nonlinear curve fit: how to optimize?

I have a custom model which I want to fit to my data. The model works manually, i.e. when I know approximately the fit paramaters. But now I need to optimize this solution, so that it works for similar curves (the one that I will give here is only a perfect noise free data), so please consider this problem in a general case.

The fit model is:

function [x,errorfitted] = fit1d_ABCpara(q,psd1d)
x0 = [2e-10,6e-4, 2.4];
lb = [1e-11, 3e-04,2];
ub = [Inf,3e-3,3];
fun = @(x,xdata)0.5e14 * x(1) .* (1+((x(2).*q).^2)).^-((x(3)/2));
[x,errorfitted] = lsqcurvefit(fun,x0,q,psd1d,lb,ub);

This is the curve for original data points:

This is the fit I get from the code above for my data in log-log space:

But this is what I want and I could get the fit by manually changing my fit parameters:

How can I optimize my 3 parameters?

Thanks in advance!

[Torsten]

Try what happens if you multiply your ydata and your function by a factor of, say, 1e14.

Best wishes

Torsten.

[Mona Mahboob Kanafi]

Thanks a lot! It helped a little bit, but didn't fix the problem. Now, I can get some kind of fit, but it's not optimizing! I edited the question and now added further explanation.

[Torsten]

To get the curve you want, you will have to introduce different weights for different data points. Deviances between data and model for data with high x-values must be weighted more than deviances between data and model for data with low x-values.

Use lsqnonlin instead of lsqcurvefit, define the F_i as

F_i = (y_func(xdata(i))-ydata(i))/ydata(i)

and see what you get.

Best wishes

Torsten.

[Mona Mahboob Kanafi]

Dear Torsen, Thank you very much! I got the idea but sorry to say that I can't implement it! I didn't get where to apply F_i? (F_i is the weight, isn't it?)

Could you please give me some step-wise structure how to proceed and when/where to apply the weighting in the lsqnonlin? Thanks!

[Torsten]

function [x,errorfitted] = fit1d_ABCpara(q,psd1d)
x0 = [2e-10,6e-4, 2.4];
lb = [1e-11, 3e-04,2];
ub = [Inf,3e-3,3];
fun=@(x)(0.5 * x(1) .* (1+((x(2).*q).^2)).^-((x(3)/2))-psd1d)./psd1d;
[x,errorfitted]=lsqnonlin(fun,x0,lb,ub);

Best wishes

Torsten.

[Mona Mahboob Kanafi]

Dear Torsen, Thank you so much! Weighting solved the problem!! I would really appreciate if you could rewrite your comment as a separate answer, so that I can accept your answer and close this trend. Thanks for the help again!

Answer 1

[Torsten]

To get the curve you want, you will have to introduce different weights for different data points. Deviances between data and model for data with high x-values must be weighted more than deviances between data and model for data with low x-values.

Here is a suggestion on how to modify your code:

function [x,errorfitted] = fit1d_ABCpara(q,psd1d)
x0 = [2e-10,6e-4, 2.4];
lb = [1e-11, 3e-04,2];
ub = [Inf,3e-3,3];
fun=@(x)(0.5 * x(1) .* (1+((x(2).*q).^2)).^-((x(3)/2))-psd1d)./psd1d;
[x,errorfitted]=lsqnonlin(fun,x0,lb,ub);

Best wishes

Torsten.

P.S. Nice to hear that it worked :-)