Clustering is one of the most important task in pattern recognition. For most of partitional clustering algorithms, a partition that represents as much as possible the structure of the data is generated. In this paper, we adress the problem of finding the optimal number of clusters from data. This can be done by introducing an index which evaluates the validity of the generated fuzzy c-partition. We propose to use a criterion based on the fuzzy combination of membership values which quantifies the l-order overlap and the intercluster separation of a given pattern.


Fuzzy Cluster Membership Degree Validity Index Cluster Validity Fuzzy Partition 
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.


  1. 1.
    Bensaid, A.M., Hall, L.O., Bezdek, J.C., Clarke, L.P., Silbiger, M.L., Arrington, J.A., Murtagh, R.F.: Validity-guided (re)clustering with applications to image segmentation. IEEE Transactions on Fuzzy Systems 4(2), 112–123 (1996)CrossRefGoogle Scholar
  2. 2.
    Bezdek, J.C.: Cluster validity with fuzzy sets. Journal of Cybernetics 3, 58–72 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bezdek, J.C.: Pattern Recognition with fuzzy objective function algorithm. Plenum Press (1981)Google Scholar
  4. 4.
    Bezdek, J.C., Pal, N.R.: Some new indexes of cluster validity. IEEE Transactions on Systems, Man and Cybernetics 23(3), 301–315 (1998)CrossRefGoogle Scholar
  5. 5.
    Blake, C., Merz, C.: Uci repository of machine learning databases (1998),
  6. 6.
    Dave, R.N.: Validating fuzzy partitions obtained through c-shells clustering. Pattern Recognition Letters 17(6), 613–623 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fukuyama, Y., Sugeno, M.: A new method for choosing the number of clusters for the fuzzy c-means method. In: Proc. 5th Fuzzy Systems Symposium, pp. 247–250 (1989)Google Scholar
  8. 8.
    Gustafson, D.E., Kessel, W.C.: Fuzzy clustering with fuzzy covariance matrix. In: Proc. IEEE Conference on Decision and Control, San Diego, California, pp. 761–766 (1979)Google Scholar
  9. 9.
    Kim, D.-W., Lee, K.H., Lee, D.: On cluster validity index for estimating the optimal number of fuzzy clusters. Pattern Recognition 37(3), 2009–2025 (2004)CrossRefGoogle Scholar
  10. 10.
    Klement, E.P., Mesiar, R.: Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms. Elsevier, Amsterdam (2005)zbMATHGoogle Scholar
  11. 11.
    Mascarilla, L., Berthier, M., Frélicot, C.: A k-order fuzzy or operator for pattern classification with k-order ambiguity rejection. Fuzzy Sets and Systems 159(15), 2011–2029 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pal, N.R., Bezdek, J.C.: On cluster validity for the fuzzy c-means model. IEEE Transactions on Fuzzy Systems 3(3), 370–379 (1995)CrossRefGoogle Scholar
  13. 13.
    Wang, W., Zhang, Y.: On fuzzy cluster validity indices. Fuzzy Sets and Systems 158(19), 2095–2117 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wu, K.L., Yang, M.S.: A cluster validity index for fuzzy clustering. Pattern Recognition Letters 26(9), 1275–1291 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Xie, X.L., Beni, G.: A validity measure for fuzzy clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence 13(8), 841–847 (1991)CrossRefGoogle Scholar
  16. 16.
    Zarandi, M.H.F., Neshat, E., Türksen, I.B.: A new cluster validity index for fuzzy clustering based on similarity measure. In: 11th International Conference on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, pp. 127–135 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Hoel Le Capitaine
    • 1
  • Carl Frélicot
    • 1
  1. 1.MIA LaboratoryUniversity of La RochelleLa RochelleFrance

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