Graphical models and its estimation in time series analysis

Abstract

Graphical time series models encode the dynamic relationships among the variables in multivariate time series in graphs, in which nodes represent the variables and edges characterize the conditional dependence. In applications, the graph structure is not known in advance, and it is of interest to estimate and determine the graph based on samples. To determine graphical time series models, we propose two estimation methods based on sparse vector autoregressive models. An alternating maximization method is introduced to estimate sparse vector autoregressive models under sparsity constraints on both the autoregressive coefficients and the inverse noise covariance matrix. This alternating method estimates sparse vector autoregressive models by considering the maximum likelihood estimation with the sparsity constraints as a biconcave problem. Such optimization problem is concave when either the autoregressive coefficients or the inverse covariance matrix is fixed. Simulation experiments study the estimation performance of the alternating method and compare with other non-linear optimization methods. We also introduce two approaches in determining the sparsity constraints. These two methods are studied by simulation studies for comparisons. Real data examples are provided as illustrations. The sparsity constraints in the alternating maximization method, however, require being identified before the estimation procedure. A penalized likelihood estimation for vector autoregressive models is proposed to encourage sparsity on both the autoregressive coefficients and the inverse noise covariance matrix. This penalization method implements penalty terms on the autoregressive coefficients and the off-diagonal elements of the inverse covariance matrix to achieve parsimonious models. The finite sample properties of the penalized likelihood estimator are investigated by simulation experiments. A real dataset application is presented for demonstration.