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Integrated Uncertainty Management and Applications pp 103–113Cite as

Entropy of Fuzzy Measure

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Part of the book series: Advances in Intelligent and Soft Computing ((AINSC,volume 68))

Abstract

A definition for the entropy of fuzzy measures defined on set systems is proposed. The underlying set is not necessarily the whole power set, but satisfy a condition of regularity. This definition encompasses the classical definition of Shannon for probability measures, as well as the definition of Marichal et al. for classical fuzzy measures, and may have applicability to most fuzzy measures which appear in applications. We give several characterizations of this entropy which are natural and understandable as the entropy.

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References

  1. Choquet, G.: Theory of capacities. Ann Inst Fourier 5, 131–295 (1953)

    MathSciNet  Google Scholar 

  2. Dukhovny, A.: General entropy of general measures. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 10, 213–225 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Faddeev, D.K.: The notion of entropy of finite probabilistic schemes (Russian). Uspekhi Mat. Nauk 11, 15–19 (1956)

    Google Scholar 

  4. Grabisch, M., Labreuche, C.: Bi-capacities – Part I: definition, Mobius transform and interaction. Fuzzy Sets and Systems 151, 211–236 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Grabisch, M., Labreuche, C.: Bi-capacities – Part II: the Choquet integral. Fuzzy Sets and Systems 151, 237–259 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Honda, A., Grabisch, M.: Entropy of capacities on lattices. Information Sciences 176, 3472–3489 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Honda, A., Grabisch, M.: An axiomatization of entropy of capacities on set systems. European Journal of Operational Research 190, 526–538 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kojadinovic, I., Marichal, J.-L., Roubens, M.: An axiomatic approach to the definition of the entropy of a discrete Choquet capacity. Information Sciences 172, 131–153 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Marichal, J.-L.: Entropy of discrete Choquet capacities. Eur. J. of Oper. Res. 137, 612–624 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Marichal, J.-L.: Roubens, M.: Entropy of discrete fuzzy measure. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 8, 625–640 (2000)

    MATH  MathSciNet  Google Scholar 

  11. Shannon, C.E.: A mathematical theory of communication. Bell System Tech. Journ. 27, 374–423, 623–656 (1948)

    MathSciNet  Google Scholar 

  12. Sugeno, M.: Theory of fuzzy integrals and its applications. PhD Thesis, Tokyo Institute of Technology (1974)

    Google Scholar 

  13. Sugeno, M.: Fuzzy measures and fuzzy integrals: a survey. In: Gupta, M.M., Saridis, G.N., Gains, B.R. (eds.) Fuzzy automata and decision processes, pp. 89–102. North Holland, Amsterdam (1977)

    Google Scholar 

  14. Wenxiu, H.: Entropy of fuzzy measure spaces. Bulletin for Studies and Exchanges on Fuzziness and its Applications 50 (1992)

    Google Scholar 

  15. Yager, R.R.: On the entropy of fuzzy measures. IEEE Transaction on Fuzzy Systems 8, 453–461 (2000)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Kyushu Institute of Technology, 680-4 Kawazu, Iizuka city, Fukuoka, 820-8502, Japan

    Aoi Honda

  2. Centre d’Économie de la Sorbonne, University of Paris I, 106-112, Bd de l’Hôpital, 75013, Paris, France

    Michel Grabisch

Authors
  1. Aoi Honda
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  2. Michel Grabisch
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Editor information

Editors and Affiliations

  1. School of Knowledge Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, 923-1292, Ishikawa, Japan

    Van-Nam Huynh  & Yoshiteru Nakamori  & 

  2. Department of Engineering Mathematics, University of Bristol, Queen’s Building, University Walk, BS8 1TR, Bristol, United Kingdom

    Jonathan Lawry

  3. Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, 560-8531, Osaka, Japan

    Masahiro Inuiguchi

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Honda, A., Grabisch, M. (2010). Entropy of Fuzzy Measure. In: Huynh, VN., Nakamori, Y., Lawry, J., Inuiguchi, M. (eds) Integrated Uncertainty Management and Applications. Advances in Intelligent and Soft Computing, vol 68. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11960-6_11

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  • DOI: https://doi.org/10.1007/978-3-642-11960-6_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11959-0

  • Online ISBN: 978-3-642-11960-6

  • eBook Packages: EngineeringEngineering (R0)

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