Log double-centering

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

It is assumed that the original data in X are strictly positive. As is evident from Table 31.7 both the row-means m and the column-means of the transformed table Z are equal to zero. 

The biplot of Fig. 31.9 shows that both the centroids of the compounds and of the methods coincide with the origin (the small cross in the middle of the plot). The first two latent variables account for 83 and 14% of the inertia, respectively. Three percent of the inertia is carried by higher order latent variables. In this biplot we can only make interpretations of the bipolar axes directly in terms of the original data in X. Three prominent poles appear on this biplot DMSO, methylene-dichloride and ethylalcohol. They are called poles because they are at a large distance from the origin and from one another. They are also representative for the three clusters that have been identified already on the column-standardized biplot in Fig. 31.7. 

Atmospheric data from Table 31.1, after log double-centering 

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. 

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The first bipolar axis (DMSO/ethanol) accounts for the contrast between compounds with NO2 substitutions and those without. Compounds with a NO2 substituent systematically obtain higher scores on this bipolar axis than others. The second bipolar axis (methylenedichloride/ethanol) seems to produce an order of the substituents according to their electronic properties. To emphasize this point we have reproduced the log double-centered biplot again in Fig. 31.10. The dashed line near the middle separates the class of NO2 substituted chalcones from the other compounds. Further, we have joined substituents by line segments according to the sequence CF3, F, H, methyl, ethyl, I -propyl, t-butyl, methoxy, phenyl and di-methylamine. The electronic properties of these substituents vary progressively from electron acceptors to electron donors [ 11 ] in accordance with their scores on the second bipolar axis. 

Fig. 31.9. Biplot of chromatographic retention times in Table 31.2, after log double-centering. Two bipolar axes have been drawn through the representation of the methods DMSO, methylenedichloride...

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 transformation by log double-centering has received various names among which spectral mapping [13], logarithmic analysis [14], saturated RC association model [15], log-bilinear model [16] and spectral map analysis or SMA for short [17]. 

The effect of double-closure is shown in Table 31.8. For convenience, we have subtracted a constant value of one from all the elements of Z in order to emphasize the analogy of the results with those obtained by log double-centering in Table 31.7. The marginal means in the table are average values for the relative deviations from expectations and thus must be zero. 

The analysis of Table 31.2 by CFA is shown in Fig. 31.11. As can be seen, the result is very similar to that obtained by log double-centering in Figs. 31.9 and 31.10. The first latent variable expresses a contrast between NO2 substituted chalcones and the others. The second latent variable seems to be related to the electronic properties of the substituents. The contributions of the two latent variables to the total inertia is 96%. The double-closed biplot of Fig. 31.11 does not allow a direct interpretation of unipolar and bipolar axes in terms of the original data X. The other rules of interpretation are similar to those of the log double-centered biplot in the previous subsection. Compounds and methods that seem to have moved away from the center and in the same directions possess a positive interaction (attraction). Those that moved in opposite directions show a negative interaction (repulsion). 

Fig. 31.15. Scree-plot, representing the residual variance V as a function of the number of factors r that has been extracted. The diagram is based on a factor analysis of Table 31.2 after log double-centering. A break point occurs after the second factor, which suggests the presence of only two structural factors, the residual factors being attributed to noise and artefacts in the data.

Validation results obtained from factor analysis of Table 31.2, containing the retention times of 23 chalcones in 8 chromatographic methods, after log double-centering and global normalization. The results are used in the Malinowski s f-test and in cross-validation by PRESS. 

Fig. 37.6. PLS biplot obtained from the pharmacological data in Table 37.9, after log double-centering and analysis by two-block PLS [56]. Circles represent 17 reference neuroleptic compounds, squares denote tests. Areas of circles and squares are proportional to the potencies of the compounds and the sensitivities of the tests, respectively. Reproduced with permission of E.J. Kaijalainen.

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. 

In the early method proposed by Broto et al., 222 of these atom-centered clusters were verified by a test set of 1868 measured log Poct values of structures containing no intramolecular hydrogen bonds. The residual standard deviation by regression analysis was 0.43 log units. In a set of 500 solutes thought to contain intramol-H-bonds, the standard deviation doubled, which the authors attributed to variation in FI-bond strength. 

The result is a semicircle having a radius equal to R 2, with its center on the x axis and displaced from the origin of coordinates by R + RJ2. Each point on the semicircle in Fig. lOG represents a measurement at a given frequency. At very high frequencies, the fara-daic resistance is effectively shorted out by the double-layer capacitance, leaving the solution resistance in series as the only measured quantity. At very low frequency the opposite occurs, namely, the capacitive impedance becomes very high and one measures the sum of the two resistors in series. 

Metallic americium has a face-centered cubic structure at its melting point and a double hexagonal closed-packed structure at temperatures below its melting point. The isotope americium-241 emits a-particles and y-rays in its radioactive decay, and is a source of y-radiation, used to measure the thickness of metals, coatings, degree of soil compaction, sediment concentration, and so on. The same isotope, mixed with beryllium, is used as a neutron source in oilwell logging and other applications. Americium-241... 

Figure 17.5 shows the size distributions of the different fractions (LCFq i, LCFq-o.i, and LCFo.i i). The fillers size distributions have been determined by light scattering with a particles size analyzer. LCFo i presents a double granulometric distribution, a population centered on 50 pm and another one on 700 pm. After sieving, we obtain two log-normal curves, i.e., two homogeneous populations. A first distribution is centered on 50 pm and another one oti 630 pm. 

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