Conclusions

This paper revisited the observation by Butler and Woronow (1986) that traditional statistical analyses of geochemical data is flawed because it ignores the effects of data-closure. Since the work of Aichison (1982, 1986), it is possible to account and correct for the constant-sum constraint by transforming the data to log-ratio space. Butler and Woronow (1986) then went on to do a principal component analysis. The present paper instead uses the log-ratio method for the related, albeit different technique of discriminant analysis.

First, a number of popular discrimination diagrams were revisited. Many of these historically important diagrams were not derived from a real discriminant analysis sensu Fisher (1936), but were instead obtained by drawing decision boundaries by eye. A positive side-effect of this is that the resulting diagrams are much less affected by the constant-sum constraint discussed before. A negative consequence remains, however, that all statistical rigor is lost. Nevertheless, it is not the intention of this paper to discredit the discriminantion diagrams of Pearce and Cann (1973), Wood (1980), Shervais (1982), Meschede (1986) and others. Rather, the paper merely explains how to perform discriminant analysis of geochemical data in a statistically more rigorous manner.

After revisiting these historically important discrimination diagrams, an exhaustive exploration was done of all possible linear and quadratic discriminant analyses using a dataset of 756 IABs, MORBs and OIBs. The best overall performance was given by the Si-Ti-Sr (LDA) and Na-Nb-Sr (QDA) systems. The best LDA and QDA using only immobile elements are the Ti-V-Sc and Ti-V-Sm systems, respectively. One of the features of the old discrimination diagrams was a field of ``not classifiable'' compositions. If an unkown sample plotted outside the pre-defined fields tectonic affinity fields, it would be labeled as ``other''. The revisited discriminant analyses discussed above do not have this feature. On the one hand, it might be considered a positive thing that the method no longer ``breaks down'' when encountering ``difficult'' samples. On the other hand, one might wonder what would happen if we were to plot a rock of very different affinity on the discrimination diagrams. To mitigate this ``garbage in, garbage out'' effect, we might want to opt for a hybrid solution, and only accept results for data that plot inside the old (hand-drawn) affinity fields, or within the clouds of training data shown on all discrimination diagrams in this paper (Figures 11 - 22 and 31 - 35).

Historically, discrimination diagrams and discriminant analysis have been the method of choice for geochemists to statistically classify rocks of different environments. However, discriminant analysis is not the only ``data mining'' method that can be used for this purpose. For examples, Vermeesch (in press) introduces classification trees as a potentially very useful tool for tectonic classification. Some of the advantages of classification trees over discriminant analysis are that the former (a) do not make any distributional assumptions, (b) can handle an unlimited number of geochemical species, isotopic ratios or other features, while still being easily interpretable as a two-dimensional graph and (c) can still be used if some of these features are not available. Two trees were constructed using the same training data as in the present paper: one tree using 51 elements and isotopic ratios and one using only 23 High Field Strength (HFS) elements and isotopic ratios. Both trees were evaluated with the same test data used on the discrimination diagrams. The full tree misclassifies 23 and the HFS tree 41 out of the 182 test data. Presently, the Si-Ti-Sr and Eu-Lu-Sr LDAs, and the Na-Nb-Sr and Ti-V-Sm QDAs introduced in this paper still outperform the trees of Vermeesch (in press). However, this is likely to change for trees created from a larger training set. Whereas discriminant analysis does not gain much from using exceedingly large training sets, classification trees continue to improve with growing sets of training data. Furthermore, the classification trees succeeded in classifying all 182 test data, even for samples missing several geochemical features. None of the discrimination diagrams achieved this. Therefore, it is probably a good idea to use a combination of both methods.