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1、AnIntroductiontoIndependentComponentAnalysis(ICA)吳育德陽明大學放射醫學科學研究所台北榮總整合性腦功能實驗室ThePrincipleofICA:acocktail-partyproblemx1(t)=a11s1(t)+a12s2(t)+a13s3(t)x2(t)=a21s1(t)+a22s2(t)+a12s3(t)x3(t)=a31s1(t)+a32s2(t)+a33s3(t)IndependentComponentAnalysisReference:A.Hyvär

2、inen,J.Karhunen,E.Oja(2001)JohnWiley&Sons.IndependentComponentAnalysisCentrallimittheoremThedistributionofasumofindependentrandomvariablestendstowardaGaussiandistributionObservedsignal=IC1IC2ICnm1+m2….+mntowardGaussianNon-GaussianNon-GaussianNon-GaussianCentr

3、alLimitTheoremPartialsumofasequence{zi}ofindependentandidenticallydistributedrandomvariablesziSincemeanandvarianceofxkcangrowwithoutboundask,considerinsteadofxkthestandardizedvariablesThedistributionofykaGaussiandistributionwithzeromeanandunitvariancewhenk

4、.Partialsumofasequence{zi}ofindependentandidenticallydistributedrandomvariablesziHowtoestimateICAmodelPrincipleforestimatingthemodelofICAMaximizationofNonGaussianityMeasuresforNonGaussianityKurtosisSuper-Gaussiankurtosis>0Gaussiankurtosis=0Sub-Gaussiankurto

5、sis<0Kurtosis:E{(x-)4}-3*[E{(x-)2}]2kurt(x1+x2)=kurt(x1)+kurt(x2)kurt(x1)=4kurt(x1)AssumemeasurementWhiteningprocessiszeromeanandThenisawhiteningmatrixxEDVxzT21-==Iss=}{TEsxA=TEDV21-=TTTEEVxxVzz}{}{=2121--=EDEDEEDTTI=TTEEDExx=}{LetDandEbetheeigenvaluesand

6、eigenvectormatrixofcovariancematrixofx,i.e.ImportanceofwhiteningForthewhiteneddataz,findavectorwsuchthatthelinearcombinationy=wTzhasmaximumnongaussianityundertheconstrainMaximize

7、kurt(wTz)

8、underthesimplerconstraintthat

9、

10、w

11、

12、=1ThenConstrainedOptimizationmaxF(w)

13、,

14、

15、w

16、

17、2=1Atthestablepoint,thegradientofF(w)mustpointinthedirectionofw,i.e.equaltowmultipliedbyascalar.0),(=¶¶wwlL0]2[)(=+¶¶ÞwwwlF1),1()(),(22==-+=wwwwwwTFLllwww-=¶¶Þl2)(FGradientofkurtosis}]){([3}){()()(224zwzwzwwTTTEEkurtF-==wwwzwwzwzwww¶-¶=¶-¶=¶¶å=21424)(3)

18、)((1}}){(3}){()(TTtTTTtTEEF))((2*3)]()[(413wwwwzwz+-=å=TTtTttT]3}][{))[((423wwzwzzw-=TTEkurtsignå==TttTE1)(1}{yyQFixed-pointalgorithmusingkurtosiswk+1=wk+Notethataddingthegradienttowkdoe