Abstract
Most surrogate models for computer experiments are interpolators, and the most common interpolator is a Gaussian process (GP) that deliberately omits a small-scale (measurement) error term called the nugget. The explanation is that computer experiments are, by definition, “deterministic”, and so there is no measurement error. We think this is too narrow a focus for a computer experiment and a statistically inefficient way to model them. We show that estimating a (non-zero) nugget can lead to surrogate models with better statistical properties, such as predictive accuracy and coverage, in a variety of common situations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ababou, R., Bagtzoglou, A.C., Wood, E.F.: On the condition number of covariance matrices in kriging, estimation, and simulation of random fields. Math. Geol. 26(1), 99–133 (1994)
Ankenman, B., Nelson, B., Staum, J.: Stochastic kriging for simulation metamodeling. Oper. Res. 58(2), 371–382 (2010)
Bastos, L., O’Hagan, A.: Diagnostics for Gaussian process emulators. Technometrics 51(4), 425–438 (2009)
Friedman, J.H.: Multivariate adaptive regression splines. Ann. Stat. 19(1), 1–67 (1991)
Gillespie, D.: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115(4), 1716–1733 (2001)
Gramacy, R.B.: Bayesian treed Gaussian process models. Ph.D. thesis, University of California, Santa Cruz (2005)
Gramacy, R.B.: tgp: An R package for Bayesian nonstationary, semiparametric nonlinear regression and design by treed Gaussian process models. J. Stat. Softw. 19, 9 (2007)
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 (2008a)
Gramacy, R.B., Lee, H.K.H.: Gaussian processes and limiting linear models. Comput. Stat. Data Anal. 53, 123–136 (2008b)
Gramacy, R.B., Lee, H.K.H.: Adaptive design and analysis of supercomputer experiment. Technometrics 51(2), 130–145 (2009)
Henderson, D.A., Boys, R.J., Krishnan, K.J., Lawless, C., Wilkinson, D.J.: Bayesian emulation and calibration of a stochastic computer model of mitochondrial DNA deletions in substantia nigra neurons. J. Am. Stat. Assoc. 104(485), 76–87 (2009)
Johnson, L.: Microcolony and biofilm formation as a survival strategy for bacteria. J. Theor. Biol. 251, 24–34 (2008)
Kennedy, M., O’Hagan, A.: Bayesian calibration of computer models (with discussion). J. R. Stat. Soc. B 63, 425–464 (2001)
Martin, J., Simpson, T.: Use of kriging models to approximate deterministic computer models. AIAA J. 43(4), 853–863 (2005)
Neal, R.M.: Monte Carlo implementation of Gaussian process models for Bayesian regression and classification. Tech. Rep. 9702, Department of Statistics, University of Toronto (1997)
O’Hagan, A., Kennedy, M.C., Oakley, J.E.: Uncertainty analysis and other inference tools for complex computer codes. In: Bernardo, J.M., Berger, J.O., Dawid, A., Smith, A. (eds.) Bayesian Statistics 6, pp. 503–524. Oxford University Press, Oxford (1999)
Pepelyshev, A.: The role of the nugget term in the Gaussian process method. In: MODA 9—Advances in Model-Oriented Design and Analysis, pp. 149–156. Springer, Berlin (2010)
Perlin, K.: Improving noise. ACM Trans. Graph. 21, 681–682 (2002)
Ranjan, P., Haynes, R., Karsten, R.: Gaussian process models and interpolators for deterministic computer simulators. Department of Mathematics and Statistics, Acadia University (2010)
Rogers, S.E., Aftosmis, M.J., Pandya, S.A., Chaderjian, N.M., Tejnil, E., Ahmad, J.U.: Automated CFD parameter studies on distributed parallel computers. In: 16th AIAA Computational Fluid Dynamics Conference (2003). AIAA Paper 2003-4229
Rougier, J., Guillas, S., Maute, A., Richmond, A.: Expert knowledge and multivariate emulation: The thermosphere–ionosphere electrodynamics general circulation model (TIE-GCM). Technometrics 51(4), 414–424 (2009)
Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4, 409–435 (1989)
Santner, T.J., Williams, B.J., Notz, W.I.: The Design and Analysis of Computer Experiments. Springer, New York (2003)
Stein, M.L.: Interpolation of Spatial Data. Springer, New York (1999)
Taddy, M., Lee, H.K.H., Gray, G.A., Griffin, J.D.: Bayesian guided pattern search for robust local optimization. Tech. Rep. ams2008-02, University of California, Santa Cruz, Department of Applied Mathematics and Statistics (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Part of this work was done while R.B.G. was at the Statistical Laboratory, University of Cambridge.
Rights and permissions
About this article
Cite this article
Gramacy, R.B., Lee, H.K.H. Cases for the nugget in modeling computer experiments. Stat Comput 22, 713–722 (2012). https://doi.org/10.1007/s11222-010-9224-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11222-010-9224-x