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.)

Abstract

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.

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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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