Document details for 'Changing reference measure in Bayes spaces and its effects in statistical processing of density functions'

Authors Talská, R., Menafoglio, A., Hron, K., Egozcue, J.J. and Palarea Albaladejo, J.
Publication details In "Book of Abstracts of the 8th International Workshop on Compositional Data Analysis (CoDaWork2019)", 24. Eds. M.I. Ortego. Universitat Politecnica de Catalunya-BarcelonaTECH, Barcelona, Spain.
Publisher details Universitat Politecnica de Catalunya-BarcelonaTECH, Barcelona, Spain
Abstract Probability density functions (PDFs) can be understood as functional data carrying relative information. As such, standard methods of functional data analysis (FDA) (Ramsay et al. 2005) are not appropriate for their statistical processing. They are typically designed for the $L^2$ space (with Lebesgue reference measure), thus they cannot be directly applied to densities as the metrics of $L^2$ does not correspond to their geometric properties. This has recently motivated the construction of the so-called Bayes Hilbert spaces (Boogaart et al. 2014), which result from the generalization of the Aitchison geometry for compositional data to the infinite dimensional setting. More precisely, if we focus on PDFs restricted to a bounded support $I \subset R$, which is typically used in practical applications, they can be represented with respect to the Lebesgue reference measure using the Bayes space of positive real functions with square-integrable logarithm. The reference measure can be easily changed through the well-known chain rule and interpreted as a weighting technique in Bayes spaces. Moreover, this has an impact on the geometry of the Bayes spaces and results in so-called weighted Bayes spaces. The aim of this contribution is to show the effects of changing the reference measure from the Lebesgue measure to a general probability measure focusing on its practical implications for the Simplicial Functional Principal Component Analysis (SFPCA) (Hron et al. 2016). A centered log-ratio transformation is proposed to map weighted Bayes spaces into an unweighted $L^2$ space (i.e. with Lebesgue reference measure), hence enabling the application of standard statistical methods on PDFs.
ISBN 978-84-947240-1-5
Last updated 2019-08-06

Unless explicitly stated otherwise, all material is copyright © Biomathematics and Statistics Scotland.

Biomathematics and Statistics Scotland (BioSS) is formally part of The James Hutton Institute (JHI), a registered Scottish charity No. SC041796 and a company limited by guarantee No. SC374831. Registered Office: JHI, Invergowrie, Dundee, DD2 5DA, Scotland