Principal component analysis of three-mode data by means of alternating least squares algorithms
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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.
- 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.
- Harshman, R. A.Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multimode factor analysis (Working Papers in Phonetics No. 16). Los Angeles: University of California, 1970.
- Jennrich, R.A generalization of the multidimensional scaling model of Carroll & Chang (Working Papers in Phonetics No. 22). Los Angeles: University of California, 1972.
- Kroonenberg, P. M. & de Leeuw, J.TUCKALS2: A principal component analysis of three mode data (Res. Bull. RB 001-77). Leiden: Department of Data Theory, University of Leiden, 1977.
- Levin, J.Three-mode factor analysis (Unpublished doctoral thesis). Urbana, Ill.: University of Illinois, 1963.
- 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.
- 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.
- Carroll, J. D. & Wish, M. Models and methods for three-way multidimensional scaling. In D. H. Krantz, R. D. Luce, R. C. Atkinson, & P. Suppes (Eds.),Contemporary developments in mathematical psychology (Vol. II). San Francisco: W. H. Freeman, 1974.
- d'Esopo, D. A. A convex programming procedure.Naval Research Logistics Quarterly, 1959,11, 33–42.
- Hubert, L. J. & Baker, F. B. Evaluating the symmetry of a proximity matrix.Quality & Quantity, 1979,13, 77–84.
- Israelsson, A. Three-way (or second order) component analysis. In H. Wold & E. Lyttkens (Eds.), Nonlinear iterative partial least-squares (NIPALS) estimation procedures.Bulletin of the International Statistical Institute, 1969,43, 29–51.
- Meyer, R. R. The validity of a family of optimization methods.SIAM Journal of Control and Optimization, 1970,15, 699–715.
- 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.
- Osgood, C. E., Suci, G. J. & Tannenbaum, T. H.The measurement of meaning. Urbana, Ill.: University of Illinois Press, 1957.
- Ostrowski, A. M.Solution of equations and systems of equations. New York: Academic Press, 1966.
- Penrose, R. On the best approximate solutions of linear matrix equations.Proceedings of the Cambridge Philosophical Society, 1955,51, 406–413.
- Rutishauser, H. Computational aspects of F. L. Bauer's simultaneous iteration method.Numerische Mathematik, 1969,13, 4–13.
- 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.
- Schwartz, H. R., Rutishauser, H. & Stiefel, E.Numerik. Symmetrischer matrizen. Stuttgart: Teubner, 1968.
- Shepard, R. N. Psychological representation of speech sounds. In E. E. David & P. B. Denes (Eds.),Human communication. A unified view. New York: McGraw Hill, 1972.
- Shepard, R. N. Representation of structure in similarity data: Problems and prospects.Psychometrika, 1974,39, 373–421.
- Smith, P. T. Feature-testing models and their application to perception and memory for speech.Quarterly Journal of Experimental Psychology, 1973,25, 511–534.
- 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.
- 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.
- 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.
- Tucker, L. R. Implications of factor analysis of three-way matrices for measurement of change. In C. W. Harris (Ed.),Problems in measuring change. Madison, Wis.: University of Wisconsin Press, 1963.
- Tucker, L. R. The extension of factor analysis to three-dimensional matrices. In H. Gulliksen & N. Frederiksen (Eds.),Contributions to mathematical psychology. New York: Holt, Rinehardt & Winston, 1964.
- Tucker, L. R. Some mathematical notes on three-mode factor analysis.Psychometrika, 1966,31, 279–311.
- Tucker, L. R. Relations between multidimensional scaling and three-mode factor analysis.Psychometrika, 1972,37, 3–27.
- Tucker, L. R. & Messick, S. An individual difference model for multidimensional scaling.Psychometrika, 1963,28, 333–367.
- 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.
- Wainer, H., Gruvaeus, G. & Snijder, F. TREMOD: A 360/75 program for three-mode factor analysis.Behavioral Science, 1971,16, 421–422.
- Walsh, J. A. An IBM 709 program for factor analyzing three-mode matrices.Educational and Psychological Measurement, 1964,24, 669–773.
- Walsh, J. A. & Walsh, R. A revised Fortran program for three-mode factor analysis.Educational and Psychological Measurement, 1976,36, 169–170.
- Young, F. W., de Leeuw, J. & Takane, Y. Quantifying qualitative data. In E. D. Lantermann & H. Feger (Eds.),Similarity and choice. Bern: Huber, 1980 (in press).
- Principal component analysis of three-mode data by means of alternating least squares algorithms
Volume 45, Issue 1 , pp 69-97
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- three-mode principal component analysis
- alternating least squares
- factor analysis
- multidimensional scaling
- individual differences scaling
- simultaneous iteration
- confusion of consonants
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