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Gennclus: New Models for General Nonhierarchical Clustering Analysis

Published online by Cambridge University Press:  01 January 2025

Wayne S. Desarbo
Wayne S. Desarbo*
Affiliation:
Bell Laboratories
*
Requests for reprints should be sent to: Wayne S. DeSarbo, Bell Laboratories, Room 2C-479, 600 Mountain Avenue, Murray Hill, N.J., 07974.
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Abstract

A general class of nonhierarchical clustering models and associated algorithms for fitting them are presented. These (metric) clustering models generalize the Shepard-Arabie Additive Clusters model in allowing for: (1). either overlapping or nonoverlapping clusters; (2). either symmetric (one-way clustering) or nonsymmetric (two-way clustering) proximities (input data); and, (3). either symmetric or diagonal weights. The GENNCLUS algorithms utilize alternating least-squares methods combining ordinary and constrained least-squares, nonlinear constrained mathematical programming, and combinatorial optimization techniques in estimating model parameters. In addition to developing the mathematical bases of these models, a comprehensive set of Monte Carlo simulations of the different models is reported. Two applications concerning brand-switching data and celebrity-brand proximities are discussed. Finally, extensions to three-way models, nonmetric analyses, and other model specifications are provided.

Keywords

cluster analysistwo-way clusteringoverlapping clusteringADCLUS
Type
Original Paper
Information
Psychometrika , Volume 47 , Issue 4 , December 1982 , pp. 449 - 475
Copyright
Copyright © 1982 The Psychometric Society

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Footnotes

Wayne S. DeSarbo is a Member of Technical Staff in the Mathematics and Statistics Research Center at Bell Laboratories in Murray Hill, N.J. I wish to thank R. Gnanadesikan, J. D. Carroll, and P. Arabie for their comments on a previous draft of this paper. I also wish to acknowledge the computer assistance provided by Linda Clark. Finally, I wish to thank the reviewers and editor for their very complete reviews and comments.

References

Reference Notes

Harshman, R. Models for analysis of asymmetrical relationships amongN objects or stimuli, 1978, Canada: University of Western Ontario.Google Scholar
DeSarbo, W. S. & Harshman, R. A. Celebrity-brand congruence analysis, 1981, Murray Hill, N. J.: Bell Laboratories.Google Scholar
Carrol, J. D. Non-parametric multidimensional analysis of paired comparisons data, 1964, Murry Hill, N.J.: Bell Laboratories.CrossRefGoogle Scholar
Rao, V. R., Sabavala, D. J., & Langfeld, P. A. Alternative measures for partitioning analysis based on brand switching data, Working Paper, Cornell University, 1977.Google Scholar
Carroll, J. D. & Arabie, P. A. INDCLUS: a three-way approach to clustering. Presented at theMeeting of the Psychometric Society, Monterey, Cal., 1979.Google Scholar
Pruzansky, S., Tversky, A., & Carroll, J. D. Spatial versus tree Representations of proximity data, 1980, Murry Hill, N.J.: Bell Laboratories.Google Scholar
DeSarbo, W. S., Hausman, R., Lin, S., & Thompson, W. Discrete canonical correlation, 1981, Murry Hill, N.J.: Bell Laboratories.Google Scholar
Bibby, J. Some effects of rounding optimal estimates, 1980, England: The Open University.Google Scholar
Thompson, W. C. Rounding as an approach to constrained multivariate analysis, Working Paper, Bell Laboratories, 1981.Google Scholar

References

Addelman, S. Orthogonal main-effect plans for asymmetrical factorial experiments. Technometrics, 1962, 4, 21–46.CrossRefGoogle Scholar
Arabie, P. A. & Carroll, J. D. MAPCLUS: A mathematical programming approach to fitting the ADCLUS model. Psychometrika, 1980, 45, 211–235.CrossRefGoogle Scholar
Arabie, P. A., Carroll, J. D., DeSarbo, W. S. & Wind, Y. Overlapping clustering: a new method for product positioning. Journal of Marketing Research, 1981, 18, 310–317.CrossRefGoogle Scholar
Bass, F. M., Pessimier, E. A., & Lehman, D. R. An experimental study of relationships between attitudes, brand preference, and choice. Behavioral Science, 1972, 17, 532–541.CrossRefGoogle Scholar
Bass, F. M. Theory of stochastic preference and brand switching. Journal of Marketing Research, 1974, 11, 1–20.CrossRefGoogle Scholar
Cooper, L. & Steinberg, D. Introduction to Methods of Optimization, 1970, Philadelphia, PA: W. B. Saunders Co..Google Scholar
Courant, R. Differential and Integral Calculus, 2nd edition, New York: J. W. Wiley & Sons, 1965.Google Scholar
Ekman, G. A. Dimensions of color vision. Journal of Psychology, 1954, 38, 467–474.CrossRefGoogle Scholar
Ekman, G. A. A direct method for multidimensional ratio scaling. Psychometrika, 1963, 28, 33–41.CrossRefGoogle Scholar
Green, P. E. Analyzing Multivariate Data, 1978, Hinsdale, Ill.: Holt, Rinehart & Winston.Google Scholar
Hartigan, J. A. Clustering Algorithms, 1975, New York: J. W. Wiley & Sons.Google Scholar
Himmelblau, D. M. Applied Nonlinear Programming, 1972, New York: McGraw Hill, Inc..Google Scholar
Johnston, J. Econometric Methods, 2nd edition, New York: McGraw Hill, 1972.Google Scholar
Kruskal, J. B. Multidimensional scaling by optimizing goodness to fit to a nonmetric hypothesis. Psychometrika, 1964, 29, 1–27.CrossRefGoogle Scholar
Kruskal, J. B. Nonmetric multidimensional scaling: a numerical method. Psychometrika, 1964, 29, 115–129.CrossRefGoogle Scholar
Lawson, C. L. & Hanson, R. J. Solving Least Squares Problems, 1974, Englewood Cliffs, N.J.: Prentice-Hall, Inc..Google Scholar
Lin, S. & Kernighan, M. An effective heuristic algorithm for the traveling salesman problem. Operations Research, 1973, 21, 489–516.CrossRefGoogle Scholar
Luenberger, D. G. Introduction to Linear and Nonlinear Programming, 1973, Reading, Mass.: Addison-Wesley Co..Google Scholar
McCormick, W. T., Schweitzer, P. J. & White, T. W. Problem decomposition and data reorganization by a clustering technique. Operations Research, 1972, 20, 993–1009.CrossRefGoogle Scholar
Pendyck, R. S. & Rubinfeld, D. L. Econometric Models and Economic Models, 2nd ed., New York: McGraw Hill, 1981.Google Scholar
Rosen, J. The gradient projection method for nonlinear programming, I. linear constraints. Journal of the Society of Industrial and Applied Mathematics, 1960, 8, 181–217.CrossRefGoogle Scholar
Rosen, J. The gradient projection method for nonlinear programming II. nonlinear constraints. Journal of the Society of Industrial and Applied Mathematics, 1961, 9, 514–532.CrossRefGoogle Scholar
Sarle, W. S. Cluster analysis by least-squares, Proceedings of the 1982 SAS Users Group International, Cary, North Carolina, 1982.Google Scholar
Shepard, R. N. & Arabie, P. Additive clustering: representation of similarities as combination of discrete overlapping properties. Psychological Review, 1979, 86, 87–123.CrossRefGoogle Scholar
Snedecor, G. W. & Cochran, W. G. Statistical Methods, 7th edition, Ames, Iowa: Iowa State University Press, 1981.Google Scholar
Sokal, R. R. & Michener, C. D. A statistical method for evaluating systematic relationships. University of Kansas Scientific Bulletin, 1958, 38, 1409–1438.Google Scholar
Sonquist, J. A. Multivariate Model Building: The Validation of a Search Strategy, 1971, Ann Arbor, Michigan: Institute for Social Research, University of Michigan.Google Scholar
Taha, H. Integer Programming: Theory Applications and Computations, 1975, New York: Academic Press.Google Scholar
Tyron, R. C. & Bailey, D. E. Cluster Analysis, 1970, New York: McGraw Hill.Google Scholar