gaussian mixture model (gmm)：高斯混合模型（gmm）

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1、GaussianMixtureModel(GMM)andHiddenMarkovModel(HMM)SamudravijayaKTataInstituteofFundamentalResearch,Mumbaichief@tifr.res.in09-JAN-20091of88PatternRecognitionModelTrainingGenerationInputSignalProcessingTestingPatternOutputMatchingGMM:staticpatternsHMM:sequentialpatte

2、rns2of88BasicProbabilityJointandConditionalprobabilityp(A,B)=p(A

3、B)p(B)=p(B

4、A)p(A)Bayes’rulep(A

5、B)=IfAisaremutuallyexclusiveevents,p(B)=ip(A

6、B)=p(B

7、A)p(A)p(B)p(B

8、Ai)p(Ai)p(B

9、A)p(A)ip(B

10、Ai)p(Ai)3of88NormalDistributionManyphenomenonaredescribedbyGaussianpdfp(x

11、θ)=√2π

12、σ211exp−2σ2(x−µ)2(1)pdfisparameterisedbyθ=[µ,σ2]wheremean=µandvariance=σ2.Aconvenientpdf:secondorderstatisticsissufficient.Example:HeightsofPygmies⇒Gaussianpdfwithµ=4ft&std-dev(σ)=1ftOR:Heightsofbushmen⇒Gaussianpdfwithµ=6ft&std-dev(σ)=1ftQuestion:Ifwearbitrarilypickap

13、ersonfromapopulation⇒whatistheprobabilityoftheheightbeingaparticularvalue?4of88IfIpickarbitrarilyaPygmy,sayx,thenPr(Heightofx=4’1”)=√2π.111exp−2.1(4′1′′−4)2Note:Heremeanandvariancesarefixed,onlytheobservations,x,change.Alsosee:Pr(x=4′1′′)≫Pr(x=5′)Pr(x=4′1′′)≫Pr(x=3′

14、)5of88(2)ANDConversely:Givenaperson’sheightis4′1′′⇒Personismorelikelytobeapygmythanbushman.Ifweobserveheightsofmanypersons–say3′6′′,4′1′′,3′8′′,4′5′′,4′7′′,4′,6′5′′andallarefromsamepopulation(i.e.eitherpygmyorbushmen.)⇒thenmorecertainwearethatthepopulationispygmy.More

15、theobservations⇒betterwillbeourdecision6of88LikelihoodFunctionx[0],x[1],...,x[N−1]⇒setofindependentobservationsfrompdfparameterisedbyθ.PreviousExample:x[0],x[1],...,x[N−1]areheightsobservedandθisthemeanofdensitywhichisunknown(σ2assumedknown).NL(X;θ)=p(x0...xN−1;θ)=i=

16、0p(xi;θ)=1(2πσ2)N

17、21exp−2σ2Ni=0(xi−θ)2(3)L(X;θ)isafunctionofθandiscalledLikelihoodFunctionGiven:x0...xN−1,⇒whatcanwesayaboutvalueofθ,i.e.bestestimateofθ.7of88MaximumLikelihoodEstimationExample:Weknowheightofapersonx[0]=4′4′′.Mostlikelytohavecomefromwhichpdf⇒θ=3′

18、,4′6′′or6′?MaximumofL(x[0];θ=3′),L(x[0];4′6′′)andL(x[0];θ=6′)⇒chooseθ=4′6′′.Ifθisjustaparameter,wewillchooseargmaxL(x