In this section, we demonstrate our proposed monitoring schemes with a real data set. A manufacturing process of aluminum electrolytic capacitors (AEC’s), which was first described in Qiu et al. (2010), is a process that transforms raw materials, such as anode and cathode aluminum foil, guiding pin, electrolyte sheet, plastic
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AEC Data
Temperature
Dissipation Factor
(a)
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Smoothed AEC Data
Temperature
Dissipation Factor
(b)
Figure 3.10: The original and the smoothed profiles of the three of the AEC data.
in low-leakage circuits and are well adapted to a wide range of environmental temperatures. The whole manufacturing process consists of multiple operations, e.g., clenching, rolling, soaking, etc., and a careful quality monitoring is required before packing. The dissipation factor (DF), which is measured automatically by an electronic device, is regarded as an important characteristic in monitoring the quality of AEC’s. However, the DF is affected significantly by the temperature of the environment and hence the profile of the DF as a function of temperature is a characteristic of interest regarding the quality of the AEC. To monitor the adaptability, the sampled AEC’s are put in a container in which the temperature can be controlled. The temperature in the container is gradually increased from
−26◦F to 78◦F and recorded by a sensor. In this process, the measurements of DF and the corresponding temperature are taken at 53 equally-spaced time points for a total of 144 AEC profiles. Note that the actual temperature measured in a container at each time point is fluctuant but around its nominal level at each observation time. Therefore, the temperature records are different from profile to profile although the differences are small.
To cope with the non-consistent temperature recording problem and filter out
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AEC Data (truncated)
Temperature
Dissipation Factor
(a)
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Smoothed AEC Data (truncated)
Temperature
Dissipation Factor
(b)
Figure 3.11: The first 15 records of the three profiles and their corresponding smoothing estimates.
the noise as well, the local linear smoothing technique is applied to the data before analysis. Figure 3.10 depicts three AEC profiles and their smoothing estimates (using bandwidth h = 6.54 from GCV). According to the Q-Q plots of data at each set point, neither the original nor the smoothed 144 AEC profiles with 53 set points follow the multivariate normal distribution. To overcome this problem, we choose only a segment of the AEC profiles to alleviate the effect of non-normality.
According to the Q-Q plots for each of the first 15 set points (not shown here) and the p-value (about 0.085) of the multivariate normality test proposed by Mardia (1970), the evidences are not strong enough to reject the normality. Therefore, the 144 profiles with 15 set points are used to demonstrate our proposed methodology.
Three of the original and smoothed profiles after truncation are shown in Figure 3.11.
Qiu et al. (2010) regarded the first 96 profiles as the training data in Phase I analysis and the rest as testing data in Phase II analysis. To use the CS chart in Phase I analysis, we apply PCA to the sample covariance matrix of the first 96 smoothed profiles, which leads to choose K = 3 because the first three PCs
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explain 91.96% of the total variation. As a result, none of the profiles is regarded as an OC profile by our proposed CS chart for Phase I application. Thus all the 96 profiles are used to estimate the mean and covariance matrix for Phase II.
Let µ0 and Σ0 be the sample mean vector and sample variance-covariance matrix of the 96 IC profiles (after smoothing). The effect-visualizing plots are also used to interpret the effect of the three PCs and shown in Figure 3.12. From these plots, we observe that the mean curve is mound-shaped; the first two PCs explain the variation from the temperature less than −5◦ and greater than −18◦, respectively; and the two crosses at temperatures −18◦ and −4◦ on the plot for the third PC indicate that the third PC explains the variation of the profiles in the rate of declining from the top.
Next, the rest of the 48 AEC profiles are used to demonstrate our proposed Phase II monitoring scheme. By setting λ = 0.2, the values of the T2-type and EWMA-type statistics corresponding to T02 and T12 are shown in Figure 3.13. Set the ARL0 at 200. Then the control limits in the CS chart is χ23,α′ and χ212,α′ for T02 and T12 charts, respectively, where α′ = 1−√
1− 0.005. The parameters γ0
and γ1 in equations (3.9) and (3.10) are chosen to be 3.38 and 3.004, respectively, and the control limits of CE charts are then L0 = 5.76 and L1 = 16.91. From Figure 3.13(a), based on the T02 statistic, the Shewhart chart detects only the 139th profile, whereas the EWMA chart regards the profiles from 114th to 125th and 139th to 144th as OC cases. For the T12-based charts, Figure 3.13(b) shows
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T02 statistic
index
values of statistics
CE CS
(a)
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T12 statistic
index
values of statistics
CE CS
(b)
Figure 3.13: The values of the charting statistics based on T02 (panel (a)) and T12 (panel (b)) for the AEC data.
that the results of the two charts are fairly consistent — the 105th, 115th, 131st, and 137th profiles are detected by the Shewhart chart and a little bit more by the EWMA chart. The OC cases considered by the CS (CE) chart is the union of the OC cases claimed by the two Shewhart (EWMA) charts based on the T02 and T12 statistics.
Figure 3.14 shows the effect-visualizing plots with respect to the first three RPCs. From these plots, the three RPCs explain the variation in the front, middle, and end areas (with some overlaps) of the profiles, respectively. To search for the
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source of the process shift, one can examine the scores of the primary PCs or the RPCs for each OC case detected by the T02-based charts. The plots of the scores of the first three original PCs as well as the RPCs are shown in Figure 3.15. For each OC-flagged profile, at least one of the scores is abnormally large. For example, the 115th profile, detected by both T02 and T12 of the CE chart and only T12 of the CS chart, has a large value on the second PC score indicating the shift could be in the last 2/3 part of the profile and may be quite different from the general pattern of the IC profiles; the 139th profile, detected by both T02 of the CS and CE charts, has an unusually large value on the score of the first original PC as well as the first RPC, exhibiting a pattern quite different from others at the front part of the profile; the 140th profile has a large score value on the second RPC, indicating that the change affects mainly on the middle 2/3 of the profile and it may have an unusually high or low peak. Note that a large score value appears on the first PC at point 112, but it was not signaled by either the CS or CE chart.
A closer look shows that the T02 statistics on both CS and CE charts are very close to the control limits. In addition, the EWMA chart exhibits an increasing trend beginning at the 112nd profile. It indicates that some OC conditions may have occurred in the process at or before the 112nd but not signaled until the 114th profile. The aforementioned OC profiles are shown in Figure 3.16 and their patterns match what we observe from the PC or RPC scores.
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PC Scores
index
squared standardized score
PC 1 PC 2 PC 3
112 115 139
140
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Rotated PC Scores
index
squared standardized score RPC 1
RPC 2 RPC 3
115
140 112
139
Figure 3.15: The scores of the first three rotated PCs for the AEC data.