Simplicial geometry for compositional data

The main features of the Aitchison geometry of the simplex of D parts are reviewed. Compositions are positive vectors in which the relevant information is contained in the ratios between their components or parts. They can be represented in the simplex of D parts by closing them to a constant sum, e.g. percentages, or parts per million. Perturbation and powering in the simplex of D parts are respectively an internal operation, playing the role of a sum, and of an external product by real numbers or scalars. These operations impose the structure of ( D − 1)-dimensional vector space to the simplex of D parts. An inner product, norm and distance, compatible with perturbation and powering, complete the structure of the simplex, a structure known in mathematical terms as a Euclidean space. This general structure allows the representation of compositions by coordinates with respect to a basis of the space, particularly, an orthonormal basis. The interpretation of the so-called balances, coordinates with respect to orthonormal bases associated with groups of parts, is stressed. Subcompositions and balances are interpreted as orthogonal projections. Finally, log-ratio transformations (alr, clr and ilr) are considered in this geometric context.