Abstract
In this paper a new kind of Choquet integral with respect to fuzzy measure on fuzzy σ–algebra is introduced, and some elementary properties of this kind of Choquet integral are studied. Convergence theorems for sequences of Choquet integrals are shown. A transformation theorem is given, which reveals the relation between this kind of Choquet integral on fuzzy sets and the Choquet integral on classical crisp sets. Finally, a new set function defined by this kind of Choquet integral is discussed. This new set function preserves many structural characteristics of the original set function.
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References
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)
Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1955)
Grabisch, M., Murofushi, T., Sugeno, M. (eds.): Fuzzy Measures and Integrals: Theory and Applications. Physica-Verlag, Heidelberg (2000)
Leung, K., Wong, M., Lam, W., Wang, Z., Xu, K.: Learning nonlinear multiregression networks based on evolutionary computation. IEEE Trans. Syst. Man. Cybern. 32, 630–644 (2002)
Xu, K., Wang, Z., Heng, P., Leung, K.: Classification by nonlinear integral projections. IEEE Trans. Fuzzy Systems 11, 187–201 (2003)
Hocaoglu, A.K., Gader, P.D.: Domain learning using Choquet integral-based morphological shared weight neural networks. Image and Vision Computing 21, 663–673 (2003)
Denneberg, D.: Conditional expectation for monotone measures, the discrete case. Journal of Mathematical Economics 37, 105–121 (2002)
De Waegenaere, A., Kast, R., Lapied, A.: Choquet pricing and equilibrium. Insurance: Mathematics and Economics 32, 359–370 (2003)
Bolaños, M.J., et al.: Convergence properties of the monotone expectation and its application to the extension of fuzzy measures. Fuzzy Sets and Systems 33, 201–212 (1989)
Narukawa, Y., Murofushi, T.: Conditions for Choquet integral representation of the comonotonically additive and monotone functional. J. Math. Anal. Appl. 282, 201–211 (2003)
Wang, Z., Klir, G.J., Wang, W.: Monotone set functions defined by Choquet integral. Fuzzy Sets and Systems 81, 241–250 (1996)
Wang, Z.: Convergence theorems for sequences of Choquet integrals. Int. J. General Systems 26, 133–143 (1997)
Sugeno, M.: The theory of fuzzy integral and its applications. Ph.D Dissertation. Tokyo Institute of Technology (1974)
Qiao, Z.: On fuzzy measure and fuzzy integral on fuzzy set. Fuzzy Sets and Systems 37, 77–92 (1990)
Wang, Z., Qiao, Z.: Transformation theorems for fuzzy integrals on fuzzy sets. Fuzzy Sets and Systems 34, 355–364 (1990)
Pap, E.: Null-Additive Set Functions. Kluwer Academic, Dordrecht (1995)
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Huang, Y., Wu, C. (2006). Choquet Integrals with Respect to Fuzzy Measure on Fuzzy σ–Algebra. In: Yeung, D.S., Liu, ZQ., Wang, XZ., Yan, H. (eds) Advances in Machine Learning and Cybernetics. Lecture Notes in Computer Science(), vol 3930. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11739685_35
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DOI: https://doi.org/10.1007/11739685_35
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-33584-9
Online ISBN: 978-3-540-33585-6
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