Endmember graphics

Thek points inp-space corresponding to a specified set ofk (k<p) linearly independent endmember estimates associated with ap-variate (n×p) compositional datasetX, define the vertices of a (k?1) dimensional simplexH. Then estimated mixtures which together account for the systematic variation in the datasetX, are each convex combinations of thek fixed endmember estimates. Accordingly, then-points inp-space which represent these mixtures must be interior points of the simplexH. The purpose of this paper is to describe a simple graphical procedure for examining the positions of the estimated mixtures relative tok=3 or 4 putative endmembers, and for assessing their compliance or otherwise with the convexity constraints. Since the mixture coefficients must be estimated first in order to obtain the mixture estimates, the paper begins with a review of the least-squares partitioning procedure.