Hostname: page-component-7f64f4797f-tldsr Total loading time: 0.001 Render date: 2025-11-07T10:46:59.773Z Has data issue: false hasContentIssue false
  • English
  • Français

Spatial Versus Tree Representations of Proximity Data

Published online by Cambridge University Press:  01 January 2025

Sandra Pruzansky ,
Amos Tversky  and
J. Douglas Carroll
Sandra Pruzansky*
Affiliation:
Bell Laboratories
Amos Tversky
Affiliation:
Stanford University
J. Douglas Carroll
Affiliation:
Bell Laboratories
*
Requests for reprints should be sent to Sandra Pruzansky, Bell Laboratories, 2C-552, Murray Hill, New Jersey 07974.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we investigated two of the most common representations of proximities, two-dimensional euclidean planes and additive trees. Our purpose was to develop guidelines for comparing these representations, and to discover properties that could help diagnose which representation is more appropriate for a given set of data. In a simulation study, artificial data generated either by a plane or by a tree were scaled using procedures for fitting either a plane (KYST) or a tree (ADDTREE). As expected, the appropriate model fit the data better than the inappropriate model for all noise levels. Furthermore, the two models were roughly comparable: for all noise levels, KYST accounted for plane data about as well as ADDTREE accounted for tree data. Two properties of the data proved useful in distinguishing between the models: the skewness of the distribution of distances, and the proportion of elongated triangles, which measures departures from the ultrametric inequality, Applications of KYST and ADDTREE to some twenty sets of real data, collected by other investigators, showed that most of these data could be classified clearly as favoring either a tree or a two-dimensional representation.

Keywords

multidimensional scalingclusteringtree structuresadditive trees

Information

Type
Original Paper
Information
Psychometrika , Volume 47 , Issue 1 , March 1982 , pp. 3 - 24
Copyright
Copyright © 1982 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.)

Article purchase

Temporarily unavailable

Footnotes

A portable PASCAL program implementing the Sattath and Tversky [1977] ADDTREE algorithm is available from J. Corter, Department of Psychology, Stanford University, Stanford, California 94305.

References

Reference Notes

Furnas, G. W. The construction of random, terminally labelled, binary trees. Unpublished paper, Bell Laboratories, 1981.Google Scholar
Kruskal, J. B., Young, F. W., & Seery, J. B. How to use KYST, a very flexible program to do multidimensional scaling and unfolding. Unpublished paper, Bell Laboratories, 1973.Google Scholar
Mervis, C. B., Rips, L. J., Rosch, E., Shoben, E. J., & Smith, E. E. Personal communication, 1980.Google Scholar
Kraus, V. Personal communication, 1976.Google Scholar
Hutchinson, W. & Lockhead, G. Personal communication, 1979.Google Scholar
Gati, I. Personal communication, 1980.Google Scholar
Terbeek, D. A cross-language multidimensional scaling study of vowel perception. UCLA Working Papers in Phonetics, 1977, 37.Google Scholar
Bricker, P. D. & Pruzansky, S. A comparison of sorting and pair-wise similarity judgment techniques for scaling auditory stimuli. Unpublished paper, Bell Laboratories, 1970.CrossRefGoogle Scholar

References

Carroll, J. D. Spatial, nonspatial and hybrid models for scaling. Psychometrika, 1976, 41, 439–463.CrossRefGoogle Scholar
Carroll, J. D. & Pruzansky, S. Discrete and hybrid scaling models. In Lantermann, E.D. & Feger, H. (Eds.), Similarity and Choice, 1980, Bern: Hans Huber.Google Scholar
Fillenbaum, S. & Rapoport, A. Structures in the subjective lexicon, 1971, New York: Academic Press.Google Scholar
Gati, I. & Tversky, A. Representations of qualititive and quantitative dimensions. Journal of Experimental Psychology : Human Perception and Performance, 1982, 8, 325–340.Google Scholar
Ghosh, B. Random distances within a rectangle and between two rectangles. Calcutta Mathematical Society Bulletin, 1951, 43, 17–24.Google Scholar
Graef, J., & Spence, I. Using distance information in the design of large multidimensional scaling experiments. Psychological Bulletin, 1979, 86, 60–66.CrossRefGoogle Scholar
Henley, N. M. A psychological study of the semantics of animal terms. Journal of Verbal Learning and Behavior, 1969, 8, 176–184.CrossRefGoogle Scholar
Indow, T. & Uchizono, T. Multidimensional mapping of Munsell colors varying in hue and chroma. Journal of Experimental Psychology, 1960, 59, 321–329.CrossRefGoogle ScholarPubMed
Indow, T. & Kanazawa, K. Multidimensional mapping of Munsell colors varying in hue, chroma and value. Journal of Experimental Psychology, 1960, 59, 330–336.CrossRefGoogle ScholarPubMed
Kuennapas, T., & Janson, A. J. Multidimensional similarity of letters. Perceptual and Motor Skills, 1969, 28, 3–12.CrossRefGoogle ScholarPubMed
Sattath, S., & Tversky, A. Additive Similarity Trees. Psychometrika, 1977, 42, 319–345.CrossRefGoogle Scholar
Schwarz, G., & Tversky, A. On the reciprocity of proximity relations. Journal of Mathematical Psychology, 1980, 22, 157–175.CrossRefGoogle Scholar
Shepard, R. N. Representation of structure in similarity data: Problems and prospects. Psychometrika, 1974, 39, 373–421.CrossRefGoogle Scholar
Tversky, A. Features of Similarity. Psychological Review, 1977, 84, 327–352.CrossRefGoogle Scholar