We assume that for the j th random sample collected over time, we have the observations ( x i , y ij ), i = 1, 2, …, n .

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We assume that for the j th random sample collected over time, we have the observations ( x i , y ij ), i = 1, 2, …, n .

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The Monitoring of Linear ProfilesKeun Pyo Kim Mahmoud A. MahmoudWilliam H. WoodallVirginia Tech Blacksburg, VA 24061-0439(Send request for paper, submitted to JQT, to bwoodall@vt.edu)

- 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)

It is assumed that when the process is in statistical control, the underlying model isi = 1, 2, …, n, where the ’s are independent, identically distributed (i.i.d.) N(0, ).

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

Phase II

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

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.

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.

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.

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).

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.)

- 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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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.

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.

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.