Title: Flexible regression for probability densities in Bayes spaces
Authors: Almond Stoecker - Humboldt University of Berlin (Germany) [presenting]
Eva-Maria Maier - LMU Munich (Germany)
Sonja Greven - LMU Munich (Germany)
Abstract: A flexible regression framework is presented for functional compositional data, i.e. functional additive regression models (FAMs) for the case that functional response variables or covariates are probability density functions (PDFs). The special nature of PDFs - in particular their property to integrate to one - prohibits direct application of usual functional regression models. Instead, we formulate FAMs for PDFs in a Bayes Hilbert space. The isometry given by the so called centered log-ratio transform allows us to carry over the flexibility of previous FAMs to Bayes space models. Thus, we are able to provide a wide range of different types of categorical, scalar or functional covariate effects. For instance, in an application to the annual birth distributions in Germany from 1950 to 2016, we include a smooth scalar effect for the year and a categorical effect for sex to model the birth PDFs, while also accounting for the cyclic nature of the response PDFs. For estimation, we consider two different procedures: a penalized least squares approach and component-wise gradient boosting, which also yields inherent model selection.