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Uniqueness of Three-Mode Factor Models with Sparse Cores: The 3 × 3 × 3 Case

Published online by Cambridge University Press:  01 January 2025

Henk A. L. Kiers ,
Jos M. F. ten Berge  and
Roberto Rocci
Henk A. L. Kiers*
Affiliation:
University of Groningen
Jos M. F. ten Berge
Affiliation:
University of Groningen
Roberto Rocci
Affiliation:
University la Sapienza, Rome
*
Requests for reprints should be sent to Henk A. L. Kiers, Grote Kruisstraat 2/1, 9712 TS Groningen, THE NETHERLANDS.
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Abstract

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.

Keywords

three-way methodsPARAFAC
Type
Original Paper
Information
Psychometrika , Volume 62 , Issue 3 , September 1997 , pp. 349 - 374
Copyright
Copyright © 1997 The Psychometric Society

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Footnotes

This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author. Part of this research has been presented at the first conference on ThRee-way methods In Chemistry (TRIC), a meeting of Psychometrics and Chemometrics, Epe, The Netherlands, August 1993. The authors are obliged to Age Smilde for stimulating this research, and two anonymous reviewers for many helpful suggestions.

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