Seite - 373 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 pointofE, itwill suffice tocheckthat foranygeodesicγ startingfrom Y¯, ddt|t=0E(γ(t))=0 [31] (p.76). Note that: d dt |t=0E(γ(t))= ∫ Pm d dt |t=0d(γ(t),Z)p(Z|Y¯,σ)dv(Z) (26) where forallZ = Y¯ [32]: d dt |t=0d(γ(t),Z)=−gY¯(logY¯(Z),γ′(0))d(Y¯,Z)−1 The integral inEquation(26) is,uptoaconstant, d dt |t=0 ∫ Pm p(Z|γ(t),σ)dv(Z)=0 since ∫ Pm p(Z|γ(t),σ)dv(Z)=1. (ii)Differentiating ∫ Pm exp(− d(Z,Y¯) σ )dv(Z)= ζm(σ)withrespect toσ, it comes that: σ2×d logζm(σ)/dσ=σ2ζ ′ m(σ) ζm(σ) = ∫ Pm d(Z,Y¯)p(Z|Y¯,σ)dv(Z)=E(Y¯|Y¯,σ) whichproves (ii). Proposition 3 (Consistency of YˆN). Let Y1,Y2, · · · be independent samples from a Laplace distribution G(Y¯,σ). The empiricalmedianYˆN ofY1, . . . ,YN convergesalmost surely to Y¯, asN→∞ . ProofofProposition3. Corollary3.5 in [33] (p. 49) states that if (Yn) is a sequenceof i.i.d. random variablesonPmwith lawπ, thentheRiemannianmedian YˆN ofY1, · · · ,YN convergesalmostsurely as N →∞ to Yˆπ, theRiemannianmedianof π definedbyEquation (12). Applying this result to π=L(Y¯,σ)andusing Yˆπ = Y¯,which follows fromitem(i) ofLemma1, shows that YˆN converges almostsurely to Y¯. 4.MixturesofLaplaceDistributions There are severalmotivations for consideringmixtures of distributions in general. Themost naturalapproach is toenvisageadatasetasconstitutedofseveral subpopulations.Anotherapproach is the fact that there is a support for the argument thatmixtures of distributions provide a good approximationtomostdistributions inaspirit similar towavelets. Thepresent section introduces the classofprobabilitydistributions that arefinitemixturesof RiemannianLaplacedistributionsonPm. Theseconstitute themain theoretical tool, tobeusedfor the targetapplicationof thepresentpaper,namelytheproblemof textureclassification incomputervision, whichwillbe treated inSection5. AmixtureofRiemannianLaplacedistributions isaprobabilitydistributiononPm,whosedensity withrespect to theRiemannianvolumeelementEquation(13)has the followingexpression: p(Y|( μ,Y¯μ,σμ)1≤μ≤M)= M ∑ μ=1 μ×p(Y|Y¯μ,σμ) (27) where μarenonzeroweights,whosesumisequal toone, Y¯μ∈Pmandσμ∈ (0,σm) forall1≤μ≤M, andtheparameterM is calledthenumberofmixturecomponents. Section4.1describesanewEMalgorithm,whichcomputes themaximumlikelihoodestimates of themixtureparameters ( μ,Y¯μ,σμ)1≤μ≤M, basedonindependentobservationsY1, . . . ,YN fromthe mixturedistributionEquation(27). 373