How to fit a nonlinear function with parameter-dependent constraint?

2013 3 18
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Hello,
I want to fit the nonlinear function
y(x)=a(1)*(C1/x)^a(2)
to experimental data. Here, a(1) and a(2) are the parameters to be optimized. C1 is a known constant.
In order to avoid singularity at x=0, I'd like to manipulate the fitting function and want to set
y(x=0)=a(1)*(a(2)+1)*(C1/C2)^a(2)
where C2 ist another known constant. Since y(x=0) depends on a(1) and a(2), I don't see any way to implement this.
Do you have any suggestions?
Thanks in advance!

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Alan Weiss
Alan Weiss 2013-3-18
编辑：Alan Weiss 2013-3-18

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Why not take a logarithm?
log y(x) = log(a(1)) + a(2)*log(C1/x)
Have log(a(1)) and a(2) be the variables to find, you now have a linear regression. Solve it using LinearModel.fit or just plain \ (mldivide).
In detail, your data points are:
Independent variable: measurements of log(C1/x)
Response variable: measurements of log(y)
Make a matrix XX with two columns, the independent variable in column 2, and ones in column 1. Make a matrix YY with the response variable.
Your model is XX*a = YY.
Least squares solution: ahat = XX \ YY.
Alan Weiss
MATLAB mathematical toolbox documentation

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Markus
Markus 2013-3-18
Thanks for you post, Alan. Taking the logarithm is a good idea, I obviously forgot about.
However, I still don't see how to implement the function manipulation at x=0 that I stated in the question. Do I overlook something in your suggestion?
Alan Weiss
Alan Weiss 2013-3-19
Do you have data with x = 0? If so, is y = Inf there? If x = 0 and y is not = Inf, then your model is no good, and you need to come up with a better model. If you have data with x = 0 and y = Inf, then simply throw those points away, they add nothing to the fitting of the parameters.
If you have no data with x = 0, then don't worry about it.
Alan Weiss
MATLAB mathematical toolbox documentation
Markus
Markus 2013-3-19
To answer your questions: I do have data at x=0 which is not equal to Inf. And yes, my model is not matching the envelope of my experimental data very well, but still its the one I need to fit.
Inspired by your post I solved the problem with "fmincon". I added a third coloumn to XX, with its first row element equal to one, all others equal to zero. Additionally, I set
p(3)=log10(a(2)+1);
so my parameters are:
p(1)=log10(a(1));
p(2)=a(2);
p(3)=log10(a(2)+1);
My modelfunction is:
f_temp=XX*p;
f=sqrt(sum(log10(YY)-f_temp)^2);
where f ist my objective function I aim to minimize with fmincon like
x=fmincon(@modelfunction,x0,[],[],[],[],[],[],@coneq,opt);
with a nonlinear constraint to p(3) given by
function [c,ceq]=coneq(x)
c=[];
ceq=log10(1+p(2))-p(3);
end
This seems to work for me.

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