Electronic Journal of Statistics

Sensitivity analysis for multidimensional and functional outputs

Abstract

Let $X:=(X_{1},\ldots,X_{p})$ be random objects (the inputs), defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$ and valued in some measurable space $E=E_{1}\times\ldots\times E_{p}$. Further, let $Y:=Y=f(X_{1},\ldots,X_{p})$ be the output. Here, $f$ is a measurable function from $E$ to some Hilbert space $\mathbb{H}$ ($\mathbb{H}$ could be either of finite or infinite dimension). In this work, we give a natural generalization of the Sobol indices (that are classically defined when $Y\in\mathbb{R}$), when the output belongs to $\mathbb{H}$. These indices have very nice properties. First, they are invariant under isometry and scaling. Further they can be, as in dimension $1$, easily estimated by using the so-called Pick and Freeze method. We investigate the asymptotic behaviour of such an estimation scheme.

Article information

Source
Electron. J. Statist., Volume 8, Number 1 (2014), 575-603.

Dates
First available in Project Euclid: 20 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1400592265

Digital Object Identifier
doi:10.1214/14-EJS895

Mathematical Reviews number (MathSciNet)
MR3211025

Zentralblatt MATH identifier
1348.62106

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties

Citation

Gamboa, Fabrice; Janon, Alexandre; Klein, Thierry; Lagnoux, Agnès. Sensitivity analysis for multidimensional and functional outputs. Electron. J. Statist. 8 (2014), no. 1, 575--603. doi:10.1214/14-EJS895. https://projecteuclid.org/euclid.ejs/1400592265

References

• [1] Borgonovo, E., A new uncertainty importance measure. Reliability Engineering & System Safety, 92(6):771–784, 2007.
• [2] Campbell, K., McKay, M. D., and Williams, B. J., Sensitivity analysis when model outputs are functions. Reliability Engineering & System Safety, 91(10):1468–1472, 2006.
• [3] Fort, J.-C., Klein, T., and Rachdi, N., New sensitivity analysis subordinated to a contrast. ArXiv e-prints, Accepted in Communications in Statistics – Theory and Methods, 2014.
• [4] Fort, J. C., Klein, T., Lagnoux, A., and Laurent, B., Estimation of the Sobol indices in a linear functional multidimensional model. Journal of Statistical Planning and Inference, 143(9):1590–1605, September 2013.
• [5] Gamboa, F., Janon, A., Klein, T., Lagnoux-Renaudie, A., and Prieur, C., Statistical inference for Sobol pick freeze monte carlo method. Preprint available at http://hal.inria.fr/hal-00804668/en, 2013.
• [6] Janon, A., Klein, T., Lagnoux-Renaudie, A., Nodet, M., and Prieur, C., Asymptotic normality and efficiency of two Sobol index estimators. ESAIM P & S, Vol. eFirst month 6, ISSN:1262-3318, 2013, doi:10.1051/ps/2013040.
• [7] Lamboni, M., Monod, H., and D. Makowski, Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models. Reliability Engineering & System Safety, 96(4):450–459, 2011.
• [8] Ledoux, M., The Concentration of Measure Phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001.
• [9] Ledoux, M. and Talagrand, M., Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin, 1991.
• [10] Owen, A., Dick, J., and Chen, S., Higher order Sobol’ indices. Information and Inference, 3(1):59–81, 2014. doi:10.1093/imaiai/iau001.
• [11] Owen, A. B., Variance components and generalized Sobol’ indices. SIAM/ASA Journal on Uncertainty Quantification, 1(1):19–41, 2013.
• [12] Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., and Tarantola, S., Global Sensitivity Analysis: The Primer. Wiley Online Library, 2008.
• [13] Sobol, I. M., Sensitivity estimates for nonlinear mathematical models. Math. Modeling Comput. Experiment, 1(4):407–414 (1995), 1993.
• [14] van der Vaart, A. W., Asymptotic Statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 1998.
• [15] Xu, C. and Gertner, G. Z., Reliability of global sensitivity indices. Journal of Statistical Computation and Simulation, 81(12):1939–1969, 2011.