Statistical Modelling 15 (2) (2015), 175–190
Tools for compositional data with a total
Vera Pawlowsky-Glahn
Departament d’Informàtica,
Matemàtica Aplicada i Estadística,
Universitat de Girona,
Girona,
Spain
e-mail: vera.pawlowsky@udg.edu
Juan José Egozcue
Departament de Matemàtica Aplicada III,
Universitat Politécnica de Catalunya,
Barcelona,
Spain
David Lovell
CSIRO Computational Informatics,
Canberra,
Australia
Abstract:
Compositional data analysis usually deals with relative information between parts where the total (abundances, mass, amount, etc.) is unknown or uninformative. This article addresses the question of what to do when the total is known and is of interest. Tools used in this case are reviewed and analysed, in particular the relationship between the positive orthant of D-dimensional real space, the product space of the real line times the D-part simplex, and their Euclidean space structures. The first alternative corresponds to data analysis taking logarithms on each component, and the second one to treat a log-transformed total jointly with a composition describing the distribution of component amounts. Real data about total abundances of phytoplankton in an Australian river motivated the present study and are used for illustration.
Keywords:
Aitchison geometry; Euclidean isometry; ilr coordinates; product space; simplex.
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