Two-way component and regression models

Important multi-way component and regression models have been described in this chapter. PARAFAC and Tucker3 are the best-known methods which can both be viewed as extensions of ordinary two-way PCA. PARAFAC is an extension in the sense that it provides the bestfitting rank R component model of a three-way data set, and Tucker3 is an extension of PCA... 

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. 

This chapter discusses two-way models and serves as a introduction for the chapters to come. A distinction is made between component models and regression models. 

Likewise, a PARAFAC model for X can be assumed and y can subsequently be regressed on the proper PARAFAC components [Bro 1997, Gel adi etui. 1998], Obviously, the number of PARAFAC components or the number of components in the Tucker3 model has to be found. This can be done, e.g., by cross-validation (see Chapter 7). The procedure of Equations (4.21)-(4.23) resembles principal component regression and can be generalized to higher-way X and two-way Y. 

In step (ii) any multi-way regression model may be used and tested. Usually, different model types (e.g. Af-PLS and Tucker3-based regression on scores model), or models with a different number of components (e.g. a two-component Af-PLS model and a three-component W-PLS model) are tested. To have complete independence of and y, the matrices involved in building the model have to be preprocessed based on interim calibration data each time step (ii) is entered. 

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