Simplicial principal component analysis for density functions in Bayes spaces
Wednesday 2nd July 2014
Hron, K.; Menafoglio, A.; Templ, M.; Hruzova K.; Filzmoser, P.
Probability density functions are frequently used to characterize the distributional properties of large-scale database systems. As functional compositions, densities carry primarily relative information. As such, standard methods of functional data analysis (FDA) are not appropriate for their statistical processing. The specific features of density functions are accounted for in Bayes spaces, which result from the generalization to the infinite dimensional setting of the Aitchison geometry for compositional data. The aim of the paper is to build up a concise methodology for functional principal component analysis of densities. We propose the simplicial functional principal component analysis (SFPCA), which is based on the geometry of the Bayes space B^2 of functional compositions. We perform SFPCA by exploiting the centred log-ratio transform, an isometric isomorphism between B^2 and L^2 which enables one to resort to standard FDA tools. Advances of the proposed approach are demonstrated using a real-world example of population pyramids in Upper Austria.