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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Prognoseintervall</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="Inferenzstatistik" class="mw-redirect" title="Inferenzstatistik">Inferenzstatistik</a> ist ein <b>Prognoseintervall</b> (auch <b>Vorhersageintervall</b> oder <b>Prädiktionsintervall</b>) ein Bereich um die Vorhersage eines Modells, in dem eine <a href="Ex_ante" title="Ex ante">zukünftige</a> Realisierung einer Messung mit hoher <a href="Wahrscheinlichkeit" title="Wahrscheinlichkeit">Wahrscheinlichkeit</a> (z. B. 95 %) anzutreffen ist.
</p><p>Prognoseintervalle ähneln <a href="Konfidenzintervall" title="Konfidenzintervall">Konfidenzintervallen</a>, sind jedoch aufgrund ihrer Eigenschaften nicht mit ihnen zu verwechseln. Beispielsweise beschreibt das Konfidenzintervall für einen Schätzer des bedingten Erwartungswertes <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 {\hat {E}}[Y|X=x]={\hat {Y}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\hat {E}}[Y|X=x]={\hat {Y}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb0a8372040c5e36ef01e23657e1e2954686d3b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.769ex; height:3.343ex;" alt="{\displaystyle {\hat {E}}[Y|X=x]={\hat {Y}}}" loading="lazy"></span>, wie unsicher dieser Erwartungswert-Schätzer ist. Das Prognoseintervall beschreibt dagegen die Streuung des <a href="Prognosefehler" class="mw-redirect" title="Prognosefehler">Prognosefehlers</a>, weswegen <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 {\hat {Var}}[(Y-{\hat {Y}})|X=x]}">
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<annotation encoding="application/x-tex">{\displaystyle {\hat {Var}}[(Y-{\hat {Y}})|X=x]}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d66fbbbdd5369e1befe1d4e736a74a6af8060722.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.611ex; height:3.343ex;" alt="{\displaystyle {\hat {Var}}[(Y-{\hat {Y}})|X=x]}" loading="lazy"></span> von zentraler Bedeutung ist.
</p><p>Vorhersageintervalle können gegebenenfalls mit dem <a href="Standardfehler_der_Regression" title="Standardfehler der Regression">Standardfehler der Regression</a> berechnet werden.
</p><p>Aus dem <a href="Verzerrung-Varianz-Dilemma" title="Verzerrung-Varianz-Dilemma">Verzerrung-Varianz-Dilemma</a> folgt, dass <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 Var[(Y-{\hat {Y}})|X=x]}">
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<annotation encoding="application/x-tex">{\displaystyle Var[(Y-{\hat {Y}})|X=x]}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbb1c0af3ced6f87270cb8146cd588648f25d82e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.611ex; height:3.343ex;" alt="{\displaystyle Var[(Y-{\hat {Y}})|X=x]}" loading="lazy"></span> nicht kleiner sein kann als die Streuung <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 \sigma ^{2}=Var[Y|X=x]}">
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<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=Var[Y|X=x]}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52eb91a601b122770759729c591e5a2b4460e459.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.671ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}=Var[Y|X=x]}" loading="lazy"></span> der Messwerte selbst. Für eine erwartungstreue Schätzung dieser Varianz <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 {\hat {Var}}[(Y-{\hat {Y}})|X=x]}">
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<annotation encoding="application/x-tex">{\displaystyle {\hat {Var}}[(Y-{\hat {Y}})|X=x]}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d66fbbbdd5369e1befe1d4e736a74a6af8060722.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.611ex; height:3.343ex;" alt="{\displaystyle {\hat {Var}}[(Y-{\hat {Y}})|X=x]}" loading="lazy"></span> folgt daher, dass sie ebenfalls nicht kleiner 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 \sigma ^{2}=Var[Y|X=x]}">
<semantics>
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<mi>σ<!-- σ --></mi>
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<mi>X</mi>
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<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=Var[Y|X=x]}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52eb91a601b122770759729c591e5a2b4460e459.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.671ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}=Var[Y|X=x]}" loading="lazy"></span> sein kann. Daher bedeutet das für korrekt kalibrierte Prognoseintervalle, dass ihre Minimalgröße durch die Breite der Verteilung der Messwerte <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}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> vorgegeben ist.
</p><p>Das Prognoseintervall ist vom Toleranzintervall abzugrenzen.
</p>
<div class="mw-heading mw-heading2"><h2 id="Einfaches_Beispiel">Einfaches Beispiel</h2></div>
<p>Gegeben sei ein sechsseitiger Würfel mit Augenzahlen 1 bis 6. Das <a href="Konfidenzintervall" title="Konfidenzintervall">Konfidenzintervall</a> für den geschätzten Erwartungswert der Augenzahl wird um 3,5 liegen und mit mehr Stichproben enger werden. Das Prognoseintervall für den nächsten Wurf wird jedoch näherungsweise von 1 bis 6 reichen, auch bei beliebig vielen bisher gesehenen Stichproben.
</p>
<div class="mw-heading mw-heading2"><h2 id="Lineares_Modell">Lineares Modell</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Modell">Modell</h3></div>
<p>In der <a href="Multiple_lineare_Regression" title="Multiple lineare Regression">multiplen linearen Regression</a> ergibt sich das Prognosemodell durch
</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 \mathbf {y} _{0}=\mathbf {X} _{0}{\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }}_{0}}">
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{0}=\mathbf {X} _{0}{\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }}_{0}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9aed10344d55873395684f6c770a722491752b24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.296ex; height:2.676ex;" alt="{\displaystyle \mathbf {y} _{0}=\mathbf {X} _{0}{\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }}_{0}}" loading="lazy"></span>,</dd></dl>
<p>wobei
</p>
<ul><li><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 {y} _{0}}">
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{0}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8eb2432f5c3eb38b1b7de0296044928fa637f5f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.465ex; height:2.176ex;" alt="{\displaystyle \mathbf {y} _{0}}" loading="lazy"></span> den <a href="Vektor" title="Vektor">Vektor</a> zukünftiger abhängiger Variablen darstellt und</li>
<li><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} _{0}}">
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} _{0}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d04867e75d5ff1848a97e3b18a843209bd55b6e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.074ex; height:2.509ex;" alt="{\displaystyle \mathbf {X} _{0}}" loading="lazy"></span> die <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> der erklärenden Variablen zum Zeitpunkt <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 T_{0}}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}" loading="lazy"></span>.</li></ul>
<p>Die Prognose wird dargestellt 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 {\hat {\mathbf {y} }}_{0}=\mathbf {X} _{0}\mathbf {b} }">
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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {y} }}_{0}=\mathbf {X} _{0}\mathbf {b} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe2d2de4a4f35c160aa58f051c8bad9f038dc224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.123ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {y} }}_{0}=\mathbf {X} _{0}\mathbf {b} }" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading3"><h3 id="Prognoseintervall">Prognoseintervall</h3></div>
<p>Wichtig für die Berechnung eines Prognoseintervalls ist die <a href="Varianz" title="Varianz">Varianz</a> des <a href="Prognosefehler" class="mw-redirect" title="Prognosefehler">Prognosefehlers</a>, welche die <a href="Variation_(Statistik)" class="mw-redirect" title="Variation (Statistik)">Variation</a> des Prognosefehlers und somit die Zuverlässigkeit der Prognose wiedergibt.
</p><p>Sie ist in der <a href="Einfache_lineare_Regression" class="mw-redirect" title="Einfache lineare Regression">linearen Einfachregression</a> gegeben durch:
</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 \sigma _{0}^{2}=\operatorname {Var} ({\hat {y}}_{0}-y_{0})=\sigma ^{2}\left(1+{\frac {1}{n}}+{\frac {(x_{0}-{\bar {x}})^{2}}{\sum \limits _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mn>2</mn>
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</msubsup>
<mo>=</mo>
<mi>Var</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
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<mi>y</mi>
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mo>−<!-- − --></mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mrow>
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<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
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</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
</mrow>
<mrow>
<munderover>
<mo movablelimits="false">∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{0}^{2}=\operatorname {Var} ({\hat {y}}_{0}-y_{0})=\sigma ^{2}\left(1+{\frac {1}{n}}+{\frac {(x_{0}-{\bar {x}})^{2}}{\sum \limits _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b422286e477d010a0340fa7b5b92999227ac4970.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:50.178ex; height:12.843ex;" alt="{\displaystyle \sigma _{0}^{2}=\operatorname {Var} ({\hat {y}}_{0}-y_{0})=\sigma ^{2}\left(1+{\frac {1}{n}}+{\frac {(x_{0}-{\bar {x}})^{2}}{\sum \limits _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}\right)}" loading="lazy"></span>.</dd></dl>
<p>Mithilfe der Varianz des Prognosefehlers erhält man dann 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 (1-\alpha )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (1-\alpha )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1dcc25f05dca60e358d4d22e8342fad5ad7affbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.3ex; height:2.843ex;" alt="{\displaystyle (1-\alpha )}" loading="lazy"></span>-Prognoseintervall für den prognostizierten Wert 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 y_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y_{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d943dbbb0b56ca750c4d62c5b54b4ae29a773da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{0}}" loading="lazy"></span><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><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 {\hat {y}}_{0}\pm t_{(1-\alpha /2,n-2)}\cdot {\sqrt {{\hat {\operatorname {Var} }}({\hat {y}}_{0}-y_{0})}}={\hat {y}}_{0}\pm t_{(1-\alpha /2,n-2)}\cdot {\sqrt {{\hat {\sigma }}^{2}\left(1+{\frac {1}{n}}+{\frac {(x_{0}-{\bar {x}})^{2}}{\sum \limits _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}\right)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>y</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>±<!-- ± --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo>,</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>Var</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
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</mrow>
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<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>y</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</msqrt>
</mrow>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>y</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>±<!-- ± --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo>,</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>σ<!-- σ --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<munderover>
<mo movablelimits="false">∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {y}}_{0}\pm t_{(1-\alpha /2,n-2)}\cdot {\sqrt {{\hat {\operatorname {Var} }}({\hat {y}}_{0}-y_{0})}}={\hat {y}}_{0}\pm t_{(1-\alpha /2,n-2)}\cdot {\sqrt {{\hat {\sigma }}^{2}\left(1+{\frac {1}{n}}+{\frac {(x_{0}-{\bar {x}})^{2}}{\sum \limits _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}\right)}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d0f3c9740d3a4fd45d61297439af4fccddd88ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:84.607ex; height:13.509ex;" alt="{\displaystyle {\hat {y}}_{0}\pm t_{(1-\alpha /2,n-2)}\cdot {\sqrt {{\hat {\operatorname {Var} }}({\hat {y}}_{0}-y_{0})}}={\hat {y}}_{0}\pm t_{(1-\alpha /2,n-2)}\cdot {\sqrt {{\hat {\sigma }}^{2}\left(1+{\frac {1}{n}}+{\frac {(x_{0}-{\bar {x}})^{2}}{\sum \limits _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}\right)}}}" loading="lazy"></span>.</dd></dl>
<p>Beachte, dass die Breite des Prognoseintervalls an den Rändern des Trägers der Trainingsdaten typischerweise zunimmt, da nicht nur die konstante Varianz der Residuuen <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 {\hat {\sigma }}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>σ<!-- σ --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ad9d89160c9e63c0aa4c158282cb75a894de56f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle {\hat {\sigma }}^{2}}" loading="lazy"></span> einfließt, sondern auch die Unsicherheit bei der Schätzung des Modells. Letztere ist an den Rändern typischerweise größer.
</p>
<div class="mw-heading mw-heading2"><h2 id="Bootstrap">Bootstrap</h2></div>
<p><a href="Bootstrap_(Statistik)" class="mw-redirect" title="Bootstrap (Statistik)">Bootstrapping</a><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> kann zum Erzeugen mehrerer Regressionsmodelle benutzt werden, deren Streuung und Residuen dann zur Konstruktion von Bootstrap-Prognoseintervallen verwendet werden können.
</p>
<div class="mw-heading mw-heading2"><h2 id="Bayessches_Prognoseintervall">Bayessches Prognoseintervall</h2></div>
<p>Die <a href="Posterior_predictive_distribution" title="Posterior predictive distribution">Posterior predictive distribution</a> kann zur Konstruktion von Bayesschen Prognoseintervallen verwendet werden.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Conformal_Prediction">Conformal Prediction</h2></div>
<p>Conformal prediction kann unter Annahme von <a href="Austauschbarkeit_(Statistik)" class="mw-redirect" title="Austauschbarkeit (Statistik)">Austauschbarkeit</a> zur Konstruktion von Prognoseintervallen benutzt werden. Konforme Punkte sind jene Punkte, welche „ähnlich“ zu den bisher beobachteten Punkten sind. Bei der Split Conformal Prediction<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> wird der Datensatz in einen Trainings- und Validierungsdatensatz aufgeteilt. Die Non-conformity wird beispielsweise mit der absoluten Abweichung vom (<a href="Ausgleichungsrechnung" title="Ausgleichungsrechnung">angepassten</a>) Modell gemessen: je größer die Abweichung eines Punktes umso weniger konform ist dieser Punkt. Die Non-conformity wird für alle Punkte im Validierungsdatensatz ermittelt. Für einen neuen Testpunkt <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},y)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (x_{i},y)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9463696740ba7fac0d0efda580a7e7eedd1f3f93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.128ex; height:2.843ex;" alt="{\displaystyle (x_{i},y)}" loading="lazy"></span> werden alle möglichen <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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="./_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"></span> durchgetestet (eventuell diskretisiert). Das 95% Prognoseintervall ist dann jene Menge von y-Werten, bei denen der Non-conformity Score des Test-Punktes einen kleineren Rang in der aufsteigend sortierten Liste der Non-conformity Scores des Validierungsdatensatz hatte als 95 % der Validierungspunkte.
</p>
<div class="mw-heading mw-heading2"><h2 id="Quantilsregression">Quantilsregression</h2></div>
<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Quantilsregression" title="Quantilsregression">Quantilsregression</a></i></div>
<p>Werden das 97.5 und das 2.5 Quantil geschätzt, so kann daraus ein 95-%-Prognoseintervall konstruiert werden.
</p>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Von Auer: <i>Ökonometrie. Eine Einführung.</i> 6. Auflage, S. 135.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">L. Fahrmeir, R. Künstler u. a.: <i>Statistik. Der Weg zur Datenanalyse.</i> 8. Auflage. Springer 2016, S. 448.</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Stine, Robert A. <a rel="nofollow" class="external text" href="https://doi.org/10.2307/2288570">“Bootstrap Prediction Intervals for Regression.”</a>, Journal of the American Statistical Association, vol. 80, no. 392, 1985, pp. 1026–31. JSTOR, abgerufen am 27. Januar 2025.</span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Agresti, A. (2015). <a rel="nofollow" class="external text" href="https://books.google.de/books?id=mgIzBgAAQBAJ&pg=PA339">Foundations of Linear and Generalized Linear Models.</a> Deutschland: Wiley, Seite 339. Die Seite wird von books.google.de nicht immer angezeigt, am 27. Januar 2025.</span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Xing Han, Ziyang Tang, Joydeep Ghosh, Qiang Liu: <a rel="nofollow" class="external text" href="https://arxiv.org/abs/2206.13092v2">Split Localized Conformal Prediction</a>, arxiv.org, 2022.</span>
</li>
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