Deming regression: Difference between revisions

Revision as of 03:26, 23 January 2022

Revision as of 03:27, 23 January 2022

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We seek to find the line of "best fit"

: <math>y^* = \beta_0 + \beta_1 x^*,</math>

such that the weighted sum of squared residuals of the model is minimized:{{sfn|Fuller|1987|atloc=Ch. 1.3.3}}

: <math>SSR = \sum_{i=1}^n\bigg(\frac{\varepsilon_i^2}{\sigma_\varepsilon^2} + \frac{\eta_i^2}{\sigma_\eta^2}\bigg) = \frac{1}{\sigma_\varepsilon^2} \sum_{i=1}^n\Big((y_i-\beta_0-\beta_1x^*_i)^2 + \delta(x_i-x^*_i)^2\Big) \ \to\ \min_{\beta_0,\beta_1,x_1^*,\ldots,x_n^*} SSR</math>

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==Orthogonal regression==

For the case of equal error variances, i.e., when <math>\delta=1</math>, Deming regression becomes ''orthogonal regression'': it minimizes the sum of squared [[distance from a point to a line|perpendicular distances from the data points to the regression line]]. In this case, denote each observation as a point ''z''_''j'' in the complex plane (i.e., the point (''x''_''j'', ''y''_''j'') is written as ''z''_''j'' = ''x''_''j'' + ''iy''_''j'' where ''i'' is the [[imaginary unit]]). Denote as ''Z'' the sum of the squared differences of the data points from the [[centroid]] (also denoted in complex coordinates), which is the point whose horizontal and vertical locations are the averages of those of the data points. Then:{{sfn|Minda|Phelps|2008|atloc=Theorem 2.3}}

*If ''Z'' = 0, then every line through the centroid is a line of best orthogonal fit.

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===Application===

In the case of three [[Line (geometry)|non-collinear]] points in the plane, the [[triangle]] with these points as its [[vertex (geometry)|vertices]] has a unique [[Steiner inellipse]] that is tangent to the triangle's sides at their midpoints. The [[Ellipse#Elements of an ellipse|major axis of this ellipse]] falls on the orthogonal regression line for the three vertices.{{sfn|Minda|Phelps|2008|atloc=Corollary 2.4}} The quantification of a biological cell's intrinsic [[cellular noise]] can be quantified upon applying Deming regression to the observed behavior of a two reporter [[synthetic biological circuit]].{{sfn|Quarton|2020}}

==See also==

* [[Line fitting]]

==Notes==

{{Reflist}}

==References==

==;Notes==

{{Reflist|30em}}

;Bibliography

* {{cite journal|last=Adcock|first=R. J.|year=1878|title=A problem in least squares|journal=The Analyst|volume=5|issue=2|pages=53–54|doi=10.2307/2635758|doi-access=free|jstor=2635758|jstor-access=free}}

* {{cite journal|author=Coolidge|first=J. L.|year=1913|title=Two geometrical applications of the mathematics of least squares|journal=The American Mathematical Monthly|volume=20|issue= 6|pages=187–190|doi=10.2307/2973072|jstor=2973072}}