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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 |

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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**[Help-gsl] Weighted Levenberg Marquardt**,
*Tom Banwell* **<=**