Global Sensitivity Analysis: a new generation of mighty estimators based on rank statistics

Abstract

We propose a new statistical estimation framework for a large family of global sensitivity analysis methods. Our approach is based on rank statistics and uses an empirical correlation coefficient recently introduced by Sourav Chatterjee. We show how to apply this approach to compute not only the Cramér-von-Mises indices, which are directly related to Chatterjee's notion of correlation, but also Sobol indices at any order, higher-order moment indices, and Shapley effects. We establish consistency of the resulting estimators and demonstrate their numerical efficiency, especially for small sample sizes.

Fairness seen as Global Sensitivity Analysis

“…The second one circumvents the limitations of working with variance-decomposition. We finish this section by merging these two alternatives, inspired by the works of [1,16] and [5].…”

“…Many other criteria to compare the conditional distribution of the output knowing some of the inputs to the distribution of the output have been proposed -by the means of divergences, or measures of dissimilarity between distributions for example. We recall here the Cramér-von-Mises indices [16], an answer to this issue that will be of use later in a fairness framework -see Section 3.…”

“…We denote by (X i , i = 1, • • • , p) the inputs and by (X j , Y j ), j = 1, • • • , n a n-sample of both inputs and outputs. They are defined (see [16]), for every input i, as follow:…”

“…As before, these indices extend to the multivariate case. Simple estimators have been proposed [5,16], and are based on permutations and rankings. We will recall the estimation in a following section.…”

“…Various indicators, usually taking the form of indices between 0 and 1, allow the understanding of how much a feature is important. Multiple set of indices have been proposed over the years such as Sobol' indices, Cramér-von-Mises indices, HSIC -see [23,10,22,18,16] and references therein. The flexibility in the choice allows for deep understanding in the relationship between a feature and the outcome of an algorithm.…”