吴育德阳明大学放射医学科学研究所台北荣总整合性脑功能研

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1、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研究室IntroductionToPrincipalComponentAnalysisPrincipleComponentAnalysisPrincipleComponentAnalysisseeksaprojectionthatbestrepresentsthedata.Theeigenspacesthatretainthemostsignificantamountofinformationarethosethatcorrespondtothelargesteignevalue.PrincipleComponentAnalysisLetX

2、denoteanm-dimensionalrandomvectorrepresentingtheenvironmentofinterest.WeassumethattherandomvectorXhaszeromean:Letqdenoteaunitvectoralsoofdimensionm,ontowhichthevectorXistobeprojected.ThisprojectionisdefinedbytheinnerproductofthevectorsXandq,asshownbysubjecttotheconstraintPrincipleComponentAnalysisU

3、ndertheassumptionthattherandomvectorXhaszeromean,itfollowsthatthemeanvalueoftheprojectionAiszerotoo:ThevarianceofAisthereforethesameasitsmean-squarevalue,andsowemaywritePrincipleComponentAnalysisThem-by-mmatrixRisthecorrelationmatrixoftherandomvectorX,formallydefinedastheexpectationoftheouterproduc

4、tionofthevectorXwithitself,asshownbyWeobservethatthecorrelationmatrixRissymmetric,whichmeansthatthevarianceoftheprojectionAisafunctionoftheunitvectorq;wemaythuswriteonthebasisofwhichwemaythinkofasavarianceprobe.Tobeconsideredisthatoffindingthoseunitvectoralongwhichhasextremalvalues(localmaximaormin

5、ima):Ifqisaunitvectorsuchthatthevarianceprobehasanextremalvalue,thenforanysmallperturbationoftheunitvectorq,wefindthat,toafirstorderin,,i.e.PrincipleComponentAnalysisFromthedefinitionofthevarianceprobeHence---(1)PrincipleComponentAnalysisTheEuclideannormoftheperturbationvectorremainsequaltounity;th

6、atis:orequivalently,Thismeansthattheperturbationsmustbeorthogonaltoq,andthereforeonlyachangeinthedirectionofqispermitted.---(2)PrincipleComponentAnalysisIfweintroduceascalingfactorintotheequation(2)withthesamedimensionsastheentriesinthecorrelationmatrixRof(1).Wemaythencombine(1)and(2)intoorequivale

7、ntly,itisnecessaryandsufficienttohaveThisistheequationthatgovernstheunitvectorsqforwhichthevarianceprobehasexternalvalues.PrincipleComponentAnalysisisrecognizedastheeigenvalueproblem,commonlyencounteredinli