PROCESS CAPABILITY ANALYSIS AND ESTIMATION SCHEME FOR AUTOCORRELATED DATA

J Syst Sci Syst Eng (Mar 2010) 19(1): 105-127ISSN: 1004-3756 (Paper) 1861-9576 (Online)DOI: 10.1007/s11518-010-5127-9CN11-2983/NPROCESS CAPABILITY ANALYSIS AND ESTIMATION SCHEME FORAUTOCORRELATED DATA*Jing SUNShengxian WANGZhihui FUResearch Center for Contemporary Management, Key Research Institute of Humanities and Social Sciences atUniversities, School of Economics and Management, Tsinghua Universit, Bejing 100084, P.R. Chinasunj3@sem. tsinghua.edu.cn (网)AbstractAutocorrelation is prevalent in continuous production processes, such as the processes in thechemical and pharmaceutical industries. With the development of measurement technology and dataacquisition technology, sampling frequency is getting higher and the existence of autocorrelationcannot be ignored. This paper analyzes five estimation schemes of process capability for autocorrelateddata. Comparisons among these schemes are discussed for small sample and large sample. Inconclusion, this paper gives a procedure of process capability analysis for autocorrelated data.Keywords: Autocorrelation, estimation, process capability analysis, statitical process control1. Introductionactually varying. The index Cp was designed toProcess capability indices (PCI) establish themeasure the magnitude of the overall processrelationships between the potential processvariation relative to the manufacturing tolerance,performancethemanufacturingwhich is to be used for processes based on dataspecifications. The frst process capability indexthat are normal, independent, and in theto appear in academic literature is the index Cp,statistical control. Clearly, the index measuresdefined by Kane (1986) as:only the potential of a process to provide anUSL- LSLacceptable product and does not take intoaccount whether the process is centered or not.where USL is the upper specification limit, LSLIn order to account for the deviations ofis the lower specification limit, and σ is the .process mean from the target value, severalprocess standard deviation. The numerator ofCp indices, similar in nature to Cp, have beenprovides the range over which the processproposed over the years. These indices attemptmeasurements are acceptable. The denominatorto account for the magnitude of process variancegives the range over which the process ias well as for the process departures from the中国煤化工* This paper is part of Project supported by National Natural Sa.MYHCNMHG'772019, 70621061)and National Social Science Foundation of China (08BTJ002).◎Sytems Engineering Society of China & Springer-Verlag Berlin Heidelberg 2010Sun et al: Process Capability Analyis and Estimation Scheme for Autocorrelated Data106JSyst Sci Syst Engtarget value. One of such indices is Cpx definedindependent, and in the statistical control. Manyas:kinds of control charts could be used as toolsCpk =min[USL-μ μ-LSL]together with out-of-control criteria to evaluate3σwhether process is in control. The range betweenwhere μ is the process mean, USL, LSL andhe 99.865% and 0.135% percentile ofσ as above. Revealing relationships betweenobservations could be used as the denominatorCp and Cp are discussed by Barnett (1990), Kotzinstead of 6σ for the definition of Cp, provided& Johnson (1999). The definition of the indexby ISO 3534-2:2006. For a normal distribution,Cpk can altermatively be written as,the definition is kept the same. In practice, it is ,common to test if observations are collctedd-|μ- m|Cpt=5 m=C,(--5")from in-control process and follow normaldwhere d=(USL- LSL)/2 is half of the lengthdistribution when calculating these indices.One of the most essential assumptions is thatofhespecificationintervalandm=(USL+ LSL)/2 is the mid-point betweenobservations are statistically independent.However, there are many processes, particularlythe lower and the upper specification limit.in chemical industries, where the data areHowever, these indices do not take intoinherently correlated. With the development ofaccount the cost of failing to meeting customers'measurement technology and data acquisitionrequirement. On the other hand, a famousJapanese quality expert, G Taguchi, focuses ontechnology in recent years, sampling frequencythe loss in a product's worth when one of itsis getting higher, and the existence ofautocorrelation cannot be ignored. Therefore,characteristics departs from customers' idealthesindices mayindicate inappropriatevalue. To handle this situation, Hsiang andconclusions if the correlation effort is not takenTaguchi (1985) introduced the index Cpm,independently proposed by Chan et al. (1988). Itinto account, because the variance of subgroupconcentrates on measuring the ability of themean is larger for autocorrelation observationsthan for noncorrelated ones and the expectedprocess to cluster around the target value T,value of variance is smaller than the actualwhich reflects the degrees of process targetingprocess variance. Zhang (1998) studied the(centering). It is defined asindices Cp and Cpk for autocorrelated data.USL - LSLUSL- LSLGuevara & Vargas (2007) extended Zhang'sCpm6σr6、02 +(μ-T)°study to the capacity indices Cpm and Cpmk andmade a general comparison of these four indices.whereo咋=o +(u-7)2 =E[(x-n)?]Wang & Sun (2006) compared processincorporatestwovariationcomponents: .capabilityindicesp, Cpk andprocessvariation with respect to the process mean andperformance indices Pp, Ppk based on individualdeviation of the process mean from the value T.observation for processes based on data that areObviously, these indices could be used for中国煤化工，the stistcalprocesses based on data that are normal, co1HC N M H Gprocess standardSun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated DataJ Syst Sci Syst Eng107deviation is estimated byR,1d2 where R, isdifferent schemes for process capability indicesthe average moving range and d2 is functionsCp, Cpt and Cpm for autocorrelated data will beof subgroup size to be 2. For Pp and Ppks processdiscussed in the way similar to processvariation is estimated to provide the totalmonitoring with autocorrelation. Consideringvariation by sample variance of all observationsthe number of observations, comparison amongas follows:these schemes will be, organized for smallsample and large sample respectively in terms of8=g(x-邓. (1)MSE. In the end, a procedure of process日n-1capability analysis for autocorrelated data iswhere n is the number of all observations. It isgiven.concluded that process performance index isrecommended as more reliable. Sun et al (2009)2. AR(1) Modelenhanced the comparison of Cp, Cpk and Pp, PpkAn AR(1) process is given by Box et al.fromindiyidual observation0 group(1994)observations. For Cp and Cpk, process standardX;-μ=q(X_ -4)+&;(2)deviation is estimated from the averagewhere X, is value of observation, μ is thewithin-subgroup variability. For Pp and Ppt,process mean and E is a white noise processprocess variation is estimated from the totalwithzeromean andvariancevariability including within subgroup variabilityε, ~ NID(0,σZ),NIDmeansnormaland subgroup-between-subgroup variability,independent distribution. It is also assumed thatgiven by formula (1). Both the STD and MSE of-1<φ<1. For the simplicity of the formula, letPp and Ppk are smaller than that of Cp and Cpt. ItZ;=X,-μ, thenis intended that sample standard deviation of allz; =φZ_ +&observations is an alternative method to workfor the assessment of process capability indices,For AR(1) model, the autocorrelation coefficientwhich has been used to analyze the statistical between X, and Xr-k isditribution of the estimators of Cp, Cpk and CpmPr=φk ,k=1,2,..(3)forautocorrelateddataandcoverageprobabilities of some confidence intervals forParameters of the model are unknown underthese estimates provided by Guevata & Vargasmost circumstances. It is necessary to estimate(2007). In this paper, this method is denoted asthese parameters from massive data, forexample:Scheme with Observations (Scheme O).There are two methods, a model-based and aμ=x==Zx(4)mode-free approach, which deal withautocorrelations in the literature for processThe estimation of φ and o depend on thecontrol. As the asssment of process capabilityestimation method of AR(I) model. Commonbegins after the evaluation of the process in ametl!YH中国煤化工on include Yulestate of statistical control, in this paper fiveWallCNMHG;Method(LSM)Sun et aL: Process Capability Analyis and Estimation Scheme for Autocorrelated Data108J Syst Sci Syst Engand Maximum Likelihood Estimation (MLE).process, i.e. φ=0, the variance of X is theUsing LSM for AR(1) model, samplecommon formulastatistics 应and p are often used to estimatethe autocovariance rh and the autocorrelationVar[X]=9nP:&=j4x=-(Z[x,()(x.+-=]k=.1Let f(n,P2)=1+=2(n-k)pr , then then台varianceof X isp=p_=i_σ2Then the estimator ofσ2 isVar[X]=o f(n,px)(7)og=2s3.2 Parameter Estimation for Processwhere Qs -2[z,-02Z_B.Standard DeviationWhen the process capability index like Cp isFor a large n, there is lttle differenceused for processes based on data that are normalbetween LSM and Yule-Walker estimation butand independent, the assessment of processLSM is more precise, given by He (2003). Forcapability begins after control issues in both theAR(p) models LSM and MLE estimation are theX and R charts have been resolved. σ issame. In this paper, LSM method is used forestimated fom the average within-subgroupparameter estimation.variability and is given by R/d2. Ifthe X ands charts have been used for process control, o3. Estimators of These Indices withis given by 51c4 to show the averageAutocorrelated Observationswithin-subgroup variability where d2 and C4are functions of subgroup size, widely tabulated3.1 Parameter Estimation for Processin quality control books and literature based onMeanthe assumption of normality and independence.The estimation of process mean μ is firstlyAn unbiased estimator s2 is often used toanalyzed. According to (4), an estimator of μestimate total process variance σ2 for theis X.And又is an unbiased estimatorof μ.evaluation of process performance indices, P,E[x]=μ(5)and Ppb, which is defined by formula (1). Fromthe study on comparison of process capabilityvar[幻]=[+字(1-k)pindices (Cp and Cpx) and process performancenk=indices (P。and Pp) provided by Sun et al._o[.. 20(n-1-np+o")](2009), it is intended that total process variation1+-n(1-p)2(6)by formula (1) is an altermnative method to workor!中国煤化工pability indices.When autocorrelation does not exist in theCNMHGandformula(1)Sun et aL: Process Capabiliy Analysis and Estimation Scheme for Autocorrelated DataJ Syst Sci Syst Eng109is chosen as the estimation of σ，these are the3.3 Estimating Cp with Autocorrelatedfollowing conclusions (Zhang 1998):DataTo distinguish from traditional index, Cpr isused to denote Cpr for autocorrelated data. Theestimator of Cpr isUSL- LSLq(n-1-mq+p°)](8)Cm=6S(12)n(n-1)(1-)ThenVa[s2]=-2o4+210-)2(n-1)1Va([C.]=(U5ucSI) Va[s5][+(n-)-0]It is difficult to get precise expressions ofE[s- ] and Var[s-1]. However, according- z∞(-1-)00})(9)to Kotz & Johnson (1993)， approximaten i=0j=0expressions. can be derived by statisticalLetdifferential method (Scagliarini 2002),g(n,px)=1--2(-)p)E[s-"]=[E($)]3 Var(s2)8[E(5)] ]G(n,Pn)=-{n+2)(n-k)pkn-1[3G(n,p)p2σ[g(n,)]"I[ 4(n-1)[g(n,p,)]n+222((n-k)Pek=lVar[s3]G(n,p,)-3E艺(n-i-)p0Va[5"]=-4[E(s2]2o2(n-1)[8(n,p)]ni=0j=0Then the expected value and varianceof s2，3G(n,p;)can be rewriten as:E(Cm)=C7[(n,p)'"l" 4(n-()E(n,.)] ]e[s]=o'g(n,p,)(10)(13)V=[s]=5_nG(n,Px) (11)中国煤化工(14)1YHCN M HGO)]Sun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated Daia110J Syst Sci Syst EngThe extent of relative offset on expectedE|x-m|value of estimator Cpr compared to Cpr isσ、Pex[等]+tn~denoted by B(Cpr,n,p;)B(p,p)=(.)/c-1+-3G(n,p)Let ξ三E|灭-m/σ , then[g(n,p1)]"2 [ 4(n-)[g(,p)] 」E[C°a](15)For AR(1) model, autocorrelation coefficient=[-1u-m+1u-m-l18m]E[s"]between X and X1-k is Pk=φ^ ,k=1,2,.,so expected relative offset of estimator of Cpr isalso denoted as B[(C,n,p).4-1](心]3.4 Estimating Cpk with Autocorrelated[Co些李C.,m)+]DataLet Cpr denote the index Cpk withLet B(rp.,n,k.p)=E[Cpr]/Cp-l,autocorrelated data. The estimator of Cpo isthend-|x-m|Cpo=sS(16)B(ot.n,.,p,)= B(r.,",)whereasaboved =(USL-LSL)/2 and+(-(5(5~(,m.)+] (7)m =(USL + LSL )/2. According to Zhang (1998)appendix A, it can be inferred that theSimilarly,correlation coefficient between| 刘and S isnear to zero, andVa[Cm]ECm]=E[~|1- m]xe[s~]C咖 f(n,P),G(n,p)8g(n,p:)[ 9mCpx 2(n-1)g2(n,p,)]From (7) it can be inferred that(18)文-m~ N[u-m,o2f(n,p)/r].Letf=f(n,p;), then3.5 Estimating Cpm with Autocorrelated|& - m=σ应EXP_n(4-m)户 ILet the index Com with autocorrelated datann2o2fbe CpmrUSL- LSLCpmr=+(μ-m)|中国煤化工+(u-T)Let k =(μ-m)/σ , thenMHCNMHGSun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated DataJ Syst Sci Syst Eng111Cm =USL- LSL(19)7-1/:Var(s})6SrE[5s)]=[(S}]"1+3where(E[57]_(x.-T)=[2(+43)]'13G(n,k2,P.)4n(1+经)」Var[$s']=-Var(Ss})=-G(n,k2,P.)then(E(S)] 22no[+招]e[sz]=o2 +(u-7)} =σ[I+陉]ThenVar[Sξ]=°x3G(n,k2,p;)nE[m]=Cm| 1+4n(1+经户n+2步(n-i)p? +2k气(+(+(0-0)]=Cm[+B(rw,n,.P)] (20)where k =(μ-T)1σ.LetB(Cpmr,n,k2,p;) stands for theG(n,k2,P.)=1+2(n-i)pfn问difference from true value of CpmrG(n,k2,P)Var[Cmr]=Cmr2n(1+超)(21)Va[s}]=20-G(n,6,p)4. Five Schemes of Process CapabilityIt can be inferred that expected value andIndices Estimation forvariance of the estimator are:Autocorrelated DataThere are two main methods, a model-basedand a mode-free approach, to deal withE[Cm]=USL-[$5e[sr]6autocorrelations in the literature of processcontrol. As the assessment of process capabilitybegins after the evaluation of the process in aIt is difficult to get a precise expression ofstate of statistical control, five schemes areanalyzed in the way similar to processE[s] and Va[ s;]. However, we can getmon中国煤化工MSE is used toan approximateexpression by statisticalmaknodel-based anddifferential method (Kotz & Johnson, 1993)MHCNMH(is defined asmodc-l ouiJlv. sui 一pr,Sun et a: Process Capability Analysis and Estimation Scheme for Autocorrelated Data112JSyst Sci Syst Enganalyzing confidence intervals of these indices.MISe(E(a-_C.0]4.2 Modifed Scheme with Observations=va(c,)+ E[(C,)Cm ]2)For Scheme O, autocorelated observationsSimiarly, MsSE(Cpo ) and MSE(Cmm ) couldare applied directly to analyze process capabilitybased on total process variability. Then it isbe given.considered to modify Scheme O with the extentOne way to control autocorrelated process isto which a sample PCI is over- 01to adjust control limits of control chart in orderunder- estimating the true PCI, which is calledto take non-random variation caused bymodified scheme with observations, denoted asautocorrelation into account based on processSchemeA. Use Cpr as an example,time series model. From the view point 0methodology, it isrealizable to adjust(23)specification limit for process capability analysisCm=.xowith autocorrelated data. However, in practice itmay cause misunderstanding for the customers.In section 33, B(Cp,n,p)= E(Cm)/Cm-1Specification limits are usually establishedhas been defined to show the difference betweenbased on intermnational standards， nationalthe expected value of estimator and the truestandards, industrial standards, designers'value of index. Thenrequirements, customers' requirements and so on,in order to determine whether the products areCmx=/(.m,)+1) (24)conforming. The application of specificationForCp and Cpmr， B(Cpxo;n,h,p}) andlimits is kept unchanged even for autocorrelatedprocesses. Thus, the way to adjust specificationB(Cmr,n,k2,P) have been defined in sectionlimits will not be discussed. Subsequently, five3.4 and 3.5, which could be used directly to giveschemes will be discussed.CporA and Cpmra.For AR(p) model, LSE and MLE of p are4.1 Scheme with Observationsthe same. p; is unrelated with x and s2 ，Using the formula (1), process capability canwhich is proved in Erik (2001). These B(*)be assessed based on the statistic s2 as the.values are still MLE, thenestimate of total process variance. This schemeis denoted as Scheme 0, then Cpro， Cpro， MSE(C%mx)= Var(prn)Cpmo will be used to identify the indices Cp,Cp，Cpmr with Scheme 0. Indeed, a fiting=B(rwnm,p)+1] Va[C,](2)modelfor process observationswithautocorrelation is not necessary for theSimilady, Msp(.) and MsSE(Cmn)中国煤化工calculation of indices, but it is essential whencoulYHCNMHGSun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated DataJ Syst Sci Syst Eng4.3 Scheme with Residualscalculated directly using tradition definitionsFor autocorrelated process control, one of theunder the independence assumption. Thismost popular methods, called residual chart, ismethod is denoted as Scheme C, then Cprc，to monitor residuals which could be obtainedCpxc and CpmrC will be used to idenifly thefrom the observations based on the process timeseries model. Using residuals, 02 , the varianceindices Cpr, Cpo, Cpom with Scheme C.of error in AR(1) model, could be estimated andThe essence of leapingly sarmpling method isbe used for the assessment of process capability,to design sarmpling interval 1. The rule forwhich is denoted as Scheme B.batch-mean method mentioned by Runger &Willemain (1995) could be used, i.e. to select an(26)ls.t. pr<0.1. In AR(1) model, P; =φ' . WhenCm=.(-叭”2φ=0.9,l=22. It appears that theco=d-18-m(1-@y'"(27)determination of sampling interval l requires30。knowledge of autocorrelation coefficient. Thev1/rule in the literature discussed above is to have acmo_只_ 1-口(28)sampling interval l larger than 20 or even larger30.(+)k2(1-42)when data are suficient. Obviously a largeamount of data is absolutely necessary in thismethod to ensure sufficient sub-samples andwhere d=(USL - LSL)/2 andreliable assessment for process capability.k'=(x-r)102.If o2=(1-})s2, then4.5 Batch-mean MethodMSE(CpmB)= MSE(Cp) and the sameBatch-meanmethodoriginates fromUnweighted Batch Means Chart by Runger &conclusions for Cpr，Cpmr.Willemain(1995).Shore (1997) hasIf o2=-2[(x,n-X)-*(x,-J)] ,recommended a model-free approach usingn-1台unweighted batch means in terms of boththen MSE(mB)= MSE(Cp ), and the sameperformance andconvenience. Batch-meanfor C. In this paper, o should bemethod is denoted as Scheme D. For allobservations of the autocorrelated process, everyobtainedby σ =(1-q*})s2.n1 observations are regarded as a batch. There isno overlap between 严and j+1 batch. The户4.4 Leapingly Sampling Methodbatch mean isLeapingly sampling method is a methodwhich subtracts sub sample at intervals of time l-=-+x-1.+2-12.... (29)from total sample to meet the independenceassumption of process capability analysis. Based中国煤化工on observations of sub-sample, these indices are.MYHCN MH Ges. Then,Sun et aL: Process Capability Analysis and Estimation Scheme, for Autocorrelated Data114J Syst Sci Syst EngE(y)=μmean is less than 0.10. Smallest sizes of batchfor different AR(1) models are given in Table 1.Var(y,)=f(m],p.)Table 1 Smallest batch size of AR(1) modelThe average and variation of batch mean are4nσy 1σPr1σy1σused to estimate process mean and variance.0.1 2 0.7416-0.10.67080.2 3 0.6566-0.20.6325β=Y=_Zy,(30)03 4 0.6232-0.3 30.46900.4 6 0.5722-0.4 4 0.3633d=;.0.5 8 0.5592-0.5 6 0.26020.5304- -0.6 80.19610.5288 .120.13430 = n1xdf[f(m,pn)]'(31)0.8 20.5278- -0.8180.0876Calculate CprD,0.9 57 0.5273-0.94(0.04030.95 117 0.5272-0.9580.1990f(n,p)(32)Cmo=v↓When n1 is determined, n2 batches aresubtracted. The value of n2 is required aswhere Cpy =d/3&y . Thentraditional condition, which is usually largerMsSE(Cp)than 25. Then the number of total observations isn=n1xn2. For AR(1) model in which φ=0.8, nshould be at least 675.In the literature of process ”control withml(33)autocorrelations, there are two main methods, amodel-based and a model-free approach. As aSimilar definitions of the other two indicesmodel-based approach, to adjust control limitsCpkrD and CpmrD can be generated. However,and to adjust statistics of control chart are twoit is difficult to get precise analytical formulas ofalternativemethodsformonitoringMSE. As a result, Monte-Carlo simulation isautocorrelated processes. For process capabilityused when comparing MSE of different schemes.analysis, the way to adjust specification limitsObviously, an autocorrelation coefficient iswill not be discussed as mentioned above.needed to determine batch size for batch-meanScheme with residuals is a popular and commonmethod, even though a fitting model is notway to control process and analyze capabilitynecessary. Take AR(1) model for example,for autocorrelated observations. As a model-freeRunger & Willemain (1995) proposed a detailapproach, Leapingly sampling method andanalysis of batch size. They suggested that batchBatch-mean method are typical ways forsize should be large enough so that the中国煤化Iprocesses.Forfirst- order autocorrelation coefficient of batchprot:YHCNMHGautocorrelation,Sun et aL: Process Capability Analysis and Estimation Scheme for Autocorrelated Data」Syst Sci Syst Eng115Leapingly sampling method (Scheme C) andTable 2 Parameter configuration of simulationBatch-mean method (Scheme D) are suggested,analysesbut the determination of sampling interval I andParameterValuebatch size somehow also requires knowledge ofthe assumed AR(1) model. Indeed among theSpecificationUSL=3, LSL= -3, m=7=0above five schemes of process capability indices2'X25, i=0(1)7estimation for autocorrelated data, the only oneP-0.75, -0.25, 0.25, 0.75that is truly model-free is Scheme withobservations (Scheme O)， process capabilityu0, 0.5indices based on total variation estimation0.5without making any adjustments to the index or(*) denotes incremental amountsampling mechanism. All the others rely to acertain extent on model knowledge.5.1 Comparison among Scheme O, A andB for Small Sample5. Comparisons of Five SchemesTables A1 and A2 in appendix show MSEs ofSimulation is used to compare MSE ofCpr, Cpr and Cpmr for Scheme O, A and Bdifferent schemes. Without loss of generality,with φ= 0.75, 0.25, -0.25, -0.75 in case ofp=0parameter configuration is shown in Table 2.Among these parameters the number of totaland 0.5 under the conditions of small sample,observations n should be given special attentionwhere the lowest MSE among these differentschemes is bold-faced for givenμ, q, n, and thebecause Scheme C and D requires a greatnumber of observations while Scheme O, A andindex.●As |q| increases, MSEs with each SchemeB do not have specific requirement of sampleincrease for any given μ , n, and the index.size. As a result, when n is no larger than 200,●For given μ and φ，as the number ofonly Scheme 0, A and B are compared.observations n increases, MSEs withIn Table 2, when μ=0, then μ= m and nocertain Scheme decrease and the differenceshift between process mean and midpoint ofof MSEs among different Schemes alsospecification limits happens. Cpr =Cphr = 2, thatdecreases, no matter which index isis, the true value of Cpr and Cpor is the same.studied.When μ= 0, then μ= T and target value is●For given μ, φ, n, and the index, MSEsprocess mean. Cpr=Cpmr= 2, that is, the truewith Scheme B never become the lowestvalue of Cpr and Cpmr is the same. Thusamong Scheme O, A and B. For Scheme B,Cpr =Cpo=Cpmr=2. When μ=0.5, then μ≠o2, the variance of error in AR(1) model,m and there exists shift between process meantogether with φ should be estimated fromand midpoint of specification limits. Meanwhilethesimulatedobservationsryμ≠T and target value is not process mean.中国煤化工n s2 and φCpr=2, Cpo=1.67, Cpmr= 1.414.YHC NMH Gtly as knownSun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated Data116 。」Syst Sci Syst Engparameters. Compare with Scheme 0 andμ= 0.5. Thus the modification ofB, it is obviously concluded that Scheme 0B(*) is valuable for positive andwill result in a more reliable estimationweak negative φ.with lower MSE since it is not necessary toIn conclusion, when there is strong negativeestimate φ for Scheme 0.autocorrelation, it is recommended to adoptFor given μ，φ, n, and the index, mostScheme O. When there is positive or weakMSEs with Scheme A become the lowest,negative autocorrelation,SchemeA isoccasionally MSEs with Scheme 0 becomerecommended, especially for no more than 100the lowest.observations.■When μ =0.5, ifφ= 0.75, 0.25, -0.25,Although all of Scheme O, A and B are usedScheme A is the best choice, if φ=for small sample, a fitting model is not-0.75, Scheme 0 is the best choice.necessary to estimate Cpr ，Coo and Cpmr■ When μ=0, ifφ= 0.75, 0.25, -0.25,with Scheme 0 when confidence interval of theScheme A is the best choice exceptindex is not discussed. However Scheme A andCpor, if φ= -0.75, Scheme 0 is theB must be analyzed based on the process modelbest choice except Cpmr 。As nwith autocorrelation, especially for Scheme A,increases, the best choice has changedthe B(*) values are determined on the truefrom Scheme A to Scheme 0, forvalues of Cpr， Cplr and Cpmr, theexample, for Cor, Scheme O alwaysautocorrelation coefficient φ，the number ofrepresents better than Scheme Aobservations n, the shift between process meanespecially whenn> 100, for Cpmr, ifapd midpoint of specification limits, and theφ = -0.75，Scheme 0 representsdifference between process mean and targetbetter than Scheme A when n> 100.value. Thus Scheme A is not easy for use in■ When μ = 0, no shift betweenpractice. Fortunately when n is moderately large,process mean and midpoint offor example, n= =200, Scheme A has litlespecification limits exists, target valuedifference from Scheme O. The B(*) values tendis process mean, and Cpr = Cpr=to near zero as sample size n increases. SchemeCpm=2, thus Cor should be used asO is recommended to estimate these indices forthe representative for processa large number of observations.capability analysis for autocorrelatedobservations. For Cor when μ= Q5.2 Comparison among O, B, C, and Difφ= 0.75, 0.25, -0.25, Scheme A is .for a Large Number of Observationsthe best choice; if φ= -0.75, SchemeFor Scheme C, sampling interval I is0 is the best choice.determined to be 10. For Scheme D, batch size■It is concluded that the result fromn1 is中国煤化工;A3 and A4 inCpr when μ=0 is the same as when apper:YHC N MH GFpr and CpmrSun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated DataJ Syst Sci Syst Eng117for Scheme 0, B, C and D with φ= 0.75, 0.25,6. Procedure and Example of Process-0.25, -0.75 in case of μ=0 and 0.5 under theAnalysis for Autocorrelated Dataconditions of a large number of observations,where the lowest MSE among these different6.1 Procedure of Process Analysis forAutocorrelated Dataschemes is bold-faced for given μ, o, n, and theProcess analysis for autocorrelated data isindex.●As |中| increases, MSEs with each Schemegiven as follows,1. Data analysis. Many tools, such asincrease for any given μ , n,and the index.descriptive statistics, histogram, box-plot,●Whatever μ =0 and 0.5, MSEs of Schemerun chart and so on, can be used to give theO and Scheme B are much better than thatlocation, variability and the shape of theof Scheme C and Scheme D. As thedistribution, interpret the occurrence ofnumber of observations n increases, MSEsoutliers, and examine if the measured dataof Scheme 0 and B tend to become. thein time -ordered sequence follow a definitesame values. When μ = 0.5, Scheme 0 ispatterm to point the right direction fora better choice. When μ= 0, Scheme O isfurther work.a better choice for Cpr and Cpmr;2. Test if the data are normally distributed.MethodsincludeShapiro-Wilk,Scheme B is a better choice with lttleKolmogorov- Smimov, Cramer-von Misesimprovement in case of φ= 0.25 andand Anderson-Darling.-0.25 for Cp . For process capability3. Test if the process is in state of statisticalanalysisfor a larger number ocontrol. If the data are correlated,autocorrelated observations, Scheme 0 isautocorrelatedcofficient and partialthe best choice for Cpr when μ= 0 andcorrelation coefficientareusedfor Cpr， Cpr and Cpmr when μ= 0.5.determine the derivative order of the●Compare with Scheme C and D for anycharacteristics of autocorrelation model.given φ. When μ =0, Scheme C is aSome control charts corresponding tobetter choice. When μ= 0.5, for Cpr andautocorrelated processes, such as residualCpo，Scheme C is a better choice; forcontrol chart, the autoregressive T chart(Apley & Tsung, 2002),， batch-meansComr, Scheme D is a better choice forcontrol chart (Runger & Willemain, 1996)almost all n, especially larger than 400.and so on, could be used to test if theHowever, MSEs of Scheme C and D areprocess is in state of statistical control.dependent with the design of sampling4. Processcapability analysis. Chooseinterval l and batch size n1, and the way torelevant schemes according to sample size.determine appropriate I and n1 for Scheme5. Make decision based on the estimation ofCandDtogetthebestMSEwillnotbe中国煤化工swhether todiscussed in this paper.FYHC N M H Gss capability orSun et aL: Process Capability Anabysis and Estimation Scheme for Autocorrelated Data118JSyst Sci Syst Engmake progress on process capabilitythrough the way to allocate resources.21,]6.2 Example网.An axletree's diameter is measured every1cm. Specification limits required by thecustomer are (47mm, 53mm) with a target valueof 50mm. From the historical data of long term,↑:020030040050060T6086C100100it is known that an autoregressive process ofNorder oneis fitted with autocorrelationFigure 2Runchartof X2cofficient φ=0.5, μ= 50, and o =0.75.For showing the procedure of process analysiswith small sample and large sample respectively,6.2.2 Test if the Data Are Normallytwo groups of AR(1) data are generated byDistributedsimulation, that is, Group 1 X with a sampleFrom Figure 3 it can be referred that Xsize of 100 and Group2 X2 with a sample sizeand Xz are approximately normally distributed.of 1000.Thus the assumption of normal distribution isSuppose that Group 1 and Group 2 aresatisfied.provided to the personnel without any furtherdetail on process model. The procedure of6.2.3 Test if the Process Is in State ofprocess analysis could be deployed as follow.Statistical ControlConsider X} at first, autocorrelation6.2.1 Data Analysiscofficient and partial correlation coefficient areDrawrunchartofX|andX2。Itisshown in Figure 4.illustrated in^ Figure 1 and 2 that the dataIt can be inferred from Figure 4 thatfluctuate around the target value 50 with noautocorrelation cofficient has' tailing effecttendency or periodicity.while partial correlation coefficient hasfirst-order truncation effect. So AR(1) model canbe used to fit this series. The model of X is*21 ;estimated as followX+1-50.18926-0.52961(X, - 50.18926)+Ewhere o =0.747559. Both white noise test ofresidual and significance test of parameters aresatisfied. Autocorrelation coefficient of residualsand partial correlation coefficient of residualsare中国煤化工Figure 1 Runchartof X“HCNMH G x The modelSun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated DataJSyst Sci Syst Eng119of X2 isestimate closest to 1 is bolded. Scheme A seemsX+1 - 49.98894-=0.49507(X, - 49.98894)+eto be a considerable choice for most cases,which is coincident with the analytical resultswhere ot = 0.730563.based on comparison of MSEs for small sample.Control charts with the consideration ofThe difference among different schemes isautocorrelation could be used to test if thewithin 5% and the difference among Scheme O,process is in state of statistical control. Since itA and B is even smaller. It is shown that theis not the topic of this paper, use the conclusionestimates of Cpr,Cpo and Cpmr withdirectly.Scheme D are always smaller than the true value.But it should be noted that the results presented6.2.4 Process Capability AnalysisThe estimates of Cpr， Cpo and Cpmr forin Table 3 are obtained from just one simulateddiferent schemes are shown in Table 3. Sincedata set, and therefore should not be overthe true valueof Cpr, Cpr and Cpmr is1, thegeneralized.9.99四0十:474849505520.01-46.4748495051525354X2Figure 3 Nomal probability plotsLO.0a88a.eaef.Q4a202alhntttTTTa-----0a2a例160.8-Q-.02468101214161820224中国煤化工工之2Figure 4 Atorelation coffcient and pJYHCNMHGSun et al: Process Capability Analysis and Estimation Scheme for Autocororelated Data120J Syst Sci Syst Eng08ne40.22.-0.212.-Q4104n6-08418-10-.02468101214161820224.2468101214161820221Figure 5 Atorelation coflcient and partial crelation cffcnt of residualsTable 3 Process capability analysis for diferent schemesIndexABCD1001.141861.113191.14399Cpr10001.01 8001.015731.018811.03098 .0.982681.072761.052731.07477Cpo1.014131.022881.014941.001730.978941.123491.107881.12551Cpmr1.018441.017181.019251.032160.982616.2.5 Take Actionssummarized for process capability estimation ofIf process capability indices are lower thanthe index Cpr，Cpo" and Cpmr， that is,customers' expectation, try to reduce mean shiftScheme with observations (Scheme 0),or process variation in order to improve processmodified Scheme with observations (Scheme A),capability. If the indices are met with customers'Scheme with residuals (Scheme B), leapinglyexpectation,maintainingcurrentprocesssampling method (Scheme C) and batch-meancapability will be an essential task.method (Scheme D).Scheme with observations (Scheme 0) is7. Conclusionintrc中国煤化Iof the study toFor AR(1) process, five schemes aredealY片C N M H Gobservations asSun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated DataJSyst SciSyst Eng121the estimator of process variability. Theindices for a large number of observations.statistical characteristics have been discussed for3. For large sample, MSEs of Scheme 0 andheindex Cpr，Cpr and CpmrforB are much better than that of Scheme Cautocorrelated data with Scheme 0. Scheme A isand D. Scheme 0 is the best choice forprovided with the modification of the B(*)Cpr when Cpr =Cph=Cpmr and for Cpr,values on Scheme 0. Scheme B is provided withCpo and Cpmr when true values of Cpr,the consideration of residuals instead ofCpr and Cpmr are different.observations. However, Scheme A and B areProcess capability indices establish thedeeply dependent on the ftting model of processrelationships between the process performanceobservations and face the risk of an inaccurateand the manufacturing specifications to measureestimate of process model. Under the conditionthe potential of a process to provide anof large sample, Scheme C eliminates theacceptable product. Sometimes specificationsautocorrelation by using the leaping sample withcannot be revised no matter if processthe risk of massive data omitted and Scheme Dobservations are correlated. Thus the focus fordeals with the man-made batches to form batchprocess capability analysis with autocorrelationmeans instead of observations to avoidis to measure the potential process variation.autocorrelation with the risk to break the series.Obviously, within- subgroup variation, which isFrom comparison of MSEs of these schemes, itcommonly used for the traditional processcan be inferred thatcapability analysis under the assumption of1. For small sample, when there is strongindependence, will seriously underestimate thenegativeautocorrelation,isactual process variation.It is certainlyrecommended to adopt Scheme O, Wheninteresting from comparison among differentthere is positive or weak negativeschemes in terms of MSEs to know that Schemeautocorrelation, Scheme A is recommendedO，calculating Cpr， Cpo and Cpmr directlyespecially for no more than 100based on sample variation of all observations,observations.performs reasonably well for AR(1) model.2. For Scheme A，the B(*) values areWhether the same conclusion holds for moredetermined by true values of Cpr,complex time series model will be discussed irCpr and Cpmr , autocorrelation coefficientfuture work.φ, the number of observations n, the shiftbetween process mean and midpoint ofAcknowledgementspecification limits, and the differenceThe authors would like to thank anonymousbetween process mean and target value.referees for their constructive comments toFortunately when n is moderately large, forimprove the quality of the paper.example, n= 200，Scheme A has lttledifference from Scheme 0. Then Scheme中国煤化工0 is recommended to estimate theseMYHCNMHGSun et aL: Process Capability Analysis and Estimation Scheme. for Autocorrelated Data122」Syst Sci Syst EngAppendixTable A1 Comparison of MSEs among Scheme 0, A and B for small sample in case of μ=0Nindices2500020000.550060.215190.088180.040410.75Cpr_A0.525880.207200.084630.03927B0.666310.240250.094260.041830.409640.171110.075330.03695CpaA0.471100.192610.082140.039330.496880.189580.079410.037820.342060.161850.074910.03711Cpmr0.318110.158850.074760.037090.401550.177130.078900.038140.123340.053540.024560.011700.113400.050960.023890.011530.250.145810.058680.025610.012020.109290.049850.024030.011710.108310.050180.024370.011870.124040.052650.024360.011780.110370.050500.023960.011500.100890.048200.023400.011360.127790.054730.024850.108890.049340.023600.01158-0.250.100470.047620.023160.011460.126530.052980.024470.011800.102610.047800.023330.01156Cptr0.098830.047450.023390.011610.115840.050120.023760.011630.107780.049040.023620.011590.098660.047100.023110.125720.0528801183O0.321160.154490.074750.03639-0.750.327040.157710.076650.036820.389470.170020.078300.037140.309130.150060.073400.035950.323900.156540.076400.036790.374090.164730.076700.036630.337970.159170.075840.036690.328050.1571 |中国煤化工0.409750.175MHCNMHG-Sun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated DataJ Syst Sci Syst Eng123Table A2 Comparison of MSEs among Scheme 0, A and B for small sample in case ofμ=0.5Nindices2550100200C0.567620.211040.089800.040520.75CprA0.534260.202460.086470.03983B0.692720.237010.095230.041710.434960.166160.070760.032420.400860.155990.067670.031750.523520.184550.074590.033310.267740.115430.051220.02405C pmr0.221840.101830.047670.023140.296310.121270.052440.0243600.123170.052400.024370.011960.250.111340.049900.023770.011840.146590.057260.025430.012250.094830.040630.018850.00926Cplr0.085510.038690.018400.009170.111660.044210.019600.009460.056320.025380.011720.00582Cpmr0.050100.023940.011390.005740.061390.026590.011950.005880.110490.050480.024400.011880.101900.048290.01177-0.250.127600.054840.025360.012070.079850.036460.017580.008550.073720.034910.017250.008480.091850.039510.018270.008680.027130.012520.006120.002990.025340.012050.006000.002970.029260.013080.006250.003010.326970.150010.074970.037460.332210.154280.075950.03790-0.750.394690.165170.078760.038270.227510.104910.052290.026080.231070.107830.052970.026390.274470.115470.054920.026640.033420.017770.009190.004810.040860.019中国煤化工7E0.03930TYHCNMHGSun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated Data124J Syst Sci Syst EngTable A3 Comparison of MSEs among Scheme 0, B, C and D for a large number of observations in case ofμ=0NPIndex400003200Cpr0.018550.009080.004500.00228B0.018890.009140.004520.00229C0.057330.027630.012850.006450.125820.049630.022950.010910.018040.009200.004690.002380.018190.054300.027240.012920.00653D0.118270.047690.022420.010720.017770.008880.004450.00226Cpmr0.018020.008920.004470.055330.027170.012770.00642016950.047900.022550.0108050.005690.002860.001440.000690.005760.002870.026720.012960.003120.042370.019880.009840.004760.005830.002980.001520.00074Cpor .0.005810.002970.001510.250.026190.013070.006630.003210.041100.019500.009760.004750.005640.00285c pmr0.005700.026230.012900.006500.003110.041610.019710.009800.005790.001420.000710.005850.002880.001430.026730.012190.006390.003080.054570.024950.012130.00595Cpir0.002900.000730.005820.002890.00072-0.250.026280.012330.006570.003190.053540.024600.012020.005920.005800.005860.026360.012160.006370.054220.024860.012110.018320.008910.004380.018510.008980.004400.057240.026370.013000.006440.205950.079040.037640.01744Cpar0.018150.008870.004360.002250.00893-0.750.054200.026040.012980.006580.203770.078460.037420.017370.018390.004390.01860中国煤化工- 0.002260.05521:" 0.006410.20552sYHC N M H G0.01744Sun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated DataJ Syst Sci Syst Eng125Table A4 Comparison of MSEs among Scheme 0, B, C and D for a large number of observations in case ofμ=0.5NPIndex40080016003200Cpr0.018830.009200.004520.002200.019170.009280.004540.056450.026550.012890.00637D0.122660.051010.022700.010850.015180.007350.003650.00176Cp0.015420.007410.003660.001770.042280.020050.009760.004810.087520.036550.016260.007780.011590.005550.002790.00136Cpmr0.011660.005570.002800.021650.010630.005200.002530.021990.010300.004920.00242Cpm0.005680.002870.001440.000720.005740.002880.001450.026420.013130.006260.003200.043910.020080.009660.004720.04430.002240.001120.000550.004470.002250.250.019730.009820.004710.002400.030990.014120.006800.003320.002840.001420.000700.000340.002850.00340.009680.004870.002370.00120_0.007150.003400.00168。0.000830.005710.002920.005760.002940.001430.026870.013000.006400.003150.055310.025270.012030.004120.002110.001020.00051CporB0.004160.002120.001030.0051-0.250.009710.00474D_0.038590.017580.008380.00409Camr0.000730.000360.000180.001460.00180.009720.004730.002320.001190.006900.003290.001630.000820.017780.009160.00470.017990.009210.004490.057440.027320.013140.201010.085040.037850.017910.012390.006390.003120.001570.012540.006420.00313-0.750.042990.020600.009850.139620.059060.026320.01244Cpw0.002340.001220.000600.002352.00030C0.021920.010中国煤化工1002550.017900.008MYHCNMHG.00217Sun et al: Process Capability Analysts and Estimation Scheme for Autocorrelated Data126J Syst Sci Syst EngReferences[11] Kotz, S. & Johnson, N.L. (1999). Delicate[l]Apley，D.W. & Tsung, F. (2002). The .relations among the basic process capabilityautoregressive T chart for monitoringindices Cp, Cpk, Cpm and their modifications.univariate autocorrelated processes. JoumalCommunications Statistics - Theory andofQuality Technology, 34 (1): 80-96Method, 28 (3): 849-861[2] Barmett, N.S. (1990). Process control and[12] Runger, G.C. & Willemain, T.R. (1995).product quality: the Cp and Cpk revisited.Model-based and model-free control ofInternational Jourmal of Quality andautocorrelated process. Joumal of QualityReliability Management, 7 (5): 34-43Technology, 27 (4): 283-292[3] Box, G.E.P., Jenkins, G.M. & Reinsel, G.C.[13] Runger, G.C. & Willemain, T.R. (1995).(1994). Time Series Analysis: ForecastingBatch-meanscontrolchartsorcan Control. Prentice Hall, Englewood Cliffs,autocorrelated data. IIE Transactions, 28 (6):NJ483[4] Chan, LK, Cheng, S.W. & Spiring, F.A.[14] Scagliarini, M. (2002). Estimation ofCp for(1988). A new measure of process capability:autocorrelated data and measurement errors.Cpm. Journal of Quality Technology, 20 (3):Communications in Statistics 一Theory and162-175Methods, 31 (9): 1647-1664[5] Erik, W. (2001). Confidence limits for the[15] Shore, H. (1997). Process capabilityprocess capabilityindexCpk foranalysis when data are autocorrelated.autocorrelated quality characteristics. In:Quality Engineering, 9 (4): 615-626Joachim, H, Theodor, P. (eds.), Froptiers in[16] Sun, J, Wang, S.X. & Fu, Z. (2009).Statistical Quality Control 6. Baker & TaylorComparison of process capability indices andBooks, Wiurzburg, Germanyprocess performance indices. In: Proceedings[6] Guevara, R.D. & Vargas, J.A. (2007).of the 8 Korea-China Quality Symposium,Comparison of process capability indicesPusan, Korea, 2009underautocorrelateddata.Revista[17] Wang, S.X. & Sun, J: (2006). ComparisonColombiana de Estadistica, 30: 301-316of process capability index and process[7] He, S. (2003). Applied Time Series Analysis.performance index based on individualPekingUniversity Press, Beijing (Inobservations. Journal of Tsinghua UniversityChinese)(Science and Technology)， 46 (12):[8] Hsiang, T.C. & Taguchi, G. (1985). A2049 2052 (In Chinese)tutorial on quality control and' assurance -[18] Zhang, NF. (1998). Estimating processthe Taguchi methods. ASA Annual Meeting,capability indexes for autocorrelated data.Las Vegas, Nevada, 1985Journal of Applied Statistics, 25 (4): 559-574[9] Kane，V.E. Process capability indices.Journal of Quality Technology, 18(1): 41-52Jing SUN is an Associate professor at School of[10] Kotz, S. & Johnson, NL. (1993). ProcessMH中国煤化工nghua Universit,Capability Indices. Chapman&Hall, Londonrec |C N M H Gagement scienceSun et al: Process Capability Analysis and Estimation Scheme for Autocorrelated Data」Syst Sci Syst Eng127and engineering from Beijing University ofProcess Capability Analysis, Data StatisticalAeronautics and Astronautics and in 1999 berAnalysis and Decision.Master and Bachelor degrees in systemengineering from Tianjin University in 1994,Zhihui FU works at China Construction Bank,1991. Her current research and teaching interestsreceived his Master degree in managementare Statistical Quality Control, Total Qualityscience and engineering from TsinghuaManagement, Innovation and Strategic Decision.University in 2009, and his Bachelor degree indouble majors, Information Management andShengxian WANG works at Xinhua NewsSystem (IMS) and Economics, from PekingAgency, received his Master degree inUniversity in 2007. His interest is Managementmanagement science and engineering fromInformation System, Financial Data analysis andTsinghua University in 2007. His interest isdecision.中国煤化工MYHCNMHG