An invitation to intrinsic compositional data analysis using projective geometry and Hilbert’s metric

Résumé

We propose to study Compositional Data (CoDa) from the projec-tive geometry viewpoint. Indeed, CoDa, as equivalence classes of propor-tional vectors, corresponds to projective points in a projective space, and thus can be studied using the tools, language and framework of projec-tive geometry. Combined with the partial order structure induced by the non-negativity of CoDa, the projective viewpoint highlights the inherent geometrical and structural properties CoDa, irrespective of a particular representation and an arbitrarily chosen coordinate system. This intrinsic approach helps to clarify the relationships and offers much needed geomet-ric insight between other competing coordinate-based approaches, such as Aitchison’s log-ratio, Watson’s spherical, or plain aﬀine representations in the simplex. Our first objective is thus to give a tutorial on projective geometry geared towards compositional data analysis. In addition, owing to the projective or ordering structures of CoDa, the positive CoDa space can be endowed with an intrinsic metric, Hilbert’s projective metric, which is independent of any metrization of any ambi-ent space. Such non-smooth metric is well-suited with the principles of compositional analysis (in particular, subcompositional coherence) and is compatible with both Aitchison’s vector space geometry in log coordinates and the straight aﬀine geometry of the simplex. In view of statistical ap-plications, a smooth and strictly convex approximation of Hilbert’s metric is constructed and is shown to share most properties of the original metric. Our second objective is then to establish the firsts steps of such an intrinsic statistical analysis of CoDa, based on Hilbert’s metric and the projective viewpoint. To that regards, we show how Hilbert metric and its smooth surrogate allow to build extrinsic and intrinsic measures of location and spread, in particular Fréchet means and variance, how to construct analogues of the Gaussian/Laplace distribution and how to per-form nonparametric regression. All three examples are supplemented by numerical simulations. We close by drawing some perspectives for fur-ther research, inviting for new directions for CoDa analysis based on the intrinsic projective viewpoint.