Inference for Gaussian Processes with Matern Covariogram on Compact Riemannian Manifolds
Authors: Didong Li, Wenpin Tang, Sudipto Banerjee; Volume 24(101):1−26, 2023.
Abstract
Gaussian processes are widely used as versatile modeling and predictive tools in spatial statistics, functional data analysis, computer modeling, and various machine learning applications. They have been extensively studied in Euclidean spaces, where they are specified using covariance functions or covariograms to model complex dependencies. However, there is a growing interest in Gaussian processes over Riemannian manifolds to develop more flexible inferential frameworks for non-Euclidean data. While numerical approximations through graph representations have been well studied for the Matern covariogram and heat kernel, there has been relatively limited attention to the behavior of asymptotic inference on the covariogram parameters. In this paper, we focus on the asymptotic behavior of Gaussian processes constructed over compact Riemannian manifolds. Using a recently introduced Matern covariogram on a compact Riemannian manifold, we employ formal notions and conditions to derive the identifiable parameter, known as the microergodic parameter, and formally establish the consistency of the maximum likelihood estimate and the asymptotic optimality of the best linear unbiased predictor. We also provide a specific example of the circle as a compact Riemannian manifold and conduct numerical experiments to illustrate and validate the theory.
[Abstract]