Log column-centering

The transformation by log column-centering consists of taking logarithms followed by column-centering. The choice of the base of the logarithms has no effect on the interpretation of the result, but decimal logs will be used throughout. 

In this case it is required that the original data in X are strictly positive. The effect of the transformation appears from Table 31.6. Column-means are zero, while column-standard deviations tend to be more homogeneous than in the case of simple column-centering in Table 31.4 as can be seen by inspecting the corresponding values for Na and Cl. 

With log column-centering we obtain unipolar axes by substituting eq. (31.46) in eq. (31.35)  

Atmospheric data from Table 31.1, after log column-centering. 

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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.

A bipolar axis through columns j and/ can be interpreted in the same way as in the log column-centered case (eq. (31.48)) since the terms nij and cancel out. The first (close to horizontal) axis between DMSO and ethanol represents the (log)ratios of the corresponding retention times. They can be read off by vertical projection of the compounds on this scale. Note that the scale is divided logarithmically. In the same way, one can read off the (log)ratios of methylenedichloride and ethanol from the second (close to vertical) axis on Fig. 31.9. Graphical estimation of these contrasts for the dimethylamine-N02 substituted chalcone produces 9.5 on the DMSO/ethanol axis and 6.2 on the methylenedichloride/ ethanol axis of Fig. 31.9. The exact ratios from Table 31.2 are 10.00 and 6.14, respectively. 

Preprocessing by log double-centering consists of first taking logarithms, and then to center the data both by rows and by 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 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. 

Table 11.1 gives the results from the application of PCA on column mean-centered data, on column autoscaled data, and on log-transformed column mean-centered data. Using five PCs, the amount of variance explained was 84.0, 46.1, and 69.6%, respectively. The results for the column mean-centered data were nearly identical to those obtained for the nonmean-centered raw data. The reason for this is that the means of... 

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