eprintid: 48475 rev_number: 19 eprint_status: archive userid: 1482 importid: 105 dir: disk0/00/04/84/75 datestamp: 2023-12-22 11:01:00 lastmod: 2024-11-04 12:56:12 status_changed: 2024-11-04 12:56:12 type: monograph metadata_visibility: show creators_name: Faugeras, Olivier Paul creators_id: olivier.faugeras@tse-fr.eu creators_idrefppn: 135534518 creators_affiliation: Toulouse School of Economics creators_halaffid: 1002422 title: An invitation to intrinsic compositional data analysis using projective geometry and Hilbert’s metric ispublished: pub subjects: subjects_ECO abstract: 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. date: 2023-12 date_type: published publisher: TSE Working Paper official_url: http://tse-fr.eu/pub/128861 faculty: tse divisions: tse language: en has_fulltext: TRUE view_date_year: 2023 full_text_status: public monograph_type: working_paper series: TSE Working Paper volume: 23-1496 place_of_pub: Toulouse pages: 58 institution: Université Toulouse Capitole department: Toulouse School of Economics book_title: TSE Working Paper oai_identifier: oai:tse-fr.eu:128861 harvester_local_overwrite: creators_name harvester_local_overwrite: pending harvester_local_overwrite: creators_idrefppn harvester_local_overwrite: creators_halaffid harvester_local_overwrite: abstract harvester_local_overwrite: department harvester_local_overwrite: place_of_pub harvester_local_overwrite: institution harvester_local_overwrite: pages harvester_local_overwrite: creators_id oai_lastmod: 2024-04-29T08:27:43Z oai_set: tse site: ut1 citation: Faugeras, Olivier Paul (2023) An invitation to intrinsic compositional data analysis using projective geometry and Hilbert’s metric. TSE Working Paper, n. 23-1496, Toulouse document_url: https://publications.ut-capitole.fr/id/eprint/48475/1/wp_tse_1496.pdf