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MACPLUS: A Mathematical Programming Approach to Fitting the ADCLUS Model

Published online by Cambridge University Press:  01 January 2025

Phipps Arabie  and
J. Douglas Carroll
Phipps Arabie*
Affiliation:
University of Minnesota and Bell Laboratories
J. Douglas Carroll*
Affiliation:
Bell Laboratories
*
Requests for reprints should be sent to either Phipps Arabic, Department of Psychology, Elliott Hall, University of Minnesota, Minneapolis, Minnesota 55455, or to J. Douglas Carroll, Room 2C-553, Bell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974.
Requests for reprints should be sent to either Phipps Arabic, Department of Psychology, Elliott Hall, University of Minnesota, Minneapolis, Minnesota 55455, or to J. Douglas Carroll, Room 2C-553, Bell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974.
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Abstract

We present a new algorithm, MAPCLUS (MAthematical Programming CLUStering), for fitting the Shepard-Arabie ADCLUS (for ADditive CLUStering) model. MAPCLUS utilizes an alternating least squares method combined with a mathematical programming optimization procedure based on a penalty function approach, to impose discrete (0,1) constraints on parameters defining cluster membership. This procedure is supplemented by several other numerical techniques (notably a heuristically based combinatorial optimization procedure) to provide an efficient general-purpose computer implemented algorithm for obtaining ADCLUS representations. MAPCLUS is illustrated with an application to one of the examples given by Shepard and Arabie using the older ADCLUS procedure. The MAPCLUS solution uses half as many clusters to achieve nearly the same level of goodness-of-fit. Finally, we consider an extension of the present approach to fitting a three-way generalization of the ADCLUS model, called INDCLUS (INdividual Differences CLUStering).

Keywords

additive clusteringnonhierarchical clusteringalternating least squares
Type
Original Paper
Information
Psychometrika , Volume 45 , Issue 2 , June 1980 , pp. 211 - 235
Copyright
Copyright © 1980 The Psychometric Society

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Footnotes

We are indebted to Scott A. Boorman, W. K. Estes, J. A. Hartigan, Lawrence J. Hubert, Carol L. Krumhansl, Joseph B. Kruskal, Sandra Pruzansky, Roger N. Shepard, Edward J. Shoben, Sigfrid D. Soli, and Amos Tversky for helpful discussions of this work, as well as the anonymous referees for their suggestions and corrections on an earlier version of this paper. We are also grateful to Pamela Baker and Dan C. Knutson for technical assistance. The research reported here was supported in part by LEAA Grant 78-NI-AX-0142 and NSF Grants SOC76-24512 and SOC76-24394.

References

References Notes

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