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RELATIVE PRINCIPLE COMPONENT AND RELATIVE PRINCIPLE COMPONENT ANALYSIS ALGORITHM

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RELATIVE PRINCIPLE COMPONENT AND RELATIVE PRINCIPLE COMPONENT ANALYSIS ALGORITHM

RELATIVE PRINCIPLE COMPONENT

    AND RELATIVE PRINCIPLE

    COMPONENT ANALYSIS ALGORITHM

    V_01.24No.1JOURNALOFELECTRONICS(CHINAJanuary2007

    RELTIVEPRINCIPLECOMPONENTANDRELTIVEPRINCIPLE

    COMPONENTANAISISALG0RITHM

    W_enChenglinWangTianzhen'HuJing

    (DepartmentofAutomation,HangzhouDianziUniversity,Hangzhou310018,China) '(DepartmentofElectricalAutomation,ShanghaiMaritimeUniversity,Shanghai200135,China)

    "(DepartmentofComputerandInformationEngineering,HenanUniversity,n9475001,China)

    AbstractInthisletter,thenewconceptofRelativePrincipleComponent(RPC)andmethodofRPC

    Analysis(RPCA)areputforward.Meanwhile,theconceptssuchasRelativeTransform(RT),Ro-

    tundityScatter(RS)andsoonareintroduced.Thisnewmethodcanovercomesomedisadvantagesof

    theclassicalPrincipleComponentAnalysis(PCA)whendataarerotundityscatter.TheRPCselected

    byRPCAaremorerepresentative,andtheirsignificanceofgeometryismorenotable,sothatthe

    applicationofthenewalgorithmwillbeveryextensive.Theperformanceandeffectivenessaresimply

    demonstratedbythegeometricalinterpretationproposed.

    KeywordsRelativePrincipleComponent(RPC);RelativeTransform(RT);RotundityScatt

er(RS)

    CLCindex0231

    DOI10.1007/s11767-006——0097-2

    I.Introduction

    TheclassicalPrincipleComponentAnalysis (PCA)isoneofthemostimportantmethodsfor statisticalcontrolofmultivariateprocess.whichis implementedbyadatamatrixwithafinitese- quenceofprocessmultivariate.PCAcallreflect informationfromtheoriginaldatamatrixbyuseof afewnewdata,whicharecalledPrincipleCom- ponentfPC).EveryPCisalinearcombinationof theserowdatafromthedatamatrix,andtheir essentialfunctionistoexplainthevariance.co- variancestructureofthedatamatrix.W_ecannot oilllycompressdataandanalyzedatabyuseofPAC, butalsoapplyittofaultdiagnose,signalprocess, patternidentificationandsoon.

    However,listedinthefollowingaresome

    problemsintheclassicalPCA.

    (1)ThesePCsareobtainedbasedontheei

    genvaluesandeigenvectorsofcovaxiancematrixof Manuscriptreceiveddate:April17,2006;reviseddate: May18,2006.

    SupportedbytheNationalNaturalScienceFoundationof China(No.60434020,No.60374020),InternationalCoop- erationItemofHenanProvince(No.0446650006),and HenanProvinceOutstandingYouthScienceFund (No.0312001900).

    Communicationauthor:W_enChenglin,bornin1963.

    male,Ph.D.,professor.DepartmentofAutomation. HangzhouDianziUniversity,Hangzhou310018,China. Emall:wencl~hziee.edu.an.

    afinitesequencewithprocessmultivariate,butthe orderandthesignificanceofthesePCsonlydepend onthemagnitudeoftheseeigenvalueserea~the magnitudeofeigenvaluestightlycorrelatewith errororabsolutevalueofvariable,atthesametime, theerrororabsolutevalueofavariablerelatewith itsdimension(forexampleonemeterequals100 centimeter).Inaword.itisnotalwaysthatthe biggerthemagnitudeinallerrors,themoreim- portantthecorrespondingvariable.

    (2)ThenumberofPCsandtheircapability

    orpowertotakeorpossessinformationcontalned inoriginaldatamatrixoritscovariancematrixlie onthecorrelationbetweenprocessmultivariate andthedifferencedegreebetweenbiggereigen- valuesandsmallereigenvalues,

    becausethese

    processesmaynotbealwayscorrelatedinreal systemsandthesedatafromafinitesequencewith processmultivariatecanfallintoanapproximative hyperball,itmightbedi~culttoselectsignificant PCsbyusingclassicalPCA.

    II?PrincipleComponentAnalysis(PCA)

    Usually,theessentialcharacteristicsandmost variabilityofacomplexdynamicsystemwhichis perfectlydescribedbyuseofalotofprocess variablescanbeacquiredbyafewPCs.

Consideringthevariablesofadynamicsystem

    x(k)三【()z2(k),.()T?R(1)

WENeta1.RelativePrincipleComponentandRelativePrincipleComponentAnalysisAlgo

    rithm

    amatrixcomposedofafinitesequencewiththe

    processvariablesis

    x(k,k+N1)[X(k)X(k+1)x(k+N1)]

    (2)

    Ifthe/-throwofthematrixx(k,k+N1)is

    denotedby

    Xi:[z()z(+1)zt(+N1)],

    i=1,2,,n

    thenx(k,k+N1)canberewrittenas

    x(k,k+N1)

    (3)

    (4)

    ThecovariancematrixExofamatrixXcan becomputedby

    =

    E{(,+N1)E{X(k,+?一1))

    .(,k+N1)-E{X(k,k+?一1))T)(5)

    Thenitseigenvaluesandcorrespondingeigen- vectorofcovariancematrixcanbealso computedrespectivelyasfollows and

    where

    

    I=0

【入一】:0,i=1,,n

    et=I(1)(2)…岛(n)Ir1?

    andsupposel2.

    WitheifromEq.(7),wehave

    =

    (e1)X=e1(1)五十e1(2)++e1()

    tJ2=()x=e2(1)X1+e2(2)++e2(n)

    (6)

    (7)

    Vn=()X=e~(1)X1+(2)++(n)

    (8)

    andselectm(m<n)vectorsVl,tJ2,,tobe

    PCs.

    III.RelativePrincipleComponent Analysis(RPCA)

    TosolvetheseproblemsinPCA,theconceptof

    relativeprinciplecomponent(RPC)andthe methodofRelativePrincipleComponentAnaly8i8

    (RPCA)areputforwardinthissection. 1.RelativeTransform(RT)

    Giventhefollowingmatrix

    X(1,N)=

    -(1)Xl(2)Xl(?)

    (1)(2)(?)

    ;i;

    (1)(2)(?)

    Definition1RelativeTransform(RT) Let

    X:MX

    (9)

Xl(1)X1(2)Xl(?)

    (1)(2)z2(N)

    Xn(1)(2)(?)

    z(1)z(2)(?)

    (1)(2)(?)

    xff(1)z(2)R(?)

    (10)

    ThenEq.(10)iscalledRT,Mistheoperatorof relativetransform.andistherelativematrixof matrixX

    where

    (=M,x(J)(11)

    Mi=

    maxlx,(j)l'.

    ?

    i:1,2,,n(12)

    InEq.(12),istheproportioncoefficientwhich reflectstheimportancedegreeofthevariable )?1/I)Ii8thestandardizationfactor

    correspondingtoeachoriginalvariable()or

    vector.Theobjectiveofthestandardization factorgetsridofdimensionofeverydatainthe matrixX,themagnitudeofeveryproportionco

    efficientshowsthesignificanceoftheoriginal variable().TheprocessofRTisshowninFig.1. NotedthatRTcannotchangethesystem relativity,becauseitisalineartransformation. Definition2RotundityScatter(RS)

    IftheseeigenvaluesAI,,,.fromEq.(6)are

    00

:::

    2

    0M0

    0;0

    ;

110JOURNALOFELECTRONICS(CHINA),Vo1.24No.1,January2007

    approximatelyequa1.thematrixcomposedofa finitesequencewithprocessmultivariatex(k)is referredtoasRS.

    一一一一一.一一一.一一一.一一一一一...一一一

    Fig.1Therelativetrans~rmmodel

    CommentIfprocessmultivariatesequence matrixisanRS,thesevectors(1)x(2)

    (?)willformahyperballinR.

    CommentTherearefollowingpropertiesfor anRT

    (1)RSofamultivariatesequencematrix canbesuccessfullyadjustedbyuseofanappro- priaterelativetransform,SOthatitsrelativema- trixXRhasbetterperformancethanthematrix x.

    (2)ItiseasytoselectRPCsbyuseofthe covariancematrixofwhichmayhavestronger abilitytorepresentactualsystemthanPSc. 2.ComputingRPCs

    TheseRPCs,,,canbegainedbythe

    followingsteps.

    (1)Computingthecovariancematrix offromEq.(10,

    xE{[-E(X)儿一E(X)n(13) (2)Calculatingrelativeeigenvaluesand

    itseigenvectorrespectivelyby

    

    x=0

    and

    [一】e=0,i=1,,

    Where

    --

    --

    [e~(1)(2)).r

    suppose.

    (3)ObtainingtheRPCs Giventhefollowingtransformation

    eaR(1)e1~(2)()

    (1)(2)()

    (1)(2))

    x

    x

    x:

    (14)

    (15)

    (16)

    :e(17)

    thenselectm(m<)vectorsVl",,,Ras RPCs.

    SimilartoPCA,theeffectofRPCt,is

    JF{:

    IV.AnExample

?i=1

    ×100%(18)

    Thislettergivesasimulationexampletopre- sentPCAandRPCAwhenthemultivariatese- quencematrixofsystemisRS.Parameterset

    tingandsimulationresultareasshowninTab.1 and6.2respectively.Fig.2showsthecompara- tivegeometricalrepresentationofPCAandRPCA. Fig.2(a)istheoriginalmultivariatesequencema- trixofsystemwithRS,isobtainedby

    PCA,theellipseisjustlikeround,cannot

    "replace"theoriginalsystemmatrixWhen RPCAiSused,withRSiSchangedthroughRT, asshowninFig.2(b),where>isobtained andP1can"replace"theoriginalsystemmatrix Tab.1Parameter~ttmg

    Systemmatrix

    ThenumberofmultivariatesequenceN Thenumberofvariable

    E{X)

    Theproportioncoefficient

    Theproportioncoefficient

    RS

    62

    2

    0.5,0.5

    1

    3

    0.0765

    O.O5o4

0.6221

    0.0573

    6O.2

    91.57%

    3

    2

    l

    0

    -

    10123

    (a)PCA,^J---x2(b)RPCA,>

    Fig.2ThecomparativegeometricalrepresentationofPCAand

    RPCA

    R

    2515O5l

    L0O

    ;

WENetaLRelativePrincipleComponentandRel~ivePrincipleComponentAnalysisAlgor

    ithm111

    V.Conclusion

    Inthisletter.theconceptofRPChasbeen successfullyintroduced,andthemethodofRPCA hasbeeneffectivelyimplemented.Theseachieve- meritsareowingtotheseproblemshappenedin classicalPCA.suchasRSaboutmatrixX.The approachofRPCresolvessuccessfullytheabove problemsandpossessesthefollowingadvantages. (1)RPCAcanavoidtheshortcomingthat, thebiggertheabsolutevalueofasystemvariableis,

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