Graphical LASSO based Model Selection for Time Series

We propose a novel graphical model selection scheme for high-dimensional stationary time series or discrete time processes. The method is based on a natural generalization of the graphical LASSO algorithm, introduced originally for the case of i.i.d. samples, and estimates the conditional independence graph of a time series from a finite length observation. The graphical LASSO for time series is defined as the solution of an l
₁
-regularized maximum (approximate) likelihood problem. We solve this optimization problem using the alternating direction method of multipliers. Our approach is nonparametric as we do not assume a finite dimensional parametric model, but only require the process to be sufficiently smooth in the spectral domain. For Gaussian processes, we characterize the performance of our method theoretically by deriving an upper bound on the probability that our algorithm fails. Numerical experiments demonstrate the ability of our method to recover the correct conditional independence graph from a limited amount of samples.