, Volume 62, Issue 3, pp 349–374 | Cite as

Uniqueness of three-mode factor models with sparse cores: The 3 × 3 × 3 case

  • Henk A. L. Kiers
  • Jos M. F. ten Berge
  • Roberto Rocci


Three-Mode Factor Analysis (3MFA) and PARAFAC are methods to describe three-way data. Both methods employ models with components for the three modes of a three-way array; the 3MFA model also uses a three-way core array for linking all components to each other. The use of the core array makes the 3MFA model more general than the PARAFAC model (thus allowing a better fit), but also more complicated. Moreover, in the 3MFA model the components are not uniquely determined, and it seems hard to choose among all possible solutions. A particularly interesting feature of the PARAFAC model is that it does give unique components. The present paper introduces a class of 3MFA models in between 3MFA and PARAFAC that share the good properties of the 3MFA model and the PARAFAC model: They fit (almost) as well as the 3MFA model, they are relatively simple and they have the same uniqueness properties as the PARAFAC model.

Key words

three-way methods PARAFAC 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Carroll, J. D., & Chang, J.-J. (1970). Analysis of individual differences in multidimensional scaling via ann-way generalization of “Eckart-Young” decomposition.Psychometrika, 35, 283–319.Google Scholar
  2. Harshman, R. A. (1970).Foundations of the PARAFAC procedure: models and conditions for an “explanatory” multi-mode factor analysis, (UCLA Working Papers in Phonetics No. 16, pp. 1–84). Los Angeles: University of California at Los Angeles.Google Scholar
  3. Harshman, R. A. (1972).Determination and proof of minimum uniqueness conditions for PARAFAC1, (UCLA Working Papers in Phonetics No. 22, pp. 111–117). Los Angeles: University of California at Los Angeles.Google Scholar
  4. Harshman, R. A., & Lundy, M. E. (1984). The PARAFAC model for three-Way factor analysis and multidimensional scaling. In H. G. Law, C. W. Snyder, J. A. Hattie, and R. P. McDonald (Eds.),Research methods for multimode data analysis (pp. 122–215). New York: Praeger.Google Scholar
  5. Kiers, H. A. L. (1992). Tuckals core rotations and constrained Tuckals modelling.Statistica Applicata, 4, 659–667.Google Scholar
  6. Kiers, H. A. L. (in press). Three-mode Orthomax rotation. Psychometrika.Google Scholar
  7. Kroonenberg, P. M. (1983).Three mode principal component analysis: Theory and applications. Leiden: DSWO press.Google Scholar
  8. Kroonenberg, P. M., & de Leeuw, J. (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms.Psychometrika, 45, 69–97.Google Scholar
  9. 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.Google Scholar
  10. Kruskal, J. B. (1988, June).Simple structure for three-way data: A new method to intermediate between 3-mode factor analysis and PARAFAC/CANDECOMP. Paper presented at the 53rd Annual meeting of the Psychometric Society, Los Angeles.Google Scholar
  11. Kruskal, J. B. (1989). Rank, decomposition, and uniqueness for 3-way andN-way arrays. In: R. Coppi & S. Bolasco (Eds.),Multiway data analysis (pp. 7–18). Amsterdam: Elsevier Science Publishers.Google Scholar
  12. Leurgans, S., & Ross, R. T. (1992). Multilinear models: Applications in Spectroscopy.Statistical Science, 7, 289–319.Google Scholar
  13. Lundy, M. E., Harshman, R. A., & Kruskal, J. B. (1985). A two-stage procedure incorporating good features of both trilinear and quadrilinear models. In R. Coppi & S. Bolasco (Eds.),Multiway data analysis (pp. 123–130). Amsterdam: Elsevier Science Publishers.Google Scholar
  14. Rocci, R. (1992). Three-mode factor analysis with binary core and orthonormality constraints.Journal of the Italian Statistical Society, 3, 413–422.Google Scholar
  15. Smilde, A. K., Wang, Y., & Kowalski, B. R. (1994). Theory of medium-rank second-order calibration with restricted-Tucker models.Journal of Chemometrics, 8, 21–36.Google Scholar
  16. Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis.Psychometrika, 31, 279–311.Google Scholar

Copyright information

© The Psychometric Society 1997

Authors and Affiliations

  • Henk A. L. Kiers
    • 1
  • Jos M. F. ten Berge
    • 1
  • Roberto Rocci
    • 2
  1. 1.University of GroningenThe Netherlands
  2. 2.University la SapienzaRome

Personalised recommendations