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A Model and Algorithm for Multidimensional Scaling with External Constraints on the Distances

Published online by Cambridge University Press:  01 January 2025

Ingwer Borg  and
James C. Lingoes
Ingwer Borg
Affiliation:
Rheinisch-Westfälische Technische Hochschule
James C. Lingoes*
Affiliation:
The University of Michigan
*
Requests for reprints should be sent to James C. Lingoes, The University of Michigan, 1005 North University Building, Ann Arbor, Michigan 48109.
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Abstract

A method for externally constraining certain distances in multidimensional scaling configurations is introduced and illustrated. The approach defines an objective function which is a linear composite of the loss function of the point configuration X relative to the proximity data P and the loss of X relative to a pseudo-data matrix R. The matrix R is set up such that the side constraints to be imposed on X’s distances are expressed by the relations among R’s numerical elements. One then uses a double-phase procedure with relative penalties on the loss components to generate a constrained solution X. Various possibilities for constructing actual MDS algorithms are conceivable: the major classes are defined by the specification of metric or nonmetric loss for data and/or constraints, and by the various possibilities for partitioning the matrices P and R. Further generalizations are introduced by substituting R by a set of R matrices, Ri, i = 1, …r, which opens the way for formulating overlapping constraints as, e.g., in patterns that are both row- and column-conditional at the same time.

Keywords

constrained multidimensional scalinghypothesis testinggeometric modelsnonlinear optimization
Type
Original Paper
Information
Psychometrika , Volume 45 , Issue 1 , March 1980 , pp. 25 - 38
Copyright
Copyright © 1980 The Psychometric Society

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References

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