Page - 376 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 In this section, theorderM inEquation(34) is consideredasknown.Thetrainingphaseof these dataconsistsof learning its structureasa familyofMdisjointclassesCμ,μ=1, · · · ,M. Tobemore precise, depending on the family ( μ), someof these classesmaybe empty. Training is done by applying theEMalgorithmdescribed inSection4.1. Asa result, each classCμ is representedbya triple (ˆμ,Yˆμ, σˆμ)correspondingtomaximumlikelihoodestimatesof ( μ,Yμ,σμ). Eachobservation Yn is now associatedwith the class Cμ∗ where μ∗ = argmaxμω(Yn, νˆ) (recall the definition from Equation(28)). In thisway,{Y1, · · · ,YN} is subdividedintoMdisjointclasses. Theclassificationphaserequiresaclassificationrule. Following[15], theoptimalrule (in thesense ofaBayesianriskcriteriongivenin[35])consistsofassociatinganynewdataYt totheclassCμ∗ where: μ∗=argmaxμ { Nˆμ×p(Yt|Yˆμ, σˆμ) } (35) Here, Nˆμ is thenumberof elements inCμ. Replacing Nˆμ withN× ˆμ, Equation (35)becomes argmaxμˆμ×p(Yt|Yˆμ, σˆμ).Note thatwhentheweights μ inEquation(34)areassumedtobeequal, this rulereduces toamaximumlikelihoodclassificationrulemaxμ p(Yt|Yˆμ, σˆμ).Aquick lookat the expressionEquation(17) showsthatEquation(35)canalsobeexpressedas: μ∗=argminμ { − log ˆμ+ log ζ(σˆμ)+ d(Yt ,Yˆμ) σˆμ } (36) TheruleEquation(36)willbecalledtheLaplaceclassificationrule. It favorsclustersCμ having a largernumberofdatapoints (theminimumcontains− log ˆμ)orasmallerdispersionawayfrom themedian(theminimumcontains log ζ(σˆμ)).Whenchoosingbetweentwoclusterswith thesame numberofpointsandthesamedispersion, this rule favors theonewhosemedian iscloser toYt . If the numberofdatapoints insideclustersandtherespectivedispersionsareneglected, thenEquation(36) reduces to thenearestneighborrule involvingonly theRiemanniandistance introducedin[2]. TheanalogousrulesofEquation(36) for theRiemannianGaussiandistribution(RGD)[15]andthe Wishartdistribution(WD)[17]onPm canbeestablishedbyreplacingp(Yt|Yˆμ, σˆμ) inEquation(35)with theRGDandtheWDandthenfollowingthesamereasoningasbefore. Recall thataWDdependson anexpectationΣ∈Pmandanumberofdegreesof freedomn [29]. For theWD,Equation(36)becomes: μ∗=argminμ {−2log ˆ(μ)− nˆ(μ)(logdet(Σˆ−1(μ)Yt)− tr(Σˆ−1(μ)Yt))} Here, ˆ(μ), Σˆ(μ)and nˆ(μ)denotemaximumlikelihoodestimatesof the trueparameters (μ), Σ(μ)andn(μ),whichdefinethemixturemodel (theseestimatescanbecomputedas in [36,37]). 5.2.Application toTextureClassification This sectionpresents an application of themixture of Laplace distributions to the context of textureclassificationontheMITVisionTexture (VisTex)database [38]. Thepurposeof thisexperiment is to classify the textures, by taking into consideration thewithin-class diversity. In addition, the influenceofoutliersontheclassificationperformances isanalyzed. Theobtainedresults for theRLD arecomparedto thosegivenbytheRGD[15]andtheWD[17]. TheVisTexdatabase contains40 images, consideredasbeing40different texture classes. The databaseusedfortheexperiment isobtainedafterseveralsteps. Firstofall,eachtextureisdecomposed into169patchesof128×128pixels,withanoverlapof32pixels,givingatotalnumberof6760textured patches. Next, somepatches are corrupted, in order to introduce abnormal data into the dataset. Therefore, their intensity ismodifiedbyapplyingagradientof luminosity. Foreachclass,between zeroand60patchesaremodifiedinorder tobecomeoutliers.AnexampleofaVisTex texturewithone of itspatchesandanoutlierpatchareshowninFigure1. 376