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Geological Society, London, Special Publications; 2006; v. 264; p. 145-159;
DOI: 10.1144/GSL.SP.2006.264.01.11
© 2006 Geological Society of London

General Theory and Methods

Simplicial geometry for compositional data

J. J. Egozcue1 & V. Pawlowsky-Glahn2

1 Department Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Jordi Girona Salgado 1-3, C2, E-08034 Barcelona, Spain juan.jose.egozcue{at}upc.edu
2 Department Informàtica i Matemàtica Aplicada, Universitat de Girona, Campus Montilivi, P4, E-17071 Girona, Spain

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.





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