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Nonmetric Multidimensional Scaling: A Numerical Method

Published online by Cambridge University Press:  01 January 2025

J. B. Kruskal*
Affiliation:
Bell Telephone Laboratories

Abstract

We describe the numerical methods required in our approach to multi-dimensional scaling. The rationale of this approach has appeared previously.

Type
Original Paper
Copyright
Copyright © 1964 Psychometric Society

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References

Barton, D. E. and Mallows, C. L. The randomization bases of the amalgamation of weighted means. J. roy. statist. Soc., 1961, 23, 423433.CrossRefGoogle Scholar
Bartholomew, D. J. A test of homogeneity for ordered alternatives. Biometrika, 1959, 46, 3648.CrossRefGoogle Scholar
Bartholomew, D. J. A test of homogeneity of means under restricted alternatives (with discussion). J. roy. statist. Soc., 1961, 23, 239281.CrossRefGoogle Scholar
Hardy, G. H., Littlewood, J. E., and Polya, G. Inequalities (2nd ed.), Cambridge, Eng.: Cambridge Univ. Press, 1952.Google Scholar
Hildebrandt, P. and Isbitz, H. Radix-exchange—An internal sorting method for digital computers. J. Assoc. computing Machinery, 1959, 6, 156163.CrossRefGoogle Scholar
Kolmogorov, A. N. and Fomin, S. V. In Boron, Leo F. (Eds.), Elements of the theory of functions and functional analysis. Vol. 1.Metric and normed spaces, Rochester, N. Y.: Graylock Press, 1957.Google Scholar
Kruskal, J. Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika, 1964, 29, 128.CrossRefGoogle Scholar
Miles, R. E. The complete amalgamation into blocks, by weighted means, of a finite set of real numbers. Biometrika, 1959, 46, 317327.CrossRefGoogle Scholar
Shepard, R. N. The analysis of proximities: Multidimensional scaling with an unknown distance function. Psychometrika, 1962, 27, 125139.CrossRefGoogle Scholar
Spang, H. A. III. A review of minimization techniques for nonlinear functions. SIAM Rev., 1962, 4, 343365.CrossRefGoogle Scholar
van Eeden, C. Maximum likelihood estimation of partially or completely ordered parameters, I. Proc. Akademie van Wetenschappen, 1957, 60, 128136.Google Scholar
van Eeden, C. Note on two methods for estimating ordered parameters of probability distributions. Proc. Akademie van Wetenschappen, 1957, 60, 506512.Google Scholar