Compositional data: the sample space and its structure

Juan José Egozcue ^[1] ; Vera Pawlowsky-Glahn ^[2]
1. [1] Universitat Politècnica de Catalunya
  
  Universitat Politècnica de Catalunya
  
  Barcelona, España
2. [2] Universitat de Girona
  
  Universitat de Girona
  
  Gerona, España
Localización: Test: An Official Journal of the Spanish Society of Statistics and Operations Research, ISSN-e 1863-8260, ISSN 1133-0686, Vol. 28, Nº. 3, 2019, págs. 599-638
Idioma: inglés
DOI: 10.1007/s11749-019-00670-6
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Resumen
- The log-ratio approach to compositional data (CoDa) analysis has now entered a mature phase. The principles and statistical tools introduced by J. Aitchison in the eighties have proven successful in solving a number of applied problems. The algebraic–geometric structure of the sample space, tailored to those principles, was developed at the beginning of the millennium. Two main ideas completed the J. Aitchison’s seminal work: the conception of compositions as equivalence classes of proportional vectors, and their representation in the simplex endowed with an interpretable Euclidean structure. These achievements allowed the representation of compositions in meaningful coordinates (preferably Cartesian), as well as orthogonal projections compatible with the Aitchison distance introduced two decades before. These ideas and concepts are reviewed up to the normal distribution on the simplex and the associated central limit theorem. Exploratory tools, specifically designed for CoDa, are also reviewed. To illustrate the adequacy and interpretability of the sample space structure, a new inequality index, based on the Aitchison norm, is proposed. Most concepts are illustrated with an example of mean household gross income per capita in Spain.