Time-adaptive functional Gaussian Process regression

TL;DR

This paper introduces a novel time-adaptive functional Gaussian Process regression method on manifolds, utilizing spectral analysis and invariance properties to improve spatiotemporal data modeling.

Contribution

It develops a new formulation of functional Gaussian Process regression on manifolds using an Empirical Bayes approach and spectral techniques for dimension reduction.

Findings

Demonstrates finite sample and asymptotic properties through simulations

Effective dimension reduction via spectral truncation

Applicable to synthetic and real spatiotemporal data

Abstract

This paper proposes a new formulation of functional Gaussian Process regression in manifolds, based on an Empirical Bayes approach, in the spatiotemporal random field context. We apply the machinery of tight Gaussian measures in separable Hilbert spaces, exploiting the invariance property of covariance kernels under the group of isometries of the manifold. The identification of these measures with infinite-product Gaussian measures is then obtained via the eigenfunctions of the Laplace-Beltrami operator on the manifold. The involved time-varying angular spectra constitute the key tool for dimension reduction in the implementation of this regression approach, adopting a suitable truncation scheme depending on the functional sample size. The simulation study and synthetic data application undertaken illustrate the finite sample and asymptotic properties of the proposed functional regression predictor.