Hostname: page-component-8678677fb7-pvr6d Total loading time: 0 Render date: 2026-10-06T16:08:14.140Z Has data issue: false hasContentIssue false
  • English
  • Français

Additive Similarity Trees

Published online by Cambridge University Press:  01 January 2025

Shmuel Sattath  and
Amos Tversky
Shmuel Sattath
Affiliation:
Hebrew University of Jerusalem
Amos Tversky*
Affiliation:
Hebrew University of Jerusalem
*
Requests for reprints should be sent to Prof. Amos Tversky, Dept. of Psychology, The Hebrew University, Jerusalem, Israel.
Get access Rights & Permissions [Opens in a new window]

Abstract

Similarity data can be represented by additive trees. In this model, objects are represented by the external nodes of a tree, and the dissimilarity between objects is the length of the path joining them. The additive tree is less restrictive than the ultrametric tree, commonly known as the hierarchical clustering scheme. The two representations are characterized and compared. A computer program, ADDTREE, for the construction of additive trees is described and applied to several sets of data. A comparison of these results to the results of multidimensional scaling illustrates some empirical and theoretical advantages of tree representations over spatial representations of proximity data.

Keywords

proximityclusteringmultidimensional scaling

Information

Type
Original Paper
Information
Psychometrika , Volume 42 , Issue 3 , September 1977 , pp. 319 - 345
Copyright
Copyright © 1977 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We thank Nancy Henley and Vered Kraus for providing us with data, and Jan deLeeuw for calling our attention to relevant literature. The work of the first author was supported in part by the Psychology Unit of the Israel Defense Forces.

References

Reference Notes

Holman, E. W. A test of the hierarchical clustering model for dissimilarity data. Unpublished manuscript, University of California at Los Angeles, 1975.Google Scholar
Cunningham, J. P. Finding an optimal tree realization of a proximity matrix. Paper presented at the mathematical psychology meeting, Ann Arbor, August, 1974.Google Scholar
Cunningham, J. P. Discrete representations of psychological distance and their applications in visual memory. Unpublished doctoral dissertation, University of California at San Diego, 1976.Google Scholar
Carroll, J. D. & Pruzansky, S. Fitting of hierarchical tree structure (HTS) models, mixture of HTS models, and hybrid models, via mathematical programming and alternating least squares. paper presented at the US-Japan Seminar on Theory, Methods, and Applications of Multidimensional Scaling and related techniques, San Diego, August, 1975.Google Scholar
Kraus, personal communication, 1976.Google Scholar

References

Buneman, P. The recovery of trees from measures of dissimilarity. In Hodson, F. R. Kendall, D. G. & Tautu, P.(Eds.), Mathematics in the Archaeological and Historical Sciences, 1971, Edinburgh: Edinburgh University Press.Google Scholar
Buneman, P. A note on the metric properties of trees. Journal of Combinatorial Theory, 1974, 17, 48–50.CrossRefGoogle Scholar
Carroll, J. D. Spatial, non-spatial and hybrid models for scaling. Psychometrika, 1976, 41, 439–463.CrossRefGoogle Scholar
Carroll, J. D. & Chang, J. J. 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, 81st Annual Convention, American Psychological Association, 1973, 8, 1097–1098.Google Scholar
Dobson, J. Unrooted trees for numerical taxonomy. Journal of Applied Probability, 1974, 11, 32–42.CrossRefGoogle Scholar
Fillenbaum, S. & Rapoport, A. Structures in the subjective lexicon, 1971, New York: Academic Press.Google Scholar
Guttman, L. A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 1968, 33, 469–506.CrossRefGoogle Scholar
Hakimi, S. L. & Yau, S. S. Distance matrix of a graph and its realizability. Quarterly of Applied Mathematics, 1964, 22, 305–317.CrossRefGoogle Scholar
Henley, N. M. A psychological study of the semantics of animal terms. Journal of Verbal Learning and Verbal Behavior, 1969, 8, 176–184.CrossRefGoogle Scholar
Holman, E. W. The relation between hierarchical and Euclidean models for psychological distances. Psychometrika, 1972, 37, 417–423.CrossRefGoogle Scholar
Jardine, N. Sibson, R. Mathematical taxonomy, 1971, New York: Wiley.Google Scholar
Johnson, S. C. Hierarchical clustering schemes. Psychometrika, 1967, 32, 241–254.CrossRefGoogle ScholarPubMed
Kendall, M. G., & Moran, M. A. Geometrical Probability, 1963, New York: Hafner Publishing Company.Google Scholar
Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 1964, 29, 1–27.CrossRefGoogle Scholar
Kuennapas, T., & Janson, A. J. Multidimensional similarity of letters. Perceptual and Motor Skills, 1969, 28, 3–12.CrossRefGoogle ScholarPubMed
Lingoes, J. C. An IBM 360/67 program for Guttman-Lingoes smallest space analysis-PI. Behavioral Science, 1970, 15, 536–540.Google Scholar
Patrinos, A. N., & Hakimi, S. L. The distance matrix of a graph and its tree realization. Quarterly of Applied Mathematics, 1972, 30, 255–269.CrossRefGoogle Scholar
Shepard, R. N. Representation of structure in similarity data: Problems and prospects. Psychometrika, 1974, 39, 373–421.CrossRefGoogle Scholar
Sneath, P. H. A., & Sokal, R. R. Numerical taxonomy: the principles and practice of numerical classification, 1973, San Francisco: W. H. Freeman.Google Scholar
Turner, J., & Kautz, W. H. A survey of progress in graph theory in the Soviet Union. Siam Review, 1970, 12 Supplement1–68.CrossRefGoogle Scholar
Tversky, A. Features of similarity. Psychological Review, 1977, 84, 327–352.CrossRefGoogle Scholar