Space-Time Models based on Random Fields with Local Interactions

TL;DR

This paper introduces a novel approach for deriving space-time covariance functions using effective equations of motion and linear response theory, enabling more physically motivated models for complex phenomena.

Contribution

It proposes a new method to generate space-time covariance functions by solving effective equations of motion based on Hamiltonian dynamics and Langevin equations.

Findings

Derived new forms of space-time covariance functions.

Linked covariance functions to physical Hamiltonian models.

Extended Gaussian field theory with curvature term.

Abstract

The analysis of space-time data from complex, real-life phenomena requires the use of flexible and physically motivated covariance functions. In most cases, it is not possible to explicitly solve the equations of motion for the fields or the respective covariance functions. In the statistical literature, covariance functions are often based on mathematical constructions. We propose deriving space-time covariance functions by solving "effective equations of motion", which can be used as statistical representations of systems with diffusive behavior. In particular, we propose using the linear response theory to formulate space-time covariance functions based on an equilibrium effective Hamiltonian. The effective space-time dynamics are then generated by a stochastic perturbation around the equilibrium point of the classical field Hamiltonian leading to an associated Langevin equation. We employ a Hamiltonian which extends the classical Gaussian field theory by including a curvature term and leads to a diffusive Langevin equation. Finally, we derive new forms of space-time covariance functions.