Graphical models of autoregressive processes

Abstract

We consider the problem of fitting a Gaussian autoregressive model to a time series, subject to conditional independence constraints. This is an extension of the classical covariance selection problem to time series. The conditional independence constraints impose a sparsity pattern on the inverse of the spectral density matrix, and result in nonconvex quadratic equality constraints in the maximum likelihood formulation of the model estimation problem. We present a semidefinite relaxation, and prove that the relaxation is exact when the sample covariance matrix is block-Toeplitz. We also give experimental results suggesting that the relaxation is often exact when the sample covariance matrix is not block-Toeplitz. In combination with model selection criteria the estimation method can be used for topology selection. Experiments with randomly generated and several real data sets are also included. Introduction Graphical models give a graph representation of relations between random variables. The simplest example is a Gaussian graphical model , in which an undirected graph with n nodes is used to describe conditional independence relations between the components of an n -dimensional random variable x ~ N (0, ∑). The absence of an edge between two nodes of the graph indicates that the corresponding components of x are independent, conditional on the other components. Other common examples of graphical models include contingency tables , which describe conditional independence relations in multinomial distributions, and Bayesian networks , which use directed acyclic graphs to represent causal or temporal relations.