Principal Component Analysis and Related Methods

One definition of factor analysis is given by Malinowsky in [28] Factor analysis is a multivariate technique for reducing matrices of data to their lowest dimensionality by the use of orthogonal factor space and transformations that yield predictions and/or recognizable factor . 

In this expression S is a p x p diagonal matrix whose elements are the square roots of the eigenvalues, sorted in decreasing order of magnitude. 

Usually, not all of the p factors are needed to explain the features of the spectra and only ny factors can report data variation other than noise. Then the factor model can be compressed into  

Once the number of meaningful factors, is determined, a transformation step can allow the recognition of chemical factors and in turn find the concentration of the different chemical components. For this to be accomplished, some different techniques are available and discussed in the literature [28]. 

---