Matrix Autoregressive Model with Vector Time Series Covariates for Spatio-Temporal Data

TL;DR

This paper introduces MARAC, a novel matrix autoregressive model incorporating auxiliary vector data for spatio-temporal forecasting, with reduced parameters and smooth spatial coefficients, validated through simulations and geophysical data.

Contribution

The paper proposes a new MARAC model that combines matrix autoregression with auxiliary covariates, featuring a tensor approach and RKHS-based smoothness, along with asymptotic theory and practical validation.

Findings

MARAC effectively forecasts TEC with high accuracy.

The model reduces parameter complexity compared to traditional methods.

Simulation results confirm robustness and efficiency.

Abstract

We develop a new methodology for forecasting matrix-valued time series with historical matrix data and auxiliary vector time series data. We focus on a time series of matrices defined on a static 2-D spatial grid 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 on a spatial grid from a Reproducing 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.