In Phase II monitoring, we use the true mean function µ(·) and the sam-ple covariance matrix Σ of the above 199 profiles as the in-control process parameters to perform our simulation. Let µ be the 19 × 1 vector com-puted by µ(x) = 1 + 15e−1.5(x−1)2 at x = 0.64, 0.8, . . . , 3.52 . We then generate in-control profiles from MV N (µ, Σ). Consider I-shift from µI to µI + α × σI, α = 0, 0.25, . . . , 3; M-shift from µM to µM + β × σM, β = 0, 0.25, . . . , 3; N-shift from µN to µN + γ × σN, γ = 0, 0.25, . . . , 3. Fig-ures 21-23 display the curves of f (x) = I + MeN (x−1)2 with different value of I, M, N respectively. Figure 24 display the plot of the first four eigenvectors of Σ. And Figures 25-28 illustrate respectively the corresponding features captured by the first four principal components, by showing in each plot the mean profile and the profiles corresponding to the largest score and smallest score of the principal component. Now for each shift in I, M, N , we generate 200,000 profiles to computed an ARL estimate. Then repeat this 1000 times to get a more accurate estimate along with its standard error as before. Ta-bles 4-6 display the ARL values for the shifts in I, M, N , respectively. We can see from Table 4 that PC3 and PC1 can capture the shift in I. PC3 has the best power for detecting I-shift. PC2 and PC4 hardly have any power on detecting I-shift. Also, we can see from Table 5 that PC2 and PC1
out-perform the other two with PC2 slightly better than PC1 in capturing the shift in M . And for detecting N-shift, we can see from Table 6 that PC2 and PC1 are the best two. But as the shift tends to 3 units of σN, PC2 still has the best power and PC3 becomes the second best for detecting N-shift.
In fact, we can obtain the exact ARL values as follows. Let P = (P1, P2,
· · · , P4)0, where Pi is the i-th eigenvector of Σ. Then P projects profiles to the first four principal components. Let λ1, · · · , λ4 be the corresponding eigenvalues of Σ. Since the in-control profile Y ∼ MV N (µ, Σ) and P ΣP0 = diag(λ1, · · · , λ4) ≡ Λ4, we have the score vector P Y ∼ MV N(P µ, Λ4).
Denote the shifted profile by ˜Y and the shifted mean profile by ˜µ. Then P ˜Y ∼ MV N(P ˜µ, Λ4). Let δ = ˜µ − µ. The probability of detecting the shift of the i-th principal component chart is
p = P (|P0i(Y − µ)
where Z is the standard normal variate. The value 1/p is the actual ARL of the i-th principal-component chart. Tables 7-9 show respectively the actual ARL for each of the I, M, N shifts. By comparing Tables 4-9, we can see that our simulated ARLs are all very close to the actual ARLs, which verifies the correctness of the simulation.
Since each of the I, M, N shifts is captured by more than one principal-component chart, we recommend using the combined chart to monitor these shifts. A combined chart means we monitor the process by more than one chart and the combined chart signals out of control if any of the charts
signals. Set the overall false-alarm rate at α = 0.005. Since these scores are independent, the individual false-alarm rate is α0 = 1 − (0.995)1/4 for each of the four principal component scores. Then we can have the control limits of these four scores as shown below:
(P01µ − Z1−(0.995)1/4
If one of the principal component scores is out of the control limits, then this profile is claimed as out of control. That is, only when all four principal component scores of that profile are within the control limits simultaneously, then it will be treated as an in-control profile. In order to do the comparison the above, we also simulate 200,000 profiles each time to get an ARL estimate.
Then repeat 1000 times to get the final ARL estimate and its standard error.
Tables 10-12 display the simulation results for ARL comparison for I, M, N shifts, respectively. Also, we can derive the exact control limits and compute the exact ARL. Tables 13-15 show the ARL computed from the exact control limits with shifts in I, M, N , respectively. By comparing the simulated ARL with the exact ARL for our combined chart, we can see that they are very closed to each other. Also, we plot the ARL comparison plots for PC1, PC2, PC3, PC4 and the combined chart with I, M, N shifts in Figures 29-31 respectively. As we can see for I-shifts, PC3 dominates primarily. And for the combined chart scheme, it was the second best in monitoring the shifts in I. As the shifts gets bigger, the difference in ARL between PC3 and the combined chart gets smaller. Also, for M-shifts, PC2 is the best scheme to monitor it. For the second best one, the PC1 and the combined chart are comparable. Note that two ARL curves intersect at about 1.25σM shift. That is, for shifts smaller than 1.25σM, PC1 has better power. But the combined
chart scheme outperforms PC1 as the shift size tends larger than 1.25σM. We can see the same phenomenon in N. From Figure 31, PC2 is still the best one to monitor shifts in N. PC1 and the combined chart scheme are comparable. Also the two ARL curves intersect at about 1.1σN shift.
Also, we show the ARL values for simultaneous shifts of I and M, I and N, and M and N, in Tables 16-18, respectively. We can see from Table 16 that PC1 takes the major job of monitoring simultaneous shifts in I and M while PC3 serves as the second best. And from Table 17, we can observe that PC2 dominates the simultaneous shifts in I and N while PC1 serves as the second best. Also from Table 18, we can see that PC1 plays the major role in monitoring simultaneous shifts in M and N while PC2 performs the second best. But notice that the power for PC2 here is not comparable with PC1.
Finally, we consider shifts in the variance of I, M, and N. Recall that the in-control situation is: I ∼ N(1, 0.22), M ∼ N(15, 1), N ∼ N(−1.5, 0.32).
Now for variance shifts in I, we consider σI = 0.3, 0.4, . . . , 1. For each shift, we simulate 200,000 profiles to get an ARL estimate and then repeat the procedure for 1000 times to get the mean and the corresponding standard deviation. The ARL results for each principal component chart and the com-bined chart are shown in Table 19. And Figure 32 shows the corresponding ARL curves. As we can see that PC3 dominates for the shifts in variance of I while the combined chart scheme also performs very well in monitoring this kind of shifts.
And for variance shifts in M, we consider σM = 1.5, 2, . . . , 5. The simula-tion results are shown in Table 20 and Figure 33. Likewise, we can see that PC2 dominates for the shifts in variance of M. But here, the combined chart scheme is comparable with the PC2. As σM gets bigger, the combined chart scheme performs even better than PC2. At last, for shifts in variance of N,
we consider σN = 0.4, 0.5, . . . , 1.1. Table 21 and Figure 34 shows the simula-tion results. From these simulasimula-tion results, we can see that PC3 has the best detecting power for variance shifts in N. The combined chart scheme also has great detecting power for monitoring shifts in σN. Notice that as σN tends bigger than 0.7, the profile shape will be greatly affected. Therefore we can see that all these four principal components are sensitive to it and they all have great detecting power. While it is difficult to figure out which principal component dominates the shifts, we recommend the combined chart scheme for monitoring.
Since the model under study is a random effect model, we are curious of the power in monitoring variance shifts. From the above simulations, we have shown that the combined chart scheme is a very good choice for monitoring random effect model. So here, in order to compare the power of our combined chart in detecting mean shifts and variance shifts, we use the ARL of the combined chart scheme. Then, use our comparison basis vef to quantify the difference between the shifted profiles and the reference profile.
Hence we can have the plot that compares the ARL of detecting mean shifts and variance shifts; see Figure 35.
We can see from Figure 35 that for both I, M, N shifts, our monitoring scheme performs better in detecting variance shifts than mean shifts. We can observe that there exist big differences for all I, M, N shifts at small values of comparison basis. And as the value of comparison basis tends bigger, the difference between mean shifts and variance shifts tends smaller. When the value of comparison basis getting large enough, there exist almost no differences between mean shifts and variance shifts. Also from Figure 35, we can see that our monitoring scheme does the best job in detecting variance shifts in N while the second best one is variance shifts in I.
5 A Case Study - VDP Example
5.1 Phase I monitoring
The VDP data set contains n = 24 profiles, each was measured at p = 314 set points. Figure 2 is the plot of the VDP data. We fit these 24 profiles by B-splines with 16 degrees of freedom. See Figure 36 for the plot of the fitted B-splines. Also Figures 37-40 respectively plot principal components 1 to 4 showing the modes of variation they capture. And the first four principal components account for 0.8535, 0.1084, 0.0190, 0.0084 of variation in the profiles, respectively. The total is 0.9893. Figure 41 is the plot of the corresponding eigenvectors. Now for Phase I monitoring, we use the same method as for the aspartame example. Figures 42 and 43 are the control charts of T12 and T22 respectively. We can see that there are no out-of-control profiles in the VDP data.
5.2 Phase II monitoring
Since, in the aspartame example, the first four principal components cannot distinguish the shifts in I, M, N very clearly, we suggest using the combined chart scheme to monitor these shifts. Now, we are going to demonstrate an example that its principal components can capture the shapes of shifts very clearly by simulation.
For demonstrating Phase I monitoring, we need to generate new VDP data. Applying PCA to the original VDP data, we then have the corre-sponding eigenvectors {ρ1, · · · , ρ314} and eigenvalues {λ1, · · · , λ314}. Since the first four principal components can explain 97.57 percent of variation, we only use these four principal components to regenerate the new VDP data.
And these first four principal components account for 0.8402, 0.1072, 0.0192, 0.0091 of variation in these profiles, respectively.
We use the following steps to regenerate a set of new VDP data.
1. Generate i.i.d. ²j,q from the normal distribution with zero mean and variance λq, 1 ≤ j ≤ 24, 1 ≤ q ≤ 4.
2. Generate ˜Xj,k = µ(tk) +P4
q=1²j,q· ρq(tk), 1 ≤ j ≤ 24, 1 ≤ k ≤ 314.
Note that λ1, λ2, λ3, λ4 are the four largest eigenvalues and ρ1, ρ2, ρ3, ρ4 are the corresponding eigenvectors with length 314. And µ(·) is the mean profile of the 24 original VDP curves where tk denotes the k-th set point.
Figure 44 displays the simulated VDP data. We can see that the simulated data do capture the peaks and the shapes of the original VDP data.
Then, we apply our estimation method to the generated data to get the simulated mean profile, eigenvalues, and eigenvectors. Figure 45 displays the mean profile of the real data and the simulated data. We can see that these two curves are very close. Figures 46-49 show the first four eigenvectors of the real data and that of the simulated data, respectively. Likewise, from these four figures we can see that the simulated eigenvectors are all close to that of the real data. Also, Figure 50 shows that the simulated eigenvalues are fairly close to the eigenvalues of the original profile data. Therefore, we can say that the data generation method we adopt here can really capture the shapes and variations of the original data.
In Phase II monitoring, we treat the average profile vector µ of the 24 smoothed VDP profiles and the sample covariance matrix Σ as the in-control process parameters to perform our simulation. Here, µ = (µ(t1), · · · , µ(t314))0 is a 314 × 1 vector and the covariance matrix Σ is a 314×314 matrix. We can generate the in-control profiles by Y ∼ MV N (µ, Σ). For out-of-control conditions, we shift the profiles in their first two principal components. That is, generate new profile data by
Y ∼ k˜ p
λi× vi+ MV N (µ, Σ) ,
where k=0, 0.25, . . . , 2 and vi is the i-th eigenvector of Σ, i=1, 2. We sim-ulated 200,000 profiles to compute an ARL estimate for each out-of-control conditions considered. Then we repeat the procedure 1000 times to get our final ARL estimate along with its standard error. Tables 22 and 23 report the ARL results for shifts in principal component 1 and principal component 2 respectively.
As we can see from Table 22 that shifts in principal component 1 are solely captured by the first principal component score. Likewise, from Ta-ble 23, shifts in principal component 2 are captured by the second principal component score. The other three principal component scores make no con-tributions to the power of detection.
6 Conclusions
The monitoring of process or product profiles is a very popular and promising area of research in statistical process control in recent years. In this study, we discuss monitoring schemes for the random effect nonliner profiles. We use the principal component analysis to analyze the covariance matrix of the profile data and use the corresponding principal component scores that capture the main features of these profile data for process monitoring.
In the study of the aspartame example, our simulation shows that prin-cipal component 3 performs the best over the entire range of I shifts. And principal component 2 is the best for monitoring M shifts while principal component 1 also plays an important role in monitoring M shifts. And for shifts in N, principal component 2 performs the best over the entire range.
Likewise, principal component 1 also plays an important role in monitoring N shifts. Note that as the shift tends bigger in N shift, all those four prin-cipal components are good at catching the out-of-control profiles. And for
each shift, we can also use the combined chart to perform monitoring. We have shown that although this monitoring scheme may not be the best for each shift, it still has comparably good power to monitor these shifts. More-over, we have displayed in the tables the results when two of I, M, N shift simultaneously.
Using the example of vertical density profiles, we demonstrate that when the shift corresponds to a mode of variation that a particular principal ponent represents, then we can directly use the score of that principal com-ponent to perform monitoring.
So if a principal component can clearly identify the shift, a situation may be rare in real applications, then we recommend monitoring the score of that principal component. Otherwise we recommend the combined chart scheme because it still has comparably good power to monitor these shifts.
Moreover, we use a mean-squares-error-like (MSE-like) measure as a com-parison basis to compare the ARL performance in mean shifts and variance shifts. Simulation results indicate that our monitoring scheme performs bet-ter in detecting variance shifts. And in the selection of number of principal components, we adopt the cross-validation method for its popularity and simplicity. Also, in the use of cross-validation, we have observed that the se-lection is affected by the number of groups that the data set is divided into.
By our simulation result, we can see that as the number of groups increases, the selected number of principal components decreases. Further studies are needed on this issue.
Finally in the studies of Phase I monitoring, we compare T12 and T22. And we show by simulation that T12 has a better overall performance than T22 under temporal shifts.
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One shift set at the 100th curve among 200 curves chart\ I shifts 5 × σI 7 × σI 10 × σI
(vef) (1.543) (1.925) (2.554) T12 power 0.4506 0.8213 0.9842 false-alarm 0.0068 0.0062 0.0059 T22 power 0.1467 0.4567 0.869 false-alarm 0.0028 0.0026 0.0025
Two shifts set at the 66th and 133th curve among 200 curves chart\ I shifts 5 × σI 7 × σI 10 × σI
(vef) (1.543) (1.925) (2.554) T12 power 0.2924 0.52 0.7339 false-alarm 0.0062 0.0059 0.0054 T22 power 0.0726 0.1746 0.333 false-alarm 0.0026 0.0025 0.0022
Three shifts set at the 50th, 100th and 150th curve among 200 curves chart\ I shifts 5 × σI 7 × σI 10 × σI
(vef) (1.543) (1.925) (2.554) T12 power 0.1972 0.325 0.4515 false-alarm 0.0059 0.0054 0.005
T22 power 0.0392 0.0745 0.1207 false-alarm 0.0025 0.0023 0.002
Table 1: The performance assessed by detecting power and false-alarm rate for T12 and T22 to detect I shifts with temporal shifts.
(i.e. Shift from µI=1 to 1+k × σI and vef (in parentheses), represents the impact of the shift on the whole function variation.)
One shift set at the 100th curve among 200 curves chart\ M shifts 5 × σM 7 × σM 10 × σM
(vef) (3.358) (4.580) (6.445) T12 power 0.716 0.9637 0.9989 false-alarm 0.007 0.0068 0.0067 T22 power 0.3077 0.7331 0.9747 false-alarm 0.003 0.0029 0.003
Two shifts set at the 66th and 133th curve among 200 curves chart\ M shifts 5 × σM 7 × σM 10 × σM
(vef) (3.358) (4.580) (6.445) T12 power 0.4449 0.671 0.8207 false-alarm 0.0067 0.0066 0.0064 T22 power 0.1209 0.2622 0.4298 false-alarm 0.0029 0.003 0.0028
Three shifts set at the 50th, 100th and 150th curve among 200 curves chart\ M shifts 5 × σM 7 × σM 10 × σM
(vef) (3.358) (4.580) (6.445) T12 power 0.2854 0.4197 0.52 false-alarm 0.0065 0.0064 0.0063
T22 power 0.0583 0.1025 0.1502 false-alarm 0.0028 0.0028 0.0027
Table 2: The performance assessed by detecting power and the false-alarm rate for T12 and T22 to detect M shifts with temporal shifts.
(i.e. Shift from µM=15 to 15+k × σM and vef (in parentheses), represents the impact of the shift on the whole function variation.)
One shift set at the 100th curve among 200 curves chart\ N shifts 2 × σN 2.5 × σN 3 × σN
(vef) (2.284) (2.979) (3.923) T12 power 0.1587 0.4858 0.8961 false-alarm 0.006 0.0045 0.0028 T22 power 0.0396 0.2023 0.6566 false-alarm 0.0024 0.0018 9.5612e-004 Two shifts set at the 66th and 133th curve among 200 curves
chart\ N shifts 2 × σN 2.5 × σN 3 × σN
(vef) (2.284) (2.979) (3.923) T12 power 0.1152 0.3208 0.6172 false-alarm 0.005 0.0031 0.0016 T22 power 0.0243 0.092 0.2507 false-alarm 0.002 0.0012 5.0625e-004
Three shifts set at the 50th, 100th and 150th curve among 200 curves chart\ N shifts 2 × σN 2.5 × σN 3 × σN
(vef) (2.284) (2.979) (3.923) T12 power 0.0906 0.2097 0.3781 false-alarm 0.0042 0.0025 0.0011 T22 power 0.017 0.0451 0.0946 false-alarm 0.0016 8.8298e-004 3.1702e-004
Table 3: The performance assessed by detecting power and the false-alarm rate for T12 and T22 to detect N shifts with temporal shifts.
(i.e. Shift from µN= -1.5 to -1.5+k × σN and vef (in parentheses), represents the impact of the shift on the whole function variation.)
α
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.002) (1.007) (1.015) (1.027) (1.042) (1.060) PC1 200.38 196.64 187.86 174.44 157.88 140.51 123.51 0.2004 0.1970 0.1795 0.1630 0.1394 0.1226 0.0940 PC2 200.12 200.14 200.16 200.21 199.77 199.26 198.53 0.2003 0.1961 0.1981 0.1962 0.2006 0.1979 0.1987 PC3 200.13 172.04 119.20 76.094 48.049 30.992 20.497 0.2027 0.1669 0.0933 0.0487 0.0239 0.0118 0.0066 PC4 200.40 200.34 200.27 199.69 199.23 198.58 197.87 0.2031 0.1922 0.2027 0.2006 0.1964 0.1984 0.1957
chart 1.75 2 2.25 2.5 2.75 3
(1.081) (1.105) (1.131) (1.160) (1.190) (1.223) PC1 107.43 93.12 80.410 69.369 59.919 51.721
0.0796 0.0643 0.0489 0.0415 0.0310 0.0268 PC2 198.53 197.7 196.91 196.16 195.29 194.32 0.1965 0.1950 0.1936 0.1915 0.1858 0.1976 PC3 13.999 9.859 7.1627 5.3581 4.1247 3.2674 0.0036 0.0020 0.0012 0.0008 0.0005 0.0003 PC4 197.71 196.6 195.58 194.84 193.55 192.47 0.1951 0.1950 0.1900 0.1981 0.1839 0.1929 Table 4: ARL comparison for I-shift (µI = 1 + 0.2 × α).
β
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.019) (1.062) (1.126) (1.207) (1.302) (1.408) PC1 200.6 177.7 131.5 89.18 59.37 39.81 27.25
0.2032 0.1735 0.1068 0.0601 0.0322 0.0116 0.0099 PC2 200.1 168.9 112.9 69.49 42.81 27.02 17.66
0.1943 0.1549 0.0871 0.0411 0.0201 0.0098 0.0050 PC3 200.3 199.0 196.8 192.8 187.7 181.3 174.3
0.1958 0.2064 0.1942 0.1884 0.1759 0.1689 0.1611 PC4 199.8 199.8 197.9 195.5 191.9 187.5 182.1
0.1963 0.1986 0.2041 0.1906 0.1901 0.1841 0.1679
chart 1.75 2 2.25 2.5 2.75 3
(1.524) (1.647) (1.775) (1.908) (2.044) (2.183) PC1 19.06 13.65 10.023 7.5328 5.7976 4.5616
0.0058 0.0034 0.0021 0.0014 0.0009 0.0006 PC2 11.93 8.344 6.0404 4.5196 3.492 2.7828 0.0028 0.0016 0.0009 0.0006 0.0004 0.0003 PC3 166.2 157.9 149.08 140.33 131.89 123.37 0.1489 0.1378 0.1276 0.1192 0.1080 0.0961 PC4 176.4 169.6 163.41 156.41 149.01 142.01 0.1699 0.1562 0.1475 0.1354 0.1294 0.1229 Table 5: ARL comparison for M-shift (µM = 15 + 1 × β).
γ
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.045) (1.123) (1.233) (1.377) (1.552) (1.761) PC1 200.6 175.46 123.33 77.235 46.416 27.765 16.783 0.2045 0.1576 0.0990 0.0480 0.0221 0.0103 0.0047 PC2 200.07 170.31 109.92 61.319 32.571 17.17 9.2111 0.1943 0.1512 0.0825 0.0339 0.0132 0.0051 0.0018 PC3 200.33 195.5 188.38 184.87 188.98 198.38 192.89 0.1958 0.1895 0.1835 0.1750 0.1877 0.1956 0.1812 PC4 199.79 196.79 191.12 189.78 194.81 199.71 184.19 0.1963 0.2030 0.1898 0.1835 0.1934 0.1935 0.1758
chart 1.75 2 2.25 2.5 2.75 3
(2.004) (2.284) (2.607) (2.979) (3.413) (3.923) PC1 10.336 6.5317 4.2695 2.9118 2.0902 1.5929
0.0023 0.0011 0.0005 0.0003 0.0001 0.00009 PC2 5.133 3.0404 1.9639 1.4142 1.1482 1.0383
0.0007 0.0003 0.0001 0.00006 0.00003 0.00001 PC3 138.36 64.003 21.627 6.4314 2.1517 1.1355
0.1115 0.0351 0.0068 0.0011 0.0002 0.00003 PC4 126.66 61.374 23.666 8.422 3.222 1.5783
0.1013 0.0342 0.0081 0.0017 0.0003 0.00009 Table 6: ARL comparison for N-shift (µN = −1.5 + 0.3 × γ).
α
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.002) (1.007) (1.015) (1.027) (1.042) (1.060) PC1 200.0 196.78 187.65 174.04 157.74 140.46 123.39 PC2 200.0 199.96 199.84 199.64 199.35 198.99 198.55 PC3 200.0 172.05 119.16 75.951 47.999 30.926 20.493 PC4 200.0 199.94 199.77 199.49 199.09 198.58 197.96
chart 1.75 2 2.25 2.5 2.75 3
(1.081) (1.105) (1.131) (1.160) (1.190) (1.223) PC1 107.48 93.109 80.42 69.38 59.87 51.73 PC2 198.03 197.44 196.7 196.0 195.2 194.3 PC3 13.999 9.8623 7.162 5.358 4.126 3.267 PC4 197.24 196.41 195.5 194.4 193.3 192.1
Table 7: Real ARL comparison for I-shift (µI = 1 + 0.2 × α).
β
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.019) (1.062) (1.126) (1.207) (1.302) (1.408) PC1 200.0 177.55 131.31 89.198 59.356 39.824 27.235 PC2 200.0 168.89 112.77 69.516 42.801 27.032 17.646 PC3 200.0 199.19 196.78 192.89 187.67 181.32 174.07 PC4 200.0 199.45 197.83 195.17 191.56 187.09 181.87
chart 1.75 2 2.25 2.5 2.75 3
(1.524) (1.647) (1.775) (1.908) (2.044) (2.183) PC1 19.056 13.655 10.02 7.5334 5.7958 4.5617 PC2 11.926 8.3429 6.04 4.5196 3.4921 2.7826 PC3 166.15 157.78 149.2 140.47 131.87 123.46 PC4 176.04 169.71 163.0 156.08 149.01 141.88
Table 8: Real ARL comparison for M-shift (µ = 15 + 1 × β).
γ
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.045) (1.123) (1.233) (1.377) (1.552) (1.761) PC1 200.0 175.24 123.25 77.172 46.405 27.762 16.779 PC2 200.0 170.23 109.89 61.338 32.564 17.173 9.2121 PC3 200.0 195.66 188.26 184.73 188.84 198.28 192.84 PC4 200.0 196.55 191.12 189.61 194.36 199.99 184.32
chart 1.75 2 2.25 2.5 2.75 3
(2.004) (2.284) (2.607) (2.979) (3.413) (3.923) PC1 10.333 6.5314 4.27 2.9119 2.0901 1.5928 PC2 5.1331 3.0408 1.964 1.4141 1.1483 1.0383 PC3 138.29 63.968 21.63 6.4329 2.1517 1.1355 PC4 126.59 61.296 23.66 8.422 3.2224 1.578
Table 9: Real ARL comparison for N-shift (µN = −1.5 + 0.3 × γ).
α
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.002) (1.007) (1.015) (1.027) (1.042) (1.060) Combined 200.49 189.43 160.22 124.07 89.755 62.165 42.299 0.2053 0.1911 0.1427 0.0988 0.0622 0.0350 0.0191
chart 1.75 2 2.25 2.5 2.75 3
(1.081) (1.105) (1.131) (1.160) (1.190) (1.223) Combined 28.733 19.734 13.784 9.8526 7.2125 5.4224
0.0106 0.0061 0.0036 0.0021 0.0013 0.0008 Table 10: Combined chart ARL for I-shift (µI = 1 + 0.2 × α).
β
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.019) (1.062) (1.126) (1.207) (1.302) (1.408) Combined 200.49 181.31 139.05 95.667 62.436 40.059 25.88
0.2053 0.1695 0.1162 0.0647 0.0353 0.0017 0.0091
chart 1.75 2 2.25 2.5 2.75 3
(1.524) (1.647) (1.775) (1.908) (2.044) (2.183) Combined 17.02 11.46 7.9553 5.689 4.2035 3.2099
0.0048 0.0027 0.0015 0.0009 0.0005 0.0003 Table 11: Combined chart ARL for M-shift (µM = 15 + 1 × β).
γ
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.045) (1.123) (1.233) (1.377) (1.552) (1.761) Combined 200.49 179.56 131.96 83.949 48.194 25.846 13.451
0.2053 0.1616 0.1052 0.0530 0.0237 0.0089 0.0033
chart 1.75 2 2.25 2.5 2.75 3
(2.004) (2.284) (2.607) (2.979) (3.413) (3.923) Combined 7.026 3.819 2.2347 1.4543 1.1021 1.0055 0.0012 0.0005 0.0002 0.00007 0.00002 0.00001 Table 12: Combined chart ARL for N-shift (µN = −1.5 + 0.3 × γ).
α
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.002) (1.007) (1.015) (1.027) (1.042) (1.060) Combined 200.0 188.98 160.3 124.03 89.657 62.159 42.295
chart 1.75 2 2.25 2.5 2.75 3
(1.081) (1.105) (1.131) (1.160) (1.190) (1.223) Combined 28.738 19.729 13.786 9.849 7.213 5.421
Table 13: Real combined chart ARL for I-shift (µI = 1 + 0.2 × α).
β
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.019) (1.062) (1.126) (1.207) (1.302) (1.408) Combined 200.0 181.28 138.9 95.578 62.398 40.086 25.892
chart 1.75 2 2.25 2.5 2.75 3
(1.524) (1.647) (1.775) (1.908) (2.044) (2.183) Combined 17.022 11.469 7.953 5.689 4.203 3.21
Table 14: Real combined chart ARL for M-shift (µM = 15 + 1 × β).
γ
chart 0 0.25 0.5 0.75 1 1.25 1.5
(1) (1.045) (1.123) (1.233) (1.377) (1.552) (1.761) Combined 200.0 179.26 131.93 83.893 48.162 25.855 13.446
chart 1.75 2 2.25 2.5 2.75 3
(2.004) (2.284) (2.607) (2.979) (3.413) (3.923) Combined 7.025 3.8194 2.235 1.454 1.102 1.006
Table 15: Real combined chart ARL for N-shift (µN = −1.5 + 0.3 × γ).
α\β 1.5 1.75 2 2.25 2.5 1.5 PC1 12.9351 11.5453 10.3478 9.2981 8.3825
0.0031 0.0027 0.0022 0.0019 0.0015 PC2 19.1699 19.4443 19.6996 19.9855 20.2791
0.0058 0.0058 0.0061 0.0063 0.0065 PC3 30.4043 20.1576 13.7867 9.7243 7.0723 0.0111 0.0063 0.0035 0.0020 0.0013 PC4 192.091 192.6035 193.8697 195.1105 195.8213
0.1893 0.1842 0.1964 0.1832 0.1872 1.75 PC1 9.5331 8.587 7.7562 7.0278 6.3915 0.0020 0.0017 0.0014 0.0013 0.0010 PC2 12.8624 13.0283 13.1982 13.3635 13.5408
0.0031 0.0032 0.0031 0.0033 0.0033 PC3 32.538 21.4606 14.6218 10.259 7.4253 0.0125 0.0066 0.0039 0.0023 0.0013 PC4 186.8353 188.369 189.5544 191.5863 192.4798
0.1808 0.1826 0.1786 0.1837 0.1966 2 PC1 7.1927 6.5333 5.9532 5.4399 4.9876 0.0012 0.0011 0.0010 0.0008 0.0007 PC2 8.938 9.0445 9.1467 9.2587 9.3672 0.0018 0.0018 0.0018 0.0019 0.0018 PC3 34.8378 22.924 15.5161 10.8357 7.802
0.0146 0.0076 0.0042 0.0024 0.0015 PC4 181.7749 183.6495 185.2074 186.4019 188.2948
0.1767 0.1735 0.1716 0.1767 0.1765 Table 16: ARL comparison for simultaneous I(row) and M(column) shifts.
(µI = 1 + 0.2 × α , µM = 15 + 1 × β)
α\γ 0.25 0.5 0.75 1 1.25 1.5 PC1 14.8805 13.2395 11.8106 10.5787 9.4991
0.0038 0.0033 0.0029 0.0023 0.0020 PC2 9.108 9.0018 8.8951 8.7957 8.6955 0.0019 0.0018 0.0018 0.0018 0.0017 PC3 146.9559 96.3573 60.8213 38.7273 25.3012
0.1283 0.0654 0.0335 0.0165 0.0089 PC4 186.3605 187.5394 189.4829 190.5522 192.2426
0.1789 0.1850 0.1811 0.1892 0.1908 1.75 PC1 9.2862 8.3691 7.5679 6.8644 6.2446 0.0018 0.0016 0.0014 0.0012 0.0010
PC2 5.0842 5.036 4.9869 4.941 4.894
0.0009 0.0007 0.0007 0.0007 0.0007 PC3 89.8152 56.6432 36.1729 23.7163 16.0212
0.0630 0.0319 0.0148 0.0081 0.0044 PC4 128.9885 131.0008 133.2422 135.5495 138.0885
0.1019 0.1035 0.1079 0.1089 0.1143 2 PC1 5.9526 5.4386 4.9852 4.5818 4.2257 0.0009 0.0008 0.0007 0.0006 0.0005 PC2 3.0185 2.9961 2.9734 2.9521 2.9305
0.1019 0.1035 0.1079 0.1089 0.1143 2 PC1 5.9526 5.4386 4.9852 4.5818 4.2257 0.0009 0.0008 0.0007 0.0006 0.0005 PC2 3.0185 2.9961 2.9734 2.9521 2.9305