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We propose a flexible yet computationally efficient approach for building Gaussian process models for computer experiments with both qualitative and quantitative factors. This approach uses the hypersphere parameterization to model the correlations of the qualitative factors, thus avoiding the need of directly solving optimization problems with positive definite constraints. The effectiveness of the proposed method is successfully illustrated by several examples.

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A growing trend in engineering and science is to use multiple computer codes with different levels of accuracy to study the same complex system. We propose a framework for sequential design and analysis of a pair of high-accuracy and low-accuracy computer codes. It first runs the two codes with a pair of nested Latin hypercube designs (NLHDs). Data from the initial experiment are used to fit a prediction model. If the accuracy of the fitted model is less than a prespecified threshold, the two codes are evaluated again with input values chosen in an elaborate fashion so that their expanded scenario sets still form a pair of NLHDs. The nested relationship between the two scenario sets makes it easier to model and calibrate the difference between the two sources. If necessary, this augmentation process can be repeated a number of times until the prediction model based on all available data has reasonable accuracy. The effectiveness of the proposed method is illustrated with several examples. Matlab codes are provided in the online supplement to this article.

Large-scale computer experiments are becoming increasingly important in science. A multi-step procedure is introduced to statisticians for modeling such experiments, which builds an accurate interpolator in multiple steps. In practice, the procedure shows substantial improvements in overall accuracy, but its theoretical properties are not well established. We introduce the terms nominal and numeric error and decompose the overall error of an interpolator into nominal and numeric portions. Bounds on the numeric and nominal error are developed to show theoretically that substantial gains in overall accuracy can be attained with the multi-step approach.Comment: Published in at http://dx.doi.org/10.1214/11-AOS929 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

New types of designs called nested space-filling designs have been proposed for conducting multiple computer experiments with different levels of accuracy. In this article, we develop several approaches to constructing such designs. The development of these methods also leads to the introduction of several new discrete mathematics concepts, including nested orthogonal arrays and nested difference matrices.Comment: Published in at http://dx.doi.org/10.1214/09-AOS690 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org