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Register a new userSensitivity Analysis and Generalized Chaos Expansions. Lower Bounds for Sobol indices
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Added by
Olivier Roustant
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
Authors
O Roustant
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The so-called polynomial chaos expansion is widely used in computer experiments. For example, it is a powerful tool to estimate Sobol sensitivity indices. In this paper, we consider generalized chaos expansions built on general tensor Hilbert basis. In this frame, we revisit the computation of the Sobol indices and give general lower bounds for these indices. The case of the eigenfunctions system associated with a Poincar{e} differential operator leads to lower bounds involving the derivatives of the analyzed function and provides an efficient tool for variable screening. These lower bounds are put in action both on toy and real life models demonstrating their accuracy.
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Gaelle Chastaing
, Clementine Prieur (- Methodes dAnalyse Stochastique desn Codes et Traitements Numeriques
2013
The hierarchically orthogonal functional decomposition of any measurable function f of a random vector X=(X_1,...,X_p) consists in decomposing f(X) into a sum of increasing dimension functions depending only on a subvector of X. Even when X_1,..., X_p are assumed to be dependent, this decomposition is unique if 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=f(X), this decomposition leads to the definition of generalized sensitivity indices able to quantify the uncertainty of Y with respect to the dependent inputs X. 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.
In the context of computer code experiments, sensitivity analysis of a complicated input-output system is often performed by ranking the so-called Sobol indices. One reason of the popularity of Sobols approach relies on the simplicity of the statistical estimation of these indices using the so-called Pick and Freeze method. In this work we propose and study sensitivity indices for the case where the output lies on a Riemannian manifold. These indices are based on a Cramer von Mises like criterion that takes into account the geometry of the output support. We propose a Pick-Freeze like estimator of these indices based on an $U$--statistic. The asymptotic properties of these estimators are studied. Further, we provide and discuss some interesting numerical examples.
Sensitivity indices are commonly used to quantity the relative inuence of any specic group of input variables on the output of a computer code. In this paper, we focus both on computer codes the output of which is a cumulative distribution function and on stochastic computer codes. We propose a way to perform a global sensitivity analysis for these kinds of computer codes. In the rst setting, we dene two indices: the rst one is based on Wasserstein Fr{e}chet means while the second one is based on the Hoeding decomposition of the indicators of Wasserstein balls. Further, when dealing with the stochastic computer codes, we dene an ideal version of the stochastic computer code thats ts into the frame of the rst setting. Finally, we deduce a procedure to realize a second level global sensitivity analysis, namely when one is interested in the sensitivity related to the input distributions rather than in the sensitivity related to the inputs themselves. Several numerical studies are proposed as illustrations in the dierent settings.
In this paper, we introduce new indices adapted to outputs valued in general metric spaces. This new class of indices encompasses the classical ones; in particular, the so-called Sobol indices and the Cram{e}r-von-Mises indices. Furthermore, we provide asymptotically Gaussian estimators of these indices based on U-statistics. Surprisingly, we prove the asymp-totic normality straightforwardly. Finally, we illustrate this new procedure on a toy model and on two real-data examples.
To estimate direct and indirect effects of an exposure on an outcome from observed data strong assumptions about unconfoundedness are required. Since these assumptions cannot be tested using the observed data, a mediation analysis should always be accompanied by a sensitivity analysis of the resulting estimates. In this article we propose a sensitivity analysis method for parametric estimation of direct and indirect effects when the exposure, mediator and outcome are all binary. The sensitivity parameters consist of the correlation between the error terms of the mediator and outcome models, the correlation between the error terms of the mediator model and the model for the exposure assignment mechanism, and the correlation between the error terms of the exposure assignment and outcome models. These correlations are incorporated into the estimation of the model parameters and identification sets are then obtained for the direct and indirect effects for a range of plausible correlation values. We take the sampling variability into account through the construction of uncertainty intervals. The proposed method is able to assess sensitivity to both mediator-outcome confounding and confounding involving the exposure. To illustrate the method we apply it to a mediation study based on data from the Swedish Stroke Register (Riksstroke).
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