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Some Uniqueness Results for PARAFAC2

Published online by Cambridge University Press:  01 January 2025

Jos M. F. ten Berge  and
Henk A. L. Kiers
Jos M. F. ten Berge*
Affiliation:
University of Groningen
Henk A. L. Kiers
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Jos M. F. ten Berge, Department of Psychology, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, THE NETHERLANDS.
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Abstract

Whereas the unique axes properties of PARAFAC1 have been examined extensively, little is known about uniqueness properties for the PARAFAC2 model for covariance matrices. This paper is concerned with uniqueness in the rank two case of PARAFAC2. For this case, Harshman and Lundy have recently shown, subject to mild assumptions, that PARAFAC2 is unique when five (covariance) matrices are analyzed. In the present paper, this result is sharpened. PARAFAC2 is shown to be usually unique with four matrices. With three matrices it is not unique unless a certain additional assumption is introduced. If, for instance, the diagonal matrices of weights are constrained to be non-negative, three matrices are enough to have uniqueness in the rank two case of PARAFAC2.

Keywords

three-way analysisstationary component analysis
Type
Original Paper
Information
Psychometrika , Volume 61 , Issue 1 , March 1996 , pp. 123 - 132
Copyright
© 1996 The Psychometric Society

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Footnotes

The authors are obliged to Richard Harshman for stimulating this research, and to the Associate Editor and reviewers for suggesting major improvements in the presentation.

References

Carroll, J. D., Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition. Psychometrika, 35, 23–38.CrossRefGoogle Scholar
Carroll, J. D., Wish, M. (1974). Models and methods for three-way multidimensional scaling. In Krantz, D. H., Atkinson, R. C., Luce, R. D., Suppes, P. (Eds.), Contemporary developments in mathematical psychology, Vol. II: Measurement, psychophysics, and neural information processing (pp. 57–105). San Francisco: Freeman.Google Scholar
Harshman, R. A. (1972). PARAFAC2: Mathematical and technical notes. UCLA Working Papers in Phonetics, 22, 31–44.Google Scholar
Harshman, R. A. (1972). Determination and proof of minimum uniqueness conditions for PARAFAC1. UCLA Working Papers in Phonetics, 22, 111–117.Google Scholar
Harshman, R. A., Lundy, M. E. (1984). The PARAFAC model for three-way factor analysis and multi-dimensional scaling. In Law, H. G., Snyder, C. W. Jr., Hattie, J. A., McDonald, R. P. (Eds.), Research methods for multimode data analysis (pp. 122–215). New York: Praeger.Google Scholar
Harshman, R. A., & Lundy, M. E. (1996). Uniqueness proof for a family of models sharing features of Tucker's three-mode factor analysis and PARAFAC/CANDECOMP. Psychometrika, 61, Copy Editor: Provide page numbers..CrossRefGoogle Scholar
Jacob, H. G., Bailey, D. W. (1971). Linear Algebra, New York: Houghton Mifflin.Google Scholar
Kiers, H. A. L. (1993). An alternating least squares algorithm for PARAFAC2 and three-way DEDICOM. Computational Statistics and Data Analysis, 16, 103–118.CrossRefGoogle Scholar
Kruskal, J. B. (1977). Three-way Arrays: Rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics. Linear Algebra and its Applications, 18, 95–138.CrossRefGoogle Scholar
Kruskal, J. B. (1989). Rank, decomposition, and uniqueness for 3-way and n-way arrays. In Coppi, R., Bolasco, S. (Eds.), Multiway data analysis (pp. 7–18). Amsterdam: North-Holland.Google Scholar