Electronic Journal of Statistics

Parseval inequalities and lower bounds for variance-based sensitivity indices

Abstract

The so-called polynomial chaos expansion is widely used in computer experiments. For example, it is a powerful tool to estimate Sobol’ sensitivity indices. In this paper, we consider generalized chaos expansions built on general tensor Hilbert basis. In this frame, we revisit the computation of the Sobol’ indices with Parseval equalities and give general lower bounds for these indices obtained by truncation. The case of the eigenfunctions system associated with a Poincaré differential operator leads to lower bounds involving the derivatives of the analyzed function and provides an efficient tool for variable screening. These lower bounds are put in action both on toy and real life models demonstrating their accuracy.

Article information

Source
Electron. J. Statist., Volume 14, Number 1 (2020), 386-412.

Dates
Received: June 2019
First available in Project Euclid: 22 January 2020

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1579662085

Digital Object Identifier
doi:10.1214/19-EJS1673

Citation

Roustant, Olivier; Gamboa, Fabrice; Iooss, Bertrand. Parseval inequalities and lower bounds for variance-based sensitivity indices. Electron. J. Statist. 14 (2020), no. 1, 386--412. doi:10.1214/19-EJS1673. https://projecteuclid.org/euclid.ejs/1579662085

References

• [1] Allaire, G. (2007)., Numerical analysis and optimization: an introduction to mathematical modelling and numerical simulation. Oxford University Press.
• [2] Allaire, G. (2015). A review of adjoint methods for sensitivity analysis, uncertainty quantification and optimization in numerical codes., Ingénieurs de l’Automobile 836 33–36.
• [3] Antoniadis, A. (1984). Analysis of variance on function spaces., Statistics: A Journal of Theoretical and Applied Statistics 15 59–71.
• [4] Bakry, D., Gentil, I. and Ledoux, M. (2014)., Analysis and geometry of Markov diffusion operators, volume 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Cham.
• [5] Bakry, D. and Mazet, O. (2003). Characterization of Markov semigroups on $\mathbbR$ associated to some families of orthogonal polynomials. In, Séminaire de Probabilités XXXVII 60–80. Springer.
• [6] Bonnefont, M., Joulin, A. and Ma, Y. (2016). A note on spectral gap and weighted Poincaré inequalities for some one-dimensional diffusions., ESAIM: Probability and Statistics 20 18–29.
• [7] Ciric, C., Ciffroy, P. and Charles, S. (2012). Use of sensitivity analysis to identify influential and non-influential parameters within an aquatic ecosystem model., Ecological Modelling 246 119–130.
• [8] Crestaux, T., Maître, O. L. and Martinez, J.-M. (2009). Polynomial chaos expansions for uncertainties quantification and sensitivity analysis., Reliability Engineering and System Safety 94 1161–1172.
• [9] Cukier, R., Levine, H. and Shuler, K. (1978). Nonlinear sensitivity analysis of multiparameter model systems., Journal of Computational Physics 26 1–42.
• [10] Da Veiga, S. and Gamboa, F. (2013). Efficient estimation of sensitivity indices., Journal of Nonparametric Statistics 25 573–595.
• [11] Da Veiga, S., Wahl, F. and Gamboa, F. (2009). Local polynomial estimation for sensitivity analysis on models with correlated inputs., Technometrics 51 452–463.
• [12] Efron, B. and Stein, C. (1981). The jackknife estimate of variance., The Annals of Statistics 9 586–596.
• [13] Ernst, O. G., Mugler, A., Starkloff, H.-J. and Ullmann, E. (2012). On the convergence of generalized polynomial chaos expansions., ESAIM: Mathematical Modelling and Numerical Analysis 46 317–339.
• [14] Ghanem, R. G. and Spanos, P. D. (1991)., Stochastic finite elements – A spectral approach. Springer, Berlin.
• [15] Giné, E. and Nickl, R. (2008). A simple adaptive estimator of the integrated square of a density., Bernoulli 14 47–61.
• [16] Halmos, P. R. (2012)., A Hilbert space problem book 19. Springer Science & Business Media.
• [17] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution., Ann. Math. Statist. 19 293–325.
• [18] Homma, T. and Saltelli, A. (1996). Importance measures in global sensitivity analysis of non linear models., Reliability Engineering and System Safety 52 1–17.
• [19] Iooss, B., Janon, A. and Pujol, G. (2019). sensitivity: Global Sensitivity Analysis of Model Outputs R package version, 1.17.0.
• [20] Iooss, B. and Lemaitre, P. (2015). A review on global sensitivity analysis methods. In, Uncertainty management in Simulation-Optimization of Complex Systems: Algorithms and Applications (C. Meloni and G. Dellino, eds.) 101–122. Springer.
• [21] Iooss, B., Popelin, A.-L., Blatman, G., Ciric, C., Gamboa, F., Lacaze, S. and Lamboni, M. (2012). Some new insights in derivative-based global sensitivity measures. In, Proceedings of the PSAM11 ESREL 2012 Conference 1094–1104.
• [22] Iooss, B. and Saltelli, A. (2017). Introduction: Sensitivity analysis. In, Springer Handbook on Uncertainty Quantification (R. Ghanem, D. Higdon and H. Owhadi, eds.) 1103–1122. Springer.
• [23] Janon, A., Klein, T., Lagnoux, A., Nodet, M. and Prieur, C. (2014). Asymptotic normality and efficiency of two Sobol index estimators., ESAIM: Probability and Statistics 18 342–364.
• [24] Kucherenko, S. and Iooss, B. (2017). Derivative-based global sensitivity measures. In, Springer Handbook on Uncertainty Quantification (R. Ghanem, D. Higdon and H. Owhadi, eds.) 1241–1263. Springer.
• [25] Kucherenko, S., Rodriguez-Fernandez, M., Pantelides, C. and Shah, N. (2009). Monte Carlo evaluation of derivative-based global sensitivity measures., Reliability Engineering and System Safety 94 1135–1148.
• [26] Kucherenko, S. and Song, S. (2016). Derivative-based global sensitivity measures and their link with Sobol’ sensitivity indices. In, Monte Carlo and Quasi-Monte Carlo Methods (R. Cools and D. Nuyens, eds.) 455–469. Springer International Publishing, Cham.
• [27] Kuo, F. Y., Sloan, I. H., Wasilkowski, G. W. and Woźniakowski, H. (2010). On decompositions of multivariate functions., Mathematics of Computation 79 953–966.
• [28] Lamboni, M., Iooss, B., Popelin, A. L. and Gamboa, F. (2013). Derivative-based global sensitivity measures: General links with Sobol’ indices and numerical tests., Mathematics and Computers in Simulation 87 45–54.
• [29] Laurent, B. (1996). Efficient estimation of integral functionals of a density., The Annals of Statistics 24 659–681.
• [30] Laurent, B. and Massart, P. (2000). Adaptive estimation of a quadratic functional by model selection., The Annals of Statistics 28 1302–1338.
• [31] Prieur, C. and Tarantola, S. (2017). Variance-based sensitivity analysis: Theory and estimation algorithms. In, Springer Handbook on Uncertainty Quantification (R. Ghanem, D. Higdon and H. Owhadi, eds.) 1217–1239. Springer.
• [32] Pronzato, L. (2019). Sensitivity analysis via Karhunen-Loève expansion of a random field model: Estimation of Sobol’ indices and experimental design., Reliability Engineering and System Safety 187 93–109.
• [33] Roustant, O., Barthe, F. and Iooss, B. (2017). Poincaré inequalities on intervals – application to sensitivity analysis., Electron. J. Statist. 11 3081–3119.
• [34] Serfling, R. J. (2009)., Approximation theorems of mathematical statistics 162. John Wiley & Sons.
• [35] Sobol’, I. (1969)., Multidimensional quadrature formulas and Haar functions. Izdat “Nauka”, Moscow.
• [36] Sobol’, I. (1993). Sensitivity estimates for non linear mathematical models., Mathematical Modelling and Computational Experiments 1 407–414.
• [37] Sobol’, I. and Gershman, A. (1995). On an alternative global sensitivity estimator. In, Proceedings of SAMO 1995 40–42.
• [38] Sobol’, I. M. and Kucherenko, S. (2009). Derivative based global sensitivity measures and their links with global sensitivity indices., Mathematics and Computers in Simulation 79 3009–3017.
• [39] Song, S., Zhou, T., Wang, L., Kucherenko, S. and Lu, Z. (2019). Derivative-based new upper bound of Sobol’ sensitivity measure., Reliability Engineering & System Safety 187 142–148.
• [40] Sudret, B. (2008). Global sensitivity analysis using polynomial chaos expansion., Reliability Engineering and System Safety 93 964–979.
• [41] Sudret, B. and Mai, C. V. (2015). Computing derivative-based global sensitivity measures using polynomial chaos expansions., Reliability Engineering & System Safety 134 241–250.
• [42] Tissot, J.-Y. (2012). Sur la décomposition ANOVA et l’estimation des indices de Sobol’. Application à un modèle d’écosystème marin, PhD thesis, Grenoble, University.
• [43] Wiener, N. (1938). The homogeneous chaos., American Journal of Mathematics 60 897–936.