I'm not at all surprised that the fitting programs have trouble!
Here are some comments/questions that might give you food for
thought...
Are you sure the "theory" is reasonable and correct? Doesn't the
theory give you any of the exponents in this complicated function?
How does the theory lead to such a complicated,
strange form? It's very rare to see such a convoluted set of
trigonometric, polynomial, and exponential functions coming out of
a theory.
In what sense is the formula empirical? Usually, an empirical form
is derived from the data's behavior, or from some very general
(and often arbitrary) representation, or from dimensional analysis.
In most of these cases, a simpler form or approach should be available.
How many points are in your dataset? Do you have enough data and
good enough coverage of the parameter space to have any hope of
determining the parameters?
What is the domain of your data? Are the data points' ranges of values
consistent with the function? Your f(x,y) can't exceed 1 in value,
if I'm reading your formula (and it's typed) correctly, and if
I've made reasonable guesses about the signs of the variables' values
(all positive?). Is that ok according to the dataset you're trying
to fit? (Or if I've made the wrong assumption, does a similar
thought process, correctly applied, help determine the suitability
of the functional form?)
Have you thought about the asymptotic behavior? Is it reasonable
for the situation you're trying to describe? I assume your arc tan
is in the interval of 0 - 2*pi? If so, and assuming g & h are
1 or greater (do you have any bounds on them?), the exponential
term will be pretty quick at killing off any result from the
arc tan (still assuming positive values). The resulting f(x,y) will
quickly approach 1 whenever the exponent of the exp is less
than (say) -4.
Are you sure there's nothing simpler? I'd certainly spend some time
looking at simpler forms before I tried to fit data with
something so bizarre. I'd also try looking at various simple
plots of the data to see if the data can justify a simple or
complex form. Is your data relatively free of noise? The combination
of such a nonlinear function and noisy data could turn your
problem from awful into impossible!
If you are strongly convinced the function is the one you need to use,
try some plots using various hypothetical values of several of the
parameters to try to get some intuitive connection between the
formula and the data. See any promising correspondence?
Lastly, do you have any expert(s) closer at hand who can help you
check out the derivation of the function? At first look, I'm
far more concerned about the functional form than about finding
a fitting program that could handle it.
Good luck!
William C. Mead
wcm@ansr.com
Visit "Adaptive Network Solutions Research, Inc." on the web
at http://www.ansr.com !