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K. Hron, A. Menafoglio, M. Templ, K. Hruzova, P. Filzmoser:
"Simplicial principal component analysis for density functions in Bayes spaces";
Report No. MOX-report 25/2014, 2014; 20 pages.

Probability density functions are frequently used to characterize the distribu-
tional properties of large-scale database systems. As functional compositions,
densities carry primarily relative information. As such, standard methods of func-
tional 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 pro-
pose the simplicial functional principal component analysis (SFPCA), which is
based on the geometry of the Bayes space
B2
of functional compositions. We
perform SFPCA by exploiting the centred log-ratio transform, an isometric iso-
morphism between B2
and L2
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.

compositional data; Bayes spaces; centred log-ratio transformation; func- tional principal component analysis

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