On Prediction Properties of Kriging: Uniform Error Bounds and Robustness

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 and universal 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 Matérn correlation functions, the relationship between space-filling designs and kriging models, and the robustness of the Matérn correlation functions. Supplementary materials for this article are available online.

Keywords:

• Gaussian process modeling
• Radial basis functions
• Space-filling designs
• Spatial statistics
• Uniform convergence

Acknowledgments

The authors are grateful to an AE and referees for very helpful comments.

Additional information

Funding

Tuo’s work is supported by NSF grant DMS 1564438 and NSFC grants 11501551, 11271355, and 11671386. Wu’s work is supported by NSF grant DMS 1564438.