The VDP data set described in Section 1 contains n = 24 profiles, each was measured at p = 314 set points. Figure 4.4(a) is the plot of the VDP data.
First de-noise these profiles by smoothing splines using statistical package R.
See Figure 4.4(b) for the plot of the smoothed profiles. Apply PCA to the sample covariance matrix of the smoothed profiles. And the first four principal components account for 85.26%, 10.83%, 1.90%, 0.84% of the total variation in the profiles, respectively. We select K = 3 principal components for Phase I process monitoring because the total variation explained by the first three PCs is already as high as 97.99%. Figure 4.5(a) is the plot of the first three eigenvectors. Figures 4.5(b)-4.5(d) show the modes of variation they capture by
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
35404550556065
(a) 24 original VDP profiles
depth
density
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
35404550556065
(b) 24 smoothed VDP profiles
depth
density
Figure 4.4: (a) 24 VDP profiles (b) 24 smoothed VDP profiles.
0.0 0.1 0.2 0.3 0.4 0.5 0.6
−0.50.00.5
(a) First three eigenvecors
depth
plotting the mean vector µµµ and µµµ ± 10 vvvr, r = 1, 2, 3. The figures show that: (i) PC1 represents the variation in the ground level of the VDP profiles, especially the bottom part of the “bathtub”; (ii) PC2 can catch some variation in the roundness of the bottom part of the bathtub; and (iii) PC3 may be able to describe the variation in the roundness on the two ends and the central part of the bathtub bottom. All these interpretations can also be seen from the three eigenvectors shown in Figure 4.5(a). As to Phase I monitoring,the T2 control chart (not shown) indicates no out-of-control profiles in the VDP 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 ∼ M V 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 + M V N (µ, Σ) ,
where k=0, 0.25, . . . , 2 and vi is the i-th eigenvector of Σ, i=1, 2. We simulated 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.
ARL results for shifts in principal component 1 and 2 are listed separately
in Table(4.1) and Table(4.2). As we can see that shifts in principal component 1 are solely captured by the first principal component score. Likewise, shifts in principal component 2 are captured by the second principal component score.
The other three principal component scores make no contributions to the power of detection.
4.4 Concluding Remarks
In this study, we propose and discuss monitoring schemes for nonlinear pro-files. We use the principal components analysis to analyze the covariance matrix of the profiles and then utilizing the principal component scores that capture the main features of the profile data for process monitoring.
In addition to the individual PC score charts, we study two charts that uti-lize the overall information contained in the K effective principal components, namely, the combined chart and T2 chart. The T2 chart performs somewhat better than the combined chart in terms of the average run length, but not too far off. However, by providing charts for all of the effective components, the combined chart gives more clues for finding assignable causes than the T2 chart.
When the shift corresponds to a mode of variation that a particular princi-pal component represents, then it would be ideal to use the individual PC-score chart for process monitoring because this particular individual chart will have
the best power among the charts under study. Unfortunately, this ideal situa-tion is rare in practice. Moreover, by just monitoring one individual PC chart, one is running a risk of not being able to detect other types of process changes.
One may argue that we should monitor the process with all K individual charts simultaneously in order to catch all potential changes. But with the same α for each of the individual charts, the overall false-alarm rate is greatly increased to 1 − (1 − α)K, which is about K times of the original false-alarm rate. Thus, for being more practical and conservative, we recommend using the T2 chart or the combined chart scheme, because they still have comparably good power to monitor these particular types of shifts and have a lot better power than the individual chart for general types of shifts.
It is noted that the degree of smoothness used in the data smoothing step has a great impact on the result of the subsequent PCA step. The high to-tal explanation power of the first few principal components demonstrated in the paper is in fact caused by the high degree of smoothness in the smoothed curves. This argument is supported by our finding that if B-spline regression is used for smoothing, the number of B-spline bases used is exactly the num-ber of the principal components with nonzero eigenvalues. So if the underlying profiles (i.e., with no noises) are fairly smooth as what we have in the hypothet-ical aspartame example (in which a data profile is a 3-parameter exponential function plus noises), then the data dimension can be well reduced by applying PCA to the smoothed curves. The situation in the VDP data is similar. On
the other hand, if the underlying profiles are not very smooth and data pro-files are not too much over-smoothed, then it might take quite a few principal components to explain good enough proportion of variation. We remark that, regardless of which K, the number of effective principal components, is chosen, the false-alarm rate for each of our schemes stays at α. However, the diagnosis of out-of-control conditions would likely become more complicated when K is large.
The degree of smoothness may affect the effectiveness of the Phase II process monitoring as well. If the noise levels are about the same for both Phase I and Phase II profile data, we suggest applying the same degree of smoothness to them. In this way, the results of Phase II monitoring are somehow not that sensitive to the extent of smoothing. However, when profiles are over-smoothed to the extent that some local features are lost, then the process changes associated with these vanished local features cannot be detected. On the other hand, when profiles are under-smoothed to an extent, some spurious features may appear in the fitted curves. Unfortunately, these spurious features may not appear at the same place and may not have the same form across profiles. Then the estimated in-control model obtained from Phase I data may not suitable for effective Phase II monitoring. We remark that even for the case that the in-control process is known or appropriately characterized, the spurious features in the “smoothed” Phase II profiles caused by under-smoothing will cause more false alarms to signal.
In practice, when one employs the T2 chart or the combined chart scheme and detects significant shifts, it is desirable to find the sources responsible for the shifts. For this, we suggest to rank the standardized PC-scores and investigate the corresponding principal components in order, starting from the largest PC score. With the help of the plots like Figures 4.2(a)-4.2(d) and 4.5(a)-4.5(d) and engineers’ expertise, the characteristics of the principal components sometimes can reveal potential root causes of the shifts.
The monitoring of process or product profiles has become a popular and promising area of research in statistical process control in recent years. At the same time, functional data analysis (FDA) is also gaining lots of attentions and applications. We believe many techniques developed for FDA may be extended to developing new profile monitoring techniques in SPC.
k
chart 0 0.25 0.5 0.75 1
PC1 201.141 157.7244 91.6269 50.1074 28.1935 0.6537 0.4263 0.1869 0.0773 0.0339 PC2 202.5093 201.1011 202.7793 201.3586 202.3857
0.6897 0.6687 0.6295 0.6704 0.6432 PC3 202.2185 202.1651 201.4955 203.475 202.0911
0.6937 0.6648 0.6495 0.6207 0.6461 PC4 202.4143 200.7331 201.7661 202.0241 202.1318
0.6594 0.6526 0.6303 0.6551 0.6248
chart 1.25 1.5 1.75 2
PC1 16.7264 10.4605 6.8874 4.7689 0.0153 0.0071 0.0037 0.0020 PC2 202.5093 201.1011 202.7793 201.3586
0.6897 0.6687 0.6295 0.6704 PC3 202.2185 202.1651 201.4955 203.475
0.6937 0.6648 0.6495 0.6207 PC4 202.4143 200.7331 201.7661 202.0241
0.6594 0.6526 0.6303 0.6551
Table 4.1: ARL comparison for shifts in principal component 1.
k
chart 0 0.25 0.5 0.75 1
PC1 203.1434 202.2938 202.2304 201.9739 202.0307 0.6599 0.6496 0.6643 0.6826 0.6423 PC2 202.3857 157.2349 91.368 50.138 28.2533
0.6432 0.4475 0.1938 0.0783 0.0318 PC3 202.0911 201.3132 202.4239 202.4192 201.6323
0.6461 0.6695 0.6455 0.6580 0.6622 PC4 202.1318 201.4323 201.5589 203.1134 201.6549
0.6248 0.6797 0.6407 0.6532 0.6349
chart 1.25 1.5 1.75 2
PC1 200.6637 202.6417 202.2779 201.5532 0.6327 0.6492 0.6344 0.6624 PC2 16.75 10.455 6.8888 4.771
0.0141 0.0073 0.0038 0.0020 PC3 201.7072 202.043 202.6156 202.3369
0.6355 0.6418 0.6803 0.6310 PC4 203.0749 202.5458 202.4285 201.5443
0.6780 0.6515 0.6505 0.6427
Table 4.2: ARL comparison for shifts in principal component 2.
Chapter 5
Conclusion
5.1 Symmetric Quantile Coverage Interval
Symmetric quantile coverage interval performs better than empirical quantile coverage interval in terms of the followings:
a. symmetric quantile coverage interval can cover more of the high density part of distribution function, when the underlying distribution is asymmetric.
b. symmetric quantile coverage interval is more robust against outliers than empirical quantile coverage interval.
c. Even when the underlying distribution is symmetric, symmetric quantile coverage interval is with smaller variance when the coverage percentage is with large 1 − α, which is the common practices of most applications of coverage interval.
5.2 Multivariate Control Chart by Symmetric Quan-tile
An application of coverage interval is a control chart, with the upper and lower control limits constructed by the two ends of a coverage interval. As for constructing a control chart with interests on the multiple characteristics of one distribution, multivariate control chart by symmetric quantiles is proposed and its asymptotic theorem is derived.
When the underlying distribution is with heavy tails, the proposed symmet-ric quantile control chart performs better than the empisymmet-rical quantile control chart, proposed by Grimshaw and Alt(1997).
5.3 Monitoring Nonlinear Profiles by Nonparametric Regression
PCA score based T2 monitoring schemes is proposed to deal with nonlin-ear profile data in phase-1 monitoring. For phase-II monitoring, three PCA score based monitoring schemes(Individual score charts, T2chart, the combined chart) are investigated and compared. T2 chart is recommended in practical uses with the plotting of
µµµ ± L vvvr .
Data smoothing is recommended to be deone before PCA analysis, for it has great impacts on the effectiveness of the construction of phase-I control schemes and the effectiveness of phase-II monitoring.
5.4 Future Research
PAT(Part Average Testing) is the post-process defined in AEC-Q001, where Automotive Eletronics Council(AEC) was established for the purpose of es-tablishing common part-qualification and quality system standards. PAT is primarily for detecting outlier parts which tend to be higher contributors to long term quality and reliability problems. Current static PAT limits= (Ro-bust Mean) ± 6*(Ro(Ro-bust Sigma) where the Ro(Ro-bust Mean is the median, and the Robust Sigma qual to (Q3 − Q1)/1.35.
PAT has recently been adopted by a number of semiconductor companies, with the typical applications on wafer sort and final test data. Most semicon-ductor wafer sort and final test parameters are not normal and with heavy tails or asymmetric. Application of symmetric qunatile coverage interval as outlier detection tool can be investigated and compared with the one proposed by AEC standard.
Tolerance analysis is adopted by semiconductor companies to assist design-ing specs. The output performance transfer function and component tolerance intervals are the inputs into tolerance analysis and output performance
toler-ance interval is the desired result for important references in spec definition.
However, when major output performance is a I-V curve under specific condi-tions, the solution to the tolerance region of the I-V curve is a topic which needs to be explored. Curve/Profile tolerance region will contribute an important I-V curve performance measure in semiconductor IC performance evaluation.
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