Abstract
In global sensitivity analysis, the well-known Sobol’ sensitivity indices aim to quantify how the variance in the output of a mathematical model can be apportioned to the different variances of its input random variables. These indices are based on the functional variance decomposition and their interpretation becomes difficult in the presence of statistical dependence between the inputs. However, as there are dependencies in many application studies, this drawback enhances the development of interpretable sensitivity indices. Recently, the Shapley values that were developed in the field of cooperative games theory have been connected to global sensitivity analysis and present good properties in the presence of dependencies. Nevertheless, the available estimation methods do not always provide confidence intervals and require a large number of model evaluations. In this paper, a bootstrap resampling is implemented in existing algorithms to assess confidence intervals. We also propose to consider a metamodel in substitution of a costly numerical model. The estimation error from the Monte-Carlo sampling is combined with the metamodel error in order to have confidence intervals on the Shapley effects. Furthermore, we compare the Shapley effects with existing extensions of the Sobol’ indices in different examples of dependent random variables.
Highlights
In the last decades, computational models have been increasingly used to approximate physical phenomenons
For the linear Gaussian model we found a relation between the Shapley effects and the sensitivity indices obtained with the Rosenblatt Transformation (RT) method
We present a methodology for estimating the Shapley effects through a kriging surrogate model taking into account both the Monte Carlo error and the surrogate model error
Summary
Computational models have been increasingly used to approximate physical phenomenons. Among GSA methods, variance-based approaches are a class of probabilistic ones that measure the part of variance of the model output which is due to the variance of a particular input These methods were popularized by [29] who introduced the well-known first order Sobol’ indices. The total Sobol’ indices have been introduced by [7] taking advantage of [12] These sensitivity indices are based on the functional ANalysis Of VAriance (ANOVA), the decomposition of which is unique only if the input random variables are assumed independent. The paper’s outline is as follows: Section 1 recalls the basic concept of Sobol’ indices in the independent and dependent configuration; Section 2 introduces the Shapley values and their links with sensitivity analysis; Section 3 theoretically compares the Sobol’ indices and the Shapley effects for two toy examples; Section 4 studies the quality of the estimated Shapley effects and their confidence intervals; Section 5 introduces the kriging model and how the kriging and Monte-Carlo errors can be separated from the overall error; Section 6 compares the indice performances using a kriging model on two toy examples; Section 7 synthesizes this work and suggests some perspectives
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