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<title>Prädiktionsmatrix</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Prädiktionsmatrix</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">In der <a href="Statistik" title="Statistik">Statistik</a> ist die Prädiktionsmatrix (<a href="Englische_Sprache" title="Englische Sprache">englisch</a> prediction matrix) eine <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrische</a> und <a href="Idempotenz" title="Idempotenz">idempotente</a> Matrix und damit eine <a href="Projektionsmatrix_(Statistik)" title="Projektionsmatrix (Statistik)">Projektionsmatrix</a>. Die Prädiktionsmatrix wird gelegentlich Hut-Matrix oder Dach-Matrix genannt, da sie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {y}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dc8de3d8ea01304329ef9518fad7a6d196c4c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\hat {y}}}" loading="lazy"> abbildet. Dementsprechend wird sie entweder mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

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

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</math><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"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f017b876ed763037d8818ec5dfbbdc6703e0f683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {H} }" loading="lazy"> notiert. Der Begriff „Prädiktionsmatrix“ bzw. „Vorhersagematrix“ wurde von Hoaglin & Welsh (1978)<a href="#cite_note-1">[1]</a> sowie Chatterjee & Hadi (1986)<a href="#cite_note-Chatterjee&Hadi-2">[2]</a> geprägt und rührt daher, dass wenn man die Matrix auf die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">-Werte anwendet sie die vorhergesagten Werte (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {y}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dc8de3d8ea01304329ef9518fad7a6d196c4c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\hat {y}}}" loading="lazy">-Werte) generiert.<a href="#cite_note-Chatterjee&Hadi-2">[2]</a> Eine weitere in der Statistik wichtige Matrix ist die <a href="Residuenerzeugende_Matrix" class="mw-redirect" title="Residuenerzeugende Matrix">Residualmatrix</a>, die durch die Prädiktionsmatrix definiert wird und ebenfalls eine Projektionsmatrix ist.

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

Gegeben ein typisches <a href="Multiple_lineare_Regression" title="Multiple lineare Regression">multiples lineares Regressionsmodell</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d0cb6f6c07811a61ec22849b4521fd0363878ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.133ex; height:2.509ex;" alt="{\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }}}" loading="lazy">, mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}}">

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

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</math><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"> dem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\times 1}">

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<annotation encoding="application/x-tex">{\displaystyle p\times 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9b3ff9128b8bc9ccf1c3b9a3ba1d253b95f5754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p\times 1}" loading="lazy"> Vektor der unbekannten <a href="Regressionsparameter" title="Regressionsparameter">Regressionsparameter</a>, der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times p}">

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<annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43ad58cdd60e9b0ab2bec828151c740accf92028.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}" loading="lazy"> <a href="Versuchsplanmatrix" class="mw-redirect" title="Versuchsplanmatrix">Versuchsplanmatrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><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">, dem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times 1}">

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<annotation encoding="application/x-tex">{\displaystyle n\times 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d24148f103e1cccb60addeeb0a64cb1c3d5622e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle n\times 1}" loading="lazy"> Vektor der abhängigen Variablen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb25a040b592282dc2a254c3117e792c3c81161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }" loading="lazy"> und dem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times 1}">

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<annotation encoding="application/x-tex">{\displaystyle n\times 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d24148f103e1cccb60addeeb0a64cb1c3d5622e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle n\times 1}" loading="lazy"> Vektor der <a href="St%C3%B6rgr%C3%B6%C3%9Fe_und_Residuum" title="Störgröße und Residuum">Störgrößen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\varepsilon }}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8445af5ff7da70714382bc35e78bedcacf68e825.svg" 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 {\boldsymbol {\varepsilon }}}" loading="lazy">. Dann ist die Prädiktionsmatrix definiert durch

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} \equiv \mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\quad }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee9393214ef14cf912356fc96d6793d8a4dc439d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.81ex; height:3.843ex;" alt="{\displaystyle \mathbf {P} \equiv \mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\quad }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \mathbf {P} \in \mathbb {R} ^{n\times n}}">

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<annotation encoding="application/x-tex">{\displaystyle \quad \mathbf {P} \in \mathbb {R} ^{n\times n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97c571f54e8d0873acd556773e91ebbadb72f76e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.151ex; height:2.343ex;" alt="{\displaystyle \quad \mathbf {P} \in \mathbb {R} ^{n\times n}}" loading="lazy">.</dd></dl>

Die Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} ^{+}=\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} ^{+}=\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/623ee35f5209ff6168bf19950652418065939ade.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.171ex; height:3.843ex;" alt="{\displaystyle \mathbf {X} ^{+}=\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }}" loading="lazy"> wird auch <a href="Moore-Penrose-Inverse" class="mw-redirect" title="Moore-Penrose-Inverse">Moore-Penrose-Inverse</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><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"> genannt.

Die mithilfe der <a href="Methode_der_kleinsten_Quadrate" title="Methode der kleinsten Quadrate">Methode der kleinsten Quadrate</a> geschätzte Regressions(hyper)ebene ist dann gegeben durch die <a href="Stichproben-Regressionsfunktion" title="Stichproben-Regressionsfunktion">Stichproben-Regressionsfunktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {y} }}={\widehat {\operatorname {E} (\mathbf {y} )}}=\mathbf {X} {\hat {\boldsymbol {\beta }}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {y} }}={\widehat {\operatorname {E} (\mathbf {y} )}}=\mathbf {X} {\hat {\boldsymbol {\beta }}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98275c4771e0aeec7b95389d7efc822dfe0eb440.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.001ex; height:3.676ex;" alt="{\displaystyle {\hat {\mathbf {y} }}={\widehat {\operatorname {E} (\mathbf {y} )}}=\mathbf {X} {\hat {\boldsymbol {\beta }}}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\beta }}}=\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\mathbf {y} }">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\beta }}}=\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\mathbf {y} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90731ac231a74b2370427e0db0db3ad31cf95d60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.623ex; height:3.843ex;" alt="{\displaystyle {\hat {\boldsymbol {\beta }}}=\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\mathbf {y} }" loading="lazy"> der Kleinste-Quadrate-Schätzvektor ist. Die Prädiktionsmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

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

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</math><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"> ist die Matrix der <a href="Orthogonalprojektion#Lineare_Algebra" title="Orthogonalprojektion">Orthogonalprojektion</a> auf den Spaltenraum von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><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"> und hat maximal den Rang <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" 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}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=k+1}">

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<annotation encoding="application/x-tex">{\displaystyle p=k+1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1be43960847699b78dbedcfe85bc22ac5d359245.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.571ex; height:2.509ex;" alt="{\displaystyle p=k+1}" loading="lazy"> ist die Anzahl der Parameter des Regressionsmodells). Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} }">

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

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</math><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"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n\times p)}">

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<annotation encoding="application/x-tex">{\displaystyle (n\times p)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3510e16b6a508efeed780b3a67fc4b283f819be4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.214ex; height:2.843ex;" alt="{\displaystyle (n\times p)}" loading="lazy"> Matrix mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Rang} (\mathbf {X} )=p}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Rang} (\mathbf {X} )=p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76b481b952d20707327121edf544faa1833eae84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.425ex; height:2.843ex;" alt="{\displaystyle \operatorname {Rang} (\mathbf {X} )=p}" loading="lazy"> ist, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Rang} (\mathbf {P} )=p}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Rang} (\mathbf {P} )=p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac6ba790d065ce3c1f3a91aa5c2765876fb97be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.232ex; height:2.843ex;" alt="{\displaystyle \operatorname {Rang} (\mathbf {P} )=p}" loading="lazy">. Da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

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

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</math><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"> eine Projektionsmatrix ist, gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Rang} (\mathbf {P} )=\operatorname {Spur} (\mathbf {P} )=p}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Rang} (\mathbf {P} )=\operatorname {Spur} (\mathbf {P} )=p}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ef269c22bf8591dbdfdc38c813d500068ad48b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.756ex; height:2.843ex;" alt="{\displaystyle \operatorname {Rang} (\mathbf {P} )=\operatorname {Spur} (\mathbf {P} )=p}" loading="lazy">. Die Idempotenz- und die Symmetrieeigenschaft (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} \cdot \mathbf {P} =\mathbf {P} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8aa5abf6e80e500422fe9646df3483070a5d4c0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.258ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} \cdot \mathbf {P} =\mathbf {P} }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ^{\top }=\mathbf {P} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b9a087e209eda2d3e32f5dacaa9722be3aa5bf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.263ex; height:2.676ex;" alt="{\displaystyle \mathbf {P} ^{\top }=\mathbf {P} }" loading="lazy">) implizieren, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

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

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</math><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"> ein <a href="Orthogonaler_Projektor" class="mw-redirect" title="Orthogonaler Projektor">orthogonaler Projektor</a> auf den Spaltenraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {X} )=S(\mathbf {P} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15936ad8687c697aad53344db904f4f166f85d4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.562ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {X} )=S(\mathbf {P} )}" loading="lazy"> ist.<a href="#cite_note-3">[3]</a> Die Projektionsrichtung ergibt sich aus der Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {I} -\mathbf {P} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/599f59346714ff1cabe276694342e7acd2646516.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.49ex; height:2.843ex;" alt="{\displaystyle (\mathbf {I} -\mathbf {P} )}" loading="lazy">, deren Spalten senkrecht auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\mathbf {X} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c417bb3820fa689dd5a33a0462a3a6043d180dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle S(\mathbf {X} )}" loading="lazy"> stehen. Die Matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

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

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</math><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"> wird Prädiktionsmatrix genannt, da sich die Vorhersagewerte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {y} }}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3826fc591cfe42348f58b15e4c23152f30931ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.676ex;" alt="{\displaystyle {\hat {\mathbf {y} }}}" loading="lazy"> durch die linksseitige Multiplikation des Vektors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb25a040b592282dc2a254c3117e792c3c81161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }" loading="lazy"> mit dieser Matrix ergeben. Dies kann durch Einsetzen des KQ-Parameterschätzers wie folgt gezeigt werden:<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {y} }}=\mathbf {X} {\hat {\boldsymbol {\beta }}}=\underbrace {\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }} _{=\mathbf {P} }\mathbf {y} =\mathbf {P} \mathbf {y} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a94739e6175bea68504cc0a0d7a00be8caae264.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:34.507ex; height:7.343ex;" alt="{\displaystyle {\hat {\mathbf {y} }}=\mathbf {X} {\hat {\boldsymbol {\beta }}}=\underbrace {\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }} _{=\mathbf {P} }\mathbf {y} =\mathbf {P} \mathbf {y} }" loading="lazy">.</dd></dl>

Die Vorhersagewerte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> (die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {y}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dc8de3d8ea01304329ef9518fad7a6d196c4c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\hat {y}}}" loading="lazy">-Werte) können also als eine Funktion der beobachteten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">-Werte verstanden werden. Zahlreiche statistische Resultate lassen sich auch mit der Prädiktionsmatrix darstellen. Beispielsweise lässt sich der Residualvektor mittels der Prädiktionsmatrix darstellen als: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\boldsymbol {\varepsilon }}}=\mathbf {y} -{\hat {\mathbf {y} }}=\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}}=(\mathbf {I} -\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top })\mathbf {y} =(\mathbf {I} -\mathbf {P} )\mathbf {y} }">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\boldsymbol {\varepsilon }}}=\mathbf {y} -{\hat {\mathbf {y} }}=\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}}=(\mathbf {I} -\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top })\mathbf {y} =(\mathbf {I} -\mathbf {P} )\mathbf {y} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c016b46ced702fa5dd5a98f2e06246f8237847cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:58.854ex; height:3.843ex;" alt="{\displaystyle {\hat {\boldsymbol {\varepsilon }}}=\mathbf {y} -{\hat {\mathbf {y} }}=\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}}=(\mathbf {I} -\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top })\mathbf {y} =(\mathbf {I} -\mathbf {P} )\mathbf {y} }" loading="lazy">.<a href="#cite_note-5">[5]</a> Die (nichttriviale) <a href="Kovarianzmatrix" title="Kovarianzmatrix">Kovarianzmatrix</a> des Residualvektors lautet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Cov} ({\hat {\boldsymbol {\varepsilon }}})=\sigma ^{2}(\mathbf {I} -\mathbf {P} )}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Cov} ({\hat {\boldsymbol {\varepsilon }}})=\sigma ^{2}(\mathbf {I} -\mathbf {P} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03567648effc23a2f6e8daa2c275dbf93e9232e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.269ex; height:3.176ex;" alt="{\displaystyle \operatorname {Cov} ({\hat {\boldsymbol {\varepsilon }}})=\sigma ^{2}(\mathbf {I} -\mathbf {P} )}" loading="lazy"> und spielt für die Analyse von <a href="#Hebelwerte">Hebelwerten</a> eine Rolle.

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

<div class="mw-heading mw-heading3"><h3 id="Idempotenz">Idempotenz</h3></div>

Die Prädiktionsmatrix ist idempotent. Dies kann so interpretiert werden, dass „zweimaliges Anwenden der Regression zum gleichen Ergebnis führt“. Die Idempotenzeigenschaft der Prädiktionsmatrix kann wie folgt gezeigt werden:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{2}=\mathbf {P} \cdot \mathbf {P} =\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }=\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {I} \mathbf {X} ^{\top }=\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }=\mathbf {P} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fc10ee8faba1e6dc34e0242af64a8bc75816d9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:88.795ex; height:3.843ex;" alt="{\displaystyle \mathbf {P} ^{2}=\mathbf {P} \cdot \mathbf {P} =\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }=\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {I} \mathbf {X} ^{\top }=\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }=\mathbf {P} }" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} }">

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</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a458c8aeb096ce732abf346ae8edf3e4f53a126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }" loading="lazy"> die <a href="Einheitsmatrix" title="Einheitsmatrix">Einheitsmatrix</a> ist.

<div class="mw-heading mw-heading3"><h3 id="Symmetrie">Symmetrie</h3></div>

Die Prädiktionsmatrix ist symmetrisch. Die Symmetrieeigenschaft der Prädiktionsmatrix kann wie folgt gezeigt werden

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{\top }=\left(\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\right)^{\top }=\left(\left(\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\right)\left(\mathbf {X} ^{\top }\right)\right)^{\top }=\ \left(\mathbf {X} ^{\top }\right)^{\top }\left(\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\right)^{\top }=\ \mathbf {X} \left(\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\right)^{\top }\mathbf {X} ^{\top }=\ \mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }=\mathbf {P} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e2f6458c983ace7fa5aff4633db433325c3f3bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:129.386ex; height:5.176ex;" alt="{\displaystyle \mathbf {P} ^{\top }=\left(\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }\right)^{\top }=\left(\left(\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\right)\left(\mathbf {X} ^{\top }\right)\right)^{\top }=\ \left(\mathbf {X} ^{\top }\right)^{\top }\left(\mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\right)^{\top }=\ \mathbf {X} \left(\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\right)^{\top }\mathbf {X} ^{\top }=\ \mathbf {X} \left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {X} ^{\top }=\mathbf {P} }" loading="lazy"></dd></dl>

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

Die Diagonalelemente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{ii}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb3ec9e30b6a7d6c6bb9150d55855583a7ee24c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.626ex; height:2.009ex;" alt="{\displaystyle p_{ii}}" loading="lazy"> der Prädiktionsmatrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }">

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

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</math><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"> können als Hebelwerte interpretiert werden und spielen in der <a href="Regressionsdiagnostik" title="Regressionsdiagnostik">Regressionsdiagnostik</a> eine große Rolle. Sie sind gegeben durch

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<annotation encoding="application/x-tex">{\displaystyle p_{ii}=\mathbf {x} _{i}^{\top }\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {x} _{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa04a03d12fee3f0bed8dce67091bf6ed386444b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:20.869ex; height:3.843ex;" alt="{\displaystyle p_{ii}=\mathbf {x} _{i}^{\top }\left(\mathbf {X} ^{\top }\mathbf {X} \right)^{-1}\mathbf {x} _{i}}" loading="lazy">.</dd></dl>

Diese Hebelwerte werden bei der Berechnung des <a href="Cook-Abstand" title="Cook-Abstand">Cook-Abstands</a> verwendet und können genutzt werden, um einflussreiche Beobachtungen zu identifizieren. Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n}}\leq p_{ii}\leq {\frac {1}{r}}}">

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}\leq p_{ii}\leq {\frac {1}{r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dca551dadccd6c874463d2e68fdd4b599db4b9b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.963ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{n}}\leq p_{ii}\leq {\frac {1}{r}}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> die Anzahl der Zeilen in der Versuchsplanmatrix <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><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"> darstellt, die unterschiedlich sind. Wenn alle Zeilen unterschiedlich sind, dann gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{n}}\leq p_{ii}\leq 1}">

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{n}}\leq p_{ii}\leq 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2efed730c60d3353061fbab4ce983ded92bc3f03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.127ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{n}}\leq p_{ii}\leq 1}" loading="lazy">.<a href="#cite_note-6">[6]</a>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> David C. Hoaglin & Roy E. Welsch: The Hat Matrix in Regression and ANOVA. In: <a href="The_American_Statistician" title="The American Statistician">The American Statistician</a>, 32(1), 1978, S. 17–22, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080/00031305.1978.10479237">10.1080/00031305.1978.10479237</a>, <a href="JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/2683469">2683469</a>.

</li>

<li id="cite_note-Chatterjee&Hadi-2">↑ <a href="#cite_ref-Chatterjee&Hadi_2-0">a</a> <a href="#cite_ref-Chatterjee&Hadi_2-1">b</a> Samprit Chatterjee & Ali S. Hadi: Influential observations, high leverage points, and outliers in linear regression. In: Statistical Science, 1(3), 1986, S. 379–393, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214/ss%2F1177013622">10.1214/ss/1177013622</a>, <a href="JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/2245477">2245477</a>.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Wilhelm Caspary: <a rel="nofollow" class="external text" href="https://books.google.de/books?id=REjpBQAAQBAJ&pg=PA124&lpg=PA124&dq=spur+projektionsmatrix&source=bl&ots=A5niKmFenc&sig=z05KBQUJ_9cqmVvkovRlTQ7-ONw&hl=de&sa=X&ved=0ahUKEwjI8PXL-8rUAhXGKVAKHXgYCJIQ6AEIQzAF#v=onepage&q=spur%20projektionsmatrix&f=false">Fehlertolerante Auswertung von Messdaten</a>, S. 124

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> <a href="Rainer_Schlittgen" title="Rainer Schlittgen">Rainer Schlittgen</a>: Regressionsanalysen mit R., ISBN 978-3-486-73967-1, S. 27 (abgerufen über <a href="Verlag_Walter_de_Gruyter" class="mw-redirect" title="Verlag Walter de Gruyter">De Gruyter</a> Online).

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> <a href="Ludwig_Fahrmeir" title="Ludwig Fahrmeir">Ludwig Fahrmeir</a>, <a href="Thomas_Kneib" title="Thomas Kneib">Thomas Kneib</a>, Stefan Lang, Brian Marx: Regression: models, methods and applications. Springer Science & Business Media, 2013, ISBN 978-3-642-34332-2, S. 122.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx: Regression: models, methods and applications. Springer Science & Business Media, 2013, ISBN 978-3-642-34332-2, S. 108.

</li>

</ol>

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

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

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

<a class="mw-selflink selflink">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 href="Korrelationsmatrix" title="Korrelationsmatrix">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>

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