A0401
Title: Functional principal component analysis for univariate and multivariate density data
Authors: Karel Hron - Palacky University (Czech Republic) [presenting]
Adela Czolkova - Palacky University Olomouc (Czech Republic)
Sonja Greven - Humboldt University of Berlin (Germany)
Abstract: The Bayes space provides a Hilbert space structure for the analysis of probability density functions (PDFs), equipping them with a geometry that respects their relative and constrained nature. A key tool in this framework is the centered logratio (clr) transformation, which establishes an isometric isomorphism between the Bayes space and the classical $L^2$ space. This enables the application of functional data analysis (FDA) techniques - in the context of dimension reduction, particularly functional principal component analysis (FPCA) - to both univariate and multivariate density data. In the multivariate setting, embedding PDFs in the Bayes space allows for an orthogonal decomposition into independent and interactive components; furthermore, the independent part can be decomposed into mutually orthogonal geometric marginals. This structure yields a deeper understanding of the sources of variation in multivariate densities and has direct consequences for interpreting the eigenfunctions and scores resulting from FPCA. It is shown that applying FPCA directly to multivariate densities is equivalent to applying multivariate FPCA to their decomposed form and that the resulting eigenfunctions and scores decompose accordingly. The theoretical results are illustrated with an application to empirical data, demonstrating the interpretability and practical value of this approach.
