Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-08T09:32:55.069Z Has data issue: false hasContentIssue false
  • English
  • Français

Exploring the Set-Theoretical Structure of Objects by Additive Trees

Published online by Cambridge University Press:  01 January 2025

Math J. J. M. Candel
Math J. J. M. Candel*
Affiliation:
Wageningen Agricultural University
*
Requests for reprints should be sent to Math J. J. M. Candel, Department of Marketing and Marketing Research, Wageningen Agricultural University, Hollandseweg 1, 6706 KN Wageningen, THE NETHERLANDS.
Get access Rights & Permissions [Opens in a new window]

Abstract

Arrangements of feature sets that have been proposed to represent qualitative and quantitative variation among objects are shown to generate identical sets of set-symmetric distances. The set-symmetric distances for these feature arrangements can be represented by path lengths in an additive linear tree. Imperfect versions of these feature arrangements are proposed, which also are indistinguishable by the set-symmetric distance model. The distances for the imperfect versions can be represented by path lengths in an additive imperfectly linear tree. When dissimilarities are defined by the more general contrast model and a constant may be added to proximity data, then for both the perfect and imperfect arrangements an additive tree analysis obtains a perfect fit with an imperfectly linear tree. However, in the case of the contrast model also the distinction between the perfect and imperfect arrangements disappears in that also for the perfect arrangements the resulting tree need no longer be linear.

Keywords

proximitydissimilarityRestle feature systemset-symmetric distancecontrast model(imperfect) nesting(imperfectly) linear arraychainadditive treepath lengthtree distance(imperfectly) linear tree
Type
Original Paper
Information
Psychometrika , Volume 62 , Issue 1 , March 1997 , pp. 119 - 131
Copyright
Copyright © 1997 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

The author is grateful to Mathieu Koppen for detailed comments on an earlier version of this article.

References

Barthélemy, J. P., Guénoche, A. (1991). Trees and proximity representations, Chichester: Wiley.Google Scholar
Buneman, P. (1971). 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 (pp. 387–395). Edinburgh: Edinburgh University Press.Google Scholar
Candel, M. J. J. M. (1992). The additive random feature model for unfolding and the feature structure of choice objects. Mathematical Social Sciences, 23, 311–342.CrossRefGoogle Scholar
Carroll, J. D. (1976). Spatial, nonspatial and hybrid models for scaling. Psychometrika, 41, 439–463.CrossRefGoogle Scholar
Carroll, J. D., 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 Foundation, 8, 1097–1098.Google Scholar
Carroll, J. D., Pruzansky, S. (1980). Discrete and hybrid scaling models. In Lantermann, E. D., Feger, H. (Eds.), Similarity and choice (pp. 108–139). Bern: Hans Huber.Google Scholar
Corter, J. E. (1982). ADDTREE/P: A PASCAL program for fitting additive trees based on Sattath and Tversky's ADDTREE algorithm. Behavior Research Methods and Instrumentation, 14, 353–354.CrossRefGoogle Scholar
Corter, J. E., Tversky, A. (1986). Extended similarity trees. Psychometrika, 51, 429–451.CrossRefGoogle Scholar
Cunningham, J. P. (1978). Free trees and bidirectional trees as representations of psychological distance. Journal of Mathematical Psychology, 17, 165–188.CrossRefGoogle Scholar
De Soete, G. (1983). A least squares algorithm for fitting additive tress to proximity data. Psychometrika, 48, 621–626.CrossRefGoogle Scholar
De Soete, G. (1983). On some properties of dissimilarities satisfying Restle's symmetric set difference model. Psychologica Belgica, 13, 135–139.CrossRefGoogle Scholar
Eisler, H. (1964). A choice model for paired comparison data based on imperfectly nested sets. Psychometrika, 29, 363–370.CrossRefGoogle Scholar
Gati, I., Tversky, A. (1982). Representations of qualitative and quantitative dimensions. Journal of Experimental Psychology: Human Perception and Performance, 8, 325–340.Google ScholarPubMed
Johnson, S. C. (1967). Hierarchical clustering schemes. Psychometrika, 32, 241–254.CrossRefGoogle ScholarPubMed
Pruzansky, S., Tversky, A., Carroll, J. D. (1982). Spatial versus tree representations of proximity data. Psychometrika, 47, 3–24.CrossRefGoogle Scholar
Restle, F. (1959). A metric and an ordering on sets. Psychometrika, 24, 207–220.CrossRefGoogle Scholar
Restle, F. (1961). The psychology of judgement and choice: A theoretical essay, New York: Wiley.Google Scholar
Sattath, S., Tversky, A. (1977). Additive similarity trees. Psychometrika, 42, 319–347.CrossRefGoogle Scholar
Sattath, S., Tversky, A. (1987). On the relation between common and distinctive feature models. Psychological Review, 94, 16–22.CrossRefGoogle Scholar
Schwarz, G., Tversky, A. (1980). On the reciprocity of proximity relations. Journal of Mathematical Psychology, 22, 157–175.CrossRefGoogle Scholar
Tversky, A. (1977). Features of similarity. Psychological Review, 84, 327–352.CrossRefGoogle Scholar
Tversky, A., Gati, I. (1982). Similarity, separability and the triangle inequality. Psychological Review, 89, 123–154.CrossRefGoogle ScholarPubMed
Tversky, A., Hutchinson, J. W. (1986). Nearest neighbor analysis of psychological spaces. Psychological Review, 93, 3–22.CrossRefGoogle Scholar
Tversky, A., Rinott, Y., Newman, C. M. (1983). Nearest neighbor analysis of point processes: Applications to multidimensional scaling. Journal of Mathematical Psychology, 27, 235–250.CrossRefGoogle Scholar