nonlinear curve fit: how to optimize?

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Mona Mahboob Kanafi
Mona Mahboob Kanafi 2015-10-15
回答： Torsten 2015-10-19
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!
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Torsten
Torsten 2015-10-16
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 2015-10-19
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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Torsten
Torsten 2015-10-19
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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