Estimating and quantifying uncertainties on level sets using the Vorob'ev expectation and deviation with Gaussian process models

Clément Chevalier (1, 2, 3) , David Ginsbourger (1) , Julien Bect (4) , Ilya Molchanov (1)

1 IMSV - Institute of Mathematical Statistics and Actuarial Science [Bern]
2 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
3 IRSN - Institut de Radioprotection et de Sûreté Nucléaire
4 E3S - Supélec Sciences des Systèmes

Several methods based on Kriging have been recently 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 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.

Mots clés: Set estimation; Kriging; Vorob'ev expectation

Domaines: Modélisation et simulation; Méthodologie [stat.ME]; Machine Learning [stat.ML]; Probabilités [math.PR]

Clément Chevalier, David Ginsbourger, Julien Bect, Ilya Molchanov. Estimating and quantifying uncertainties on level sets using the Vorob'ev expectation and deviation with Gaussian process models. Dariusz Ucinski, Anthony C. Atkinson, Maciej Patan. mODa 10 - Advances in Model-Oriented Design and Analysis Proceedings of the 10th International Workshop in Model-Oriented Design and Analysis Held in Łagów Lubuski, Poland, June 10-14, 2013, Springer International Publishing, pp.35-43, 2013, Contributions to Statistics, 978-3-319-00217-0. ⟨10.1007/978-3-319-00218-7⟩. ⟨hal-00731783v2⟩