Spatial Models with Applications in Computer Experiment
Ke Wang , Ph.D. Candidate, Department of Statistics, Colorado State University
3:00 p.m.; 006 Statistics
Often, a deterministic computer response is modeled as a realization from a stochastic process such as a Gaussian random field. In this talk, we describe this stochastic modeling approach and will focus on the use of a stochastic heteroscedastic process (SHP), a stationary non-Gaussian process with non-stationary covariance function. By conditioning on a latent process, the SHP is a non-stationary Gaussian process. As such, the sample paths of this process exhibit greater variability and hence offer more modeling flexibility than those produced by a traditional Gaussian process (GP) model. We use maximum likelihood for inference, 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 predictors. Prediction error variances are also obtained. In examples with simulated and real computer experiment data, the SHP model is superior to traditional Gaussian process models. In addition, the SHP model can be used in an active learning context to select new locations that provide improved estimates of the response surface. Implementing active learning via the SHP model appears to work better than other traditional approaches.
The SHP model can be 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 similar strategy to develop an importance sampling technique to compute 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.