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Some Additional Results on Principal Components Analysis of Three-Mode Data by Means of Alternating Least Squares Algorithms

Published online by Cambridge University Press:  01 January 2025

Jos M. F. ten Berge ,
Jan de Leeuw  and
Pieter M. Kroonenberg
Jos M. F. ten Berge*
Affiliation:
University of Groningen
Jan de Leeuw
Affiliation:
University of Leiden
Pieter M. Kroonenberg
Affiliation:
University of Leiden
*
Requests for reprints should be sent to Jos M. F. ten Berge, Subfakulteit Psychologie, RU Groningen, Grote Markt 32, 9712 HV Groningen, THE NETHERLANDS.
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Abstract

Kroonenberg and de Leeuw (1980) have developed an alternating least-squares method TUCKALS-3 as a solution for Tucker's three-way principal components model. The present paper offers some additional features of their method. Starting from a reanalysis of Tucker's problem in terms of a rank-constrained regression problem, it is shown that the fitted sum of squares in TUCKALS-3 can be partitioned according to elements of each mode of the three-way data matrix. An upper bound to the total fitted sum of squares is derived. Finally, a special case of TUCKALS-3 is related to the Carroll/Harshman CANDECOMP/PARAFAC model.

Keywords

partitioning of least-squares fitrank-constrained regressionCandecompParafac
Type
Original Paper
Information
Psychometrika , Volume 52 , Issue 2 , March 1987 , pp. 183 - 191
Copyright
Copyright © 1987 The Psychometric Society

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