Reduced-rank spatio-temporal models

Author

Dulari Dineshika Liyana Arachchillage, Southern Illinois University CarbondaleFollow

Date of Award

8-1-2025

Degree Name

Master of Science

Department

Mathematics

First Advisor

Samadi, Yaser

Abstract

Time and space are two of the most significant and complex dimensions underlying real-world phenomena. Although we often overlook them in daily reasoning, they are crucial for understanding real-world circumstances such as crime rate patterns in neighborhoods, climate shifts, disease outbreaks, and urban traffic. To model these processes accurately, it is essential to consider their spatial and temporal dependencies simultaneously. Spatio-temporal models provide this framework by incorporating spatial structure with temporal evolution. allowing for a more comprehensive understanding of how processes evolve across both dimensions. Unlike time series and spatial models, spatio-temporal models capture both cross-sectional dependencies and dynamic trends over time.Many existing models use spatial weight matrices to parameterize the coefficient matrices. But construction of these matrices requires more information about different locations/ units, can be challenging in high-dimensional settings, and may not reflect true underlying relationships. To address these limitations, a novel approach that directly captures spatial and temporal dependencies through a low-rank structure on coefficient matrices is proposed here.In comparison to vector autoregressive (VAR) models, spatio-temporal models allow the number of cross-sectional units (locations) to diverge. This flexibility leads to the curse of dimensionality and the failure of standard estimation techniques due to the complex, nonlinear structure in the coefficients. To estimate model parameters, a quasi-likelihood estimation method is developed, and a ridge-type ratio estimator is introduced for selecting the appropriate rank. Reduced-rank spatio-temporal models effectively reduce dimensionality, facilitate interpretation, and enhance estimation performance in high-dimensional environments.