We develop a new methodology for forecasting matrix-valued time series with historical matrix data and auxiliary vector time series data. We focus on time series of matrices with observations distributed on a fixed 2-D spatial grid, i.e., the spatio-temporal data, 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，用于结合历史矩阵数据和辅助向量时间序列数据对矩阵值时间序列进行预测。我们重点关注分布在固定二维空间网格上的矩阵时间序列观测（即时空数据），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模型（Matrix AutoRegression with Auxiliary Covariates, MARAC），包含一个用于历史矩阵预测变量的自回归分量，以及一个通过张量-向量乘积将辅助向量预测变量映射为矩阵响应变量的加性分量。自回归分量采用Chen等人（2021）提出的双线性变换框架，可显著减少参数数量。辅助分量假设将非空间预测变量映射到空间响应变量的张量系数包含空间光滑的矩阵系数切片，这些切片是可再生核希尔伯特空间（RKHS）中光滑函数的离散评估。我们建议在惩罚最大似然估计框架下，结合交替最小化算法来估计模型参数。我们建立了固定维数和高维情形下自回归参数与张量参数的联合渐近性质。通过大量模拟实验以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模型的性能。