Science China Information Sciences

, Volume 57, Issue 11, pp 1–8 | Cite as

Fuzziness parameter selection in fuzzy c-means: The perspective of cluster validation

  • KaiLe Zhou
  • Chao Fu
  • ShanLin Yang
Research Paper


Fuzzy c-means (FCM) algorithm is an important clustering method in pattern recognition, while the fuzziness parameter, m, in FCM algorithm is a key parameter that can significantly affect the result of clustering. Cluster validity index (CVI) is a kind of criterion function to validate the clustering results, thereby determining the optimal cluster number of a data set. From the perspective of cluster validation, we propose a novel method to select the optimal value of m in FCM, and four well-known CVIs, namely XB, VK, VT, and SC, for fuzzy clustering are used. In this method, the optimal value of m is determined when CVIs reach their minimum values. Experimental results on four synthetic data sets and four real data sets have demonstrated that the range of m is [2, 3.5] and the optimal interval is [2.5, 3].


clustering fuzziness parameter fuzzy c-means (FCM) cluster validation cluster validity index 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hartigan J A. Clustering Algorithms. New York: Wiley, 1975zbMATHGoogle Scholar
  2. 2.
    Yue S H, Wu T, Cui L J, et al. Clustering mechanism for electric tomography imaging. Sci China Inf Sci, 2012, 55: 2849–2864MathSciNetCrossRefGoogle Scholar
  3. 3.
    Jain A K. Data clustering: 50 years beyond k-means. Pattern Recogn Lett, 2010, 31: 651–666CrossRefGoogle Scholar
  4. 4.
    Xu R, Wunsch II D. Survey of clustering algorithms. IEEE Trans Neural Networ, 2005, 16: 645–678CrossRefGoogle Scholar
  5. 5.
    Hu C X, Liu Y M, Li G, et al. Improved FOCUSS method for reconstruction of cluster structured sparse signals in radar imaging. Sci China Ser F-Inf Sci, 2012, 55: 1776–1788MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ni W W, Chong Z H. Clustering-oriented privacy-preserving data publishing. Knowl-based Syst, 2012, 35: 264–270CrossRefGoogle Scholar
  7. 7.
    Dunn C. A fuzzy relative of the ISODATA process and its use in detecting compact, well-separated clusters. J Cybern, 1974, 3: 32–57MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bezdek J C. Pattern recognition with fuzzy objective function algorithms. New York: Plenum Press, 1981CrossRefzbMATHGoogle Scholar
  9. 9.
    Pal N R, Bezdek J C. On cluster validity for the fuzzy c-mean model. IEEE Trans Fuzzy Syst, 1995, 3: 370–379CrossRefGoogle Scholar
  10. 10.
    Hall L L, Bensaid A M, Clarke L P. A comparison of neural network and fuzzy clustering techniques in segmenting magnetic resonance images of the brain. IEEE Trans Neural Networ, 2002, 3: 672–682CrossRefGoogle Scholar
  11. 11.
    Cannon R L, Dave J V, Bezdek J C. Efficient implementation of the fuzzy c-means clustering algorithms. IEEE Trans Pattern Anal, 1986, PAMI-8: 248–255CrossRefGoogle Scholar
  12. 12.
    Shen Y, Shi H, Zhang J Q. Improvement and optimization of a fuzzy c-means clustering algorithm. In: Proceedings of the 18th Instrumentation and Measurement Technology Conference, IEEE Computer Society 2001. 1430–1433Google Scholar
  13. 13.
    Bezdek J C. A physical interpretation of fuzzy ISODATA. IEEE Trans Syst Man Cy B, 1976, SMC-6: 387–390MathSciNetGoogle Scholar
  14. 14.
    Bezdek J C, Hathaway R. Convergence theory for fuzzy c-means: Counterexamples and repairs. IEEE Trans Syst Man Cy B, 1987, 17: 873–877CrossRefzbMATHGoogle Scholar
  15. 15.
    Chan K P, Cheung Y S. Clustering of clusters. Pattern Recogn, 1992, 25: 211–217CrossRefGoogle Scholar
  16. 16.
    Choe H, Jordan J B. On the optimal choice of parameters in a fuzzy c-means algorithm. In: Proceedings of IEEE International Conference on Fuzzy Systems, IEEE Computer Society 1992. 349–354CrossRefGoogle Scholar
  17. 17.
    Ozkan I, Turksen I B, Entropy assessment for type-2 fuzziness. In: Proceedings of IEEE International Conference on Fuzzy Systems, IEEE Computer Society 2004. 1111–1115Google Scholar
  18. 18.
    Ozkan I, Turksen I B. Upper and lower values for the level of fuzziness in FCM. Inform Sci, 2007, 177: 5143–5152CrossRefzbMATHGoogle Scholar
  19. 19.
    Wu K L. Analysis of parameter selections for fuzzy C-means. Pattern Recogn, 2012, 45: 407–415CrossRefzbMATHGoogle Scholar
  20. 20.
    Huang M, Xia Z, Wang H, et al. The range of the value for the fuzzifier of the fuzzy c-means algorithm. Pattern Recogn Lett, 2012, 33: 2280–2284CrossRefGoogle Scholar
  21. 21.
    Hwang C, Rhee F C H. Uncertain fuzzy clustering: Interval type-2 fuzzy approach to c-means. IEEE Trans Fuzzy Syst, 2007, 15: 107–120CrossRefGoogle Scholar
  22. 22.
    Yu J. On the fuzziness index of the FCM algorithms. Chin J Comput, 2003, 26: 965–973Google Scholar
  23. 23.
    Yu J, Cheng Q, Huang H. Analysis of the weighting exponent in the FCM. IEEE Trans Syst Man Cy B, 2004, 34: 634–639CrossRefGoogle Scholar
  24. 24.
    Fadili M J, Ruan S, Bloyet D, et al. On the number of clusters and the fuzziness index for unsupervised FCA application to BOLD fMRI time series. Med Image Anal, 2001, 5: 55–67CrossRefGoogle Scholar
  25. 25.
    Devijver P A, Kittler J. Pattern Recognition: A Statistical Approach. London: Prentice-Hall, 1982zbMATHGoogle Scholar
  26. 26.
    Hoppner F, Klawon F, Kruse R, et al. Fuzzy Cluster Analysis: Methods for Classifications Data Analysis and Image Recognition. New York: Wiley, 1999Google Scholar
  27. 27.
    Kim M, Ramakrishna R S. New indices for cluster validity assessment. Pattern Recogn Lett, 2005, 26: 2353–2363CrossRefGoogle Scholar
  28. 28.
    Wang W, Zhang Y. On fuzzy cluster validity indices. Fuzzy Set Syst, 2007, 158: 2095–2117CrossRefzbMATHGoogle Scholar
  29. 29.
    Xie X L, Beni G, A validity measure for fuzzy clustering. IEEE Trans Pattern Anal, 1991, 13: 841–847CrossRefGoogle Scholar
  30. 30.
    Kwon S H. Cluster validity index for fuzzy clustering. Electron Lett, 1998, 34: 2176–2177CrossRefGoogle Scholar
  31. 31.
    Tang Y, Sun F, Sun Z. Improved validation index for fuzzy clustering. In: Proceedings of the 2005 American Control Conference, IEEE Computer Society, 2005. 1120–1125Google Scholar
  32. 32.
    Bensaid A M, Hall L O, Bezdek J C, et al. Validity-guided (Re)clustering with applications to image segmentation. IEEE Trans Fuzzy Syst, 1996, 4: 112–123CrossRefGoogle Scholar
  33. 33.
    Bezdek J C, Ehrlish R, Full W. FCM: The fuzzy c-means clustering algorithm. Comput Geosci-UK, 1984, 10: 191–203CrossRefGoogle Scholar
  34. 34.
    Frank A, Asuncion A. UCI Machine Learning Repository. California: University of California, 2010Google Scholar
  35. 35.
    Zhang Y J, Wang W N, Zhang X N, et al. A cluster validity index for fuzzy clustering. Info Sci, 2008, 178: 1205–1218CrossRefzbMATHGoogle Scholar
  36. 36.
    Jegatha Deborah L, Baskaran R, Kannan A. A survey on internal validity measure for cluster validation. Int J Computer Sci Eng Surv, 2010, 1: 85–102CrossRefGoogle Scholar
  37. 37.
    Guerra L, Robles V, Bielza C. et al. A comparison of clustering quality indices using outliers and noise. Intell Data Anal, 2012, 16: 703–715Google Scholar
  38. 38.
    Arbelaitz O, Gurrutxaga I, Muguerza J, et al. An extensive comparative study of cluster validity indices. Pattern Recogn, 2013, 46: 243–256CrossRefGoogle Scholar
  39. 39.
    Gurrutxaga I, Muguerza J, Arbelaitz O, et al. Towards a standard methodology to evaluate internal cluster validity indices. Pattern Recogn Lett, 2011, 32: 505–515CrossRefGoogle Scholar
  40. 40.
    Zalik K R, Zalik B. Validity index for clusters of different sizes and densities. Pattern Recogn Lett, 2011, 32: 221–234CrossRefGoogle Scholar
  41. 41.
    Zalik K R. Cluster validity index for estimation of fuzzy clusters of different sizes and densities. Pattern Recogn, 2010, 43: 3374–3390CrossRefzbMATHGoogle Scholar
  42. 42.
    Geva A B, Steinberg Y, Bruckmair S, et al. A comparison of cluster validity criteria for a mixture of normal distributed data. Pattern Recogn Lett, 2000, 21: 511–529CrossRefGoogle Scholar
  43. 43.
    Dimitriadou E, Dolňicar S, Weingessel A. An examination of indexes for determining the number of clusters in binary data sets. Psychometrika, 2002, 67: 137–159MathSciNetCrossRefGoogle Scholar
  44. 44.
    Maulik U, Bandyopadhyay S. Performance evaluation of some clustering algorithms and validity indices. IEEE Trans Pattern Anal, 2002, 24: 1650–1654CrossRefGoogle Scholar
  45. 45.
    Pal N R, Bezdek J C. Correction to on cluster validity for the fuzzy c-means model. IEEE Trans Fuzzy Syst, 1997, 5: 152–153CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of ManagementHefei University of TechnologyHefeiChina
  2. 2.Key Laboratory of Process Optimization and Intelligent Decision-MakingMinistry of EducationHefeiChina

Personalised recommendations