Page - 378 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 algorithm is repeated 10 times, and the resultmaximizing the log-likelihood function is retained. Finally, theclassification isperformedbyassigningeachelementYt∈P2 in the testingset to theclass of theclosest clusterμ∗, givenby: μ∗=argminμ { − log ˆμ+ F ∑ f=1 log ζ(σˆμ,f)+ F ∑ f=1 d(Yt ,Yˆμ,f) σˆμ,f } (39) This expression isobtainedstarting fromEquations (36) and (37), knowing thatF featuresare extractedforeachpatch. Theclassification results of theproposedmodel (solid red line), expressed in termsofoverall accuracy, showninFigure2,arecomparedto thosegivenbyafixednumberofmixturecomponents (that is, for M = 3, dashed red line) andwith those givenwhen thewithin-class diversity is not considered(that is, forM=1,dottedredline). Inaddition, theclassificationperformancesgivenby theRGDmodel (displayed inblack)proposed in [15]andtheWDmodel (displayed inblue)proposed in [17] are also considered. For each of thesemodels, the number ofmixture components is first computedusingtheBIC,andnext, it isfixedtoM=3andM=1. Forallof theconsideredmodels, theclassificationrate isgivenasa functionof thenumberofoutliers,whichvariesbetweenzeroand 60foreachclass. Figure2.Classificationresults. It is shownthat,as thenumberofoutliers increases, theRLDgivesprogressivelybetter results than theRGDand theWD.The results are improvedbyusing theBIC criterion for choosing the suitablenumberofclusters. Inconclusion, themixtureofRLDscombinedwith theBICcriterionto estimate thebestnumberofmixturecomponents canminimize the influenceofabnormal samples present in thedataset, illustratingtherelevanceof theproposedmethod. 6.Conclusions Motivatedbytheproblemofoutliers instatisticaldata, thispaperintroducesanewdistributionon thespacePmofm×msymmetricpositivedefinitematrices,calledtheRiemannianLaplacedistribution. 378