25.4.1 Formulation
Assuming that the process is stable,{yt} can be described by a stationary ARMA process as given in equation (25.1). In SPC, our objective is to detect changes in mean as early as possible. Recall that the sum or average of all the measurements since the change occurred is the best indicator of process change, if we know or can guess the time of occurrence. This intuitive claim can be supported by the hypothesis testing problem shown in (25.2) and (25.3).
In the presence of autocorrelation, underH0in (25.2), we have
¯ yn−j+1
n− j+1∼N
0,
|h|<n−j+1
1− |h| n− j+1
γ(h)
, (25.16)
where
¯ yn−j+1=
n
i=jyi
n−j+1
and γ(ã) is the autocovariance function of {yt, t=1, 2, . . .}.26 The variance of
¯ yn−j+1√
n−j+1 approaches υ2=∞
h=−∞γ(h) as n becomes large. Using (25.16), we can therefore test (25.2) against (25.3) using the test statistic
z= y¯n−j+1
√n− j+1
|h|<n−j+1
(1− |h|/n−j+1)γ(h). (25.17)
With the inclusion of the autocorrelation coefficient, the development is similar to that of the i.i.d. case. We reject the null hypothesis (25.2) in favor of (25.3) when the z-statistic in (25.17) exceeds a certain critical valuezα, that is,
¯ y√
n− j+1
|h|<n−j+1(1− |h|/n− j+1)γ(h) >zα. (25.18) As the exact time of occurrence of change is not known a priori, to implement the scheme on an on-line basis, we need to calculate n z-statistics,
zj = y¯n−j+1
√n−j+1
|h|<n−j+1(1− |h|/n− j+1)γ(h), j =1,2, . . . ,n. (25.19) Similarly, when azjexceeds a pre-specified control limit, sayz* (to be obtained from simulation with a desired ARL value) we conclude that there has been a change in
392 CUSUM and Backward CUSUM for Autocorrelated Observations
mean; and for the same reason as in i.i.d. case,zαin (25.18) is not used as theα-value cannot be interpreted as the probability of eventually obtaining a false alarm.27
Note that (25.19) can also be written as n
i=j
yi >zα
(n− j+1)
|h|<n−j+1
1− |h| n−j+1
γ(h), j =1,2, . . . ,n.
The left-hand side is just the sum of then− j+1 most recent observations, which is the (BCUSUM) shown in the previous section. We can therefore test the presence of a step shift in mean by applying the following scheme:
BCUSUMnj ≥z*υn−j+1√
n− j+1, j =1,2, . . . ,n, (25.20) where
υn2−j+1 =
|h|<n−j+1
1− |h| n−j+1
γ(h).
Similar to (25.9), to detect both an increase and a decrease in mean, the following scheme may be implemented:
BCUSUMnj ≥z*υn−j+1
√n−j+1, BCUSUMnj ≤ −z*υn−j+1
√n−j+1, j =1,2, . . . ,n. (25.21) Graphically, the control limits for the BCUSUM in (25.20) and (25.21) resemble a parabolic mask. Note that in implementing the above BCUSUM schemes, one only needs an estimate of the autocovariance function of the series being monitored.
25.4.2 Mask representation
Implementing the BCUSUM from either equation (25.20) or (25.21) for the on-line detection of mean shift implies that we need to calculatenBCUSUMs at every time periodt. In addition, note thatn increases as more observations become available.
The BCUSUM procedure is therefore computationally intensive when implemented on-line. Interestingly, we can express the BCUSUM decision rules shown in equations (25.20) and (25.21) using the typical CUSUM representation, which has been shown Section 25.3 (see the derivations of equation (25.13)) to be as follows:
CUSUMnj ≤ −z*υn−j
n−j+CUSUMnn, j =1,2, . . . ,n, (25.22) and
CUSUMnj ≥z*υn−j
√n− j+CUSUMnn, CUSUMnj ≤ −z*υn−j
√n− j+CUSUMnn, j=1,2, . . . ,n, (25.23) respectively. Similar to the BCUSUM scheme, thez* in equation (25.22) or (25.23) is chosen such that the CUSUM chart achieves a pre-specified in-control ARL. The parabolic CUSUM scheme for monitoring a sequence of i.i.d. normal random variables shown in Section 25.3 is a special case of equation (25.23).
In using equation (25.22) or (25.23), we also assume that CUSUM00=0 as in the i.i.d.
case. Similarly, for the parabolic masks equation (25.22) or (25.23), the first observation
CUSUM Scheme for Autocorrelated Observations 393 will signal a change wheny1>z*υ1. The assumption that CUSUM00=0 is crucial to the use of the V-mask in which the initial observation is used in decision making applies only when initial CUSUM value is zero and for a stationary process with a zero mean. When the process is not centered at zero, one monitors the following the following CUSUM scheme:
CUSUMnj = j
i=1
(yi−μ0),
whereμ0represents the mean of the process.
25.4.3 Sensitivity analysis
In analyzing the performance of the proposed CUSUM scheme, we focus our attention on the detection of changes in the mean of an AR(1) process. The importance of an AR(1) process in SPC has been emphasized in the literature.8,11,12In the following, we describe a simple graphical procedure that can facilitate the choice of the constraint z*. Subsequently, we compare the performance of the proposed CUSUM scheme with the other established procedures for monitoring autocorrelated processes. We then address one of the major criticisms in implementing a CUSUM in mask form (i.e. how far should we extend the CUSUM mask arms?).
25.4.3.1 Choice of z*
Determining the CUSUM parameterz* through simulation may take a lot of time specially when the initial guess forz* is far from the required value. One simple approach is to construct a monogram similar to that shown in Figure 25.3.
3.5 3.3 3.1 2.9 2.7 2.5 2.3 2.1 1.9 1.7
1.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
z*
1000 400 200 100
30 ARL0=20
50
φ
Figure 25.3 Choice ofz* for various combinations of ARL andφ.
394 CUSUM and Backward CUSUM for Autocorrelated Observations
Letδφ represent the error in estimating the AR(1) parameterφ. Using Figure 25.3 and given a target in-control ARL, one can see that the effect ofδφon the appropriate z* is less significant forφ near 0 compared to whenφis near 1. This signifies that misspecification ofφwould not result in a significantly differentz* when the auto- correlation is weak. However, for a highly positively autocorrelated process, small misspecification ofφwould result in a significantly differentz*.
Since the parabolic mask that dictates the properties of the proposed CUSUM scheme is characterized not only byz* but also by the autocovariance structure of the assumed model, it would be interesting to see the different parabolic masks for vari- ous values ofφthat would produce a given in-control ARL. The parabolic masks that would provide an in-control ARL of 370 for a two-sided CUSUM scheme (orARL0= 740 for a one-sided CUSUM scheme) various values ofφare shown in Figure 25.4.
From Figure 25.4, one can see that asφapproaches 1 the change in the angle of the arms of the parabola becomes faster. Thus, one can conclude that the proposed CUSUM scheme is less sensitive to misspecification ofφwhen the process is weakly positively autocorrelated compared to when the process is highly positively autocorrelated.
Note that whenφis overestimated, the resulting parabola has a wider envelope than expected. The overestimation ofφis therefore expected to produce a CUSUM scheme
60
40
20
−20
−40
−60 0
φ=0.00
=0.25
=0.50
=0.75
=0.90
Figure 25.4 Parabolic masks for selected AR processes (φ=0.00,0.25,0.50,0.75 and 0.90);
in-control ARL=370.
CUSUM Scheme for Autocorrelated Observations 395 that has a much larger ARL than expected. Conversely, whenφis underestimated, the resulting parabola has a narrower envelope than expected. This will result in a smaller than expected ARL. This property of the proposed CUSUM is similar to the properties of the CUSUM or exponentially weighted moving average applied to forecast residuals27).
25.4.3.2 ARL comparisons
For various levels of autocorrelationφ1and shift in meanδ, Atienza et al.28compare the ARL performance of the proposed CUSUM with the SCR and CUSUMR. Their results are replicated in Table 25.4. The shift in mean is measured in terms of bothσε
andσy. For comparison purposes, the control chart parameters for SCR, CUSUMR and the proposed CUSUM are chosen such that the in-control ARL for the one-sided scheme is approximately 740.
From Table 25.4, it can be seen that the performance of the proposed CUSUM is very close to the performance of the CUSUM on residuals. For large process shifts, the proposed CUSUM outperforms both the SCR and CUSUMR. In detecting small shifts for highly positively autocorrelated processes, CUSUMR is more sensitive than the proposed CUSUM. Note, however, that forφ1 close to 1 (i.e. highly positively autocorrelated processes) a process shift measured in term ofσεis small compared to when the shift is measured in terms ofσy. For example, a 2σε shift for an AR(1) process withφ1=0.75 is equal to only a 1.323σyshift. Similarly, whenσ1=0.9, a 2σε
shift in mean is equivalent to only a 0.872σyshift. By this token, we can expect that the proposed CUSUM will outperform CUSUMR in a wider shift range when the shift is measured in terms ofσy.
25.4.3.3 Length of the mask arm
Ideally, in implementing the CUSUM in a mask scheme, one needs to extend the mask arms to cover all the data points. Thus, when the process is stable, we will often need to store a large amount data for process monitoring. To determine the effect of the length of mask arms on the sensitivity of the proposed CUSUM in detecting a shift in mean, Atienza et al.28 analyze the performance of the CUSUM when the analysis is based only on the latest mw (moving window) of process observations. For a particular mw size, the value ofz* is adjusted such that the in-control ARL will be approximately 740. The results are taken from Atienza et al.28and shown in Table 25.5. Note that the ARL figures in the table are calculated using the same approach as described in the previous section.
From Table 25.5, we can see that for small values ofφ1, we may focus our attention on the latest 50 observations without severely compromising the sensitivity of the CUSUM scheme. For larger values ofφ1, we need to concentrate on the latest 100 observations. For all cases, the performance of the CUSUM scheme utilizing only the latest 200 observations is already quite close to the performance of the CUSUM that uses all process observations. From these results, it is quite clear that one does not need to indefinitely extend CUSUM mask arm in order to effectively detect shifts in process mean.
Table25.4ARL(SARL)Comparisonforone-sidedSCR,CUSUMRandCUSUMschemes(fromTable1ofAtienzaetal.28). δ(inσε) 1ControlChartChartParameter(s)0.000.250.500.751.001.502.003.00 0.00δ(inσx)0.000.250.500.751.001.502.003.00 SCRUCL=+3.00740.78335.60161.0481.8043.9614.976.302.00 (740.26)(335.09)(160.54)(81.30)(43.45)(14.46)(5.78)(1.41) CUSUMRh=4.78,k=0.50740.03123.9835.2916.199.935.523.862.49 (734.93)(118.74)(29.00)(10.67)(5.30)(2.22)(1.25)(0.64) CUSUMz*=3.391740.92102.4534.4717.2710.695.443.461.88 (737.41)(77.14)(22.21)(10.33)(5.94)(2.73)(1.59)(0.79) 0.25δ(inσx)0.000.240.480.730.971.451.942.9 SCRUCL=+3.00740.78406.64230.38134.5880.9431.7013.593.21 (740.26)(406.34)(230.31)(134.73)(81.29)(32.33)(14.29)(3.42) CUSUMRh=6.00,k=0.375739.96156.0451.1324.7215.298.355.673.43 (730.15)(148.43)(41.64)(16.45)(8.47)(3.63)(2.05)(0.98) CUSUMz*=3.316740.74151.5753.6527.1316.528.074.712.22 (741.44)(120.94)(37.66)(17.87)(10.12)(4.78)(2.73)(1.21)
396
0.50δ(inσx)0.000.220.430.650.871.301.732.60 SCRUCL=+3.00740.78494.54334.51229.00158.3877.3437.988.48 (740.26)(494.50)(335.09)(230.30)(160.49)(81.04)(42.62)(11.81) CUSUMRh=8.02,k=0.25740.02211.5583.8344.0928.0915.4410.335.98 (728.30)(193.50)(69.66)(31.34)(16.77)(7.22)(4.13)(1.90) CUSUMz*=3.205741.20235.2296.4950.4131.7314.698.553.43 (736.83)(200.84)(73.08)(36.20)(21.68)(10.13)(5.94)(2.48) 0.75δ(inσx)0.000.170.330.500.660.991.321.98 SCRUCL=+3.00740.78603.67492.95402.88328.96216.40136.4941.90 (740.26)(603.94)(494.49)(406.30)(334.95)(229.41)(156.97)(64.67) CUSUMRh=12.11,k=0.125739.97330.54175.15107.6373.9843.1329.3616.95 (710.16)(300.28)(145.46)(79.15)(48.70)(23.14)(13.56)(6.56) CUSUMz*=3.017741.57394.21217.06127.7987.0344.2025.5410.33 (744.00)(370.38)(192.81)(105.02)(70.65)(34.51)(20.56)(9.28) 0.90δ(inσx)0.000.110.220.330.440.650.871.31 SCRUCL=+3.00740.78681.58626.47574.53524.77427.91330.28145.22 (740.26)(682.09)(628.84)(580.03)(535.20)(455.07)(381.89)(228.02) CUSUMRh=17.75,k=0.05739.99499.22355.22265.14206.18137.1399.6661.27 (691.52)(437.77)(297.32)(206.46)(146.85)(86.80)(57.11)(28.91) CUSUMz*=2.734740.53542.05409.33318.23230.90142.6797.9046.37 (736.47)(533.93)(405.81)(308.90)(215.76)(132.31)(90.66)(45.47)
397
Table25.5EffectofmwontheARL(SRL)performanceoftheproposedCUSUMscheme(fromTable2Atienzaetal.28). Shiftinmean(inσε) φ1mwz*0.000.250.500.751.001.502.002.503.00 0.00103.342740.92179.5952.1420.6411.355.323.352.381.84 (740.34)(175.14)(48.26)(17.06)(7.96)(2.75)(1.57)(1.05)(0.78) 253.372741.20140.7738.9317.2810.405.393.392.401.86 (743.31)(135.06)(31.38)(11.54)(5.98)(2.68)(1.58)(1.07)(0.79) 503.383740.21121.7735.0417.2610.585.483.422.431.88 (744.49)(110.26)(24.64)(10.46)(6.02)(2.73)(1.61)(1.08)(0.78) 1003.388740.14109.5634.2817.2210.625.473.432.441.88 (742.00)(93.19)(22.31)(10.41)(5.97)(2.73)(1.61)(1.09)(0.78) 2003.390740.92103.7334.4317.2710.685.443.422.421.88 (739.60)(81.23)(22.16)(10.37)(5.94)(2.73)(1.59)(1.08)(0.79) All3.391740.92102.4534.4717.2710.695.443.462.431.88 (737.41)(77.14)(22.21)(10.33)(5.94)(2.73)(1.59)(1.08)(0.79) 0.25103.261740.96238.8388.1838.1719.887.954.642.982.14 (741.18)(231.27)(85.46)(35.73)(16.66)(5.29)(2.77)(1.72)(1.20) 253.293740.21204.9966.4329.5016.698.044.723.042.18 (747.92)(197.08)(59.27)(23.18)(11.42)(4.80)(2.73)(1.71)(1.20) 503.306741.24183.2057.8827.0416.238.024.683.102.21 (744.47)(171.59)(48.04)(18.51)(10.12)(4.78)(2.73)(1.76)(1.22) 1003.312740.99166.0754.2326.8516.548.024.693.082.22 (743.18)(149.69)(40.46)(17.69)(10.19)(4.75)(2.75)(1.76)(1.22) 2003.314740.21155.3853.4327.1116.468.064.723.092.22 (739.16)(132.76)(37.46)(17.91)(10.08)(4.75)(2.74)(1.78)(1.24) All3.316740.74151.5753.6527.1316.528.074.713.122.22 (741.44)(120.94)(37.66)(17.87)(10.12)(4.78)(2.73)(1.78)(1.21)
398
ShiftinMean(inσε) φ1mwz*0.000.250.500.751.001.502.002.503.00 0.50103.153740.92322.19150.3178.8343.9016.818.695.123.28 (749.28)(310.89)(151.31)(75.15)(41.57)(15.09)(7.27)(4.02)(2.45) 253.182741.24296.54125.8063.7935.1214.748.285.133.34 (747.78)(287.54)(123.97)(57.70)(30.39)(10.60)(5.83)(3.73)(2.48) 503.194741.43275.27111.0755.2732.1114.628.325.123.39 (743.17)(262.46)(104.11)(46.75)(24.49)(9.96)(5.76)(3.68)(2.50) 1003.200741.43256.69100.4351.4631.6014.728.505.193.40 (742.65)(238.85)(85.67)(38.74)(21.79)(10.10)(5.90)(3.76)(2.49) 2003.203740.96244.3897.0850.4331.6114.738.545.213.43 (739.25)(218.18)(76.38)(36.30)(21.73)(10.10)(5.96)(3.75)(2.49) All3.205741.20235.2296.4950.4131.7314.698.555.203.43 (736.83)(200.84)(73.08)(36.20)(21.68)(10.13)(5.94)(3.78)(2.48) 0.75102.969740.21440.64274.63185.16122.6158.6230.3217.9411.04 (735.20)(434.81)(275.77)(184.30)(120.52)(59.31)(31.19)(17.85)(11.57) 252.991740.34433.78260.48168.44111.2851.7727.2215.9810.33 (744.89)(433.13)(262.38)(167.34)(108.08)(41.63)(26.28)(14.75)(10.01) 503.004740.14419.40248.51151.96100.9047.0025.5515.9210.21 (743.44)(410.78)(247.73)(148.20)(95.85)(41.43)(21.91)(13.66)(9.39) 1003.012740.21408.27233.90138.9992.2044.7025.3516.0910.21 (740.45)(398.56)(229.58)(125.88)(83.24)(36.43)(20.59)(13.74)(9.29) 2003.015741.24403.04223.93131.2186.9344.1925.4016.1810.34 (745.10)(388.40)(211.27)(112.79)(72.07)(34.57)(20.58)(13.81)(9.29) All3.017741.57394.21217.06127.7987.0344.2025.5416.2410.33 (744.00)(370.38)(192.81)(105.02)(70.65)(34.51)(20.56)(13.73)(9.28) (Continued)
399
Table25.5EffectofmwontheARL(SRL)performanceoftheproposedCUSUMscheme(fromTable2Atienzaetal.28).(Contineued) ShiftinMean(inσε) φ1mwz*0.000.250.500.751.001.502.002.503.00 0.90102.709740.53555.85426.97338.67254.69162.43114.0271.7251.33 (749.76)(553.89)(435.00)(345.92)(257.55)(173.15)(117.10)(76.82)(55.73) 252.716740.49557.07425.04337.30254.26160.80113.3871.0649.88 (755.12)(553.57)(429.05)(348.23)(256.05)(168.92)(116.09)(75.86)(53.27) 502.723740.53554.90424.90332.30250.08156.41109.4368.8348.26 (747.07)(549.57)(430.99)(342.59)(249.30)(161.72)(113.50)(72.70)(50.64) 1002.729740.96551.67421.02329.14243.27151.39104.4266.4645.98 (739.88)(546.87)(425.45)(337.90)(240.56)(152.94)(103.74)(66.43)(46.28) 2002.732740.65547.06414.75323.74237.52145.41100.5965.2845.52 (738.82)(540.14)(414.57)(326.70)(231.39)(143.04)(97.07)(63.36)(44.92) All2.734740.53542.05409.33318.23230.90142.6797.9065.9846.37 (736.47)(533.93)(405.81)(308.90)(215.76)(132.31)(90.66)(63.16)(45.47)
400
CUSUM Scheme for Autocorrelated Observations 401
−4
−3
−2
−1 0 1 2 3 4 5
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 Time (t)
yt
Figure 25.5 The simulated AR(1) series withφ1= 0.75,μ=0, andσε=1.
25.4.4 An example
To illustrate the application of the proposed CUSUM scheme, Atienza et al.28simulated the AR(1) series shown in Figure 25.5. The first 50 observations represent the in-control process state. Under this in-control state, the observations were generated from an AR(1) process withφ1=0.75,μ=0 andσε=1. Starting att=51, a mean shift of size 2σε or, equivalently, 0.875σy was introduced. Incidentally, for this example, model estimation using the first 50 observations shows that ˆφ1=0.746≈0.75, ˆμ=0.0,and σˆε=0.98.
The corresponding Shewhart chart on residuals detected the shift att=90 (see Figure 25.6a). A CUSUM on residuals for the series in Figure 25.5 was constructed, using the optimal CUSUMR parameters whenφ1=0.75 suggested by Runger et al.12 From Figure 25.6b, we can see that the corresponding CUSUMR detected the shift at t=75.
For an AR(1) series, the autocovariance function is given by
γ(0)= 1 1−φ12
σε2, γ(1)= φ1
1−φ12σε2,
γ(h)=φ1γ(h−1), h ≥2.
From here, it is easy to calculate the quantity υn−2 j =
|h|<n−j
1− |h|
n− j
γ(h), j=1,2,3, . . . ,n,
402 CUSUM and Backward CUSUM for Autocorrelated Observations
0 20 40 60
(a)
80 100
−4
−3
−2
−1 0 1 2 3 4
Time (t) Residual
UCL
LCL
0 20 40 60 80
−15
−10
−5 0 5 10 15 20
Time (t) (b)
CUSUMR
h = −12.11 h = 12.11
Figure 25.6 Shewhart and CUSUM (k=0.125) charts on residuals for the series in Figure 25.5.
needed for calculating the control limits for the CUSUM. Note that sinceγ(h)=γ(−h), thenυn2−jmay also be written as
υn−2 j =
⎧⎪
⎪⎨
⎪⎪
⎩
0, j =n,
γ(0), j =n−1,
γ(0)+2n−j−1
i=1
1−n−ij
γ(i), j =n−2,n−3, . . . ,1.
The above expression can be implemented using standard spreadsheet packages such as Microsoft Excel and Lotus 1-2-3. For illustration purposes, assume that we want to establish a CUSUM mask that focuses only on the latest 10 observations (i.e.mw= 10). This means we only need to calculate the first 10 υn2−j values (i.e., for j =n, n−1, . . . ,n−9). For an AR(1) process withφ1=0.75,μ=0, andσε=1 these values are given in Table 25.6. Once theυn−2 j values are available, we just need to specify
CUSUM Scheme for Autocorrelated Observations 403 Table 25.6 CUSUM mask for an AR(1) process withφ1=0.75,μ=0, andσε=1 (mw=10).
n− j γ(n−j) υn−2 j Lower arm Upper arm
9 0.172 10.823 −29.303 29.303
8 0.229 10.362 −27.032 27.032
7 0.305 9.829 −24.628 24.628
6 0.407 9.209 −22.070 22.070
5 0.542 8.484 −19.338 19.338
4 0.723 7.632 −16.405 16.405
3 0.964 5.429 −13.236 13.236
2 1.286 4.000 −9.783 9.783
1 1.714 2.286 −5.938 5.938
0 2.286 0.000 0.000 0.000
the required constantz* to establish the CUSUM mask given by equation (25.22) or (25.23). Assuming an ARL0 of 370 for a two-sided CUSUM scheme (i.e. an ARL0of approximately 740 for a one-sided scheme) is required, we can see from Table 25.5 that we need az*-value equals to 2.969. The summary of the calculations done using a spreadsheet is given in Table 25.6, while the resulting CUSUM mask is shown in Figure 25.7.
In monitoring the process, we just need to superimpose the vertex of the CUSUM mask in Figure 25.7 on the latest CUSUM value. Figure 25.8 shows how the CUSUM mask in Figure 25.7 detected the change in the mean of the process att=60, or 10 observations after the change was introduce. Since our mask is based only on mw= 10, it is not necessary for us to plot all the historical CUSUM values. The same result will be obtained if we are only maintaining the latest 10 observations.
−40 9 8 7 6 5 4 3 2 1 0
−30
−20
−10 0 10 20 30 40
n−j Unit
Figure 25.7 Plot of the CUSUM mask in Table 25.6.
404 CUSUM and Backward CUSUM for Autocorrelated Observations
0 10 20 30 40 50 60
−20
−10 0 10 20 30 40 50
TIME (t) CUSUM
Figure 25.8 The CUSUM chart for from Atienza et al.28