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FUZZY CLUSTERING OF FEATURE VECTORS WITH SOME ORDINAL VALUED ATTRIBUTES USING GRADIENT DESCENT FOR LEARNING

ROELOF K. BROUWER

Visiting Professor, Department of Mechanical and Mechatronic Engineering, University of Stellenbosch, South Africa

Senior member of IEEE. Presently at School of Advanced Technologies and Mathematics, Thompson Rivers University, 900 McGill Road, PO Box 3010, Kamloops, BC, Canada V2C 5N3.

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https://doi.org/10.1142/S0218488508005133Cited by:0 (Source: Crossref)

Abstract

There are well established methods for fuzzy clustering especially for the cases where the feature values are numerical of ratio or interval scale. Not so well established are methods to be applied when the feature values are ordinal or nominal. In that case there is no one best method it seems. This paper discusses a method where unknown numeric variables are assigned to the ordinal values. Part of minimizing an objective function for the clustering is to find numeric values for these variables. Thus real numbers of interval scale and even ratio scale for that matter are assigned to the original ordinal values. The method uses the same objective function as used in fuzzy c-means clustering but both the membership function and the ordinal to real mapping are determined by gradient descent. Since the ordinal to real mapping is not known it cannot be verified for its legitimacy. However the ordinal to real mapping that is found is best in terms of the clustering produced. Simulations show the method to be quite effective.

Keywords:

References

A. K. Jain and R. C. Dubes , Algorithms for Clustering Data ( Prentice Hall , Upper Saddle River, NJ , 1988 ) . Google Scholar
J. Bezdek , Pattern Recognition with Fuzzy Objective Function Algorithms ( Plenum , 1981 ) . Crossref, Google Scholar
J. C. Dunn, Journal of Cybernetics 3, 32 (1973), DOI: 10.1080/01969727308546046. Crossref, Google Scholar
M. S. Kamel and S. Z. Selim, Pattern Recognition 27, 585 (1994), DOI: 10.1016/0031-3203(94)90118-X. Web of Science, Google Scholar
A. Kandel , Fuzzy Mathematical Techniques with Applications ( Addison Wesley , Reading, MA , 1986 ) . Google Scholar
S. S. Stevens, Science 103(2684), 677 (1946), DOI: 10.1126/science.103.2684.677. Crossref, Web of Science, Google Scholar
M. F. Janowitz and B. Schweizer, Mathematical Social Sciences 18, 135 (1989), DOI: 10.1016/0165-4896(89)90043-7. Crossref, Web of Science, Google Scholar
J. W. T. Lee and D.-Z. Liu , Induction of ordinal decision trees , International Conference on Machine Learning and Cybernetics ( Beijing, China , 2002 ) . Google Scholar
J. W. T. Lee and D.-Z. Liu , Induction of ordinal decision trees , Proceedings of International Conference on Machine Learning and Cybernetics ( 2002 ) . Google Scholar
J. W. Owsinski and S. Zadrozny, Control and Cybernetics 15, 183 (1986). Web of Science, Google Scholar
M. Sato-Ilic, Dynamic clustering model for ordinal similarity, 1998 Conference of the North American Fuzzy Information Processing Society - NAFIPS (Cat. No.98TH8353) (1998) pp. 91–95. Google Scholar
D. Barbara, J. Couto and Y. Li, Information and Knowledge Management 582 (2002). Google Scholar
V. Ganti, J. Gehrke and R. Ramakrishman, CACTUS-Clustering Categorical Data Using Summaries, 5th ACM SGKDD International Conference on Knowledge Discovery and Data Mining (1999) pp. 73–88. Google Scholar
F. Hoppner et al. , Fuzzy Cluster Analysis ( John Wiley and Sons , 1999 ) . Google Scholar
C. Doring, C. Borgelt and R. Kruse, Fuzzy clustering of quantitative and qualitative data, Annual Meeting of the North American Fuzzy Information Processing Society NAFIPS '04 (2004) pp. 84–89. Google Scholar
R. K. Brouwer, Clustering without Use of Prototypes, Thompson Rivers University, Kamloops TR TRU-CS-CIG-2006-01, May 1 2006 . Google Scholar
L. Kaufman and P. J. Rousseeuw , Finding Groups in Data: An Introduction to Cluster Analysis ( John Wiley and Sons , New York , 1990 ) . Crossref, Google Scholar
R. K. Brouwer , Clustering with Mixed Attributes Consisting of Intervalued and Ordinal Valued Attributes , CIRAS ( 2005 ) . Google Scholar
J. Zhang, Selecting typical instances in instance-based learning, Ninth International Machine Learning Conference (1992) pp. 470–479. Google Scholar
W. H. Wolberg and O. L. Mangasarian, National Academy of Sciences 9193 (1990), DOI: 10.1073/pnas.87.23.9193. Google Scholar