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200799:100026788(2007)0920142206EWMA1,2,2(11,221008;21,710072):EWMA,Markovchain,,.:EWMA;;;:O21311:AATwo2dimensionalEWMAControlChartwithVariableSamplingIntervalsJIMing2ming1,SUNHao2,TANGWei2guang2(11SchoolofMathematics&PhysicsScience,XuzhouInstituteofTechnology,Xuzhou221008,China;21DeptofAppliedMathematics,NorthwesternPolytechnicalUniversity,Xian710072,China)Abstract:Atwo2dimensionalexponentiallyweightedmovingaveragecontrolchartwithvariablesamplingintervalsisproposed.TheMarkovchainmethodisusedtocalculatetheaveragetimetosignal.Thedatumshowthatthisproposedchartismoreefficientindetectingshiftsthanthefixedsamplingintervalcontrolchart.Keywords:two2dimensionalEWMAcontrolchart;variablesamplingintervals;averagetimetosignal;Markovchain:2006205231:(79970022);(02J53079);(N5CS0002):(1982-),,,,,:.1,,,Reynoldsetal[1]Shewhart,.(CUSUM)[2](EWMA)[3,4].,,.EWMA,Markovchain,,,.2MEWMAXt=(Xt1,Xt2,,Xtp)T(t=1,2,)p(p2)t,Xt,,XtNp0,X,0,X.EWMA,MEWMA[5]:Wt=r(Xt-0)+(1-r)Wt-1,W0p,r,0r1.Qt=WTt-1WtWtH,,H0ARL(AverageRunLength).r=1,MEWMAHotellings2.WtWt,:Wt={r[1-(1-r)2t]P(2-r)}X,Wt[rP(2-r)]X,X0,X,Lowryetal[6],MEWMAARL(-0)T-1X(-0).X,Y=-1P2X(X-0),Y-1P2X(-0),.MEWMAARL(-0)T-1X(-0),X,,u=(2-r)Pr,Qt=uWt2.p=2EWMA.3EWMA311EWMAp=2,Xt=(Xt1,Xt2)T,Wt=(Wt1,Wt2)T,Qt=uWt2=uW2t1+W2t2,EWMAARL,qt=Wt=W2t1+W2t2Qt=uWt2H,,,Wtu-1P2H1P2,qtu-1P2H1P2,.qt,UCLUCL=u-1P2H1/2qtUCL,.,UCL,1.1EWMA312(VariableSamplingIntervals,VSI),:,,,,,,,,,,.,(FixedSamplingInterval,FSI)[7].,d1d2,d1d2,3419EWMA,d1;,d2;,,.,ARL,,ARL,,ARL.ARL,,ARL,Reynoldsetal[1]:(AverageTimeToSignal,ATS):.(AverageNumberofSamplesToSignal,ANSS):.d1d2VSI,idi(i=1,2),d0,d0=di(i=1,2).ATS,ANSS,1,2:ANSS=1+1+2ATS=d0+d11+d22=1+1ANSS,d0=d11ANSS,d0=d2ATSANSS:ATS=dANSS,d=d1+d2(1-),d,d.,ANSS,VSIFSI,,ATS,ATS.ATS,Markovchain,Montecarlo,MarkovchainVSIEWMAATS.313VSIEWMA,,UWL=u-1P2H1P2(0HH),,2.2VSIEWMAd1d2,d1d20,:qt,d1;qt,d2,n.VSIEWMA.31311,m+1g=UCLP(m+1),3,Markovchain,E0,E1,,Em,Em+1,.b=(b0,b1,b2,,bm)T,bi(i=0,1,2,,m)qtEi,Ei,bi=d1,,bi=d2.Rungeretal[8]m+1,EiEjpij,441200793VSIEWMApij=P{(j-015)2g2Pr22(2,c)(j+015)2g2Pr2},j0P{2(2,c)0152g2Pr2},j=0c=[(1-r)igPr]2,2(2,c)2,c2.[8].Markovchainm+1P=[pij](m+1)(m+1).M=(I-P)-1=[mij](m+1)(m+1),Im+1,EWMA,W0=(0,0)T,E0,EWMAATSATS=mj=0m0jbj31312,0=(0,0)T=(,0)T,1,X1,X2.4VSIEWMA,w12m1+1,Wt12m1+1Markovchain,g1,g1=2UCLP(2m1+1),h1,h2,,h2m1+1.,w22m2+1g2=2UCLP(2m2+1),Wt22m2+1Markovchain,v1,v2,,v2m2+1.,,4,Markovchain,E1,E2,,Ek,k(2m1+1)(2m2+1).Ei(i=1,2,,k)w1ix(ix(h1,h2,,h2m1+1))w2iy(iy(v1,v2,,v2m2+1)),Ei(ix,iy).b=(b1,b2,,bk)T,Ei(i=1,2,,k),bi=d1,,bi=d2.Ei(i=1,2,,k)Ej(j=1,2,,k)(ix,iy)(jx,jy)pij,pij=p(i,j)=p[(ix,iy),(jx,jy)]=h(ix,jx)v(iy,jy)h(i,j)v(i,j).h(i,j)=P(Wt1j|Wt-1,1i)=P(-UCL+(j-1)g1rXt1+(1-r)Wt-1,1-UCL+jg1|Wt-1,1=c1i)=P(-UCL+(j-1)g1rXt1+(1-r)Wt-1,1c1i-UCL+jg1)=P[(-UCL+(j-1)g1-(1-r))c1iPr-Xt1-(-UCL+jg1-(1-r)c1iPr-]Xt1-N(0,1)a1=(-UCL+(j-1)g1-(1-r)c1i)Pr-a2=(-UCL+jg1-(1-r)c1i)Pr-5419EWMAh(i,j)=a2a112e-x22dx,c1i=-UCL+(i-015)g1v(i,j)=P(Wt2j|Wt-2,1i)=P(-UCL+(j-1)g2rXt2+(1-r)Wt-1,2-UCL+jg2|Wt-1,2=c2i)=P(-UCL+(j-1)g2rXt2+(1-r)c2i-UCL+jg2)=P[(-UCL+(j-1)g2-(1-r)c2i)PrXt2(-UCL+jg2-(1-r)c2i)Pr]Xt2N(0,1)b1=(-UCL+(j-1)g2-(1-r)c2i)Prb2=(-UCL+jg2-(1-r)c2i)Prv(i,j)=b2b112e-x22dx,c2i=-UCL+(i-015)g2MarkovchainkP=[pij]kk,MarkovChainEk0(k0(1,2,,k)),M=(1-P)-1=(mij)kk,EWMAATSATS=kj=1mk0jbjX1=0,,,.4VSIFSIEWMA,,,ATS.VSIFSIrUCL,ANSS,,ANSS.=0,rH,d1d2,VSIFSI,.0ATS,ATS,.1H=9,H=4,rX1VSIFSIEWMAATS,0,0.,0=0,d=1.,r,(=0)ATS,(0)VSIFSI.,r=012,d1d2,ATS,d1=11050,d2=01586,ATS130173,=015,ATSVSI:28105,FSI:29173.,,,VSIFSI.d1d2,d1d2,.,,EWMA,VSIFSI,,,,.641200791VSIEWMAFSIEWMAATSr=011r=012r=013r=014r=015(d1,d2)(11050,01600)(d1,d2)(11050,01586)(d1,d2)(11050,01588)(d1,d2)(11050,01592)(d1,d2)(11050,01595)VSIFSIVSIFSIVSIFSIVSIFSIVSIFSI01017018617018713017313017311310411310310317010316998134981340152619929112281052917330160311983316234182361953810211091491017181559164817991829156101551016711167115516861414168513341425105414651074165513121041124161312831702196313721843123218031223102176310421142139118821111173119311631181:[1]ReynoldsMRJr,AminRW,ArnoldJC,etal.X2barchartswithvariablesamplingintervals[J].Technometrics,1988,30(2):181-192.[2]ArnoldsJC,ReynoldsMRJr.CUSUMcontrolchartswithvariablesamplesizesandsamplingintervals[J].JournalofQualityTechnology,2001,33(1):66-81.[3]SaccucciMS,AminRW,LucasJM.Exponentiallyweightedmovingaveragecontrolschemeswithvariablesamplingintervals[J].CommunicationsinStatistics2SimulationandComputation,1992,21(3):627-657.[4]ReynoldsMRJr,ArnoldJC.EWMAcontrolchartswithvariablesamplesizesandvariablesamplingintervals[J].IIETransactions,2001,32(6):511-530.[5]PrabhuSS,RungerGC.DesigningamultivariateEWMAcontrolchart[J].JournalofQualityTechnology,1997,29(1):8-15.[6]LowryCA,WoodallWH.Amultivariateexponentiallyweightedmovingaveragecontrolchart[J].Technometrics,1992,34(1):46-53.[7].[J].,2002,18(3):316-333.WangZhaojun.Thedesigntheoryofadaptivecontrolcharts[J].ChineseJournalofAppliedProbabilityandStatistics,2001,18(3),316-333.[8]RungerGC,PrabhuSS.Amarkovchainmodelforthemultivariateexponentiallyweightedmovingaveragecont