Page - 375 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 Givenasetofobservations{Y1, · · · ,YN}arisingfromEquation(27)whereM isunknown, the BICconsistsofchoosingtheparameter: M¯=argmaxMBIC(M) where: BIC(M)=LL− 1 2 ×DF× log(N) (31) Here,LL is the log-likelihoodgivenby: LL= N ∑ n=1 log ( M ∑ k=1 ˆkp(Yn|Yˆk, σˆk) ) (32) andDF is thenumberofdegreesof freedomof thestatisticalmodel: DF=M×m(m+1) 2 +M+M−1 (33) InFormula (32), (ˆk,Yˆk, σˆk)1≤k≤M areobtained fromanEMalgorithmas stated inSection4.1 assumingtheexactdimension isM. Finally,note that inFormula (33),M×m(m+1)2 (respectivelyM andM−1)corresponds to thenumberofdegreesof freedomassociatedwith (Yˆk)1≤k≤M (respectively (σˆk)1≤k≤M and (ˆk)1≤k≤M). 5.ApplicationtoClassificationofDataonPm Recently,severalapproacheshaveusedtheRiemanniandistanceingeneralasthemaininnovation in imageorsignalclassificationproblems[2,15,34]. It turnsout that theuseof thisdistance leads to moreaccurate results (incomparison, forexample,with theEuclideandistance). This sectionproposes anapplicationthat followsasimilarapproach,but inadditionto theRiemanniandistance, italsorelies onastatisticalapproach. It considers theapplicationof theRiemannianLaplacedistribution(RLD) to theclassificationofdata inPm andgivesanoriginalLaplaceclassificationrule,whichcanbeusedto carryout thetaskofclassification,eveninthepresenceofoutliers. Italsoapplies thisclassificationrule to theproblemof textureclassification incomputervision, showingthat it leads to improvedresults in comparisonwithrecent literature. Section5.1considers, fromthepointofviewofstatistical learning, theclassificationofdatawith values inPm. GivendatapointsY1, · · · ,YN∈Pm, thisproceeds in twosteps, calledthe learningphase andtheclassificationphase, respectively. The learningphaseuncovers theclassstructureof thedata, byestimatingamixturemodelusing theEMalgorithmdeveloped inSection4.1. Once training is accomplished,datapointsaresubdividedintodisjointclasses.Classificationconsistsofassociating eachnewdatapoint to themostsuitableclass. For this,anewclassificationrulewillbeestablished andshowntobeoptimal. Section5.2 is the implementationof theLaplaceclassificationrule togetherwith theBICcriterion to textureclassification incomputervision. Ithighlights theadvantageof theLaplacedistribution in thepresenceofoutliersandshowsitsbetterperformancecomparedtorecentapproaches. 5.1. ClassificationUsingMixturesofLaplaceDistributions Assumetobegivenasetof trainingdataY1, · · · ,YN. Thesearenowmodeledasarealizationofa mixtureofLaplacedistributions: p(Y)= M ∑ μ=1 μ×p(Y|Y¯μ,σμ) (34) 375