Local Interaction Autoregressive Model for High Dimension Time Series Data

High-dimensional matrix and tensor time series often exhibit local dependency, where each entry interacts mainly with a small neighborhood.Accounting for local interactions in a prediction model can greatly reduce the dimensionality of the parameter space, leading to more efficient inference and more accurate predictions.We propose a Local Interaction Autoregressive (LIAR) framework for high-dimensional matrix and tensor time series forecasting problems, and study Separable LIAR, a matrix variant with shared row and column components.We derive a scalable parameter estimation algorithm via parallel least squares with a BIC-type neighborhood selector.Theoretically, we show consistency of neighborhood selection and derive error bounds for kernel and auto-covariance estimation.Numerical simulations show that the BIC selector recovers the true neighborhood with high success rates, the LIAR achieves small estimation errors, and the forecasts outperform matrix time-series baselines.In a Total Electron Content (TEC) application, the proposed method identifies localized spatio-temporal propagation patterns and achieves improved prediction compared with non-local time series prediction models.