Varimax rotated factor matrices

Table 1.6 Principal component (PC) analysis of descriptors of chemical structures for a set of diverse organic compounds (a) > 80% of the explained variance is expressed in the first four PCs (b) the loadings of the original descriptor variables in the VARIMAX rotated factor matrix reflect the grouping of the parameters (i.e. high loadings in the same PC indicate high intercorrelations between the descriptors).

Table 2. Varimax rotated factor matrix after rotation with Kaiser normalization, for the group of 6 - 12-year-olds (loadings >. 45 are underlined)...

The columns of V are the abstract factors of X which should be rotated into real factors. The matrix V is rotated by means of an orthogonal rotation matrix R, so that the resulting matrix F = V R fulfils a given criterion. The criterion in Varimax rotation is that the rows of F obtain maximal simplicity, which is usually denoted as the requirement that F has a maximum row simplicity. The idea behind this criterion is that real factors should be easily interpretable which is the case when the loadings of the factor are grouped over only a few variables. For instance the vector f, =[000 0.5 0.8 0.33] may be easier to interpret than the vector = [0.1 0.3 0.1 0.4 0.4 0.75]. It is more likely that the simple vector is a pure factor than the less simple one. Returning to the air pollution example, the simple vector fi may represent the concentration profile of one of the pollution sources which mainly contains the three last constituents. 

PAHs introduced in Section 34.1. A PCA applied on the transpose of this data matrix yields abstract chromatograms which are not the pure elution profiles. These PCs are not simple as they show several minima and/or maxima coinciding with the positions of the pure elution profiles (see Fig. 34.6). By a varimax rotation it is possible to transform these PCs into vectors with a larger simplicity (grouped variables and other variables near to zero). When the chromatographic resolution is fairly good, these simple vectors coincide with the pure factors, here the elution profiles of the species in the mixture (see Fig. 34.9). Several variants of the varimax rotation, which differ in the way the rotated vectors are normalized, have been reviewed by Forina et al. [2]. 

Tab. 9-5. Factor loading matrix of the Flettstedt soil profile after varimax rotation (loadings less than 0.5 in absolute value are set to zero)...

Table 10 The rotated factor loading matrix, retaining only the first two factors and using varimax...

A special application of FA in QS AR work is its use as a data pre-processing step in multiple regression [1, 50, 51] and other analyses [212], The ta matrix to be analysed then contains the biological potency to be considered as well as all molecular parameters to be checked as descriptors. After varimax rotation to obtain a simple stmcture the following can be deduced from the factor pattern ... 

These criteria lead to different numeric transformation algorithms. The main distinction between them is orthogonal and oblique rotation. Orthogonal rotations save the structure of independent factors. Typical examples are the varimax, quartimax, and equi-max methods. Oblique rotations can lead to more useful information than orthogonal rotations but the interpretation of the results is not so straightforward. The rules about the factor loadings matrix explained above are not observed. Examples are oblimax and oblimin methods. 

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