Deming regression

In statistics , Deming regression is used to determine a best-fit straight line for a finite set of metrically scaled data pairs ( ) using the least squares method. It is a variant of linear regression . With Deming regression, the residuals (measurement errors) for both the and the values ​​are included in the model. ${\displaystyle x_{i},y_{i}}$ ${\displaystyle x}$ ${\displaystyle y}$

Deming regression is thus a special case of regression analysis ; it is based on a maximum likelihood estimate of the regression parameters , in which the residuals of both variables are assumed to be independent and normally distributed and the quotient of their variances is assumed to be known. ${\displaystyle \delta }$

The Deming regression is based on a work by CH Kummell (1879); In 1937 the method was taken up again by TC Koopmans and made known in a more general context by WE Deming in 1943 for technical and economic applications.

The orthogonal regression is an important special case of Deming regression; she's handling the case . Deming regression is a special case of York regression . ${\displaystyle \delta =1}$

Calculation path

The measured values and are understood as the sums of the “ true values ” or and the “error” or , i. H. The data pairs ( ) lie on the straight line to be calculated. and be independent with a known quotient of the error variances . ${\displaystyle x_{i}}$ ${\displaystyle y_{i}}$ ${\displaystyle x_{i}^{*}}$ ${\displaystyle y_{i}^{*}}$ ${\displaystyle \eta _{i}}$ ${\displaystyle \varepsilon _{i}}$ ${\displaystyle (x_{i},y_{i})=(x_{i}^{*}+\eta _{i},y_{i}^{*}+\varepsilon _{i})}$ ${\displaystyle x_{i}^{*},y_{i}^{*}}$ ${\displaystyle \eta _{i}}$ ${\displaystyle \varepsilon _{i}}$ ${\displaystyle \delta ={\frac {\sigma _{\varepsilon }^{2}}{\sigma _{\eta }^{2}}}}$

It will be a straight line

${\displaystyle y=\beta _{0}+\beta _{1}x}$

that minimizes the weighted residual sum of squares:

${\displaystyle SQR=\sum _{i=1}^{n}{\bigg (}{\frac {\eta _{i}^{2}}{\sigma _{\eta }^{2}}}+{\frac {\varepsilon _{i}^{2}}{\sigma _{\varepsilon }^{2}}}{\bigg )}={\frac {1}{\sigma _{\varepsilon }^{2}}}\sum _{i=1}^{n}{\Big (}(y_{i}-\beta _{0}-\beta _{1}x_{i}^{*})^{2}+\delta (x_{i}-x_{i}^{*})^{2}{\Big )}\ \to \ \min _{\beta _{0},\beta _{1},x_{i}^{*}}SQR}$

The following auxiliary values ​​are required for the further calculation:

${\displaystyle {\overline {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$     ( arithmetic mean of ) ${\displaystyle x_{i}}$

${\displaystyle {\overline {y}}={\frac {1}{n}}\sum _{i=1}^{n}y_{i}}$     (arithmetic mean of ) ${\displaystyle y_{i}}$

${\displaystyle s_{x}^{2}={\tfrac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}$     ( Sample variance of ) ${\displaystyle x_{i}}$

${\displaystyle s_{y}^{2}={\tfrac {1}{n-1}}\sum _{i=1}^{n}(y_{i}-{\overline {y}})^{2}}$     (Sample variance of ) ${\displaystyle y_{i}}$

${\displaystyle s_{xy}={\tfrac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})(y_{i}-{\overline {y}})}$     ( Sample covariance of ). ${\displaystyle (x_{i},y_{i})}$

This gives the parameters for solving the minimization problem:

${\displaystyle \beta _{1}={\frac {s_{y}^{2}-\delta s_{x}^{2}+{\sqrt {(s_{y}^{2}-\delta s_{x}^{2})^{2}+4\delta s_{xy}^{2}}}}{2s_{xy}}}}$

${\displaystyle \beta _{0}={\overline {y}}-\beta _{1}{\overline {x}}}$.

The coordinates are calculated with ${\displaystyle x_{i}^{*}}$

${\displaystyle x_{i}^{*}=x_{i}+{\frac {\beta _{1}}{\beta _{1}^{2}+\delta }}(y_{i}-\beta _{0}-\beta _{1}x_{i})}$.

Individual evidence

1. ↑ CH Kummell: Reduction of observation equations Which contain more than one-observed quantity . In: Annals of Mathematics (Ed.): The Analyst . 6, No. 4, 1879, pp. 97-105. doi : 10.2307 / 2635646 .
2. ^ TC Koopmans: Linear regression analysis of economic time series . DeErven F. Bohn, Haarlem, Netherlands, 1937.
3. ^ WE Deming : Statistical adjustment of data . Wiley, NY (Dover Publications edition, 1985), 1943, ISBN 0-486-64685-8 .
4. ^ P. Glaister: Least squares revisited . The Mathematical Gazette . Vol. 85 (2001), pp. 104-107, JSTOR 3620485 .