Mathematical Foundations of Compositional Data Ana- lysis

Mathematical Foundations of Compositional Data Analysis ´ ´ Carles Barcelo-Vidal, Josep A. Mart´ın-Fernandez and Vera Pawlowsky-Glahn ` ` Departament d’Informatica i Matematica Aplicada, Universitat de Girona, Campus de Montilivi, E-17071 Girona, Spain. e-mail: [email protected] Summary. The definition of composition like a vector whose components are subject to the restriction of constant sum is revised from a mathematical point of view. Starting from the definition real space 
 of an equivalence’s relation on the positive orthant of the multidimensional 
 IR , the compositions become equivalence classes of elements of IR , and the set of  them —i.e., the corresponding quotient set— the compositional space . In this way, the simplex becomes one, among many, of the possible representations of this quotient space.  The logarithmic transformation defines a one-to-one transformation between and a suitable defined on the multidimensional real space. This transformation vector quotient space  allows to transfer the Euclidean structure easily defined on  to the compositional space. It is showed that, from a mathematical point of view, the methodology introduced by Aitchison is fully compatible with the nature of compositional data, and is independent of the representation used to manage the compositional data. Keywords: Compositional data, closed data

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