This article adresses the problem of assessing how close two strict and/or fuzzy partitions are. A new index based on a measurement of the sparsity of the contingency matrix crossing the partitions is proposed that satisfies the required properties formulated within the paper and presents a low complexity. It is compared to well-known existing indices of the literature, such as the Rand and the Jaccard indices, the transfert distance and some of their recent fuzzy counterparts.


Cluster analysis Rand index Jaccard Index Transfert distance Sparsity measure Fuzzy residual implications 


  1. 1.
    Pal, N., Bezdek, J.: On cluster validity for the fuzzy c-means model. IEEE Trans. on Fuzzy Systems 3(3) (1995)Google Scholar
  2. 2.
    Wang, W., Zhang, Y.: On fuzzy cluster validity indices. Fuzzy Sets and Systems 158(19) (2007)Google Scholar
  3. 3.
    Borgelt, C.: Prototype-based classification and clustering Habilitation Thesis. Habilitation Thesis, Fakultat fur Informatik der Otto von Guericke, Universitat Magdeburg (2005)Google Scholar
  4. 4.
    Patrikainen, A.: Methods for comparing subspace clusterings. PhD thesis, Helsinki University of Technology (2005)Google Scholar
  5. 5.
    Hurley, N., Rickard, S.: Comparing measures of sparsity. IEEE Trans. on Information Theory 55(10) (2009)Google Scholar
  6. 6.
    Mas, M., Monserrat, M., Torrens, J., Trillas, E.: A survey on fuzzy implication functions. IEEE Trans. on Fuzzy Systems 15(6) (2007)Google Scholar
  7. 7.
    Albatineh, A., Niewiadomska-Bugaj, M., Mihalko, D.: On similarity indices and correction for chance agreement. J. of Classification 23 (2006)Google Scholar
  8. 8.
    Charon, I., Denoeud, L., Guenoche, A., Hudry, O.: Maximum transfer distance between partitions. J. of Classification 23(1) (2006)Google Scholar
  9. 9.
    Anderson, D., Bezdek, J., Popescu, M., Keller, J.: Comparing fuzzy, probabilistic, and possibilistic partitions. IEEE Trans. on Fuzzy Systems 18(5) (2010)Google Scholar
  10. 10.
    Quéré, R., Capitaine, H.L., Fraisseix, N., Frélicot, C.: On normalizing fuzzy coincidence matrices to compare fuzzy and/or possibilistic partitions with the rand index. In: 10th IEEE International Conference on Data Mining, pp. 977–982 (2010)Google Scholar
  11. 11.
    Hüllermeier, E., Rifqi, M.: A fuzzy variant of the rand index for comparing clustering structures. In: 13th IFSA World Congress (2009)Google Scholar
  12. 12.
    Campello, R.: Generalized external indexes for comparing data partitions with overlapping categories. Pattern Recognition Letters 31(9) (2010)Google Scholar
  13. 13.
    Brouwer, R.: Extending the rand, adjusted rand and jaccard indices to fuzzy partitions. J. of Intelligent Information Systems 32(3) (2009)Google Scholar
  14. 14.
    Klement, E., Mesiar, R.: Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms. Elsevier (2005)Google Scholar
  15. 15.
    Karvanen, J., Cichocki, A.: Measuring sparseness of noisy signals. In: 4th Int. Symp. on Independent Component Analysis and Blind Signal Separation (2003)Google Scholar
  16. 16.
    Le Capitaine, H., Frélicot, C.: Classification with reject options in a logical framework: a fuzzy residual implication approach. In: 13th IFSA World Congress (2009)Google Scholar
  17. 17.
    Grabisch, M., Marichal, J., Mesiar, R., Pap, E.: Aggregation Functions. Encyclopedia of Mathematics and its Applications. Cambridge University Press (2009)Google Scholar
  18. 18.
    Ceccarelli, M., Maratea, A.: A Fuzzy Extension of Some Classical Concordance Measures and an Efficient Algorithm for their Computation. In: Lovrek, I., Howlett, R.J., Jain, L.C. (eds.) KES 2008, Part III. LNCS (LNAI), vol. 5179, pp. 755–763. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Frank, A., Asuncion, A.: UCI machine learning repository (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Romain Quéré
    • 1
  • Carl Frélicot
    • 1
  1. 1.Mathématiques, Image et ApplicationsUniversité de La RochelleFrance

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