aspects of multivariate statistical theory 2ed（下）

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1、AspectsofMultivanateStatisticalTheowROBEI.MUlRHEADCopyright81982.2WSbyJohnWiley&Sons.I~C.'CHAPTER9PrincipalComponentsandRelatedTopics9.1.INTRODUCTIONInmanypracticalsituationsobservationsaretakenonalargenumberofcorrelatedvariablesandinsuchcasesitisnaturaltolookatvariouswaysinwhi

2、chthedimensionoftheproblem(thatis,thenumberofvariablesbeingstudied)mightbereduced,withoutsacrificingtoomuchoftheinformationaboutthevariablescontainedinthecovariancematrix.Onesuchexplora-torydata-analytictechnique,developedbyHotelling(1933),isprincipalcomponentsanalysis.Inthisan

3、alysisthecoordinateaxes(representingtheoriginalvariables)arerotatedtogiveanewcoordinatesystemrepresentingvariableshavingcertainoptimalvarianceproperties,Thisisequivalenttomakingaspecialorthogonaltransformationoftheoriginalvariables.Thefirstprincipalcomponent(thatis,thefirstvari

4、ableinthetransformedset)isthenormalizedlinearcombinationoftheoriginalvariableswithmaximumvariance;thesecondprincipalcoinponentisthenormalizedlinearcombina-tionhavingmaximumvarianceoutofalllinearcombinationsuncorrelatedwiththefirstprincipalcomponent,andsoon.Henceprincipalcompone

5、ntsanalysisisconcernedwithattemptingtocharacterizeorexplainthevariabil-ityinavectorvariablebyreplacingitbyanewvariablewithasmallernumberofcomponentswithlargevariance.ItwillbeseeninSection9.2thatprincipalcomponentsanalysisisconcernedfundamentallywiththeeigenstructureofcovariance

6、matrices,thatis,withtheirlatentrootsandeigenvectors.Thecoefficientsinthefirstprincipalcomponentare,infact,thecomponentsofthenormalizedeigen-vectorcorrespondingtothelargestlatentroot,andthevarianceofthefirstprincipalcomponentisthislargestroot.Acommonandoftenvalidcrili-cismofthet

7、echniqueisthatitisnotinvariantunderlineartransformations3moPopuluaonPrincipulComponents38Iofthevariablessincesuchtransformationschangetheeigenstructureofthecovariancematrix.Becauseofthis,thechoiceofaparticularcoordinatesystemorunitsofmeasurementofthevariablesisveryimportant;pri

8、ncipalcomponentsanalysismakesmuchmoresenseifallthevari