In this studies, the eigenvalues/eigenvectors of the smoothed VDP data are treated as the true population parameters. The performance of the control charts are estimated in terms
of ARLs. Table 2-4 presents the ARL values of the charts under study on PC1, PC2, and PC23,respectively. Figure 4-6 display the corresponding detecting power of charts. The results are summarized as below:
• Since the T2 charts and the combined chart treats each principal component equally important, there is no difference in performance among all principal components for them.
• The T2 chart using only the effective principal components, T12, performs better than the T2 chart using all principal components, T11, for all PC scores.
• The combined chart performs in between the two T2 charts, but fairly close to the T2 chart with only four components for the first four PCs.
• The Satterwaite’s approximation indeed is not good enough for process monitor-ing, which leads to an unsuitable in-control ARL. Hence, this control limit is not suggested.
• The TSS chart using the empirical (1 − α) quantile as the control limit perform better than the T2 charts and the combined chart for PC1.
• The detecting power of the TSS chart drops very quickly after the first component.
This confirms the expectation that statistic T3 is insensitive to “unimportant” PCs.
But, the problem is that the TSS chart has very little detecting power for changes in other principal components.
Tables 5-19 show the ARL values when eigenvalues and eigenvectors need to be esti-mated from historical m profiles. The corresponding power curves are given in Figures 7-21. The results of this simulation study are summarized as below:
• For m not large enough, the ARL0 of all the T2 charts are disired This indicates the estimation error does play an important role in the performance of the charts. We would need a fairly large historical data set to get the estimation accurate enough.
• For the T2 charts, the one using only effective components has ARL0 values closer to 370.4, then the one using all components. The former also has a better detecting power than the latter.
• Again, the performance of the combined chart falls in between the two T2 chart.
• As to the TSS chart, using sample eigenvalues to construct the control limit (T33) has ARL0 value than the one using the “true” control limit (T31). In fact, they are not too far from the nominal value of 370.4. This indicates that this chart (T33) may be useful in practice.
5 Conclusion
In recent years, the profile monitoring has become a popular area of research in statistical process control. In this study, we discuss profile monitoring schemes with nonparametric regression models. We use the principal components analysis to analyze the covariance structure of the profiles and use the principal component scores that capture special fea-tures of profiles for process monitoring.
In the thesis, we compare three statistics. The first statistic is the usual T2. T2 treats each principal component equally important but has poor power with an increasing the number of principal components. Thus, we only select the first K principal components according the proportion of the variation they explain. The second statistic is the max-imum score, which corresponds to the combined chart that combines all the PC score charts. It monitors each component of a profile and also treats each principal component equally important. The third statistics is the TSS. It is sensitive to “important” principal components and insensitive to the “unimportant” principal components.
Using the VDP data as an illustrative example, our simulation shows that TSS performs
However, the effects of shifts on those components with little contribution to the overall profile may be statistically significant but in fact have no practical significance in quality.
Wasting some power on “unimportant” components, classical T2 statistics have less power in detecting changes in the “important” component. On the other hand, the statistic we propose giving more weights to more “important”principal components scores is sensitive to the change of the “important” principal components. Then, when shifts occur in the
“important” principal component, the statistic we propose can detect more quickly than classical T2 statistics. It is very insensitive to those “unimportant” components since almost no weights are given to them.
However, our simulation shows TSS only performs well in PC1, the power of TSS decreases very quickly after PC2. It has almost no power for PC3 and on. This is because the eigenvalues of the VDP data drop so quickly from. λ1=0.8451 to λ2=0.1076, then to λ3=0.0192, and so on. It seems the weights need to be adjusted if the process changes are bound to be captured by some principal components other than PC1. This could be a potential future research topic.
References
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[2] Castro, P. E., Lawton, W. H., and Sylvester, E. A. (1986). “Principal Models of Variation for Processes With Continuius Sample Curves”. Technormetrics 28, pp.
329-337.
[3] Jensen, W. A., Birch, J. B., and Woodall W. H. (2006a). “Profile Monitoring via Nonlinear Mixed Models”, Techniquical Report. Virginia Polytechnic Institute and State University.
[4] Jensen, W. A., Jones-Farmer, L. A., Champ, C. W., and Woodall W. H.
(2006b).“Effects of Parameter Estimation on Control Chart Properties: A Litera-ture Review”,Journal of Quality Technology 38, pp. 349-366.
[5] Kang, L. and Albin, S. L. (2000).“On-Line Monitoring when the Process Yields a Linear Profile”. Journal of Quality Technology 32, pp. 418-426.
[6] Kim, K., Mahmoud, M. A., and Woodal, W. H. (2003).“On the Monitoring of Linear Profiles”. Journal of Quality Technology 35, pp. 317-328.
[7] Mestek, O., Pavlik, J., and Suchanek, M. (1994). “Multivariate Control Charts: Con-trol Charts for Calibration Curves”. Fresenius Journal of Analytical Chemistry 350, pp. 344-351.
[8] Rice, J. A. and Silverman, B. W. (1991) “Estimating the Mean and Covariance Struc-ture Nonparametrically When the Data Are Curves”. Journal of the Royal Statistical Society, Ser. B, 53, pp. 233-243.
[9] Ramsay, J. O. and Silverman, B. W. (2002) Applied Functional Data Analysis: Meth-ods and Case Studies. Springer, New York.
[10] Ramsay, J. O. and Silverman, B. W. (2005) Functinal Data Analysis, 2nd. Springer,
[11] Satterthwaite, F. W. (1941) “Synthesis of Variance”, Psychometrik, 6, 309-316.
[12] Shiau, J.-J. H. and Weng, Z.-P (2004) “Profile Monitoring by Nonparametric Regres-sion”. Techniqual Report. Institute of Statistics, National Chiao Tung University.
[13] Walker, E. and Wright, S. (2002) Comparing Curves Using Additive Models, Journal of Quality Technology 34, pp. 118-129.
[14] Williams, J. D., Woodall, W. H., and Birch, J. B. (2003), “Phase I Analysis of Non-linear Product and Process Quality Profiles”, Technical Report. Virginia Polytechnic Institute and State University.
[15] Woodall, W. H., Spitzner, D. J., Montgomery, D. C., and Gupta, S. (2004). “Using Control Charts to Monitor Process and Product Quality Profiles”, Journal of Quality Technology 36, pp. 309-320.
Table 1: Eigenvalues for smoothed VDP data (proportion of explaining variation)
k λk k λk k λk
1 899.1265 9 1.6551 17 0.5106 (0.8451) (0.0015) (0.0005) 2 114.4609 10 1.3654 18 0.3880
(0.1075) (0.0012) (0.0003) 3 20.4119 11 1.2072 19 0.3788
(0.0191) (0.0011) (0.0003) 4 9.4104 12 1.0337 20 0.2879
(0.0088) (0.0009) (0.0002) 5 3.2126 3 0.8993 21 0.2800
(0.0030) (0.0008) (0.0002) 6 2.7177 14 0.6758 22 0.2529
(0.0025) (0.0006) (0.0002) 7 2.3202 15 0.6040 23 0.1787
(0.0021) (0.0005) (0.0001) 8 1.9736 16 0.5144
(0.0018) (0.0004)
Table 2: ARL for PC1
T11 T12 T2 T31 T32
0 370.3703 370.3703 370.3703 370.3703 287.3563 1 217.3913 103.7344 221.2389 45.2488 36.2318 2 60.9756 15.3280 28.4414 6.3259 5.6357 3 14.6842 3.5868 4.9677 2.0023 1.8912 4 4.3821 1.5755 1.7818 1.1791 1.1565 5 1.9051 1.1044 1.1418 1.0222 1.0187 6 1.2214 1.0111 1.0155 1.0014 1.0011 7 1.0384 1.0005 1.0000 1.0000 1.0000 8 1.0037 1.0000 1.0000 1.0000 1.0000 9 1.0002 1.0000 1.0000 1.0000 1.0000 10 1.0001 1.0000 1.0000 1.0000 1.0000 11 1.0000 1.0000 1.0000 1.0000 1.0000 12 1.0000 1.0000 1.0000 1.0000 1.0000
Table 3: ARL for PC2
T11 T12 T2 T31 T32
0 370.3703 370.3703 370.3703 370.3704 287.3563 1 217.3913 103.7344 221.2389 294.1176 242.7184 2 60.9756 15.3280 28.4414 303.0303 238.0952 3 14.6842 3.5860 4.9677 194.5525 143.6781 4 4.3821 1.5755 1.7818 108.4598 80.7754 5 1.9051 1.1044 1.1418 44.6827 33.9673 6 1.2214 1.0111 1.0155 15.8177 12.0569 7 1.0384 1.0005 1.0008 5.25762 4.12269 8 1.0037 1.0002 1.0000 2.0786 1.7925 9 1.0002 1.0000 1.0000 1.2426 1.1713 10 1.0001 1.0000 1.0000 1.0372 1.0227 11 1.0000 1.0000 1.0000 1.0028 1.0014 12 1.0000 1.0000 1.0000 1.0000 1.0000
Table 4: ARL for PC23
T11 T2 T31 T32
0 370.3703 370.3703 333.3333 260.4166 1 217.3913 221.2389 384.6153 267.3796 2 60.9756 28.4414 381.6793 297.6190 3 14.6842 4.9677 335.5704 268.8172 4 4.3821 1.7818 406.5040 292.3976 5 1.9051 1.1418 406.5040 335.5704 6 1.2214 1.0155 387.5968 292.3976 7 1.0384 1.0008 359.7122 260.4166 8 1.0037 1.0000 359.7122 268.8172 9 1.0002 1.0000 373.1343 277.7777 10 1.0001 1.0000 387.5968 284.0909 11 1.0000 1.0000 359.7122 287.3563 12 1.0000 1.0000 347.3152 288.14523
Table 5: ARL for PC1( m=200)
T11 T12 T2 T31 T32 T33 T34
0 155.3097 236.9668 112.4590 214.5922 263.1578 367.6470 256.4102 1 34.4827 58.7544 74.9625 24.5700 36.6300 35.5114 28.1056 2 13.8927 9.96618 16.6444 4.3335 5.7730 5.4042 4.6576 3 5.2675 2.8088 3.7147 1.6500 1.9043 1.8492 1.7088 4 2.3243 1.4020 1.5549 1.1160 1.1671 1.1584 1.1281 5 1.3730 1.0685 1.0948 1.01268 1.0209 1.0189 1.0144 6 1.0842 1.0060 1.00888 1.0006 1.0011 1.0004 1.0006 7 1.0116 1.0002 1.0003 1.0000 1.0000 1.0002 1.0002 8 1.0008 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 9 1.0002 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Table 6: ARL for PC2( m=200)
T11 T12 T2 T31 T32 T33 T34
0 154.9451 235.8491 114.4165 201.6129 274.7253 337.8378 228.3105 1 40.2901 81.1688 88.0282 188.6792 265.9574 335.5705 226.2443 2 16.8237 13.5355 24.5459 153.8462 234.7418 264.5503 182.4818 3 6.0518 3.4729 5.1020 102.8807 149.2537 177.9359 125.6281 4 2.5431 1.5520 1.8230 58.6166 84.4595 104.6025 71.2251 5 1.4504 1.1000 1.1514 26.8528 37.3692 47.1698 32.4675 6 1.1048 1.0107 1.0163 9.5475 14.6071 17.0823 11.4784 7 1.0163 1.0006 1.0008 3.3697 4.8866 5.6218 3.9364 8 1.0014 1.0000 1.0000 1.6070 2.0075 2.1865 1.7538 9 1.0001 1.0000 1.0000 1.1237 1.2262 1.2752 1.1630 10 1.0000 1.0000 1.0000 1.0156 1.0345 1.0446 1.0224 11 1.0000 1.0000 1.0000 1.0009 1.0026 1.0036 1.0014 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0001 1.0001
Table 7: ARL for PC23( m=200)
T11 T2 T31 T32 T33 T34
0 154.1712 236.8224 116.0784 276.4167 364.8780 228.3105 1 39.2157 63.2111 200.0000 294.1176 352.1127 241.5459 2 18.7758 18.1159 196.8504 287.3563 320.5128 227.2727 3 7.3206 4.9796 204.9180 294.1176 352.1127 256.4103 4 3.1504 2.0962 204.0816 306.7485 316.4557 242.7184 5 1.6904 1.2916 206.6116 285.7143 375.9398 247.5248 6 1.1905 1.0657 210.0840 280.8989 384.6154 255.1020 7 1.0371 1.0094 200.0000 304.8780 328.9474 234.7418 8 1.0041 1.0006 195.3125 289.0173 344.8276 227.2727 9 1.0003 1.0000 217.3913 263.1579 333.3333 251.2563 10 1.0000 1.0000 207.2487 251.2884 324.1854 228.2159 11 1.0000 1.0000 219.1488 285.1587 334.7575 258.6783
Table 8: ARL for PC1( m=300)
T11 T12 T2 T31 T32 T33 T34
0 220.7729 403.2258 269.4915 359.7122 284.0909 375.9398 427.3504 1 81.6993 108.6957 132.2751 41.8410 39.9361 43.8982 47.8927 2 31.8066 17.1350 32.1750 6.2555 6.1013 6.4591 6.8213 3 10.0746 3.9448 5.5371 1.9874 1.9676 2.0188 2.0761 4 3.5865 1.6472 1.8963 1.1849 1.1807 1.1916 1.2044 5 1.7561 1.1258 1.1716 1.0232 1.0222 1.0242 1.0264 6 1.1930 1.0145 1.0195 1.0013 1.0011 1.0014 1.0015 7 1.0340 1.0007 1.0012 1.0000 1.0000 1.0000 1.0000 8 1.0035 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 9 1.0002 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Table 9: ARL for PC2( m=300)
T11 T12 T2 T31 T32 T33 T34
0 222.2494 416.6667 260.7717 362.3188 280.8989 400.0000 462.9630 1 82.3723 124.6883 149.2537 340.1361 242.7184 359.7122 406.5041 2 32.9598 20.2429 38.9408 248.7562 196.8504 270.2703 312.5000 3 10.7735 4.5106 6.7168 171.8213 141.6431 185.8736 205.7613 4 3.9711 1.7859 2.1251 92.5926 70.9220 99.6016 111.6071 5 1.8934 1.1570 1.2178 39.7772 31.0559 42.8816 48.9716 6 1.2369 1.0200 1.0279 14.1643 11.2587 15.3186 17.3792 7 1.0459 1.0009 1.0016 4.6155 3.8652 4.9417 5.5636 8 1.0052 1.0000 1.0000 1.9427 1.7212 2.0232 2.1824 9 1.0002 1.0000 1.0000 1.2076 1.1544 1.2294 1.2699 10 1.0000 1.0000 1.0000 1.0310 1.0200 1.0353 1.0438
Table 10: ARL for PC23( m=300)
T11 T2 T31 T32 T33 T34
0 225.3132 286.5671 381.2048 276.2430 402.5806 389.7122 1 82.3723 83.6120 312.5488 297.6190 328.9473 373.1343 2 24.5218 15.0966 324.6753 280.8984 337.8378 373.1343 3 8.2877 3.9491 342.4657 322.5806 359.7122 403.2258 4 2.9751 1.6767 349.6503 299.4011 373.1343 423.7288 5 1.5691 1.1436 328.9473 289.0173 354.6099 403.2255 6 1.1336 1.0203 364.9635 261.7801 375.9398 409.8360 7 1.0209 1.0015 375.9398 303.0303 393.7007 442.4778 8 1.0017 1.0001 316.4556 279.3296 340.1360 400.5156 9 1.0000 1.0000 328.4822 289.4891 348.5842 431.0344 10 1.0000 1.0000 357.8843 236.8997 316.1584 403.5941 11 1.0000 1.0000 318.2562 289.4879 369.4988 389.2971 12 1.0000 1.0000 316.158 325.5498 365.2594 408.4997
Table 11: ARL for PC1( m=400)
T11 T12 T2 T31 T32 T33 T34
0 222.2494 324.2152 374.2160 340.4494 274.7253 352.1127 374.8252 1 82.3723 53.5906 92.4214 20.4666 23.3333 33.2882 41.6967 2 32.9598 8.9977 14.6413 3.8772 4.2676 5.9552 5.2827 3 10.7735 2.5788 3.2757 1.5708 1.8194 1.9418 1.6448 4 3.9711 1.3339 1.4532 1.0989 1.1498 1.1763 1.1137 5 1.8934 1.0537 1.0739 1.0102 1.0172 1.0218 1.0122 6 1.2369 1.0080 1.0111 1.0006 1.0009 1.0012 1.0007 7 1.0459 1.0002 1.0005 1.0000 1.0001 1.0000 1.0000 8 1.0052 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 9 1.0002 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 1.0002 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 11 1.0002 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Table 12: ARL for PC2( m=400)
T11 T12 T2 T31 T32 T33 T34
0 250.1923 330.4147 390.8397 336.6120 290.6977 370.3704 368.3502 1 71.3267 71.7360 125.9446 119.9041 145.7143 285.4713 318.3488 2 27.2777 13.2240 25.3421 96.8992 190.1141 122.7391 271.2494 3 8.7199 3.4151 4.7985 65.3595 81.3333 133.1502 183.9672 4 3.1447 1.5249 1.7506 34.6741 43.5689 95.4198 72.9367 5 1.6138 1.1008 1.1387 15.3374 19.2110 31.4684 46.9442 6 1.2538 1.0184 1.0260 8.6790 11.1383 10.4745 16.5441 7 1.0505 1.0013 1.0019 3.1393 3.7222 3.2345 5.6668 8 1.0055 1.0000 1.0000 1.5305 1.6959 2.0870 1.6758 9 1.0002 1.0000 1.0000 1.1043 1.1453 1.2454 1.1384 10 1.0000 1.0000 1.0000 1.0116 1.0191 1.0395 1.0182 11 1.0000 1.0000 1.0000 1.0005 1.0015 1.0027 1.0009 12 1.0000 1.0000 1.0000 1.0000 1.0001 1.0001 1.0000
Table 13: ARL for PC23( m=400)
T11 T2 T31 T32 T33 T34
0 354.4165 386.5672 334.0483 282.4859 337.8378 367.2241 1 76.6871 83.6120 138.8889 271.7391 438.5965 372.4138 2 28.3607 15.0966 133.6898 260.4167 384.6154 360.7717 3 8.8778 3.9491 149.7006 247.5248 357.1429 284.5018 4 3.1702 1.6767 142.8571 299.4012 362.3188 280.5054 5 1.6202 1.1436 153.3742 271.7391 409.8361 289.3939 6 1.3185 1.0203 217.3913 322.5806 359.7122 255.1020 7 1.0639 1.0016 239.2344 310.5590 446.4286 271.7391 8 1.0081 1.0001 211.8644 301.2048 364.9635 251.2563 9 1.0005 1.0000 205.7613 271.7391 340.1361 250.4840 10 1.0000 1.0000 283.4983 284.4985 357.4894 275.1859
Table 14: ARL for PC1( m=500)
T11 T12 T2 T31 T32 T33 T34
0 287.2659 384.6154 338.0952 326.2443 292.3977 378.7879 367.3797 1 98.8142 80.6452 135.5014 27.6091 31.1865 35.6174 40.0559 2 33.1345 12.2070 20.7383 4.6777 5.3781 5.0411 5.9378 3 8.6957 3.1180 4.0806 1.7151 1.8186 1.9446 1.7826 4 3.0460 1.4529 1.6074 1.1272 1.1544 1.1765 1.1413 5 1.5534 1.0798 1.1037 1.0132 1.0183 1.0210 1.0154 6 1.1574 1.0103 1.0142 1.0010 1.0008 1.0015 1.0012 7 1.0239 1.0004 1.0008 1.0000 1.0000 1.0000 1.0000 8 1.0021 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Table 15: ARL for PC2( m=500)
T11 T12 T2 T31 T32 T33 T34
0 280.5054 423.7288 351.2563 309.2050 292.3977 354.6099 355.1020 1 126.5823 123.4568 188.6792 178.5714 213.7801 316.4557 261.6752 2 45.3309 19.3199 40.3551 154.3210 208.3333 292.1176 294.3077 3 13.0993 4.4045 6.4994 96.3391 149.2537 112.0104 173.8668 4 4.2878 1.7434 2.0464 55.9910 74.9625 68.4036 99.3060 5 1.9722 1.1476 1.2018 25.9875 31.0907 33.9484 46.6656 6 1.1535 1.0091 1.0139 8.7935 11.9646 11.4847 15.3973 7 1.0232 1.0003 1.0004 3.1620 4.0254 4.9039 3.8601 8 1.0020 1.0000 1.0000 1.5452 1.7687 2.0227 1.7422 9 1.0001 1.0000 1.0000 1.1108 1.1689 1.2353 1.1599 10 1.0000 1.0000 1.0000 1.0124 1.0228 1.0350 1.0204 11 1.0000 1.0000 1.0000 1.0005 1.0018 1.0029 1.0014
Table 16: ARL for PC23( m=500)
T11 T2 T31 T32 T33 T34
0 297.6285 341.5459 392.3077 259.0674 326.7974 336.9668 1 129.5337 176.0563 210.9705 270.2703 390.6250 248.7562 2 49.9002 37.9939 207.4689 318.4713 326.7974 243.9024 3 14.4051 7.1388 219.2982 271.7391 393.7008 257.7320 4 4.9544 2.4116 199.2032 294.1176 326.7974 235.8491 5 2.1429 1.3365 206.6116 290.6977 318.4713 238.0952 6 1.2341 1.0275 250.0000 271.7391 393.7008 310.5590 7 1.0431 1.0017 210.9705 337.8378 342.4658 268.8172 8 1.0046 1.0000 229.3578 308.6420 431.0345 312.5000 9 1.0002 1.0000 235.4984 277.7778 364.9635 282.4859 10 1.0000 1.0000 208.4894 278.4874 372.4566 298.4894 11 1.0000 1.0000 248.8772 236.4898 369.4894 267.4897 12 1.0000 1.0000 225.5152 258.4898 356.4897 258.4897
Table 17: ARL for PC1( m=600)
T11 T12 T2 T31 T32 T33 T34
0 377.3585 386.1004 338.0952 327.2727 363.1579 367.6471 304.8780 1 221.2389 93.4579 156.7398 32.0102 33.6022 39.5236 44.0930 2 32.2165 13.5722 23.9006 5.2214 5.3214 5.4078 6.9538 3 8.9381 3.3174 4.5082 1.7981 1.8310 1.9998 1.9180 4 3.1534 1.5094 1.6862 1.1475 1.1553 1.1911 1.1749 5 1.5932 1.0946 1.1261 1.0168 1.0166 1.0239 1.0208 6 1.1384 1.0096 1.0128 1.0008 1.0009 1.0013 1.0011 7 1.0203 1.0005 1.0007 1.0000 1.0000 1.0000 1.0000 8 1.0019 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 9 1.0001 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Table 18: ARL for PC2( m=600)
T11 T12 T2 T31 T32 T33 T34
0 380.2281 349.6503 325.2252 320.2643 348.7562 381.6794 322.5806 1 253.1646 120.1923 185.1852 207.4689 271.7391 270.6753 224.2703 2 42.1941 17.7873 33.9905 169.4915 199.2032 221.3797 267.2389 3 11.7288 4.0064 5.6664 111.1111 140.0560 151.5714 178.0574 4 3.9250 1.6501 1.8967 65.1890 79.1139 88.0664 107.4956 5 1.8258 1.1206 1.1654 28.0112 39.0320 39.2144 47.0625 6 1.2090 1.0152 1.0213 9.9483 13.2485 13.8634 16.7893 7 1.0372 1.0006 1.0010 3.4732 4.3948 4.3648 5.5372 8 1.0036 1.0000 1.0000 1.6342 1.8930 1.1329 2.9168 9 1.0001 1.0000 1.0000 1.1288 1.1972 1.2607 1.2025 10 1.0000 1.0000 1.0000 1.0160 1.0280 1.0417 1.0296 11 1.0000 1.0000 1.0000 1.0010 1.0017 1.0035 1.0022 12 1.0000 1.0000 1.0000 1.0001 1.0001 1.0001 1.0001
Table 19: ARL for PC23(m=600)
T11 T2 T31 T32 T33 T34
0 375.9398 326.2443 325.2252 384.0909 340.1361 385.7143 1 210.9705 147.9290 232.5581 320.5128 354.6099 310.5590 2 40.6174 27.0856 221.2389 287.3563 373.1343 310.5590 3 12.3365 5.2598 231.4815 265.9574 387.5969 310.5590 4 4.0700 1.8858 233.6449 265.9574 347.2222 299.4012 5 1.8669 1.1814 208.3333 312.5000 314.4654 273.2240 6 1.2212 1.0236 227.2727 271.7391 367.6471 312.5000 7 1.0398 1.0018 226.2443 273.2240 370.3704 322.5806 8 1.0046 1.0000 209.2050 273.2240 359.7122 290.6977 9 1.0001 1.0000 236.9668 255.1020 384.6154 314.4654 10 1.0000 1.0000 286.4894 225.7587 357.5278 323.7755 11 1.0000 1.0000 254.4568 279.5630 324.7858 312.5278
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
35404550556065
Depth
Density
Figure 1: Original 24 VDP-profiles
0.0 0.1 0.2 0.3 0.4 0.5 0.6
4045505560
Smooth VDP
0
0
●
0 1000 2000 3000 4000
0.00000.00040.00080.0012
Density
T3
c*Chisq(df)
Figure 3: Density for T3
●
Figure 4: Power of PC1 (true VDP)
●
Figure 5: Power of PC2 (true VDP)
●
Figure 6: Power of PC23 (true VDP)
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC1 (m=200)
shift
Figure 7: Power for PC1 with m=200
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC2 (m=200)
shift
Figure 8: Power for PC2 with m=200
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC23 (m=200)
shift
Figure 9: Power for PC23 with m=200
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC1 (m=300)
shift
Figure 10: Power for PC1 with m=300
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC2 (m=300)
shift
Figure 11: Power for PC2 with m=300
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC23 (m=300)
shift
Figure 12: Power for PC23 with m=300
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC1 (m=400)
shift
Figure 13: Power for PC1 with m=400
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC2 (m=400)
shift
Figure 14: Power for PC2 with m=400
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC23 (m=400)
shift
Figure 15: Power for PC23 with m=400
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC1 (m=500)
shift
Figure 16: Power for PC1 with m=500
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC2 (m=500)
shift
Figure 17: Power for PC2 with m=500
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC23 (m=500)
shift
Figure 18: Power for PC23 with m=500
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC1 (m=600)
shift
Figure 19: Power for PC1 with m=600
●
Figure 20: Power for PC2 with m=600
●
0 2 4 6 8 10 12
0.00.20.40.60.81.0
Power for PC23 (m=600)
shift
Figure 21: Power for PC23 with m=600