nonlinear curve fit: how to optimize?

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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!
  6 Comments
Torsten
Torsten on 16 Oct 2015
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
Mona Mahboob Kanafi on 19 Oct 2015
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!

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Accepted Answer

Torsten
Torsten on 19 Oct 2015
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 :-)

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