The profile monitoring is a very useful and promising area of research. In this thesis, we focus on the non-linear profile monitoring. Three profile monitoring approaches are proposed. And from the simulation studies, all approaches appear to perform well for the exponential profile under study.
Comparisons among these approaches are made. For I, M, and N shifts in the exponential profile, the EWMA chart performs the best over entire range of shifts, while the Absolute-value metric has secondly smaller ARL. And for standard deviation shifts, the EWMSD chart is a good choice to detect, while we also can use the R chart to monitor the process. And for linear profiles, our simulation study shows that new methods proposed by us are slightly effective than the methods of Kang and Albin (2000) and Kim et al. (2003). As a design issue, it is observed that increasing n is helpful in reducing the out-of-control ARL values. How to properly choose the number of bases is worth further studies.
This study extends the framework of statistical process control to more general applications. More statistical methods, models, and ideas are needed to extend the framework to a more complete profile monitoring strategy.
References
Crowder, S. V. and Hamilton, M. D. (1992). “An EWMA for Monitoring a Process Standard Deviation”. Journal of Quality Technology, 24, 12-21.
de Boor, C. (1978). A Practical Guide to Splines. Springer, Berlin.
Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications.
Chapman & Hall.
Gardner, M. M., Lu, J. C., Gyurcsik, R. S., Wortman, J. J., Hornung, B. E., Heinisch, H. H., Rying, E. A., Rao, S., Davis, J. C., and Mozumder, P. K. (1997).
“Equipment Fault Detection Using Spatial Signatures”. IEEE Transactions on Components, Packaging, and Manufacturing Technology – Part C, 20, 295-304.
Gisela, E. M. and Frank, U. (1996). Numerical Algorithms with C. Springer-Verlag, New York, 325-330.
Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. New York: Chapman.
Jin, J. and Shi, J. (1999). “Feature-Preserving Data Compression of Stamping Tonnage Information Using Wavelets”. Technometrics, 41, 327-339.
Jin, J. and Shi, J. (2001). “Automatic Feature Extraction of Waveform Signals for In-Process Diagnostic Performance Improvement”. Journal of Intelligent Manufacturing, 12, 257-268.
Jones, M. C. and Rice, J. A. (1992). “Displaying the Important Features of Large Collections of Similar Curves”. The American Statistican, 46, 140-145.
Kang, L. and Albin, S. L. (2000). “On-Line Monitoring When the Process Yields a Linear Profile”. Journal of Quality Technology, 32, 418-426.
Kim, K., Mahmoud, M. A., and Woodall, W. H. (2003). “On The Monitoring of Linear Profiles”. To appear in the Journal of Quality Technology.
Lucas, J. M. and Saccucci, M. S. (1990). “Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements”. Technometrics, 32, 1-29.
Mestek, O., Pavlik, J., and Suchánek, M. (1994). “Multivariate Control Charts:
Control Charts for Calibration Curves”. Fresenius’ Journal of Analytical Chemistry, 350, 344-351.
Montgomery, D. C. (2001). Introduction to Statistical Quality Control. 4th Edition, John Wiley & Sons, New York.
Myers, R. H. (1990). Classical and Modern Regression with Applications. 2nd Edition, PWS-Kent Publishing Company, Boston, MA.
Ryan, T. P. (1997). Modern Regression Methods. John Wiley & Sons, New York.
Simonoff, J. S. (1996). Smoothing Methods in Statistics. Springer-Verlag, New York.
Stover, F. S. and Brill, R. V. (1998). “Statistical Quality Control Applied to Ion Chromatography Calibrations”. Journal of Chromatography, 804, 37-43.
Walker, E. and Wright, S. P. (2002). “Comparing Curves Using Additive Models”.
Journal of Quality Technology, 34, 118-129.
0 5 10 15
Figure1. Milligrams of aspartame dissolved per liter of water from four samples.
3 4 5 6 7 8
0.00.20.40.6
B1,4 B2,4 B3,4 B4,4 B5,4 B6,4
B7,4
B0,4
Figure 2. 8 B-spline bases; each basis is of order 4.
x
y
0 1 2 3 4
051015
Figure 3. The reference profile m x( ) 1 15= + e− −(x1)2.
2
2
( 1)
( 1)
solid line :1 15 dotted line : 3 15
x
x
e e
− −
− −
+ +
x
y
0 1 2 3 4
05101520
Figure 4. The reference curve (solid line) and the curve affected by I shift (dotted line).
2
2
( 1)
( 1)
solid line :1 15 dotted line :1 18
x
x
e e
− −
− −
+ +
x
y
0 1 2 3 4
05101520
Figure 5. The reference curve (solid line) and the curve affected by M shift (dotted line).
2
2
( 1)
1.5( 1)
solid line :1 15 dotted line :1 15
x
x
e e
− −
− −
+ +
x
y
0 1 2 3 4
05101520
Figure 6. The reference curve (solid line) and the curve affected by N shift (dotted line).
x
0 1 2 3 4
0.00.20.40.6
Figure 7. Thirteen cubic B-splines bases.
x
y
0 1 2 3 4
051015
Figure 8. Scatter plot is the reference curve and the dashed line is the B-spline estimate.
Table 1. Simulation results to illustrate boundary effects; Underlines are larger deviations.
Basis 1 2 3 4 5
Cc 2.874264 6.157306 11.611126 16.178782 16.183834
Z1-C -0.0437711 0.0042971 0.00457416 0.00045094 -0.0013554 Z2-C 0.06077714 0.00684893 -0.0002403 -0.0109680 0.00645003 Z3-C 0.08942159 -0.0136533 0.01052427 -0.0160640 0.01881415
Basis 6 7 8 9 10
C 11.603308 6.176568 2.767467 1.423282 1.070967
Z1-C -0.0031249 -0.0001978 0.00197017 -0.0013328 -0.0054912
Z2-C -0.0079634 0.01199316 -0.0198097 0.0244964 -0.0223153
Z3-C 0.00019966 -0.0030393 0.00455043 0.01870481 -0.0077123
Basis 11 12 13
U 1.008571 1.000276 1.003893
Z1-U 0.00677231 0.00502447 -0.0936737 Z2-U 0.00303271 0.01045864 -0.1506641
Z3-U -0.0079037 0.02057199 -0.2117598
Table 2. ARL comparisons under I shifts fromI0 ToI0+ασ
EWMA 200.42
(0.613)
Squared metric 200.70
(0.622) Absolute value metric
Figure 9. ARL comparisons under I shifts of sizeα. α
Table 3. ARL comparisons under M shifts fromM0 ToM0+βσ
Squared metric 201.30
(0.623) Absolute value metric
β
Figure 10. ARL comparisons under M shifts of sizeβ .
Table 4. ARL comparisons under N shifts fromN0 ToN0+γσ
Squared metric 201.05
(0.638) Absolute value metric
Figure 11. ARL comparisons under N shifts of sizeγ . γ
Table 5. ARL comparisons under standard deviation shifts fromσ Toλσ
Squared metric 200.23
(0.632)
EWMV 200.08
(0.618) Absolute value metric EWMV
Figure 12. ARL comparisons under standard deviation shifts of sizeλ λ
Table 6. ARL comparisons under combinations of M and N shifts
Table 7. ARL comparisons under combinations of I and M shifts.
Table 8. ARL comparisons under combinations of I and N shifts.
Table 9. ARL comparisons of T2 chart with different number of bases for I shifts.
Figure 13. ARL comparisons of T2 chart with different number of bases for I shifts.
Table 10. ARL comparisons of T2 chart with different n’s for I shifts.
Figure 14. ARL comparisons of T2 chart with different n’s for I shifts.
n=50
Figure 15. The reference curve (solid line) and the fitted B-spline curve with 5 bases and n=50.
Table 11. ARL comparisons of EWMA chart with different n’s for M shifts.
Figure 16. ARL comparisons of the EWMA chart with different n’s for M shifts.
n=5
Table 12. ARL comparisons of Squared metric with different n’s for standard deviation shifts.
Chart λ
deviation shifts.
Table 13. ARL comparisons of Absolute-value metric with different n’s for I shifts
Figure 18. ARL comparisons of Absolute-value metric with different n’s for I shifts.
Table 14. ARL comparisons under intercept shifts of a linear profile.
Chart λ
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
EWMA/R 198.85
(0.619)
Squared metric 201.43
(0.638) Absolute value metric
λ
Figure19. ARL comparisons under intercept shifts of a linear profile.
Table 15. ARL comparisons under slope shifts of a linear profile.
Chart β
0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250
EWMA/R 198.75
(0.625)
Squared metric 199.50
(0.627) Absolute value metric
Figure 20. ARL comparisons under slope shifts of a linear profile β
Table 16. ARL comparisons under standad deviation shifts of a linear profile.
Chart γ
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
EWMA/R 198.55
(0.623)
Squared metric 201.27
(0.638)
EWMV 199.83
(0.629) Absolute value metric EWMV
γ
Figure 21. ARL comparisons under standad deviation shifts of a linear profile.