Page - 366 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 parameters Y¯∈Pm andσ>0,anditsdensitywithrespect to theRiemannianvolumeformdv(Y)of Pm (seeFormula (13) inSection2) is: 1 Zm(σ) exp [ −d 2(Y,Y) 2σ2 ] (2) whereZm(σ) isanormalizingfactordependingonlyonσ (andnoton Y¯). For theGaussian distribution Equation (2), themaximum likelihood estimate (MLE) for the parameter Y¯basedonobservationsY1, · · · ,Yn correspondstothemeanEquation(1). In[15],adetailed studyofstatistical inference for thisdistributionwasgivenandthenappliedto theclassificationof data inPm, showingthat ityieldsbetterperformance, incomparisontorecentapproaches [2]. Whenadatasetcontainsextremevalues (oroutliers),becauseof the impactof thesevaluesond2, themeanbecomes lessuseful. It isusuallyreplacedwith theRiemannianmedian: Median(Y1, · · · ,Yn)=argminY∈Pm n ∑ i=1 d(Y,Yi) (3) DefinitionEquation(3)correspondsto thatof themedianinstatisticsbasedonorderingof the values of a sequence. However, this interpretationdoes not continue to hold onPm. In fact, the Riemannian distance onPm is not associatedwith any norm, and it is therefore only possible to comparedistancesofasetofmatrices toareferencematrix. In thepresenceofoutliers, theGaussiandistributiononPm also loses its robustnessproperties. Themaincontributionof thepresentpaper is toremedythisproblembyintroducingtheRiemannian Laplacedistributionwhilemaintainingthesameone-to-onerelationbetweenMLEandtheRiemannian median. Thiswillbeshowntoofferconsiderable improvement indealingwithoutliers. Thispaper isorganizedas follows. Section 2 reviews theRiemanniangeometryofPm,when thismanifold is equippedwith the RiemannianmetricknownastheRao–Fisheroraffine invariantmetric [10,11]. Inparticular, itgives analytic expressions for geodesic curves, Riemanniandistance and recalls the invariance ofRao’s distanceunderaffinetransformations. Section3introducestheLaplacedistributionL(Y¯,σ) throughitsprobabilitydensityfunctionwith respect to thevolumeformdv(Y): p(Y|Y,σ)= 1 ζm(σ) exp [ −d(Y,Y) σ ] here, σ lies in an interval ]0,σmax[withσmax<∞. This is because thenormalizing constant ζm(σ) becomes infinite forσ≥ σmax. Itwill be shownthat ζm(σ)dependonlyonσ (andnoton Y¯) forall σ<σmax. This important fact leads tosimpleexpressionsofMLEsofYandσ. Inparticular, theMLE of Y¯basedona familyofobservationsY1, · · · ,YN sampled fromL(Y¯,σ) isgivenby themedianof Y1, · · · ,YN definedbyEquation(3)whered isRao’sdistance. Section4focusesonmixturesofRiemannianLaplacedistributionsonPm. Adistributionof this kindhasadensity: p(Y|(ωμ,Yμ,σμ)1≤μ≤M)= M ∑ μ=1 μp(Y|Yμ,σμ) (4) withrespect tothevolumeform dv(Y).Here, M is thenumberofmixturecomponents, μ > 0, Yμ∈ Pm,σμ>0 forall 1≤μ≤Mand∑Mμ=1 μ=1.AnewEM(expectation-maximization)algorithmthat computesmaximumlikelihoodestimatesof themixtureparameters ( μ,Y¯μ,σμ)1≤μ≤M isprovided. Theproblemof theorderselectionof thenumberM inEquation(4) isalsodiscussedandperformed usingtheBayesian informationcriterion(BIC) [16]. 366