🤖 AI Summary
This paper addresses matrix-valued time series forecasting on static two-dimensional grids. We propose a joint modeling framework that simultaneously incorporates historical matrix sequences and non-spatial auxiliary vector sequences. Our method innovatively unifies a bilinear low-parameter autoregressive structure—with explicit spatiotemporal autocorrelation modeling for matrix sequences—with a reproducing kernel Hilbert space (RKHS)-regularized tensor mapping, which ensures high-dimensional interpretability and spatial smoothness in mapping auxiliary variables to matrix responses. The model is estimated via tensor-vector products and penalized maximum likelihood, solved by an alternating minimization algorithm. We rigorously establish joint asymptotic consistency and convergence rates of the estimators. Extensive simulations and real-world global ionospheric total electron content (TEC) forecasting demonstrate substantial improvements over state-of-the-art baselines.
📝 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 from a Reproducible 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.
