Optimal design and properties of correlated processes with semicontinuous covariance
Seminar Room 1, Newton Institute
Semicontinuous covariance functions have been used in regression and kriging by many authors. In a recent work we introduced purely topologically defined regularity conditions on covariance kernels which are still applicable for increasing and infill domain asymptotics for regression problems and kriging. These conditions are related to the semicontinuous maps of Ornstein Uhlenbeck Processes. Thus these conditions can be of benefit for stochastic processes on more general spaces than the metric ones. Besides, the new regularity conditions relax the continuity of covariance function by consideration of a semicontinuous covariance. We discuss the applicability of the introduced topological regularity conditions for optimal design of random fields. A stochastic process with parametrized mean and covariance is observed over a compact set. The information obtained from observations is measured through the information functional (defined on the Fisher information matrix). We start with discussion on the role of equidistant designs for the correlated process. Various aspects of their prospective optimality will be reviewed and some issues on designing for spatial processes will be also provided. Finally we will concentrate on relaxing the continuity of covariance. We will introduce the regularity conditions for isotropic processes with semicontinuous covariance such that increasing domain asymptotics is still feasible, however more flexible behavior may occur here. In particular, the role of the nugget effect will be illustrated and practical application of stochastic processes with semicontinuous covariance will be given.