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

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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.

Keywords

  • Gaussian Process
  • Bottom Left Plot
  • Infill Asymptotics
  • Adaptive Design Strategy
  • Gaussian Process Framework

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Correspondence to Clément Chevalier .

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Chevalier, C., Ginsbourger, D., Bect, J., Molchanov, I. (2013). Estimating and Quantifying Uncertainties on Level Sets Using the Vorob’ev Expectation and Deviation with Gaussian Process Models. In: Ucinski, D., Atkinson, A., Patan, M. (eds) mODa 10 – Advances in Model-Oriented Design and Analysis. Contributions to Statistics. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00218-7_5

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