Journal of Classification

, Volume 5, Issue 2, pp 163–180 | Cite as

Convergence of the majorization method for multidimensional scaling

  • Jan de Leeuw
Authors Of Articles

Abstract

In this paper we study the convergence properties of an important class of multidimensional scaling algorithms. We unify and extend earlier qualitative results on convergence, which tell us when the algorithms are convergent. In order to prove global convergence results we use the majorization method. We also derive, for the first time, some quantitative convergence theorems, which give information about the speed of convergence. It turns out that in almost all cases convergence is linear, with a convergence rate close to unity. This has the practical consequence that convergence will usually be very slow, and this makes techniques to speed up convergence very important. It is pointed out that step-size techniques will generally not succeed in producing marked improvements in this respect.

Keywords

Multidimensional scaling Convergence Step size Local minima 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BORG, I. (1981),Anwendungsorientierte Multidimensionale Skalierung, Berlin, West Germany: Springer.Google Scholar
  2. DEFAYS, D. (1978), “A Short Note on a Method of Seriation,”British Journal of Mathematical and Statistical Psychology, 31, 49–53.Google Scholar
  3. DE LEEUW, J. (1977), “Applications of Convex Analysis to Multidimensional Scaling,” inRecent Developments in Statistics, eds. J.R. Barra, F. Brodeau, G. Romier and B. van Cutsem, Amsterdam: North Holland, 133–145.Google Scholar
  4. DE LEEUW, J. (1984), “Differentiability of Kruskal's Stress at a Local Minimum.”Psychometrika, 49, 111–113.Google Scholar
  5. DE LEEUW, J., and HEISER, W. J. (1980), “Multidimensional Scaling with Restrictions on the Configuration,” inMultivariate Analysis, Vol. V, ed. P. R. Krishnaiah, Amsterdam: North Holland.Google Scholar
  6. DE LEEUW, J., and STOOP, I. (1984), “Upper Bounds for Kruskal's Stress,”Psychometrika, 49, 391–402.Google Scholar
  7. GUTTMAN, L. (1968), “A General Nonmetric Technique for Finding the Smallest Coordinate Space for a Configuration of Points,”Psychometrika, 33, 469–506.Google Scholar
  8. HARTMANN, W. (1979),Geometrische Modelle zur Analyse empirischer Daten, Berlin: Akademie Verlag.Google Scholar
  9. HEISER, W. J. (1981),Unfolding Analysis of Proximity Data, Unpublished Doctoral Dissertation, University of Leiden.Google Scholar
  10. HUBERT, L., and ARABIE, P. (1986), “Unidimensional Scaling and Combinatorial Optimization,” inMultidimensional Data Analysis, eds. J. de Leeuw, et al, Leiden: DSWO-Press, 181–196.Google Scholar
  11. KRUSKAL, J. B. (1964a), “Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypotheses,”Psychometrika, 29, 1–28.Google Scholar
  12. KRUSKAL, J. B. (1964b), “Nonmetric Multidimensional Scaling: A Numerical Method,”Psychometrika, 29, 115–129.Google Scholar
  13. KRUSKAL, J. B. (1971), “Monotone Regression: Continuity and Differentiability Properties,”Psychometrika, 36, 57–62.Google Scholar
  14. KRUSKAL, J. B., and WISH, M. (1978),Multidimensional Scaling, Newbury Park, CA: Sage.Google Scholar
  15. LINGOES, J. C., and ROSKAM, E. E. (1973), “A Mathematical and Empirical Comparison of Two Multidimensional Scaling Algorithms,”Psychometrika, 38, Monograph Supplement.Google Scholar
  16. ORTEGA, J. M., and RHEINBOLDT, W. C. (1970),Iterative Solution of Nonlinear Equations in Several Variables, New York: Academic Press.Google Scholar
  17. OSTROWSKI, A. M. (1966),Solution of Equations and Systems of Equations, New York: Academic Press.Google Scholar
  18. STOOP, I., and DE LEEUW, J. (1983),The Stepsize in Multidimensional Scaling Algorithms, Paper presented at the Third European Meeting of the Psychometric Society, Jouy-en-Josas, France, July 5–8, 1983.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Jan de Leeuw
    • 1
  1. 1.Departments of Psychology and MathematicsUniversity of California Los AngelesLos AngelesUSA

Personalised recommendations