Historical introduction to multi-way models

A short historical introduction of multi-way analysis is given here. Most of this historical work comes from psychometrics and the most important references related to the models presented in the following sections are given. The pioneering work began in the middle of the twentieth century and ended around 1980, when the most important multi-way models and their algorithms had been introduced. 

Some of the first ideas on multi-way analysis were published by Raymond Cattell [1944,1952], Thurstone s principle of parsimony states that a simple structure should be found to describe a data matrix or its correlation matrix with the help of factors [Thurstone 1935], For the simultaneous analysis of several matrices together, Cattell proposed to use the principle of parallel proportional profiles [Cattell 1944], The principle of parallel proportional profiles states that a set of common factors should be found that can be fitted with different dimension weights to many data matrices at the same time. This is the same as finding a common set of factors for a stack of matrices, a three-way array. To quote Cattell  

The principle of parsimony, it seems should not demand Which is the simplest set of factors reproducing this particular correlation matrix but rather Which set of factors will be most parsimonious at once to this and other matrices considered together  

The criterion is then no longer that the rotation shall offer fewest factor loadings for any matrix but that it shall offer fewest dissimilar (and therefore fewest total) loadings in all matrices together. 

Multi-way Analysis With Applications in the Chemical Sciences. A. Smilde, R. Bro and R Geladi 2004 John Wiley Sons, Ltd ISBN 0-471-98691-7 

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