An exhaustive exploration of binary and ternary discriminant analyses

Some of the popular discrimination diagrams discussed in Section 5 use a choice of elements that is based on petrological reasons (e.g., Shervais, 1982). However, more often the reasons are entirely statistical, i.e. those features are used that result in a ``good'' classification. If a database of N elements is used, there are $\binom{N}{2}$=N(N-1)/2 possible binary diagrams and $\binom{N}{3}$=N(N-1)(N-2)/6 possible ternary diagrams. For the database of 11 major oxides, this corresponds to 55 binary and 165 ternary diagrams, whereas the database of 45 elements yields 990 binary and 14,190 ternary diagrams. To efficiently summarize the results of these thousands of discrimination diagrams, a matrix visualization was used.