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Transforming Three-Way Arrays to Maximal Simplicity

Published online by Cambridge University Press:  01 January 2025

Roberto Rocci  and
Jos M. F. ten Berge
Roberto Rocci
Affiliation:
University of Molise
Jos M. F. ten Berge*
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Jos M.F. ten Berge, Heÿmans Institute of Psychological Research, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, THE NETHERLANDS. E-Mail: j.m.f.ten.berge@ppsw.rug.nl
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Abstract

Transforming the core array in Tucker three-way component analysis to simplicity is an intriguing way of revealing structures in between standard Tucker three-way PCA, where the core array is unconstrained, and CANDECOMP/PARAFAC, where the core array has a generalized diagonal form. For certain classes of arrays, transformations to simplicity, that is, transformations that produce a large number of zeros, can be obtained explicitly by solving sets of linear equations. The present paper extends these results. First, a method is offered to simplifyJ ×J × 2 arrays. Next, it is shown that the transformation that simplifies anI ×J ×K array can be used to also simplify the (complementary) arrays of order (JK −I) ×J ×K, of orderI × (IK −J) ×K and of orderI ×J × (IJ −K). Finally, the question of what constitutes the maximal simplicity for arrays (the maximal number of zero elements) will be considered. It is shown that cases of extreme simplicity, considered in the past, are, in fact, cases of maximal simplicity.

Keywords

three-way PCAcore arraysimplicity
Type
Articles
Information
Psychometrika , Volume 67 , Issue 3 , September 2002 , pp. 351 - 365
Copyright
Copyright © 2002 The Psychometric Society

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References

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.CrossRefGoogle Scholar
Gurden, S.P., Westerhuis, J.A., Bijlsma, S., & Smilde, A. (2001). Modelling of spectroscopic batch process data using grey models to incorporate external information. Journal of Chemometrics, 15, 101–121.3.0.CO;2-V>CrossRefGoogle Scholar
Harshman, R.A. (1970). Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multi-mode factor analysis. UCLA Working Papers in Phonetics, 16, 1–84.Google Scholar
Horn, R.A., & Johnson, C.R. (1993). Matrix analysis. Cambridge, MA: Cambridge University Press.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. (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 (pp. 563–574). Tokyo, Japan: Springer-Verlag.CrossRefGoogle Scholar
Kiers, H.A.L. (1998). Three-way SIMPLIMAX for oblique rotation of the three-mode factor analysis core to simple structure. Computational Statistics & Data Analysis, 28, 307–324.CrossRefGoogle Scholar
Kiers, H.A.L. (2000). Towards a standard notation and terminology in multiway analysis. Journal of Chemometrics, 14, 105–122.3.0.CO;2-I>CrossRefGoogle Scholar
Kiers, H.A.L., ten Berge, J.M.F., & Rocci, R. (1997). Uniqueness of three-mode factor models with sparse cores: The 3 × 3 × 3 case. Psychometrika, 62, 349–374.CrossRefGoogle 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
Lutkepohl, H. (1996). Handbook of matrices. New York, NY: John Wiley & Sons.Google Scholar
Murakami, T. (1999). A simple core does not necessarily facilitate interpretations in three-mode principal components analysis. Nagoya, Japan: Nagoya University.Google Scholar
Murakami, T., ten Berge, J.M.F., & Kiers, H.A.L. (1998). A case of extreme simplicity of the core matrix in three-mode principal components analysis. Psychometrika, 63, 255–261.CrossRefGoogle Scholar
Rocci, R. (2001). Core matrix rotation to natural zeros in three-mode factor analysis. In Borra, S., Rocci, R., Vichi, M., & Schader, M. (Eds.), Advances in classification and data analysis (pp. 161–168). Berlin, Germany: Springer-Verlag.CrossRefGoogle Scholar
ten Berge, J.M.F. (1991). Kruskal's polynomial for 2 × 2 × 2 arrays and a generalization to 2 ×n ×n arrays. Psychometrika, 56, 631–636.CrossRefGoogle Scholar
ten Berge, J.M.F., & Kiers, H.A.L. (1999). Simplicity of core arrays in three-way principal component analysis and the typical rank ofP ×Q × 2 arrays. Linear Algebra and Its Applications, 294, 169–179.CrossRefGoogle Scholar
ten Berge, J.M.F., Kiers, H.A.L., Murakami, T., & van der Heijden, R. (2000). Transforming three-way arrays to multiple orthonormality. Journal of Chemometrics, 14, 275–284.3.0.CO;2-R>CrossRefGoogle Scholar
Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311.CrossRefGoogle ScholarPubMed