Multi-element discriminant function analysis

As illustrated by Figure 2, LDA offers the possibility of projecting a dataset onto a subspace of lower dimensionality. As explained in Section 2 this procedure is related to, but quite different from PCA. Therefore, it is somewhat puzzling why Butler and Woronow (1986) performed a PCA on a dataset of Zr, Ti, Y and Sr analyses of oceanic basalts. These authors were the first to note the significance of the constant sum constraint to the problem of tectonic discrimination, but they stopped short of doing a full discriminant analysis. Figure 21 does exactly that. The two linear discriminant functions (ld1 and ld2) are:

$$=$$ ld1 -0.016 log(Zr/Ti) - 2.961 log(Y/Ti) + 1.500 log(Sr/Ti)

$$=$$ ld2 -1.474 log(Zr/Ti) + 2.143 log(Y/Ti) + 1.840 log(Sr/Ti) (7)

Note that the training data cluster quite well, that the clusters are of approximately equal size, and that they are well separated, resulting in a misclassification rate of only 8%.

Butler and Woronow (1986) were the first ones to note the potential importance of data-closure in the context of tectonic discimination of oceanic basalts. However, as said before, they did not use the log-ratio transformation to improve discriminant analysis, but performed a PCA instead, the implications of which are unclear. On the other hand, Pearce (1976) did perform a traditional multi-element discriminant analysis, but since his paper predated the work of Aitchison (1982, 1986), he was unaware of the effects of closure. Figure 22 shows the results of a re-analysis of the major element abundances (except FeO) used by Pearce (1976). The two linear discriminant functions are:

$$= \mbox{0.555 log(TiO$_2$/SiO$_2$) + 3.822 log(Al$_2$O$_3$/SiO$_2$) + 0.522 log(CaO/SiO$_2$) +}$$ ld1

$$ \mbox{1.293 log(MgO/SiO$_2$) - 0.531 log(MnO/SiO$_2$) - 0.145 log(K$_2$O/SiO$_2$) -}$$

$$ \mbox{0.399 log(Na$_2$O/SiO$_2$)}$$

$$= \mbox{3.796 log(TiO$_2$/SiO$_2$) + 0.008 log(Al$_2$O$_3$/SiO$_2$) - 2.868 log(CaO/SiO$_2$) +}$$ ld2

$$ \mbox{0.313 log(MgO/SiO$_2$) + 0.650 log(MnO/SiO$_2$) + 1.421 log(K$_2$O/SiO$_2$) -}$$

$$ \mbox{3.017 log(Na$_2$O/SiO$_2$)}$$ (8)

This discriminant analysis performs about as well as the Ti-Zr-Y-Sr diagram of Figure 21, although it uses many more elements. The benefits of multi-element LDA are clearly a decrease in misclassification rate. This comes at the expense of interpretability, because the linear discriminant functions (ld1 and ld2) have no easily interpretable meaning, in contrast with their binary and ternary counterparts.