BUSCADOR RECOLECTA

The research in this paper has benefited from discussions with Damjan Skulj, Barbara
Vantaggi, Davide Petturiti, Jasper de Bock and Max Nendel. We also acknowledge the financial support by
project PID2022-140585NB-I00 from the Spanish Ministry of Science and Innovation.

The present paper presents some results that allow us to perform with High Relative Accuracy linear algebra operations with correlation and covariance matrices of Gaussian Markov Random Fields over graphs of paths. Some numerical experiments are carried out showing the computational benefits of this approach.

Gaussian Markov Random Fields are a popular statistical model that has been used successfully in many fields of application. Recent work has studied conditions under which the covariance matrix of a Gaussian Markov Random Field over a graph of paths is totally positive. In such case, many linear algebra operations concerning the covariance matrix can be performed with High Relative Accuracy (the relative error is of order of machine precision). Unfortunately, classical estimators of the covariance matrix do not necessarily yield a totally positive matrix, even when the population covariance matrix is totally positive. Essentially, this inconvenience prevents the available High Relative Accuracy methods to be used with real-life data. Here, we present a method for the estimation of the covariance matrix of a Gaussian Markov Random Field over a graph of paths assuring the estimated covariance matrix (or its inverse) is totally positive.