On Prediction Properties of Kriging: Uniform Error Bounds and Robustness

TL;DR

This paper provides uniform error bounds and robustness analysis for kriging predictors, enhancing understanding of their convergence, design strategies, and performance under covariance misspecification in Gaussian process modeling.

Contribution

It derives uniform error bounds for kriging predictions applicable to scattered points in any dimension, including cases with covariance misspecification, and analyzes their convergence and robustness.

Findings

Derived uniform error bounds for kriging predictors.

Analyzed convergence rates for Gaussian and Matérn covariance functions.

Explored robustness of kriging under covariance misspecification.

Abstract

Kriging based on Gaussian random fields is widely used in reconstructing unknown functions. The kriging method has pointwise predictive distributions which are computationally simple. However, in many applications one would like to predict for a range of untried points simultaneously. In this work we obtain some error bounds for the (simple) kriging predictor under the uniform metric. It works for a scattered set of input points in an arbitrary dimension, and also covers the case where the covariance function of the Gaussian process is misspecified. These results lead to a better understanding of the rate of convergence of kriging under the Gaussian or the Mat\'ern correlation functions, the relationship between space-filling designs and kriging models, and the robustness of the Mat\'ern correlation functions.