Hostname: page-component-7dd5485656-wjfd9 Total loading time: 0 Render date: 2025-10-30T16:59:39.159Z Has data issue: false hasContentIssue false
  • English
  • Français

Differentiability of Kruskal's Stress at a Local Minimum

Published online by Cambridge University Press:  01 January 2025

Jan De Leeuw
Jan De Leeuw*
Affiliation:
Leiden University
*
Requests for reprints should be sent to Jan de Leeuw, Department of Data Theory FSW/RUL, Middelstegracht 4, 2312 TW Leiden, Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that Kruskal's multidimensional scaling loss function is differentiable at a local minimum. Or, to put it differently, that in multidimensional scaling solutions using Kruskal's stress distinct points cannot coincide.

Keywords

multidimensional scalingloss functions

Information

Type
Original Paper
Information
Psychometrika , Volume 49 , Issue 1 , March 1984 , pp. 111 - 113
Copyright
Copyright © 1984 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

References

Reference Notes

De Leeuw, J. (1974). Smoothness properties of nonmetric loss functions. Unpublished paper, Bell Telephone Labs.Google Scholar
De Leeuw, J. (1981). Linear convergence of multidimensional scaling algorithms. Unpublished paper, Department of Data Theory, Leiden University.Google Scholar

References

Carroll, J. D. INDSCAL (1981). In Schiffman, S. S., Reynolds, M. L., & Young, F. W. (Eds.), Introduction to multidimensional scaling, New York: Academic Press.Google 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, Amsterdam: North Holland Publishing Co..Google Scholar
De Leeuw, J. & Heiser, W. (1980). Multidimensional scaling with restrictions on the configuration. In Krishnaiah, P. R. (Eds.), Multivariate analysis, New York: Academic Press.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. (1971). Monotone regression: continuity and differentiability properties. Psychometrika, 36, 57–62.CrossRefGoogle Scholar
Kruskal, J. B. (1977). Multidimensional scaling and other methods for discovering structure. In Enslein, K., Ralston, A. & Wilf, H. S. (Eds.), Mathematical methods for digital computers, New York: Wiley.Google Scholar
Ramsay, J. O. (1978). MULTISCALE: four programs for multidimensional scaling by the method of maximum likelihood, Chicago: National Educational Resources Inc..Google 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, 7–67.CrossRefGoogle Scholar