Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T21:51:46.742Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic normality and efficiency of two Sobol indexestimators

Published online by Cambridge University Press:  03 October 2014

Alexandre Janon ,
Thierry Klein ,
Agnès Lagnoux ,
Maëlle Nodet  and
Clémentine Prieur
Alexandre Janon
Affiliation:
Laboratoire Jean Kuntzmann, Université Joseph Fourier, INRIA/MOISE, 51 rue des Mathématiques, BP 53, 38041 Grenoble cedex 9, France. alexandre.janon@imag.fr
Thierry Klein
Affiliation:
Laboratoire de Statistique et Probabilités, Institut de Mathématiques Université Paul Sabatier (Toulouse 3), 31062 Toulouse cedex 9, France
Agnès Lagnoux
Affiliation:
Laboratoire de Statistique et Probabilités, Institut de Mathématiques Université Paul Sabatier (Toulouse 3), 31062 Toulouse cedex 9, France
Maëlle Nodet
Affiliation:
Laboratoire Jean Kuntzmann, Université Joseph Fourier, INRIA/MOISE, 51 rue des Mathématiques, BP 53, 38041 Grenoble cedex 9, France. alexandre.janon@imag.fr
Clémentine Prieur
Affiliation:
Laboratoire Jean Kuntzmann, Université Joseph Fourier, INRIA/MOISE, 51 rue des Mathématiques, BP 53, 38041 Grenoble cedex 9, France. alexandre.janon@imag.fr
Get access

Abstract

Many mathematical models involve input parameters, which are not precisely known. Globalsensitivity analysis aims to identify the parameters whose uncertainty has the largestimpact on the variability of a quantity of interest (output of the model). One of thestatistical tools used to quantify the influence of each input variable on the output isthe Sobol sensitivity index. We consider the statistical estimation of this index from afinite sample of model outputs: we present two estimators and state a central limittheorem 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 isreplaced by a noisy version.

Keywords

Sensitivity analysissobol indicesasymptotic efficiencyasymptotic normalityconfidence intervalsmetamodellingsurface response methodology
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 342 - 364
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

G.E.P. Box and N.R. Draper, Empirical model-building and response surfaces. John Wiley and Sons (1987).
R. Carnell, lhs: Latin Hypercube Samples (2009). R package version 0.5.
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).
Cukier, R.I., Levine, H.B. and Shuler, K.E., Nonlinear sensitivity analysis of multiparameter model systems. J. Comput. Phys. 26 (1978) 1–42. Google Scholar
Da Veiga, S. and Gamboa, F., Efficient estimation of sensitivity indices. J. Nonparametric Statist. 25 (2013) 573–595. Google Scholar
G.M. Dancik, mlegp: Maximum Likelihood Estimates of Gaussian Processes (2011). R package version 3.1.2.
T. Hayfield and J.S. Racine, Nonparametric econometrics: The np package. J. Statist. Softw. 27 (2008).
Helton, J.C., Johnson, J.D., Sallaberry, C.J. and Storlie, C.B., Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliab. Eng. Syst. Saf. 91 (2006) 1175–1209. Google Scholar
Homma, T. and Saltelli, A., Importance measures in global sensitivity analysis of nonlinear models. Reliab. Eng. Syst. Saf. 52 (1996) 1–17. Google Scholar
I.A. Ibragimov and R.Z. Has’ Minskii, Statistical estimation–asymptotic theory. Vol. 16 of Appl. Math. Springer−Verlag, New York (1981).
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.
Janon, A., Nodet, M. and Prieur, C., Certified reduced-basis solutions of viscous Burgers equations parametrized by initial and boundary values. ESAIM: M2AN 47 (2013) 317–348. Google Scholar
Janon, A., Nodet, M. and Prieur, C., Uncertainties assessment in global sensitivity indices estimation from metamodels. Internat. J. Uncert. Quantification 4 (2014) 21–36. Google Scholar
Kahan, W., Pracniques: further remarks on reducing truncation errors. Commun. ACM 8 (1965) 40. Google Scholar
Madych, W.R. and Nelson, S.A., Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J. Approx. Theory 70 (1992) 94–114. Google Scholar
Marrel, A., Iooss, B., Laurent, B. and Roustant, O., Calculations of sobol indices for the gaussian process metamodel. Reliab. Eng. Syst. Saf. 94 (2009) 742–751. Google Scholar
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.
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).
N.C. Nguyen, K. Veroy and A.T. Patera, Certified real-time solution of parametrized partial differential equations. Handbook Mater. Model. (2005) 1523–1558.
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.
Racine, J., An efficient cross-validation algorithm for window width selection for nonparametric kernel regression. Commun. Stat. Simul. Comput. 22 (1993) 1107–1107. Google Scholar
A. Saltelli, K. Chan and E.M. Scott, Sensitivity analysis. Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester (2000).
A. Saltelli, S. Tarantola, F. Campolongo and M. Ratto, Sensitivity analysis in practice: a guide to assessing scientific models (2004).
T. J. Santner, B. Williams and W. Notz, The Design and Analysis of Computer Experiments. Springer−Verlag (2003).
R. Schaback, Mathematical results concerning kernel techniques. In Prep. 13th IFAC Symposium on System Identification, Rotterdam. Citeseer (2003) 1814–1819.
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.
Sobol, I.M., Sensitivity estimates for nonlinear mathematical models. Math. Modeling Comput. Experiment 1 (1995) 407–414, 1993. Google Scholar
Sobol, I.M., Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul. 55 (2001) 271–280. Google Scholar
Storlie, C.B., Swiler, L.P., Helton, J.C. and Sallaberry, C.J., Implementation and evaluation of nonparametric regression procedures for sensitivity analysis of computationally demanding models. Reliab. Eng. Syst. Saf. 94 (2009) 1735–1763. Google Scholar
Sudret, B., Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93 (2008) 964–979. Google Scholar
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).
A.W. van der Vaart, Asymptotic statistics. Vol. 3 of Cambr. Series Statist. Probab. Math. Cambridge University Press, Cambridge (1998).