Parameter modification for clustering criteria. (English) Zbl 0662.62066

The more ways there are of understanding a clustering technique, the more effectively the results can be analyzed and used. I will give a general procedure, called parameter modification, to obtain from a clustering criterion a variety of equivalent forms of the criterion. These alternative forms reveal aspects of the technique that are not necessarily apparent in the original formulation. This procedure is successful in improving the understanding of a significant number of clustering techniques.
The insight obtained will be illustrated by applying parameter modification to partitioning, mixture and fuzzy clustering methods, resulting in a unified approach to the study of these methods and a general algorithm for optimizing them.


62H30 Classification and discrimination; cluster analysis (statistical aspects)
65C99 Probabilistic methods, stochastic differential equations
Full Text: DOI


[1] BEZDEK, J. C. (1981),Pattern Recognition with Fuzzy Objective Function Algorithms, New York: Plenum. · Zbl 0503.68069
[2] BEZDEK, J. C., HATHAWAY, R. J., HOWARD, R. E., WILSON, C. A., and WINDHAM, M. P. (1986), ”Local Convergence Analysis of a Grouped Variable Version of Coordinate Descent,”Journal of Optimization and Applications, (in press). · Zbl 0597.90076
[3] BOCK, H. H. (1974),Automatische Klassifikation, Göttingen: Vandenhoeck & Ruprecht.
[4] BOCK, H. H. (1979), ”Clusteranalyse mit unscharfen Partitionen,” inKlassifikation und Erkenntnis III: Numerische Klassifikation, ed. H.H. Bock, Frankfurt: Gesellschraft für Klassifikation, e.V., 137–163.
[5] BRYANT, P., and WILLIAMSON, J. A. (1978), ”Asymptotic Behaviour of Classification Maximum Likelihood Estimates,”Biometrika, 65, 273–281. · Zbl 0393.62011
[6] CARROLL, J. D., and ARABIE, P. (1980), ”Multidimensional Scaling,”Annual Review of Psychology, 31, 607–649.
[7] DIDAY, E., and SIMON, J. C. (1976), ”Clustering Analysis,” inDigital Pattern Recognition, ed. K.S. Fu, Berlin: Springer-Verlag, 47–94. · Zbl 0331.62043
[8] FRIEDMAN, H. P., and RUBIN, J. (1967), ”On Some Invariant Criteria for Grouping Data,”Journal of the American Statistical Association, 62, 1152–1178.
[9] LUENBERGER, D. G. (1984),Linear and Nonlinear Programming (2nd ed.), Reading, MA: Addison-Wesley. · Zbl 0571.90051
[10] MARRIOTT, F. H. C. (1975), ”Separating Mixtures of Normal Distributions,”Biometrics, 31, 767–769. · Zbl 0308.62050
[11] MARRIOTT, F. H. C. (1982), ”Optimization Methods of Cluster Analysis,”Biometrika, 69, 417–421.
[12] MCLACHLAN, G. J. (1982), ”The Classification and Mixture Maximum Likelihood Approaches to Cluster Analysis,” inHandbook of Statistics, Vol. 2, eds. P.R. Krishnaiah and L.N. Kanal, Amsterdam: North Holland, 199–208. · Zbl 0513.62064
[13] REDNER, R. A., and WALKER, H. F. (1984), ”Mixture Densities, Maximum Likelihood and the EM Algorithm,”SIAM Review, 26, 195–239. · Zbl 0536.62021
[14] SCOTT, A. J., and SYMONS, M. J. (1971), ”Clustering Methods Based on Likelihood Ratio Criteria,”Biometrics, 27, 387–397.
[15] SELIM, S. Z., and ISMAIL, M. A. (1984), ”k-Means-type Algorithms: A Generalized Convergence Theorem and Characterization of Local Optimality,”IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6, 81–87. · Zbl 0546.62037
[16] SPÄTH, H. (1985),Cluster Dissection and Analysis, Chichester: Ellis Horwood. · Zbl 0584.62094
[17] SYMONS, M. J. (1981), ”Clustering Criteria and Multivariate Normal Mixtures,”Biometrics, 37, 35–43. · Zbl 0473.62048
[18] TUCKER, L. R. (1964), ”The Extension of Factor Analysis to Three-dimensional Matrices,” inContributions to Mathematical Psychology, eds. N. Frederiksen and H. Gulliksen, New York: Holt, Rinehart & Winston, 109–127.
[19] WINDHAM, M. P. (1985), ”Numerical Classification of Proximity Data with Assignment Measures,”Journal of Classification, 2, 157–172.
[20] WINDHAM, M. P. (1986), ”A Unification of Optimization-based Numerical Classification Algorithms,” inClassification as a Tool of Research, eds. W. Gaul and M. Schader, Amsterdam: North Holland, 447–452. · Zbl 0587.62126
[21] WOLFE, J. H. (1970), ”Pattern Clustering by Multivariate Mixture Analysis,”Multivariate Behavioral Research, 5, 329–350.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.