Properties of Tucker3 models

In order to decide on the size of the Tucker3 core-array the following is a useful result. If (P, Q, R) is the dimensionality vector of X (see Section 2.6), then X can be modeled exactly with a (P, Q, R) Tucker3 model [Kruskal 1989], This means that it is always useful to calculate the dimensionality vector, which can be done by matricizing X (7 x J x K) in the three different two-way matrices X(/x/x), X(Jx K) and X(KxU) (see notation section) and subsequently use two-way tools for establishing the ranks of these matrices. Methods to establish the size of the Tucker3 core-array based on this principle are given by Timmerman and Kiers [2000], 

An alternative procedure for establishing the size of the core-array is by extending the principle of cross-validation to three-way arrays which is explained in Chapter 7. 

A result which sheds light on the properties of a Tucker3 model is the following. Suppose that a (P, Q, R) Tucker3 models fits exactly a three-way array X of rank S. Then the core-array G of the Tucker3 model also has rank S. Hence, the rank of the three-way array G inherits the rank of X [Carroll et al. 1980], 

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