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Bivariate densities in Bayes spaces: orthogonal decomposition and spline representation

Author

Listed:
  • Karel Hron

    (Faculty of Science, Palacký University)

  • Jitka Machalová

    (Faculty of Science, Palacký University)

  • Alessandra Menafoglio

    (Politecnico di Milano)

Abstract

A new orthogonal decomposition for bivariate probability densities embedded in Bayes Hilbert spaces is derived. It allows representing a density into independent and interactive parts, the former being built as the product of revised definitions of marginal densities, and the latter capturing the dependence between the two random variables being studied. The developed framework opens new perspectives for dependence modelling (e.g., through copulas), and allows the analysis of datasets of bivariate densities, in a Functional Data Analysis perspective. A spline representation for bivariate densities is also proposed, providing a computational cornerstone for the developed theory.

Suggested Citation

  • Karel Hron & Jitka Machalová & Alessandra Menafoglio, 2023. "Bivariate densities in Bayes spaces: orthogonal decomposition and spline representation," Statistical Papers, Springer, vol. 64(5), pages 1629-1667, October.
    • Handle: RePEc:spr:stpapr:v:64:y:2023:i:5:d:10.1007_s00362-022-01359-z
    DOI: 10.1007/s00362-022-01359-z
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    References listed on IDEAS

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    1. 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.
    2. Freedman, David & Lane, David, 1983. "A Nonstochastic Interpretation of Reported Significance Levels," Journal of Business & Economic Statistics, American Statistical Association, vol. 1(4), pages 292-298, October.
    3. Nerini, David & Ghattas, Badih, 2007. "Classifying densities using functional regression trees: Applications in oceanology," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4984-4993, June.
    4. Talská, R. & Menafoglio, A. & Machalová, J. & Hron, K. & Fišerová, E., 2018. "Compositional regression with functional response," Computational Statistics & Data Analysis, Elsevier, vol. 123(C), pages 66-85.
    5. 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.
    6. J. Machalová & K. Hron & G.S. Monti, 2016. "Preprocessing of centred logratio transformed density functions using smoothing splines," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(8), pages 1419-1435, June.
    7. Delicado, P., 2011. "Dimensionality reduction when data are density functions," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 401-420, January.
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