Asymptotic normality and efficiency of two Sobol index estimators

Alexandre Janon; Thierry Klein; Agnès Lagnoux; Maëlle Nodet; Clémentine Prieur

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 342-364
  • ISSN: 1292-8100

Abstract

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Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest (output of the model). One of the statistical tools used to quantify the influence of each input variable on the output is the Sobol sensitivity index. We consider the statistical estimation of this index from a finite sample of model outputs: we present two estimators and state a central limit theorem for each. We show that one of these estimators has an optimal asymptotic variance. We also generalize our results to the case where the true output is not observable, and is replaced by a noisy version.

How to cite

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Janon, Alexandre, et al. "Asymptotic normality and efficiency of two Sobol index estimators." ESAIM: Probability and Statistics 18 (2014): 342-364. <http://eudml.org/doc/273644>.

@article{Janon2014,
abstract = {Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest (output of the model). One of the statistical tools used to quantify the influence of each input variable on the output is the Sobol sensitivity index. We consider the statistical estimation of this index from a finite sample of model outputs: we present two estimators and state a central limit theorem for each. We show that one of these estimators has an optimal asymptotic variance. We also generalize our results to the case where the true output is not observable, and is replaced by a noisy version.},
author = {Janon, Alexandre, Klein, Thierry, Lagnoux, Agnès, Nodet, Maëlle, Prieur, Clémentine},
journal = {ESAIM: Probability and Statistics},
keywords = {sensitivity analysis; sobol indices; asymptotic efficiency; asymptotic normality; confidence intervals; metamodelling; surface response methodology; Sobol indices},
language = {eng},
pages = {342-364},
publisher = {EDP-Sciences},
title = {Asymptotic normality and efficiency of two Sobol index estimators},
url = {http://eudml.org/doc/273644},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Janon, Alexandre
AU - Klein, Thierry
AU - Lagnoux, Agnès
AU - Nodet, Maëlle
AU - Prieur, Clémentine
TI - Asymptotic normality and efficiency of two Sobol index estimators
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 342
EP - 364
AB - Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest (output of the model). One of the statistical tools used to quantify the influence of each input variable on the output is the Sobol sensitivity index. We consider the statistical estimation of this index from a finite sample of model outputs: we present two estimators and state a central limit theorem for each. We show that one of these estimators has an optimal asymptotic variance. We also generalize our results to the case where the true output is not observable, and is replaced by a noisy version.
LA - eng
KW - sensitivity analysis; sobol indices; asymptotic efficiency; asymptotic normality; confidence intervals; metamodelling; surface response methodology; Sobol indices
UR - http://eudml.org/doc/273644
ER -

References

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  1. [1] G.E.P. Box and N.R. Draper, Empirical model-building and response surfaces. John Wiley and Sons (1987). Zbl0614.62104MR861118
  2. [2] R. Carnell, lhs: Latin Hypercube Samples (2009). R package version 0.5. 
  3. [3] W. Chen, R. Jin and A. Sudjianto, Analytical variance-based global sensitivity analysis in simulation-based design under uncertainty. Vol. 127 of Transactions-American Society Of Mechanical Engineers Journal Of Mechanical Design (2005). 
  4. [4] R.I. Cukier, H.B. Levine and K.E. Shuler, Nonlinear sensitivity analysis of multiparameter model systems. J. Comput. Phys.26 (1978) 1–42. Zbl0369.65023MR465355
  5. [5] S. Da Veiga and F. Gamboa, Efficient estimation of sensitivity indices. J. Nonparametric Statist.25 (2013) 573–595. Zbl06231487MR3174285
  6. [6] G.M. Dancik, mlegp: Maximum Likelihood Estimates of Gaussian Processes (2011). R package version 3.1.2. 
  7. [7] T. Hayfield and J.S. Racine, Nonparametric econometrics: The np package. J. Statist. Softw. 27 (2008). Zbl1183.62200
  8. [8] J.C. Helton, J.D. Johnson, C.J. Sallaberry and C.B. Storlie, Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliab. Eng. Syst. Saf.91 (2006) 1175–1209. 
  9. [9] T. Homma and A. Saltelli, Importance measures in global sensitivity analysis of nonlinear models. Reliab. Eng. Syst. Saf.52 (1996) 1–17. 
  10. [10] I.A. Ibragimov and R.Z. Has’ Minskii, Statistical estimation–asymptotic theory. Vol. 16 of Appl. Math. Springer−Verlag, New York (1981). Zbl0467.62026
  11. [11] T. Ishigami and T. Homma, An importance quantification technique in uncertainty analysis for computer models, in Proc. of First International Symposium on Uncertainty Modeling and Analysis, 1990. IEEE (1990) 398–403. 
  12. [12] A. Janon, M. Nodet and C. Prieur, Certified reduced-basis solutions of viscous Burgers equations parametrized by initial and boundary values. ESAIM: M2AN 47 (2013) 317–348. Zbl1272.35016MR3021689
  13. [13] A. Janon, M. Nodet and C. Prieur, Uncertainties assessment in global sensitivity indices estimation from metamodels. Internat. J. Uncert. Quantification4 (2014) 21–36. MR3249675
  14. [14] W. Kahan, Pracniques: further remarks on reducing truncation errors. Commun. ACM 8 (1965) 40. 
  15. [15] W.R. Madych and S.A. Nelson, Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J. Approx. Theory70 (1992) 94–114. Zbl0764.41003MR1168377
  16. [16] A. Marrel, B. Iooss, B. Laurent and O. Roustant, Calculations of sobol indices for the gaussian process metamodel. Reliab. Eng. Syst. Saf.94 (2009) 742–751. 
  17. [17] H. Monod, C. Naud and D. Makowski, Uncertainty and sensitivity analysis for crop models, in Chap. 4 of Working with Dynamic Crop Models: Evaluation, Analysis, Parameterization, and Applications. Edited by D. Wallach, D. Makowski and J. W. Jones. Elsevier (2006) 55–99. 
  18. [18] V.I. Morariu, B.V. Srinivasan, V.C. Raykar, R. Duraiswami and L.S. Davis, Automatic online tuning for fast gaussian summation, in Advances in Neural Information Processing Systems, NIPS (2008). 
  19. [19] N.C. Nguyen, K. Veroy and A.T. Patera, Certified real-time solution of parametrized partial differential equations. Handbook Mater. Model. (2005) 1523–1558. 
  20. [20] R Development Core Team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2011). ISBN 3-900051-07-0. 
  21. [21] J. Racine, An efficient cross-validation algorithm for window width selection for nonparametric kernel regression. Commun. Stat. Simul. Comput.22 (1993) 1107–1107. 
  22. [22] A. Saltelli, K. Chan and E.M. Scott, Sensitivity analysis. Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester (2000). Zbl1152.62071MR1886391
  23. [23] A. Saltelli, S. Tarantola, F. Campolongo and M. Ratto, Sensitivity analysis in practice: a guide to assessing scientific models (2004). Zbl1049.62112MR2048817
  24. [24] T. J. Santner, B. Williams and W. Notz, The Design and Analysis of Computer Experiments. Springer−Verlag (2003). Zbl1041.62068MR2160708
  25. [25] R. Schaback, Mathematical results concerning kernel techniques. In Prep. 13th IFAC Symposium on System Identification, Rotterdam. Citeseer (2003) 1814–1819. 
  26. [26] M. Scheuerer, R. Schaback and M. Schlather, Interpolation of spatial data – a stochastic or a deterministic problem? Universität Göttingen (2011) http://num.math.uni-goettingen.de/schaback/research/papers/IoSD.pdf. Zbl06248582
  27. [27] I.M. Sobol, Sensitivity estimates for nonlinear mathematical models. Math. Modeling Comput. Experiment 1 (1995) 407–414, 1993. Zbl1039.65505MR1335161
  28. [28] I.M. Sobol, Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul.55 (2001) 271–280. Zbl1005.65004MR1823119
  29. [29] C.B. Storlie, L.P. Swiler, J.C. Helton and C.J. Sallaberry, Implementation and evaluation of nonparametric regression procedures for sensitivity analysis of computationally demanding models. Reliab. Eng. Syst. Saf.94 (2009) 1735–1763. 
  30. [30] B. Sudret, Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf.93 (2008) 964–979. 
  31. [31] J.Y. Tissot and C. Prieur, A bias correction method for the estimation of sensitivity indices based on random balance designs. Reliab. Eng. Syst. Saf. (2010). 
  32. [32] A.W. van der Vaart, Asymptotic statistics. Vol. 3 of Cambr. Series Statist. Probab. Math. Cambridge University Press, Cambridge (1998). Zbl0910.62001MR1652247

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