GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Korrelationsmatrix</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Korrelationsmatrix</span></h1>

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>In der <a href="Stochastik" title="Stochastik">Stochastik</a> ist die <b>Korrelationsmatrix</b> eine <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrische</a> und <a href="Definitheit" title="Definitheit">positiv semidefinite</a> Matrix, die die <a href="Korrelation" title="Korrelation">Korrelation</a> zwischen den Komponenten eines <a href="Zufallsvektor" title="Zufallsvektor">Zufallsvektors</a> erfasst. Die Korrelationsmatrix kann aus der <a href="Varianz-Kovarianzmatrix" class="mw-redirect" title="Varianz-Kovarianzmatrix">Varianz-Kovarianzmatrix</a> erhalten werden und umgekehrt.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Die Korrelationsmatrix als <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> aller paarweisen <a href="Korrelationskoeffizient" class="mw-redirect" title="Korrelationskoeffizient">Korrelationskoeffizienten</a> der Elemente eines <a href="Zufallsvektor" title="Zufallsvektor">Zufallsvektors</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} =(X_{1},X_{2},\dotsc ,X_{n})^{\top }}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =(X_{1},X_{2},\dotsc ,X_{n})^{\top }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ac18db369fdb6f13891071f4a8c0b6bee50e1c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.75ex; height:3.176ex;" alt="{\displaystyle \mathbf {X} =(X_{1},X_{2},\dotsc ,X_{n})^{\top }}" loading="lazy"></span> enthält Informationen über die <a href="Korrelation" title="Korrelation">Korrelationen</a> zwischen seinen Komponenten.<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> Analog zur <a href="Varianz-Kovarianzmatrix" class="mw-redirect" title="Varianz-Kovarianzmatrix">Varianz-Kovarianzmatrix</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Sigma } }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\Sigma } }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90f99b56fe6ada781ecd0f8a45b6e787b6dfed56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {\Sigma } }" loading="lazy"></span> ist die <i>Korrelationsmatrix</i> definiert als<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>

</p>

</mrow>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} \equiv \operatorname {Corr} (\mathbf {X} )={\begin{pmatrix}\rho _{11}&\rho _{12}&\cdots &\rho _{1n}\\\\\rho _{21}&\rho _{22}&\cdots &\rho _{2n}\\\\\vdots &\vdots &\ddots &\vdots \\\\\rho _{n1}&\rho _{n2}&\cdots &\rho _{nn}\end{pmatrix}}={\begin{pmatrix}1&\rho _{12}&\cdots &\rho _{1n}\\\\\rho _{21}&1&\cdots &\rho _{2n}\\\\\vdots &\vdots &\ddots &\vdots \\\\\rho _{n1}&\rho _{n2}&\cdots &1\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5903c931994f637d4fda56eb03327c90075b4187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.505ex; width:68.763ex; height:24.176ex;" alt="{\displaystyle \mathbf {P} \equiv \operatorname {Corr} (\mathbf {X} )={\begin{pmatrix}\rho _{11}&\rho _{12}&\cdots &\rho _{1n}\\\\\rho _{21}&\rho _{22}&\cdots &\rho _{2n}\\\\\vdots &\vdots &\ddots &\vdots \\\\\rho _{n1}&\rho _{n2}&\cdots &\rho _{nn}\end{pmatrix}}={\begin{pmatrix}1&\rho _{12}&\cdots &\rho _{1n}\\\\\rho _{21}&1&\cdots &\rho _{2n}\\\\\vdots &\vdots &\ddots &\vdots \\\\\rho _{n1}&\rho _{n2}&\cdots &1\end{pmatrix}}}" loading="lazy"></span>,</dd></dl>

<p>wobei <span class="mwe-math-element mwe-math-element-inline"><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 _{ij}=\operatorname {Cov} (X_{i},X_{j})/{\sqrt {\operatorname {Var} (X_{i})\operatorname {Var} (X_{j})}}=\sigma _{ij}/\sigma _{i}\sigma _{j}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msqrt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{ij}=\operatorname {Cov} (X_{i},X_{j})/{\sqrt {\operatorname {Var} (X_{i})\operatorname {Var} (X_{j})}}=\sigma _{ij}/\sigma _{i}\sigma _{j}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4816015b3e29339cbf41df269258c27bf1ab2eac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.36ex; height:4.843ex;" alt="{\displaystyle \rho _{ij}=\operatorname {Cov} (X_{i},X_{j})/{\sqrt {\operatorname {Var} (X_{i})\operatorname {Var} (X_{j})}}=\sigma _{ij}/\sigma _{i}\sigma _{j}}" loading="lazy"></span> der <a href="Korrelationskoeffizient" class="mw-redirect" title="Korrelationskoeffizient">Korrelationskoeffizient</a> zwischen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><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_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{j}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.834ex; height:2.843ex;" alt="{\displaystyle X_{j}}" loading="lazy"></span> ist.

</p><p>Beispielsweise beinhaltet die zweite Zeile von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"></span> die <a href="Korrelation" title="Korrelation">Korrelation</a> von <span class="mwe-math-element mwe-math-element-inline"><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_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ad47c14b8a092f182512e76c96638aea6e3bea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{2}}" loading="lazy"></span> mit jeder anderen <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" 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}" loading="lazy"></span>-Variablen.

Die Korrelationsmatrix in der <a href="Grundgesamtheit" title="Grundgesamtheit">Grundgesamtheit</a> wird als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} _{\rho }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} _{\rho }}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a22f478497d70987c9cdefbd84c519d7b10a30bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.909ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} _{\rho }}" loading="lazy"></span> bzw. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"></span> und die <a href="#Stichproben-Korrelationsmatrix">Stichproben-Korrelationsmatrix</a> als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"></span> bezeichnet. Wenn man die <a href="Diagonalmatrix" title="Diagonalmatrix">Diagonalmatrix</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {D} =\left(\operatorname {diag} ({\boldsymbol {\Sigma }})\right)^{1/2}=\operatorname {diag} (\sigma _{1},\sigma _{2},\dotsc ,\sigma _{n})}">

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {D} =\left(\operatorname {diag} ({\boldsymbol {\Sigma }})\right)^{1/2}=\operatorname {diag} (\sigma _{1},\sigma _{2},\dotsc ,\sigma _{n})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca225e1e8a88852f7749a98f255335fd50927624.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.354ex; height:3.509ex;" alt="{\displaystyle \mathbf {D} =\left(\operatorname {diag} ({\boldsymbol {\Sigma }})\right)^{1/2}=\operatorname {diag} (\sigma _{1},\sigma _{2},\dotsc ,\sigma _{n})}" loading="lazy"></span> definiert, dann erhält man <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"></span> durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Sigma }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Sigma }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8532511177f5a2d2dc2b8c1ea37d483c70266911.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Sigma }}}" loading="lazy"></span> und umgekehrt:

</p>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =\mathbf {D} ^{-1}\,{\boldsymbol {\Sigma }}\,\mathbf {D} ^{-1}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dee6689fad160b62c51fb2e95d8cfe62b8aeb1f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.396ex; height:2.676ex;" alt="{\displaystyle \mathbf {P} =\mathbf {D} ^{-1}\,{\boldsymbol {\Sigma }}\,\mathbf {D} ^{-1}}" loading="lazy"></span></dd></dl>

<p>oder äquivalent

</p>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\Sigma }}=\mathbf {D} \,\mathbf {P} \,\mathbf {D} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e85e1dfcc3a0615de81487dcf3550670fbb9141.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.73ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Sigma }}=\mathbf {D} \,\mathbf {P} \,\mathbf {D} }" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

<ul><li>Sind alle Komponenten des Zufallsvektors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f75966a2f9d5672136fa9401ee1e75008f95ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }" loading="lazy"></span> <a href="Lineare_Unabh%C3%A4ngigkeit" title="Lineare Unabhängigkeit">linear unabhängig</a>, so ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"></span> <a href="Positiv_definit" class="mw-redirect" title="Positiv definit">positiv definit</a>.</li>

<li>Auf der <a href="Hauptdiagonale" title="Hauptdiagonale">Hauptdiagonalen</a> wird die Korrelation der Größen mit sich selbst berechnet. Da der Zusammenhang der Größen strikt linear ist, ist die Korrelation auf der Hauptdiagonalen immer eins.</li>

<li>Bei Stichprobenziehung aus einer <a href="Mehrdimensionale_Normalverteilung" title="Mehrdimensionale Normalverteilung">mehrdimensionalen Normalverteilung</a> ist die Stichproben-Korrelationsmatrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"></span> <a href="Maximum-Likelihood-Methode" title="Maximum-Likelihood-Methode">Maximum-Likelihood-Schätzer</a> der Korrelationsmatrix in der Grundgesamtheit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }" loading="lazy"></span>.<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></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Stichproben-Korrelationsmatrix">Stichproben-Korrelationsmatrix</h2></div>

<p>Eine <i><a href="Sch%C3%A4tzmethode_(Statistik)" title="Schätzmethode (Statistik)">Schätzung</a> der Korrelationsmatrix in der Grundgesamtheit</i> <span class="mwe-math-element mwe-math-element-inline"><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 {\mathbf {P} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbf {P} }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbe845488e68e8ad9d44ad84f33a7fbc7d3d3ebe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.843ex;" alt="{\displaystyle {\widehat {\mathbf {P} }}}" loading="lazy"></span> erhält man, indem man die Korrelationskoeffizienten in der Grundgesamtheit <span class="mwe-math-element mwe-math-element-inline"><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 _{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{ij}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc5b03a345b5d54671b7c896627555d4aa5245be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.679ex; height:2.343ex;" alt="{\displaystyle \rho _{ij}}" loading="lazy"></span> durch die <a href="Korrelationskoeffizient#Empirischer_Korrelationskoeffizient" class="mw-redirect" title="Korrelationskoeffizient">empirischen Korrelationskoeffizienten</a> (ihre empirischen Gegenstücke) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{ij}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/857845aef8b93395ad10279211c6c49180bb8791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.526ex; height:2.343ex;" alt="{\displaystyle r_{ij}}" loading="lazy"></span> ersetzt. Dies führt zur <i>Stichproben-Korrelationsmatrix</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"></span>

</p>

<mtr>

<mtd>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mrow>

</mover>

</mrow>

</mtd>

<mtd>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {R} ={\widehat {\mathbf {P} }}={\widehat {\operatorname {Corr} (\mathbf {X} )}}&={\begin{pmatrix}1&r_{12}&\cdots &r_{1k}\\\\r_{21}&1&\cdots &r_{2k}\\\\\vdots &\vdots &\ddots &\vdots \\\\r_{k1}&r_{k2}&\cdots &1\end{pmatrix}}\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb90ba6a60b6975d631443258671e5b5488889ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.505ex; width:46.012ex; height:24.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {R} ={\widehat {\mathbf {P} }}={\widehat {\operatorname {Corr} (\mathbf {X} )}}&={\begin{pmatrix}1&r_{12}&\cdots &r_{1k}\\\\r_{21}&1&\cdots &r_{2k}\\\\\vdots &\vdots &\ddots &\vdots \\\\r_{k1}&r_{k2}&\cdots &1\end{pmatrix}}\end{aligned}}}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Kovarianzmatrix" title="Kovarianzmatrix">Kovarianzmatrix</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="Ludwig_Fahrmeir" title="Ludwig Fahrmeir">Ludwig Fahrmeir</a>, <a href="Thomas_Kneib" title="Thomas Kneib">Thomas Kneib</a>, Stefan Lang, Brian Marx: <i>Regression: models, methods and applications.</i> Springer Science & Business Media, 2013, ISBN 978-3-642-34332-2, S. 646 ff.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Rencher, Alvin C., und G. Bruce Schaalje: <a rel="nofollow" class="external text" href="https://www.utstat.toronto.edu/~brunner/books/LinearModelsInStatistics.pdf"><i>Linear models in statistics.</i></a> John Wiley & Sons, 2008, S. 77.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Rencher, Alvin C., und G. Bruce Schaalje: <a rel="nofollow" class="external text" href="https://www.utstat.toronto.edu/~brunner/books/LinearModelsInStatistics.pdf"><i>Linear models in statistics.</i></a> John Wiley & Sons, 2008, S. 247.</span>

</li>

</ol>

/* start https://de.wikipedia.org/ */

.mw-parser-output div.klappleiste{border:1px solid var(--dewiki-rahmenfarbe1);clear:both;font-size:95%;box-sizing:border-box;margin-top:1.5em;padding:2px}.mw-parser-output div.klappleiste:after{clear:both;content:"";display:block}.mw-parser-output div.klappleiste-bild{float:left;padding:2px}.mw-parser-output div.klappleiste-kopf{background:var(--dewiki-hintergrundfarbe5);color:var(--color-base,#202122);text-align:center;font-weight:bold}.mw-parser-output div.klappleiste.mw-collapsed .klappleiste-bild{display:none}.mw-parser-output div.klappleiste+div.klappleiste,.mw-parser-output div.klappleiste+link+div.klappleiste,.mw-parser-output div.klappleiste+link+link+div.klappleiste,.mw-parser-output div.klappleiste+link+style+div.klappleiste,.mw-parser-output div.klappleiste+style+div.klappleiste,.mw-parser-output div.klappleiste+style+style+div.klappleiste,.mw-parser-output div.klappleiste+style+link+div.klappleiste{margin-top:-1px}@media screen{html.skin-theme-clientpref-night .mw-parser-output .klappleiste-bild span[typeof="mw:File"]:not(.skin-invert-image) img{background-color:#c8ccd1}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .klappleiste-bild span[typeof="mw:File"]:not(.skin-invert-image) img{background-color:#c8ccd1}}

/* end https://de.wikipedia.org/ */

</style>

<div class="klappleiste-kopf">Spezielle Matrizen in der Statistik</div>

<p><a href="Datenmatrix" title="Datenmatrix">Datenmatrix</a> |

<a href="Produktsummenmatrix" title="Produktsummenmatrix">Produktsummenmatrix</a> |

<a href="Pr%C3%A4diktionsmatrix" title="Prädiktionsmatrix">Prädiktionsmatrix</a> |

<a href="Residuenerzeugende_Matrix" class="mw-redirect" title="Residuenerzeugende Matrix">residuenerzeugende Matrix</a> |

zentrierende Matrix |

<a href="Kovarianzmatrix" title="Kovarianzmatrix">Kovarianzmatrix</a> |

<a class="mw-selflink selflink">Korrelationsmatrix</a> |

<a href="Pr%C3%A4zisionsmatrix" class="mw-redirect" title="Präzisionsmatrix">Präzisionsmatrix</a> |

Gewichtsmatrix |

Restriktionsmatrix |

<a href="Fisher-Informationsmatrix" class="mw-redirect" title="Fisher-Informationsmatrix">Fisher-Informationsmatrix</a> |

Bernoulli-Matrix |

<a href="Leslie-Matrix" title="Leslie-Matrix">Leslie-Matrix</a> |

<a href="Zufallsmatrix" title="Zufallsmatrix">Zufallsmatrix</a> |

<a href="%C3%9Cbergangsmatrix" title="Übergangsmatrix">Übergangsmatrix</a>

</p>

</div></div></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-03-05" href="https://de.wikipedia.org/wiki/?title=Korrelationsmatrix&oldid=253932658">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>