Geometrical Properties of Compositional Data

Abstract

For an appropriate statistical processing it is essential to consider the inherent geometrical properties of the sample space of observations. In case of compositional data, this space is represented by equivalence classes of proportional vectors, possibly represented on the simplex, endowed with the Aitchison geometry. Its Euclidean vector space structure enables to construct coordinates with respect to a basis, eventually coefficients of a generating system. Here, isometric logratio coordinates, real coordinates with respect to an orthonormal basis in the Aitchison geometry, are preferable. As their name indicates, they are isometric with the Aitchison geometry, which makes it possible to proceed with standard statistical analyses in a meaningful way. For interpretation purposes, pivot coordinates that extract relative information about a compositional part in just one coordinate are taken as first option. In addition, also other alternatives are considered: symmetric pivot coordinates, which are suitable for a bivariate analysis, and particularly balance coordinates, which are interpretable in the sense of balances between groups of compositional parts. They can be intuitively constructed using sequential binary partitioning, and they form a family of general isometric logratio coordinates; also the preferable pivot coordinates can be taken as a special case. 3.1 Motivation A frequent argument for the necessity of a special treatment of compositional data is that this kind of data is not coherent with the usual Euclidean geometry. Rather, compositional data follow the so-called Aitchison geometry on the simplex, see Sect. 3.2. But how is the "usual" Euclidean geometry defined, and what is the simplex? In analytical geometry, a Euclidean space is associated with a vector space. Starting from an origin in the Euclidean space, one can reach a specific point by a vector in terms of an arrow, connecting the origin with this point. It is then possible to measure distances and angles in the Euclidean space with the help of the lengths of the arrows and the angles between them. This generates a vector space with a scalar product.