From: Ethan M. <merritt@u.washington.edu> - 2011-05-24 00:08:17
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On Monday, May 23, 2011 01:12:23 am pl...@pi... wrote: > > if there is an alternative approach for the case when x > > uncertainty can't be ignored, keep the discussion going and maybe we can > > add such a feature to the list of items we'd like to add in the future. > > > > Dan > > Yes , I would love to propose that. Some kind of total least squares may > be an option I don't know enough about that to make a concrete proposal. It is standard in my field to use instead a maximum likelihood residual that allows for separate error distributions on x and y. The maximum likelihood treatment reduces to a weighted least squares treatment if and only if the distribution of errors on both x and y are (1) Gaussian and (2) of equal magnitude. Gnuplot allows for input of precalculated non-uniform weights on y, which partially addresses (2). But there is no provision for non-Gaussian errors on y, and no provision for any error model at all on x. If the errors in your data do not follow the simple Gaussian model, then almost certainly you would be better off using maximum likelihood rather than least-squares. But in order to do so you need first a model for the error distributions. That may be obvious for any particular experiment or source of data, but it's difficult to impossible for a general-purpose program to figure it out for you. You're the one who knows the source of the data, so it's up to you to provide an appropriate explicit error model along with the data. The point is that switching to a better minimization residual is more than just a matter of changing the internal code. It would require a more complex description of your data that includes error models for both the independent and dependent variables. NB: When I say "x", I really mean the full set of independent variables [x1,x2,x3,...]. "y" is the single dependent variable estimated by f(x1,x2,x3,...) Ethan -- Ethan A Merritt Biomolecular Structure Center, K-428 Health Sciences Bldg University of Washington, Seattle 98195-7742 |