Abstract: The vector auto-regressive (VAR) model is commonly used to model multivariate time series, and there are many penalized methods to handle high dimensionality. However for spatio-temporal data, most of these methods do not consider the spatial and temporal structure of the data, which may lead to unreliable network detection and inaccurate forecasts. This paper proposes a data-driven weighted l₁ regularized approach for spatio-temporal VAR models. Extensive simulation studies compare the proposed method with five existing methods for high-dimensional VAR models, demonstrating advantages of our method over others in terms of parameter estimation, network detection, and out-of-sample forecasts. We also apply our method to a traffic data set to evaluate its performance in a real application. In addition, we explore the theoretical properties of the l₁ regularized estimation of the VAR model under a weakly sparse scenario, in which exact sparsity can be viewed as a special case. To the best of our knowledge, this is the first study to do so. For a general stationary VAR process, we derive the nonasymptotic upper bounds on the l₁ regularized estimation errors, provide the conditions for estimation consistency, and further simplify these conditions for a special VAR(1) case.

Key words and phrases: l₁ regularization, spatio-temporal structure, vector autoregressive model, weak sparsity.