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Normally, linear regression asks for a pair of parameters m,b such that for a set of given points $\{x_i,y_i\}$ the variance of $y-m\cdot x-b$ is minimized (this minimizes the distance in y-direction only).

Instead, I would like to find a line $y'=m\cdot x'+b$ such that the pairs $(x_i,y_i)$ have minimal perpendicular distance from that line.

I strongly suspect that the parameters (m,b) turn out to be the same in either case, but I can't find anything on the web, and when I tried the calculation by myself I found me totally out of practice with one error for each 3 lines of my calculations.

Gyro Gearloose
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1 Answers1

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With appropriate method names, you will be able to find a lot of references: you seem to be looking for Deming regression, orthogonal regression, orthogonal distance regression (ODR), errors-in-variables (EIV) modeling, or more often total least squares (TLS). It is illustrated below in the bivariate case:

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The model was originally introduced by R. J. Adcock, 1878, A problem in least squares, refined by C. H. Kummell, 1879, Reduction of observation equations which contain more than one observed quantity, and revived by W. E. Deming, 1943, Statistical adjustment of data.

Laurent Duval
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