×

Parseval inequalities and lower bounds for variance-based sensitivity indices. (English) Zbl 1437.62702

The authors discuss the role played by the functional basis bounds for global sensitivity analysis, and the suggest using generalized polynomial chaos expansions in Hilbert spaces to obtain lower, making use of Parseval’s inequality. This provides a tool for variable screening. They choose a special Hilbert space basis obtained by diagonalizing the Poincaré differential operators. The authors discuss the relation to the principal component expansion: The two expansions coincide in case of the normal or the gamma distribution. It is shown how to choose the orthonormal functions in the generalized chaos expansion in order to avoid weightings in the derivative-based sensitivity analysis. Several examples are discussed, and the authors consider two applications: A model for the cost of floods and a predator-prey model. For these models, the authors give numerical results, where bootstrap is used to obtain confidence intervals.

MSC:

62R10 Functional data analysis
62H25 Factor analysis and principal components; correspondence analysis
62G05 Nonparametric estimation
60E15 Inequalities; stochastic orderings
62P20 Applications of statistics to economics

Software:

sensitivity
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] Allaire, G. (2007)., Numerical analysis and optimization: an introduction to mathematical modelling and numerical simulation. Oxford University Press. · Zbl 1120.65001
[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. · Zbl 0544.62066 · doi:10.1080/02331888408801747
[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. · Zbl 1376.60002
[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. · Zbl 1060.33014
[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. · Zbl 1355.60103 · doi:10.1051/ps/2015019
[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. · Zbl 0369.65023 · doi:10.1016/0021-9991(78)90097-9
[10] Da Veiga, S. and Gamboa, F. (2013). Efficient estimation of sensitivity indices., Journal of Nonparametric Statistics 25 573-595. · Zbl 1416.62195 · doi:10.1080/10485252.2013.784762
[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. · Zbl 0481.62035 · doi:10.1214/aos/1176345462
[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. · Zbl 1273.65012 · doi:10.1051/m2an/2011045
[14] Ghanem, R. G. and Spanos, P. D. (1991)., Stochastic finite elements - A spectral approach. Springer, Berlin. · Zbl 0722.73080
[15] Giné, E. and Nickl, R. (2008). A simple adaptive estimator of the integrated square of a density., Bernoulli 14 47-61. · Zbl 1155.62025 · doi:10.3150/07-BEJ110
[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. · Zbl 0032.04101 · doi:10.1214/aoms/1177730196
[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. · Zbl 1305.62147 · doi:10.1051/ps/2013040
[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. · Zbl 1356.65010
[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. · Zbl 1196.41022 · doi:10.1090/S0025-5718-09-02319-9
[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. · Zbl 1490.62035
[29] Laurent, B. (1996). Efficient estimation of integral functionals of a density., The Annals of Statistics 24 659-681. · Zbl 0859.62038 · doi:10.1214/aos/1032894458
[30] Laurent, B. and Massart, P. (2000). Adaptive estimation of a quadratic functional by model selection., The Annals of Statistics 28 1302-1338. · Zbl 1105.62328 · doi:10.1214/aos/1015957395
[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. · Zbl 1454.60030 · doi:10.1214/17-EJS1310
[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. · Zbl 0195.16903
[36] Sobol’, I. (1993). Sensitivity estimates for non linear mathematical models., Mathematical Modelling and Computational Experiments 1 407-414. · Zbl 1039.65505
[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. · Zbl 1167.62005 · doi:10.1016/j.matcom.2009.01.023
[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. · Zbl 0019.35406 · doi:10.2307/2371268
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.