We develop a new methodology for forecasting matrix-valued time series with historical matrix data and auxiliary vector time series data. We focus on a time series of matrices defined on a static 2-D spatial grid and an auxiliary time series of non-spatial vectors. The proposed model, Matrix AutoRegression with Auxiliary Covariates (MARAC), contains an autoregressive component for the historical matrix predictors and an additive component that maps the auxiliary vector predictors to a matrix response via tensor-vector product. The autoregressive component adopts a bi-linear transformation framework following Chen et al. (2021), significantly reducing the number of parameters. The auxiliary component posits that the tensor coefficient, which maps non-spatial predictors to a spatial response, contains slices of spatially smooth matrix coefficients that are discrete evaluations of smooth functions from a Reproducible Kernel Hilbert Space (RKHS). We propose to estimate the model parameters under a penalized maximum likelihood estimation framework coupled with an alternating minimization algorithm. We establish the joint asymptotics of the autoregressive and tensor parameters under fixed and high-dimensional regimes. Extensive simulations and a geophysical application for forecasting the global Total Electron Content (TEC) are conducted to validate the performance of MARAC.

翻译：本文提出了一种新的方法论，用于预测包含历史矩阵数据与辅助向量时间序列数据的矩阵值时间序列。我们聚焦于定义在静态二维空间网格上的矩阵时间序列，以及非空间向量的辅助时间序列。所提出的模型——带辅助协变量的矩阵自回归（MARAC）——包含历史矩阵预测变量的自回归分量，以及通过张量-向量积将辅助向量预测变量映射为矩阵响应的加性分量。自回归分量采用Chen等人（2021）的双线性变换框架，显著减少了参数数量。辅助分量假设：用于将非空间预测变量映射到空间响应的张量系数，包含空间平滑矩阵系数切片，这些系数是从再生核希尔伯特空间（RKHS）中光滑函数的离散评估。我们提出在惩罚最大似然估计框架下结合交替最小化算法估计模型参数。分别在固定维度与高维度条件下建立了自回归参数与张量参数的联合渐近性。通过大量数值模拟及一项用于预测全球总电子含量（TEC）的地球物理应用，验证了MARAC的性能。