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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Linearer Prädiktor</span></h1>
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<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="Statistik" title="Statistik">Statistik</a> und dort insbesondere in der parametrischen <a href="Regressionsanalyse" title="Regressionsanalyse">Regressionsanalyse</a> ist ein <b>linearer Prädiktor</b> eine <a href="Linearkombination" title="Linearkombination">Linearkombination</a> einer Reihe von <a href="Koeffizient" title="Koeffizient">Koeffizienten</a> (<a href="Regressionskoeffizient" class="mw-redirect" title="Regressionskoeffizient">Regressionskoeffizienten</a>) und erklärenden Variablen (<a href="Unabh%C3%A4ngige_Variable" class="mw-redirect" title="Unabhängige Variable">unabhängige Variablen</a>), deren Wert zur Vorhersage (Prädiktion) einer <a href="Antwortvariable" class="mw-redirect" title="Antwortvariable">Antwortvariablen</a> verwendet wird. Diese <i>additiv-lineare systematische Komponente</i> ist ein Hauptbestandteil von <a href="Lineare_Regression" title="Lineare Regression">linearen Regressionsmodellen</a>.
</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>In der parametrischen Regressionsanalyse wird mittels mehrerer Regressionsparameter ein Suchraum aus potenziellen Regressionsfunktionen gebildet. Im Anschluss soll diejenige Parameterkonfiguration bestimmt werden, die die höchste <a href="Anpassungsg%C3%BCte" title="Anpassungsgüte">Anpassungsgüte</a> für die beobachteten Werte der Antwortvariablen und erklärenden Variablen liefert. Die wichtigsten Modellklassen der parametrischen Regressionsanalyse sind zum einen die Klasse der <a href="Lineares_Modell" title="Lineares Modell">linearen Modelle</a> und zum anderen die Klasse der <a href="Verallgemeinerte_lineare_Modelle" title="Verallgemeinerte lineare Modelle">verallgemeinerten linearen Modelle</a>. Das Beiwort „linear“ resultiert daraus, dass die beiden Modellklassen auf dem <i>linearen Prädiktor</i> aufbauen, der wie folgt definiert ist
</p>
<dl><dd><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 \eta _{i}\colon =x_{i0}\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\ldots +x_{ik}\beta _{k}=\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}}">
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<annotation encoding="application/x-tex">{\displaystyle \eta _{i}\colon =x_{i0}\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\ldots +x_{ik}\beta _{k}=\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5dc875773ffa54a83529446e69a85ef72f8f8e21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:55.765ex; height:4.676ex;" alt="{\displaystyle \eta _{i}\colon =x_{i0}\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\ldots +x_{ik}\beta _{k}=\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}}" loading="lazy"></span>.</dd></dl>
<p>Dieser linearen Prädiktor wird aus den erklärenden Variablen <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_{i0},x_{i1},\ldots ,x_{ik}}">
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<annotation encoding="application/x-tex">{\displaystyle x_{i0},x_{i1},\ldots ,x_{ik}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/241a1e313ea64b13b965c96fcd2773d83796c058.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.101ex; height:2.009ex;" alt="{\displaystyle x_{i0},x_{i1},\ldots ,x_{ik}}" loading="lazy"></span> und den festen, aber unbekannten <a href="Regressionsparameter" title="Regressionsparameter">Regressionsparametern</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 \beta _{0},\beta _{1},\beta _{2},\ldots ,\beta _{k}}">
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<annotation encoding="application/x-tex">{\displaystyle \beta _{0},\beta _{1},\beta _{2},\ldots ,\beta _{k}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8708d330b81f038fe858ee2b04615c91fa89f60d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.761ex; height:2.509ex;" alt="{\displaystyle \beta _{0},\beta _{1},\beta _{2},\ldots ,\beta _{k}}" loading="lazy"></span> gebildet, 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 x_{i0}}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92b936b313824ef06d94c317308bc63fcdbb028d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.951ex; height:2.009ex;" alt="{\displaystyle x_{i0}}" loading="lazy"></span> für gewöhnlich gleich eins gesetzt wird (<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_{i0}\equiv 1}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb8341ad8d0b498a728a3460731a1777490a2c9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.212ex; height:2.509ex;" alt="{\displaystyle x_{i0}\equiv 1}" loading="lazy"></span>). Der Parameter <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 \beta _{0}}">
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<annotation encoding="application/x-tex">{\displaystyle \beta _{0}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40b42f71f244103a8fca3c76885c7580a92831c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{0}}" loading="lazy"></span> ist somit der Achsenabschnitt der Regressionsgerade bzw. genauer „Regressions<a href="Hyperebene" title="Hyperebene">hyperebene</a>“. Er bestimmt das Niveau des linearen Prädiktors und wird folglich auch Niveauparameter genannt. In der Regressionsanalyse geht es darum den Achsenabschnitt <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 \beta _{0}}">
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<annotation encoding="application/x-tex">{\displaystyle \beta _{0}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40b42f71f244103a8fca3c76885c7580a92831c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.37ex; height:2.509ex;" alt="{\displaystyle \beta _{0}}" loading="lazy"></span>, die Steigungsparameter <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 \beta _{1},\beta _{2},\ldots ,\beta _{k}}">
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<annotation encoding="application/x-tex">{\displaystyle \beta _{1},\beta _{2},\ldots ,\beta _{k}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fe16a967fc6c291172e7373b97c8be03b58c27c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.357ex; height:2.509ex;" alt="{\displaystyle \beta _{1},\beta _{2},\ldots ,\beta _{k}}" loading="lazy"></span> und die Varianz der Störgrößen zu schätzen.<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>
</p>
<div class="mw-heading mw-heading2"><h2 id="Lineare_Modelle_vs._verallgemeinerte_lineare_Modelle">Lineare Modelle vs. verallgemeinerte lineare Modelle</h2></div>
<p><a href="Lineares_Modell" title="Lineares Modell">Lineare Modelle</a> gehen vom folgenden Zusammenhang zwischen der Regressionsfunktion und dem linearen Prädiktor aus
</p>
<dl><dd><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 f(x_{i1},x_{i2},\ldots ,x_{ik})=\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}=\eta _{i}}">
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<annotation encoding="application/x-tex">{\displaystyle f(x_{i1},x_{i2},\ldots ,x_{ik})=\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}=\eta _{i}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b67b504029d255a59435262c0eacecf96b73c33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:38.125ex; height:4.676ex;" alt="{\displaystyle f(x_{i1},x_{i2},\ldots ,x_{ik})=\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}=\eta _{i}}" loading="lazy"></span>.</dd></dl>
<p><a href="Verallgemeinerte_lineare_Modelle" title="Verallgemeinerte lineare Modelle">Verallgemeinerte lineare Modelle</a> dagegen gehen von aus, dass der <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a> der <a href="Einflussgr%C3%B6%C3%9Fe_und_Zielgr%C3%B6%C3%9Fe" class="mw-redirect" title="Einflussgröße und Zielgröße">Antwortvariablen</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 \mu =\operatorname {E} (Y_{i})}">
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<annotation encoding="application/x-tex">{\displaystyle \mu =\operatorname {E} (Y_{i})}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f427f48cc8cd11e8dd48db8888eefadaf6f44bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.043ex; height:2.843ex;" alt="{\displaystyle \mu =\operatorname {E} (Y_{i})}" loading="lazy"></span> erst durch eine geeignete invertierbare <a href="Kopplungsfunktion" title="Kopplungsfunktion">Kopplungsfunktion</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 g(\cdot )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(\cdot )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/421d5ab89ea154ac75586bdfb687db2160d1ea33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.572ex; height:2.843ex;" alt="{\displaystyle g(\cdot )}" loading="lazy"></span> die Form eines linearen Prädiktors annimmt<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>
<dl><dd><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 g(\mu )=\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}=\eta _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msubsup>
<mo movablelimits="false">∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msubsup>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>η<!-- η --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(\mu )=\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}=\eta _{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a06ebfb390f6c3edc25658f87eb9b872633d4a8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:24.264ex; height:4.676ex;" alt="{\displaystyle g(\mu )=\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}=\eta _{i}}" loading="lazy"></span>.</dd></dl>
<p>Mit der <a href="Umkehrfunktion" title="Umkehrfunktion">Umkehrfunktion</a> der Kopplungsfunktion, der <a href="Kopplungsfunktion#Antwortfunktion" title="Kopplungsfunktion">Antwortfunktion</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 h(\cdot )=g^{-1}(\cdot )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo stretchy="false">(</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h(\cdot )=g^{-1}(\cdot )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42353f038af01180cec42faab67e8e379c3437bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.801ex; height:3.176ex;" alt="{\displaystyle h(\cdot )=g^{-1}(\cdot )}" loading="lazy"></span> ergibt sich für die Regressionsfunktion in diesem Fall
</p>
<dl><dd><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 f(x_{i1},x_{i2},\ldots ,x_{ik})=h\left(\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}\right)=h(\eta _{i})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>h</mi>
<mrow>
<mo>(</mo>
<mrow>
<msubsup>
<mo movablelimits="false">∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msubsup>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>h</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>η<!-- η --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x_{i1},x_{i2},\ldots ,x_{ik})=h\left(\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}\right)=h(\eta _{i})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e68ab5f7528b144e09666cca2bf6589000c4db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.775ex; height:4.843ex;" alt="{\displaystyle f(x_{i1},x_{i2},\ldots ,x_{ik})=h\left(\sum \nolimits _{j=0}^{k}x_{ij}\beta _{j}\right)=h(\eta _{i})}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Vektor-Matrix-Schreibweise">Vektor-Matrix-Schreibweise</h2></div>
<p>Mittels <a href="Multiple_lineare_Regression#Das_klassische_Modell_der_linearen_Mehrfachregression" title="Multiple lineare Regression">Vektor-Matrix-Schreibweise</a> lässt sich der lineare Prädiktor wie folgt schreiben:
</p>
<dl><dd><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 \eta _{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}\quad }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>η<!-- η --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mn>2</mn>
</mrow>
</msub>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<mo>…<!-- … --></mo>
<mo>+</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">⊤<!-- ⊤ --></mi>
</mrow>
</msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">β<!-- β --></mi>
</mrow>
<mspace width="1em"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \eta _{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}\quad }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5cedd86fab7536432ee9364c359518ea4891e122.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.418ex; height:3.176ex;" alt="{\displaystyle \eta _{i}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}\quad }" loading="lazy"></span>, 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 \quad \mathbf {x} _{i}^{\top }=(1,x_{i1},\ldots ,x_{ik})_{(k+1\times 1)}\quad }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="1em"></mspace>
<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">⊤<!-- ⊤ --></mi>
</mrow>
</msubsup>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo>×<!-- × --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msub>
<mspace width="1em"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \quad \mathbf {x} _{i}^{\top }=(1,x_{i1},\ldots ,x_{ik})_{(k+1\times 1)}\quad }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72bdbab7b4df972df51f5a2db6aa7efb1c87e616.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.356ex; height:3.343ex;" alt="{\displaystyle \quad \mathbf {x} _{i}^{\top }=(1,x_{i1},\ldots ,x_{ik})_{(k+1\times 1)}\quad }" 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 \quad {\boldsymbol {\beta }}=(\beta _{0},\beta _{1},\ldots ,\beta _{k})_{(k+1\times 1)}^{\top }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="1em"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">β<!-- β --></mi>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>β<!-- β --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msubsup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo>×<!-- × --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">⊤<!-- ⊤ --></mi>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \quad {\boldsymbol {\beta }}=(\beta _{0},\beta _{1},\ldots ,\beta _{k})_{(k+1\times 1)}^{\top }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d95a4f3bc755d80dae5c7a4d93c10db1bab74a68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:28.69ex; height:3.676ex;" alt="{\displaystyle \quad {\boldsymbol {\beta }}=(\beta _{0},\beta _{1},\ldots ,\beta _{k})_{(k+1\times 1)}^{\top }}" loading="lazy"></span>.</dd></dl>
<p>Hierbei 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 {\boldsymbol {\beta }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">β<!-- β --></mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/702cafc420cc00c54896f6d125112820956aaf6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}" loading="lazy"></span> ein <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 (k+1)\times 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (k+1)\times 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb0324bd0379f701eeac993029b3df27555519ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.026ex; height:2.843ex;" alt="{\displaystyle (k+1)\times 1}" loading="lazy"></span>-<a href="Spaltenvektor" class="mw-redirect" title="Spaltenvektor">Spaltenvektor</a> 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 \mathbf {x} _{i}^{\top }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">⊤<!-- ⊤ --></mi>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{i}^{\top }}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0365d061e61345d67d4ca911d5f6ac0bf224ea7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.922ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} _{i}^{\top }}" loading="lazy"></span> ist ein <a href="Transponierte_Matrix" title="Transponierte Matrix">transponierter</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 (k+1)\times 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (k+1)\times 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb0324bd0379f701eeac993029b3df27555519ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.026ex; height:2.843ex;" alt="{\displaystyle (k+1)\times 1}" loading="lazy"></span>-Spaltenvektor, sodass das Produkt <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} _{i}^{\top }{\boldsymbol {\beta }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">⊤<!-- ⊤ --></mi>
</mrow>
</msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">β<!-- β --></mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8718cd6300d70c27c8b5d38584676968a99b7d5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.456ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}}" loading="lazy"></span> eine <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 1\times 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>×<!-- × --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1\times 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b4bf91a527dc01af9ef6ace81199becf1308e00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 1\times 1}" loading="lazy"></span>-Matrix bzw. ein <a href="Skalar_(Mathematik)" title="Skalar (Mathematik)">Skalar</a> ergibt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Verwendung_in_der_linearen_Regression">Verwendung in der linearen Regression</h2></div>
<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→&nbsp;</span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Lineare_Regression" title="Lineare Regression">Lineare Regression</a></i></div>
<p>Ein Beispiel für die Verwendung eines linearen Prädiktors ist die <a href="Lineare_Regression" title="Lineare Regression">lineare Regression</a>, bei der jeder die Beziehung zwischen erklärenden Variablen und Antwortvariablen durch eine additive Störgröße überlagert wird. In der <a href="Multiple_lineare_Regression" title="Multiple lineare Regression">multiple lineare Regression</a> lässt sich der Zusammenhang wie folgt schreiben:
</p>
<dl><dd><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 Y_{i}=\eta _{i}+\varepsilon _{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}+\varepsilon _{i}}">
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<mi>Y</mi>
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<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle Y_{i}=\eta _{i}+\varepsilon _{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}+\varepsilon _{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98f8c69b22ec8487d87a4a6dbac518e21fb66bc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.205ex; height:3.176ex;" alt="{\displaystyle Y_{i}=\eta _{i}+\varepsilon _{i}=\mathbf {x} _{i}^{\top }{\boldsymbol {\beta }}+\varepsilon _{i}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Torsten Becker et al.: <i>Stochastische Risikomodellierung und statistische Methoden.</i> Springer Spektrum, 2016. S.&nbsp;288.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Torsten Becker et al.: <i>Stochastische Risikomodellierung und statistische Methoden.</i> Springer Spektrum, 2016. S.&nbsp;288.</span>
</li>
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