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<title>Deming-Regression</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Deming-Regression</h1>

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In der <a href="Statistik" title="Statistik">Statistik</a> wird mit der Deming-Regression eine <a href="Ausgleichsgerade" class="mw-redirect" title="Ausgleichsgerade">Ausgleichsgerade</a> für eine endliche Menge <a href="Skalenniveau" title="Skalenniveau">metrisch skalierter</a> Datenpaare (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i},y_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2dccea2bb6a826b389fd2d042508c3d30443dd77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.102ex; height:2.009ex;" alt="{\displaystyle x_{i},y_{i}}" loading="lazy">) nach der <a href="Methode_der_kleinsten_Quadrate" title="Methode der kleinsten Quadrate">Methode der kleinsten Quadrate</a> bestimmt. Es handelt sich um eine Variante der <a href="Lineare_Regression" title="Lineare Regression">linearen Regression</a>. Bei der Deming-Regression werden die <a href="St%C3%B6rgr%C3%B6%C3%9Fe_und_Residuum" title="Störgröße und Residuum">Residuen</a> (Messfehler) sowohl für die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">- als auch für die <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 in das Modell einbezogen.

Die Deming-Regression ist somit ein Spezialfall der <a href="Regressionsanalyse" title="Regressionsanalyse">Regressionsanalyse</a>; sie beruht auf einer <a href="Maximum-Likelihood-Methode" title="Maximum-Likelihood-Methode">Maximum-Likelihood-Schätzung</a> der <a href="Regressionsparameter" title="Regressionsparameter">Regressionsparameter</a>, bei der die Residuen beider Variablen als unabhängig und <a href="Normalverteilung" title="Normalverteilung">normalverteilt</a> angenommen werden und der Quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> ihrer <a href="Varianz_(Stochastik)" title="Varianz (Stochastik)">Varianzen</a> als bekannt unterstellt wird.

Die Deming-Regression geht auf eine Arbeit von C.H. Kummell (1879) zurück;<a href="#cite_note-1">[1]</a> 1937 wurde die Methode von T.C. Koopmans wieder aufgegriffen<a href="#cite_note-2">[2]</a> und in allgemeinerem Rahmen 1943 von <a href="William_Edwards_Deming" title="William Edwards Deming">W. E. Deming</a> für technische und ökonomische Anwendungen bekannt gemacht.<a href="#cite_note-3">[3]</a>

Die <a href="Orthogonale_Regression" title="Orthogonale Regression">orthogonale Regression</a> ist ein wichtiger Spezialfall der Deming-Regression; sie behandelt den Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta =1}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30c5c1a63ccd876384e6cf0a31296e3aee31ac84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta =1}" loading="lazy">. Die Deming-Regression wiederum ist ein Spezialfall der York-Regression.

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

Die gemessenen Werte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"> werden als Summen der „<a href="Wahrer_Wert" title="Wahrer Wert">wahren Werte</a>“ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}^{*}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d4aad224eaab545bb11af7a02363e59addaad6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:2.843ex;" alt="{\displaystyle x_{i}^{*}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{*}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b358e19039c3610c53a3e38a67e3a4509561b3dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.215ex; height:2.843ex;" alt="{\displaystyle y_{i}^{*}}" loading="lazy"> und der „Fehler“ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b336ed51c970728280e38f5a131ac52f69833c67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.955ex; height:2.176ex;" alt="{\displaystyle \eta _{i}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00e1b6ad3cbad4af49bf21a3ad2dc379ff045079.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{i}}" loading="lazy"> aufgefasst, d. h. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i},y_{i})=(x_{i}^{*}+\eta _{i},y_{i}^{*}+\varepsilon _{i})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{i},y_{i})=(x_{i}^{*}+\eta _{i},y_{i}^{*}+\varepsilon _{i})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57dd838652c2ce6d1c2a25ae3ddb391e6cdcd73e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.971ex; height:3.009ex;" alt="{\displaystyle (x_{i},y_{i})=(x_{i}^{*}+\eta _{i},y_{i}^{*}+\varepsilon _{i})}" loading="lazy"> Die Datenpaare (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}^{*},y_{i}^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}^{*},y_{i}^{*}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44dccab631b1f7c6cd9791c988e8c251feb6676c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.633ex; height:2.843ex;" alt="{\displaystyle x_{i}^{*},y_{i}^{*}}" loading="lazy">) liegen auf der zu berechnenden Geraden. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b336ed51c970728280e38f5a131ac52f69833c67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.955ex; height:2.176ex;" alt="{\displaystyle \eta _{i}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00e1b6ad3cbad4af49bf21a3ad2dc379ff045079.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{i}}" loading="lazy"> seien unabhängig mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\eta }^{2}:=Var(\eta )}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{\eta }^{2}:=Var(\eta )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00355439793983cb9944825dc9a5b380ee4deb96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.176ex; height:3.176ex;" alt="{\displaystyle \sigma _{\eta }^{2}:=Var(\eta )}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\varepsilon }^{2}:=Var(\varepsilon )}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{\varepsilon }^{2}:=Var(\varepsilon )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e780523c7949fe9b373c6b3a6648e0e454f77973.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.089ex; height:3.009ex;" alt="{\displaystyle \sigma _{\varepsilon }^{2}:=Var(\varepsilon )}" loading="lazy">. Bekannt sei zumindest der Quotient der <a href="Erwartungstreue_Sch%C3%A4tzung_der_Varianz_der_St%C3%B6rgr%C3%B6%C3%9Fen" title="Erwartungstreue Schätzung der Varianz der Störgrößen">Fehlervarianzen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ={\frac {\sigma _{\eta }^{2}}{\sigma _{\varepsilon }^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \delta ={\frac {\sigma _{\eta }^{2}}{\sigma _{\varepsilon }^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29f53edb91a6d88655de17a6f65bb282bc7842a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:7.37ex; height:6.676ex;" alt="{\displaystyle \delta ={\frac {\sigma _{\eta }^{2}}{\sigma _{\varepsilon }^{2}}}}" loading="lazy">.

Es wird eine Gerade

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<annotation encoding="application/x-tex">{\displaystyle y=\beta _{0}+\beta _{1}x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e648dba6a90fd70d788feff29f96e316a5c40ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.164ex; height:2.509ex;" alt="{\displaystyle y=\beta _{0}+\beta _{1}x}" loading="lazy"></dd></dl>

gesucht, die die <a href="Residuenquadratsumme#Gewichtete_Residuenquadratsumme" title="Residuenquadratsumme">gewichtete Residuenquadratsumme</a> minimiert:

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<annotation encoding="application/x-tex">{\displaystyle SQR=\sum _{i=1}^{n}{\bigg (}{\frac {\eta _{i}^{2}}{\sigma _{\eta }^{2}}}+{\frac {\varepsilon _{i}^{2}}{\sigma _{\varepsilon }^{2}}}{\bigg )}={\frac {1}{\sigma _{\varepsilon }^{2}}}\sum _{i=1}^{n}{\Big (}(y_{i}-\beta _{0}-\beta _{1}x_{i}^{*})^{2}+\delta (x_{i}-x_{i}^{*})^{2}{\Big )}\ \to \ \min _{\beta _{0},\beta _{1},x_{i}^{*}}SQR}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/424b8f99770972d0777e96d22d0488517dedf5c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:86.446ex; height:7.009ex;" alt="{\displaystyle SQR=\sum _{i=1}^{n}{\bigg (}{\frac {\eta _{i}^{2}}{\sigma _{\eta }^{2}}}+{\frac {\varepsilon _{i}^{2}}{\sigma _{\varepsilon }^{2}}}{\bigg )}={\frac {1}{\sigma _{\varepsilon }^{2}}}\sum _{i=1}^{n}{\Big (}(y_{i}-\beta _{0}-\beta _{1}x_{i}^{*})^{2}+\delta (x_{i}-x_{i}^{*})^{2}{\Big )}\ \to \ \min _{\beta _{0},\beta _{1},x_{i}^{*}}SQR}" loading="lazy"></dd></dl>

Für die weitere Rechnung werden die folgenden Hilfswerte benötigt:

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<annotation encoding="application/x-tex">{\displaystyle {\overline {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c43d447f94845fbf2c2e04d507a55e0e3ae7a106.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.032ex; height:6.843ex;" alt="{\displaystyle {\overline {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}" loading="lazy">     (<a href="Arithmetisches_Mittel" title="Arithmetisches Mittel">arithmetisches Mittel</a> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy">)</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\overline {y}}={\frac {1}{n}}\sum _{i=1}^{n}y_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d619412f6b4c04092d4b70fd14496a0d5cdd2f94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.678ex; height:6.843ex;" alt="{\displaystyle {\overline {y}}={\frac {1}{n}}\sum _{i=1}^{n}y_{i}}" loading="lazy">     (arithmetisches Mittel der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy">)</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle s_{x}^{2}={\tfrac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c5a689f96929bbcd1fe805ca21413d209117234.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.304ex; height:6.843ex;" alt="{\displaystyle s_{x}^{2}={\tfrac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}" loading="lazy">     (<a href="Empirische_Varianz" title="Empirische Varianz">Stichprobenvarianz</a> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy">)</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle s_{y}^{2}={\tfrac {1}{n-1}}\sum _{i=1}^{n}(y_{i}-{\overline {y}})^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/464612cd0942bd6e40516a076b979618291a062e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.832ex; height:6.843ex;" alt="{\displaystyle s_{y}^{2}={\tfrac {1}{n-1}}\sum _{i=1}^{n}(y_{i}-{\overline {y}})^{2}}" loading="lazy">     (Stichprobenvarianz der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy">)</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle s_{xy}={\tfrac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})(y_{i}-{\overline {y}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79d3170fc7344967c19b1f9ea4a67e3beff36780.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.937ex; height:6.843ex;" alt="{\displaystyle s_{xy}={\tfrac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})(y_{i}-{\overline {y}})}" loading="lazy">     (<a href="Stichprobenkovarianz" title="Stichprobenkovarianz">Stichprobenkovarianz</a> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{i},y_{i})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{i},y_{i})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6dbb919b91ccacf17ed47898048428a1baf9703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.912ex; height:2.843ex;" alt="{\displaystyle (x_{i},y_{i})}" loading="lazy">).</dd></dl>

Damit ergeben sich die Parameter zur Lösung des Minimierungsproblems:<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle \beta _{1}={\frac {s_{y}^{2}-\delta s_{x}^{2}+{\sqrt {(s_{y}^{2}-\delta s_{x}^{2})^{2}+4\delta s_{xy}^{2}}}}{2s_{xy}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45127c3b1082eb5b41f3d6942500a99c2be76662.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.057ex; height:8.343ex;" alt="{\displaystyle \beta _{1}={\frac {s_{y}^{2}-\delta s_{x}^{2}+{\sqrt {(s_{y}^{2}-\delta s_{x}^{2})^{2}+4\delta s_{xy}^{2}}}}{2s_{xy}}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \beta _{0}={\overline {y}}-\beta _{1}{\overline {x}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6da31a85e3be36f472093ac41d9f1ad418c875af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.405ex; height:2.676ex;" alt="{\displaystyle \beta _{0}={\overline {y}}-\beta _{1}{\overline {x}}}" loading="lazy">.</dd></dl>

Die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}^{*}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d4aad224eaab545bb11af7a02363e59addaad6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:2.843ex;" alt="{\displaystyle x_{i}^{*}}" loading="lazy">-Koordinaten berechnet man mit

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<annotation encoding="application/x-tex">{\displaystyle x_{i}^{*}=x_{i}+{\frac {\beta _{1}}{\beta _{1}^{2}+\delta }}(y_{i}-\beta _{0}-\beta _{1}x_{i})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9af09e58b5f3b66c0506d6d5ca315ecf4895467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.867ex; height:6.343ex;" alt="{\displaystyle x_{i}^{*}=x_{i}+{\frac {\beta _{1}}{\beta _{1}^{2}+\delta }}(y_{i}-\beta _{0}-\beta _{1}x_{i})}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Erweiterung_York-Regression">Erweiterung York-Regression</h2></div>

York-Regression erweitert die Deming-Regression, da es korrelierte x- und y-Fehler erlaubt<a href="#cite_note-5">[5]</a>.

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Charles H. Kummell: Reduction of observation equations which contain more than one observed quantity. In: <a href="The_Analyst" title="The Analyst">The Analyst</a>. Band 6, Nummer 4, 1879, S. 97–105, <a href="JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/2635646">2635646</a>.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a href="Tjalling_C._Koopmans" title="Tjalling C. Koopmans">Tjalling Koopmans</a>: Linear regression analysis of economic time series (= Publications of the Netherland Economic Institute. 20). De Erven F. Bohn, Haarlem 1937.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> <a href="William_Edwards_Deming" title="William Edwards Deming">W. Edwards Deming</a>: Statistical adjustment of data. Wiley u. a., New York NY 1943, (Unabriged and corrected republication. Dover Publications, New York NY 1985, ISBN 0-486-64685-8).

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Paul Glaister: Least squares revisited. In: <a href="The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a>. Band 85, Nummer 502, 2001, S. 104–107, <a href="JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/3620485">3620485</a>.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Derek York, Norman M. Evensen, Margarita López Martı́nez, Jonás De Basabe Delgado: Unified equations for the slope, intercept, and standard errors of the best straight line. In: <a href="American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a>. Band 72, 2004, S. 367–375, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119/1.1632486">10.1119/1.1632486</a>.

</li>

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