As calculated in the previous section, there are 990 ways to choose
three out of 11 major oxides, and 14,190 ways to choose three out of
45 major, minor and trace elements. Although all these possibilities
were explored in the framework of this research, it is not practical
to visually show all the results in this paper, even using the highly
compact matrix visualization. Therefore, only an (important) subset is
shown of all ternary diagrams using Ti. As discussed before, many of
the most effective bivariate discriminant analyses use Ti. In
addition to being an excellent discriminator, Ti is also highly
immobile, in contrast with for example Sr, which is another powerful
discriminator. For these reasons, only the results of ternary LDAs
and QDAs using Ti are shown in Figures 27,
28, 29 and 30.
The resubstitution errors of all 14,190 ternary LDAs (i.e., not only
those using Ti) were ranked to find the best combinations of elements.
Table 1 shows the 100 best LDAs. Only those
diagrams for which at least 100 IABs, 100 MORBs and 100 OIBs of the
training data had been analysed for all three elements were used.
2,333 out of 14,190 possible combinations fulfilled this requirement.
The best ternary LDA uses the Si-Ti-Sr system. It has an overall
resubstitution error of 6.2%, (2.7% for IABs, 2.8% for MORBs and
2.7% for OIBs), using nearly all the training data (221/256 IABs,
211/241 MORBs and 192/259 OIBs). Figure 31 shows
the Si-Ti-Sr LDA in detail. Another powerful ternary diagram using
minor and trace elements is the Eu-Lu-Sr system, which ranks third
among all the ternary LDAs of Table 1. This
diagram is shown on Figure 32. Many if not most
of the best performing ternary QDAs use Sr as one of the elements.
However, as discussed before, Sr is quite mobile during processes of
alteration and metamorphism, potentially affecting the reliability of
the discrimination diagrams using it. The Ti-V-Sc diagram, ranking
28th in Table 1, suffers much less from this
problem and still has an overall misclassification rate of only 10.4%
while using 374 out of 756 training data. Figure 33
shows the Ti-V-Sc diagram in detail. Table
2 lists the best performing (lowest
resubstitution error) ternary LDAs, using the following 25
incompatible elements: Ti, La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm,
Yb, Lu, Sc, V, Cr, Y, Zr, Nb, Hf, Ta, Pb, Th, and U.
Table 3 shows the 100 best-performing ternary QDAs. The Na-Nb-Sr system performs the best, with an overall resubstitution error of only 5%. As shown on Figure 34, this diagram misclassifies only 22 out of 425 training samples. However, Na is a very mobile element and not much faith can be had in a classification that uses it for basalt samples that are not perfectly fresh. The Ti-V-Sm diagram (Figure 35) is the best-performing QDA using only relatively immobile elements. It is ranked 33rd in Table 3. Notice that both for LDA and QDA, the best-performing ternary discrimination diagrams using immobile elements contain both Ti and V, apparently confirming the effectiveness of the approach used by Shervais (1982). The latter author selected Ti and V for mostly petrological reasons, while the present paper arrived at the same elements using an entirely statistical method. The compatibility of both approaches lends more credibility to the results. Table 4 lists the best performing LDAs using ternary combinations of the 25 incompatible elements listed in the previous paragraph for which at least 100 training samples of each tectonic affinity were represented.