Statistics and Its Interface

Volume 17 (2024)

Number 2

Special issue on statistical learning of tensor data

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

Pages: 275 – 290

DOI: https://dx.doi.org/10.4310/23-SII812

Authors

Nan-Jung Hsu (Institute of Statistics, National Tsing-Hua University, Hsinchu, Taiwan)

Hsin-Cheng Huang (Institute of Statistical Science, Academia Sinica, Taipei, Taiwan)

Ruey S. Tsay (Booth School of Business, University of Chicago, Illinois, U.S.A.)

Tzu-Chieh Kao (Institute of Statistics, National Tsing-Hua University, Hsinchu, Taiwan)

Abstract

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.

Keywords

alternating direction method of multipliers, Bayesian information criterion, Kronecker product, low-rank approximation, matrix-variate time series, nonstationary spatial model, singular value decomposition

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Nan-Jung Hsu’s research was supported by ROC National Science and Technology Council grant 110-2118-M-007-003-MY2.

Hsin-Cheng Huang’s research was supported by Academia Sinica Investigator Award AS-IA-109-M05, and by ROC National Science and Technology Council grant 111-2118-M-001-011-MY3.

The research of R. Tsay was supported in part by the Booth School of Business, University of Chicago, and he also acknowledges the hospitality received while visiting the Institute of Statistics, National Tsing-Hua University.

Received 25 October 2022

Accepted 12 August 2023

Published 1 February 2024