Full Rank Cholesky Factorization for the Re-Weighted Least Squares Applications

Abstract

Factorization of matrices are required frequently in sciences research, in order to simplify computations when solving linear or non-linear systems of equations. In particular, these systems arise in data analysis and also after modeling industrial process, playing an important role in the final solution. When the matrices involved are full rank and squared classical factorizations can be used efficiently. However, when one has to deal with rectangular and/or rank deficient matrices, extensions of these factorizations should be considered, if it is possible with the same good properties. Objective: In this work a full rank Cholesky factorization for rank deficient matrices is described, along with its use in a particular application arising in Statistics, where usually one has to solve overdetermined systems of equations via least squared method or its extensions. Methodology: We describe an algorithm to generate this factorization in an iteratively reweighed least squared method applied to maximum likelihood parameters estimation. Results: Preliminary experiments are presented over synthetic data, supporting the use of this decomposition for rectangular or rank deficient matrices arising in similar applications. Conclusions: The proposal of using the full rank Cholesky factorization in the iteratively reweighted least squared to estimate maximum likehood parameters is promising and could be used to improve CPU time in the solution of similar problems.