Page - 379 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 Denotedthroughout thepaperbyL(Y¯,σ),where Y¯∈Pm andσ>0are the indexingparameters, this distributionmaybethoughtofasspecifyingthe lawofa familyofobservationsonPm concentrated around the location Y¯ andhavingdispersion σ. If ddenotes Rao’s distance onPm and dv(Y) its associatedvolumeform, thedensityofL(Y¯,σ)withrespect todv(Y) isproportional toexp(−d(Y,Y¯σ )). Interestingly, thenormalizingconstantdependsonlyonσ (andnoton Y¯). Thisallowsustodeduce exactexpressions formaximumlikelihoodestimatesof Y¯andσ relyingontheRiemannianmedian onPm. Theseestimatesarealso computednumericallybymeansof sub-gradient algorithms. The estimationofparameters inmixturemodelsofLaplacedistributionsarealsoconsideredandperformed usinganewexpectation-maximizationalgorithm. Finally, themaintheoretical resultsare illustrated byanapplication to textureclassification. Theproposedexperimentconsistsof introducingabnormal data (outliers) intoa set of images fromtheVisTexdatabaseandanalyzing their influenceson the classificationperformances. Each image ischaracterizedbyasetof2×2covariancematricesmodeled asmixturesofRiemannianLaplacedistributions in thespaceP2. Thenumberofmixtures isestimated usingtheBICcriterion. Theobtainedresultsarecomparedto thosegivenbytheRiemannianGaussian distribution, showingthebetterperformanceof theproposedmethod. Acknowledgments: This studyhasbeen carriedoutwithfinancial support fromtheFrenchState,managed by theFrenchNationalResearchAgency (ANR) in the frameof the“Investments for the future”Programme initiatived’excellence (IdEX)Bordeaux-CPU(ANR-10-IDEX-03-02). AuthorContributions: HatemHajriandSalemSaidcarriedout themathematicaldevelopmentandspecified thealgorithms. IoanaIleaandLionelBombrunconceivedanddesignedtheexperiments. YannickBerthoumieu gavethecentral ideaof thepaperandmanagedthemaintasksandexperiments.HatemHajriwrote thepaper. All theauthorsreadandapprovedthefinalmanuscript. Conflictsof Interest: Theauthorsdeclarenoconflictof interest. Appendix: ProofsofSomeTechnicalPoints Thesubsectionsbelowprovideproofs (usingthesamenotations)ofcertainpoints in thepaper. A.DerivationofEquation (16) fromEquation (14) ForU ∈ O(m) and r = (r1, · · · ,rm) ∈ Rm, letY(r,U) = U†diag(er1, · · · ,erm)U. OnO(m), consider theexteriorproductdet(θ)= ∧ i<j θij,whereθij=∑kUjkdUik. Proposition4. Forall test functions f :Pm→R, ∫ Pm f(Y)dv(Y)=(m!2m)−1 × 8m(m−1)4 ∫ O(m) ∫ Rm f(Y(r,U))det(θ)∏ i<j sinh (|ri−rj| 2 ) m ∏ i=1 dri This proposition allows one to deduce Equation (16) from Equation (14), since∫ O(m)det(θ)= 2mπm 2/2 Γm(m/2) (see [29],p. 70). Sketchof theproofofProposition4. Inadifferential form, theRao–FishermetriconPm is: ds2(Y)= tr[Y−1dY]2 ForU∈O(m)and (a1, · · · ,am)∈ (R∗+)m, letY=U†diag(a1, · · · ,am)U. Then: ds2(Y)= m ∑ j=1 da2j a2j +2 ∑ 1≤i<j≤m (ai−aj)2 aiaj θ2ij 379