The Annals of Applied Statistics

Composite Gaussian process models for emulating expensive functions

Shan Ba and V. Roshan Joseph

Full-text: Open access


A new type of nonstationary Gaussian process model is developed for approximating computationally expensive functions. The new model is a composite of two Gaussian processes, where the first one captures the smooth global trend and the second one models local details. The new predictor also incorporates a flexible variance model, which makes it more capable of approximating surfaces with varying volatility. Compared to the commonly used stationary Gaussian process model, the new predictor is numerically more stable and can more accurately approximate complex surfaces when the experimental design is sparse. In addition, the new model can also improve the prediction intervals by quantifying the change of local variability associated with the response. Advantages of the new predictor are demonstrated using several examples.

Article information

Ann. Appl. Stat., Volume 6, Number 4 (2012), 1838-1860.

First available in Project Euclid: 27 December 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Computer experiments functional approximation kriging nugget nonstationary Gaussian process


Ba, Shan; Joseph, V. Roshan. Composite Gaussian process models for emulating expensive functions. Ann. Appl. Stat. 6 (2012), no. 4, 1838--1860. doi:10.1214/12-AOAS570.

Export citation


  • Ababou, R., Bagtzoglou, A. C. and Wood, E. F. (1994). On the condition number of covariance matrices in kriging, estimation, and simulation of random fields. Math. Geol. 26 99–133.
  • Anderes, E. B. and Stein, M. L. (2008). Estimating deformations of isotropic Gaussian random fields on the plane. Ann. Statist. 36 719–741.
  • Ankenman, B., Nelson, B. L. and Staum, J. (2010). Stochastic kriging for simulation metamodeling. Oper. Res. 58 371–382.
  • Ba, S. and Joseph, V. R. (2011). Multi-layer designs for computer experiments. J. Amer. Statist. Assoc. 106 1139–1149.
  • Banerjee, S., Charlin, B. P. and Gelfand, A. E. (2003). Hierarchical Modeling and Analysis for Spatial Data. Chapman & Hall/CRC, Boca Raton, FL.
  • Cressie, N. A. C. (1991). Statistics for Spatial Data. Wiley, New York.
  • Currin, C., Mitchell, T., Morris, M. and Ylvisaker, D. (1991). Bayesian prediction of deterministic functions, with applications to the design and analysis of computer experiments. J. Amer. Statist. Assoc. 86 953–963.
  • Fang, K.-T., Li, R. and Sudjianto, A. (2006). Design and Modeling for Computer Experiments. Chapman & Hall/CRC, Boca Raton, FL.
  • Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc. 102 359–378.
  • Gramacy, R. B. and Lee, H. K. H. (2008). Bayesian treed Gaussian process models with an application to computer modeling. J. Amer. Statist. Assoc. 103 1119–1130.
  • Gramacy, R. B. and Lee, H. K. H. (2012). Cases for the nugget in modeling computer experiments. Statist. Comput. 22 713–722.
  • Haaland, B. and Qian, P. Z. G. (2011). Accurate emulators for large-scale computer experiments. Ann. Statist. 39 2974–3002.
  • Higdon, D. M., Swall, J. and Kern, J. (1999). Non-stationary spatial modeling. In Bayesian Statistics 6, Proceedings of the Sixth Valencia International Meeting 761–768. Oxford Univ. Press, London.
  • Huang, W., Wang, K., Breidt, F. J. and Davis, R. A. (2011). A class of stochastic volatility models for environmental applications. J. Time Series Anal. 32 364–377.
  • Joseph, V. R. (2006). Limit kriging. Technometrics 48 458–466.
  • Joseph, V. R., Hung, Y. and Sudjianto, A. (2008). Blind kriging: A new method for developing metamodels. ASME Journal of Mechanical Design 130 031102–1–8.
  • Joseph, V. R. and Kang, L. (2011). Regression-based inverse distance weighting with applications to computer experiments. Technometrics 53 254–265.
  • Paciorek, C. J. and Schervish, M. J. (2006). Spatial modelling using a new class of nonstationary covariance functions. Environmetrics 17 483–506.
  • Peng, C. Y. and Wu, C. F. J. (2012). Regularized kriging. Unpublished manuscript.
  • Qian, P. Z. G., Seepersad, C. C., Joseph, V. R., Allen, J. K. and Wu, C. F. J. (2006). Building surrogate models with detailed and approximate simulations. ASME Journal of Mechanical Design 128 668–677.
  • Ranjan, P., Haynes, R. and Karsten, R. (2011). A computationally stable approach to Gaussian process interpolation of deterministic computer simulation data. Technometrics 53 366–378.
  • Sacks, J., Welch, W. J., Mitchell, T. J. and Wynn, H. P. (1989). Design and analysis of computer experiments. Statist. Sci. 4 409–423.
  • Sampson, P. D. and Guttorp, P. (1992). Nonparametric estimation of nonstationary spatial covariance structure. J. Amer. Statist. Assoc. 87 108–119.
  • Santner, T. J., Williams, B. J. and Notz, W. I. (2003). The Design and Analysis of Computer Experiments. Springer, New York.
  • Schmidt, A. M. and O’Hagan, A. (2003). Bayesian inference for nonstationary spatial covariance structure via spatial deformations. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 745–758.
  • Wackernagel, H. (2003). Multivariate Geostatistics, 3rd ed. Springer, New York.
  • Welch, W. J., Buck, R. J., Sacks, J., Wynn, H. P., Mitchell, T. J. and Morris, M. D. (1992). Screening, predicting, and computer experiments. Technometrics 34 15–25.
  • Xiong, Y., Chen, W., Apley, D. W. and Ding, X. (2007). A non-stationary covariance-based kriging method for metamodelling in engineering design. Internat. J. Numer. Methods Engrg. 71 733–756.
  • Yamamoto, J. K. (2000). An alternative measure of the reliability of ordinary Kriging estimates. Math. Geol. 32 489–509.