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Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 220))

Abstract

In this chapter, we review some families of fuzzy measures and their use in fuzzy integrals. We will also review the determination of fuzzy measures from examples in the case of the Choquet integral.

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References

  1. Benvenuti, P., Mesiar, R., Vivona, D., (2002), Monotone set functions-based integrals, in E. Pap (Ed.), Handbook of Measure Theory, North-Holland, 1329–1379.

    Google Scholar 

  2. Calvo, T., Mayor, G., Mesiar, R., (Eds.) (2002), Aggregation Operators, Physica-Verlag.

    Google Scholar 

  3. Choquet, G., (1953/54), Theory of capacities, Ann. Inst. Fourier, 5, 131–295.

    Google Scholar 

  4. Dubois, D., Modèles mathĂ©matiques de l’imprĂ©cis et de l’incertain, en vue d’applications aux techniques d’aide ‘a la dĂ©cision, Thèse d’Etat, Univ. de Grenoble, 1983.

    Google Scholar 

  5. Dubois, D., Prade, H., (1982), A class of fuzzy measures based on triangular norms - A general framework for the combination of uncertain information, Int. J. of General Systems, 8, 43–61.

    Article  MATH  MathSciNet  Google Scholar 

  6. Edwards, W., (1953), Probability-Preferences in Gambling, American Journal of Psychology 66, 349–364.

    Article  Google Scholar 

  7. Edwards, W., (1962), Subjective probabilities inferred from decisions, Psychological review 69, 109–135.

    Article  Google Scholar 

  8. Grabisch, M., (1997), k-order additive discrete fuzzy measures and their representation, Fuzzy sets and systems 92:2, 167–189.

    Google Scholar 

  9. Klement, E. P., Mesiar, R., Pap, E., (2000), Triangular Norms, Kluwer Academic Publisher.

    Google Scholar 

  10. Imai, H., Torra, V., (2003), On a modeling of decision making with a twofold integral, Proc. EUSFLAT 2003, 714–717.

    Google Scholar 

  11. Mori, T., Murofushi, T., (1989), An analysis of evaluation model using fuzzy measure and the Choquet integral, Proc. of the 5th Fuzzy System Symposium, 207–212, in Japanese.

    Google Scholar 

  12. Murofushi, T., Sugeno, M., (1991), Fuzzy t-conorm integral with respect to fuzzy measures: Generalization of Sugeno integral and Choquet integral, Fuzzy Sets and Systems, 42, 57–71.

    Article  MATH  MathSciNet  Google Scholar 

  13. Narukawa, Y., Torra, V., (2004), Twofold integral and Multi-step Choquet integral, Kybernetika, 40:1 39–50.

    MathSciNet  Google Scholar 

  14. Narukawa, Y., Torra, V., (2005), Fuzzy measure and probability distributions: distorted probabilities, IEEE Trans. on Fuzzy Systems, 13:5, 617–629.

    Article  MathSciNet  Google Scholar 

  15. Rota, G.-C., (1964), On the Foundations of Combinatorial Theory. I. Theory of Möbius Functions, Z. Wahrscheinlichkeitstheorie 2, 340–368.

    Article  MATH  MathSciNet  Google Scholar 

  16. Sugeno, M., (1972), Fuzzy Measures and Fuzzy Integrals, Trans. of the Soc. of Instrument and Control Engineers, 8:2, 218–226.

    Google Scholar 

  17. Sugeno, M., (1974), Theory of fuzzy integrals and its applications, Ph. D. Dissertation, Tokyo Institute of Technology, Tokyo, Japan.

    Google Scholar 

  18. Torra, V., (1997), The Weighted OWA operator, Int. J. of Intel. Syst., 12, 153–166.

    Article  MATH  Google Scholar 

  19. Torra, V., (1999), On hierarchically S-decomposable fuzzy measures, Int. J. of Intel. Syst., 14:9, 923–934.

    Article  MATH  Google Scholar 

  20. Torra, V., (2000), Learning weights for Weighted OWA operators, Proc. of the IEEE Int. Conf. on Industrial Electronics, Control and Instrumentation (IECON 2000), 2530–2535.

    Google Scholar 

  21. Torra, V., (2003), La integral doble o twofold integral: Una generalitzaciĂ³ de les integrals de Choquet i Sugeno, Butlletì de l’AssociaciĂ³ Catalana d’IntelÄ‹ligencia Artificial 29 (2003) 13–19. (Preliminary version in English: Twofold integral: A generalization of Choquet and Sugeno integral, IIIA Technical Report TR-2003-08).

    Google Scholar 

  22. Torra, V., (2004), OWA operators in data modeling and reidentification, IEEE Trans. on Fuzzy Systems, 12:5, 652–660.

    Article  Google Scholar 

  23. Torra, V., Narukawa, Y., (2006), Modeling decisions: Information fusion and aggregation operators, forthcoming.

    Google Scholar 

  24. Vitali, G., (1925), Sulla definizione di integrale delle funzioni di una variabile, Annali di Matematica Serie IV, Tomo II 111–121.

    Google Scholar 

  25. Weber, S., (1986), Two integrals and some modified versions – critical remarks, Fuzzy Sets and Systems 20, 97–105.

    Article  MATH  MathSciNet  Google Scholar 

  26. Yager, R. R., (1988), On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Trans. on Systems, Man and Cybernetics, 18, 183–190.

    Article  MATH  MathSciNet  Google Scholar 

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Torra, V. (2008). On the Construction of Models Based on Fuzzy Measures and Integrals. In: Bustince, H., Herrera, F., Montero, J. (eds) Fuzzy Sets and Their Extensions: Representation, Aggregation and Models. Studies in Fuzziness and Soft Computing, vol 220. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73723-0_5

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  • DOI: https://doi.org/10.1007/978-3-540-73723-0_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73722-3

  • Online ISBN: 978-3-540-73723-0

  • eBook Packages: EngineeringEngineering (R0)

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