Column-centered biplot

The size component which may be strongly present (as in this chromatographic application) is eliminated by the operation of double-centering. Hence, double-centered latent variables only express contrasts. In column-centered biplots one may find that one latent variable expresses mainly size and the others mainly contrasts. In general, none of the latter is a pure component of size or of contrasts. If we want to see size and some contrasts represented in a biplot, column-centering... 

Sometimes it is claimed that the double-centered biplot of latent variables 1 and 2 is identical to the column-centered biplot of latent variables 2 and 3. This is only the case when the first latent variable coincides with the main diagonal of the data space (i.e. the line that makes equal angles with all coordinate axes). In the present application of chromatographic data this is certainly not the case and the results are different. Note that projection of the compounds upon the main diagonal produces the size component. 

Fig. 31.6. Biplot of chromatographic retention times in Table 31.2, after column-centering of the data. Two unipolar axes and one bipolar axis have been drawn through the representations of the methods DMSO and methylenedichloride (CH2CI2). The projections of three selected compounds are indicated by dashed lines. TTie values read off from the unipolar axes reproduce the retention times in the corresponding columns. The values on the bipolar axis reproduce the differences between retention times.

Fig. 31.8. Biplot of chromatographic retention times in Table 31.2, after log column-centering of the data. The values on the bipolar axis reproduce the (log) ratios between retention times in the two corresponding columns.

One can also state that the log double-centered biplot shows interactions between the rows and columns of the table. In the context of analysis of variance (ANOVA), interaction is the variance that remains in the data after removal of the main effects produced by the rows and columns of the table [12], This is precisely the effect of double-centering (eq. (31.49)). 

The theory of the non-linear PCA biplot has been developed by Gower [49] and can be described as follows. We first assume that a column-centered measurement table X is decomposed by means of classical (or linear) PCA into a matrix of factor scores S and a matrix of factor loadings L ... 

The logarithmic transformation prior to column- or double-centered PCA (Section 31.3) can be considered as a special case of non-linear PCA. The procedure tends to make the row- and column-variances more homogeneous, and allows us to interpret the resulting biplots in terms of log ratios. 

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