Construction of bootstrap confidence intervals on sensitivity indices computed by polynomial chaos expansion

Sensitivity analysis aims at quantifying influence of input parameters dispersion on the output dispersion of a numerical model. When the model evaluation is time consuming, the computation of Sobol' indices based on Monte Carlo method is not applicable and a surrogate model has to be used. Among all approximation methods, polynomial chaos expansion is one of the most efficient to calculate variance-based sensitivity indices. Indeed, their computation is analytically derived from the expansion coefficients but without error estimators of the meta-model approximation. In order to evaluate the reliability of these indices, we propose to build confidence intervals by bootstrap re-sampling on the experimental design used to estimate the polynomial chaos approximation. Since the evaluation of the sensitivity indices is obtained with confidence intervals, it is possible to find a design of experiments allowing the computation of sensitivity indices with a given accuracy.

[1]  Emanuele Borgonovo,et al.  Sampling strategies in density-based sensitivity analysis , 2012, Environ. Model. Softw..

[2]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[3]  C. Mallows.  Some Comments on Cp , 2000, Technometrics.

[4]  B. Sudret,et al.  An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis , 2010 .

[5]  B. Efron,et al.  Bootstrap confidence intervals , 1996 .

[6]  de Maurice Lemaire.  Fiabilité des structures , 2006 .

[7]  Jon C. Helton,et al.  Implementation and evaluation of nonparametric regression procedures for sensitivity analysis of computationally demanding models , 2009, Reliab. Eng. Syst. Saf..

[8]  Pierre-André Cornillon,et al.  Régressión :: théorie et applications , 2007 .

[9]  Ilya M. Sobol,et al.  Sensitivity Estimates for Nonlinear Mathematical Models , 1993 .

[10]  I. Sobola,et al.  Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates , 2001 .

[11]  Bruno Sudret,et al.  Adaptive sparse polynomial chaos expansion based on least angle regression , 2011, J. Comput. Phys..

[12]  Emanuele Borgonovo,et al.  Global sensitivity measures from given data , 2013, Eur. J. Oper. Res..

[13]  Alexandre Janon,et al.  Confidence intervals for sensitivity indices using reduced-basis metamodels , 2011, 1102.4668.

[14]  Sharif Rahman,et al.  Global sensitivity analysis by polynomial dimensional decomposition , 2011, Reliab. Eng. Syst. Saf..

[15]  Michel Salaün,et al.  A new adaptive response surface method for reliability analysis , 2013 .

[16]  B. Efron.  Better Bootstrap Confidence Intervals , 1987 .

[17]  I. Sobol.  Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates , 2001 .

[18]  Joseph A. C. Delaney.  Sensitivity analysis , 2018, The African Continental Free Trade Area: Economic and Distributional Effects.

[19]  Roger G. Ghanem,et al.  Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure , 2005, SIAM J. Sci. Comput..

[20]  Bruno Sudret,et al.  Efficient computation of global sensitivity indices using sparse polynomial chaos expansions , 2010, Reliab. Eng. Syst. Saf..

[21]  N. Gayton,et al.  CQ2RS: a new statistical approach to the response surface method for reliability analysis , 2003 .

[22]  G. Blatman,et al.  Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis , 2009 .

[23]  Nicolas Gayton,et al.  RPCM: a strategy to perform reliability analysis using polynomial chaos and resampling , 2010 .

[24]  Robert Tibshirani,et al.  An Introduction to the Bootstrap , 1994 .

[25]  T. Klein,et al.  Asymptotic normality and efficiency of two Sobol index estimators , 2013, 1303.6451.

[26]  Bruno Sudret,et al.  Global sensitivity analysis using polynomial chaos expansions , 2008, Reliab. Eng. Syst. Saf..

[27]  R. Tibshirani,et al.  Least angle regression , 2004, math/0406456.