Mathematical Foundations of Compositional Data Analysis

The definition of composition like a vector whose components are subject to the restriction of constant sum is revised from a mathematical point of view. Starting from the definition of an equivalence’s relation on the positive orthant of the multidimensional real space IR , the compositions become equivalence classes of elements of IR , and the set of them —i.e., the corresponding quotient set— the compositional space . In this way, the simplex becomes one, among many, of the possible representations of this quotient space. The logarithmic transformation defines a one-to-one transformation between and a suitable vector quotient space defined on the multidimensional real space. This transformation allows to transfer the Euclidean structure easily defined on to the compositional space. It is showed that, from a mathematical point of view, the methodology introduced by Aitchison is fully compatible with the nature of compositional data, and is independent of the representation used to manage the compositional data.