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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

Information

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

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