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Upper Bounds for Kruskal's Stress

Published online by Cambridge University Press:  01 January 2025

Jan De Leeuw  and
Ineke Stoop
Jan De Leeuw*
Affiliation:
Leiden University
Ineke Stoop
Affiliation:
Leiden University
*
Requests for reprints should be sent to Jan De Leeuw, Department of Datatheory, FSW/RUL, Middelstegracht 4, 2312 TW Leiden, The Netherlands.
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Abstract

In this paper the relationships between the two formulas for stress proposed by Kruskal in 1964 are studied. It is shown that stress formula one has a system of nontrivial upper bounds. It seems likely that minimization of this loss function will be liable to produce solutions for which this upper bound is small. These are regularly shaped configurations. Even though stress formula two yields less equivocal results, it seems to be expected that minimization of this loss function will tend to produce configurations in which the points are clumped. These results give no clue as to which of the two loss functions is to be preferred.

Keywords

nonmetric scalingmultidimensional scalingdistance geometry
Type
Original Paper
Information
Psychometrika , Volume 49 , Issue 3 , September 1984 , pp. 391 - 402
Copyright
Copyright © 1984 The Psychometric Society

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Footnotes

This study has been supported by the Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Netherlands Organization for the Advancement of Pure Research), under grant 56-146.

Comments by Willem Heiser and Frank Critichley have been very helpful.

The second author presently is employed by the Netherlands Central Bureau of Statistics, Voorburg.

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