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3.4.7 A Statistical Control Chart for Stationary Process Data
Nien Fan Zhang Statistical Engineering Division, ITL Statistical process control (SPC) techniques are widely used in industry for process monitoring and quality improvement. Traditional SPC methodology is based on a fundamental assumption that process data are statistically independent. Process data, however, are not always statistically independent from each other. In the continuous industries such as the chemical industry, most process data are autocorrelated. Under such conditions, traditional SPC procedures are not effective and appropriate for monitoring, controlling, and improving process quality. To accommodate autocorrelated data, some SPC methodologies have been developed in recent years. One approach is to use a process residual chart. This procedure requires one to model the process data and obtain the process residuals. Assuming a true model, the residuals are statistically uncorrelated to each other. Then, traditional SPC charts such as X charts (Shewhart individual charts), CUSUM charts, and exponentially weighted moving average (EWMA) charts can be applied to the residuals. Use of a residual chart has the advantage that it can be applied to any autocorrelated data even if the data are from nonstationary processes. However, the residual charts do not have the same properties as the traditional charts. In addition, time series modeling is often awkward in the SPC environment. Although automatic-modeling algorithms can be used to obtain the process residuals, this approach often requires much effort in practice. The user of a residual chart must check the validity of the model over time to reduce the mixed effects of modeling error and process change. In this article, I propose a new SPC chart, the EWMAST chart, for stationary process data. The chart is constructed by charting the traditional EWMA statistic. The control limits are based on the approximate standard deviation of the EWMA, which has been analytically derived. This EWMAST chart does not require any time series modeling effort. I compare the EWMAST chart with the residual X chart and other charts via the average run length. Simulation study shows that the EWMAST chart performs better than the residual X chart, X chart, and other charts when the process autocorrelation is not very positively strong and the mean shifts are small to medium. In the accompanying figure, ARL of EWMAST chart with parameter 0.2 and the residual chart for AR(1) processes with phi = 0.25, 0.5, 0.75, and 0.95 are plotted when the mean shifts are 0, 0.5, 1, 2, and 3 in the unit of process standard deviation. Solid lines with circles indicate the ARL of EWMAST charts while dot lines with asterisks indicate the ARL of the residual charts. The ARL are on a logarithmic scale with base = 10. For a very wide range of autocorrelated data including uncorrelated data as a special case, I recommend to use the EWMAST chart with 3-sigma control limits and parameter 0.2 to monitor the process mean. This paper is published in Technometrics (1998), 40, 24-38.
Figure 31: This figure shows the ARLs of the EWMAST and residual charts applied to AR(1) processes.
Date created: 7/20/2001 |