Re: Nonlinear Least-Squares curve-fitting


cafeinst@xxxxxxx wrote:
Old Mac User wrote:
Cafei...

If you are finding approximately the same residual sum of squares for
various combinations of the fitting parameters, then you have very poor
estimates of those parameters. In other words, the confidence intervals
on the parameters are very wide.

It always goes back to the "design of the experiments" that spawned the
data. Attempting to fit a "complex" nonlinear model to just any old set
of accumulated data is a gross waste of time. Estimating parameters in
a linear model can be a loser game if the "independent variables" are
poorly arranged in their space. Attempting to do this with a "complex"
nonlinear model is many times worse. Worse in the sense of wasting time
and effort.

Multiple minima... or what seem to be multiple minima... many of which
have approx. the same residual sum of squares... suggests your data is
poorly conditioned for the model you are attempting to fit.

You may be right. Let me describe what I did in more detail, so that
perhaps you can explain why this happened:

The data that I got came from physics experiments. The functions to fit
the data came from standard physics equations. The parameters in these
equations were actually real physics quantities which could be
experimentally measured. When I applied the algorithm which I described
of gradually adding data points, the parameters that the algorithm
output were much closer to the experimentally observed values for the
parameters than when I did not do it gradually.

I'll let Old Mac User answer your question to him.

Meanwhile, my comment to your preceding paragraph is this:

Why didn't you both playing with nonlinear estimation algorithms
when you know nothing about nonlinear models and nonlinear
estimation methods?

Why didn't you just EXPERIMENTALLY MEAFURE those
parameters?


You last two lines certainly SUGGEST that you think it's a good
thing for the numerical algorithm to come close to the physically
measurable estimates of the same parameters, which in turn
suggest that you think the "experimentally observed values for
the parameters" are more accurate or reliable than the ones
you staistically estimated via your nonlinear estimation model.

But HOW do you know which one is more reliable or accurate?


When I co-authored the textbook "Conversational Statistics"
with Harry Roberts, he had an anecdotal tale that according
a sampling God (sorry I can't remember his name), whose
belief coincided with Harry's, that modern sampling methods for
a population can be more accurate than a CENSUS (counting
everyone) of the same population.

Imbedded in Harry's belief is of course the well-known fact
that for the US census done every 10 years, there has NEVER
been a true census because people move around all the time,
and nobody can catch all the street people (living off grocery
carts and cardboard boxes), so the CENSUS is never a TRUE
CENSUS.

So this was what Harry and the sampling God believed:

There is an unknown parameter p about the US population.

We can get an estimate of p by counting everyone, as in a
census. Call that pc-hat. We can also estimate the same
p by counting only a small portion of the population, via
finely stratified samples and with all the statistical gadgets
to fill the void, and call that estimate ps-hat.

Conjecture: | p - ps-hat | < | p - pc-hat |.

Before exercising my veto-power <G>, I asked Harry the
following question: If we DON'T know what p is, then HOW
do we know which of the estimates, pc or ps , is closer to
the unknown p ?

Perhaps Harry was too busy to argue, he graciously withdrew
that anecdotal tale from the manuscript without any dissent, and
I had not heard him mentionng that anecdotal "urban legend"
ever since. :-)

So, that would be the same question I have for teh OP. If you
DON'T know what the parameter values are, in the physics
problem, where it can be physically measure or statistically
estimated from a nonlinear model, how do you know which one
is a "better" estimate?

-- Reef Fish Bob.

.