Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-08T14:28:06.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Principal Component Analysis of Three-Mode Data by Means of Alternating Least Squares Algorithms

Published online by Cambridge University Press:  01 January 2025

Pieter M. Kroonenberg  and
Jan de Leeuw
Pieter M. Kroonenberg*
Affiliation:
University of Leiden
Jan de Leeuw
Affiliation:
University of Leiden
*
Requests for reprints should be sent to Pieter M. Kroonenberg, Vakgroep W.E.P., Subfakulteit der Pedagogische en Andragogische Wetenschappen, Schuttersveld 9 (5e verd.), 2316 XG Leiden, THE NETHERLANDS.
Get access Rights & Permissions [Opens in a new window]

Abstract

A new method to estimate the parameters of Tucker’s three-mode principal component model is discussed, and the convergence properties of the alternating least squares algorithm to solve the estimation problem are considered. A special case of the general Tucker model, in which the principal component analysis is only performed over two of the three modes is briefly outlined as well. The Miller & Nicely data on the confusion of English consonants are used to illustrate the programs TUCKALS3 and TUCKALS2 which incorporate the algorithms for the two models described.

Keywords

three-mode principal component analysisalternating least squaresfactor analysismultidimensional scalingindividual differences scalingsimultaneous iterationconfusion of consonants
Type
Original Paper
Information
Psychometrika , Volume 45 , Issue 1 , March 1980 , pp. 69 - 97
Copyright
Copyright © 1980 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.)

References

Reference Notes

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, Princeton, N. J., March 1972.Google Scholar
Harshman, R. A. Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multimode factor analysis, 1970, Los Angeles: University of California.Google Scholar
Jennrich, R. A generalization of the multidimensional scaling model of Carroll & Chang, 1972, Los Angeles: University of California.Google Scholar
Kroonenberg, P. M. & de Leeuw, J. TUCKALS2: A principal component analysis of three mode data, 1977, Leiden: Department of Data Theory, University of Leiden.Google Scholar
Levin, J. Three-mode factor analysis, 1963, Urbana, Ill.: University of Illinois.Google Scholar
Wish, M. An INDSCAL analysis of the Miller & Nicely consonant confusion data. Paper presented at meetings of the Acoustical Society of America. Houston, November, 1970.CrossRefGoogle Scholar

References

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
Carroll, J. D. & Wish, M. Models and methods for three-way multidimensional scaling. In Krantz, D. H., Luce, R. D., Atkinson, R. C. & Suppes, P. (Eds.), Contemporary developments in mathematical psychology (Vol. II), 1974, San Francisco: W. H. Freeman.Google Scholar
d’Esopo, D. A. A convex programming procedure. Naval Research Logistics Quarterly, 1959, 11, 33–42.CrossRefGoogle Scholar
Hubert, L. J. & Baker, F. B. Evaluating the symmetry of a proximity matrix. Quality & Quantity, 1979, 13, 77–84.CrossRefGoogle Scholar
Israelsson, A. Three-way (or second order) component analysis. Bulletin of the International Statistical Institute, 1969, 43, 29–51.Google Scholar
Meyer, R. R. The validity of a family of optimization methods. SIAM Journal of Control and Optimization, 1970, 15, 699–715.CrossRefGoogle Scholar
Miller, G. A. & Nicely, P. E. An analysis of perceptual confusion among some English consonants. Journal of the Acoustical Society of America, 1955, 27, 338–352.CrossRefGoogle Scholar
Osgood, C. E. & Suci, G. J., Tannenbaum, T. H. The measurement of meaning, 1957, Urbana, Ill.: University of Illinois Press.Google Scholar
Ostrowski, A. M. Solution of equations and systems of equations, 1966, New York: Academic Press.Google Scholar
Penrose, R. On the best approximate solutions of linear matrix equations. Proceedings of the Cambridge Philosophical Society, 1955, 51, 406–413.CrossRefGoogle Scholar
Rutishauser, H. Computational aspects of F. L. Bauer’s simultaneous iteration method. Numerische Mathematik, 1969, 13, 4–13.CrossRefGoogle Scholar
Sands, R. & Young, F. W. Component models for three-way data: ALSCOMP3, an alternating least squares algorithm with optimal scaling features. Psychometrika, 1980, 45, 39–67.CrossRefGoogle Scholar
Schwartz, H. R. & Rutishauser, H., Stiefel, E. Numerik. Symmetrischer matrizen, 1968, Stuttgart: Teubner.Google Scholar
Shepard, R. N. Psychological representation of speech sounds. In David, E. E. & Denes, P. B. (Eds.), Human communication. A unified view, 1972, New York: McGraw Hill.Google Scholar
Shepard, R. N. Representation of structure in similarity data: Problems and prospects. Psychometrika, 1974, 39, 373–421.CrossRefGoogle Scholar
Smith, P. T. Feature-testing models and their application to perception and memory for speech. Quarterly Journal of Experimental Psychology, 1973, 25, 511–534.CrossRefGoogle ScholarPubMed
Smith, P. T., & Jones, K. F. Some hierarchical scaling methods for confusion matrix analysis II. Application to large matrices. British Journal of Mathematical and Statistical Psychology, 1975, 28, 30–45.CrossRefGoogle Scholar
Soli, S. D. & Arabie, P. Auditory versus phonetic accounts of observed confusions between consonant phonemes. Journal of the Acoustical Society of America, 1979, 66, 46–59.CrossRefGoogle ScholarPubMed
Takane, Y., Young, F. W. & 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
Tucker, L. R. Implications of factor analysis of three-way matrices for measurement of change. In Harris, C. W. (Eds.), Problems in measuring change, 1963, Madison, Wis.: University of Wisconsin Press.Google Scholar
Tucker, L. R. The extension of factor analysis to three-dimensional matrices. In Gulliksen, H., Frederiksen, N. (Eds.), Contributions to mathematical psychology, 1964, New York: Holt, Rinehardt & Winston.Google Scholar
Tucker, L. R. Some mathematical notes on three-mode factor analysis. Psychometrika, 1966, 31, 279–311.CrossRefGoogle ScholarPubMed
Tucker, L. R. Relations between multidimensional scaling and three-mode factor analysis. Psychometrika, 1972, 37, 3–27.CrossRefGoogle Scholar
Tucker, L. R. & Messick, S. An individual difference model for multidimensional scaling. Psychometrika, 1963, 28, 333–367.CrossRefGoogle Scholar
Wainer, H., Gruvaeus, G. & Blair, M. TREBIG: A 360/75 FORTRAN program for three-mode factor analysis for big data sets. Behavioral Research Methods and Instrumentation, 1974, 6, 53–54.CrossRefGoogle Scholar
Wainer, H., Gruvaeus, G. & Snijder, F. TREMOD: A 360/75 program for three-mode factor analysis. Behavioral Science, 1971, 16, 421–422.Google Scholar
Walsh, J. A. An IBM 709 program for factor analyzing three-mode matrices. Educational and Psychological Measurement, 1964, 24, 669–773.CrossRefGoogle Scholar
Walsh, J. A. & Walsh, R. A revised Fortran program for three-mode factor analysis. Educational and Psychological Measurement, 1976, 36, 169–170.CrossRefGoogle Scholar
Young, F. W., de Leeuw, J. & Takane, Y. Quantifying qualitative data. In Lantermann, E. D., Feger, H. (Eds.), Similarity and choice, 1980, Bern: Huber.Google Scholar