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Least Squares Metric, Unidimensional Scaling of Multivariate Linear Models

Published online by Cambridge University Press:  01 January 2025

Keith T. Poole
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Abstract

The squared error loss function for the unidimensional metric scaling problem has a special geometry. It is possible to efficiently find the global minimum for every coordinate conditioned on every other coordinate being held fixed. This approach is generalized to the case in which the coordinates are polynomial functions of exogenous variables. The algorithms shown in the paper are linear in the number of parameters. They always descend and, at convergence, every coefficient of every polynomial is at its global minimum conditioned on every other parameter being held fixed. Convergence is very rapid and Monte Carlo tests show the basic procedure almost always converges to the overall global minimum.

Keywords

city block scalingmetric unfoldingconstrained coordinates
Type
Original Paper
Information
Psychometrika , Volume 55 , Issue 1 , March 1990 , pp. 123 - 149
Copyright
Copyright © 1990 The Psychometric Society

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Footnotes

The author thanks Ivo Molenaar, three anonymous referees, and Howard Rosenthal for their many helpful comments.

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