Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes

TL;DR

This paper investigates how the regularity of spatial sampling affects the accuracy of covariance parameter estimation in Gaussian processes, demonstrating that stronger perturbations improve estimation accuracy.

Contribution

It provides an asymptotic analysis of covariance parameter estimators under perturbed regular grids, revealing the impact of sampling regularity on estimation quality.

Findings

Stronger perturbations lead to better covariance parameter estimation.

Asymptotic covariance matrices depend deterministically on the regularity parameter.

Using groups of closely spaced points benefits covariance estimation.

Abstract

Covariance parameter estimation of Gaussian processes is analyzed in an asymptotic framework. The spatial sampling is a randomly perturbed regular grid and its deviation from the perfect regular grid is controlled by a single scalar regularity parameter. Consistency and asymptotic normality are proved for the Maximum Likelihood and Cross Validation estimators of the covariance parameters. The asymptotic covariance matrices of the covariance parameter estimators are deterministic functions of the regularity parameter. By means of an exhaustive study of the asymptotic covariance matrices, it is shown that the estimation is improved when the regular grid is strongly perturbed. Hence, an asymptotic confirmation is given to the commonly admitted fact that using groups of observation points with small spacing is beneficial to covariance function estimation. Finally, the prediction error, using a consistent estimator of the covariance parameters, is analyzed in details.