Robust Parameter Design With Computer Experiments Using Orthonormal Polynomials

Robust parameter design with computer experiments is becoming increasingly important for product design. Existing methodologies for this problem are mostly for finding optimal control factor settings. However, in some cases, the objective of the experimenter may be to understand how the noise and control factors contribute to variation in the response. The functional analysis of variance (ANOVA) and variance decompositions of the response, in addition to the mean and variance models, help achieve this objective. Estimation of these quantities is not easy and few methods are able to quantity the estimation uncertainty. In this article, we show that the use of an orthonormal polynomial model of the simulator leads to simple formulas for functional ANOVA and variance decompositions, and the mean and variance models. We show that estimation uncertainty can be taken into account in a simple way by first fitting a Gaussian process model to experiment data and then approximating it with the orthonormal polynomial model. This leads to a joint normal distribution for the polynomial coefficients that quantifies estimation uncertainty. Supplementary materials for this article are available online.