In this paper, we propose a new model for forecasting time series data distributed on a matrix-shaped spatial grid, using the historical spatio-temporal data together with auxiliary vector-valued time series data. We model the matrix time series as an auto-regressive process, where a future matrix is jointly predicted by the historical values of the matrix time series as well as an auxiliary vector time series. The matrix predictors are associated with row/column-specific autoregressive matrix coefficients that map the predictors to the future matrices via a bi-linear transformation. The vector predictors are mapped to matrices by taking mode product with a 3D coefficient tensor. Given the high dimensionality of the tensor coefficient and the underlying spatial structure of the data, we propose to estimate the tensor coefficient by estimating one functional coefficient for each covariate, with 2D input domain, from a Reproducing Kernel Hilbert Space. We jointly estimate the autoregressive matrix coefficients and the functional coefficients under a penalized maximum likelihood estimation framework, and couple it with an alternating minimization algorithm. Large sample asymptotics of the estimators are established and performances of the model are validated with extensive simulation studies and a real data application to forecast the global total electron content distributions.

翻译：本文提出一种新型模型，用于预测分布在矩阵形空间网格上的时间序列数据，该模型融合历史时空数据与辅助向量值时间序列数据。我们将矩阵时间序列建模为自回归过程，其中未来矩阵由矩阵时间序列的历史值及辅助向量时间序列共同预测。矩阵预测因子通过行/列特异性自回归矩阵系数，经由双线性变换映射至未来矩阵；而向量预测因子则通过三维系数张量的模积转化为矩阵形式。鉴于张量系数的高维性与数据固有的空间结构，我们提出从再生核希尔伯特空间估计每个协变量的函数系数（其输入域为二维），从而实现张量系数的估计。在惩罚最大似然估计框架下，我们联合估计自回归矩阵系数与函数系数，并耦合交替最小化算法进行求解。建立了估计量的大样本渐近性质，通过大量仿真实验及全球总电子含量分布预测的真实数据应用验证了模型性能。