1 Graphical models of autoregressive processes

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 times 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 blockToeplitz. 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.