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TITLE: Removing the Bumps: Brownian-Integrated Covariance Functions for Gaussian Process Modeling
Abstract: Gaussian process (GP) models have become the de facto choice for response surface modeling of computer simulation data. The covariance functions that are most often used for this purpose – Gaussian, power exponential, and Matérn – are localized in the sense that the covariance at two input locations decays to zero as the locations move further apart. We contend that such covariance models are inherently poor choices for most real physical systems because they imply a what-goes-up-must-come-down GP behavior and tend to result in bumpy fitted response surfaces, whereas most physical systems may be better represented by a GP model that exhibits what-goes-up-may-stay-up behavior. To achieve the latter behavior, we propose a class of covariance models that can be viewed as incorporating an integrator into any standard stationary GP model, analogous to the integrator in an ARIMA time series model. Specifically, in the white noise integral representation of a fractional Brownian field (FBF), we replace the white noise by any stationary GP model and refer to the result as a Brownian-integrated GP. We show that this generalization inherits the desirable what-goes-up-may-stay-up behavior of FBFs without inheriting the undesirable, overly rough behavior that makes FBFs unsuitable for most deterministic response surfaces. We also discuss fundamental differences between Brownian-integrated vs. standard GP covariance models, such as a sigmoidal versus localized nature of their associated basis functions. Remarkably, for every real physical system that we have considered so far, the Brownian-integrated model has performed better than the standard covariance models.
Bio: Dan Apley is Professor of Industrial Engineering & Management Sciences at Northwestern University. His research and teaching interests are at the interface of engineering modeling, statistical analysis, and predictive analytics, with particular emphasis on understanding sources of variation in manufacturing and other enterprise systems. His work has been supported by numerous industries and government agencies. He received the NSF CAREER award in 2001, the IIE Transactions Best Paper Award in 2003, and the Technometrics Wilcoxon Prize in 2008. He was formerly Editor-in-Chief of the Journal of Quality Technology and is currently Editor-Elect of Technometrics. He has also served as Chair of the Quality, Statistics & Reliability Section of INFORMS and Director of the Manufacturing and Design Engineering Program at Northwestern.
