Proportional marginal effects for global sensitivity analysis

TL;DR

This paper introduces proportional marginal effects (PME), a new variance-based sensitivity index that better distinguishes exogenous variables from endogenous ones in models with correlated inputs, improving upon Shapley effects.

Contribution

The paper extends the proportional allocation to variance-based GSA, proposing PME indices that differentiate exogenous variables even when correlated with endogenous inputs.

Findings

PME effectively identifies exogenous variables in correlated input scenarios.

PME provides a clearer distinction between exogenous and endogenous inputs.

Comparative analysis shows PME's advantages over Shapley effects in various cases.

Abstract

Performing (variance-based) global sensitivity analysis (GSA) with dependent inputs has recently benefited from cooperative game theory concepts.By using this theory, despite the potential correlation between the inputs, meaningful sensitivity indices can be defined via allocation shares of the model output's variance to each input. The ``Shapley effects'', i.e., the Shapley values transposed to variance-based GSA problems, allowed for this suitable solution. However, these indices exhibit a particular behavior that can be undesirable: an exogenous input (i.e., which is not explicitly included in the structural equations of the model) can be associated with a strictly positive index when it is correlated to endogenous inputs. In the present work, the use of a different allocation, called the ``proportional values'' is investigated. A first contribution is to propose an extension of this allocation, suitable for variance-based GSA. Novel GSA indices are then proposed, called the ``proportional marginal effects'' (PME). The notion of exogeneity is formally defined in the context of variance-based GSA, and it is shown that the PME allow the distinction of exogenous variables, even when they are correlated to endogenous inputs. Moreover, their behavior is compared to the Shapley effects on analytical toy-cases and more realistic use-cases.