Regularized Estimation in High-Dimensional Vector Auto-Regressive Models using Spatio-Temporal Information

TL;DR

This paper introduces a weighted l1 regularized method for high-dimensional spatio-temporal VAR models, improving network detection and forecasting by leveraging spatial-temporal data structure, with theoretical guarantees and real-world application.

Contribution

It proposes a novel data-driven weighted l1 regularization approach for spatio-temporal VAR models, incorporating spatial-temporal structure and providing theoretical analysis under weak sparsity.

Findings

Enhanced parameter estimation accuracy

Improved network detection performance

Better out-of-sample forecast results

Abstract

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.