A0233
Title: Weighting in Bayes spaces and its effects in statistical processing of density functions
Authors: Renata Talska - Palacky University Olomouc (Czech Republic) [presenting]
Alessandra Menafoglio - Politecnico di Milano (Italy)
Karel Hron - Palacky University (Czech Republic)
Juan Jose Egozcue - Universitat Politecnica de Catalunya (Barcelona, Spain) (Spain)
Javier Palarea-Albaladejo - Biomathematics and Statistics Scotland (United Kingdom)
Abstract: Probability density functions (PDFs) can be viewed as functional data carrying relative information. The relative nature of PDFs is accounted for in Bayes spaces with Hilbert space structure, which result from generalization of the Aitchison geometry for compositional data to the infinite dimensional setting. Specifically, if the focus is 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 within the Bayes space of positive real functions with square-integrable logarithm. The reference measure can be changed and it induces a weighting effect on the domain $I$. Moreover, the weighting also impacts the geometry of the Bayes spaces and results in so-called weighted Bayes spaces. The aim of this contribution is 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). Furthermore, a centered log-ratio transformation as an isometric map between the weighted Bayes space and an unweighted $L^2$ space (i.e. with Lebesgue reference measure) is proposed, which enables statistical processing of PDFs using standard statistical tools provided by Functional Data Analysis (FDA).
