Title: Functional regression analysis with compositional response
Authors: Renata Talska - Palacky University Olomouc (Czech Republic) [presenting]
Alessandra Menafoglio - Politecnico di Milano (Italy)
Karel Hron - Palacky University (Czech Republic)
Eva Fiserova - Palacky University (Czech Republic)
Jitka Machalova - Palacky University (Czech Republic)
Abstract: Regression analysis is a key statistical tool to model a linear relationship between a response variable and a set of covariates. In functional data analysis (FDA), methods to perform linear regression with functional response and scalar predictors have been widely discussed. More delicate appears the situation in which the response variable is represented as a probability density, since the L2 space (of square integrable functions), usually employed in FDA, does not account for the inherent constraints of densities. The aim is to introduce functional regression model with distributional response using the Bayes space approach, i.e. a geometric viewpoint that allows capturing the inherent features of distributional data. Indeed, densities primarily carry relative information, and the unit integral constraint represents just one of its possible equivalent representations. Accordingly, densities can be considered as elements of a Bayes Hilbert space, whose geometry is designed to precisely capture the specific properties of densities (e.g. scale invariance, relative scale). In order to apply functional regression tools for L2 data, particularly those based on B-spline representations, the centred logratio transformation - mapping the Bayes Hilbert space into L2 - is considered. The methodological developments are illustrated with a real-world example.