Abstract
This chapter tries to answer the fundamental question of what main contributions of fuzzy clustering to the theory of cluster analysis from theoretical viewpoints. While fuzzy clustering is thought to be clearly useful by users of this technique, others think that the concept of fuzziness is not needed in clustering. Thus the usefulness of fuzzy clustering is not trivial. The discussion here is divided into two: one is on fuzzy c-means which is best-known fuzzy method of clustering. However, there is another techniques, discussed by Zadeh, in hierarchical clustering which is equivalent to the old technique of the single linkage. This chapter overviews the both techniques, beginning from basic discussion of fuzzy c-means, and introducing the fundamental concept of fuzzy classifiers and its usefulness. A concept of inductive clustering is introduced which means that a result of clustering can be extended to a partition of the whole space. Moreover hierarchical fuzzy clustering is briefly discussed where the transitive closure gives a simple algebraic form of clusters.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
S. Basu, I. Davidson, K.L. Wagstaff, Constrained Clustering, CRC Press, Boca Raton, 2009.
J.C. Bezdek, Fuzzy Mathematics in Pattern Classification, Ph.D. Thesis, Cornell Univ., Ithaca, NY, 1973.
J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, 1981.
J.C. Bezdek, J. Keller, R. Krishnapuram, N.R. Pal, Fuzzy Models and Algorithms for Pattern Recognition and Image Processing, Kluwer, Boston, 1999.
O. Chapelle, B. Schölkopf, A. Zien, eds., Semi-Supervised Learning, MIT Press, Cambridge, Massachusetts, 2006.
R.N. Davé, R. Krishnapuram, Robust clustering methods: a unified view, IEEE Trans. on Fuzzy Systems, Vol. 5, pp. 270–293, 1997.
D. Dumitrescu, B. Lazzerini, L.C. Jain, Fuzzy Sets and Their Application to Clustering and Training, CRC Press, Boca Raton, Florida, 2000.
R.O. Duda, P.E. Hart, Pattern Classification and Scene Analysis, John Wiley & Sons, 1973.
J.C. Dunn, A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, J. of Cybernetics, Vol. 3, pp. 32–57, 1974.
J.C. Dunn, Well-separated clusters and optimal fuzzy partitions, J. of Cybernetics, Vol. 4, pp. 95–104, 1974.
M. Ester, H.-P. Kriegel, J. Sander, X.W. Xu, A density-based algorithm for discovering clusters in large spatial databases with noise, Proc. of 2nd Intern. Conf. on Knowledge Discovery and Data Mining (KDD-96), AAAI Press, pp. 226–231, 1996.
B.S. Everitt, Cluster Analysis, 3rd Ed., Arnold, London, 1993.
M. Girolami, Mercer kernel based clustering in feature space, IEEE Trans. on Neural Networks, Vol. 13, No. 3, pp. 780–784, 2002.
E.E. Gustafson, W.C. Kessel, Fuzzy clustering with a fuzzy covariance matrix, IEEE CDC, San Diego, California, pp. 761–766, 1979.
R.J. Hathaway, J.C. Bezdek, Switching regression models and fuzzy clustering, IEEE Trans. on Fuzzy Systems, Vol. 1, No. 3, pp. 195–204, 1993.
F. Höppner, F. Klawonn, R. Kruse, T. Runkler, Fuzzy Cluster Analysis, Jhon Wiley & Sons, 1999
H. Ichihashi, K. Honda, N. Tani, Gaussian mixture PDF approximation and fuzzy c-means clustering with entropy regularization, Proc. of Fourth Asian Fuzzy Systems Symposium, Vol. 1, pp. 217–221, 2000.
H. Ichihashi, K. Miyagishi, K. Honda, Fuzzy c-means clustering with regularization by K-L information, Proc. of 10th IEEE International Conference on Fuzzy Systems, Vol. 2, pp. 924–927, 2001.
A.K. Jain, R.C. Dubes, Algorithms for Clustering Data, Prentice Hall, Englewood Cliffs, NJ, 1988.
L. Kaufman, P.J. Rousseeuw, Finding Groups in Data: An Introduction to Cluster Analysis, Wiley, New York, 1990.
T. Kohonen, Self-Organizing Maps, 2nd Ed., Springer, Berlin, 1997.
R. Krishnapuram, J. M. Keller, A possibilistic approach to clustering, IEEE Trans. on Fuzzy Systems, Vol. 1, pp. 98–110, 1993.
R.-P. Li and M. Mukaidono, A maximum entropy approach to fuzzy clustering, Proc. of the 4th IEEE Intern. Conf. on Fuzzy Systems (FUZZ-IEEE/IFES’95), Yokohama, Japan, March 20–24, 1995, pp. 2227–2232, 1995.
J.B. MacQueen, Some methods of classification and analysis of multivariate observations, Proc. of 5th Berkeley Symposium on Math. Stat. and Prob., pp. 281–297, 1967.
G. McLachlan, D. Peel, Finite Mixture Models, Wiley, New York, 2000.
S. Miyamoto, Fuzzy Sets in Information Retrieval and Cluster Analysis, Kluwer, Dordrecht, 1990.
S. Miyamoto, M. Mukaidono, Fuzzy \(c\)-means as a regularization and maximum entropy approach, Proc. of the 7th International Fuzzy Systems Association World Congress (IFSA’97), June 25–30, 1997, Prague, Czech, Vol. II, pp. 86–92, 1997.
S. Miyamoto, Introduction to Cluster Analysis, Morikita-Shuppan, Tokyo, 1999 (in Japanese).
S. Miyamoto, D. Suizu, Fuzzy \(c\)-means clustering using kernel functions in support vector machines, Journal of Advanced Computational Intelligence and Intelligent Informatics, Vol. 7, No. 1, pp. 25–30, 2003.
S. Miyamoto, H. Ichihashi, K. Honda, Algorithms for Fuzzy Clustering, Springer, Berlin, 2008.
S. Miyamoto, Statistical and non-statistical models in clustering: an introduction and recent topics, A. Okada, D. Vicari, G. Ragozini, Eds., Analysis and Modelling of Complex Data in Behavioural and Social Sciences, JCS-CLADAG 12, Anacapri, Italy, Sept. 3–4, 2012, Cleup, Padova, ISBN 978-88-6129-916-0, pp. 3–6 (Web and USB Proc.) 2012.
S. Miyamoto, An Overview of Hierarchical and Non-hierarchical Algorithms of Clustering for Semi-supervised Classification, V. Torra et al. (Eds.): MDAI 2012, LNAI 7647, pp. 1–10, 2012.
S. Miyamoto, Inductive and Non-inductive Methods of Clustering, Proc. of 2012 IEEE International Conference on Granular Computing, Aug. 11–12, Hangzhou, China, pp. 12–17, 2012.
R.A. Redner, H.F. Walker, Mixture densities, maximum likelihood and the EM algorithm, SIAM Review, Vol. 26, No. 2, pp. 195–239, 1984.
K. Rose, E. Gurewitz, and G. Fox, “A deterministic annealing approach to clustering,” Pattern Recognition Letters, Vol. 11, pp. 589–594, 1990.
B. Schölkopf, A.J. Smola, Learning with Kernels, the MIT Press, 2002.
N. Shental, A. Bar-Hillel, T. Hertz, D. Weinshall, Computing Gaussian mixture models with EM using equivalence constraints, In: Advances in Neural Information Processing Systems, Vol. 16, 2004.
V.N. Vapnik, Statistical Learning Theory, Wiley, New York, 1998.
V.N. Vapnik, The Nature of Statistical Learning Theory: 2nd Ed., Springer, New York, 2000.
N. Wang, X. Li, X. Luo, Semi-supervised Kernel-based Fuzzy \(c\)-Means with Pairwise Constraints, Proc. of WCCI 2008, pp. 1099–1103, 2008.
Wishart, D.: Mode analysis: a generalization of nearest neighbour which reduces chaining effects, In: A.J. Cole, ed., Numerical Taxonomy, Proc. Colloq., in Numerical Taxonomy, Univ. of St. Andrews, pp. 283–311, 1968.
L.A. Zadeh, Similarity relations and fuzzy orderings, Information Sciences, Vol. 3, pp. 177–200, 1971.
X. Zhu, A.B. Goldberg, Introduction to Semi-Supervised Learning, Morgan and Claypool, 2009.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Miyamoto, S. (2017). Contributions of Fuzzy Concepts to Data Clustering. In: Torra, V., Dahlbom, A., Narukawa, Y. (eds) Fuzzy Sets, Rough Sets, Multisets and Clustering. Studies in Computational Intelligence, vol 671. Springer, Cham. https://doi.org/10.1007/978-3-319-47557-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-47557-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-47556-1
Online ISBN: 978-3-319-47557-8
eBook Packages: EngineeringEngineering (R0)