🤖 AI Summary
This study addresses the challenge of effectively estimating spatiotemporal dynamic dependencies in high-dimensional multivariate vector autoregressive (VAR) models, where parameter proliferation impedes reliable inference. The authors propose a novel decomposition of the transition matrix into inter-variable dependency coefficients and a spatial transition matrix, the latter constrained by a predefined spatial graph. Structured estimation is achieved via weighted ℓ₁-regularized least squares, embedding spatial graph information—hitherto unexplored in high-dimensional VAR modeling—directly into the estimation framework. The resulting biconvex optimization problem is efficiently solved using the alternating direction method of multipliers (ADMM), with theoretical guarantees established under stability and restricted eigenvalue conditions. Empirical results demonstrate that the proposed method significantly outperforms existing two-stage ℓ₁ approaches in both support recovery and estimation accuracy, successfully uncovering interpretable variable dependency networks and inter-regional spatial interaction patterns in North American climate 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.