Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction.

*(English)*Zbl 0785.62025Summary: 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.

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 |