Maximum-likelihood estimation of autoregressive models with conditional independence constraints

We propose a convex optimization method for maximum likelihood estimation of autoregressive models, subject to conditional independence constraints. This problem is an extension to times series of the classical covariance selection problem in graphical modeling. The conditional independence constraints impose quadratic equalities on the autoregressive model parameters, which makes the maximum likelihood estimation problem nonconvex and difficult to solve. We formulate a convex relaxation and prove that it is exact when the sample covariance matrix is block-Toeplitz. We also observe experimentally that in practice the relaxation is exact under much weaker conditions. We discuss applications to topology selection in graphical models of time series, by enumerating all possible topologies, and ranking them using information-theoretic model selection criteria. The method is illustrated by an example of air pollution data.