Uniqueness proof for a family of models sharing features of Tucker's Three-Mode Factor Analysis and PARAFAC/CANDECOMP

Psychometrika

Volumn 61, Issue 1, 1996, Pages 133-154

  Harshman, Richard A ^a   Lundy, Margaret E ^b  

^a UNIVERSITY OF WESTERN ONTARIO   (Canada)

^b NONE

Author keywords

Cattell; DEDICOM; Factor rotation problem; Intrinsic axes; Multidimensional scaling; Oblique confactor; PARAFAC2; Parallel proportional profiles; Principal components; Quadrilinear models; Trilinear models

Indexed keywords

EID: 0030521152     PISSN: 00333123     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02296963     Document Type: Article

References (32)

1. 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.
2. Carroll, J. D., & Wish, M. (1974). Models and methods for three-way multidimensional scaling. In D. H. Krantz, R. C. Atkinson, R. D. Luce, & P. Suppes (Eds.), Contemporary developments in mathematical psychology: Vol. 2 (pp. 57-105). San Francisco: W. H. Freeman.
3. Cattell, R. B. (1944). "Parallel Proportional Profiles" and other principles for determining the choice of factors by rotation. Psychometrika, 9, 267-283.
4. Cattell, R. B. (1978). The scientific use of factor analysis in behavioral and life sciences. New York: Plenum.
5. 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.
6. 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.
7. de Leeuw, J., & Pruzansky, S. (1978). A new computational method to fit the weighted Euclidean distance model. Psychometrika, 43, 479-490.
8. Dunn, T. R., & Harshman, R. A. (1982). A multidimensional scaling model for the size-weight illusion. Psychometrika, 47, 25-45.
9. 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. (University Microfilms No. 10,085)
10. Harshman, R. A. (1972a). Determination and proof of minimum uniqueness conditions for PARAFAC1. UCLA Working Papers in Phonetics, 22, 111-117. (University Microfilms No. 10,085)
11. Harshman, R. A. (1972b). PARAFAC2: Mathematical and technical notes. UCLA Working Papers in Phonetics, 22, 30-44. (University Microfilms No. 10,085)
12. Harshman, R. A. (1992). Unusual uniqueness properties of a three-way factor analysis model that incorporates "higher order" factors. Unpublished manuscript.
13. 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 H. G. Law, C. W. Snyder, Jr., J. A. Hattie, & R. P. McDonald (Eds.), Research methods for multimode data analysis (pp. 602-642). New York: Praeger.
14. Harshman, R. A., & Lundy, M. E. (1984). The PARAFAC model. In H. G. Law, C. W. Snyder, Jr., J. A. Hattie, & R. P. McDonald (Eds.), Research methods for multimode data analysis (pp. 122-215). New York: Praeger.
15. Harshman, R. A., & Lundy, M. E. (1994). PARAFAC: Parallel factor analysis. Computational Statistics & Data Analysis, 18, 39-72.
16. 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.
17. Kroonenberg, P. M. (1983). Three mode principal component analysis. Leiden: DSWO Press.
18. 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.
19. 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.
20. 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.
21. Kruskal, J. B. (1981). Multilinear models for data analysis. Behaviormetrika, 10, 1-20.
22. Kruskal, J. B. (1989). Rank, decomposition, and uniqueness for 3-way and N-way arrays. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 7-18). Amsterdam: North Holland.
23. Kruskal, J. B., Harshman, R. A., & Lundy, M. E. (1989). How 3-MFA data can cause degenerate PARAFAC solutions, among other relationships. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 115-130). Amsterdam: North Holland.
24. Leurgans, S. E. (1991). Multilinear models: Array formulas (Technical Report 473). Columbus, OH: Ohio State University, Department of Statistics.
25. 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.
26. Meredith, W. (1964). Notes on factorial invariance. Psychometrika, 29, 177-185.
27. Mitchell, B. C., & Burdick, D. S. (1994). Slowly converging PARAFAC sequences: Swamps and two-factor degeneracies. Journal of Chemometrics, 8, 155-168.
28. ten Berge, J. M. F., & Kiers, H. A. L. (1996). Some uniqueness results for PARAFAC2. Psychometrika, 61, 123-132.
29. Tucker, L. R. (1963). Implications of factor analysis of three-way matrices for measurement of change. In C. W. Harris (Ed.), Problems in measuring change (pp. 122-137). Madison, Wisconsin: University of Wisconsin Press.
30. Tucker, L. R. (1964). The extension of factor analysis to three-dimensional matrices. In N. Frederiksen & H. Gulliksen (Eds.), Contributions to mathematical psychology (pp. 109-127). New York: Holt, Rinehart, & Winston.
31. Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279-311.
32. Tucker, L. R. (1972). Relations between multidimensional scaling and three-mode factor analysis. Psychometrika, 37, 3-27.