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1、AtutorialonPrincipalComponentsAnalysisLindsayISmithFebruary26,2002Chapter1IntroductionThistutorialisdesignedtogivethereaderanunderstandingofPrincipalComponentsAnalysis(PCA).PCAisausefulstatisticaltechniquethathasfoundapplicationinfieldssuchasfacerecognitionandim
2、agecompression,andisacommontechniqueforfindingpatternsindataofhighdimension.BeforegettingtoadescriptionofPCA,thistutorialfirstintroducesmathematicalconceptsthatwillbeusedinPCA.Itcoversstandarddeviation,covariance,eigenvec-torsandeigenvalues.Thisbackgroundknowledg
3、eismeanttomakethePCAsectionverystraightforward,butcanbeskippediftheconceptsarealreadyfamiliar.Thereareexamplesallthewaythroughthistutorialthataremeanttoillustratetheconceptsbeingdiscussed.Iffurtherinformationisrequired,themathematicstextbook“ElementaryLinearAlg
4、ebra5e”byHowardAnton,PublisherJohnWiley&SonsInc,ISBN0-471-85223-6isagoodsourceofinformationregardingthemathematicalback-ground.1Chapter2BackgroundMathematicsThissectionwillattempttogivesomeelementarybackgroundmathematicalskillsthatwillberequiredtounderstandthep
5、rocessofPrincipalComponentsAnalysis.Thetopicsarecoveredindependentlyofeachother,andexamplesgiven.Itislessimportanttoremembertheexactmechanicsofamathematicaltechniquethanitistounderstandthereasonwhysuchatechniquemaybeused,andwhattheresultoftheoperationtellsusabo
6、utourdata.NotallofthesetechniquesareusedinPCA,buttheonesthatarenotexplicitlyrequireddoprovidethegroundingonwhichthemostimportanttechniquesarebased.IhaveincludedasectiononStatisticswhichlooksatdistributionmeasurements,or,howthedataisspreadout.Theothersectionison
7、MatrixAlgebraandlooksateigenvectorsandeigenvalues,importantpropertiesofmatricesthatarefundamentaltoPCA.2.1StatisticsTheentiresubjectofstatisticsisbasedaroundtheideathatyouhavethisbigsetofdata,andyouwanttoanalysethatsetintermsoftherelationshipsbetweentheindividu
8、alpointsinthatdataset.Iamgoingtolookatafewofthemeasuresyoucandoonasetofdata,andwhattheytellyouaboutthedataitself.2.1.1StandardDeviationTounderstandstandarddeviation,weneedad
