Three-way component and regression 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. 

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... 

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. 

It is necessary to decide the appropriate number of components to use in a model. The appropriate dimensionality of a model may even change depending on what the specified purpose of the model is. Hence, appropriate dimensionality of, e.g., a PARAFAC model, is not necessarily identical to the three-way pseudo-rank of the data array. Appropriate model dimensionality is not only a function of the data but also a function of the context and aim of the analysis. Hence, a suitable PARAFAC model for exploring a data set may have a rank different from a PARAFAC model where the scores are used for a subsequent regression model. 

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