Shapley effects for sensitivity analysis with dependent inputs: bootstrap and kriging-based algorithms

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.