Relationships between three-way component models

Three-way models and their properties were introduced in the previous chapters. It may be difficult for the newcomer to choose between them, just as it is difficult to choose the proper classification or regression method for two-way data. There are many three-way component and regression models to choose from. In order to decide which model to use in which situation, it is important to have a good understanding of the differences between and similarities of the models. The purpose of this chapter is to provide such an understanding. 

An important question is how the PARAFAC and Tucker3 models are related. PARAFAC models provide unique axes, while Tucker3 models do not. A Tucker model may be transformed (rotated) and simplified to look more like PARAFAC models. This can sometimes be done with little or no loss of fit. There is a hierarchy e.g. within the family of Tucker models, Tucker3, Tucker2 and Tuckerl, which is worth studying in more detail. PARAFAC models may be difficult or impossible to fit due to so-called degeneracies (Section 5.4), in which case a Tucker3 model is usually a better a choice. Further, the statistical properties of the data - noise and systematic errors - also play an important role in the choice of model. 

There is not always a clear answer or a strictly defined proper choice of a three-way model for a given application. Sometimes more than one three-way model has to be tried in order to find the proper one. In many cases, the models appear equally good and the selection may be based on practical criteria such as algorithm precision or speed. Knowing the palette of three-way models along with their properties and relationships serves as a basis for making objective choices. 

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 

Multi-way Analysis With Applications in the Chemical Sciences 

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