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A Case of Extreme Simplicity of the Core Matrix in Three-Mode Principal Components Analysis

Published online by Cambridge University Press:  01 January 2025

Takashi Murakami ,
Jos M. F. Ten Berge  and
Henk A. L. Kiers
Takashi Murakami*
Affiliation:
Nagoya University
Jos M. F. Ten Berge
Affiliation:
University of Groningen
Henk A. L. Kiers
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Takashi Murakami, Department of Educational Psychology, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-01, Japan.
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Abstract

In three-mode Principal Components Analysis, the P ×Q ×R core matrix G can be transformed to simple structure before it is interpreted. It is well-known that, when P = 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, when P = QR − 1, G can be transformed to have nearly all elements equal to values specified a priori. A closed-form solution for this transformation is offered. Theoretical and practical implications of this simple structure transformation of G are discussed.

Keywords

three-mode principal components analysiscore matrix rotationssimple structure
Type
Original Paper
Information
Psychometrika , Volume 63 , Issue 3 , September 1998 , pp. 255 - 261
Copyright
Copyright © 1998 The Psychometric Society

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Footnotes

Constructive comments from anonymous reviewers are gratefully acknowledged.

References

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
Kiers, H. A. L. (1992). TUCKALS core rotations and constrained TUCKALS modelling. Statistica Applicata, 4, 659–667.Google Scholar
Kiers, H. A. L. (1997). Three-mode Orthomax rotation. Psychometrika, 62, 579–598.CrossRefGoogle Scholar
Kiers, H. A. L. (1998). Recent developments in three-mode factor analysis: Constrained three-mode factor analysis and core rotations. In Hayashi, C., Ohsumi, N., Yajima, K., Tanaka, Y., Bock, H.-H., & Baba, Y. (Eds.), Data Science, classification and related methods, Tokyo: Springer Verlag.Google Scholar
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.CrossRefGoogle Scholar
Kroonenberg, P. M. (1983). Three-mode principal component analysis, Leiden: DSWO.Google Scholar
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
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
Kruskal, J. B. (1989). Rank, decomposition, and uniqueness for 3-way and N-way arrays. In Coppi, R., & Bolasco, S. (Eds.), Multiway data analysis, Amsterdam: Elsevier Science Publishers.Google Scholar
MacCallum, R. C. (1976). Transformations of three-mode multidimensional scaling solution to INDSCAL form. Psychometrika, 41, 385–400.CrossRefGoogle Scholar
Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311.CrossRefGoogle ScholarPubMed