Skip to main content
Log in

Complex fuzzy c-means algorithm

  • Published:
Artificial Intelligence Review Aims and scope Submit manuscript


In this paper a new clustering algorithm is presented: A complex-based Fuzzy c-means (CFCM) algorithm. While the Fuzzy c-means uses a real vector as a prototype characterizing a cluster, the CFCM’s prototype is generalized to be a complex vector (complex center). CFCM uses a new real distance measure which is derived from a complex one. CFCM’s formulas for the fuzzy membership are derived. These formulas are extended to derive the complex Gustafson–Kessel algorithm (CGK). Cluster validity measures are used to assess the goodness of the partitions obtained by the complex centers compared those obtained by the real centers. The validity measures used in this paper are the Partition Coefficient, Classification Entropy, Partition Index, Separation Index, Xie and Beni’s Index, Dunn’s Index. It is shown in this paper that the CFCM give better partitions of the data than the FCM and the GK algorithms. It is also shown that the CGK algorithm outperforms the CFCM but at the expense of much higher computational complexity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  • Abonyi J, Balasko B, Feil B, Fuzzy clustering and data analysis toolbox.

  • Babuska R (1998) Fuzzy modeling for control. Kluwer Academic Publishers, Boston

    Book  Google Scholar 

  • Babuska R, van der Veen PJ, Kaymak U (2002) Improved covariance estimation for Gustafson–Kessel clustering. IEEE Int Conf Fuzzy Syst, pp 1081–1085

  • Bensaid AM, Hall LO, Bezdek JC, Clarke LP, Silbiger ML, Arrington JA, Murtagh RF (1996) Validity-guided (Re)Clustering with applications to image segmentation. IEEE Trans Fuzzy Syst 4: 112–123

    Article  Google Scholar 

  • Bezdek JC (1981) Pattern recognition with fuzzy objective function algorithms. Plenum Press, New York

    Book  MATH  Google Scholar 

  • Dunn JC (1973) A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters. J Cybern 3: 32–57

    Article  MathSciNet  MATH  Google Scholar 

  • Gath I, Geva AB (1989) Unsupervised optimal fuzzy clustering. IEEE Trans Pattern Anal Mach Intell 7: 773–781

    Article  Google Scholar 

  • Gustafson DE, Kessel WC (1979) Fuzzy clustering with fuzzy covariance matrix. In: Proceedings of the IEEE CDC. San Diego, pp 761–766

  • Hoppner F, Klawonn F, Kruse R, Runkler T (1999) Fuzzy cluster analysis. Wiley, Chichester

    Google Scholar 

  • Kanungo T, Mount DM, Netanyahu NS, Piatko C, Silverman R, Wu AY (2002) An efficient k-means clustering algorithm: analysis and implementation. IEEE Trans Patt Anal Mach Intell 24(7): 881–892

    Article  Google Scholar 

  • Likas A, Vlassis N, Verbeek JJ (2003) The global k-means clustering algorithm. In: pattern recognition vol 36, No 2

  • Pelleg D, Moore A (1999) Accelerating exact k-means algorithms with geometric reasoning. In: Technical report CMU-CS-00105. Carnegie Mellon University, Pittsburgh

  • Pelleg D, Moore A (2000) X-means: extending k-means with efficient estimation of the number of clusters. In: Proceedings of the seventeenth international conference on machine learning. Palo Alto

  • Xie XL, Beni GA (1991) Validity measure for fuzzy clustering. IEEE Trans PAMI 3(8): 841–846

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Issam Dagher.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dagher, I. Complex fuzzy c-means algorithm. Artif Intell Rev 38, 25–39 (2012).

Download citation

  • Published:

  • Issue Date:

  • DOI: