The construction of extreme compositions

If a geochemical compositional dataset X (n×p)is a realization of a physical mixing process, then each of its sample (row) vectors will approximately be a convex combination (mixture) of a fixed set of (l×p)extreme compositions termed endmembers. The kpoints in p-space corresponding to a specified set of k (klinearly independent endmember estimates associated with a p-variate (n×p)compositional dataset X,define the vertices of a (k–1)dimensional simplex H.The nestimated mixtures X (n×p)which together account for the systematic variation in the dataset X,should each be convex combinations of the kfixed endmember estimates. Accordingly,the npoints in p-space which represent these mixtures should be interior points of the simplex H.Otherwise, for each sample point which lies outside H,at least one of the mixture coefficients (endmember contributions) will be negative. The purpose of this paper is to describe procedures for expanding H in the situation that its vertices are not a set of extreme points for the set which represents the mixtures.