Advertisement

Combination of Rough and Fuzzy Sets Based on α-Level Sets

  • Y. Y. Yao

Abstract

A fuzzy set can be represented by a family of crisp sets using its α-level sets, whereas a rough set can be represented by three crisp sets. Based on such representations, this paper examines some fundamental issues involved in the combination of rough-set and fuzzy-set models. The rough-fuzzy-set and fuzzy-rough-set models are analyzed, with emphasis on their structures in terms of crisp sets. A rough fuzzy set is a pair of fuzzy sets resulting from the approximation of a fuzzy set in a crisp approximation space, and a fuzzy rough set is a pair of fuzzy sets resulting from the approximation of a crisp set in a fuzzy approximation space. The approximation of a fuzzy set in a fuzzy approximation space leads to a more general framework. The results may be interpreted in three different ways.

Keywords

Membership Function Equivalence Relation Approximation Space Fuzzy Partition Fuzzy Similarity Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bezdek, J.C. and Harris, J.D., “Fuzzy partitions and relations: an axiomatic basis for clustering,” Fuzzy Sets and Systems, 1, pp. 111–127, 1978.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Biswas, R., “On rough sets and fuzzy rough sets,” Bulletin of the Polish Academy of Sciences, Mathematics, 42, pp. 345–349, 1994.MATHGoogle Scholar
  3. [3]
    Biswas, R., “On rough fuzzy sets,” Bulletin of the Polish Academy of Sciences, Mathematics, 42, pp. 351–355, 1994.MATHGoogle Scholar
  4. [4]
    Chanas, S. and Kuchta, D., “Further remarks on the relation between rough and fuzzy sets,” Fuzzy Sets and Systems, 47, pp. 391–394, 1992.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Dubois, D. and Prade, H., “Twofold fuzzy sets and rough sets — some issues in knowledge representation,” Fuzzy Sets and Systems, 23, pp. 3–18, 1987.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Dubois, D. and Prade, H., “Rough fuzzy sets and fuzzy rough sets,” International Journal of General Systems, 17, pp. 191–209, 1990.MATHCrossRefGoogle Scholar
  7. [7]
    Dubois, D. and Prade, H., “Putting rough sets and fuzzy sets together,” in: Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, Slowinski, R., Ed., Kluwer Academic Publishers, Boston, pp. 203–222, 1992.Google Scholar
  8. [8]
    Iwinski, T.B. “Algebraic approach to rough sets,” Bulletin of the Polish Academy of Sciences, Mathematics, 35, 673–683, 1987.MathSciNetMATHGoogle Scholar
  9. [9]
    Klir, G.J. and Yuan, B., Fuzzy Sets and Fuzzy Logic, Theory and Applications, Prentice Hall, New Jersey, 1995.MATHGoogle Scholar
  10. [10]
    Kuncheva, L.I., “Fuzzy rough sets: application to feature selection,” Fuzzy Sets and Systems, 51, pp. 147–153, 1992.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Lin, T.Y., “Topological and fuzzy rough sets,” in: Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, Slowinski, R., Ed., Kluwer Academic Publishers, Boston, pp. 287–304, 1992.Google Scholar
  12. [12]
    Lin, T.Y. and Liu, Q., “Rough approximate operators: axiomatic rough set theory,” in: Rough Sets, Fuzzy Sets and Knowledge Discovery, Ziarko, W.P., Ed., Springer-Verlag, London, pp. 256–260, 1994.Google Scholar
  13. [13]
    Mizumoto, M. and Tanaka, K., “Some properties of fuzzy numbers,” in: Advances in Fuzzy Set Theory and Applications, Gupta, M.M, Ragade, R.K., and Yager, R.R., Eds., North-Holland, New York, pp. 153–164,1979.Google Scholar
  14. [14]
    Nakamura, A., “Fuzzy rough sets,” Note on Multiple-valued Logic in Japan, 9, pp. 1–8, 1988.Google Scholar
  15. [15]
    Nakamura, A. and Gao, J.M., “A logic for fuzzy data analysis,” Fuzzy Sets and Systems, 39, pp. 127–132, 1991.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Nanda, S. and Majumdar, S., “Fuzzy rough sets,” Fuzzy Sets and Systems, 45, pp. 157–160, 1992.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Negoita, C.V. and Ralescu, D.A., “Representation theorems for fuzzy concepts,” Kybernetes, 4, pp. 169–174, 1975.MATHCrossRefGoogle Scholar
  18. [18]
    Negoita, C.V. and Ralescu, D.A., Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975.MATHGoogle Scholar
  19. [19]
    Nguyen, H.T., “A note on the extension principle for fuzzy sets,” Journal of Mathematical Analysis and Applications, 64, pp. 369–380, 1978.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Pawlak, Z., “Rough sets,” International Journal of Computer and Information Sciences, 11, pp. 341–356, 1982.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Pawlak, Z., “Rough sets and fuzzy sets,” Fuzzy Sets and Systems, 17, pp. 99–102, 1985.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    Ralescu, D.A., “A generalization of the representation theorem,” Fuzzy Sets and Systems, 51, pp. 309–311, 1992.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Uehara, K. and Fujise, M., “Fuzzy inference based on families of a-level sets,” IEEE Transactions on Fuzzy Systems, 1, pp. 111–124, 1993.CrossRefGoogle Scholar
  24. [24]
    Willaeys, D. and Malvache, N., “The use of fuzzy sets for the treatment of fuzzy information by computer,” Fuzzy Sets and Systems, 5, pp. 323–328, 1981.MATHCrossRefGoogle Scholar
  25. [25]
    Wong, S.K.M. and Ziarko, W., “Comparison of the probabilistic approximate classification and the fuzzy set model,” Fuzzy Sets and Systems, 21, pp. 357–362, 1987.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Wygralak, M., “Rough sets and fuzzy sets — some remarks on interrelations,” Fuzzy Sets and Systems, 29, pp. 241–243, 1989.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    Yao, Y.Y. “Interval-set algebra for qualitative knowledge representation,” Proceedings of the 5th International Conference on Computing and Information, pp. 370–375, 1993.Google Scholar
  28. [28]
    Yao, Y.Y., “On combining rough and fuzzy sets,” Proceedings of the CSC’95 Workshop on Rough Sets and Database Mining, Lin, T.Y. (Ed.), San Jose State University, 9 pages, 1995.Google Scholar
  29. [29]
    Yao, Y.Y., and Wong, S.K.M., “A decision theoretic framework for approximating concepts,” International Journal of Man-machine Studies, 37, pp. 793–809, 1992.CrossRefGoogle Scholar
  30. [30]
    Zadeh, L.A., “Fuzzy sets,” Information and Control, 8, pp. 338–353, 1965.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    Zadeh, L.A., “The concepts of a linguistic variable and its application to approximate reasoning,” Information Sciences, 8, pp. 199–249, 1975.MathSciNetCrossRefGoogle Scholar
  32. [32]
    Ziarko, W., “Variable precision rough set model,” Journal of Computer and System Sciences, 46, pp. 39–59, 1993.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Y. Y. Yao
    • 1
  1. 1.Department of Computer ScienceLakehead UniversityThunder BayCanada

Personalised recommendations