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The Relation between Hierarchical and Euclidean Models for Psychological Distances

Published online by Cambridge University Press:  01 January 2025

Eric W. Holman
Eric W. Holman*
Affiliation:
University of California, Los Angeles
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Abstract

In one well-known model for psychological distances, objects such as stimuli are placed in a hierarchy of clusters like a phylogenetic tree; in another common model, objects are represented as points in a multidimensional Euclidean space. These models are shown theoretically to be mutually exclusive and exhaustive in the following sense. The distances among a set of n objects will be strictly monotonically related either to the distances in a hierarchical clustering system, or else to the distances in a Euclidean space of less than n − 1 dimensions, but not to both. Consequently, a lower bound on the number of Euclidean dimensions necessary to represent a set of objects is one less than the size of the largest subset of objects whose distances satisfy the ultrametric inequality, which characterizes the hierarchical model.

Type
Original Paper
Information
Psychometrika , Volume 37 , Issue 4 , December 1972 , pp. 417 - 423
Copyright
Copyright © 1972 The Psychometric Society

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Footnotes

*

This work was supported in part by Grant GB-13588X from the National Science Foundation. I would like to thank L. M. Kelly and A. A. J. Marley for their helpful comments and suggestions.

References

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