Advertisement

Relevance of Classes in a Fuzzy Partition. A Study from a Group of Aggregation Operators

  • Fabián CastiblancoEmail author
  • Camilo Franco
  • Javier Montero
  • J. Tinguaro Rodríguez
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 831)

Abstract

This paper presents a study of the relevance property in a fuzzy partition from a fuzzy classification system. This study allows establishing a stopping criterion for the inclusion of a class in a fuzzy partition based on relevance. Such a criterion is constructed from a stable relationship on the commutative group formed by two new mappings (and the aggregation operators conjunctive and disjunctive) of the fuzzy classification system. The criterion is illustrated through an example on image analysis by the fuzzy c-means algorithm.

Keywords

Relevance Covering Overlap Classification Fuzzy partition 

Notes

Acknowledgements

This research has been partially supported by the Government of Spain (grant TIN2015-66471-P), the Government of Madrid (grant S2013/ICE-2845), and Complutense University (UCM Research Group 910149).

References

  1. 1.
    Zadeh, L.: Fuzzy sets. Inf. Control 8, 338–353 (1965)CrossRefGoogle Scholar
  2. 2.
    Bezdek, J., Harris, J.: Fuzzy partitions and relations: an axiomatic basis for clustering. Fuzzy Sets Syst. 1, 111–127 (1978)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bellman, R., Kalaba, R., Zadeh, L.: Abstraction and pattern classification. J. Math. Anal. Appl. 13, 1–7 (1966)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Pedrycz, W.: Fuzzy sets in pattern recognition: methodology and methods. Pattern Recogn. 23, 121–146 (1990)CrossRefGoogle Scholar
  5. 5.
    Del Amo, A., Montero, J., Biging, G., Cutello, V.: Fuzzy classification systems. Eur. J. Oper. Res. 156, 459–507 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Del Amo, A., Gómez, D., Montero, J., Biging, G.: Relevance and redundancy in fuzzy classification systems. Mathw. Soft Comput. 8, 203–216 (2001)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Ruspini, E.: A new approach to clustering. Inf. Control 15, 22–32 (1969)CrossRefGoogle Scholar
  8. 8.
    Dombi, J.: Basic concepts for a theory of evaluation: the aggregative operator. Eur. J. Oper. Res. 10, 282–293 (1982)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dombi, J.: A general class of fuzzy operators, the DeMorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets Syst. 8, 149–163 (1982)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Matsakis, P., Andrefouet, P., Capolsini, P.: Evaluation of fuzzy partition. Remote Sens. Environ. 74, 516–533 (2000)CrossRefGoogle Scholar
  11. 11.
    Bustince, H., Fernández, J., Mesiar, R., Montero, J., Orduna, R.: Overlap functions. Nonlinear Anal. Theory Methods Appl. 72, 1488–1499 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bustince, H., Barrenechea, E., Pagola, M., Fernández, J.: The notions of overlap and grouping functions. In: Saminger-Platz, S., Mesiar, R. (eds.) On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory, Studies in Fuzziness and Soft Computing, vol. 336, pp. 137–156. Springer, Switzerland (2016).  https://doi.org/10.1007/978-3-319-28808-6_8CrossRefGoogle Scholar
  13. 13.
    Gómez, D., Rodríguez, J., Bustince, H., Barrenechea, E., Montero, J.: n-dimensional overlap functions. Fuzzy Sets Syst. 287, 57–75 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Qiao, J., Hu, B.Q.: On interval additive generators of interval overlap functions and interval grouping functions. Fuzzy Sets Syst. 323, 19–55 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Qiao, J., Hu, B.Q.: On the migrativity of uninorms and nullnorms over overlap and grouping functions. Fuzzy Sets Syst. (2017). http://doi.org/10.1016/j.fss.2017.11.012
  16. 16.
    Klement, E., Moser, B.: On the redundancy of fuzzy partitions. Fuzzy Sets Syst. 85, 195–201 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sperber, D., Wilson, D.: Relevance: Communication and Cognition, 2nd edn. Blackwell Publishers Inc., Cambridge (1995)Google Scholar
  18. 18.
    Cutello, V., Montero, J.: Recursive connective rules. Int. J. Intell. Syst. 14, 3–20 (1999)CrossRefGoogle Scholar
  19. 19.
    Castiblanco, F., Gómez, D., Montero, J., Rodríguez, J.: Aggregation tools for the evaluation of classifications. In: 2017 Joint 17th World Congress of International Fuzzy Systems Association and 9th International Conference on Soft Computing and Intelligent Systems (IFSA-SCIS). IEEE, Otsu, pp. 1–5 (2017)Google Scholar
  20. 20.
    Mordeson, J., Bhutani, K., Rosenfeld, A.: Fuzzy Group Theory. Springer, Heidelberg (2005).  https://doi.org/10.1007/b12359CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Economic, Administrative and Accounting SciencesGran Colombia UniversityBogotáColombia
  2. 2.Department of Industrial EngineeringAndes UniversityBogotáColombia
  3. 3.Department of StatisticsComplutense UniversityMadridSpain

Personalised recommendations