Data collected from correlated processes arise in many diverse application areas including both computer and physical experiments, and studies in environmental and ecological science. Often, such data are used for prediction and optimisation of the process under study. For example, we may wish to construct an emulator of a computationally expensive computer model, or simulator, and then use this emulator to find settings of the controllable variables that maximise the predicted response.
The design of the experiment from which the data are collected may strongly influence the quality of the model fit and hence the precision and accuracy of subsequent predictions and decisions. We consider Gaussian process models that are typically defined by a correlation structure that may depend upon unknown parameters. This parametric uncertainty may affect the choice of design points, and ideally should be taken into account when choosing a design.
We consider Bayesian design for Gaussian process models, and use a new approximation to the expected loss to tackle two problems: (i) decision-theoretic optimal design for prediction; and (ii) robust optimisation of computationally expensive simulators.
The resulting designs are illustrated through a number of simple examples typical of computer experiments and spatial statistics.