Non-separable Covariance Kernels for Spatiotemporal Gaussian Processes based on a Hybrid Spectral Method and the Harmonic Oscillator

TL;DR

This paper introduces a new class of physically motivated, non-separable spatiotemporal covariance kernels for Gaussian processes, derived from a hybrid spectral approach based on the harmonic oscillator model, capturing complex space-time interactions.

Contribution

It presents a novel hybrid spectral method to derive non-separable covariance kernels rooted in physical principles, specifically from the stochastic harmonic oscillator model.

Findings

Derived explicit relations for LDHO covariance kernels in three damping regimes.

Introduced covariance kernels with both monotonic and oscillatory decay behaviors.

Illustrated the method by deriving kernels based on the Ornstein-Uhlenbeck process.

Abstract

Gaussian processes provide a flexible, non-parametric framework for the approximation of functions in high-dimensional spaces. The covariance kernel is the main engine of Gaussian processes, incorporating correlations that underpin the predictive distribution. For applications with spatiotemporal datasets, suitable kernels should model joint spatial and temporal dependence. Separable space-time covariance kernels offer simplicity and computational efficiency. However, non-separable kernels include space-time interactions that better capture observed correlations. Most non-separable kernels that admit explicit expressions are based on mathematical considerations (admissibility conditions) rather than first-principles derivations. We present a hybrid spectral approach for generating covariance kernels which is based on physical arguments. We use this approach to derive a new class of physically motivated, non-separable covariance kernels which have their roots in the stochastic, linear, damped, harmonic oscillator (LDHO). The new kernels incorporate functions with both monotonic and oscillatory decay of space-time correlations. The LDHO covariance kernels involve space-time interactions which are introduced by dispersion relations that modulate the oscillator coefficients. We derive explicit relations for the spatiotemporal covariance kernels in the three oscillator regimes (underdamping, critical damping, overdamping) and investigate their properties. We further illustrate the hybrid spectral method by deriving covariance kernels that are based on the Ornstein-Uhlenbeck model.