Bayesian Local Kriging

ABSTRACT

We consider the problem of constructing metamodels for computationally expensive simulation codes; that is, we construct interpolators/predictors of functions values (responses) from a finite collection of evaluations (observations). We use Gaussian process (GP) modeling and kriging, and combine a Bayesian approach, based on a finite set GP models, with the use of localized covariances indexed by the point where the prediction is made. Our approach is not based on postulating a generative model for the unknown function, but by letting the covariance functions depend on the prediction site, it provides enough flexibility to accommodate arbitrary nonstationary observations. Contrary to kriging prediction with plug-in parameter estimates, the resulting Bayesian predictor is constructed explicitly, without requiring any numerical optimization, and locally adjusts the weights given to the different models according to the data variability in each neighborhood. The predictor inherits the smoothness properties of the covariance functions that are used and its superiority over plug-in kriging, sometimes also called empirical-best-linear-unbiased predictor, is illustrated on various examples, including the reconstruction of an oceanographic field over a large region from a small number of observations. Supplementary materials for this article are available online.

KEYWORDS:

• Bayesian kriging
• Computer experiments
• Interpolation
• Nonstationary process
• Prediction
• Random field

Supplementary Materials

• Appendix: Likelihood, predictor and prediction variance in presence of a linear parametric trend. (pdf file)

• Matlab files for BLK: Matlab code used for the examples in the article. (GNU zipped file)

Acknowledgments

This work was partly supported by the ANR project 2011-IS01-001-01 DESIRE (DESIgns for spatial Random fiElds), joint with the Statistics Department of the JKU Universität, Linz, Austria, and by the European project DRONIC, contract nb. 611428.

We thank the two referees for their comments that helped to significantly improve the article.