Bipolar axis

Fig. 31.3. (a,b) Reproduction of distances D and angular distances 0 in a score plot (a = 1) or loading plot (p = 1) in the common factor-space (c,d) Unipolar axis through the representation of a row or column and through the origin 0 of space. Reproduction of the data X is obtained by perpendicular projection of the column- or row-pattern upon the unipolar axis (a + P = 1). (e,0 Bipolar axis through the representation of two rows or two columns. Reproduction of differences (contrasts) in the data X is obtained by perpendicular projection of the column- or row-pattern upon the bipolar axis (a + P = 1). 

A bipolar axis is defined by two rows S and s,- in Fig. 31.3e. If we project a column upon this bipolar axis, we reconstruct the difference between two elements in X. This follows readily from eq. (31.22) ... 

This bipolar axis defines a contrast between the row-variables i and i. Note that a contrast involves the difference of two quantities. 

This bipolar axis defines a contrast between the column-variables j and f. 

In Fig. 31.4b we have constructed a bipolar axis through Cl and Si. Perpendicular projections of the wind directions upon this axis are in the same order as the... 

Fig. 31.4. (a) Biplot in which the concentrations of an atmospheric trace element (Cl) are reconstructed by perpendicular projection upon a unipolar axis, (b) Biplot in which the differences (contrasts) between two atmospheric trace elements (Cl, Si) are reproduced by perpendicular projection upon a bipolar axis. 

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.

Finally, we have constmcted a bipolar axis through DMSO and methylenedichloride. By perpendicular projection of the centers of the circles upon this bipolar axis we obtain the differences in retention times obtained respectively with DMSO and with methylenedichloride. Using a similar reasoning as developed above for unipolar axes we can perform a substitution of eq. (31.42) in eq. (31.38) which leads to ... 

This shows that the bipolar axis reproduces the difference Xy - Xy- between method j and/ of the table minus a constant (/n - m ). This bipolar axis defines a clear contrast between the N02-substituted compounds and the others. 

Consequently, the bipolar axis represents the (log) ratio of columns j and / in the original data table up to the term mj - mj- which is a constant for the given bipolar axis. Note that the axis which represents the ratio of retention times with DMSO/methylenedichloride is also divided according to a logarithmic scale. 

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. 

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. 

Unipolar and bipolar axes have been discussed in Section 31.2. Briefly, a unipolar axis is defined by the origin and the representation of a row or column. A bipolar axis is drawn through the representations of two rows or through the representations of two columns. Projections upon unipolar axes reproduce the values in the transformed data table. Projections upon bipolar axes reproduce the contrasts (i.e. differences) between values in the data table. 

In the bipolar droplet, the rotation symmetry axis is referred to as the bipolar axis. The droplet... 

---