Generalized Sobol sensitivity indices for dependent variables: numerical methods

Abstract

The hierarchically orthogonal functional decomposition of any measurable function η of a random vector X=(X₁, … , X_p) consists in decomposing η(X) into a sum of increasing dimension functions depending only on a subvector of X. Even when X₁, … , X_p are assumed to be dependent, this decomposition is unique if the components are hierarchically orthogonal. That is, two of the components are orthogonal whenever all the variables involved in one of the summands are a subset of the variables involved in the other. Setting Y=η(X), this decomposition leads to the definition of generalized sensitivity indices able to quantify the uncertainty of Y due to each dependent input in X [Chastaing G, Gamboa F, Prieur C. Generalized Hoeffding–Sobol decomposition for dependent variables – application to sensitivity analysis. Electron J Statist. 2012;6:2420–2448]. In this paper, a numerical method is developed to identify the component functions of the decomposition using the hierarchical orthogonality property. Furthermore, the asymptotic properties of the components estimation is studied, as well as the numerical estimation of the generalized sensitivity indices of a toy model. Lastly, the method is applied to a model arising from a real-world problem.

Keywords:

• sensitivity analysis
• dependent variables
• extended basis
• functional decomposition
• greedy algorithm
• LARS

AMS Subject Classification::

• 62G08
• 62H99

Acknowledgements

This work has been supported by French National Research Agency (ANR) through the COSINUS program (project COSTA-BRAVA number ANR-09-COSI-015).

Funding

The authors would like to thank Loic Le Gratiet and Gilles Defaux for their help in providing and understanding the tank pressure model.