Electronic Journal of Statistics

Poincaré inequalities on intervals – application to sensitivity analysis

Olivier Roustant, Franck Barthe, and Bertrand Iooss

Full-text: Open access

Abstract

The development of global sensitivity analysis of numerical model outputs has recently raised new issues on 1-dimensional Poincaré inequalities. Typically two kinds of sensitivity indices are linked by a Poincaré type inequality, which provides upper bounds of the most interpretable index by using the other one, cheaper to compute. This allows performing a low-cost screening of unessential variables. The efficiency of this screening then highly depends on the accuracy of the upper bounds in Poincaré inequalities.

The novelty in the questions concern the wide range of probability distributions involved, which are often truncated on intervals. After providing an overview of the existing knowledge and techniques, we add some theory about Poincaré constants on intervals, with improvements for symmetric intervals. Then we exploit the spectral interpretation for computing exact value of Poincaré constants of any admissible distribution on a given interval. We give semi-analytical results for some frequent distributions (truncated exponential, triangular, truncated normal), and present a numerical method in the general case.

Finally, an application is made to a hydrological problem, showing the benefits of the new results in Poincaré inequalities to sensitivity analysis.

Article information

Source
Electron. J. Statist. Volume 11, Number 2 (2017), 3081-3119.

Dates
Received: December 2016
First available in Project Euclid: 25 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1503626422

Digital Object Identifier
doi:10.1214/17-EJS1310

Keywords
Poincaré inequality spectral gap truncated distribution Kummer’s functions Sobol-Hoeffding decomposition Sobol indices derivative-based global sensitivity measures finite elements

Rights
Creative Commons Attribution 4.0 International License.

Citation

Roustant, Olivier; Barthe, Franck; Iooss, Bertrand. Poincaré inequalities on intervals – application to sensitivity analysis. Electron. J. Statist. 11 (2017), no. 2, 3081--3119. doi:10.1214/17-EJS1310. https://projecteuclid.org/euclid.ejs/1503626422


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