Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-07T18:52:45.811Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

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

*

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

Blumenthal, L. M. Theory and applications of distance geometry, 1953, Oxford: Clarendon PressGoogle Scholar
Johnson, S. C. Hierarchical clustering schemes. Psychometrika, 1967, 32, 241–254CrossRefGoogle ScholarPubMed
Lingoes, J. C. Some boundary conditions for a monotone analysis of symmetric matrices. Psychometrika, 1971, 36, 195–203CrossRefGoogle Scholar
Shepard, R. N. Analysis of proximities: Multidimensional scaling with an unknown distance function. I. Psychometrika, 1962, 27, 125–140CrossRefGoogle Scholar
Shepard, R. N. Analysis of proximities: Multidimensional scaling with an unknown distance function. II. Psychometrika, 1962, 27, 219–246CrossRefGoogle Scholar