Analysing multivariate densities in Bayes spaces with implications for functional data analysis

Probability density functions can be embedded in the geometric framework of Bayes spaces which respect their relative nature and enable further modeling and analysis. Specifically, the Hilbert space structure of Bayes spaces (whereof compositional data are one specific instance) has several important implications for classical and Bayesian inference, as well as functional data analysis. In this contribution, an orthogonal decomposition of multivariate densities in Bayes spaces using a distributional analog of the Hoeffding-Sobol identity is constructed. The decomposition is based on a reformulation of the standard (arithmetic) margins into so-called geometric margins. These are orthogonal projections into one-dimensional space of information contained in multivariate densities and coincide with the arithmetic margins in case of independence. More generally, the decomposition contains an independent part and all possible interaction terms. The orthogonality of the decomposition results in an analog of Pythagoras' Theorem for squared norms of the decomposed densities and margin-free property of the interaction terms. Because the squared norms from the Pythagoras' decomposition are essentially compositional in nature, all tools of the log-ratio methodology can be used for their statistical processing. Theoretical results will be illustrated with empirical geochemical data. This talk is based on joint work with Christian Genest and Johanna Nešlehová from McGill University, Montréal, Canada.