×

On fuzzy cluster validity indices. (English) Zbl 1123.62046

Summary: Cluster analysis aims at identifying groups of similar objects, and helps to discover distribution of patterns and interesting correlations in large data sets. Especially, fuzzy clustering has been widely studied and applied in a variety of key areas and fuzzy cluster validation plays a very important role in fuzzy clustering. This paper introduces the fundamental concepts of cluster validity, and presents a review of fuzzy cluster validity indices available in the literature. We conducted extensive comparisons of the mentioned indices in conjunction with the fuzzy \(C\)-means clustering algorithm on a number of widely used data sets, and make a simple analysis of the experimental results.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)
65C60 Computational problems in statistics (MSC2010)
91C20 Clustering in the social and behavioral sciences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderberg, M. R., Cluster Analysis for Application (1973), Academic Press: Academic Press New York · Zbl 0299.62029
[2] Backer, E.; Jain, A. K., A clustering performance measure based on fuzzy set decomposition, IEEE Trans. Patten Anal. Mach. Intell., 3, 1, 66-74 (1981) · Zbl 0454.62057
[3] Bandyopadhyay, S.; Maulik, U., Non-parametric genetic clustering: comparison of validity indices, IEEE Trans. Systems, Man and Cybernet. Part-C, 31, 1, 120-125 (2001)
[4] Barash, Y.; Friedman, N., Context-specific Bayesian clustering for gene expression data, J. Comput. Mol. Cell Biol., 9, 2, 12-21 (2001)
[5] Bensaid, A. M.; Hall, L. O.; Bezdek, J. C.; Clarke, L. P.; Silbiger, M. L., Validity-guided (re) clustering with applications to image segmentation, IEEE Trans. Fuzzy Systems, 4, 112-123 (1996)
[6] Berry, M. J.A.; Linoff, G., Data Mining Techniques for Marketing, Sales and Customer Support (1996), Wiley: Wiley USA
[7] J.C. Bezdek, Fuzzy mathematics in pattern classification, Ph.D. Dissertation, Cornell University, Ithaca, NY, 1973.; J.C. Bezdek, Fuzzy mathematics in pattern classification, Ph.D. Dissertation, Cornell University, Ithaca, NY, 1973.
[8] Bezdek, J. C., Cluster validity with fuzzy sets, J. Cybernet., 3, 58-73 (1974) · Zbl 0294.68035
[9] Bezdek, J. C., Numerical taxonomy with fuzzy sets, J. Math. Biol., 1, 57-71 (1974) · Zbl 0403.62039
[10] Bezdek, J. C., Pattern Recognition with Fuzzy Objective Function Algorithms (1981), Plenum Press: Plenum Press New York · Zbl 0503.68069
[11] Bezdek, J. C., Pattern Recognition in Handbook of Fuzzy Computation (1998), IOP Publishing Ltd.: IOP Publishing Ltd. Boston, NY, (Chapter F6)
[12] Bezdek, J. C.; Hathaway, R. J.; Sabin, M. J., Convergence theory for fuzzy \(c\)-means: counter-examples and repairs, IEEE Trans. Systems, Man Cybernet., SMC17, 873-877 (1987) · Zbl 0653.68091
[13] Bezdek, J. C.; Pal, N. R., Cluster validation with generalized Dunn’s indices, (Kasabov, N.; Coghill, G., Proc. 1995 Second NZ Internat. Two-stream Conf. on ANNES (1995), IEEE Press: IEEE Press Piscataway, NJ)
[14] Bezdek, J. C.; Pal, N. R., Some new indices of cluster validity, IEEE Trans. Systems, Man and Cybernet., 28, 301-315 (1998)
[15] M. Bouguessa, S.R. Wang, A new efficient validity index for fuzzy clustering, in: Proc. Third Internat. Conf. on Machine Learning and Cybernetics, Shanghai, 26-29 August 2004.; M. Bouguessa, S.R. Wang, A new efficient validity index for fuzzy clustering, in: Proc. Third Internat. Conf. on Machine Learning and Cybernetics, Shanghai, 26-29 August 2004.
[16] Chen, M. Y.; Linkens, D. A., Rule-base self-generation and simplification for data-driven fuzzy models, Fuzzy Sets and Systems, 142, 243-265 (2004) · Zbl 1050.93500
[17] S.B. Cho, S.H. Yoo, Fuzzy Bayesian validation for cluster analysis of yeast cell-cycle data, Pattern Recognition (2005).; S.B. Cho, S.H. Yoo, Fuzzy Bayesian validation for cluster analysis of yeast cell-cycle data, Pattern Recognition (2005). · Zbl 1103.68751
[18] A. Chong, T.D. Gedeon, L.T. koczy, A hybrid approach for solving the cluster validity problem, IEEE Internat. Conf. on Digital Signal Processing, Vol. 2, 2002, pp. 1207-1210.; A. Chong, T.D. Gedeon, L.T. koczy, A hybrid approach for solving the cluster validity problem, IEEE Internat. Conf. on Digital Signal Processing, Vol. 2, 2002, pp. 1207-1210.
[19] Dave, R. N., Validating fuzzy partition obtained through \(c\)-shells clustering, Pattern Recognition Lett., 17, 613-623 (1996)
[20] Dave, R. N.; Bhaswan, K., Adaptive fuzzy \(c\)-shells clustering and detection of ellipses, IEEE Trans. Neural Networks, 3, 5, 643-662 (1992)
[21] Devijver, P. A.; Kittler, J., Pattern Recognition: A Statistical Approach (1982), Prentice-Hall: Prentice-Hall London · Zbl 0476.68057
[22] Dunn, J. C., A fuzzy relative of the ISODATA process and its use in detecting compact, well-separated clusters, J. Cybernet., 3, 32-57 (1974) · Zbl 0291.68033
[23] Fisher, R. A., The use of multiple measurements in taxonomic problems, Ann. Eugenics, 3, 179-188 (1936)
[24] Y. Fukuyama, M. Sugeno, A new method of choosing the number of clusters for the fuzzy \(c\); Y. Fukuyama, M. Sugeno, A new method of choosing the number of clusters for the fuzzy \(c\)
[25] Gath, I.; Geva, A. B., Unsupervised optimal fuzzy clustering, IEEE Trans. Pattern Anal. Mach. Intell., 11, 773-781 (1989) · Zbl 0709.62592
[26] M. Halkidi, Y. Batistakis, M. Vazirgiannis, Clustering algorithms and validity measures, in: Scientific and Statistical Database Management—Proc. Internat. Working Conf., 2001, pp. 3-22.; M. Halkidi, Y. Batistakis, M. Vazirgiannis, Clustering algorithms and validity measures, in: Scientific and Statistical Database Management—Proc. Internat. Working Conf., 2001, pp. 3-22. · Zbl 0998.68154
[27] Hartigan, J. A., Clustering Algorithms (1975), Wiley: Wiley New York · Zbl 0321.62069
[28] H. Hassar, A. Bensaid, Validation of fuzzy and crisp \(c\); H. Hassar, A. Bensaid, Validation of fuzzy and crisp \(c\)
[29] Jain, A. K.; Dubes, R. C., Algorithms for Clustering Data (1988), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0665.62061
[30] Kim, D. W.; Lee, K. H.; Lee, D., On cluster validity index for estimation of the optimal number of fuzzy clusters, Pattern Recognition, 37, 2009-2025 (2004)
[31] Kim, Y. I., A cluster validation index for GK cluster analysis based on relative degree of sharing, Inform. Sci., 168, 225-242 (2004) · Zbl 1075.62052
[32] Kothari, R.; Pitts, D., On finding the number of clusters, Pattern Recognition Lett., 20, 405-416 (1999)
[33] Krishnapuram, R.; Nasraoui, O.; Keller, J., The fuzzy \(c\) spherical shells algorithm: a new approach, IEEE Trans. Neural Networks, 3, 5, 663-671 (1992)
[34] Kwon, S. H., Cluster validity index for fuzzy clustering, Electron. Lett., 34, 22, 2176-2177 (1998)
[35] Kwon, S. H., Cluster validity index for fuzzy clustering, Electron. Lett., 34, 22, 2176-2178 (1998)
[36] Man, Y.; Gath, I., Detection and separation of ring-shaped clusters using fuzzy clustering, IEEE Trans. Pattern Anal. Mach. Intell., 16, 8, 855-861 (1994)
[37] Pakhira, M. K.; Bandyopadhyay, S.; Maulik, U., Validity index for crisp and fuzzy clusters, Pattern Recognition, 37, 481-501 (2004) · Zbl 1068.68135
[38] Pakhira, M. K.; Bandyopadhyay, S.; Maulik, U., A study of some fuzzy cluster validity indices, genetic clustering and application to pixel classification, Fuzzy Set and Systems, 155, 2, 191-214 (2005)
[39] Pal, N. R.; Bezdek, J. C., On cluster validity for fuzzy \(c\)-means model, IEEE Trans. Fuzzy Systems, 3, 3, 370-379 (1995)
[40] Pal, N. R.; Bezdek, J. C., Correction to “On Cluster Validity for the Fuzzy \(c\)-Means Model”, IEEE Trans. Fuzzy Systems, 5, 152-153 (1997)
[41] Rezaee, M. R.; Lelieveldt, B. P.F.; Reiber, J. H.C., A new cluster validity index for the fuzzy \(c\)-mean, Pattern Recognition Lett., 19, 237-246 (1998) · Zbl 0905.68127
[42] N.S. Rhee, K.W. Oh, A validity measure for fuzzy clustering and its use in selecting optimal number of clusters, in: IEEE Internat. Conf. on Fuzzy Systems, Vol. 2, 1996, pp. 1020-1025.; N.S. Rhee, K.W. Oh, A validity measure for fuzzy clustering and its use in selecting optimal number of clusters, in: IEEE Internat. Conf. on Fuzzy Systems, Vol. 2, 1996, pp. 1020-1025.
[43] Richardt, J.; Karl, F.; Puller, C. M., Connections between fuzzy theory, simulated annealing, and convex duality, Fuzzy Sets and Systems, 96, 307-334 (1998)
[44] Shen, J.d.; Chang, S. I.; Stanley Lee, E., Determination of cluster number in clustering microarray data, Appl. Math. Comput., 169, 1172-1185 (2005) · Zbl 1074.62043
[45] Silva, H. B.; Brito, P.; Costa, J. P.D., A partitional clustering algorithm validated by a clustering tendency index based on graph theory, Pattern Recognition, 39, 776-788 (2006) · Zbl 1122.68562
[46] Sun, H. J.; Wang, S. G.; Jiang, Q. S., FCM-based model selection algorithms for determining the number of clusters, Pattern Recognition, 37, 2027-2037 (2004) · Zbl 1056.68583
[47] Y.G. Tang, F.C. Sun, Z.Q. Sun, Improved validation index for fuzzy clustering, in: American Control Conf., June 8-10, 2005, Portland, OR, USA.; Y.G. Tang, F.C. Sun, Z.Q. Sun, Improved validation index for fuzzy clustering, in: American Control Conf., June 8-10, 2005, Portland, OR, USA.
[48] Trauwaert, E., On the meaning of Dunn’s partition coefficient for fuzzy clusters, Fuzzy Sets and Systems, 25, 217-242 (1988) · Zbl 0647.62059
[49] Tsekouras, G. E.; Sarimveis, H., A new approach for measuring the validity of the fuzzy \(c\)-means algorithm, Adv. in Eng. Software, 35, 567-575 (2004) · Zbl 1072.68097
[50] Wei, W.; Mendel, J. M., Optimality tests for the fuzzy \(c\)-means algorithm, Pattern Recognition, 27, 11, 1567-1573 (1994) · Zbl 0822.68098
[51] Windham, M. P., Cluster validity for fuzzy clustering algorithms. Fuzzy Sets and Systems, 5, 177-185 (1981) · Zbl 0456.62053
[52] Wu, K. L.; Yang, M. S., A cluster validity index for fuzzy clustering, Pattern Recognition Lett., 26, 1275-1291 (2005)
[53] Xie, X. L.; Beni, G., A validity measure for fuzzy clustering, IEEE Trans. Pattern Anal. Mach. Intell., 13, 841-847 (1991)
[54] Xie, Y.; Raghavan, V. V.; Dhatric, P.; Zhao, X. Q., A new fuzzy clustering algorithm for optimally finding granular prototypes, Approx. Reason., 40, 109-124 (2005)
[55] Yang, M. S.; Wu, K. L., Unsupervised possibilistic clustering, Pattern Recognition, 39, 5-21 (2006)
[56] J. Yu, H.K. Huang, S.F. Tian, An efficient optimality test for the fuzzy \(c\); J. Yu, H.K. Huang, S.F. Tian, An efficient optimality test for the fuzzy \(c\)
[57] Yu, J.; Li, C. X., Novel cluster validity index for FCM algorithm, J. Comput. Sci. Technol., 21, 1, 137-140 (2006)
[58] Zadeh, L. A., Fuzzy sets, Inform. and Control, 8 (1965) · Zbl 0139.24606
[59] Zahid, N.; Limouri, M.; Essaid, A., A new cluster-validity for fuzzy clustering, Pattern Recognition, 32, 1089-1097 (1999)
[60] \( \langle;\) http://www.mcm.edu.cn \(\rangle;\); \( \langle;\) http://www.mcm.edu.cn \(\rangle;\)
[61] \( \langle;\) http://www.ics.uci.edu/\( \sim;\rangle;\); \( \langle;\) http://www.ics.uci.edu/\( \sim;\rangle;\)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.