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[Help-gsl] Weighted Levenberg Marquardt

From: Tom Banwell
Subject: [Help-gsl] Weighted Levenberg Marquardt
Date: Mon, 17 Aug 2009 09:57:33 +0000


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



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