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A General Nonmetric Technique for Finding the Smallest Coordinate Space for a Configuration of Points

Published online by Cambridge University Press:  01 January 2025

Louis Guttman
Louis Guttman*
Affiliation:
The Hebrew University and The Israel Institute of Applied Social Research
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Abstract

Let A1, A2, ..., An be any n objects, such as variables, categories, people, social groups, ideas, physical objects, or any other. The empirical data to be analyzed are coefficients of similarity or distance within pairs (Ai, Ai), such as correlation coefficients, conditional probabilities or likelihoods, psychological choice or confusion, etc. It is desired to represent these data parsimoniously in a coordinate space, by calculating m coordinates {xia} for each Ai for a semi-metric d of preassigned form dij = d(|xi1 -xj1|, |xi2 -xj2|, ..., |xim - xjm|). The dimensionality m is sought to be as small as possible, yet satisfy the monotonicity condition that dij <dkl whenever the observed data indicate that Ai is “closer” to Aj than Ak is to Al. Minkowski and Euclidean spaces are special metric examples of d. A general coefficient of monotonicity μ is defined, whose maximization is equivalent to optimal satisfaction of the monotonicity condition, and which allows various options both for treatment of ties and for weighting error-of-fit. A general rationale for algorithm construction is derived for maximizing μ by gradient-guided iterations; this provides a unified mathematical solution to the basic operational problems of norming the gradient to assure proper convergence, of trading between speed and robustness against undesired stationary values, and of a rational first approximation. Distinction is made between single-phase (quadratic) and two-phase (bilinear) strategies for algorithm construction, and between “hard-squeeze” and “soft-squeeze” tactics within these strategies. Special reference is made to the rank-image and related transformational principles, as executed by current Guttman-Lingoes families of computer programs.

Keywords

Public PolicyConditional ProbabilityEuclidean SpaceEmpirical DataSocial Group
Type
Original Paper
Information
Psychometrika , Volume 33 , Issue 4 , December 1968 , pp. 469 - 506
Copyright
Copyright © 1968 Psychometric Society

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Footnotes

*

This research was supported in part by the National Science Foundation of the United States Government, through Grant No. GS 929 to the University of Michigan.

I am deeply indebted for many helpful comments on earlier drafts of this paper received from Joseph Kruskal, James Lingoes, and the managing editor.

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