Uncertainty in Gaussian Process Interpolation

  • Hilke Kracker
  • Björn Bornkamp
  • Sonja Kuhnt
  • Ursula Gather
  • Katja Ickstadt


In this article, we review a probabilistic method for multivariate interpolation based on Gaussian processes. This method is currently a standard approach for approximating complex computer models in statistics, and one of its advantages is the fact that it accompanies the predicted values with uncertainty statements. We focus on investigating the reliability of the method’s uncertainty statements in a simulation study. For this purpose we evaluate the effect of different objective priors and different computational approaches. We illustrate the interpolation method and the practical importance of uncertainty quantification in interpolation in a sequential design application in sheet metal forming. Here design points are added sequentially based on uncertainty statements.


Gaussian Process Design Point Uncertainty Statement Sequential Design Posterior Mode 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bayarri, M.J., Berger, J.O., Paulo, R., Sacks, J., Cafeo, J.A., Cavendish, J., Lin, C.H., Tu, J.: A framework for validation of computer models. Technometrics 49, 138–154 (2007) CrossRefMathSciNetGoogle Scholar
  2. Berger, J.O., de Oliveira, V., Sansó, B.: Objective Bayesian analysis of spatially correlated data. J. Am. Stat. Assoc. 96, 1361–1374 (2001) zbMATHCrossRefGoogle Scholar
  3. Carnell, R.: lhs: Latin Hypercube Samples. R Package Version 0.5 (2009) Google Scholar
  4. Cressie, N.: Statistics for Spatial Data. Wiley, New York (1993) Google Scholar
  5. Currin, C., Mitchell, T., Morris, M., Ylvisaker, D.: Bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments. J. Am. Stat. Assoc. 86, 953–963 (1991) CrossRefMathSciNetGoogle Scholar
  6. Diaconis, P.: Bayesian numerical analysis. In: Berger, J., Gupta, S. (eds.) Statistical Decision Theory and Related Topics IV, pp. 163–175. Springer, New York (1988) Google Scholar
  7. Fang, K., Li, R., Sudjianto, A.: Design and Modeling for Computer Experiments. Computer Science & Data Analysis. Chapman & Hall/CRC, New York (2006) Google Scholar
  8. Geyer, C.J.: mcmc: Markov Chain Monte Carlo. R Package Version 0.6 (2009) Google Scholar
  9. Gramacy, R.B., Lee, H.K.H.: Bayesian treed Gaussian process models with an application to computer modeling. J. Am. Stat. Assoc. 103, 1119–1130 (2008) CrossRefGoogle Scholar
  10. Gramacy, R.B., Polson, N.G.: Particle learning of Gaussian process models for sequential design and optimization. Available at (2009)
  11. Gösling, M., Kracker, H., Brosius, A., Gather, U., Tekkaya, A.: Study of the influence of input parameters on a springback prediction by FEA. In: Proceedings of IDDRG 2007 International Conference, Gyor, Hungary, 21–23 May 2007, pp. 397–404 (2007) Google Scholar
  12. Gu, C.: Smoothing Spline ANOVA Models. Springer, Berlin (2002) zbMATHGoogle Scholar
  13. Handcock, M.S., Stein, M.L.: A Bayesian analysis of kriging. Technometrics 35, 403–410 (1993) CrossRefGoogle Scholar
  14. Jones, D., Schonlau, M., Welch, W.: Global optimization of expensive black-box functions. J. Glob. Optim. 13, 455–492 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  15. Kass, R.E., Wasserman, L.: The selection of prior distributions by formal rules. J. Am. Stat. Assoc. 91, 1343–1370 (1996) zbMATHCrossRefGoogle Scholar
  16. Kenett, R., Zacks, S.: Modern Industrial Statistics: Design and Control of Quality and Reliability. Duxbury, San Francisco (1998) Google Scholar
  17. Kennedy, M., O’Hagan, A.: Bayesian calibration of computer models. J. R. Stat. Soc. B 63, 425–464 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  18. Krige, D.G.: A statistical approach to some basic mine valuation problems on the witwatersrand. J. Chem. Metall. Min. Soc. S. Afr. 52, 119–139 (1951) Google Scholar
  19. Lehman, J., Santner, T., Notz, W.: Designing computer experiments to determine robust control variables. Stat. Sin. 14, 571–590 (2004) zbMATHMathSciNetGoogle Scholar
  20. Matheron, G.: Principles of geostatistics. Econ. Geol. 58, 1246–1266 (1963) CrossRefGoogle Scholar
  21. Oakley, J.E.: Decision-theoretic sensitivity analysis for complex computer models. Technometrics 51, 121–129 (2009) CrossRefGoogle Scholar
  22. O’Hagan, A.: Curve fitting and optimal design for prediction (with discussion). J. R. Stat. Soc. B 40, 1–42 (1978) zbMATHMathSciNetGoogle Scholar
  23. O’Hagan, A.: Some Bayesian numerical analysis (with discussion). In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics 4, pp. 345–363. Oxford University Press, Oxford (1992) Google Scholar
  24. O’Hagan, A., Forster, J.: Kendall’s Advanced Theory of Statistics, vol. 2B: Bayesian Inference, 2nd edn. Arnold, London (2004) Google Scholar
  25. Paciorek, C.J., Schervish, M.J.: Nonstationary covariance functions for Gaussian process regression. In: Thrun, S., Saul, L., Schölkopf, B. (eds.) Advances in Neural Information Processing Systems, vol. 16. MIT Press, Cambridge (2004) Google Scholar
  26. Paulo, R.: Default priors for Gaussian processes. Ann. Stat. 33, 556–582 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  27. Ranjan, P., Bingham, D.: Sequential experiment design for contour estimation from complex computer codes. Technometrics 50, 527–541 (2008) CrossRefGoogle Scholar
  28. Roberts, G.O., Rosenthal, J.S.: Optimal scaling for various Metropolis-Hastings algorithms. Stat. Sci. 16, 351–367 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  29. Sacks, J., Welch, W., Mitchell, T., Wynn, H.: Design and analysis of computer experiments. Stat. Sci. 4, 409–435 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  30. Santner, T., Williams, B., Notz, W.: Design & Analysis of Computer Experiments. Springer, New York (2003) zbMATHGoogle Scholar
  31. Schonlau, M., Welch, W.: Screening the input variables to a computer model via analysis of variance and visualization. In: Dean, A., Lewis, S. (eds.) Screening, pp. 308–327. Springer, New York (2006) CrossRefGoogle Scholar
  32. Seeger, M.: Gaussian processes for machine learning. Int. J. Neural Syst. 14, 69–104 (2004) CrossRefGoogle Scholar
  33. Tokdar, S.T., Ghosh, J.K.: Posterior consistency of logistic Gaussian process priors in density estimation. J. Stat. Plan. Inference 137, 34–42 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  34. Wahba, G.: Improper priors, spline smoothing and the problem of guarding against model errors in regression. J. R. Stat. Soc. B 40, 364–372 (1978) zbMATHMathSciNetGoogle Scholar
  35. Williams, B., Santner, T., Notz, W.: Sequential design of computer experiments to minimize integrated response functions. Stat. Sin. 10, 1133–1152 (2000) zbMATHMathSciNetGoogle Scholar
  36. Xiong, Y., Chen, W., Apley, D., Ding, X.: A non-stationary covariance based kriging method for meta-modeling in engineering design. Int. J. Numer. Methods Eng. 71, 733–756 (2007) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Hilke Kracker
    • 1
  • Björn Bornkamp
    • 1
  • Sonja Kuhnt
    • 1
  • Ursula Gather
    • 1
  • Katja Ickstadt
    • 1
  1. 1.Fakultät StatistikTU DortmundDortmundGermany

Personalised recommendations