Title: Effects of changing the reference measure 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)
Abstract: Data resulting from surveys frequently occurs in form of discrete distributional observations which can be subsequently represented by probability density functions (PDFs). This motivates an increasing interest in statistical tools for their statistical processing. Although functional data analysis (FDA) consists of a wide range of such tools, they are typically designed in $L^2$ space, thus cannot be directly applied to densities, as the metrics of $L^2$ does not honor their geometric properties. This has recently motivated the construction of the so-called Bayes Hilbert spaces, which result from the generalization to the infinite dimensional setting of the Aitchison geometry for compositional data. More precisely, when focusing on PDFs with bounded domain $I \subset R$, one can consider the Bayes space of positive real functions with logarithm which is square-integrable with respect to Lebesgue reference. Nonetheless, for unbounded supports, different reference measures need to be used. The aim is to show the effects of changing the reference Lebegue measure to a general probability measure, with the emphasis on its practical implications for the Simplicial Functional Principal Component Analysis (SFPCA) which has been recently designed for dimension reduction of PDFs using the Bayes space methodology.