Bayesian Kriging Analysis and Design for Stochastic Simulations

Abstract

Kriging is an increasingly popular metamodeling tool in simulation due to its flexibility in global fitting and prediction. In the fitting of this metamodel, the parameters are often estimated from the simulation data, which introduces parameter estimation uncertainties into the overall prediction error. Traditional plug-in estimators usually ignore these uncertainties, which can be substantial in stochastic simulations. This typically leads to an underestimation of the total variability and an overconfidence in the results. In this article, a Bayesian metamodeling approach for kriging prediction is proposed for stochastic simulations to more appropriately account for the parameter uncertainties. We derive the predictive distribution under certain assumptions and also provide a general Markov Chain Monte Carlo analysis approach to handle more general assumptions on the parameters and design. Numerical results indicate that the Bayesian approach has better coverage and better predictive variance than a previously proposed modified nugget effect kriging model, especially in cases where the stochastic variability is high. In addition, we further consider the important problem of planning the experimental design. We propose a two-stage design approach that systematically balances the allocation of computing resources to new design points and replication numbers in order to reduce the uncertainties and improve the accuracy of the predictions.