Some properties of three-way component models

This chapter discusses some properties of three-way component models. First of all, hierarchical relationships exist between Tucker, constrained Tucker and PARAFAC models. These relationships have consequences for fit values, model complexity and model selection. Moreover, if there are no hierarchical relationships between different component models, then comparing fit values becomes problematic. 

In Section II we presented the standard general multilinear models, of which the bilinear and the PARAFAC and Tucker2 (T2) trilinear models are most important in spectroscopy. These models contain no information about the specimen except the linear dependence of spectral intensity on functions of each of the independent variables. However, some properties of the specimen are known, and a model that incorporates these known properties is preferred to one that does not. This is particularly true when the model is indeterminate without side conditions. In this section we discuss three settings for the application of knowledge about the specimen identifiable bilinear and T2 submodels, penalized general multilinear models, and submodels in which the dependence of the expected intensity from some components for some ways has a specific mathematical form. 

The basic hypothesis of a QSAR model is that the activity (or effect or property) can be put in relationship with the chemical, using some parameters to describe the chemical. Thus, the three main components of the QSAR model are the activity to be modeled, the chemical information, and the way to establish a link between these two components. For this, we need some suitable ways to describe the chemical and a good mathematical algorithm. 

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