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International Conference on Modeling Decisions for Artificial Intelligence

MDAI 2017: Modeling Decisions for Artificial Intelligence pp 51–62Cite as

Orness and Cardinality Indices for Averaging Inclusion-Exclusion Integrals

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10571))

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. 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. 2.

    We note however that in [7, 8], \(I(\mathbf x|A)\) is proposed to be arity-decreasing, i.e. \(I(\mathbf x|A) < I(\mathbf x|B)\) if \(B\subset A\). As noted in [9], instead using averaging functions (which are not necessarily arity-decreasing) will allow the IE-integral to be averaging.

  3. 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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Authors and Affiliations

  1. Kyushu Institute of Technology, Iizuka, Japan

    Aoi Honda

  2. Deakin University, Burwood, Australia

    Simon James & Sutharshan Rajasegarar

Authors
  1. Aoi Honda
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  2. Simon James
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  3. Sutharshan Rajasegarar
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Correspondence to Simon James .

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Editors and Affiliations

  1. University of Skövde, Skövde, Sweden

    Vicenç Torra

  2. Toho Gakuen, Kunitachi, Tokyo, Japan

    Yasuo Narukawa

  3. Kyushu Institute of Technology, Iizuka, Fukuoka, Japan

    Aoi Honda

  4. Kyushu Institute of Technology, Kitakyushu-shi, Fukuoka, Japan

    Sozo Inoue

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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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  • DOI: https://doi.org/10.1007/978-3-319-67422-3_6

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