High-Dimensional Multivariate VAR Estimation with Spatio-Temporal Structure

TL;DR

This paper introduces a structured estimation method for high-dimensional multivariate VAR models that incorporates spatial information, improving support recovery and interpretability in spatio-temporal systems.

Contribution

It develops a bi-convex optimization approach with an alternating algorithm to estimate spatial and variable dependence matrices, with theoretical guarantees and practical advantages.

Findings

Accurately recovers sparse transition structures in simulations.

Outperforms existing methods in support recovery and estimation accuracy.

Recovers interpretable climate variable networks in real data.

Abstract

High-dimensional vector autoregressive (VAR) models provide a flexible framework for characterizing dynamic dependence in multivariate spatio-temporal systems, but their unrestricted estimation becomes infeasible when multiple variables are observed over many spatial locations. This paper develops a structured estimation procedure for high-dimensional multivariate VAR processes that explicitly incorporates spatial information. We decompose each block transition matrix into a cross-variable dependence coefficient and a spatial transition matrix, and constrain the spatial transition matrices through a pre-specified spatial graph. The resulting estimator is formulated as a weighted $\ell_1$-regularized least-squares problem, where the weights encode spatial proximity or topological similarity and induce stronger shrinkage on spatially implausible interactions. Since the objective function is bi-convex, we estimate the cross-variable dependence matrix and the spatial transition matrices through an alternating convex-search algorithm implemented with ADMM. Under stability and restricted-eigenvalue-type conditions for high-dimensional VAR processes, we establish convergence to a blockwise stationary point in the subgradient sense and derive high-probability estimation error bounds for both components of the model. Simulation studies demonstrate that the proposed estimator accurately recovers sparse transition structures and improves over existing two-step $\ell_1$-regularized methods in support recovery and estimation accuracy. An application to North American climate data illustrates that the method recovers interpretable variable-dependence networks and spatial interaction patterns across different climate regions.