Large Gaussian Covariance Matrix Estimation With Markov Structures

Covariance matrix estimation for a large number of Gaussian random variables is a challenging yet increasingly common problem. A fact neglected in practice is that the random variables are frequently observed with certain temporal or spatial structures. Such a problem arises naturally in many practical situations with time series and images as the most popular and important examples. Effectively accounting for such structures not only results in more accurate estimation but also leads to models that are more interpretable. In this article, we propose shrinkage estimators of the covariance matrix specifically to address this issue. The proposed methods exploit sparsity in the inverse covariance matrix in a systematic fashion so that the estimate conforms with models of Markov structure and is amenable for subsequent stochastic modeling. The present approach complements the existing work in this direction that deals exclusively with temporal orders and provides a more general and flexible alternative to explore potential Markov properties. We show that the estimation procedure can be formulated as a semidefinite program and efficiently Computed. We illustrate the merits of these methods through simulation and the analysis of a real data example. Matlab implementation of the proposed methods is also available online as supplemental material.