On the Interpretation of Orthonormal Coordinates for Compositional Data

Abstract

The simplex with the Aitchison geometry is a natural sample space for compositional data, that is, observations carrying only relative information (especially proportions, percentages, etc., often occurring in the geosciences). For this reason, standard statistical methods that rely on Euclidean structure of the real space cannot be used directly for statistical analysis. At first, compositional data need to be expressed in coordinates of an orthonormal basis on the simplex (with respect to the Aitchison geometry). The mathematical interpretation of the orthonormal coordinates is derived from the procedure by which they are constructed (called sequential binary partition), and they act as balances between groups of compositional parts. The goal of this paper is to describe the covariance structure of coordinates and, consequently, to provide a complementary interpretation based on log-ratios of parts of the original composition. It must be noted that, in a composition, the ratios themselves contain all the relevant information. The possibilities as well as the limitations of this approach are demonstrated through illustrative examples.