Fuzzy Clustering

C-means algorithm

Published on February 13, 2020 · 3 mins read

Fuzzy Clustering

Fuzzy sets are used in fuzzy logic and can be considered as a generalisation of set theory. An element can be a member of a particular set or not in set theory, while in fuzzy set theory an element can have a gradual transition membership between sets. Hence, fuzzy clustering uses the fuzzy set to allow an instance to be in more than one cluster at the same time 1.

The most well known and used fuzzy clustering is fuzzy c-means algorithm, developed by Dunn 2 and later improved by Bezdek 3 who introduced the concept of the fuzzifier parameter \textbf{m}. This parameter, also called ‘fuzziness index’, is used to control the fuzziness of the membership of each item in the data set. Usually, m = 2 is used without any particular theoretical basis for this choice. For m = 1 the fuzzy c-means will behave as k–means algorithm, and the fuzziness of the system increases with the larger value of m parameter 4.

The fuzzy c-means algorithm has a similar approach as k–means algorithm. It requires a predefined number of clusters. Both algorithms start with random initialization of the cluster centres so c-means might have the same problem as k–means by converging to local optima. The result of the cmean algorithm is expressed as a membership percentage of each instance to the available clusters. This fuzzy membership clustering can be converted into hard clusters by choosing a cluster for each item with the highest membership ration 1.


  1. Wang, X. Y. (2006) Fuzzy Clustering in the Analysis of Fourier Transform Infrared Spectra for Cancer Diagnosis. The University of Nottingham.  2

  2. Dunn, J. C. (1974) ‘A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters’, Journal of Cybernetics, 3(3), pp. 32–57. doi: 10.1080/01969727308546046. 

  3. Jain, A. K., Murty, M. N. and Flynn, P. J. (1999) ‘Data clustering: a review’, ACM computing surveys (CSUR). ACM Press, 31(3), pp. 264–323. doi: 10.1145/331499.331504. 

  4. Bezdek, J. C. (1981) Pattern Recognition with Fuzzy Objective Function Algorithms, SIAM Review. New York: Plenum Press. doi: 10.1007/978-1-4757-0450-1. 

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