Lifted Brownian Kriging Models

ABSTRACT

Gaussian processes have become a standard framework for modeling deterministic computer simulations and producing predictions of the response surface. This article investigates a new covariance function that is shown to offer superior prediction compared to the more common covariances for computer simulations of real physical systems. This is demonstrated via a gamut of realistic examples. A simple, closed-form expression for the covariance is derived as a limiting form of a Brownian-like covariance model as it is extended to some hypothetical higher-dimensional input domain, and so we term it a lifted Brownian covariance. This covariance has connections with the multiquadric kernel. Through analysis of the kriging model, this article offers some theoretical comparisons between the proposed covariance model and existing covariance models. The major emphasis of the theory is explaining why the proposed covariance is superior to its traditional counterparts for many computer simulations of real physical systems. Supplementary materials for this article are available online.

KEYWORDS:

• Computer experiments
• Gaussian process
• Metamodeling
• Multiquadric
• Response surface modeling

Supplementary Materials

This material expands on Section 3.2 of “Lifted Brownian Kriging Models” by investigating the long range dependency between more types of segments.

Acknowledgments

The authors thank Sonja Surjanovic and Derek Bingham for collecting and archiving a set of test functions for use which formed an important subset of the functions used in Section 6 (http://www.sfu.ca/ssurjano/about.html). The authors also thank Ran Yang for help on some of the simulations. This work was supported in part by National Science Foundation Grant No. CMMI-1537641 and Air Force Office of Scientific Research Grant No. FA9550-14-1-0032-P00003. This article was submitted prior to the second author becoming Editor of Technometrics, and the review process was handled entirely by the former Editor.