欢迎来到天天文库
浏览记录
ID:7511764
大小:9.26 MB
页数:293页
时间:2018-02-17
《aspects of multivariate statistical theory 2ed(下)》由会员上传分享,免费在线阅读,更多相关内容在工程资料-天天文库。
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
此文档下载收益归作者所有