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A Generalized Majorization Method for Least Squares Multidimensional Scaling of Pseudodistances that may be Negative

Published online by Cambridge University Press:  01 January 2025

Willem J. Heiser
Willem J. Heiser*
Affiliation:
University of Leiden
*
Requests for reprints should be sent to Willem J. Heiser, Department of Data Theory, University of Leiden, PO Box 9555, 2300 RB Leiden, THE NETHERLANDS.
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Abstract

The usual convergence proof of the SMACOF algorithm model for least squares multidimensional scaling critically depends on the assumption of nonnegativity of the quantities to be fitted, called the pseudodistances. When this assumption is violated, erratic convergence behavior is known to occur. Three types of circumstances in which some of the pseudodistances may become negative are outlined: nonmetric multidimensional scaling with normalization on the variance, metric multidimensional scaling including an additive constant, and multidimensional scaling under the city-block distance model. A generalization of the SMACOF method is proposed to resolve the difficulty that is based on the same rationale frequently involved in robust fitting with least absolute residuals.

Keywords

additive constantcity-block scalingmonotone convergence
Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 1 , March 1991 , pp. 7 - 27
Copyright
Copyright © 1991 The Psychometric Society

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Footnotes

I am grateful to Patrick Groenen and Rian van Blokland-Vogelesang for their help with some of the computations, and to the anonymous referees for their very useful comments.

References

Cooper, L. G. (1972). A new solution to the additive constant problem in metric multidimensional scaling. Psychometrika, 37, 311–322.CrossRefGoogle Scholar
Defays, D. (1978). A short note on a method of seriation. British Journal of Mathematical and Statistical Psychology, 31, 49–53.CrossRefGoogle Scholar
de Leeuw, J. (1977). Applications of convex analysis to multidimensional scaling. In Barra, J. R., Brodeau, F., Romier, G., & van Cutsem, B. (Eds.), Recent developments in statistics (pp. 133–145). Amsterdam: North-Holland.Google Scholar
de Leeuw, J. (1984). Differentiability of Kruskal's Stress at a local minimum. Psychometrika, 49, 111–113.CrossRefGoogle Scholar
de Leeuw, J. (1988). Convergence of the majorization method for multidimensional scaling. Journal of Classification, 5, 163–180.CrossRefGoogle Scholar
de Leeuw, J., & Heiser, W. J. (1977). Convergence of correction matrix algorithms for multidimensional scaling. In Lingoes, J. C. (Eds.), Geometric representation of relational data (pp. 735–752). Ann Arbor: Mathesis Press.Google Scholar
de Leeuw, J., & Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In Krishnaiah, P. R. (Eds.), Multivariate analysis, Volume V (pp. 501–522). Amsterdam: North-Holland.Google Scholar
Funk, S. G., Horowitz, A. D., Lipshitz, R., & Young, F. W. (1974). The perceived structure of American ethnic groups: The use of multidimensional scaling in stereotype research. Personality and Social Psychology Bulletin, 1, 66–68.Google Scholar
Guttman, L. (1968). A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 33, 469–506.CrossRefGoogle Scholar
Heiser, W. J. (1981). Unfolding analysis of proximity data. Unpublished doctoral dissertation, University of Leiden.Google Scholar
Heiser, W. J. (1986). A majorization algorithm for the reciprocal location problem, Leiden: University of Leiden, Department of Data Theory.Google Scholar
Heiser, W. J. (1987). Notes on the LARAMP algorithm, Leiden: University of Leiden, Department of Data Theory.Google Scholar
Heiser, W. J. (1987). Correspondence analysis with least absolute residuals. Computational Statistics & Data Analysis, 5, 337–356.CrossRefGoogle Scholar
Heiser, W. J. (1989). The city-block model for three-way multidimensional scaling. In Coppi, R. & Bolasco, S. (Eds.), Multiway data analysis (pp. 395–404). Amsterdam: North-Holland.Google Scholar
Hubert, L. & Arabie, P. (1986). Unidimensional scaling and combinatorial optimization. In de Leeuw, J., Heiser, W. J., Meulman, J. J. & Critchley, F. (Eds.), Multidimensional data analysis (pp. 181–196). Leiden: DSWO Press.Google Scholar
Hubert, L. & Arabie, P. (1988). Relying on necessary conditions for optimization: Unidimensional scaling and some extensions. In Bock, H. H. (Eds.), Classification and related methods of data analysis (pp. 463–472). Amsterdam: North-Holland.Google Scholar
Kruskal, J. B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1–27.CrossRefGoogle Scholar
Kruskal, J. B. (1964). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29, 115–129.CrossRefGoogle Scholar
Kruskal, J. B. (1977). Multidimensional scaling and other methods for discovering structure. In Enslein, K., Ralston, A. & Wilf, H. S. (Eds.), Statistical methods for digital computers, Volume 3 (pp. 296–339). New York: Wiley.Google Scholar
Kruskal, J. B., & Carroll, J. D. (1969). Geometrical models and badness-of-fit functions. In Krishnaiah, P. R. (Eds.), Multivariate analysis, Volume 2 (pp. 639–671). Amsterdam: North-Holland.Google Scholar
Kruskal, J. B., Young, F. W., & Seery, J. B. (1973). How to use KYST-2: A very flexible program to do multidimensional scaling and unfolding, Murray Hill, NJ: Bell Laboratories.Google Scholar
Roskam, E. E. (1972). Multidimensional scaling by metric transformation of data. Nederlands Tijdschrift voor de Psychologie, 27, 486–508.Google Scholar
Stoop, I., & de Leeuw, J. (1982). How to use SMACOF-1B, Leiden: University of Leiden, Department of Data Theory.Google Scholar
Torgerson, W. S. (1958). Theory and methods of scaling, New York: Wiley.Google Scholar
Young, F. W. (1972). A model for polynomial conjoint analysis algorithms. In Shepard, R. N., Romney, A. K. & Nerlove, S. B. (Eds.), Multidimensional scaling, Volume I, Theory (pp. 69–104). New York: Seminar Press.Google Scholar