A Dynamical Model for Spatio-Temporal Processes Motivated by Second-Order Partial Differential Equations

TL;DR

This paper introduces a new spatio-temporal dynamical model based on second-order SPDEs, using Galerkin's method for finite approximation, with applications demonstrated through numerical experiments.

Contribution

It develops a finite-dimensional dynamical model from second-order SPDEs, enabling efficient computation and parameter estimation for complex spatio-temporal processes.

Findings

Finite-dimensional approximation accurately captures covariance structures.

Model successfully applied to wave, seismic, advection-diffusion, and wildfire processes.

Error between approximate and exact covariance quantified.

Abstract

An important class of spatio-temporal models is constructed by leveraging the hierarchical structure of dynamical (or, state-space) models. This paper proposes a new statistical dynamical model for spatio-temporal processes motivated by second-order stochastic partial differential equations (SPDE). In particular, an infinite-dimensional linear state-space representation is obtained where the state transition is governed by a proposed SDE. Then, using the Galerkin's method, a finite-dimensional approximation to the infinite-dimensional SDE is obtained, yielding a dynamical model with finite states that facilitates computation and parameter estimation. The space-time covariance of the approximated dynamical model is obtained, and the error between the approximate and exact covariance matrices is quantified. Comprehensive numerical investigations, including 2D wave equation, seismic wave propagation, advection-diffusion equations and wildfire aerosol propagation processes, are performed to demonstrate the application of the proposed model. Code is available.