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Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction. (English) Zbl 0785.62025
Summary: This article is concerned with the problem of predicting a deterministic response function $$y_ 0$$ over a multidimensional domain $$T$$, given values of $$y_ 0$$ and all of its first derivatives at a set of design sites (points) in $$T$$. The intended application is to computer experiments in which $$y_ 0$$ is an output from a computer model of a physical system and each point in $$T$$ represents a particular configuration of the input parameters. It is assumed that the first derivatives are already available (e.g., from a sensitivity analysis) or can be produced by the code that implements the model. A Bayesian approach in which the random function that represents prior uncertainty about $$y_ 0$$ is taken to be a stationary Gaussian stochastic process is used. The calculations needed to update the prior given observations of $$y_ 0$$ and its first derivatives at the design sites are given and are illustrated in a small example.
The issue of experimental design is also discussed, in particular the criterion of maximizing the reduction in entropy, which leads to a kind of $$D$$-optimality. It is shown that, for certain classes of correlation functions in which the intersite correlations are very weak, $$D$$-optimal designs necessarily maximize the minimum distance between design sites. A simulated annealing algorithm is described for constructing such maximin distance designs. An example is given based on a demonstration model of eight inputs and one output, in which predictions based on a maximin design, a Latin hypercube design, and two compromise designs are evaluated and compared.

##### MSC:
 62F15 Bayesian inference 62K05 Optimal statistical designs 65C20 Probabilistic models, generic numerical methods in probability and statistics 62P99 Applications of statistics 65C99 Probabilistic methods, stochastic differential equations
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