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Simplicial principal component analysis for density functions in Bayes spaces

Author

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  • Hron, K.
  • Menafoglio, A.
  • Templ, M.
  • Hrůzová, K.
  • Filzmoser, P.

Abstract

Probability density functions are frequently used to characterize the distributional properties of large-scale database systems. As functional compositions, densities primarily carry 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 is to build up a concise methodology for functional principal component analysis of densities. A simplicial functional principal component analysis (SFPCA) is proposed, based on the geometry of the Bayes space B2 of functional compositions. SFPCA is performed by exploiting the centred log-ratio transform, an isometric isomorphism between B2 and L2 which enables one to resort to standard FDA tools. The advantages of the proposed approach with respect to existing techniques are demonstrated using simulated data and a real-world example of population pyramids in Upper Austria.

Suggested Citation

  • Hron, K. & Menafoglio, A. & Templ, M. & Hrůzová, K. & Filzmoser, P., 2016. "Simplicial principal component analysis for density functions in Bayes spaces," Computational Statistics & Data Analysis, Elsevier, vol. 94(C), pages 330-350.
    • Handle: RePEc:eee:csdana:v:94:y:2016:i:c:p:330-350
    DOI: 10.1016/j.csda.2015.07.007
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    Cited by:

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    14. Aneiros, Germán & Cao, Ricardo & Fraiman, Ricardo & Genest, Christian & Vieu, Philippe, 2019. "Recent advances in functional data analysis and high-dimensional statistics," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 3-9.
    15. Jitka Machalová & Renáta Talská & Karel Hron & Aleš Gába, 2021. "Compositional splines for representation of density functions," Computational Statistics, Springer, vol. 36(2), pages 1031-1064, June.
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    22. Kokoszka, Piotr & Miao, Hong & Petersen, Alexander & Shang, Han Lin, 2019. "Forecasting of density functions with an application to cross-sectional and intraday returns," International Journal of Forecasting, Elsevier, vol. 35(4), pages 1304-1317.

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