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Uniqueness Proof for a Family of Models Sharing Features of Tucker's Three-Mode Factor Analysis and PARAFAC/Candecomp

Published online by Cambridge University Press:  01 January 2025

Richard A. Harshman  and
Margaret E. Lundy
Richard A. Harshman*
Affiliation:
University of Western Ontario
Margaret E. Lundy
Affiliation:
University of Western Ontario
*
Requests for reprints should be sent to Richard A. Harshman, Psychology Department, University of Western Ontario, London, Ontario, CANADA N6A 5C2.
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Abstract

Some existing three-way factor analysis and MDS models incorporate Cattell's “Principle of Parallel Proportional Profiles”. These models can—with appropriate data—empirically determine a unique best fitting axis orientation without the need for a separate factor rotation stage, but they have not been general enough to deal with what Tucker has called “interactions” among dimensions. This article presents a proof of unique axis orientation for a considerably more general parallel profiles model which incorporates interacting dimensions. The model, Xk=AADk HBDk B', does not assume symmetry in the data or in the interactions among factors. A second proof is presented for the symmetrically weighted case (i.e., where ADk=BDk). The generality of these models allows one to impose successive restrictions to obtain several useful special cases, including PARAFAC2 and three-way DEDICOM.

Keywords

Parallel proportional profilesintrinsic axesDEDICOMPARAFAC2Cattelltrilinear modelsquadrilinear modelsfactor rotation problemmultidimensional scalingprincipal componentsoblique confactor
Type
Original Paper
Information
Psychometrika , Volume 61 , Issue 1 , March 1996 , pp. 133 - 154
Copyright
© 1996 The Psychometric Society

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Footnotes

We want to express appreciation for the contributions of several colleagues: Jos M. F. ten Berge and Henk A. L. Kiers carefully went through more than one version of this article, found an important error, and contributed many improvements. J. Douglas Carroll and Shizuhiko Nishisato acted with unusual editorial preserverance and flexibility, thereby making possible the successful completion of a difficult assessment and revision process. Joseph B. Kruskal has long provided crucial mathematical insights and inspiration to those working in this area, but this is particularly true for us. His contributions to this specific article include early discussion of basic questions and careful examination of some earlier attempted proofs, finding them to be invalid. The anonymous reviewers also made useful suggestions. Some portions of this work were supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.

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, 283–319.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. 2 (pp. 57–105). San Francisco: W. H. Freeman.Google Scholar
Cattell, R. B. (1944). “Parallel Proportional Profiles” and other principles for determining the choice of factors by rotation. Psychometrika, 9, 267–283.CrossRefGoogle Scholar
Cattell, R. B. (1978). The scientific use of factor analysis in behavioral and life sciences, New York: Plenum.CrossRefGoogle Scholar
Cattell, R. B., Cattell, A. K. S. (1955). Factor rotation for proportional profiles: Analytical solution and an example. British Journal of Statistical Psychology, 8, 83–92.CrossRefGoogle Scholar
Dawson, M. R. W., Harshman, R. A. (1986). The multidimensional analysis of asymmetries in alphabetic confusion matrices: Evidence for global-to-local and local-to-global processing. Perception & Psychophysics, 40, 370–383.CrossRefGoogle ScholarPubMed
de Leeuw, J., Pruzansky, S. (1978). A new computational method to fit the weighted Euclidean distance model. Psychometrika, 43, 479–490.CrossRefGoogle Scholar
Dunn, T. R., Harshman, R. A. (1982). A multidimensional scaling model for the size-weight illusion. Psychometrika, 47, 25–45.CrossRefGoogle Scholar
Harshman, R. A. (1970). Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multi-modal factor analysis. UCLA Working Papers in Phonetics, 16, 1–84.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. (1972). PARAFAC2: Mathematical and technical notes. UCLA Working Papers in Phonetics, 22, 30–44.Google Scholar
Harshman, R. A. (1992). Unusual uniqueness properties of a three-way factor analysis model that incorporates “higher order” factors. Unpublished manuscript.Google Scholar
Harshman, R. A., DeSarbo, W. (1984). An application of PARAFAC to a small sample problem, demonstrating preprocessing, orthogonality constraints, and split-half diagnostic techniques. In Law, H. G., Snyder, C. W. Jr., Hattie, J. A., McDonald, R. P. (Eds.), Research methods for multimode data analysis (pp. 602–642). New York: Praeger.Google Scholar
Harshman, R. A., Lundy, M. E. (1984). The PARAFAC model. 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. (1994). PARAFAC: Parallel factor analysis. Computational Statistics & Data Analysis, 18, 39–72.CrossRefGoogle Scholar
Harshman, R. A., Green, P. E., Wind, Y., Lundy, M. E. (1982). A model for the analysis of asymmetric data in marketing research. Marketing Science, 1, 205–242.CrossRefGoogle Scholar
Kroonenberg, P. M. (1983). Three mode principal component analysis, Leiden: DSWO Press.Google Scholar
Kroonenberg, P. M., de Leeuw, J. (1980). Principal components analysis of three-mode data by means of alternating least-squares algorithms. Psychometrika, 5, 69–97.CrossRefGoogle Scholar
Kruskal, J. B. (1976). More factors than subjects, tests than treatments: An indeterminacy theorem for canonical decomposition and individual differences scaling. Psychometrika, 41, 281–293.CrossRefGoogle Scholar
Kruskal, J. B. (1977). Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra and Its Applications, 18, 95–138.CrossRefGoogle Scholar
Kruskal, J. B. (1981). Multilinear models for data analysis. Behaviormetrika, 10, 1–20.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
Kruskal, J. B., Harshman, R. A., Lundy, M. E. (1989). How 3-MFA data can cause degenerate PARAFAC solutions, among other relationships. In Coppi, R., Bolasco, S. (Eds.), Multiway data analysis (pp. 115–130). Amsterdam: North Holland.Google Scholar
Leurgans, S. E. (1991). Multilinear models: Array formulas, Columbus, OH: Ohio State University, Department of Statistics.Google Scholar
Leurgans, S. E., Ross, R. T., Abel, R. B. (1993). A decomposition for 3-way arrays. SIAM Journal on Matrix Analysis and Applications, 14, 1064–1083.CrossRefGoogle Scholar
Meredith, W. (1964). Notes on factorial invariance. Psychometrika, 29, 177–185.CrossRefGoogle Scholar
Mitchell, B. C., Burdick, D. S. (1994). Slowly converging PARAFAC sequences: Swamps and two-factor degeneracies. Journal of Chemometrics, 8, 155–168.CrossRefGoogle Scholar
ten Berge, J. M. F., Kiers, H. A. L. (1996). Some uniqueness results for PARAFAC2. Psychometrika, 61, 123–132.CrossRefGoogle Scholar
Tucker, L. R. (1963). Implications of factor analysis of three-way matrices for measurement of change. In Harris, C. W. (Eds.), Problems in measuring change (pp. 122–137). Madison, Wisconsin: University of Wisconsin Press.Google Scholar
Tucker, L. R. (1964). The extension of factor analysis to three-dimensional matrices. In Frederiksen, N., Gulliksen, H. (Eds.), Contributions to mathematical psychology (pp. 109–127). New York: Holt, Rinehart, & Winston.Google Scholar
Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311.CrossRefGoogle ScholarPubMed
Tucker, L. R. (1972). Relations between multidimensional scaling and three-mode factor analysis. Psychometrika, 37, 3–27.CrossRefGoogle Scholar