Estimating and Quantifying Uncertainties on Level Sets Using the Vorob’ev Expectation and Deviation with Gaussian Process Models

  • Clément Chevalier
  • David Ginsbourger
  • Julien Bect
  • Ilya Molchanov
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


Several methods based on Kriging have recently been proposed for calculating a probability of failure involving costly-to-evaluate functions. A closely related problem is to estimate the set of inputs leading to a response exceeding a given threshold. Now, estimating such a level set—and not solely its volume—and quantifying uncertainties on it are not straightforward. Here we use notions from random set theory to obtain an estimate of the level set, together with a quantification of estimation uncertainty. We give explicit formulae in the Gaussian process set-up and provide a consistency result. We then illustrate how space-filling versus adaptive design strategies may sequentially reduce level set estimation uncertainty.


Gaussian Process Bottom Left Plot Infill Asymptotics Adaptive Design Strategy Gaussian Process Framework 
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  1. Baddeley, A., Molchanov, I.: Averaging of random sets based on their distance functions. J. Math. Imaging Vis. 8, 79–92 (1998) MathSciNetCrossRefGoogle Scholar
  2. Bect, J., Ginsbourger, D., Li, L., Picheny, V., Vazquez, E.: Sequential design of computer experiments for the estimation of a probability of failure. Stat. Comput. 22, 773–793 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  3. Chevalier, C., Bect, J., Ginsbourger, D., Vazquez, E., Picheny, V., Richet, Y.: Fast parallel Kriging-based stepwise uncertainty reduction with application to the identification of an excursion set. (2012a)
  4. Chevalier, C., Picheny, V., Ginsbourger, D.: KrigInv: Kriging-based Inversion for Deterministic and Noisy Computer Experiments. R package version 1.3 (2012b)
  5. Chilès, J.P., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty. Wiley, New York (1999) zbMATHCrossRefGoogle Scholar
  6. Dubourg, V.: Métamodèles adaptatifs pour l’analyse de fiabilité et l’optimisation sous contrainte fiabiliste. Ph.D. thesis, Université Blaise Pascal et Institut Français de Mécanique Avancée (2011) Google Scholar
  7. Forrester, A.I.J., Sóbester, A., Keane, A.J.: Engineering Design via Surrogate Modelling: A Practical Guide. Wiley, Chichester (2008) CrossRefGoogle Scholar
  8. Molchanov, I.: Theory of Random Sets. Springer, London (2005) zbMATHGoogle Scholar
  9. Picheny, V., Ginsbourger, D., Roustant, O., Haftka, R.T., Kim, N.H.: Adaptive designs of experiments for accurate approximation of target regions. Journal of Mechanical Design 132 (2010). doi: 10.1115/1.4001873
  10. Ranjan, P., Bingham, D., Michailidis, G.: Sequential experiment design for contour estimation from complex computer codes. Technometrics 50, 527–541 (2008) MathSciNetCrossRefGoogle Scholar
  11. Rasmussen, C.R., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006) zbMATHGoogle Scholar
  12. Santner, T.J., Williams, B.J., Notz, W.: The Design and Analysis of Computer Experiments. Springer, Berlin (2003) zbMATHCrossRefGoogle Scholar
  13. Vazquez, E., Piera-Martinez, M.: Estimation of the volume of an excursion set of a Gaussian process using intrinsic Kriging. (2006)
  14. Vazquez, E., Piera-Martinez, M.: Estimation du volume des ensembles d’excursion d’un processus Gaussien par Krigeage intrinsèque. In: 39ème Journées de Statistiques, Angers, France (2007) Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Clément Chevalier
    • 1
  • David Ginsbourger
    • 1
  • Julien Bect
    • 2
  • Ilya Molchanov
    • 1
  1. 1.Institute of Mathematical Statistics and Actuarial ScienceUniversity of BernBernSwitzerland
  2. 2.Supélec Sciences des Systèmes, EA4454 (E3S)SUPELEC, Plateau de MoulonGif-sur-Yvette cedexFrance

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