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http://dx.doi.org/10.5391/JKIIS.2007.17.4.455

A note on distance measure and similarity measure defined by Choquet integral on interval-valued fuzzy sets  

Jang, Lee-Chae (Dept. of Mathematics and Computer Science, Konkuk University)
Publication Information
Journal of the Korean Institute of Intelligent Systems / v.17, no.4, 2007 , pp. 455-459 More about this Journal
Abstract
Interval-valued fuzzy sets were suggested for the first time by Gorzafczany(1983) and Turksen(1986). Based on this, Zeng and Li(2006) introduced concepts of similarity measure and entropy on interval-valued fuzzy sets which are different from Bustince and Burillo(1996). In this paper, by using Choquet integral with respect to a fuzzy measure, we introduce distance measure and similarity measure defined by Choquet integral on interval-valued fuzzy sets and discuss some properties of them. Choquet integral is a generalization concept of Lebesgue inetgral, because the two definitions of Choquet integral and Lebesgue integral are equal if a fuzzy measure is a classical measure.
Keywords
interval-valued fuzzy set; fuzzy measure; distance measure; similarity measure; Choquet integral;
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  • Reference
1 R.J. Aumann, 'Integrals of set-valued functions', J. Math. Anal. Appl. Vol.12 pp.1-12, 1965
2 G. Choquet, Theory of capacities, Ann. Inst. Fourior, Vol. 5, pp.131-295, 1953
3 T. Murofushi and M. Sugeno, A theory of Fuzzy measures: representations, the Choquet integral, and null sets, J. Math. Anal. and Appl. Vol.159, pp.532-549, 1991   DOI
4 G. Wang and X. Li, The applications of interval- valued fuzzy numbers and interval-distribution numbers, Fuzzy Sets and Systems Vol.98, pp.331- 335, 1998   DOI   ScienceOn
5 L.A. Zadeh, Fuzzy sets, Information and Control Vol.8, pp.338-353, 1965   DOI
6 Jin-Lum Fan, Yuan-Liang Ma and Wei-Xin Xie, On some properties of distance measures, Fuzzy Sets and Systems Vol.117, pp.355-361, 2001   DOI   ScienceOn
7 W. Zeng and Hongxing Li, Relationship between similarity measure and entropy of interval-valued fuzzy sets, Fuzzy Sets and Systems Vol. 157, pp.1477-1484, 2006   DOI   ScienceOn
8 T. Murofushi and M. Sugeno, An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure, Fuzzy Sets and Systems Vol.29, pp.201-227, 1989   DOI   ScienceOn
9 Liu Xuechang, Entropy, distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and Systems Vol.52, 1992
10 Y. Narukawa, T. Murofushi and M. Sugeno, Regular fuzzy measure and representation of comonotonically additive functional, Fuzzy Sets and Systems Vol.112, pp.177-186, 2000   DOI   ScienceOn
11 B. Turksen, Interval-valued fuzzy sets based on normal forms, Fuzzy Sets and Systems Vol.20, pp.191-210, 1986   DOI   ScienceOn
12 P. Burillo and H. Bustince, Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems Vol.78, pp.305-316, 1996   DOI   ScienceOn
13 Dug-Hun Hong and Sungho Lee, Some algebraic properties and distance measure for interval- valued fuzzy numbers, Information Sciences Vol.148, pp.1-10, 2002   DOI   ScienceOn
14 B. Gorzalczany, Approximate inference with interval- valued fuzzy sets-an outline, in Proceedings of Polish Symposium on Interval and Fuzzy Mathematics, Poznan, Poland, pp.89-95
15 Y. Narukawa, T. Murofushi and M. Sugeno, Extension and representation of comonotonically additive functionals, Fuzzy Sets and Systems Vol.121, pp.217-226, 2001   DOI   ScienceOn
16 W. Y. Zeng and Y. Shi, Note on interval-valued fuzzy sets, Lecture Notes in Artificial Intelligence 3316 pp.20-25, 2005