  A Vector Auto-Regressive (VAR) model is commonly used to model multivariate
time series, and there are many penalized methods to handle high
dimensionality. However in terms of spatio-temporal data, most methods do not
take the spatial and temporal structure of the data into consideration, which
may lead to unreliable network detection and inaccurate forecasts. This paper
proposes a data-driven weighted l1 regularized approach for spatio-temporal VAR
model. Extensive simulation studies are carried out to compare the proposed
method with four existing methods of high-dimensional VAR model, demonstrating
improvements of our method over others in parameter estimation, network
detection and out-of-sample forecasts. We also apply our method on a traffic
data set to evaluate its performance in real application. In addition, we
explore the theoretical properties of l1 regularized estimation of VAR model
under the weakly sparse scenario, in which the exact sparsity can be viewed as
a special case. To the best of our knowledge, this direction has not been
considered yet in the literature. For general stationary VAR process, we derive
the non-asymptotic upper bounds on l1 regularized estimation errors under the
weakly sparse scenario, provide the conditions of estimation consistency, and
further simplify these conditions for a special VAR(1) case.
