In article <[EMAIL PROTECTED]>, [EMAIL PROTECTED] (Vitaly Kupisk) wrote:
>Subject: Re: orthogonal linear regression >From: [EMAIL PROTECTED] (Vitaly Kupisk) >Organization: http://groups.google.com/ >Date: 3 Jan 2003 16:36:50 -0800 >Newsgroups: sci.stat.math,sci.stat.edu > >Thanks again for the comments. > >I think I need to reiterate that it's L1, not L2 regression that I am >interested in. For L2 there is the ODRPack which does Orthogonal Distance Regression and comes with several background papers on the method. The usual L1 regression can is often implemented as iteratively reweighted L2 regression. Given the rather uncertain terminology in use here, it might be the case that iterative reweighting with ODRPack will do your job as at first sight your definition does not match what I understand by the usual "orthogonal" regression. The usual "orthogonal" regression has a different rotation for each residual so the residual is orthogonal to the fitted curve. Maybe that is what you actually want but have not quite formulated it that way. Even the usual L1 suffers from degeneracy in its solution. That is a fancy way of saying that there may be mutiple solutions, each of which is equally good under the stated criteria. > >By orthogonality, Gottfried, I simply mean that whereas the "usual" L1 >regression minimizes over all possible sets of coefficients >{Ci(i=1..n), Const} the sum (over the sample points) of |Y - (C1X1 + >... + CnXn+Const)| , the orthogonal regression aims to minimize the >sum of >|Y - (C1X1 + ... + CnXn+Const)|/sqrt(C1^2+...+Cn^2+1). > >Toms algorithm 478 does the job for the "usual" L1 fit, but if I try >to apply the trick of rotating the coordinates and redoing the >algorithm, then if I use stabilization of Ci's as the stopping >criteria, the process never stops even with some 2-dimensional data >sets (flips back ad forth between 2 sets of coefficients), or, if I >continue only as long as the sum of orthogonal distances (as above) >decreases, then it stops before finding the true minimum (which I find >by "brute force", checking hyperplanes that pass through each >(n+1)-subset of data points). > >For whatever it's worth, my client is planning to apply the algorithm >to time series of market prices, but, as I mentioned, I am mostly >concerned about implementing an algorithm that'll do the minimization >and less in whether its use is appropriate to the application. > >Vitaly . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
