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Re: nonlinear regression curve fitting

In response to a recent enquiry about non-linear regression here are the
definitions for a routine that can be used.
I have not had time to set this all up correctly as a package with
contexts etc. and have only briefly tested it.
Therefore, use at your own peril.

Given a function f = F[x,y,...,a,b,...] of several variables x,y,...
which also depends on a number of parameters a,b,...  and a list of data
data = { {x1,y1,...,f1},  {x2,y2,...,f2}, ...  } the routine attempts to
find the values of a,b,... which minimizes the sum of the squares of the
The only restriction on F[] is that Mathematica must be able to calculate
symbolic derivatives of F with respect to a,b,....

An initial estimate of the paramters is required and the routine returns
a "better" estimate of the parameters. If the original estimate is good
enough, repeating the process will eventually produce a converged result.

The definitions follow:

LinearFit[list_,fun_,f0_,var_] :=Block[{n,n2,m,b},
	n = Length[list];
	n2 = Length[fun];
	m = Table[
		Sum[ (fun[[j]] fun[[k]] /. 
			Thread[var -> Drop[list[[i]],-1]]), {i,1,n}],
		{j,1,n2}, {k,1,n2}];
	b = Table[
		Sum[ (fun[[j]] (list[[i,-1]]-f0) /.
			Thread[var -> Drop[list[[i]],-1]]), {i,1,n}],

	npar = Length[par];
	sub = Thread[par -> init];
	linfun = Table[ D[ fun, par[[i]] ], {i,1,npar}] /.sub;
	f0 = fun /. sub;
	coeff = LinearFit[list,linfun,f0,var];

The routine is used as follows:

data = { {x1,y1,..,f1}, {x2,y2,...,f2}, ...};
function = F[x,y,...,a,b,...]; (* whatever function you are fitting *)
pars = {a,b,... };
initialestimate = {a0, b0, ... }; (* reasonable values for the parameters *)
vars = {x,y,...};

estimate = NonLinearFit[data,function,pars,initialestimate,vars]

estimate = NonLinearFit[data,function,pars,estimate,vars] (*keep repeating*)

Alternatively one can use FixedPoint

estimate = FixedPoint[NonLinearFit[data,function,pars,#,vars]&,initialestimate]

Leon Poladian
Theoretical Physics
University of Sydney

leon at

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