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The Monitoring of Linear Profiles 

Keun Pyo Kim Mahmoud A. Mahmoud

William H. Woodall

Virginia Tech Blacksburg, VA 24061-0439

(Send request for paper, submitted to JQT, to bwoodall@vt.edu)

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We assume that for the jth random sample collected over time, we have the observations (xi , yij), i = 1, 2, …, n.

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Applications include…• Calibration problems in analytical

chemistry (Stover and Brill, 1998)• Semiconductor manufacturing

(Kang and Albin, 2000)• Automobile manufacturing (Lawless

et al., 1999)• DOE applications (Miller, 2002 and

Nair et al. 2002)

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It is assumed that when the process is in statistical control, the underlying model is

i = 1, 2, …, n,  where the ’s are independent, identically distributed

(i.i.d.) N(0, ).

ijiij XAAY 10

ij

2

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The least squares estimators and have have a bivariate normal distribution with the mean vector    and the variance-covariance matrix

 

ja0

ja1

TAA ),( 10μ

21

201

201

20

Σ

6

xxS

x

n

222

0

1

xxS

1221

xxS

x2201

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Phase II

First we consider the Phase II case involving process monitoring with in-control values of the parameters assumed to be known.

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The first control strategy of Kang and Albin (2000) is a T2

chart based on the estimated regression coefficients

Tjjj aa ),( 10Z

)()( 12 μZΣμZ j

TjjT

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Their second control strategy is to apply an EWMA - R chart combination scheme to the residuals obtained with each sample.

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The residuals for the jth sample are

i = 1, 2, … , n.

ijijij xAAye 0

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Instead, we propose scaling the X-values to obtain the model

ijiij XBBY 10

XAAB 100

11 AB

)( XXX ii

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Since now the least squares estimators are independent, we recommend three EWMA charts in Phase II to detect sustained shifts in the parameters.

There is a chart for each regression coefficient and one for the variation about the line.

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ARL Comparisons

We use the in-control model

with error terms i.i.d. N(0, 1). The values for X are 2, 4, 6, 8.

ijiij XY 23

140.0 0.5 1.0 1.5 2.0

0

50

100

150

200

AR

L

EWMA/RT2EWMA_3

Figure 1. ARL Comparisons Under Intercept Shifts

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0.00 0.05 0.10 0.15 0.20 0.25

0

50

100

150

200

AR

L

EWMA/RT2EWMA_3

Figure 2. ARL Comparisons Under Slope Shifts

161.0 1.5 2.0 2.5 3.0

0

50

100

150

200

AR

L

EWMA/RT2EWMA_3

Figure 3. ARL Comparisons Under Standard Deviations Shifts

170.0 0.2 0.4 0.6 0.8 1.0

0

50

100

150

200

AR

L

EWMA/RT2 T2

EWMA_3

Figure 4. ARL Comparisons Under Slope Shifts

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Table 1. ARL Comparisons Under Slope Shifts in Model (10) From 1B To 1B

(Xi -values are 1, 2, 3, and 4 and In-control ARL = 200)

Chart -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1.0

EWMA/R 149.06 110.07 75.50 50.65 33.29 22.26 15.03 10.53 7.53

EMWA_3 49.07 22.85 13.13 8.87 6.63 5.27 4.38 3.78 3.32

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Our proposed method (EWMA_3) has better ARL performance than competing methods.

The interpretation is also much easier.

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Phase I

In Phase I, one has k sets of bivariate observations.

One checks for stability of the linear profiles over time and estimates parameters.

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We recommend Shewhart type charts for each regression parameter and change-point methods.

EWMA charts are not recommended in Phase I.

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Relationship to Regression-adjusted Control Charts

Monitoring linear profiles is a generalization of regression-adjusted methods studied by Mandel (1969), Zhang (1992), Wade and Woodall (1993), Hawkins (1991, 1993), and Hauck et. al (1999).

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Suppose X is an input quality variable and Y is the output quality variable with k = 1 and n = 1.

Then we have the simplest regression-adjusted chart, sometimes referred to as the cause-selecting chart. (Note X-values are random.)

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Conclusions• Monitoring linear profiles seems to be

quite useful.• Regression-adjusted methods deserve

wider application since usual methods can be misleading if output quality is affected by input quality as is often the case.

• Methods can be extended to more complicated models.

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References Albin, S. L. (2002). Personal communication. Andrews, D. W. K., Lee, I., and Ploberger, W. (1996). “Optimal Changepoint Tests for Normal Linear Regression”. Journal of Econometrics 70, pp. 9-38. Brill, R. V. (2001). “A Case Study for Control Charting a Product Quality Measure That is a Continuous Function Over Time”. Presented at the 45th Annual Fall Technical Conference, Toronto, Ontario.  Crowder, S. V., and Hamilton, M. D. (1992). “An EWMA for Monitoring a Process Standard Deviation”. Journal of Quality Technology 24, pp. 12-21. Hauck, D. J., Runger, G. C., and Montgomery, D. C. (1999). “Multivariate Statistical Process Monitoring and Diagnosis with Grouped Regression-Adjusted Variables”.Communications in Statistics - Simulation and Computation 28, pp. 309-328.

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 Hawkins, D. M. (1991). “Multivariate Quality Control Based on Regression-Adjusted Variables”. Technometrics 33, pp.61-75. Hawkins, D. M. (1993). “Regression Adjustment for Variables in Multivariate Quality Control”. Journal of Quality Technology 25, pp. 170-182.  Jin, J., and Shi, J. (2001). “Automatic Feature Extraction of Waveform Signals for In-Process Diagnostic Performance Improvement”. Journal of Intelligent Manufacturing 12, pp. 257-268.  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. Lawless, J. F., Mackay, R. J., and Robinson, J. A. (1999). “Analysis of Variation Transmission in Manufacturing Processes-Part I”. Journal of Quality Technology 31, pp. 131-142. Lucas, J. M., and Saccucci, M. S. (1990). “Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements”. Technometrics 32, pp. 1-29.

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Mandel, B. J. (1969). “The Regression Control Chart”. Journal of Quality Technology 1, pp. 1-9.

Mason, R. L., Chou, Y.-M., and Young, J. C. (2001). “Applying Hotelling’s T2 Statistic to Batch Processes”. Journal of Quality Technology 33, pp. 466-479. Miller, A. (2002). “Analysis of Parameter Design Experiments for Signal-Response Systems”. Journal of Quality Technology 34, pp. 139-151. Montgomery, D. C. (2001). Introduction to Statistical Quality Control. 4th Edition, John Wiley & Sons, New York, NY. Myers, R. H. (1990). Classical and Modern Regression with Applications. 2nd Edition, PWS-Kent Publishing Company, Boston, MA. Nair, V. N., Taam, W., and Ye, K. Q. (2002). “Analysis of Functional Responses from Robust Design Studies with Location and Dispersion Effects”. To appear in the Journal of Quality Technology.

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Ryan, T. P. (1997). Modern Regression Methods. John Wiley & Sons, New York, NY. Ryan, T. P. (2000). Statistical Methods for Quality Improvement. 2nd Edition, John Wiley & Sons, New York, NY. Stover, F. S., and Brill, R. V. (1998). “Statistical Quality Control Applied to Ion Chromatography Calibrations”. Journal of Chromatography A 804, pp. 37-43.

Wade, M. R., and Woodall, W. H. (1993). “A Review and Analysis of Cause-Selecting Control Charts”. Journal of Quality Technology 25, pp. 161-169.  

Walker, E., and Wright, S. P. (2002). “Comparing Curves Using Additive Models”. Journal of Quality Technology 34, pp. 118-129. Zhang, G. X. (1992). Cause-Selecting Control Chart and Diagnosis, Theory and Practice. Aarhus School of Business, Department of Total Quality Management, Aarhus, Denmark.