Re: [Help-gsl] Weighted Levenberg Marquardt

[John Gehman]

G'day Tom,

I may be misunderstanding your intentions, but if you want to weight  
your data differentially, you can simply incorporate this into your  
subroutines which return the residuals vector and jacobian matrix to  
the minimizer -- "sigma" in the example from the manual, from memory.  
That is, the data structure which you pass into the minimizer is a  
vector of x data values, a vector of y data values, and a vector of  
errors. The objective function to be minimized is the residuals vector  
with elements i = [ ( y_i - f(x_i,a) ) / sigma_i ], and the partial  
derivatives include this attenuating factor as well. Meanwhile, the  
covariance matrix (co-dependence of your optimized parameters in the  
fit) per se is one of the outputs from the minimization routine.

Am I off the mark?

Regards,
john

On 17/08/2009, at 7:57 PM, Tom Banwell wrote:

> Hi,
>
> I'm a computer scientist, not a mathematician and had a question  
> about the Levenberg Marquardt (LM) algorithm and hoped someone would  
> be able to help or provide some advice.  I wanted to perform  
> weighted Least-Squares and I have the following equation:
>
> [ yi - f(xi,a)]^T Vi  [ yi - f(xi,a)]
>
> where yi is the dependent variable, xi is the independent variable,  
> a are the model parameters to be estimated, and Vi is a covariance  
> matrix.
>
> I have solved the problem using an unweighted Least-squares but  
> would prefer to use weighted as some of my data have larger relative  
> uncertainties.  I have seen on the GSL reference manual that I can  
> perform weighted Least-Squares using a scalar, but I wanted to use  
> the full covariance matrix, Vi.  I did think about using the trace  
> or determinant of Vi, but not sure if that is mathematically as  
> sound so wanted to use Vi.  My problem is that I can't work out how  
> to extend my code to include the matrix weight (rather than the  
> scalar weight) and wondered if I needed to modify the internal gsl  
> LM algorithm (or maybe rewrite the algorithm myself) or can apply I  
> apply a matrix weight using the existing gsl LM.
>
> I hope someone can help with this, my email is: tombanwell * at *  
> hotmail * dot * com
>
> Thanks
>
> Tom
>
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