3.3 Simulation Studies
3.3.3 Phase II Application
We now investigate the performance of the proposed CS and CE charts in Phase II applications. For comparison, the multivariate EWMA (MEWMA) chart (Lowry et al., 1992) applying to the discretized smoothed profiles is considered. Sup-pose that each of the incoming profiles yi = (yi1, . . . , yip)′ follows Np(µ0, Σ0) and (µ0, Σ0) are assumed known. Then the charting statistic of the MEWMA control chart is defined as
TM,i2 = 2− λ
λ Zi′Σ−10 Zi,
Table 3.7: The type-I and type-II error rates and their standard errors (in parentheses) of OC Model (c) for α = 0.05 and δ1 = 0.
pI pII
δ2 T2 CS(2) CS(3) CS(4) CS(5) T2 CS(2) CS(3) CS(4) CS(5)
0.875 0.0721 0.0694 0.0670 0.0665 0.0662 0.8505 0.8455 0.8609 0.8189 0.8311 (.0005) (.0005) (.0005) (.0005) (.0004) (.0017) (.0017) (.0016) (.0019) (.0018) 1.750 0.0720 0.0687 0.0666 0.0671 0.0663 0.7788 0.7695 0.7915 0.7171 0.7348 (.0005) (.0005) (.0004) (.0005) (.0004) (.0021) (.0020) (.0019) (.0022) (.0021) 2.625 0.0714 0.0680 0.0660 0.0664 0.0661 0.7272 0.7148 0.7412 0.6539 0.6740 (.0005) (.0005) (.0005) (.0004) (.0004) (.0023) (.0022) (.0022) (.0024) (.0024) 3.500 0.0722 0.0689 0.0659 0.0671 0.0664 0.6746 0.6633 0.6906 0.5905 0.6126 (.0005) (.0005) (.0004) (.0005) (.0005) (.0023) (.0023) (.0022) (.0024) (.0024) 4.375 0.0720 0.0686 0.0666 0.0680 0.0673 0.6260 0.6188 0.6443 0.5445 0.5654 (.0005) (.0005) (.0004) (.0004) (.0005) (.0025) (.0024) (.0024) (.0024) (.0024) 5.250 0.0723 0.0681 0.0663 0.0678 0.0674 0.5900 0.5826 0.6110 0.5102 0.5287 (.0005) (.0005) (.0004) (.0005) (.0004) (.0026) (.0024) (.0025) (.0025) (.0025) 6.125 0.0718 0.0691 0.0667 0.0687 0.0675 0.5614 0.5529 0.5828 0.4772 0.4957 (.0005) (.0005) (.0004) (.0005) (.0004) (.0026) (.0025) (.0025) (.0025) (.0026) 7.000 0.0724 0.0692 0.0666 0.0685 0.0675 0.5333 0.5267 0.5530 0.4529 0.4722 (.0005) (.0005) (.0005) (.0005) (.0005) (.0026) (.0025) (.0025) (.0024) (.0024)
where 0 < λ ≤ 1 is the weighting parameter and Zi is a vector operating in a recursive form by setting Z0 = µ0, and
Zi = λyi+ (1− λ)Zi−1 for i = 1, 2, . . . .
This chart is triggered for large values of TM2. Note that we drop the term 1− (1− λ)2i from the regular EWMA form of the charting statistic for simplicity. We remark that, for a fair comparison, we apply the MEWMA chart to the smoothed profile data.
Moreover, a Phase II profile monitoring scheme proposed by Qiu et al. (2010) is also included in our comparative study. The authors considered the nonpara-metric mixed-effect (NME) model to fit the IC profiles as follows. Considering the
same model as (3.1), the authors used the following local weighted negative log for any point s ∈ [0, 1] and present time t, where λ is a weighting parameter; and ν2(x) = γ(x, x) + σ2 is the variance function of the response profile at x. The func-tion (3.14) combines the local linear kernel smoothing procedure with the EWMA scheme in time through the term (1− λ)t−i. Moreover, the heteroscedasticity of the observations is also considered through ν2(xij).
In order to obtain the local linear kernel estimator of g(s), the authors proposed minimizing (3.14) with respect to a and b. Then the solution can be expressed as
ˆ
Assume g0 is the known population mean function based on the previous knowl-edge and let ξij = yij− g0(xij) for each i and j, and ˆξt,h,λ be the estimator defined as (3.15) after replacing yij by ξij. Then, the charting statistic used for SPC is
Tt,h,λ = ct,λ
∫ [ ˆξt,h,λ(s)]2
ν2(s) Γ1(s)ds, (3.16)
where and Γ1 is some prespecified probability density function.
In practical use, the discretized version of the testing statistical quantity Tt,h,λ ≈ ct,λ
Γ1. An OC signal is triggered when the value of Tt,h,λ exceeds the control limit.
The control limit is searched for by the resampling algorithm presented in the paper. This chart is referred to as the mixed-effects nonparametric profile control (MENPC) chart hereafter.
Without imposing any distribution assumptions on the model, the methodology of MENPC scheme is distribution-free. We include the MENPC chart in our comparative study because it incorporates the within-profile correlation as what we study in this dissertation, a situation seldom considered in the literature. Note that the estimation obtained from the equation (3.15) incorporates both smoothing and EWMA schemes, thus the MENPC chart is directly applied to the raw profile data in the following simulation studies.
Consider again the example that the IC profiles are from Np(µ, Σ), in which µ and Σ are given in (3.12) and (3.13), respectively. Let S be the local linear smoother, in which the bandwidth h = 0.357 is chosen by the GCV method.
Then the variance-covariance matrix of the smoothed IC profile is SΣS′. Denote Σ0 = SΣS′. Apply the eigen-decomposition to Σ0 to obtain the corresponding eigenvalues and eigenvectors, (λ1, ν1), . . . , (λp, νp). It is found that the first three eigenvectors explain 97.32% of the total variation, so we choose K = 3 in the CS
1.0 1.5 2.0 2.5 3.0 3.5
and CE control charts. To visualize the effects of the PCs, Ramsay and Silverman (2005) proposed a technique that plots µ± cνr, where c is a suitable multiple.
Figure 3.3 presents the plots of the first three PCs. From the plots, we can observe that 75.2% of the total variation are in the vertical level shifts among profiles (excluding the tail area), which is captured by the first PC; the second PC (19.38%) explains mainly the various declining rates among the profiles, and the third PC (2.74%) captures the variation in the area after x = 2.24.
The performance of control charts in Phase II applications are usually evaluated through the ARL. For a good control chart, not only the IC ARL, denoted as ARL0, should be controlled at a nominal value, one would like to have the OC ARL as small as possible meaning that an OC signal needs to be flagged as soon as possible when the process is out of control.
In this simulation study, the ARL0 is set at 370. Then the control limits of the CS chart are chosen as χ23,α′ and χ216,α′ for T02 and T12 statistics, respectively, where α′ = 1−√
1− 1/370. In order to compare the performance of the charts involving EWMA, the weight parameter λ is set at 0.2. For the control limits of the CE chart, the Markovian approach (see Section 3.2.2) is applied to calculate the IC ARL and used to tune the control limits such that the IC ARL of the CE chart achieves 370. To get this, the parameters γ0 and γ1 in equations (3.9) and (3.10) are then chosen to be 3.783 and 3.25, respectively, for which the IC ARLs of T02
2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 3.4: ARL comparison among the CE, CS, MEWMA, and MENPC charts under Model (a) for given δ2 (top row) and δ1 (bottom row).
charts are chosen to be 29.915 and 85.678, respectively, by simulations. All the ARL values reported in this section are averages of 10,000 replications. Moreover, as suggested by Hawkins and Olwell (1998), we focus on the steady-state OC ARL of a chart, and assume that shifts occur right after time point t = 30. When computing the OC ARL, any signals occur before 30 will be ignored.
Comparing the performances between the CS and CE charts first. The CE chart outperforms the CS chart in most of the OC conditions except some extreme OC conditions (e.g. given δ1 = 3 in Models (b) and (c)). It matches our intu-ition because the Shewhart-type control charts are less efficient than their EWMA versions for small to moderate shifts of process parameters but would be more powerful for large shifts.
Consider OC Model (a), the plots of ARLs for given δ1 or δ2 (size of the mean or variance-covariance shift) are shown in Figure 3.4. We can observe that the MEWMA chart performs quite well in most of cases. The CE chart is efficient in
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 3.5: ARL comparison among the CE, CS, MEWMA, and MENPC charts under Model (b) for given δ2 (top row) and δ1 (bottom row).
detecting shifts in the variance-covariance matrix (δ2 ̸= 0) and performs the best when the shift is only on the variance-covariance matrix (δ1 = 0, δ2 ̸= 0). The CE chart is not as efficient as the MEWMA chart for moderate to large shifts on mean when δ2 is given (see upper panel in Figure 3.4) since the shift is in the primary space and only the T02 part of the charting statistics works in this case. However, when the shift size gets large, especially in the variance-covariance matrix, the CE chart outperforms the MEWMA chart. Although the MENPC chart is a distribution-free method, it is quite comparative with the others for most cases, especially in the case that only the mean shifts but not the variance-covariance matrix (i.e., δ1 ̸= 0, δ2 = 0). However, the charting statistic of MENPC considers only the variance of profiles at the design points instead of the whole covariance structure, so it is not such sensitive even for large shifts in the variance-covariance matrix.
Figure 3.5 shows the values of ARL under OC Model (b). The performance
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 3.6: ARL comparison among the CE, CS, MEWMA, and MENPC charts under Model (c) for given δ2 (top row) and δ1 (bottom row).
of the MEWMA chart outperforms the others in many cases. However, the CE chart is better than the MEWMA chart if the shift occurs only on the variance-covariance matrix. When the mean and variance-variance-covariance matrix shift at the same time, the performances of the MEWMA and CE charts are comparative.
However, under extreme cases when both the mean and variance-covariance matrix shift severely, the CE chart is more efficient in detecting OC observations. The MENPC chart is not competitive in this case, especially in detecting shifts in the variance-covariance matrix. This may be due to the facts that the MENPC chart ignores the covariance of profiles in the charting statistic and the shifts in the complementary space mainly effect the covariance structure but the variance of profiles. The CS chart is the best among the four methods under the extreme OC condition (given δ1 = 3).
Model (c) is the case when the shift is in both of the primary and complemen-tary spaces, and Figure 3.6 presents the ARL performances of the four methods
1.0 1.5 2.0 2.5 3.0 3.5
Figure 3.7: (a) Plots of the IC and OC profiles from Model (a) before (left panel) and after (right panel) smoothing.
under comparison. The results are similar to that of Model (a), but the MENPC chart is more comparable with the others. It is noted from the plot given δ1 = 3 that the CS chart becomes the most powerful one if a severe change occurs in the process; the performances of the MENPC and MEWMA charts are similar but the worst; and the CE chart performs in between.
Next, we demonstrate how the OC conditions might be diagnosed. As an example, first generate 100 IC profiles from a multivariate normal distribution with parameters (µ, Σ) as in (3.12) and (3.13) and then 100 OC profiles from Model (a) with δ1 = 5 and δ2 = 0. That is, the OC condition is the scale change in the mean function. The scale change has effects on all three PCs, but the extent could be different. Figure 3.7 shows the plots of these profiles before and after smoothing. We can clearly observe that the differences between the IC and OC profiles are mostly at the first 2/3 of the profiles; in particular, the peak of the OC profiles is much higher than the IC ones.
To diagnose the OC conditions, the PC scores are explored and presented in Figure 3.8(a). The magnitude of the first two PC scores enlarges dramatically after the 100th profile, but not the third one. Therefore, it is difficult to reveal the OC
0 50 100 150 200
010203040
PC Scores
Index
squared standardized score PC 1
PC 2 PC 3
(a)
0 50 100 150 200
0103050
Rotated PC Scores
Index
squared standardized score
PC 1 PC 2 PC 3
(b)
Figure 3.8: (a) The scores of the first three PCs and (b) the scores of the first three rotated PCs by VARIMAX rotation.
condition from the pattern of the regular PCs.
Denote ν1∗, . . . , νp∗ the rotated PCs (RPCs) found through the VARIMAX ro-tation of the IC profiles. Figure 3.9 shows the effect-visualizing plots of the first three RPCs after rotating. It is clearly seen that each of the three RPCs is in charge of the variation in about 1/3 of the region — the first part by the second RPC, the second part by the first RPC, and the third part by the third RPC.
Figure 3.8(b) shows the RPC scores of the same 100 IC profiles and the 100 OC ones. Now, the patterns for the RPCs differentiate the effects of the three PCs
1.0 1.5 2.0 2.5 3.0 3.5
on the scale change. The second RPC scores enlarge drastically after the 100th profile, indicating that the primary change in the OC profiles is on the first part of the profile, the peak area. The VARIMAX rotation provides a great aid for practitioners to search for the assignable causes when an OC signal flags. In this case, the RPCs provide more information than the original PCs in seeking the assignable causes for the OC profiles. However, for cases, the original PCs may be more helpful than the RPCs sometimes. Thus, both the original PC and the RPC scores can be considered as diagnostic aids for OC observations.
If there are OC conditions that can not be explained by the first few effective PCs or rotated PCs (e.g., Model (b)), no unusual patterns will be observed from the first few PC scores. In those cases, it indicates that the process may already be seriously changed and the profiles are no longer suitably fitted by the effective PCs.