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From: gwyn@brl-smoke.ARPA (Doug Gwyn )
Newsgroups: net.math,net.micro,net.micro.pc
Subject: Re: simplex algorithsm for curve fitting---any disadvantage ??
Message-ID: <972@brl-smoke.ARPA>
Date: Sat, 15-Feb-86 22:11:01 EST
Article-I.D.: brl-smok.972
Posted: Sat Feb 15 22:11:01 1986
Date-Received: Tue, 18-Feb-86 04:20:50 EST
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>>The authors claimed that the program could fit *any* equation with *any*
>>number of parameters and variables to experimental data.
>>Now, I am wondering if there is any disvantage or potential problems in this
>>algorithsm ???  (Up to now, we haven't met any problem yet--- Is it panacea??
>
>I don't know the algorithm, but the claim is garbage.  It is impossible to
>fit for a,b,c,d from data points to the function y=a*exp(b*x)+c*exp(d*x).
>The fitting process here is completely unstable.  (I believe this example
>is due to Wilkinson in the late 1950s.)

I don't have experience with the "simplex algorithm" for curve fitting,
but I did implement a Marquardt gradient-expansion nonlinear least-
squares fitting routine (that took error ellipses into account) years ago
and found that in practice it performed quite well, although a bit slowly.
For most functions of practical interest, a good fitting technique will
converge if one feeds it a reasonable initial guess at parameter values.
Certainly, one can invent unstable situations that will cause the fit to
diverge or to produce a bogus fit (such as, to harmonics of a periodic
function instead of to the fundamental).  Visual inspection of the fitted
curve against the empirical data set is essential if one is to have real
confidence in the result (which should of course include error estimates
for the parameters).  If the data is really noisy, a straight line may
well have a better chi-squared than any more elaborate functional form.
It is true that there is no panacea, but intelligent use of a good scheme
can work wonders.  We were able to fit M"ossbauer and TDPAC data that was
so noisy that the features measured by curve-fitting could not even be
detected by visual examination of the raw data.