×

Cases for the nugget in modeling computer experiments. (English) Zbl 1252.62098

Summary: 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.

MSC:

62M99 Inference from stochastic processes
62P99 Applications of statistics
68U99 Computing methodologies and applications

Software:

tgp
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 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) · Zbl 0970.86543
[2] Ankenman, B., Nelson, B., Staum, J.: Stochastic kriging for simulation metamodeling. Oper. Res. 58(2), 371–382 (2010) · Zbl 1342.62134
[3] Bastos, L., O’Hagan, A.: Diagnostics for Gaussian process emulators. Technometrics 51(4), 425–438 (2009)
[4] Friedman, J.H.: Multivariate adaptive regression splines. Ann. Stat. 19(1), 1–67 (1991) · Zbl 0765.62064
[5] Gillespie, D.: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115(4), 1716–1733 (2001)
[6] Gramacy, R.B.: Bayesian treed Gaussian process models. Ph.D. thesis, University of California, Santa Cruz (2005)
[7] 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)
[8] 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) · Zbl 1205.62218
[9] Gramacy, R.B., Lee, H.K.H.: Gaussian processes and limiting linear models. Comput. Stat. Data Anal. 53, 123–136 (2008b) · Zbl 1452.62064
[10] Gramacy, R.B., Lee, H.K.H.: Adaptive design and analysis of supercomputer experiment. Technometrics 51(2), 130–145 (2009)
[11] 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) · Zbl 1388.92007
[12] Johnson, L.: Microcolony and biofilm formation as a survival strategy for bacteria. J. Theor. Biol. 251, 24–34 (2008) · Zbl 1397.92097
[13] Kennedy, M., O’Hagan, A.: Bayesian calibration of computer models (with discussion). J. R. Stat. Soc. B 63, 425–464 (2001) · Zbl 1007.62021
[14] Martin, J., Simpson, T.: Use of kriging models to approximate deterministic computer models. AIAA J. 43(4), 853–863 (2005)
[15] 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)
[16] 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) · Zbl 1175.62028
[17] 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)
[18] Perlin, K.: Improving noise. ACM Trans. Graph. 21, 681–682 (2002)
[19] Ranjan, P., Haynes, R., Karsten, R.: Gaussian process models and interpolators for deterministic computer simulators. Department of Mathematics and Statistics, Acadia University (2010)
[20] 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
[21] 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)
[22] Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4, 409–435 (1989) · Zbl 0955.62619
[23] Santner, T.J., Williams, B.J., Notz, W.I.: The Design and Analysis of Computer Experiments. Springer, New York (2003) · Zbl 1041.62068
[24] Stein, M.L.: Interpolation of Spatial Data. Springer, New York (1999) · Zbl 0924.62100
[25] 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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.