Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-12T13:18:08.286Z Has data issue: false hasContentIssue false
  • English
  • Français

Least Squares Metric, Unidimensional Unfolding

Published online by Cambridge University Press:  01 January 2025

Keith T. Poole
Keith T. Poole*
Affiliation:
Graduate School of Industrial Administration, Carnegie-Mellon University
*
Requests for reprints should be addressed to Keith T. Poole, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213.
Get accessRights & Permissions [Opens in a new window]

Abstract

The partial derivatives of the squared error loss function for the metric unfolding problem have a unique geometry which can be exploited to produce unfolding methods with very desirable properties. This paper details a simple unidimensional unfolding method which uses the geometry of the partial derivatives to find conditional global minima; i.e., one set of points is held fixed and the global minimum is found for the other set. The two sets are then interchanged. The procedure is very robust. It converges to a minimum very quickly from a random or non-random starting configuration and is particularly useful for the analysis of large data sets with missing entries.

Information

Type
Original Paper
Information
Psychometrika , Volume 49 , Issue 3 , September 1984, pp. 311 - 323
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

Footnotes

This paper benefits from many conversations with and suggestions from Howard Rosenthal.

References

De Leeuw, Jan (1977). Applications of Convex Analysis to Multidimensional Scaling. In Barra, J. R. et al. (Eds.), Recent Developments in Statistics (pp. 133–145). Amsterdam: North Holland Publishing Company.Google Scholar
Eckart, Carl and Gale, Young (1936). The Approximation of One Matrix by Another of Lower Rank. Psychometrika, 1, 211–218.CrossRefGoogle Scholar
Gleason, Terry C. (1967). “A General Model for Nonmetric Multidimensional Scaling.” Michigan Mathematical Psychology Program, 3, Ann Arbor, Michigan.Google Scholar
Guttman, Louis (1968). A General Nonmetric Technique for Finding the Smallest Coordinate Space for a Configuration of Points. Psychometrika, 33, 469–506.CrossRefGoogle Scholar
Heiser, Willem and Jan De Leeuw (1979a). “Metric Multidimensional Unfolding.” Paper presented at the annual meeting of the Psychometric Society. Uppsala, Sweden, 1978.Google Scholar
Heiser, Willem and De Jan, Leeuw (1979). Multidimensional Mapping of Preference Data, The Netherlands: Department of Data Theory, University of Leyden.Google Scholar
Poole, Keith T. (1981). Dimensions of Interest Group Evaluation of the U.S. Senate, 1969–1978. American Journal of Political Science, 25, 49–67.CrossRefGoogle Scholar
Poole, Keith T. and Steven Daniels, R. (1984). “Ideology, Party and Voting in the U.S. Congress.” Working Paper #26-83-84, Graduate School of Industrial Administration, Carnegie-Mellon University.Google Scholar
Ramsay, J. O. (1977). Maximum Likelihood Estimation in Multidimensional Scaling. Psychometrika, 42, 241–266.CrossRefGoogle Scholar
Spense, Ian (1978). Multidimensional Scaling. In Colgan, Patrick W. (Eds.), Quantitative Ethology, New York: Wiley.Google Scholar