Rank-R matrix autoregressive models for modeling spatio-temporal data

We develop a matrix-variate autoregressive (MAR) model to analyze spatio-temporal data organized on a regular grid in space. The model is an extension of the bilinear MAR spatial model of Hsu, Huang and Tsay $\href{ https://doi.org/10.1080/10618600.2021.1938587 }{[10]}$ by increasing its flexibility and applicability in empirical applications. Specifically, we propose to model each autoregressive (AR) coefficient matrix of the MAR model by $R$ bilinear terms, thereby establishing a rank‑R model. The extension can be interpreted as decomposing the AR dynamics of the data into $R$ bilinear MAR components. We further incorporate a banded neighborhood structure for AR coefficient matrices and utilize a flexible nonstationary low-rank covariance model for the spatial innovation process, leading to a parsimonious model without sacrificing its flexibility. We estimate all parameters of the model by the maximum likelihood method and develop a computationally efficient alternating direction method of multipliers algorithm, involving only closed-form expressions in all steps. Applications to a wind-speed dataset and an employment dataset, as well as two simulation experiments, demonstrate the effectiveness of the proposed method in estimation, model selection, and prediction.