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Optimal Scaling by Alternating Length-Constrained Nonnegative Least Squares, with Application to Distance-Based Analysis

Published online by Cambridge University Press:  01 January 2025

Patrick J. F. Groenen ,
Bart-Jan van Os  and
Jacqueline J. Meulman
Patrick J. F. Groenen*
Affiliation:
Department of Education, Data Theory Group, Leiden University
Bart-Jan van Os
Affiliation:
Department of Education, Data Theory Group, Leiden University
Jacqueline J. Meulman
Affiliation:
Department of Education, Data Theory Group, Leiden University
*
Requests for reprints should be sent to Patrick J.E Groenen, Department of Education, Data Theory Group, Leiden University, EO. Box 9555, 2300 RB Leiden, THE NETHERLANDS. [email protected]
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Abstract

An important feature of distance-based principal components analysis, is that the variables can be optimally transformed. For monotone spline transformation, a nonnegative least-squares problem with a length constraint has to be solved in each iteration. As an alternative algorithm to Lawson and Hanson (1974), we propose the Alternating Length-Constrained Non-Negative Least-Squares (ALC-NNLS) algorithm, which minimizes the nonnegative least-squares loss function over the parameters under a length constraint, by alternatingly minimizing over one parameter while keeping the others fixed. Several properties of the new algorithm are discussed. A Monte Carlo study is presented which shows that for most cases in distance-based principal components analysis, ALC-NNLS performs as good as the method of Lawson and Hanson or sometimes even better in terms of the quality of the solution.

Keywords

nonnegative least squareslength-constraintsconstrained optimizationalternating least squaresdistance-based (principal components) analysisoptimal scalingmultidimensional scaling
Type
Original Paper
Information
Psychometrika , Volume 65 , Issue 4 , December 2000 , pp. 511 - 524
Copyright
Copyright © 2000 The Psychometric Society

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Footnotes

Supported by The Netherlands Organization for Scientific Research (NWO) by grant nr. 030-56403 for the “PIONEER” project “Subject Oriented Multivariate Analysis” to the third author. We would like to thank the anonymous referees for their valuable remarks that have improved the quality of this paper.

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