Transcription of Singular Value Decomposition & Independent Component ...
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Singular Value Decomposition & Independent Component ...
HST582 ,2005 IntroductionInthischapterwewillexamineho wwecangeneralizetheideaoftransforminga timeseriesinanalternativerepresentation, suchastheFourier(frequency)domain,tofaci li-tatesystematicmethodsofeitherremoving ( ltering)oradding(interpolating) , wewillexaminethetechniquesofPrincipalCom ponentAnalysis(PCA)usingSingularValueDec omposition(SVD),andIndependentComponentA nalysis(ICA).Bothofthesetechniquesutiliz ea representationofthedataina statisticaldomainratherthana ,thedatais projectedontoa newsetofaxesthatful llsomestatisticalcriterion,whichimplyind ependence,ratherthana thattheFouriercomponentsontowhicha datasegmentis projectedare xed, If thestructureofthedatachangesovertime,the ntheaxesontowhichthedatais essentiallya methodforseparatingthedataoutintoseparat esourceswhichwillhopefullyallowustoseeim portantstructureina ,bycalculatingthepowerspectrumofa segmentofdata, (amplitudesquared)alongcertainfrequencyv ectorsis thereforehigh,meaningwehavea strongcomponentinthesignal1atthatfrequen cy.
12.2 Matrix transformations as lters The simplest ltering of a time series involves the transformation of a discrete one dimen-sional ( ) time series , consisting of points such that
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