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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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