3.4.4 Statistical Control Charts for Monitoring the Mean of A Stationary Process

Nien Fan Zhang
Statistical Engineering Division, ITL

Statistical process control (SPC) methodologies have been developed to accommodate autocorrelated data. A common approach to detect a possible process mean shift is to use residual control charts, which are built by applying traditional SPC such as X chart, CUSUM and EWMA charts to the residuals from a time series model of the process. However, the detection capability of a X residual chart was poor for small shifts in the process mean.

Zhang (1998) proposed the EWMAST chart, which is constructed by charting the EWMA statistic for stationary processes to monitor the process mean. EWMAST charts apply to general stationary process data. The chart is simple to implement, and no time series modeling effort is required. In Zhang (1998), comparisons have been made among EWMAST chart, X residual chart, and X chart. The comparisons were based on the average run length (ARL). The EWMAST chart performs better than the X residual chart and the X chart when the process autocorrelation is not very positively strong and the mean shifts are small to medium. In Zhang (1999), the EWMAST chart is compared with CUSUM residual and EWMA residual charts as well as the X residual chart and X chart. For AR(1) processes with various parameters, $\phi$, the ARL's for the EWMAST chart with $\lambda$ = 0.2 and the EWMA residual chart with $\lambda$ = 0.2 are obtained from simulations for step mean shifts of 0, 0.5, 1, 2, and 3 in the unit of the process standard deviation. For the EWMAST chart, X chart, CUSUM residual and EWMA residual chart the control limits L$\sigma$ are adjusted to have the in-control ARL close to 370. The attached figure shows the ARL comparisons among the EWMAST chart, X residual, CUSUM residual, and EWMA residual charts for $\phi$ =0.5. The ARL's are on a logrithmic scale with base=10. The desired control chart should have large in-control (when a mean shift = 0) ARL and small out-of-control (when a mean shift > 0) ARL. From the figure, it is obvious that the EWMAST chart performs better than other charts.

The ARL comparisons based on the simulations show that the EWMAST chart performs better than the CUSUM residual and EWMA residual charts. Overall, it is also better than the X chart and X residual chart. The comparisons also show that the CUSUM residual and EWMA residual charts perform almost the same. The CUSUM residual and EWMA residual residual charts perform better than the X residual chart when the process autocorrelation is not positively strong. On the contrary, when the autocorrelation is strong, the X residual chart performs better than the other two residual charts. The X chart is better than the X residual chart when the process is positively autocorrelated. Thus, the EWMAST chart with 3$\sigma$ control limits and $\lambda$ = 0.2 is recommended to monitor the process mean for a very wide range of autocorrelated data.

References

Zhang, N. F. (1998). "A Statistical Control Chart for Stationary Process Data," Technometrics, 40, 24-38.

Zhang, N. F. (1999). "Statistical Control charts for Monitoring the Mean of A Stationary Process," submitted for publication.

Figure 19: This figure shows the ARL comparisons among the EWMAST chart, the X chart, X residual, CUSUM residual, and EWMA residual charts for the AR(1) process with $\phi$=0.5

       

Date created: 7/20/2001
Last updated: 7/20/2001
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