Abstract
Fuzzy measures on multisets are studied in this paper. We show that a class of multisets on a finite space can be represented as a subset of positive integers. Comonotonicity for multisets are defined. We show that a fuzzy measure on multisets with some comonotonicity condition can be represented by a generalized fuzzy integral.
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References
Alsina, C., Frank, M.J., Schweizer, B.: Associative Functions: Triangular Norms and Copulas. World Scientific (2006)
Benvenuti, P., Mesiar, R., Vivona, D.: Monotone set functions-based integrals. In: Pap, E. (ed.) Handbook of Measure Theory, pp. 1329–1379. Elsevier, Amsterdam (2002)
Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1953/1954)
Dellacherie, C.: Quelques commentaires sur les prolongements de capacités, Séminaire de Probabilités 1969/1970, Strasbourg. Lecture Notes in Mathematics, vol. 191, pp. 77–81 (1971)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)
Klement, E.P., Mesiar, R., Pap, E.: Integration with respect to decomposable measures, based on a conditionally distributive semiring on the unit interval. Int. J. of Unc., Fuzziness and Knowledge Based Systems 8(6), 701–717 (2000)
Ling, C.H.: Representation of associative functions. Publ. Math. Debrecen 12, 189–212 (1965)
Murofushi, T., Sugeno, M.: An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure. Fuzzy Sets and Systems 29, 201–227 (1989)
Narukawa, Y., Torra, V.: Multidimensional generalized fuzzy integral. Fuzzy Sets and Systems 160, 802–815 (2009)
Narukawa, Y., Stokes, K., Torra, V.: Fuzzy Measures and Comonotonicity on Multisets. In: Torra, V., Narakawa, Y., Yin, J., Long, J. (eds.) MDAI 2011. LNCS, vol. 6820, pp. 20–30. Springer, Heidelberg (2011)
Ralescu, D., Adams, G.: The fuzzy integral. J. Math. Anal. Appl. 75, 562–570 (1980)
Raman, R., Raman, V., Rao, S.S.: Succint indexable dictionaries with applications to encoding k-ary trees and multisets. In: Proc. of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002) (2002)
Sugeno, M.: Fuzzy measures and fuzzy integrals. Trans. of the Soc. of Instrument and Control Engineers 8(2) (1972) (in Japanese)
Sugeno, M.: Theory of fuzzy integrals and its applications, Ph. D. Dissertation, Tokyo Institute of Technology (1974)
Sugeno, M., Murofushi, T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122, 197–222 (1987)
Torra, V., Narukawa, Y., Sugeno, M.: Non-additive measures: theory and applications. Springer (2014)
Torra, V., Stokes, K., Narukawa, Y.: An extension of fuzzy measures to multisets and its relation to distorted probabilities. IEEE Trans. on Fuzzy Systems 20(6), 1032–1045 (2012)
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Narukawa, Y., Torra, V. (2014). Choquet Integral on Multisets. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2014. Communications in Computer and Information Science, vol 442. Springer, Cham. https://doi.org/10.1007/978-3-319-08795-5_29
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DOI: https://doi.org/10.1007/978-3-319-08795-5_29
Publisher Name: Springer, Cham
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