# ENBIS: European Network for Business and Industrial Statistics

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## ENBIS-18 in Nancy

*2 – 25 September 2018; Ecoles des Mines, Nancy (France)*Abstract submission: 20 December 2017 – 4 June 2018

### Designs for Characterization of Rare Events of Monotone Models

*4 September 2018, 09:40 – 10:00*

#### Abstract

- Submitted by
- Maria Joao Rendas
- Authors
- Rodrigo Cabral Farias (I3S- CNRS/UNS), Elyes Ouerghi (I3S- CNRS/UNS), Luc Pronzato (I3S- CNRS/UNS), Maria Joao Rendas (I3S- CNRS/UNS)
- Abstract
- Characterisation of rare events is a frequent goal in many distinct domains, including industrial design and environmental studies. Often, the rare event of interest is of the exceedance type, i.e. it corresponds to the fact that a function f(.) of a number of exogenous factors exceeds a given threshold η. Several authors have recently addressed the two related problems of estimation of a probability of exceedance and of identification of the configurations of exogenous factors that lead to the exceedance event of interest. These problems are particularly difficult when evaluation of the function for each input configuration is computationally costly. Efficient recent methods that resort to surrogate models based on a Gaussian Process (GP) assumption, e.g. [1,2], are specially tailored for this situation.

In some circumstances, even if the function is not exactly known, qualitative knowledge about its dependency on the input factors exists. In this presentation we exploit knowledge that f(.) is monotone in its arguments when addressing the following two related problems: (i) to determine the probability α that f(.) exceeds a given level η when upper and lower bounds for α are known; (ii) to find the input configurations and threshold η that map to the α−quantile of f(.).

Monotonicity assumptions have been explored before, in a similar context, in [3]. Our work differs in that we show that by exploiting knowledge of the probability distribution of the input factors of f(.) and about the possible range for α we can restrict the required (costly) evaluations of the function to a small subset of the entire input space. This is not only important from the point of view of reducing the (large) numerical complexity of state-of-the-art methods based on surrogate models, but it also (and this is probably equally important) increases overall robustness to the stationarity assumption that underlies these techniques. We discuss subsequent application of the adaptive method presented in [1] to the identified relevant subset, in particular concerning the choice of an initial design.

The results are illustrated numerically on the estimation of the probability of flooding using an hydro-dynamic model, enabling an appreciation of how it impacts overall efficiency and accuracy.

Bibliography

[1] Julien Bect, Ling Li, Emmanuel Vazquez. Bayesian subset simulation. SIAM/ASA Journal on Uncertainty Quantification, ASA, American Statistical Association, 2017, 5 (1), pp.762-786.

[2] C. Chevalier, J. Bect, D. Ginsbourger, E. Vazquez, V. Picheny,Y. Richet, Fast parallel kriging-based stepwise uncertainty reduction with application to the identification of an excursion set. Technometrics, 56, No.4, pp: 455—465, 2014.

[3] T. Labopin-Richard, V. Picheny, "Sequential design of experiments for estimating quantiles of black-box functions", Statistica Sinica, 2017.