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.

翻译：高维向量自回归（VAR）模型为表征多元时空系统中的动态依赖性提供了灵活框架，但当多个变量在大量空间位置上被观测时，其无约束估计变得不可行。本文提出一种显式融入空间信息的高维多元VAR过程结构化估计方法。我们将每个分块转移矩阵分解为跨变量依赖系数和空间转移矩阵，并通过预设的空间图约束空间转移矩阵。所得估计量被表述为加权$\ell_1$正则化最小二乘问题，其中权重编码空间邻近性或拓扑相似性，并对空间上不可信的交互施加更强的收缩。由于目标函数是双凸的，我们通过交替凸搜索算法（采用ADMM实现）交替估计跨变量依赖矩阵和空间转移矩阵。在高维VAR过程的稳定性和受限特征值类条件下，我们建立了次梯度意义下的分块驻点收敛性，并推导了模型两个分量的高概率估计误差界。仿真研究表明，所提估计量能准确恢复稀疏转移结构，且在支撑恢复和估计精度上优于现有两步$\ell_1$正则化方法。对北美气候数据的应用表明，该方法能恢复不同气候区域间可解释的变量依赖网络和空间交互模式。