Some Additional Results on Principal Components Analysis of Three-Mode Data by Means of Alternating Least Squares Algorithms
- 1 March 1987
- journal article
- Published by Cambridge University Press (CUP) in Psychometrika
- Vol. 52 (2) , 183-191
- https://doi.org/10.1007/bf02294233
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
This publication has 5 references indexed in Scilit:
- A Generalization of Kristof's Theorem on the Trace of Certain Matrix ProductsPsychometrika, 1983
- Principal Component Analysis of Three-Mode Data by Means of Alternating Least Squares AlgorithmsPsychometrika, 1980
- Some mathematical notes on three-mode factor analysisPsychometrika, 1966
- On best approximate solutions of linear matrix equationsMathematical Proceedings of the Cambridge Philosophical Society, 1956
- The approximation of one matrix by another of lower rankPsychometrika, 1936