curve fitting with dependencies between the coefficients

Hello everybody,

I have a problem to fit a curve to my data, because I cannot set dependencies between the coefficients of the fit!

The function is: w1*(1-exp(-a1*(t^n1))) + w2*(1-exp(-a2*(t^n2))) + w3 *(1-exp(-a3*(t^n3)))

And the secondary contion should be: w1 + w2 + w3 == 1

%% Fit:
[xData, yData] = prepareCurveData( [], conversion );

ft = fittype( 'w1*(1-exp(-a1*(t^n1)))   + w2*(1-exp(-a2*(t^n2)))   + w3 *(1-exp(-a3*(t^n3)))', 'independent', 't', 'dependent', 'y' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Display = 'Off';
opts.Lower = [1e-08 1e-08 1e-08 0.9 0.5 1.5 0 0 0];
opts.StartPoint = [0.001 0.001 0.001 1 0.5 2 0.333 0.333 0.333];
opts.Upper = [1 1 1 1.1 0.6 3.5 1 1 1];
% Fit model to data.
[fitresult, gof, output] = fit( xData, yData, ft, opts );

I also tried the optimization tool, but I couldn´t solve my problem

I would realy appreciate if somebody could help me

Answer 1

[Matt J]

Re-express your model function as,

sin(d1)^2*(1-exp(-a1*(t^n1)))   + cos(d1)^2*sin(d2)^2*(1-exp(-a2*(t^n2)))   +...
                                  cos(d1)^2*cos(d2)^2*(1-exp(-a3*(t^n3)))

and the constraint will be innately satisfied with

w1=sin(d1)^2
w2=cos(d1)^2*sin(d2)^2
w3=cos(d1)^2*cos(d2)^2

Or, use fmincon and impose linear equality constraints directly.

[Thomas Wolfinger]

Thanks a lot! It is a good solid solution. Only a mathematical topic.

Great help!

[Matt J]

You're welcome, but please Accept-click the answer if it resolved your problem.