A New Spatio-Temporal Model Exploiting Hamiltonian Equations

TL;DR

This paper introduces a novel nonparametric, non-stationary spatio-temporal model based on modified Hamiltonian equations with Gaussian process stochasticity, demonstrating improved performance over traditional Gaussian processes in simulations and real data.

Contribution

It proposes a new spatio-temporal modeling approach leveraging Hamiltonian equations with stochastic modifications, providing a flexible nonparametric framework with Bayesian inference methods.

Findings

Significant improvement over non-stationary Gaussian processes in simulations.

Model exhibits nonparametric, non-stationary, non-separable, and non-Gaussian properties.

Theoretical analysis confirms continuity and smoothness of the process.

Abstract

The solutions of Hamiltonian equations are known to describe the underlying phase space of a mechanical system. In this article, we propose a novel spatio-temporal model using a strategic modification of the Hamiltonian equations, incorporating appropriate stochasticity via Gaussian processes. The resultant spatio-temporal process, continuously varying with time, turns out to be nonparametric, non-stationary, non-separable, and non-Gaussian. Additionally, the lagged correlations converge to zero as the spatio-temporal lag goes to infinity. We investigate the theoretical properties of the new spatio-temporal process, including its continuity and smoothness properties. We derive methods for complete Bayesian inference using MCMC techniques in the Bayesian paradigm. The performance of our method has been compared with that of a non-stationary Gaussian process (GP) using two simulation studies, where our method shows a significant improvement over the non-stationary GP. Further, applying our new model to two real data sets revealed encouraging performance.