Log-Free Divergence and Covariance Matrix for Compositional Data I: The Affine/Barycentric Approach

Abstract

The presence of zeroes in Compositional Data (CoDa) is a thorny issue for Aitchison's classical log-ratio analysis. Building upon the geometric approach by (Faugeras (2023)), we study the full CoDa simplex from the perspective of affine geometry. This view allows to regard CoDa as points (and not vectors), naturally expressed in barycentric coordinates. A decomposition formula for the displacement vector of two CoDa points yields a novel family of barycentric dissimilarity measures. In turn, these barycentric divergences allow to define i) Fréchet means and their variants, ii) isotropic and  anisotropic analogues of the Gaussian distribution, and  importantly iii) variance and covariance matrices.  All together, the new tools introduced in this paper provide a log-free, direct and unified way to deal with the whole CoDa space, exploiting the linear affine structure of CoDa, and effectively handling zeroes. A strikingly related approach based on the projective viewpoint and the exterior product is studied in the separate companion paper (Faugeras (2024a)).