Robust Design Optimization With Quadratic Loss Derived From Gaussian Process Models

Abstract

Gaussian process models, which include the class of linear models, are widely employed for modeling responses as a function of control or noise factors. Using these models, the average loss at control factor settings can be estimated and compared. However, robust design optimization is often performed based on the expected quadratic loss computed as if the posterior mean were the true response function. This can give very misleading results. We propose an expected quadratic loss criterion derived by taking expectation with respect to the noise factors and the posterior predictive process. Approximate but highly accurate credible intervals for the average quadratic loss are constructed via the numerical inversion of the Lugannani–Rice saddlepoint approximation. The accuracy of the Lugannani–Rice intervals is compared with intervals constructed via moment-matching techniques on real data. This article has supplementary materials that are available online.

KEYWORDS:

• Computer experiments
• Lugannani–Rice saddlepoint approximation
• Robust parameter design.

SUPPLEMENTARY MATERIALS

Supplementary Information: Appendices B and C mentioned in the article and proofs of the results presented in Appendix A are given in the online supplementary material. A figure for Section 7.1.2 and the parameter values used for the pattern search algorithm are also given.

ACKNOWLEDGMENTS

This research is supported by NSF grant DMS 1007574 and also in part by a grant from the U.S. Army Research Office under contract number W99INF-08-10368. We thank two anonymous referees, an associate editor and the editor, for comments that helped improve the article.