ORIGINAL ARTICLE

Identifying the mean step change of UBM chart

Abbas Saghaei & Zeynab Mosanna & Hoorie Najafi

Received: 24 February 2010 /Accepted: 24 October 2011 /Published online: 16 November 2011# Springer-Verlag London Limited 2011

Abstract The traditional control charts are developed basedon the assumption that the successive observations areindependent and identically distributed. In some processes,the independence assumption is violated when there isautocorrelation between observations. To solve this problem,two methods, classified as model-based and model-free, couldbe applied.When a control chart alarms an assignable cause, itis essential to detect the process change point in order toremove the root cause. In the presence of autocorrelated data,different methods for change-point identification have beenapplied only for model-based methods. Hence, this isconsidered as the research gap and an attempt is made to fillthis gap by applying maximum likelihood function inunweighted batch mean control chart, one of the most appliedmodel-free methods. In this article, an estimator is presentedto determine the change point for the first-order autoregressiveprocess, AR(1). When a real change occurs, the performanceof proposed estimator is evaluated through simulation.

Keywords Change point .Maximum likelihood function .

Autocorrelation . ARMA . Unweighted batch mean

1 Introduction

Statistical process control (SPC) is an approach towardimproving process capability through the reduction of variabil-ity and it aims at process stability. Control chart is the one oftechniques of SPC that is applied for monitoring the variabilityin process by distinguishing between common and specialcauses. In the use of traditional control charts, one assumes thatthe independent observations are generated by an in-controlnormal process with mean μ and standard deviation σ.

The independency of observations is an important assump-tion. In the presence of autocorrelated data, the conventionalcontrol charts do not work well even if the observations exhibitlow level of autocorrelation over time. Especially, when thereis positive correlation between observations, these controlcharts will generate more false alarms. In some processes, suchas chemical processes, that we have close intervals betweensampling, high autocorrelations are often observed.

Alwan and Roberts [1] and Wardell et al. [2] assumedthat there is possibility to model the processes withautocorrelated observations by an autoregressive-moving-average (ARMA) model. There are different methods calledmodel-based and model-free for monitoring autocorrelatedprocesses. Runger and Willemain [3] have compared someof these methods that were used in AR(1) process: TheResiduals Chart, The Weighted Batch Mean (WBM), andThe Unweighted Batch Mean (UBM) Chart, and mentionedthat model-free (UBM) control charts can have superiorARL performance.

In control charts, when there is an out-of-control signal dueto an assignable cause, it is necessary to diagnose and removethe cause of change to return the process to the state ofstatistical control. Hence, it is useful to know when thechanges started. In most cases, the first observation after the

A. Saghaei (*) : Z. MosannaDepartment of Industrial Engineering,Science and Research Branch, Islamic Azad University,Tehran, Irane-mail: a.saghaei@srbiau.ac.ir

Z. Mosannae-mail: z.mosanna@srbiau.ac.ir

H. NajafiParsian Quality and Productivity Research Center,Tehran, Irane-mail: h.njf@pqprc.org

Int J Adv Manuf Technol (2012) 61:649–655DOI 10.1007/s00170-011-3728-1

change, especially when the change is very small, does notshow the out-of-control state. In small changes, it takes a longtime to get a control chart signal. Thus, determining the firstpoint that the change occurred after is not an easy task.

There are different methods to determine the changepoint in control charts that are often used in model-based techniques. Built-in change-point estimators fromthe past plots on the control chart had provided bycumulative sum and exponentially weighted movingaverage control charts suggested by Page [4] andNishina [5], respectively. Another method uses themaximum likelihood function, when the probabilitydensity function is known.

Samual et al. [6] suggested a maximum likelihoodestimator (MLE) of change point when a step changeoccurred in an x-bar control chart of normal process.Timmer and Pignatiello [7] proposed three MLEs of changepoint for the parameter of an AR(1) process. In this article,an estimator is presented to determine the mean changepoint of UBM statistics which is used to reduce theautocorrelation of AR(1) process observations.

The UBM charts were suggested by Runger andWillemain[8]. If {xi} is a sequence of autocorrelated observations,successive data will be divided into batches of size b and theaverage of the jth sample is denoted by yj as following:

yj ¼ 1

b

Xbi¼1

x j�1ð Þbþi j ¼ 1; 2; . . .

Kang and Schmeiser [9] showed how averaging withinsubgroups shrink the autocorrelation of the observations.There are empirical procedures for determining an appropri-ate batch size, suggested by Law and Carson [10] andFishman [11, 12]. In a detailed analysis of batch size for AR(1) models, is recommended that the batch size can beselected so as to decrease the lag one autocorrelation of batchmeans to approximately 0.01 by Runger and Willemain [3].

In Table 1, the minimum required batch sizes for AR(1)models, depending on the autocorrelation coefficient φ,estimated by Kang and Schmeiser [9], are presented.

First-order autoregressive process, AR(1), is the mostcommon form of process autocorrelation. Many authors havementioned the presence of AR(1) on various processes suchas, e.g., Montgomery [13] and Wardell et al. [2]). Due to thefact that in most cases we encounter AR(1) processes, thisarticle deals with this model of time series.

2 Change-point estimator for UBM chart mean

Kang and Schmeiser [9] proved that if the {xi} (i=1,2,…) issequence of a stationary AR(1) process, the sequence of samplemeans {yj}, as define above, is a stationary ARMA(1,1)process. We will assume that xi observations are generated byan AR(1) process with constant mean μ as follows:

xi � m ¼ 8 xi�1 � mð Þ þ "i

i ¼ 1; 2; . . .ð1Þ

Where "is are independently and identically distributed (i.i.d)N(0,σ2), φ is the autocorrelation coefficient and σ2 is thevariance of the white noise process. Successive data aredivided into batches of size b and the arithmetic average ofthem is calculated to reduce the autocorrelation. Consideringexplanation above, we have:

yj ¼ 1� 8ð Þmþ 8yj�1 þ "j � q "j�1

j ¼ 1; 2; . . .ð2Þ

Where "j� �

is the sequence of i.i.d, with N(0,σ*2)

distribution, and 8 is the AR parameter and q is the MAparameter.

FðBÞyj ¼ ΘðBÞ"j þ 1� 8ð Þm0 ð3ÞEquation 3 is the summary of Eq. 2 where B is an

operator that:

Bryj ¼ yj�r ; 0 � r < j

And we have:

ΘðBÞ ¼ 1� qB� �

ΦðBÞ ¼ 1� 8Bð Þ&

Alwan and Radson [14] showed that 8 ¼ 8 b. The amount

of q was calculated for some special bs [15]. Since it wasnot required here, it is not included in this paper.

Then, procedure of determining change point for meanof AR(1) process is explained. Assume that τ initialobservations are observed from an in-control ARMA(1,1)process with constant mean μ0. The mean of the process ischanged and subsequent observations τ+1, τ+2,…,n aregenerated by an ARMA(1,1) process with constant mean

φ b

0.00 1

0.10 2

0.20 3

0.30 4

0.40 6

0.50 8

0.60 12

0.70 17

0.80 27

0.90 58

0.95 115

0.99 596

Table 1 Minimum batch sizesrequired for UBM chartfor AR(1) data

650 Int J Adv Manuf Technol (2012) 61:649–655

μ1. Notice that parameter φ and σ are not changed. Assumethat μ1 is equal to μ0+dσ where d≠0. As a result of anassignable cause, observation n is the first point thatexceeds a control limit after mean change. Change-pointestimator for identifying τ, as the process mean changepoint, is derived by solving the log-likelihood function as inthe following:Assume that:

d1 ¼ 1� 8ð Þm1 d0 ¼ 1� 8ð Þm0&

For the ARMA(1,1) we have:

LnðLÞ ¼ � n

2Ln 2pð Þ � n

2Lns

»2 � 1

2s»2

�Xtt¼1

Φ Bð Þyt � d0Θ Bð Þ

� �2

� 1

2s»2

�Xnt¼tþ1

Φ Bð Þyt � d1Θ Bð Þ

� �2

ð4Þ

Notice that, τ and δ1 are two unknown parameters inEq. 4. Assuming τ to be fixed, we have:

The equation sex is the MLE for δ1 where 1≤τ<n, bysubstituting it in Eq. 4, value of τ that maximizes the log-likelihood function equals:

n� tð ÞΘ2ðBÞ

bd1 tð Þ � d0� 2

bt ¼ argmax n� tð Þ bd1 tð Þ � d0� 2

�

1≤C<n, is proposed as the change-point estimator ofARMA(1,1) process mean.

3 Performance of the change-point estimatorfor the mean of UBM statistic

The performance of the proposed estimator is evaluated byMonte Carlo simulation. The simulation results are pre-sented in Tables 2 and 3. The report formats are similar tothe previous researches in the change-point detection areasuch as, Samuel et al. [6], Timmer and Pignatiello [7],

Pignatiello and Samuel [16], and Perry et al. [17]. Here isassumed that the initial observations were from an AR(1)process with μ0=0, σ=1 and different φs.

According to autocorrelation coefficient φ, consecu-tive observations had been divided into presentedbatches size (b) in Table 1, and the arithmetic averageof the data values of each batches, yj, were calculated. Inorder to monitor yjs, individuals control chart was used inwhich the in-control average run length (ARL) for thiscontrol chart was approximately 370 observations. The

Table 2 Mean of estimated change point for μ and standard errorwith the ARMA(1,1) individuals control chart

d E(n) std(n) bt std btð Þ

0.5 51.313 0.587 49.537 3.086

φ=0.9 1.0 51.000 0.000 49.995 0.161

1.5 51.000 0.000 50.000 0.000

b=58 2.0 51.000 0.000 50.000 0.000

3.0 51.000 0.000 50.000 0.000

0.5 58.411 13.541 49.609 3.908

φ=0.6 1.0 51.486 0.837 49.594 2.952

1.5 51.025 0.161 49.883 1.393

b=12 2.0 51.000 0.014 49.983 0.398

3.0 51.000 0.000 50.000 0.000

0.5 95.992 176.382 50.219 6.411

φ=0.3 1.0 55.308 6.601 49.573 3.551

1.5 51.858 1.372 49.507 3.275

b=4 2.0 51.191 0.473 49.671 2.602

3.0 51.001 0.037 49.964 0.655

0.5 445.781 1818 54.845 20.253

φ=0.0 1.0 139.444 310.217 50.403 6.192

1.5 73.045 53.192 50.002 3.350

b=1 2.0 58.614 16.039 49.808 2.692

3.0 52.285 2.3904 49.694 2.442

0.5 210.723 569.485 50.472 6.002

φ=−0.3 1.0 64.673 36.103 49.963 1.839

1.5 53.046 3.986 49.835 1.571

b=4 2.0 51.372 0.963 49.883 0.969

3.0 51.006 0.079 49.980 0.466

0.5 75.305 68.500 50.040 2.357

φ=−0.6 1.0 51.933 1.899 49.837 1.517

1.5 51.047 0.234 49.939 0.689

b=12 2.0 51.001 0.026 49.990 0.191

3.0 51.000 0.000 50.000 0.017

0.5 51.598 1.206 49.850 1.318

φ=−0.9 1.0 51.000 0.000 49.995 0.071

1.5 51.000 0.000 50.000 0.014

b=58 2.0 51.000 0.000 50.000 0.000

3.0 51.000 0.000 50.000 0.000

C=50, N=10,000 independent simulation

Int J Adv Manuf Technol (2012) 61:649–655 651

control limits of the individuals control chart are calculat-ed as:(Upper Control Limit)

UCL ¼ yþ 3MR

d2ð7Þ

(Lower Control Limit)

LCL ¼ y� 3MR

d2ð8Þ

Where y and MR are the average of yj and the movingrang, respectively and d2 is equal to 1.128.

The change point, τ, is 50. Indeed, the first 50 yjs werefrom an ARMA(1,1) process with μ0=0 and 8 ¼ 8 b. Forthe next successive observations of AR(1) process, themean changed from μ0 to μ1, where μ1=μ0+dσ and d is0.5, 1.0, 1.5, 2.0, and 3.0. The values of the yj werecalculated for j=51,52,…,n, in which yn was the first pointthat exceeded control limit after mean changed. Weassumed that if individuals control chart signaled before

Table 3 Precision of bt estimator as the individuals control chart is used for ARMA(1,1) process

d P bt � tj j � mð Þ

m=0 1 2 3 4 5 6 7 8 9 10

0.5 0.856 0.943 0.962 0.972 0.978 0.980 0.982 0.984 0.985 0.987 0.988

φ=0.9 1.0 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

b=58 1.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

2.0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

3.0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.5 0.504 0.763 0.866 0.916 0.941 0.955 0.965 0.971 0.976 0.979 0.981

φ=0.6 1.0 0.841 0.943 0.963 0.973 0.978 0.982 0.984 0.986 0.988 0.989 0.990

b=12 1.5 0.961 0.986 0.991 0.994 0.995 0.996 0.997 0.997 0.997 0.997 0.998

2.0 0.994 0.998 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000

3.0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.5 0.252 0.472 0.601 0.689 0.756 0.804 0.840 0.867 0.890 0.905 0.922

φ=0.3 1.0 0.595 0.834 0.906 0.939 0.959 0.968 0.974 0.978 0.981 0.983 0.985

b=4 1.5 0.802 0.928 0.954 0.967 0.974 0.978 0.984 0.982 0.984 0.985 0.987

2.0 0.897 0.961 0.975 0.980 0.984 0.987 0.989 0.990 0.991 0.992 0.993

3.0 0.986 0.995 0.997 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000

0.5 0.081 0.182 0.260 0.330 0.380 0.430 0.470 0.504 0.536 0.567 0.595

φ=0.0 1.0 0.258 0.468 0.596 0.684 0.746 0.794 0.832 0.862 0.888 0.907 0.922

b=1 1.5 0.449 0.700 0.815 0.882 0.918 0.943 0.960 0.969 0.977 0.983 0.986

2.0 0.613 0.837 0.920 0.952 0.969 0.979 0.983 0.987 0.989 0.991 0.992

3.0 0.821 0.944 0.971 0.980 0.985 0.987 0.988 0.990 0.991 0.992 0.993

0.5 0.256 0.467 0.601 0.688 0.751 0.802 0.841 0.868 0.893 0.909 0.924

φ=−0.3 1.0 0.612 0.838 0.919 0.956 0.977 0.987 0.992 0.995 0.996 0.997 0.997

b=4 1.5 0.825 0.955 0.980 0.988 0.992 0.993 0.995 0.996 0.997 0.997 0.997

2.0 0.916 0.980 0.991 0.994 0.996 0.997 0.998 0.998 0.998 0.999 0.999

3.0 0.986 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.5 0.513 0.760 0.871 0.922 0.953 0.969 0.979 0.988 0.991 0.993 0.994

φ=−0.6 1.0 0.869 0.967 0.985 0.991 0.993 0.994 0.995 0.996 0.996 0.999 0.997

b=12 1.5 0.966 0.991 0.995 0.997 0.998 0.998 0.999 0.999 0.999 1.000 0.999

2.0 0.993 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

3.0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.5 0.883 0.971 0.987 0.992 0.993 0.995 0.996 0.997 0.998 0.998 0.998

φ=−0.9 1.0 0.996 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

b=58 1.5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

2.0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

3.0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

τ=50 and N=10,000 independence simulation trials

652 Int J Adv Manuf Technol (2012) 61:649–655

the τth observation, a false alarm would be occurred. Theindependent examinations were being repeated 10,000times for different values of φ and d. Since the purpose ofthis research is change-point detection, instead of signaldetection, the simulation has been designed to evaluate theaccuracy and precision of the change-point estimation. So,all the change-point estimations have been simulated onreal changes. In Table 2, the E(n) is the average of ns in10,000 replications of each simulations. std(n) is the standard

deviation of ns and bt is the average of the estimated changepoints by proposed estimator. If bt has estimated the change

point correctly, value of bt will be 50. std btð Þ is the samplestandard deviation of change-point estimations for 10,000independent simulation trials. The performance of change-point estimation is shown in Table 2. Evaluating the resultsfor a 0.5σ shift in μ, the expected number of observationresults for ARMA(1,1) individual control chart range from51.313 to 445.781 and the standard deviation of signals

range from 0.587 to 1818. The average and the standarddeviation of the 10,000 estimations of τ vary from 49.537 to54.845 and from 1.318 to 20.253, respectively.

The accuracy of the change-point estimator for μ wasevaluated by examining the probability that bt belongs toclosed interval [τ−m,τ+m] where τ is the actual changepoint and m is equal to 0, 1, …, 10. The results of thissimulation are exhibited in Table 3. The probability that theproposed change-point estimator will exactly identify theactual change point are 0.252, 0.595, 0.802, 0.897, and0.986 for step change of magnitude 0.5σ, 1.0σ, 1.5σ, 2.0σ,and 3.0σ, respectively, where φ is 0.3 and b is 4. Theprobability that the change-point estimator is within ±6observations is 0.804 for d=0.5.

4 A numerical example

In order to demonstrate the application of our method, inthis section, a numerical example was provided. Assume

Table 4 Fifty-two initial observations from an AR(1) process withφ=0.3 and σ=1

j x4j−3 x4j−2 x4j−1 x4j

1 0.6324 5.2249 4.4610 3.4909

2 5.0167 4.1014 4.7663 4.3024

3 5.3261 5.6507 4.9915 2.9431

4 4.5155 6.4476 6.4527 3.8554

5 4.5780 4.1917 3.7330 3.3855

6 4.8045 4.5120 4.9094 4.6050

7 2.9165 3.2459 3.7021 5.2228

8 2.2095 1.6403 3.5299 2.6692

9 1.4195 2.7414 4.3757 4.0206

10 5.1636 4.0536 5.0806 3.1844

11 1.8155 2.3335 3.8903 2.0509

12 3.2505 2.3737 3.2633 1.8792

13 2.6497 0.7904 1.7440 2.9853

Fig. 1 Individuals control chart of AR(1) initial observations

Fig. 2 Individuals control chart of UBM statistic

t yt τ Cτ

1 3.4523

2 4.5467 1 20.1811

3 4.7279 2 20.7861

4 5.3178 3 22.0601

5 3.9720 4 25.5788

6 4.7077 5 24.9999

7 3.7718 6 27.4355

8 2.5122 7 26.5960

9 3.1393 8 20.6228

10 4.3705 9 17.2804

11 2.5226 10 19.7610

12 2.6917 11 13.6618

13 2.0423 12 8.6377

Table 5 Example of meanchange point estimator

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that 24 autocorrelated observations, xi, are generated by anAR(1) process with φ=0.3,σ=1, and μ0=5 as you see inTable 4. An individual control chart of the consecutiveinitial observations is shown in Fig. 1 and its control limitsare computed as:

UCL ¼ xþ 3MR

d2¼ 7:2741

LCL ¼ x� 3MR

d2¼ 1:6340

Where x ¼ 124

P24i¼1

xi. The successive observations are divid-

ed into samples of size four (each row in Table 4). Thesubsequent observations are from a similar AR(1) processwith μ1=3.5 which means d=−1.5. An individual controlchart monitors the average of the batches until the controlchart alarms at observation 13.

It is shown in Fig. 2. The three sigma control limits ofindividuals control chart for average of batches are UCL=6.5535 and LCL=2.3546 and they were computed accord-ing to Eqs. 7 and 8, respectively.

4.1 To applied our proposed estimator, we need to find thevalue of τ which maximizes

Ct ¼ n� tð Þ bd1 tð Þ � d0� 2

Therefore, we need

bd1 tð Þ ¼ 13� tð Þ�1X13t¼tþ1

Φ Bð Þyt

and Cτ for τ=1,2,…,12. Working with the most resent batchmeans, we have

bd1 12ð Þ¼ 1

1y13 � 8y12ð Þ

¼ 1

12:0423� 0:34 � 2:6917� � ¼ 2:0205

bd1 11ð Þ¼ 1

2y12 � 8y11ð Þ þ y13 � 8y12ð Þð Þ

¼ 1

22:6917� 0:34 � 2:5226� �þ 2:0423� 0:34 � 2:6917

� �� �¼ 2:3459

and so on. Also we obtain

C12 ¼ 13� 12ð Þ 2:0205� 4:9595ð Þ2 ¼ 8:6377C11 ¼ 13� 11ð Þ 2:3459� 4:9595ð Þ2 ¼ 13:6618

and so on. All 12 of these values are presented in Table 5 inwhich the first and second columns contain batch index tand the batch mean value yt, respectively. Column threecontains the value of τ used in statements 6 and 7.

We know that the actual change point is τ=6, and as it isshown in Table 5, column 4, for τ=6 the amount of Cτ is

being maximized which means bt estimator estimates τ=6 asthe mean change point.

5 Conclusion

Control charts are used for detecting assignable cause inorder to keep the process in a state of statistical control.One of the essential conditions to use the control charts isthe independency of the observations. In some cases, likeprocesses with automated sampling and high frequency, inwhich there is correlation among observations, the applica-tion of the UBM statistic is one of the suitable solutions fordealing with this problem. Actually, the simplicity of thisstatistic, UBM, is its advantage in practice.

When a control chart alarms that an assignable cause ispresented, quality engineers try to determine its source andremove it immediately. The likelihood of finding the source ofthe changes is increased by knowing the exact time of theprocess change. There are different methods to find the time ofthe process change but they are often used in model-basedtechniques, not in model-free ones. We have suggested anestimator to estimate the mean change point of an AR(1)process by using log-likelihood function where autocorrela-tion of process observations is diminished by using UBMstatistic. The performance of the mean change-point estimatorwas evaluated by using Monte Carlo simulation in which theestimator was only used after real alarms. Therefore, theresults of accuracy and precision of proposed estimator arelimited to processes that false alarms could never appearbefore the mean change. The results show when the absolutevalue of the autocorrelation parameter increases the change-point detection performance increases too. Generally, whenthere is a high autocorrelation between observations, theindividuals control chart alarms more quickly and the interestof applying the proposed estimator will be little. In thepresence of a low autocorrelation and large mean shift, thestandard deviation of real alarms and change-point estimationare relatively acceptable.

Acknowledgments The authors would like to thank the referees fortheir valuable comments and suggestions that have led to theimprovement in this paper.

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