New Results on Graphical Modeling of High-Dimensional Dependent Time Series

We consider the problem of inferring the conditional independence graph (CIG) of a high-dimensional stationary multivariate Gaussian time series. A sparse-group lasso-based frequency-domain formulation of the problem has been considered in the literature where the objective is to estimate the sparse inverse power spectral density (PSD) of the data via optimization of a sparse-group lasso based penalized log-likelihood cost function that is formulated in the frequency-domain. The CIG is then inferred from the estimated inverse PSD. Optimization in the previous approach was performed using an alternating minimization (AM) approach whose performance depends upon choice of a penalty parameter. In this paper we investigate an alternating direction method of multipliers (ADMM) approach for optimization to mitigate dependence on the penalty parameter. We also investigate selection of the tuning parameters based on Bayesian information criterion, and illustrate our approach using synthetic and real data. Comparisons with the "usual" i.i.d. modeling of time series for graph estimation are also provided.