fitting curve with nonlinear function using cftool in matlab

Question

I need to fit a data with a nonlinear function just like this:

y=a+b*cos(c*x+d)+e*exp(f*x);

where a, b, c, d, e and f are the coefficients to be found. I have used the Custom Equation method in cftool, the result is bad, then how to adjust the StartPoint in cftool to get a better result? Otherwise, there are other ways to solve my problem? Thank so much.

Here is my data, each in 1*35:

    x = [1970:2004];
    y = [46808,49416,53094,57237,56677,56198,59673,61826,64158,65220,63108,60944,59543,58779,59882,60087,61825,63104,64963,66902,66443,67061,67273,67372,68679,69995,71522,73292,73932,75826,76954,78105,78439,79892,82631];

---

ok, Dan, here is my code about cftool, the function is generated by Custom Equation of cftool

% This file include two functions:
% 1. Main: testForFit.m
% 2. Function generated by cftool
function testForFit
clc; clear all;
x = 1970:2004;
y = [46808,49416,53094,57237,56677,56198,59673,61826,64158,65220,63108,60944,59543,58779,59882,60087,61825,63104,64963,66902,66443,67061,67273,67372,68679,69995,71522,73292,73932,75826,76954,78105,78439,79892,82631];
plot(x, y, 'or')
[fitresult, gof] = createFit(x, y)
end

function [fitresult, gof] = createFit(x, y)
%CREATEFIT(X,Y)
%  Create a fit.
%
%  Data for 'untitled fit 1' fit:
%      X Input : x
%      Y Output: y
%  Output:
%      fitresult : a fit object representing the fit.
%      gof : structure with goodness-of fit info.
%
%  See also FIT, CFIT, SFIT.
%  Auto-generated by MATLAB on 04-Feb-2014 00:23:03
%% Fit: curve fitting of gived form of function.
[xData, yData] = prepareCurveData( x, y );

% Set up fittype and options.
ft = fittype( 'a+b*cos(c*x+d)+e*exp(f*x)', 'independent', 'x', 'dependent', 'y' );
% ft = fittype( 'a+e*exp(f*x)', 'independent', 'x', 'dependent', 'y' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Display = 'Off';

% http://stackoverflow.com/questions/21551951/fitting-curve-with-nonlinear-function-using-cftool-in-matlab
% Firstly, give the estimation, then calculate it
% funlist={1, @(p,x) cos(p(1)*x+p(2)), @(p,x) exp(p(3)*x)};
% NLPstart = [0.605 -3159.659 0.02369];
% [cdf,abe] = fminspleas(funlist,NLPstart,x,y)
% c=cdf(1); d=cdf(2); f=cdf(3);
% a=abe(1); b=abe(2); e=abe(3);
% opts.StartPoint = [a b c d e f];

% How to give the value of StartPoint ?
opts.StartPoint = [0.957166948242946 0.485375648722841 0.8002804688888 0.141886338627215 0.421761282626275 0.915735525189067];
% opts.StartPoint = [31950 1557 0.605 -3159.659 1.183e-16 0.02369];

% Fit model to data.
[fitresult, gof] = fit( xData, yData, ft, opts );

% Plot fit with data.
figure( 'Name', 'curve fitting' );
h = plot( fitresult, xData, yData );
legend( h, 'y vs. x', 'curve fitting', 'Location', 'NorthEast' );
% Label axes
% 31950 1557 0.605 -3159.659 1.183e-16 0.02369
% title('31950+1557*cos(0.605*x-3159.659)+1.183e-16*exp(0.02369*x)')
xlabel( 'x' );
ylabel( 'y' );
grid on
% x = [1970:2004];
% y = [46808,49416,53094,57237,56677,56198,59673,61826,64158,65220,63108,60944,59543,58779,59882,60087,61825,63104,64963,66902,66443,67061,67273,67372,68679,69995,71522,73292,73932,75826,76954,78105,78439,79892,82631];
% Baidu Zhidao
% http://zhidao.baidu.com/question/198731054514100005.html?push=keyword
end

Some error will occured if the StartPoint is not proper ...

Answer 1

I don't have the CurveFit Toolbox, but supposedly there is a "Fit Options" button near where you select the curve type, leading to a dialog where you can select your own initial point.

I also recommend that you try FMINSPLEAS , which can take advantage of the fact that y has a linear dependence on 3 of your 6 parameters. You also only need to provide an initial guess for the nonlinear parameters NLPstart=[c,d,f]

funlist={1, @(p,x) cos(p(1)*x+p(2)), @(p,x) exp(p(3)*x)};

[cdf,abe] = fminspleas(funlist,NLPstart,x,y);

c=cdf(1); d=cdf(2); f=cdf(3);
a=abe(1); b=abe(2); e=abe(3);