Jürgen Pilz, Universität Klagenfurt
The interplay between random field models for Bayesian spatial prediction and the design of computer experiments
In the first part of my talk, I will give an overview of recent work with my colleagues G. Spoeck and H. Kazianka in the area of Bayesian spatial prediction and design -. The Bayesian approach not only offers more flexibility in modeling but also allows us to deal with uncertain distribution parameters, and it leads to more realistic estimates for the predicted variances. Moreover, I will demonstrate how to apply copula methodology to Bayesian spatial modeling and use it to derive predictive distributions. I will also report on recent results for determining objective priors for the crucial nugget and range parameters of the widely used Matern-family of covariance functions. Briefly, I will also consider the problem of choosing an "optimal" spatial design, i.e. finding an optimal spatial configuration of the observation sites minimizing the total mean squared error of prediction over an area of interest. Our results will be illustrated by modeling environmental phenomena and designing a hyrogeological monitoring network in Upper Austria. In the second part of my talk I will report on modifying and transferring spatial random field models to make them accessible to the analysis of complex computer code. Over the last three decades, the design of computer experiments has rapidly developed as a statistical discipline at the intersection of the well established theories of DoE, stochastic processes, stochastic simulation and statistical parameter estimation, with the aim of approximating complex computer models to reproduce the behaviour of engineering, physical, biological, environmental and social science processes. We will focus on the use of Gaussian Processes (GPs) for the approximation of computer models, thereby stepping from simple parametric setups to using GPs as basis functions of additive models. Then we discuss the numerical problems associated with the estimation of the model parameters, in particular, the second-order (variance and correlation) parameters. To overcome these problems I will highlight joint recent work with my colleague N. Vollert  using Bayesian regularization, based on objective (reference) priors for the parameters. Finally, we will consider design problems associated with the search for numerical robustness of the estimation procedures. We illustrate our findings by modeling the magnetic field of a magnetic linear position detection system as used in the automotive industry.