| Minimization of deviations from multiple curve fits. [message #35229] |
Mon, 26 May 2003 14:59  |
aaron_forster
Messages: 1
Registered: May 2003
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Junior Member |
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Hello Group,
I am utilizing IDL to analyze data obtained from adhesion tests. For
those interested, the tests are used to determine the work of adhesion
and system modulus. I measure contact area, load, and displacement
during the test. I currently use Craig Markwardt's MPFIT program to
fit two different non-linear equations to the experimental data. The
first equation (EQ1) expresses contact area as a function of load,
with the work of adhesion, system modulus, and indenter radius as
fitting parameters. The second equation (EQ2) expresses displacement
as a function of load and contact area, with the system modulus and
indenter radius as fitting parameters. It has been suggested by
others (Chin P. et al., J. Adhesion, 1997, 64 p. 145-160) that I can
increase the precision by analyzing the fit deviations from each curve
fit together. In other words, I need to minimize the function:
omega^2=sum {[EQ1_fit - EQ1i] + [EQ2_fit - EQ2i]^2}
where sum is the sum from i=1 to N of my data (EQ1, EQ2) and my fit
(EQ1_fit, EQ2_fit)
I hope the above equation is clear to everyone. Anyway, my
understanding of the regression programs I have seen in IDL is that
they will fit an equation and measure success of fit by minimizing
chi-sqr. My question is how do I both minimize chi-sqr for each
equation AND minimize omega^2, such that the fitting parameters I
obtain at the end of the day will provide satisfactory fits for EQ1
and EQ2. I hope this post is clearly written, but I am an IDL newbie
and I may have left relevant information out. If you would like to
hear more, please email me at aaron_forster@yahoo.com with questions.
Thank you in advance for all of your help. I greatly appreciate it.
Sincerely,
Aaron Forster
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| Re: Minimization of deviations from multiple curve fits. [message #35417 is a reply to message #35229] |
Tue, 27 May 2003 15:10  |
Craig Markwardt
Messages: 1869
Registered: November 1996
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Senior Member |
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aaron_forster@yahoo.com (Aaron Forster) writes:
> first equation (EQ1) expresses contact area as a function of load,
> with the work of adhesion, system modulus, and indenter radius as
> fitting parameters. The second equation (EQ2) expresses displacement
> as a function of load and contact area, with the system modulus and
> indenter radius as fitting parameters. It has been suggested by
> others (Chin P. et al., J. Adhesion, 1997, 64 p. 145-160) that I can
> increase the precision by analyzing the fit deviations from each curve
> fit together. In other words, I need to minimize the function:
Any least squares fitter can do this, including MPFIT. You are asking
for a joint fit of two data sets.
> omega^2=sum {[EQ1_fit - EQ1i] + [EQ2_fit - EQ2i]^2}
I don't really understand this formula, since (a) the first term is
not squared, and (b) what exactly is the summation over? Your
description indicates that equations 1 and 2 have different sets of
independent variables.
All MPFIT really needs is an array of residuals, and you can make that
array in any consistent manner that you choose. So, one way to
achieve what you are seeking is to compute the residuals of each
equation separately, and then join them together using IDL [res1,res2]
notation.
> My question is how do I both minimize chi-sqr for each
> equation AND minimize omega^2,
I don't think this is possible. You are asking, "what is the shortest
route from Baltimore to Philadelphia to New York, which is *also* the
shortest route from Baltimore to New York." Both constraints can't be
met.
If you are performing a joint fit, you must be willing to accept a
slightly worse fit in one data set or the other, for the sake of an
overall improvement of the aggregate. If, however, that means that
one set of data is totally ignored, then you have to reconsider your
model, or your weighting strategy.
Happy fitting,
Craig
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Craig B. Markwardt, Ph.D. EMAIL: craigmnet@cow.physics.wisc.edu
Astrophysics, IDL, Finance, Derivatives | Remove "net" for better response
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