 2009, pp. 173-199</li> <li><a rel="nofollow" class="external text" href="http://ssrn.com/abstract=1331886">Representation-Constrained Canonical Correlation Analysis: A Hybridization of Canonical Correlation and Principal Component Analyses</a> (Also provides a FORTRAN program)- in J. of Applied Economic Sciences 4(1), 2009, pp. 115-124</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" 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