       Re: FindFit & NIntegrate

• To: mathgroup at smc.vnet.net
• Subject: [mg96136] Re: FindFit & NIntegrate
• From: Bill Rowe <readnews at sbcglobal.net>
• Date: Fri, 6 Feb 2009 04:12:56 -0500 (EST)

```On 2/5/09 at 4:42 AM, hans.malissa at gmx.net (hmalissa) wrote:

>I want to use the FindFit function (or similar) to fit some
>experimental data:

>fit=FindFit[data,expr,pars,vars]

>Usually this works very well, but in this particular case the model
>expr consists of a NIntegrate over another function, like:

>model[x_,a_,b_,c_]:=NIntegrate[function[x_,t_,a_,b_,c_],{t,0,inf}];
>fit=FindFit[data,model[x,a,b,c],{a,b,c},x]

>FindFit now complains about non-numerical values; of course that
>makes sense. How can I force FindFit to do the fitting anyway?

I assume the model you are fitting is inherently non-linear,
i.e., that the model cannot be written as  Sum[a[k]
NIntegrate[...], {k...}] where a[k] are the unknown parameters
and everything being fed to NIntegrate has a known value. If you
can linearize the problem in the form I've indicated there
should be no difficulty. If this cannot be done you have a very
difficult problem.

The basic approach would be to approximate the integral with
something that exposes the parameters to FindFit. Possibly, the
integral can be approximated as a polynomial using say a series
expansion. Alternatively, you can replace NIntegrate with code
that uses say Simpson's rule to create an symbolic expression
containing the unknown parameters.

The only other approach I can see to solve this type of problem
is to find another way to model the data that did not involve
solving an integral.

```

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