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Variance-Based Cluster Selection Criteria in a K-Means Framework for One-Mode Dissimilarity Data

Published online by Cambridge University Press:  01 January 2025

J. Fernando Vera and
Rodrigo Macías
J. Fernando Vera*
Affiliation:
University of Granada
Rodrigo Macías
Affiliation:
Centro de Investigación en Matemáticas, Unidad Monterrey
*
Correspondence should be made to J. Fernando Vera, Department of Statistics and O.R., Faculty of Sciences, University of Granada, 18071 Granada, Spain. Email: jfvera@ugr.es
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Abstract

One of the main problems in cluster analysis is that of determining the number of groups in the data. In general, the approach taken depends on the cluster method used. For K-means, some of the most widely employed criteria are formulated in terms of the decomposition of the total point scatter, regarding a two-mode data set of N points in p dimensions, which are optimally arranged into K classes. This paper addresses the formulation of criteria to determine the number of clusters, in the general situation in which the available information for clustering is a one-mode \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N\times N$$\end{document} dissimilarity matrix describing the objects. In this framework, p and the coordinates of points are usually unknown, and the application of criteria originally formulated for two-mode data sets is dependent on their possible reformulation in the one-mode situation. The decomposition of the variability of the clustered objects is proposed in terms of the corresponding block-shaped partition of the dissimilarity matrix. Within-block and between-block dispersion values for the partitioned dissimilarity matrix are derived, and variance-based criteria are subsequently formulated in order to determine the number of groups in the data. A Monte Carlo experiment was carried out to study the performance of the proposed criteria. For simulated clustered points in p dimensions, greater efficiency in recovering the number of clusters is obtained when the criteria are calculated from the related Euclidean distances instead of the known two-mode data set, in general, for unequal-sized clusters and for low dimensionality situations. For simulated dissimilarity data sets, the proposed criteria always outperform the results obtained when these criteria are calculated from their original formulation, using dissimilarities instead of distances.

Keywords

dissimilaritycluster analysisK-meansSYNCLUSvariance-based criterionnumber of clusters

Information

Type
Original Paper
Information
Psychometrika , Volume 82 , Issue 2 , June 2017, pp. 275 - 294
Copyright
Copyright © 2017 The Psychometric Society

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