Multilinear modeling bilinear models

When spectroscopic intensity is linear in functions of each of k independent variables, a multilinear model can be fit to the k-v/ay array of data. For example, consider absorption measurements made on a specimen containing F components, with wavelength and some environmental variable, such as pH, being the two experimental variables. If the environmental variable alters the relative concentrations of the different components in the specimen without affecting their absorption spectra, then the expected absorbance is described by the bilinear model... 

The simplest multilinear models are bilinear models, which can be written as... 

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. 

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