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Chapter18PrincipalComponentAnalysisJi-QianFangSchoolofPublicHealthSunYat-SenUniversityE.g.Hotelling(1939)usestwomajorcomponentsforresponsesonentry-examinationtests:verbalabilityquantitativeability.Pre-knowledge:Avariablewithzerovariancedoesnotdistinguishbetweenmembersofthepopulation;onemeasureoftheamountofinformationconveyedbyeachvariableisitsvariance.18.1TheBasicConceptsofPrincipalComponentAnalysis

x2x1Example18.1AverageDevelopmentIndicesofHanYoungadults(Age19-22)in1985ChineseCensus

Howto

summarizemultiplerandomvariables?

Wewouldliketohaveasingleorafewvariablestosummarizemostoftheinformationprovidedbythe6developmentindices.

Note:Anindexwithalargevariationamongsubjectsisagoodsummaryindex.

18.2SearchingforSummaryVariable

18.2.1DefinitionofthePrincipalComponents

18.2.2PropertiesofPrincipalComponents

18.2ComputationandInterpretation

ofPrincipalComponents

18.2ComputationandInterpretation

ofPrincipalComponents

Onlythefirstfewcomponentscouldbethe“principal”ones18.3.2TheNumber

ofPrincipalComponents

18.3.3Correlationbetween

PrincipalComponentsandVariables

Example:ComprehensiveScoresforQuestionnaireofDepressionC1canbeviewedasaweightedaverageofmostofitems.C2isanindicationoflethargicorenergeticstate.C3measuresthefeelingtowardhowothersperceivehim.Example2Studyoncalculusaccumulationforsixanteriorteeth（前齒）ofthelowermandible（下顎）

Variablesarecalculusmeasureon1Rightcanine（右犬齒）2Rightlateral,incisor（右、側(cè)門牙）3Rightcentral,incisor（右、中門牙）4Leftcentral,incisor（左、側(cè)門牙）5Leftlateral,incisor（右、中門牙）6Leftcanine（左犬齒）

PC1:aweightedaverage,calculusaccumulationPC2:acomparisonofincisorswithcaninesPC3:differencesonthetwosides18.4PrincipalComponentAnalysisinRegressionAtoolnotthepurposeofdataanalysis--theintermediatestepinaresearchproject:MultivariateregressionDiscriminantanalysisFactoranalysisClusteranalysis…Co-linearityinregressionPrincipalcomponentanalysisRegressiononprincipalcomponentsSummaryPre-knowledge:Avariablewithzerovariancedoesnotdistinguishbetweenmembersofthepopulation;onemeasureoftheamountofinformationconveyedbyeachvariableisitsvariance.2.Ifwehavemorevariables,wewouldliketohaveaafewvariablestosummarizemostoftheinformationprovidedbythesevariables.PrincipalcomponentsanalysisisoneofthemethodstoSearchsuchsummaryVariables.3.Principalcomponentsarelinearcombinationsoftheoriginalvariables;thefirstprincipalcomponenthasthemaximalvariance;Thesecondprincipalcomponentistheonehasthemaximalvarianceundertheconditionofperpendiculartothefirstone;Thethirdprincipalcomponent…4.Intheory,forpvariables,onemayhavep“principalcomponents”,butonlyafewofthemcancovermostinformationofthepvariables.Howmanyis“afew”?----twoapproaches:Thevarianceofthecomponentisequalorgreaterthan1Thecumulatedvarianceisgreaterthanrequiredvalue5.Oneoftheapplicationsofprincipalcomponentsanalysisistodealwiththeco-linearityprobleminregression:Tosummarizethehighlycorrelatedvariables;WorkouttheregressionbetweenYandprincipalcomponents