Pointwise Consistency of the Kriging Predictor with Known Mean and Covariance Functions

Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


This paper deals with several issues related to the pointwise consistency of the kriging predictor when the mean and the covariance functions are known. These questions are of general importance in the context of computer experiments. The analysis is based on the properties of approximations in reproducing kernel Hilbert spaces. We fix an erroneous claim of Yakowitz and Szidarovszky (J. Multivariate Analysis, 1985) that, under some assumptions, the kriging predictor is pointwise consistent for all continuous sample paths.


Covariance Function Sample Path Reproduce Kernel Hilbert Space Lebesgue Constant Bayesian Point 
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  1. Chilès, J.-P. and P. Delfiner (1999). Geostatistics: Modeling Spatial Uncertainty. New York: Wiley.MATHGoogle Scholar
  2. De Marchi, S. and R. Schaback (2008). Stability of kernel-based interpolation. Advances in Computational Mathematics. doi: 10.1007/s10444-008-9093-4.Google Scholar
  3. Lukic, M. N. and J. H. Beder (2001). Stochastic processes with sample paths in reproducing kernel Hilbert spaces. Transactions of the American Mathematical Society 353, 3945–3969.MATHCrossRefMathSciNetGoogle Scholar
  4. Rudin, W. (1987). Real and Complex Analysis, 3rd ed. New York: McGraw-Hill.MATHGoogle Scholar
  5. Sacks, J., W. J. Welch, T. J. Mitchell, and H. P. Wynn (1989). Design and analysis of computer experiments. Statististical Science 4, 409–435.MATHCrossRefMathSciNetGoogle Scholar
  6. Santner, T. J., B. J. Williams, and W. I. Notz (2003). The Design and Analysis of Computer Experiments. New-York: Springer-Verlag.MATHGoogle Scholar
  7. Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. New York: Springer-Verlag.MATHGoogle Scholar
  8. Steinwart, I. (2001). On the influence of the kernel on the consistency of support vector machines. Journal of Machine learning Research 2, 67–93.CrossRefMathSciNetGoogle Scholar
  9. Steinwart, I., D. Hush, and C. Scovel (2006). An explicit description of the reproducing kernel Hilbert spaces of Gaussian RBF kernels. IEEE Transactions on Information Theory 52, 4635–4643.CrossRefMathSciNetGoogle Scholar
  10. Vazquez, E. and J. Bect (2009). On the convergence of the expected improvement algorithm. Preprint available on arXiv, http://arxiv.org/abs/0712.3744v2.
  11. Villemonteix, J. (2008). Optimisation de Fonctions Coûteuses. Ph. D. thesis, Université Paris-Sud XI, Faculté des Sciences d’Orsay.Google Scholar
  12. Williams, D. (1991). Probability with Martingales. Cambridge: Cambridge University Press.MATHGoogle Scholar
  13. Wu, Z. and R. Schaback (1993). Local error estimates for radial basis function interpolation of scattered data. IMA Journal of Numerical Analalysis 13, 13–27.MATHCrossRefMathSciNetGoogle Scholar
  14. Yakowitz, S. J. and F. Szidarovszky (1985). A comparison of kriging with nonparametric regression methods. Journal of Multivariate Analysis 16, 21–53.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.SUPELECGif-sur-YvetteFrance

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