Target Factor Analyses, TFA

We continue considering multivariate data sets, e.g. a series of spectra measured as a function of time, reagent addition etc. In short, a matrix of 

We have recognised in the preceding Chapter 5.1 Factor Analysis, that this path is concentrated in a much lower dimensional sub-space. Usually, for an nc component system, the sub-space has nc dimensions e.g. for a two component system, all spectra lie in a plane. Recall that, if the system is closed, the dimension of the sub-space can be further reduced by meancentring. 

To start with, we do not know the spectra A of the components in the system under investigation. Factor Analysis delivers an orthonormal system of axes V that defines the sub-space of Y and A in an optimal way. Importantly, this is done automatically, and there is no input from the chemist regarding the components in the system or their spectra. 

The basic idea of Target Factor Analysis is very simple. In order to test whether a certain compound is taking part in the process, whether its spectrum exists in the measurement, we test whether that spectrum lies in V. If such a test spectrum is outside V, there is no doubt that the component does not take part in the process under investigation. If it is in the subspace, we cannot positively conclude that the species is there the test spectrum could be a linear combination of the existing spectra. 

A typical application can be found in chromatography. A group of components elute in a strongly overlapping peak cluster. We suspect that a particular chemical, for which we know the spectrum, might be in the unknown mixture, but due to overlap, its spectrum does not appear pure in the matrix Y. 

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