5.3 Comparative Studies
5.3.2 ARL Comparisons for r-Chart, Q-Chart and DDMA-Chart
Figures 10-12 show the subgroups version of ARL curves of r-chart, Q-chart and DDMA-chart for shifts in parameters I, M, and N, respectively. According to these figures, we observe the following:
• From Figures 10-12, we note that the behavior of Q-charts are somewhat peculiar.
When the process is in control, Q-charts with subgroup sizes 2, 4, and 6 have ARL0
20.87, 21.87, 23.75, respectively, all larger than 20. Moreover, for small shift sizes (α=0.25,. . . ,1.00, β=0.25, γ=0.25), the ARL1 increases when the subgroup size (q) of the specific chart increases. This seems against our intuition. Fortunately, when shift size gets larger, the chart starts behaving as what we would expect. We have no good reasons for this but the asymptotic theory under which the control limit of the Q-chart was constructed as given in Liu (1995). The asymptotic theory requires the minimum between the reference sample size (m) and the subgroup size (q) to be infinite. But in our simulation, this q is only 2, 4, or 6, which is far from infinity.
• For all three types of shifts, the DDMA-charts perform the best among the three types of charts. Among DDMA-charts, D6 > D4 > D2, as expected. Here “>”
means “performs better than”.
• Despite the peculiar behavior of the Q-charts for small shift sizes, the ARL behaviors of all charts are pretty similar for all three types of shifts.
• When the shift size is large enough, the performance order of the charts is DDMA-charts > Q-DDMA-charts > r-chart in general, with D2 sometimes intervening with Q4 and Q6.
• The performances of these control charts are very similar for M-shift and N-shift except that ARL values drop faster for N-shift.
To summarize, the control limit for the Q-chart needs to be modified to achieve an about-right false-alarm rate. The subgroup size is also an important issue when we adopt
the Q-chart or the DDMA-chart for Phase II monitoring. Here, their performances among different subgroup sizes, q = 2, 4, 6, behave as what we would expect for the DDMA-chart.
As for the Q-chart, it starts behaving as what we would expect when shift size gets larger.
Since when using subgroups with different sizes, plotting one point on the control chart may require different number of profiles, it would be fairer to define run length as the number of profiles taken when the out-of-control alarm signals.
Figures 13-18 show the individual profiles version of ARL curves of Q-charts and DDMA-charts for shifts in parameters I, M, and N, respectively. They express the ARL in terms of the expected number of individual profiles sampled rather than the number of subgroups taken to detect a shift. Here, Figures 13-15 are for Q-charts and Figures 16-18 are for DDMA-charts. According to these figures, we observe the following:
• For Q-charts, we note that control chart with smaller subgroup size has smaller ARL values for all shift sizes from 0 to 3. Their performances do not behave as what we would expect when shift sizes are small.
• For DDMA-charts, we note that charts with different subgroup sizes (q = 2, 4, 6) perform differently (better or worse) at different shift sizes. Specifically, control chart with smaller subgroup size performs better (with smaller ARL) when the shift size becoming larger and larger. This is because the moving average has an overhead of size q at the beginning of the process monitoring.
• Figure 19 compares the average number of profiles needed to detect a shift for all the charts for I-, M-, N-shift, respectively. According to their performances, we recommend the DDMA-chart when shift size is small and the r-chart when shift size is large.
6 Conclusions
In this study, we propose and study some nonparametric schemes for Phase II profile monitoring based on the notion of data depth. We utilize data smoothing techniques to filter out random noises from raw sample profiles, principal component analysis to convert smoothed profiles to score-vectors of lower dimension, and then data-depth-based control charts to monitor incoming profiles.
According to the ARL performance comparison between our monitoring scheme and those proposed in Shiau et al. (2009) with the simulated aspartame example, we observe that our proposed monitoring scheme based on r-chart is comparable. Furthermore, the monitoring scheme based on DDMA-chart performs better than the monitoring scheme based on Q-chart and the monitoring scheme based on Q-chart performs better than the one based on r-chart in general.
The most important feature of the control schemes proposed in this paper is that they are nonparametric in the sense that no distributional assumptions are required for the profiles under monitoring. This broadens the scopes of applications of the proposed schemes. And interestingly, when compared with the profile monitoring schemes proposed in Shiau et al. (2009), the new nonparametric schemes perform equally well or even slightly better with an example satisfying the distributional assumptions of the existing schemes.
For Phase I analysis, we provide a heuristic diagnostic procedure with the objective of spotting unusual profiles in the Phase I data so that we can obtain a more reliable data set to create the reference sample for the subsequent construction of data depth control charts. The procedure is not a control chart because we cannot set a control limit that can control, say, the false-alarm rate.
Clustering depth values through different clustering methods in Phase I analysis and adopting other depth functions for Phase II profile monitoring could be interesting future studies. Another potential direction is to modify the lower control limit for the Q-chart to achieve an about-right false-alarm rate.
References
[1] Johnson, R. A. and Wichern, D. W. (2007). Applied Multivariate Statistical Analysis.
6th ed. Prentice Hall, New Jersey.
[2] Kang, L. and Albin, S. L. (2000). “On-Line Monitoring When the Process Yields a Linear Profile”. Journal of Quality Technology. 32, 418-426.
[3] Kim, K., Mahmoud, M. A., and Woodall, W. H. (2003). “On The Monitoring of Linear Profiles”. Journal of Quality Technology. 35, 317-328.
[4] Liu, R. Y. (1990). “On a Notion of Data Depth Based on Random Simplices”. The Annals of Statistics. 18, 405-414.
[5] Liu, R. Y. (1995). “Control Charts for Multivariate Processes”. Journal of the Amer-ican Statistical Association. 90, 1380-1387.
[6] Liu, R. Y. and Singh, K. (1993). “A Quality Index Based on Data Depth and Mul-tivariate Rank Tests”. Journal of the American Statistical Association. 88, 252-260.
[7] Liu, R. Y., Singh, K., and Teng, J. H. (2004). “DDMA-charts: Nonparametric Multi-variate Moving Average Control Charts Based on Data Depth”. Allgemeines Statis-tisches Archiv. 88, 235-258.
[8] Mosler, K. C. (2002). Multivariate Dispersion, Central Regions and Depth: The Lift Zonoid Approach. Springer, New York.
[9] Oja, H. (1983). “Descriptive Statistics for Multivariate Distributions”. Statistics and Probability Letters. 1, 327-332.
[10] Shiau, J.-J. H., Huang, H.-L., Lin, S.-H., and Tsai, M.-Y. (2009). “Monitoring Non-linear Profiles with Random Effects by Nonparametric Regression”. Communications in Statistics-Theory and Methods. 38, 1664-1679.
[11] Shiau, J.-J. H., Lin, S.-H., and Chen, Y.-C. (2006). “Monitoring Linear Profiles Based on a Random-effect Model”. Technical Report. Institute of statistics, National Chiao Tung University.
[12] Shiau, J.-J. H., Yen, C.-L., and Feng, Y.-W. (2006). “A New Robust Method for Phase I Monitoring of Nonlinear Profiles”. Technical Report. Institute of statistics, National Chiao Tung University.
[13] Williams, J. D., Birch, J. B., Woodall, W. H., and Ferry, N. M. (2007). “Statistical Monitoring of Heteroscedastic Dose-Response Profiles from High-throughput Screen-ing”. Journal of Agricultural, Biological and Environmental Statistics. 2, 216-235.
[14] Walker, E. and Wright, S. P. (2002). “Comparing Curves Using Additive Models”.
Journal of Quality Technology. 34, 118-129.
[15] Williams, J. D., Woodall, W. H., and Birch, J. B. (2003). “Phase I Analysis of Non-linear Product and Process Quality profiles“. Technical Report No. 03-5. Department of Statistics, Virginia Polytechnic Institute and State University.
[16] Zuo, Y. and Serfling, R. (2000). “General Notions of Statistical Depth Function“.
The Annals of Statistics. 28, 461-482.
Table 1: Diagnostic results of Bioassay Data (smoothing parameter is 0.3).
diagnosis
first second(removing first)
cumulative weeks cumulative weeks
spar PC1 PC2 PC3 PC1 PC2 PC3 PC4
0.3 0.550 0.849 20,22,24 0.616 0.811 32
34,45,46
0.3 0.550 0.849 20,22,24 0.616 0.811 0.910 26
34,45,46
0.3 0.550 0.849 20,22,24 0.616 0.811 0.910 0.980 26,32 34,45,46
0.3 0.550 0.849 0.941 34 0.598 0.868 20,22,24,45
46 0.3 0.550 0.849 0.941 34 0.598 0.868 0.942 20,22,24,26
32,45,46,48 0.3 0.550 0.849 0.941 34 0.598 0.868 0.942 0.977 32
Table 2: Diagnostic results of Bioassay Data (smoothing parameter is 0.4).
diagnosis
first second(removing first)
cumulative weeks cumulative weeks
spar PC1 PC2 PC3 PC1 PC2 PC3 PC4
0.4 0.628 0.925 20,22,24 0.701 0.884 5,13,26,32
34,45,46 44
0.4 0.628 0.925 20,22,24 0.701 0.884 0.970 26
34,45,46
0.4 0.628 0.925 20,22,24 0.701 0.884 0.970 0.992 26,32 34,45,46
0.4 0.628 0.925 0.976 24 0.663 0.920 20,22,34,45
46 0.4 0.628 0.925 0.976 24 0.663 0.920 0.974 20,22,26,34
45,46,48 0.4 0.628 0.925 0.976 24 0.663 0.920 0.974 0.989 20,22,34,45
Table 3: Diagnostic results of Bioassay Data (smoothing parameter is 0.5).
diagnosis
first second(removing first)
cumulative weeks cumulative weeks
spar PC1 PC2 PC3 PC1 PC2 PC3 PC4
0.5 0.690 0.969 20,22,24 0.763 0.941 26
34,45,46
0.5 0.690 0.969 20,22,24 0.763 0.941 0.995 26,48
34,45,46
0.5 0.690 0.969 20,22,24 0.763 0.941 0.995 0.999 16,26,32,48 34,45,46
0.5 0.690 0.969 0.995 24 0.721 0.966 20,22,34,45
46 0.5 0.690 0.969 0.995 24 0.721 0.966 0.995 20,22,26,34
45,46,48 0.5 0.690 0.969 0.995 24 0.721 0.966 0.995 0.999 20,22,45
Table 4: Diagnostic results of VDP Data with different smoothing parameters.
diagnosis
first second(removing first) cumulative profiles cumulative profiles
spar PC1 PC2 PC1 PC2
0.01 0.845 0.952 6,10 0.824 0.938 2,3,16,17,20 0.1 0.845 0.953 6,10 0.825 0.938 2,3,16,17,20 0.5 0.853 0.961 6,10 0.835 0.949 2,3,16,17,20 1 0.887 0.989 6,10 0.878 0.985 2,3,16,17,20
Table 5: ARL values and their standard errors (in parentheses) of r-chart, Q-charts, and DDMA-charts for various shift sizes of I-shift.
α r-chart Q2 Q4 Q6 D2 D4 D6
0.00 20.42 20.87 21.87 23.75 20.34 20.28 20.12 (0.093) (0.116) (0.198) (0.324) (0.105) (0.137) (0.166) 0.25 20.15 20.76 21.88 23.59 20.01 19.66 19.40
(0.091) (0.119) (0.185) (0.304) (0.101) (0.128) (0.159) 0.50 19.89 20.50 21.46 22.73 19.39 18.55 17.76
(0.090) (0.116) (0.229) (0.280) (0.099) (0.120) (0.136) 0.75 19.41 19.85 20.34 21.39 18.41 16.75 15.35
(0.089) (0.110) (0.157) (0.241) (0.091) (0.108) (0.118) 1.00 18.56 18.82 19.08 19.90 17.16 14.77 12.89
(0.081) (0.100) (0.153) (0.257) (0.084) (0.094) (0.095) 1.25 17.77 17.95 17.80 18.06 15.69 12.79 10.63
(0.076) (0.091) (0.136) (0.178) (0.076) (0.076) (0.073) 1.50 16.93 16.86 16.38 16.53 14.32 10.89 8.690
(0.072) (0.083) (0.119) (0.170) (0.065) (0.060) (0.055) 1.75 15.86 15.68 15.01 14.63 12.78 9.160 7.050
(0.064) (0.076) (0.110) (0.132) (0.056) (0.049) (0.042) 2.00 14.82 14.50 13.41 12.83 11.48 7.820 5.820
(0.060) (0.069) (0.085) (0.102) (0.050) (0.039) (0.033) 2.25 13.76 13.16 11.95 11.27 10.12 6.530 4.750
(0.055) (0.057) (0.071) (0.087) (0.040) (0.029) (0.023) 2.50 12.86 12.22 10.81 9.850 9.150 5.630 4.010
(0.051) (0.052) (0.059) (0.051) (0.037) (0.025) (0.019) 2.75 11.75 11.02 9.580 8.510 8.070 4.820 3.380
(0.042) (0.045) (0.051) (0.054) (0.031) (0.021) (0.015) 3.00 10.87 10.04 8.500 7.410 7.190 4.160 2.880
(0.039) (0.040) (0.044) (0.043) (0.026) (0.016) (0.011)
Table 6: ARL values and their standard errors (in parentheses) of r-chart, Q-charts, and DDMA-charts for various shift sizes of M-shift.
β r-chart Q2 Q4 Q6 D2 D4 D6
0.00 20.42 20.87 21.87 23.75 20.34 20.28 20.12 (0.093) (0.116) (0.198) (0.324) (0.105) (0.137) (0.166) 0.25 18.75 18.98 19.37 20.18 17.10 14.54 12.73
(0.083) (0.099) (0.166) (0.266) (0.084) (0.088) (0.094) 0.50 14.72 14.51 13.49 12.95 11.50 7.780 5.800
(0.059) (0.068) (0.093) (0.108) (0.049) (0.038) (0.030) 0.75 10.92 10.07 8.510 7.420 7.190 4.170 2.910
(0.039) (0.039) (0.042) (0.043) (0.026) (0.016) (0.011) 1.00 7.840 6.790 5.240 4.250 4.620 2.500 1.770
(0.025) (0.022) (0.021) (0.018) (0.015) (0.007) (0.004) 1.25 5.680 4.640 3.330 2.610 3.140 1.720 1.300
(0.016) (0.012) (0.010) (0.008) (0.008) (0.003) (0.002) 1.50 4.180 3.260 2.250 1.770 2.260 1.330 1.100
(0.010) (0.007) (0.005) (0.004) (0.004) (0.002) (0.001) 1.75 3.190 2.430 1.680 1.360 1.750 1.140 1.030
(0.007) (0.004) (0.003) (0.002) (0.003) (0.001) (0.000) 2.00 2.510 1.890 1.350 1.150 1.430 1.050 1.010
(0.004) (0.003) (0.002) (0.001) (0.002) (0.000) (0.000) 2.25 2.050 1.560 1.180 1.060 1.250 1.020 1.000
(0.003) (0.002) (0.001) (0.000) (0.001) (0.000) (0.000) 2.50 1.720 1.350 1.080 1.020 1.130 1.000 1.000
(0.002) (0.001) (0.000) (0.000) (0.001) (0.000) (0.000) 2.75 1.490 1.210 1.030 1.000 1.070 1.000 1.000
(0.002) (0.001) (0.000) (0.000) (0.000) (0.000) (0.000) 3.00 1.330 1.120 1.010 1.000 1.030 1.000 1.000
(0.001) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)
Table 7: ARL values and their standard errors (in parentheses) of r-chart, Q-charts, and DDMA-charts for various shift sizes of N-shift.
γ r-chart Q2 Q4 Q6 D2 D4 D6
0.00 20.42 20.87 21.87 23.75 20.34 20.28 20.12 (0.093) (0.116) (0.198) (0.324) (0.105) (0.137) (0.166) 0.25 18.88 19.19 19.35 20.19 17.65 15.25 13.41
(0.086) (0.108) (0.156) (0.243) (0.091) (0.096) (0.097) 0.50 15.15 14.80 13.97 13.41 12.00 8.230 6.180
(0.063) (0.067) (0.091) (0.110) (0.053) (0.042) (0.034) 0.75 10.81 10.01 8.390 7.350 7.150 4.110 2.870
(0.039) (0.039) (0.042) (0.041) (0.026) (0.016) (0.011) 1.00 7.240 6.140 4.660 3.750 4.200 2.260 1.610
(0.023) (0.019) (0.017) (0.015) (0.012) (0.006) (0.004) 1.25 4.790 3.800 2.660 2.080 2.600 1.470 1.170
(0.013) (0.009) (0.007) (0.005) (0.006) (0.003) (0.001) 1.50 3.170 2.400 1.660 1.340 1.730 1.140 1.030
(0.006) (0.004) (0.003) (0.002) (0.003) (0.001) (0.000) 1.75 2.180 1.660 1.220 1.080 1.300 1.030 1.000
(0.004) (0.002) (0.001) (0.001) (0.001) (0.000) (0.000) 2.00 1.600 1.280 1.050 1.010 1.100 1.000 1.000
(0.002) (0.001) (0.000) (0.000) (0.000) (0.000) (0.000) 2.25 1.270 1.090 1.010 1.000 1.020 1.000 1.000
(0.001) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) 2.50 1.100 1.020 1.000 1.000 1.000 1.000 1.000
(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) 2.75 1.030 1.000 1.000 1.000 1.000 1.000 1.000
(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) 3.00 1.000 1.000 1.000 1.000 1.000 1.000 1.000
(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)
−6 −2 2
−6 −2 2
Figure 1: DuPont Dose-Response Data for 44 weeks with their smoothing spline fittings (smoothing parameter=0.5).
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Figure 3: 7 potentially out-of-control profiles.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
35404550556065
first diagnosis VDP profiles
depth
density
6 10
Figure 4: 2 potentially out-of-control profiles from the first diagnosis.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
35404550556065
second diagnosis VDP profiles
depth
density
2 3 16 17 20
Figure 5: 5 potentially out-of-control profiles from the second diagnosis.
1.0 1.5 2.0 2.5 3.0 3.5
051015
Four hypothetical aspartame profiles
x (scaled temperature)
y (scaled amount of aspartame dissolved)
Figure 6: 4 hypothetical aspartame profiles.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
05101520
ARL comparison for I−shift with ARLo=20
shift size alpha
ARL
Oja PC1 PC2 PC3 Com Tsq
Figure 7: ARL comparisons for the I-shift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
05101520
ARL comparison for M−shift with ARLo=20
shift size beta
ARL
Oja PC1 PC2 PC3 Com Tsq
Figure 8: ARL comparisons for the M-shift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
05101520
ARL comparison for N−shift with ARLo=20
shift size gamma
ARL
Oja PC1 PC2 PC3 Com Tsq
Figure 9: ARL comparisons for the N-shift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
05101520
ARL comparison for I−shift with ARLo=20
shift size alpha
ARL
Oja Q2 Q4 Q6 D2 D4 D6
Figure 10: ARL comparisons of r, Q, and DDMA charts for the I-shift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
05101520
ARL comparison for M−shift with ARLo=20
shift size beta
ARL
Oja Q2 Q4 Q6 D2 D4 D6
Figure 11: ARL comparisons of r, Q, and DDMA charts for the M-shift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
05101520
ARL comparison for N−shift with ARLo=20
shift size gamma
ARL
Oja Q2 Q4 Q6 D2 D4 D6
Figure 12: ARL comparisons of r, Q, and DDMA charts for the N-shift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
020406080100120140
ARL comparison for I−shift with ARLo=20
shift size alpha
ARL
Q2 Q4 Q6
Figure 13: ARL (individual profiles version) comparisons of Q charts for the I-shift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
020406080100120140
ARL comparison for M−shift with ARLo=20
shift size beta
ARL
Q2 Q4 Q6
Figure 14: ARL (individual profiles version) comparisons of Q charts for the M-shift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
020406080100120140
ARL comparison for N−shift with ARLo=20
shift size gamma
ARL
Q2 Q4 Q6
Figure 15: ARL (individual profiles version) comparisons of Q charts for the N-shift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0510152025
ARL comparison for I−shift with ARLo=20
shift size alpha
ARL
D2 D4 D6
Figure 16: ARL (individual profiles version) comparisons of DDMA charts for the I-shift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0510152025
ARL comparison for M−shift with ARLo=20
shift size beta
ARL
D2 D4 D6
Figure 17: ARL (individual profiles version) comparisons of DDMA charts for the M-shift.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0510152025
ARL comparison for N−shift with ARLo=20
shift size gamma
ARL
D2 D4 D6
Figure 18: ARL (individual profiles version) comparisons of DDMA charts for the N-shift.
0.0 1.0 2.0 3.0
Figure 19: ARL (individual profiles version) comparisons of all charts for all three types of shifts.