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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-

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

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.

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

(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