The procedure for performing a discriminant analysis for ternary
systems is very similar to the binary case. For example, for the
Ti-Zr-Y system of Pearce and Cann (1973), we first impose the constant
sum constraint: x = Y/(Ti+Zr+Y), y = Zr/(Ti+Zr+Y) and z =
Ti/(Ti+Zr+Y). The log-ratio transformed variables are V = log(x/z)
and W = log(y/z). Note that this transformation only takes care of the
diagrammatic constraint x+y+z = 1. Strictly speaking, it does not
account for the physical constraint Ti+Zr+Y+(all other elements) =
100%. However, Ti+Zr+Y only amount to at most a few percent of
typical basalt compositions, thereby greatly reducing the impact of
this second type of constant sum. It would be possible to correct for
the physical constraint, for example by performing a discriminant
analysis on the following three variables: log(Ti/(106-Ti-Zr-Y)),
log(Zr/(106-Ti-Zr-Y)), and log(Y/(106-Ti-Zr-Y)). However, the
results of such an analysis can no longer be plotted on a ternary
diagram. In practice, neglecting the physical constant sum constraint
does not severely affect the performance of the classification in this
case.
Figures 15 and
16 show the results of both LDA and QDA
transformed back to the Ti-Zr-Y ternary diagram.
The raw variables of many discrimination diagrams are
multiplied by constants to improve the spread of the data. This is
equivalent to adding constants to the log-ratio transformed variables.
Either transformation does not affect the discriminant analysis. As
noted by Pearce and Cann (1973), the Ti-Zr-Y diagram is quite
good at identifying OIBs, but cannot distinguish MORBs from IABs. The
training data of the latter substantially overlap and their
resubstitution errors are quite high. The posterior probabilies of
the training data are low (
0.5 on Figure 16).
This is also the case for the Nb-Zr-Y system of Meschede (1986) (Figures 17 and 18). The high misclassification rate of both the Ti-Zr-Y and Nb-Zr-Y diagrams is largely caused by the large spread of IAB compositions, which is likely caused by the complexity of magma generation underneath island arcs, where mixing of multiple melt sources often occurs. The Th-Ta-Hf system of Wood (1980), however, achieves a much better separation between the three tectonic affinities (Figures 19 and 20). The decision boundaries of the QDA (Figure 20) are much more complicated than those of the LDA (Figure 19), without substantially improving the overall misclassification risk. Therefore, adding the extra parameters (covariances) was probably not worthwhile (see Section 7).