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Psychometrika

, Volume 63, Issue 3, pp 255–261 | Cite as

A case of extreme simplicity of the core matrix in three-mode principal components analysis

  • Takashi Murakami
  • Jos M. F. Ten Berge
  • Henk A. L. Kiers
Article

Abstract

In three-mode Principal Components Analysis, theP ×Q ×R core matrixG can be transformed to simple structure before it is interpreted. It is well-known that, whenP=QR,G can be transformed to the identity matrix, which implies that all elements become equal to values specified a priori. In the present paper it is shown that, whenP=QR − 1,G can be transformed to have nearly all elements equal to values spectified a priori. A cllsed-form solution for this transformation is offered. Theoretical and practical implications of this simple structure transformation ofG are discussed.

Key words

three-mode principal components analysis core matrix rotations simple structure 

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References

  1. Cohen, H. S. (1974, June).Three-mode rotation to approximate INDSCAL structure (TRIAS). Paper presented at the Psychometric Society Meeting, Palo Alto, CA.Google Scholar
  2. Kiers, H. A. L. (1992). TUCKALS core rotations and constrained TUCKALS modelling.Statistica Applicata, 4, 659–667.Google Scholar
  3. Kiers, H. A. L. (1997). Three-mode Orthomax rotation.Psychometrika, 62, 579–598.Google Scholar
  4. Kiers, H. A. L. (1998a). Recent developments in three-mode factor analysis: Constrained three-mode factor analysis and core rotations. In C. Hayashi, N. Ohsumi, K. Yajima, Y. Tanaka, H.-H. Bock, & Y. Baba (Eds.),Data Science, classification and related methods. Tokyo: Springer Verlag.Google Scholar
  5. Kiers, H. A. L. (1998b).Three-way simplimax for oblique rotation of the three-mode factor analysis core to simple structure. Computational Statistics and Data Analysis.Google Scholar
  6. Kroonenberg, P. M. (1983).Three-mode principal component analysis. Leiden, DSWO.Google Scholar
  7. 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.CrossRefGoogle Scholar
  8. Kruskal, J. B. (1988, June).Simple structure for three-way data: A new method intermediate between 3-mode factor analysis and PARAFAC-CANDECOMP. Paper presented at the 53rd annual Meeting of the Psychometric Society. Los Angeles.Google Scholar
  9. Kruskal, J. B. (1989). Rank, decomposition, and uniqueness for 3-way andN-way arrays. In R. Coppi & S. Bolasco (Eds.),Multiway data analysis (7–18). Amsterdam: Elsevier Science Publishers.Google Scholar
  10. MacCallum, R. C. (1976). Transformations of three-mode multidimensional scaling solution to INDSCAL form.Psychometrika, 41, 385–400.Google Scholar
  11. Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis.Psychometrika, 31, 279–311.PubMedGoogle Scholar

Copyright information

© The Psychometric Society 1998

Authors and Affiliations

  • Takashi Murakami
    • 1
  • Jos M. F. Ten Berge
    • 2
  • Henk A. L. Kiers
    • 2
  1. 1.Department of Educational PsychologyNagoya UniversityNagoyaJapan
  2. 2.University of GroningenThe Netherlands

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