A Shapley-Owen index for interaction quantification

The recent work of Owen [SIAM/ASA J. Uncertainty Quantification, 2 (2014), pp. 245--251] has introduced the Shapley value as an importance measure for global sensitivity analysis (Shapley effect, henceforth). When inputs are dependent, using Shapley effects provides a strategy to overcome conceptual difficulties related to the interpretation of Sobol' sensitivity indices. However, Shapley effects have been formulated thus far only to quantify the importance of individual model inputs, without providing information about interactions. This article extends the above-mentioned work to propose a Shapley sensitivity measure for interaction effects. We make use of the generalized Shapley value introduced by Owen [Management Sci., 18 (1972), pp. 64--79] and axiomatized later on in Grabisch and Roubens [Internat. J. Game Theory, 28 (1999), pp. 547--565]. In parallel to the work of Owen, we propose this Shapley--Owen effect as a tool for global interaction quantification in presence of dependent inputs. We show that by using this index it is also possible to gain insights on the synergistic/antagonistic nature of interactions.