Hostname: page-component-89b8bd64d-mmrw7 Total loading time: 0 Render date: 2026-05-10T08:13:52.176Z Has data issue: false hasContentIssue false
  • English
  • Français

An Efficient Alternating Least-Squares Algorithm to Perform Multidimensional Unfolding

Published online by Cambridge University Press:  01 January 2025

Michael J. Greenacre  and
Michael W. Browne
Michael J. Greenacre*
Affiliation:
University of South Africa
Michael W. Browne
Affiliation:
University of South Africa
*
Requests for reprints should be sent to Michael J. Greenacre, Department of Statistics, University of South Africa, PO Box 392, Pretoria 0001, SOUTH AFRICA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the problem of least-squares fitting of squared distances in unfolding. An alternating procedure is proposed which fixes the row or column configuration in turn and finds the global optimum of the objective criterion with respect to the free parameters, iterating in this fashion until convergence is reached. A considerable simplification in the algorithm results, namely that this conditional global optimum is identified by performing a single unidimensional search for each point, irrespective of the dimensionality of the unfolding solution.

Keywords

multidimensional unfoldingalternating least squares

Information

Type
Original Paper
Information
Psychometrika , Volume 51 , Issue 2 , June 1986 , pp. 241 - 250
Copyright
Copyright © 1986 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This work originally formed part of a doctoral thesis (Greenacre, 1978) presented at the University of Paris VI. The authors acknowledge the helpful comments of John Gower during the first author's sabbatical at Rothamsted Experimental Station. The authors are also indebted to Alexander Shapiro, who came up with the proof of the key result which the authors had long suspected, but had not proved, namely that the smallest root of function (13) provides the global minimum of function (7). The constructive comments of the referees of this paper are acknowledged with thanks. This research was supported in part by the South African Council for Scientific and Industrial Research.

References

Coombs, C. H. (1950). Psychological scaling without a unit of measurement. Psychological Review, 57, 148–158.CrossRefGoogle ScholarPubMed
Coombs, C. H., Kao, R. C. (1960). On a connection between factor analysis and multidimensional unfolding. Psychometrika, 25, 219–231.CrossRefGoogle Scholar
Fletcher, R., Powell, M. J. D. (1963). A rapidly convergent descent method for minimization. Computer Journal, 2, 163–168.CrossRefGoogle Scholar
Gold, E. M. (1973). Metric unfolding: Data requirements for unique solution and clarification of Schönemann's algorithm. Psychometrika, 38, 555–569.CrossRefGoogle Scholar
Gower, J. C. (1984). Multivariate analysis: Ordination, multidimensional scaling and allied topics. In Loyd, E. (Eds.), Handbook of Applicable Mathematics-Volume 6 (pp. 727–781). New York: Wiley.Google Scholar
Greenacre, M. J. (1978). Some objective methods of graphical display of a data matrix, Pretoria: University of South Africa, Department of Statistics.Google Scholar
Heiser, W. J. (1981). Unfolding analysis of proximity data, Leiden, The Netherlands: Department of Data Theory, Leiden University.Google Scholar
Muller, M. W. (1983). Multidimensional unfolding of preference data by maximum likelihood, Pretoria: University of South Africa.Google Scholar
Ramsay, J. O. (1975). Solving implicit equations in psychometric data analysis. Psychometrika, 40, 337–360.CrossRefGoogle Scholar
Rao, C. R. (1973). Linear Statistical Inference and its Applications, New York: Wiley.CrossRefGoogle Scholar
Ross, J., Cliff, N. (1964). A generalization of the interpoint distance model. Psychometrika, 29, 167–176.CrossRefGoogle Scholar
Schönemann, P. H. (1970). On metric multidimensional unfolding. Psychometrika, 35, 349–366.CrossRefGoogle Scholar
Schönemann, P. H., Wang, M. M. (1972). An individual difference model for the multidimensional analysis of preference data. Psychometrika, 37, 275–309.CrossRefGoogle Scholar
Takane, Y., Young, F. W., de Leeuw, J. (1977). Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features. Psychometrika, 42, 6–67.CrossRefGoogle Scholar
Wish, M., Deutsch, M., Biener, L. (1972). Differences in perceived similarity of nations. In Romney, A. K., Shepard, R. N., Nerlove, S. B. (Eds.), Multidimensional Scaling Volume II (pp. 289–313). New York: Seminar Press.Google Scholar
Young, F. W., Takane, Y., Lewyckyj, R. (1978). Three notes on ALSCAL. Psychometrika, 43, 433–435.CrossRefGoogle Scholar