Matrix Autoregressive Spatio-Temporal Models

Abstract

Matrix-variate time series are now common in economic, medical, environmental, and atmospheric sciences, typically associated with large matrix dimensions. We introduce a structured autoregressive (AR) model to characterize temporal dynamics in a matrix-variate time series by formulating the AR matrices in a bilinear form. This bilinear parameter structure reduces the model dimension and highlights dynamic interaction among columns and rows in the AR matrices, making the model highly explainable. We further incorporate spatial information and explore sparsity in the AR coefficients by introducing spatial neighborhoods. In addition, we consider a nonstationary multi-resolution spatial covariance model for innovation errors. The resulting spatio-temporal AR model is flexible in capturing heterogeneous spatial and temporal features while maintaining a parsimonious parameterization. The model parameters are estimated by maximum likelihood (ML) with a fast algorithm developed for computation. We conduct a simulation study and present an application to a wind-speed dataset to demonstrate the merits of our methodology. Supplementary files for this article are available online.

Keywords:

• Bilinear autoregression
• Dimension reduction
• Matrix-variate time series
• Maximum likelihood
• Multi-resolution spline basis functions

Supplementary Materials

The supplementary materials contain R demo code implementing the methods in this article, and the U-wind dataset used in the application.

Additional information

Funding

Nan-Jung Hsu’s research was supported by ROC Ministry of Science and Technology grant MOST 108-2118-M-007-005-MY2. Hsin-Cheng Huang’s research was supported by Academia Sinica Investigator Award AS-IA-109-M05 and by ROC Ministry of Science and Technology grant MOST 109-2118-M-001-008-MY2. The research of R. Tsay was supported in part by the Booth School of Business, University of Chicago, and he also acknowledges the hospitalities received while visiting the Institute of Statistics, National Tsing-Hua University.