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The Estimation of Ultrametric and Path Length Trees from Rectangular Proximity Data

Published online by Cambridge University Press:  01 January 2025

Geert De Soete ,
Wayne S. DeSarbo ,
George W. Furnas  and
J. Douglas Carroll
Geert De Soete*
Affiliation:
University of Ghent
Wayne S. DeSarbo*
Affiliation:
AT&T Bell Laboratories
George W. Furnas
Affiliation:
Bell Communications Research
J. Douglas Carroll
Affiliation:
AT&T Bell Laboratories
*
Requests for reprints should be sent either to G. De Soete, Department of Psychology, University of Ghent, Henri Dunantlaan 2, B-9000 Ghent, Belgium; or to W. S. DeSarbo, AT&T Bell Laboratories, Room 2C-256, 600 Mountain Avenue, Murray Hill, N.J. 07974.
Requests for reprints should be sent either to G. De Soete, Department of Psychology, University of Ghent, Henri Dunantlaan 2, B-9000 Ghent, Belgium; or to W. S. DeSarbo, AT&T Bell Laboratories, Room 2C-256, 600 Mountain Avenue, Murray Hill, N.J. 07974.
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Abstract

A least-squares algorithm for fitting ultrametric and path length or additive trees to two-way, two-mode proximity data is presented. The algorithm utilizes a penalty function to enforce the ultrametric inequality generalized for asymmetric, and generally rectangular (rather than square) proximity matrices in estimating an ultrametric tree. This stage is used in an alternating least-squares fashion with closed-form formulas for estimating path length constants for deriving path length trees. The algorithm is evaluated via two Monte Carlo studies. Examples of fitting ultrametric and path length trees are presented.

Keywords

Cluster AnalysisTrees

Information

Type
Original Paper
Information
Psychometrika , Volume 49 , Issue 3 , September 1984 , pp. 289 - 310
Copyright
Copyright © 1984 The Psychometric Society

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Footnotes

G. De Soete is “Aspirant” of the Belgian “Nationaal Fonds voor Wetenschappelijk Onderzoek” at the University of Ghent, Belgium. W. S. DeSarbo and J. D. Carroll are Members of Technical Staff at AT&T Bell Laboratories, Murray Hill, N.J. G. W. Furnas is Member of Technical Staff at Bell Communications Research, Murray Hill, N.J.

References

Addelman, S. (1962). Orthogonal main-effect plans for asymmetrical factorial experiments. Technometrics, 4, 21–46.CrossRefGoogle Scholar
Carroll, J. D. (1976). Spatial, non-spatial and hybrid models for scaling. Psychometrika, 41, 439–463.CrossRefGoogle Scholar
Carroll, J. D. and Chang, J. J. (1973). A method for fitting a class of hierarchical tree structure models to dissimilarities data, and its application to some body parts data of Miller's. Proceedings of the 81st Annual Convention of the American Psychological Association, 8, 1097–1098.Google Scholar
Carroll, J. D., Clark, L. A., and DeSarbo, W. S. (1984). The representation of three-way proximities data by single and multiple tree structure models. Journal of Classification, (in press).CrossRefGoogle Scholar
Carroll, J. D. and Pruzansky, S. (1975). Fitting of hierarchical tree structure (HTS) models, mixtures of HTS models, and hybrid models, via mathematical programming and alternating least squares. Paper presented at the U.S.-Japan Seminar on Multidimensional Scaling, La Jolla, California: University of California at San Diego.Google Scholar
Carroll, J. D. and Pruzansky, S. (1980). Discrete and hybrid scaling models. In Lantermann, E. D., Feger, H. (Eds.), Similarity and Choice, Bern: Hans Huber.Google Scholar
Coombs, C. H. (1964). A theory of data, New York: Wiley.Google Scholar
Courant, R. (1965). Differential and integral calculus 2nd edition,, New York: Wiley.Google Scholar
Cunningham, J. P. (1974). Finding the optimal tree realization of a proximity matrix. Paper presented at the Mathematical Psychology Meetings, Ann Arbor: Michigan.Google Scholar
Cunningham, J. P. (1978). Free trees and bidirectional trees as a representations of psychological distance. Journal of Mathematical Psychology, 17, 165–188.CrossRefGoogle Scholar
DeSarbo, W. S. (1982). GENNCLUS: New models for general nonhierarchical clustering analysis. Psychometrika, 47, 449–475.CrossRefGoogle Scholar
DeSoete, G. (1983). A least squares algorithm for fitting additive trees to proximity data. Psychometrika, 48, 621–626.CrossRefGoogle Scholar
Dobson, A. G. (1974). Unrooted trees for numerical taxonomy. Journal of Applied Probability, 11, 32–42.CrossRefGoogle Scholar
Farris, J. S. (1972). Estimating phylogenetic trees from distance matrices. American Naturalist, 106, 645–668.CrossRefGoogle Scholar
Furnas, G. W. (1980). Objects and their features: The metric representation of two class data. Unpublished Doctoral Dissertation, Stanford University.Google Scholar
Furnas, G. W. (1984). The construction of random, terminally labeled, binary trees. Journal of Classification, (in press).Google Scholar
Green, P. E., Tull, D. S. (1978). Research for marketing decisions 4th ed.,, Englewood, Cliffs, N.J.: Prentice-Hall.Google Scholar
Harshman, R. (1978). Models for analysis of asymmetrical relationships amongN objects or stimuli, Canada: University of Western Ontario.Google Scholar
Hartigan, J. A. (1975). Clustering algorithms, New York: Wiley.Google Scholar
Hartigan, J. A. (1976). Model blocks in definition of west coast mammals. Systematic Zoology, 25, 149–160.CrossRefGoogle Scholar
Hartigan, J. A. (1967). Representation of similarity matrices by trees. Journal of the American Statistical Association, 62, 1140–1158.CrossRefGoogle Scholar
Johnson, S. C. (1967). Hierarchical clustering schemes. Psychometrika, 32, 241–254.CrossRefGoogle ScholarPubMed
McCormick, W. T., Schweitzer, P. J., and White, T. W. (1972). Problem decomposition and data reorganization by a clustering technique. Operations Research, 20, 993–1009.CrossRefGoogle Scholar
Miller, G. A., and Nicely, P. E. (1955). An analysis of perceptual confusions among some English consonants. Journal of the Acoustical Society of America, 27, 338–352.CrossRefGoogle Scholar
Powell, M. J. D. (1977). Restart procedures for the conjugate gradient method. Mathematical Programming, 12, 241–254.CrossRefGoogle Scholar
Pruzansky, S., Tversky, A., and Carroll, J. D. (1982). Spatial versus tree representations of proximity data. Psychometrika, 47, 3–24.CrossRefGoogle Scholar
Rao, S. S. (1979). Optimization theory and applications, New York: Wiley.Google Scholar
Sattath, S. and Tversky, A. (1977). Additive similarity trees. Psychometrika, 42, 319–345.CrossRefGoogle Scholar
Shepard, R. N. (1972). Psychological representation of speech sounds. In David, E. E. and Denes, P. B. (Eds.), Human communication: A unified view, New York: McGraw Hill.Google Scholar
Shepard, R. N. and Arabie, P. (1979). Additive clustering: Representation of similarities as combination of discrete overlapping properties. Psychological Review, 86, 87–123.CrossRefGoogle Scholar
Snedecor, G. W. and Cochran, W. G. (1981). Statistical methods 7th edition,, Ames, Iowa: Iowa State University Press.Google Scholar
Sonquist, J. A. (1971). Multivariate model building: the validation of a search strategy, Ann Arbor, Michigan: Institute for Social Research, University of Michigan.Google Scholar
Tryon, R. C. and Bailey, D. E. (1970). Cluster analysis, New York: McGraw Hill.Google Scholar
Wold, H. (1966). Estimation of principal components and related models by iterative least squares. In Krishnaiah, P. R. (Eds.), Multivariate analysis, New York: Academic Press.Google Scholar