On the Optimal Designs for the Universal Kriging

Processes are so complex in many areas of science and technology that physical experimentation is often too time-consuming or expensive. Thus, computer simulations are often used to replace real experiments. These experiments rely on several runs of a computer code, which implements a simulation model of a physical system of interest, usually aimed at investigating a deterministic Input/Output (I/O) relationship. However, the request that the simulator should be accurate in describing the physical system means that the simulator itself may be rather complex. This has led to the use of surrogate models, also called metamodels, which are computationally cheaper alternatives of the simulation models. In general, these metamodels are statistical interpolators, fitted from the simulated data, which will represent a valid approximation of the original simulator and allow us to obtain efficient prediction of the I/O function at the entire design region. Several surrogate techniques have been suggested in the literature and the Kriging methodology, introduced by D.G. Krige and further developed by different authors, has become one of the most popular in the context of Computer Experiments since the pioneering paper of Sacks et al. (1989). Adopting this approach, the deterministic output of the computer code is regarded as a realization of a stationary Gaussian process, whose covariance structure is usually modeled in a parametric way to reflect the available information about the I/O function. After data have been collected, the unknown parameters of the model will be estimated and the Kriging interpolator will be derived in order to predict the deterministic output at untried sites. In this setting, in order to optimize the experiment the choice of observations to be simulated by the computer code is crucial. The reason for that lies in the diversification of the inferential aims. For instance, the choice of the observation points could be made in order to optimize i) the prediction at unobserved locations, ii) the information gain generated by the experiment and iii) the precision of parameter estimation. In this paper we study the optimal design problem for the Kriging metamodel with respect to different approaches. SACKS, J., WELCH, W.J., MITCHELL, T.J., WYNN, H.P. (1989): Design and analysis of computer experiments. Statistical Science, 4, 409-423