RE: Non-Linear implicit Curve Fitting

[Damian Harty]

> >Good Day
> >
> >I have two data sets.x and F that has about 10000 points each.
> >
> >I also have the following equation:
> >F = C_CV .* alpha.^( x - 1) .* sqrt(a .* F.^2+ b * F + c)
> >
> >I want to fit that equation on the data above. everything is unknown
> >as specified.
> >Any idea on how to accomplish this?
> >
> >I tried a few methods put nothing gave me a proper fit

I've had great results with Nelder Mead methods - fminsearch in Matlab and 
nelder_mead_min in Octave. There are some good examples here:

http://www.cogsci.ucsd.edu/~desa/109/nonlinnotes.html

It's instructive using the animation of the fit to start with because in the 
(unlikely) event of the fit failing to converge, you can use it to suggest more 
plausible start values. Use the lower part of the tutorial, the top is somewhat 
trivial. I fitted datasets of about that size without any real difficulty, and 
quite quickly.

An important hint I was given was to manipulate the functions so that all your 
unknowns are within 1 or 2 orders of magnitude. The Nelder Mead method doesn't 
like it if one of your variables is 9,872 and another one is 4.26E-5. It's easy 
to do but you'll never guess you need to do it unless someone tells you. Oh, 
and don't start with all zeros. Nelder Mead Downhill Simplex is something of a 
donkey of a fitting engine - a reliable plodder that almost always gets there 
regardless of how you load it up.

I didn't find the syntax of the call very instinctive but using the above 
tutorials I did what I needed to do.

My example was something like this:

     initial_BCDE_guess=[0.5; 0.5; 0.5; 0.5];
     estimated_BCDE = nelder_mead_min(test_fit_function4(BCDE,X,Y_DASH),ctl)

In case it's not obvious, X is the ordinate dataset (no need for them to be 
monotonic), Y_DASH is the data I'm wanting to fit - the same size as X, 
obviously. The function test_fit_function_4 was pretty simple:

    function err = test_fit_function4(par,input,data_to_fit)
    Z = par(3)*sin( par(2)*atan(par(1)*(1-par(4))*input + 
par(4)*atan(par(1)*input)) );
    err = norm(Z-data_to_fit);

My function was quite non-linear, as you can see. I'm not guaranteeing this 
code works as is - I've snipped and pasted it so it's untested - but the logic 
should get you there if you persevere a bit.

I've since used it to fit a decaying sine wave to noisy test data and got good 
results from that, too.

Damian


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