## 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].

### MSC:

 86A32 Geostatistics