A note on the choice and the estimation of Kriging models for the analysis of deterministic computer experiments (original) (raw)
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AIAA Journal, 2005
The use of kriging models for approximation and global optimization has been steadily on the rise in the past decade. The standard approach used in the Design and Analysis of Computer Experiments (DACE) is to use an Ordinary kriging model to approximate a deterministic computer model. Universal and Detrended kriging are two alternative types of kriging models. In this paper, a description on the basics of kriging is given, highlighting the similarities and differences between these three different types of kriging models and the underlying assumptions behind each. A comparative study on the use of three different types of kriging models is then presented using six test problems. The methods of Maximum Likelihood Estimation (MLE) and Cross-Validation (CV) for model parameter estimation are compared for the three kriging model types. A one-dimension problem is first used to visualize the differences between the different models. In order to show applications in higher dimensions, four two-dimension and a 5dimension problem are also given.
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Kriging is a popular analysis approach for computer experiments for the purpose of creating a cheapto-compute "meta-model" as a surrogate to a computationally expensive engineering simulation model. The maximum likelihood approach is used to estimate the parameters in the kriging model. However, the likelihood function near the optimum may be flat in some situations, which leads to maximum likelihood estimates for the parameters in the covariance matrix that have very large variance. To overcome this difficulty, a penalized likelihood approach is proposed for the kriging model. Both theoretical analysis and empirical experience using real world data suggest that the proposed method is particularly important in the context of a computationally intensive simulation model where the number of simulation runs must be kept small because collection of a large sample set is prohibitive. The proposed approach is applied to the reduction of piston slap, an unwanted engine noise due to piston secondary motion. Issues related to practical implementation of the proposed approach are discussed.
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Kriging was first introduced in the field of geostatistics. Nowadays, it is widely used to model computer experiments. Since the results of deterministic computer experiments have no experimental variability, Kriging is appropriate in that it interpolates observations at data points. Moreover, Kriging quantifies prediction uncertainty, which plays a major role in many applications. Among practitioners we can distinguish those who use Universal Kriging where the parameters of the model are estimated and those who use Bayesian Kriging where model parameters are random variables. The aim of this article is to show that the prediction uncertainty has a correct interpretation only in the case of Bayesian Kriging. Different cases of prior distributions have been studied and it is shown that in one specific case, Bayesian Kriging supplies an interpretation as a conditional variance for the prediction variance provided by Universal Kriging. Finally, a simple petroleum engineering case study presents the importance of prior information in the Bayesian approach.
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This paper investigates the use of Kriging in random simulation when the simulation output variances are not 12 constant. Kriging gives a response surface or metamodel that can be used for interpolation. Because Ordinary Kriging 13 assumes constant variances, this paper also applies Detrended Kriging to estimate a non-constant signal function, and 14 then standardizes the residual noise through the heterogeneous variances estimated from replicated simulation runs.
Recursive Co-Kriging Model for Design of Computer Experiments with Multiple Levels of Fidelity
International Journal for Uncertainty Quantification, 2014
We consider in this paper the problem of building a fast-running approximation-also called surrogate model-of a complex computer code. The co-kriging based surrogate model is a promising tool to build such an approximation when the complex computer code can be run at different levels of accuracy. We present here an original approach to perform a multi-fidelity co-kriging model which is based on a recursive formulation. We prove that the predictive mean and the variance of the presented approach are identical to the ones of the original co-kriging model. However, our new approach allows to obtain original results. First, closed-form formulas for the universal co-kriging predictive mean and variance are given. Second, a fast cross-validation procedure for the multi-fidelity co-kriging model is introduced. Finally, the proposed approach has a reduced computational complexity compared to the previous one. The multi-fidelity model is successfully applied to emulate a hydrodynamic simulator.
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We consider in this paper the problem of building a fast-running approximation-also called surrogate model-of a complex computer code. The co-kriging based surrogate model is a promising tool to build such an approximation when the complex computer code can be run at different levels of accuracy. We present here an original approach to perform a multi-fidelity co-kriging model which is based on a recursive formulation. We prove that the predictive mean and the variance of the presented approach are identical to the ones of the original co-kriging model proposed by [Kennedy, M.C. and O'Hagan, A., Biometrika, 87, pp 1-13, 2000]. However, our new approach allows to obtain original results. First, closed form formulas for the universal co-kriging predictive mean and variance are given. Second, a fast cross-validation procedure for the multi-fidelity co-kriging model is introduced. Finally, the proposed approach has a reduced computational complexity compared to the previous one. The multi-fidelity model is successfully applied to emulate a hydrodynamic simulator.
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The main issue in the analysis of computer experiments is an uncertainty of prediction and related inferences. To address the uncertainty analysis, the Bayesian analysis of deterministic computer models has been actively developed in the last decade. In the Bayesian approach, the uncertainty is expressed through a Gaussian process model. As a consequence, the resulting analysis is rather sensitive with respect to these prior assumptions. Moreover, for high dimensional data this approach leads to time consuming computations.