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Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densitiesby generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

Geologische Vereinigung; Universitat de Barcelona, Equip de Recerca Arqueomètrica; Institut d’Estadística de Catalunya; International Association for Mathematical Geology; Patronat de l’Escola Politècnica Superior de la Universitat de Girona; Fundació privada: Girona, Universitat i Futur.

Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada

Manager: Thió i Fernández de Henestrosa, Santiago
Martín Fernández, Josep Antoni
Other contributions: Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada
Author: Egozcue, Juan José
Díaz Barrero, José Luis
Abstract: Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densitiesby generalizing the Aitchison geometry for compositions in the simplex into the set probability densities
Geologische Vereinigung; Universitat de Barcelona, Equip de Recerca Arqueomètrica; Institut d’Estadística de Catalunya; International Association for Mathematical Geology; Patronat de l’Escola Politècnica Superior de la Universitat de Girona; Fundació privada: Girona, Universitat i Futur.
Document access: http://hdl.handle.net/2072/14668
Language: eng
Publisher: Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada
Rights: Tots els drets reservats
Subject: Hilbert, Àlgebra de
Anàlisi funcional
Title: Hilbert space on probability density functions with Aitchison geometry
Type: info:eu-repo/semantics/conferenceObject
Repository: Recercat

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