New Fr\'echet features for random distributions and associated sensitivity indices

TL;DR

This paper introduces new Fréchet features for random distributions using contrast-based Wasserstein costs, enabling the definition of sensitivity indices for stochastic models, extending classical Sobol indices.

Contribution

It proposes a novel framework for Fréchet features based on contrast functions, extending sensitivity analysis to random distributions with new indices.

Findings

Defined new Fréchet features using contrast-based Wasserstein costs

Extended Sobol sensitivity indices to random distribution outputs

Demonstrated the applicability of these features and indices in sensitivity analysis

Abstract

In this article we define new Fr\`Echet features for random cumulative distribution functions using contrast. These contrasts allow to construct Wasserstein costs and our new features minimize the average costs as the Fr\`Echet mean minimizes the mean square Wasserstein$_2$ distance. An example of new features is the median, and more generally the quantiles. From these definitions, we are able to define sensitivity indices when the random distribution is the output of a stochastic code. Associated to the Fr\`Echet mean we extend the Sobol indices, and in general the indices associated to a contrast that we previously proposed.