?url_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rft.relation=https%3A%2F%2Fpublications.ut-capitole.fr%2Fid%2Feprint%2F48475%2F&rft.title=An+invitation+to+intrinsic+compositional+data+analysis+using+projective+geometry+and+Hilbert%E2%80%99s+metric&rft.creator=Faugeras%2C+Olivier+Paul&rft.subject=B-+ECONOMIE+ET+FINANCE&rft.description=We+propose+to+study+Compositional+Data+(CoDa)+from+the+projec-tive+geometry+viewpoint.+Indeed%2C+CoDa%2C+as+equivalence+classes+of+propor-tional+vectors%2C+corresponds+to+projective+points+in+a+projective+space%2C+and+thus+can+be+studied+using+the+tools%2C+language+and+framework+of+projec-tive+geometry.+Combined+with+the+partial+order+structure+induced+by+the+non-negativity+of+CoDa%2C+the+projective+viewpoint+highlights+the+inherent+geometrical+and+structural+properties+CoDa%2C+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%2C+such+as+Aitchison%E2%80%99s+log-ratio%2C+Watson%E2%80%99s+spherical%2C+or+plain+a%EF%AC%80ine+representations+in+the+simplex.+Our+first+objective+is+thus+to+give+a+tutorial+on+projective+geometry+geared+towards+compositional+data+analysis.+In+addition%2C+owing+to+the+projective+or+ordering+structures+of+CoDa%2C+the+positive+CoDa+space+can+be+endowed+with+an+intrinsic+metric%2C+Hilbert%E2%80%99s+projective+metric%2C+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%2C+subcompositional+coherence)+and+is+compatible+with+both+Aitchison%E2%80%99s+vector+space+geometry+in+log+coordinates+and+the+straight+a%EF%AC%80ine+geometry+of+the+simplex.+In+view+of+statistical+ap-plications%2C+a+smooth+and+strictly+convex+approximation+of+Hilbert%E2%80%99s+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%2C+based+on+Hilbert%E2%80%99s+metric+and+the+projective+viewpoint.+To+that+regards%2C+we+show+how+Hilbert+metric+and+its+smooth+surrogate+allow+to+build+extrinsic+and+intrinsic+measures+of+location+and+spread%2C+in+particular+Fr%C3%A9chet+means+and+variance%2C+how+to+construct+analogues+of+the+Gaussian%2FLaplace+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%2C+inviting+for+new+directions+for+CoDa+analysis+based+on+the+intrinsic+projective+viewpoint.&rft.publisher=TSE+Working+Paper&rft.date=2023-12&rft.type=Monograph&rft.type=NonPeerReviewed&rft.format=text&rft.language=en&rft.identifier=https%3A%2F%2Fpublications.ut-capitole.fr%2Fid%2Feprint%2F48475%2F1%2Fwp_tse_1496.pdf&rft.identifier=++Faugeras%2C+Olivier+Paul+%3Chttps%3A%2F%2Fwww.idref.fr%2F135534518%3E++(2023)+An+invitation+to+intrinsic+compositional+data+analysis+using+projective+geometry+and+Hilbert%E2%80%99s+metric.++TSE+Working+Paper%2C+n.+23-1496%2C+Toulouse+++++&rft.relation=http%3A%2F%2Ftse-fr.eu%2Fpub%2F128861&rft.language=en