%% Optimal Fit of a Non-linear Function
% This is a demonstration of the optimal fitting of a non-linear function to a
% set of data.  It uses FMINSEARCH, an implementation of the Nelder-Mead simplex
% (direct search) algorithm, to minimize a nonlinear function of several
% variables.
%
% Copyright 1984-2002 The MathWorks, Inc.
% $Revision: 5.15 $ $Date: 2002/04/02 17:52:33 $%%
% First, create some sample data and plot it.% t = (0:.1:2)';
% y = [5.8955 3.5639 2.5173 1.9790 1.8990 1.3938 1.1359 1.0096 1.0343 ...
%      0.8435 0.6856 0.6100 0.5392 0.3946 0.3903 0.5474 0.3459 0.1370 ...
%      0.2211 0.1704 0.2636]';function [coeff,estimated_lambda] = fitdemo(XData, YData)if nargin<1,XData = (0:.1:2)';YData = [5.8955 3.5639 2.5173 1.9790 1.8990 1.3938 1.1359 1.0096 1.0343 ...0.8435 0.6856 0.6100 0.5392 0.3946 0.3903 0.5474 0.3459 0.1370 ...0.2211 0.1704 0.2636]';clc;
endif ( size(XData) ~= size(YData) )error('The two input matrices have to be of the same sizes');
endplot(XData,YData,'ro'); hold on; h = plot(XData,YData,'b'); hold off;
title('Input data');if ( size(XData, 1) == 1 )t = XData';y = YData';
elseif(size(XData, 2) == 1)t = XData;y = YData;
end%%
% The goal is to fit the following function with two linear parameters and two
% nonlinear parameters to the data:
%
%     y =  C(1)*exp(-lambda(1)*t) + C(2)*exp(-lambda(2)*t)
%
% To fit this function, we've create a function FITFUN.  Given the nonlinear
% parameter (lambda) and the data (t and y), FITFUN calculates the error in the
% fit for this equation and updates the line (h).% type fitfun%%
% Make a guess for initial estimate of lambda (start) and invoke FMINSEARCH.  It
% minimizes the error returned from FITFUN by adjusting lambda.  It returns the
% final value of lambda.start = [1;0];
options = optimset('TolX',0.1);
estimated_lambda = fminsearch('fitfun',start,options,t,y,h);
%estimated_lambda = [estimated_lambda guidata(gcf)];
coeff = guidata(h); % C1,C2
'estimated_lambda = estimated_lambda';
'coeff = coeff';

function err = fitfun(lambda,t,y,handle)
%FITFUN Used by FITDEMO.
%   FITFUN(lambda,t,y,handle) returns the error between the data and the values
%   computed by the current function of lambda.
%
%   FITFUN assumes a function of the form
%
%     y =  c(1)*exp(-lambda(1)*t) + ... + c(n)*exp(-lambda(n)*t)
%
%   with n linear parameters and n nonlinear parameters.%   Copyright 1984-2002 The MathWorks, Inc.
%   $Revision: 5.8 $  $Date: 2002/04/08 20:04:42 $A = zeros(length(t),length(lambda));
for j = 1:length(lambda)A(:,j) = exp(-lambda(j)*t);
end
c = A\y;
z = A*c;
err = norm(z-y);set(gcf,'DoubleBuffer','on');
set(handle,'ydata',z)
guidata(handle, c);
drawnow
pause(.04)