Binary discrimination diagrams

For the Ti-V system, the data were transformed to the simplex by the log-ratio transformation. Thus, two new variables were created: log(Ti/(10$^6$-Ti-V)) and log(V/(10$^6$-Ti-V)), where 10$^6$ is the constant sum of 1 million ppm. The discriminant analysis then proceeds as described in Section 2. The results are mapped back to bivariate Ti-V space using the inverse log-ratio transformation (Equation 6). Figure 11 shows the results of the LDA of the Ti-V system, whereas Figure 12 shows the QDA results. The decision boundaries look almost identical for both cases. Besides the decision boundaries, Figures 11, 12 and subsequent figures also show the training data as well as the posterior probabilities. One of the properties of many data mining algorithms, including discriminant analysis, is the ``garbage in, garbage out'' principle: any rock that was analysed for the required elements will be classified as either IAB, MORB or OIB, even continental basalts, granites or sandstones! Therefore, it is recommended to treat the classification of samples plotting far outside the range of the training data with caution.

In contrast with the Ti-V diagram, the decision boundaries of the Ti-Zr system look quite different between LDA (Figure 13) and QDA (Figure 14). The misclassification risk of the training data (i.e., the resubstitution error) of QDA is always less than that of LDA, because the former uses more parameters than the latter. However, this does not necessarily mean that QDA will perform better on future datasets. This problem will be discussed in Section 7. For now, suffice it to say that the resubstitution error can be used to compare two binary or two ternary diagrams with each other, but not to compare the performance of QDA with LDA or of a binary with a ternary diagram.