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The Approximation of Two-Mode Proximity Matrices by Sums of Order-Constrained Matrices

Published online by Cambridge University Press:  01 January 2025

Lawrence Hubert  and
Phipps Arabie
Lawrence Hubert*
Affiliation:
University of Illinois, Champaign Department of Data Theory, University of Leiden
Phipps Arabie
Affiliation:
Faculty of Management, Rutgers University
*
Requests for reprints should be sent to Lawrence Hubert, Department of Psychology, The University of Illinois, 603 East Daniel Street, Champaign, IL 61820.
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Abstract

A least-squares strategy is proposed for representing a two-mode proximity matrix as an approximate sum of a small number of matrices that satisfy certain simple order constraints on their entries. The primary class of constraints considered define Q-forms (or anti-Q-forms) for a two-mode matrix, where after suitable and separate row and column reorderings, the entries within each row and within each column are nondecreasing (or nonincreasing) to a maximum (or minimum) and thereafter nonincreasing (or nondecreasing). Several other types of order constraints are also mentioned to show how alternative structures can be considered using the same computational strategy.

Keywords

two-mode proximity matricesorder constraints(anti-)Robinson form(anti-)Q-formleast-squares matrix approximation
Type
Original Paper
Information
Psychometrika , Volume 60 , Issue 4 , December 1995 , pp. 573 - 605
Copyright
Copyright © 1995 The Psychometric Society

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References

Barthélemy, J.-P., Guénoche, A. (1991). Trees and proximity representations, Chichester: Wiley.Google Scholar
Carroll, J. D. (1980). Models and methods for multidimensional analysis of preferential choice (or other dominance) data. In Lantermann, E. D., Feger, H. (Eds.), Similarity and choice (pp. 234–289). Bern: Huber.Google Scholar
Coombs, C. H. (1964). A theory of data, New York: Wiley.Google Scholar
De Soete, G., DeSarbo, W. S., Furnas, G. W., Carroll, J. D. (1984). The estimation of ultrametric and path length trees from two-mode proximity data. Psychometrika, 49, 289–310.CrossRefGoogle Scholar
Ducamp, A., Falmagne, J. C. (1969). Composite measurement. Journal of Mathematical Psychology, 6, 359–390.CrossRefGoogle Scholar
Dykstra, R. L. (1983). An algorithm for restricted least squares regression. Journal of the American Statistical Association, 78, 837–842.CrossRefGoogle Scholar
Furnas, G. W. (1980). Objects and their features: The metric analysis of two-class data, Stanford, CA: Stanford University.Google Scholar
Gifi, A. (1990). Nonlinear multivariate analysis, Chichester: Wiley.Google Scholar
Hansen, J. C., Campbell, D. P. (1985). Manual for the SVIB-SCII 4th ed.,, Palo Alto, CA: Consulting Psychologists Press.Google Scholar
Heiser, W. J. (1981). Unfolding analysis of proximity data. Unpublished doctoral dissertation, University of Leiden.Google Scholar
Holland, J. L. (1966). The psychology of vocational choice: A theory of personality types and environmental models, New York: Ginn.Google Scholar
Holland, J. L. (1981). Making vocational choices: A theory of careers 2nd ed.,, Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
Hubert, L. J. (1974). Problems of seriation using a subject by item response matrix. Psychological Bulletin, 81, 976–983.CrossRefGoogle Scholar
Hubert, L. J., Arabie, P. (1994). The analysis of proximity matrices through sums of matrices having (anti-)Robinson forms. British Journal of Mathematical and Statistical Psychology, 47, 1–40.CrossRefGoogle Scholar
Hubert, L. J., Schultz, J. V. (1976). Quadratic assignment as a general data analysis strategy. British Journal of Mathematical and Statistical Psychology, 29, 190–241.CrossRefGoogle Scholar
Kendall, D. G. (1971). Abundance matrices and seriation in archaeology. Zeitschrift für Wahrscheinlichkeitstheorie, 17, 104–112.CrossRefGoogle Scholar
Kendall, D. G. (1971). A mathematical approach to seriation. Philosophical Transactions of the Royal Society of London, Series A, 269, 125–135.Google Scholar
Kruskal, J. B. (1964). Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika, 29, 1–27.CrossRefGoogle Scholar
Marks, W. B. (1965). Difference spectra of the visual pigments in single goldfish cones. Unpublished doctoral dissertation, Johns Hopkins University.Google Scholar
Pliner, V. (in press). Metric unidimensional scaling and global optimization. Journal of Classification.Google Scholar
Prediger, D. J. (1982). Dimensions underlying Holland's hexagon: Missing link between interests and occupations?. Journal of Vocational Behavior, 21, 259–287.CrossRefGoogle Scholar
Robinson, W. S. (1951). A method for chronologically ordering archaeological deposits. American Antiquity, 16, 293–381.CrossRefGoogle Scholar
Roskam, E. E. Ch. I. (1968). Metric analysis of ordinal data in psychology, Voorschoten: Drukkerij/Vitgeverij VAM.Google Scholar
Rounds, J. B. (1995). Vocational interests: Evaluating structural hypotheses. In Dawis, R. V., Lubinski, D. (Eds.), Assessing individual differences in human behavior: New concepts, methods, and findings (pp. 177–232). Palo Alto, CA: Consulting Psychologist Press.Google Scholar
Schiffman, H., Falkenberg, P. (1968). The organization of stimuli and sensory neurons. Physiology and Behavior, 3, 197–201.CrossRefGoogle Scholar
Schiffman, S. S., Reynolds, M. L., Young, F. W. (1981). Introduction to multidimensional scaling, New York: Academic Press.Google Scholar
Tracy, T. J., Rounds, J. (1993). Evaluating Holland's and Gati's vocational-interest models: A structural meta-analysis. Psychological Bulletin, 113, 229–246.CrossRefGoogle Scholar
Tucker, A. (1972). A structure theorem for the consecutive 1s property. Journal of Combinatorial Theory, Series B, 12, 153–162.CrossRefGoogle Scholar
Tucker, L. R. (1972). Relations between multidimensional scaling and three-mode factor analysis. Psychometrika, 37, 3–27.CrossRefGoogle Scholar