Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-07T15:57:18.602Z Has data issue: false hasContentIssue false
  • English
  • Français

A New Computational Method to Fit the Weighted Euclidean Distance Model

Published online by Cambridge University Press:  01 January 2025

Jan de Leeuw  and
Sandra Pruzansky
Jan de Leeuw
Affiliation:
University of Leiden
Sandra Pruzansky*
Affiliation:
Bell Laboratories
*
Requests for reprints should be sent to Sandra Pruzansky, Bell Laboratories, 2C-573, 600 Mountain Avenue, Murray Hill, New Jersey 07974.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper describes a computational method for weighted euclidean distance scaling which combines aspects of an “analytic” solution with an approach using loss functions. We justify this new method by giving a simplified treatment of the algebraic properties of a transformed version of the weighted distance model. The new algorithm is much faster than INDSCAL yet less arbitrary than other “analytic” procedures. The procedure, which we call SUMSCAL (subjective metric scaling), gives essentially the same solutions as INDSCAL for two moderate-size data sets tested.

Keywords

multidimensional scaling
Type
Original Paper
Information
Psychometrika , Volume 43 , Issue 4 , December 1978 , pp. 479 - 490
Copyright
Copyright © 1978 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.)

Footnotes

Comments by J. Douglas Carroll and J. B. Kruskal have been very helpful in preparing this paper.

References

Reference Notes

Bloxom, B. Individual differences in multidimensional scaling, 1968, Princeton, New Jersey: Educational Testing Service.CrossRefGoogle Scholar
Carroll, J. D., & Chang, J. J. IDIOSCAL: A generalization of INDSCAL allowing IDIOsyncratic reference systems as well as an analytic approximation to INDSCAL. Paper presented at the Spring Meeting of the Psychometric Society, March 30–31, 1972, Princeton, New Jersey.Google Scholar
Cohen, H. S. Three-mode rotation to approximate INDSCAL structure. Paper presented at the annual meeting of the Psychometric Society, Palo Alto, California, March, 1974.Google Scholar
de Leeuw, J. An initial estimate for INDSCAL. Unpublished paper, Bell Telephone Laboratories, Murray Hill, June, 1974.Google Scholar
Harshman, R. A. Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multi-model factor analysis. UCLA Working Papers in Phonetics, 1970, 16, 1–84.Google Scholar
Harshman, R. A. PARAFAC2; mathematical and technical notes. UCLA Working Papers in Phonetics, 1972, 22, 30–44.Google Scholar
Heiser, W. J. Individual differences scaling, 1975, Leiden, Holland: Psychology Department, University of Leiden.Google Scholar
Pruzansky, S. SINDSCAL: A Computer Program for Individual Differences in Multidimensional Scaling, Unpublished manuscript, Bell Laboratories, September 1975.Google Scholar

References

Bhimasankaram, P. Simultaneous reduction of several Hermitian forms. Sankhya, 1971, 33, 417–422.Google Scholar
Carroll, J. D., & Chang, J. J. Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition. Psychometrika, 1970, 35, 283–320.CrossRefGoogle Scholar
de Leeuw, J., & Heiser, W. J. Convergence of correction matrix algorithms for multidimensional scaling. In Lingoes, J. (Eds.), Geometric representations of relational data, 1977, Ann Arbor: Mathesis Press.Google Scholar
Greenstadt, J. The determination of the characteristic roots of a matrix by the Jacobi method. In Ralston, A. & Wilf, H. (Eds.), Mathematical Methods for Digital Computers, (Vol. I). New York: Wiley. 1960, 84–91.Google Scholar
Helm, C. E. A multidimensional ratio scaling analysis of perceived color relations. Journal of the Optical Society of America, 1964, 54, 256–262.CrossRefGoogle ScholarPubMed
Horan, C. B. Multidimensional scaling: Combining observations when individuals have different perceptual structures. Psychometrika, 1969, 34, 139–165.Google Scholar
Jacobi, C. G. J. Über ein leichtes Verfahren, die in der Theorie der Säkularstörungen vorkommenden Gleichungen numerisch aufzulösen. J. Reine Agnew. Math, 1846, 30, 51–95.Google Scholar
Jöreskog, K. A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 1969, 34, 183–202.CrossRefGoogle Scholar
Kruskal, J. B. More factors than subjects, tests and treatments: An indeterminacy theorem for canonical decomposition and individual differences scaling. Psychometrika, 1976, 41, 281–293.CrossRefGoogle Scholar
Kruskal, J. B. & Carroll, J. D. Geometric Models and Badness-of-Fit Functions. In Krishnaiah, P. R. (Eds.), Multivariate Analysis, (Vol. II). New York: Academic Press. 1969, 639–671.Google Scholar
MacCallum, R. C. Transformation of a three-mode multidimensional scaling solution to INDSCAL form. Psychometrika, 1976, 41, 385–400.CrossRefGoogle Scholar
Schönemann, P. H. An algebraic solution for a class of subjective metrics models. Psychometrika, 1972, 37, 441–451.CrossRefGoogle Scholar
Takane, Y., Young, F., & De Leeuw, J. Non-metric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features. Psychometrika, 1977, 42, 7–67.CrossRefGoogle Scholar
Torgerson, W. S. Theory and methods of scaling, 1958, New York: Wiley.Google Scholar
Wish, M., & Carroll, J. D. Applications of Individual Differences Scaling to Studies of Human Perception and Judgement. In Carterette, E. C. & Friedman, M. P. (Eds.), Handbook of Perception, (Vol. II)Psychophysical Judgment and Measurement,. New York: Academic Press, Inc.. 1974, 449–491.CrossRefGoogle Scholar