Document Type : Research Paper

Authors

Abstract

Estimating the optimal number of clusters in an unsupervised
partitioning of data sets has been a challenging area in recent years.
These indices usually use two criteria called compactness and
separation to evaluate the efficiency of the performed clustering. In
this paper a new separation measure for ECAS cluster validity index,
proposed by Fazel et al. [1] is identified, which uses Jaccard distance
in order to consider the whole shape of clusters. Jaccard distance uses
the size of intersection and union of fuzzy sets, giving the cluster
validity index more information about the overlap and separation of
clusters. This property results in high robustness of the proposed index
dealing with various degrees of fuzziness in comparison with ECAS.
To test the efficiency of the proposed index in comparison with nine
other indices existing in the literature, 15 data sets (3 existing datasets
and 12 artificial data sets) have been used. Computational results
indicate robustness and high capability of the proposed index in
comparison with previous indices

Keywords

1. M. H. F. Zarandi, M. R. Faraji, and M. Karbasian, An Exponential Cluster Validity
Index for Fuzzy Clustering with Crisp and Fuzzy Data, Scientia Iranica, vol. 17, no. 2,
pp. 95–110, 2010.
2. L. Zadeh, Fuzzy sets, Inf.control, no. 8, pp. 338–353, 1965.
3. J. C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact
well-separated clusters, Journal of Cybernetics, vol. 3, pp. 32–57, 1974.
4. J. C. Bezdek, Pattern recognition with fuzzy objective function algorithms. New York:
Plenum Press, 1981.
5. J. V. De Oliveira, W. Pedrycz, and others, Advances in fuzzy clustering and its
applications. Wiley Online Library, 2007.
6. Y. Zhang, W. Wang, X. Zhang, and Y. Li, A cluster validity index for fuzzy clustering,
Information Sciences, vol. 178, no. 4, pp. 1205–1218, Feb. 2008.
7. J. C. Bezdek, Cluster validity with fuzzy sets, Journal of Cybernetics, vol. 3, no. 3, pp.
58–73, 1973.
8. J. C. Bezdek, Numerical taxonomy with fuzzy sets, Journal of Mathematical Biology,
vol. 1, no. 1, pp. 57–71, 1974.
9. Y. Fukuyama and M. Sugeno, A new method of choosing the number of clusters for the
fuzzy c-means method, in Proc. 5th Fuzzy Syst. Symp, 1989, vol. 247.
10. X. L. Xie and G. Beni, A validity measure for fuzzy clustering, Pattern Analysis and
Machine Intelligence, IEEE Transactions on, vol. 13, no. 8, pp. 841–847, 1991.
11. S. H. Kwon, Cluster validity index for fuzzy clustering, Electronics Letters, vol. 34, no.
22, pp. 2176–2177, 1998.
12. W. Wang and Y. Zhang, On fuzzy cluster validity indices, Fuzzy Sets and Systems, vol.
158, no. 19, pp. 2095–2117, 2007.
13. K. Rizman Žalik, Cluster validity index for estimation of fuzzy clusters of different
sizes and densities, Pattern Recognition, vol. 43, no. 10, pp. 3374–3390, Oct. 2010.
14. E. Trauwaert, On the meaning of Dunn’s partition coefficient for fuzzy clusters, Fuzzy
Sets and Systems, vol. 25, no. 2, pp. 217–242, 1988.
15. R. N. Dave, Validating fuzzy partitions obtained through c-shells clustering, Pattern
Recognition Letters, vol. 17, no. 6, pp. 613–623, 1996.
16. K.-L. Wu and M.-S. Yang, A cluster validity index for fuzzy clustering, Pattern
Recognition Letters, vol. 26, no. 9, pp. 1275–1291, Jul. 2005.
17. D.-W. Kim, K. H. Lee, and D. Lee, On cluster validity index for estimation of the
optimal number of fuzzy clusters, Pattern Recognition, vol. 37, no. 10, pp. 2009–2025,
Oct. 2004.
18. N. R. Pal and S. K. Pal, Entropy: a new definition and its applications, Systems, Man
and Cybernetics, IEEE Transactions on, vol. 21, no. 5, pp. 1260–1270, 1991.
19. N. R. Pal and S. K. Pal, Some properties of the exponential entropy, Information
sciences, vol. 66, no. 1, pp. 119–137, 1992.
20. K. L. Wu and M. S. Yang, Alternative c-means clustering algorithms, Pattern
recognition, vol. 35, no. 10, pp. 2267–2278, 2002.
21. J. C. Bezdek, Pattern Recognition in handbook of Fuzzy computation, IOP Publishing
Ltd., Boston, NY, 1998.
22. M. K. Pakhira, S. Bandyopadhyay, and U. Maulik, Validity index for crisp and fuzzy
clusters, Pattern recognition, vol. 37, no. 3, pp. 487–501, 2004.
23. D. L. Davies and D W. Bouldin, A Cluster Separation Measure. IEEE Transactions on
Pattern Analysis and Machine Intelligence PAMI-1 (2): 224–227, 1979.
24. S. Saha and S. Bandyopadhyay, Some connectivity based cluster validity indices,
Applied Soft Computing, vol. 12, no. 5, pp. 1555–1565, May 2012.
ارائه یک شاخص اعتبار خوشه بندی جدید با... 22
25. Y. Hu, C. Zuo, Y. Yang, and F. Qu, A cluster validity index for fuzzy c-means
clustering, in System Science, Engineering Design and Manufacturing Informatization
(ICSEM), 2011 International Conference on, 2011, vol. 2, pp. 263–266