Simplicial geometry for compositional data. (English) Zbl 1156.86307

Buccianti, A. (ed.) et al., Compositional data analysis in the geosciences: from theory to practice. London: The Geological Society Publishing House (ISBN 978-1-86239-205-2/hbk). Geological Society Special Publication 264, 145-159 (2006).
Summary: 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 an 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.
For the entire collection see [Zbl 1155.86002].


86A32 Geostatistics