École Nationale Supérieure de
Techniques Avancées

Abstract

Global sensitivity analysis (GSA) of a numerical model aims at quantifying a model inputs’ im-portance regarding its output’s variability. When the inputs are assumed to be independent, theSobol’ indices allow to serve this purpose by associating to each input a percentage of the output’s variance. However, these indices loose their explanatory power when dependency is introduced.In order to circumvent this issue, an attribution tool between players, calledShapley value, in-spired from cooperative game theory has been applied to GSA. It allowed for new GSA importanceindices, calledShapley effects, which are a redistribution of the output’s variable among the in-puts. Even if these indices allow a decomposition of the output’s variance, they present a majordrawback: an exogenous variable (outside of the model) can be associated to a strictly positiveindex if it is correlated to an endogenous variable (inside if the model). Informally, this problemis better known as theShapley’s joke. The goal of this internship is to propose other indices byusing another game theory allocation rule calledproportional value. This result is not originallydefined for cost functions with null values, but an extension to these cases in proposed. Basedon these results, novel GSA importance indices are proposed: theproportional marginal effects.These indices do not fall under the Shapley’s joke, and allow to detect potentially spurious vari-ables. This behavior is proved in the general case, and the analytical behavior of the proportionalmarginal effects is studied through toy-cases. Finally, an estimation strategy is presented, andapplied to an industrial use-case of non-destructive weld study using ultrasound waves