Abstract
The inclusion-exclusion integral is a generalization of the discrete Choquet integral, defined with respect to a fuzzy measure and an interaction operator that replaces the minimum function in the Choquet integral’s Möbius representation. While in general this means that the resulting operator can be non-monotone, we have previously proposed using averaging aggregation functions for the interaction component, which under certain requirements can produce non-linear, but still averaging, operators. Here we consider how the orness of the overall function changes depending on the chosen component functions and hence propose a simplified calculation for approximating the orness of an averaging inclusion-exclusion integral.
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Notes
- 1.
It has been noted in [16] that the least squares linear approximation of a given function f (which could be used to infer the importance of each variable) actually corresponds with the Banzhaf index, a calculation similar to the Shapley index.
- 2.
- 3.
Using linear programming techniques as found, e.g. in [11]. Full details of the transformations and code used to learn fuzzy measures can be found at http://aggregationfunctions.wordpress.com.
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Honda, A., James, S., Rajasegarar, S. (2017). Orness and Cardinality Indices for Averaging Inclusion-Exclusion Integrals. In: Torra, V., Narukawa, Y., Honda, A., Inoue, S. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2017. Lecture Notes in Computer Science(), vol 10571. Springer, Cham. https://doi.org/10.1007/978-3-319-67422-3_6
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