Spatial models with applications in computer experiments
Date
2008
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Abstract
Often, a deterministic computer response is modeled as a realization from a, stochastic process such as a Gaussian random field. Due to the limitation of stationary Gaussian process (GP) in inhomogeneous smoothness, we consider modeling a deterministic computer response as a realization from a stochastic heteroskedastic process (SHP), a stationary non-Gaussian process. Conditional on a latent process, the SHP has non-stationary covariance function and is a non-stationary GP. As such, the sample paths of this process exhibit greater variability and hence offer more modeling flexibility than those produced by a, traditional GP model. We use maximum likelihood for inference in the SHP model, which is complicated by the high dimensionality of the latent process. Accordingly, we develop an importance sampling method for likelihood computation and use a low-rank kriging approximation to reconstruct the latent process. Responses at unobserved locations can be predicted using empirical best predictors or by empirical best linear unbiased predictors. In addition, prediction error variances are obtained. The SHP model can be used in an active learning context, adaptively selecting new locations that provide improved estimates of the response surface. Estimation, prediction, and adaptive sampling with the SHP model are illustrated with several examples. Our spatial model can be adapted to model the first partial derivative process. The derivative process provides additional information about the shape and smoothness of the underlying deterministic function and can assist in the prediction of responses at unobserved sites. The unconditional correlation function for the derivative process presents some interesting properties, and can be used as a new class of spatial correlation functions. For parameter estimation, we propose to use a similar strategy to develop an importance sampling technique to compute the joint likelihood of responses and derivatives. The major difficulties of bringing in derivative information are the increase in the dimensionality of the latent process and the numerical problems of inverting the enlarged covariance matrix. Some possible ways to utilize this information more efficiently are proposed.
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Subject
adaptive sampling
computer experiments
inhomogeneity
local sensitivity
metamodeling
spatial processes
stochastic heteroskedastic process
statistics