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3.4.6 Statistical Control Charts for Autocorrelated Data

Nien Fan Zhang
Statistical Engineering Division, ITL In recent years, 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 charts to the residuals from a time series model of the process. Residual X chart, which applies a X chart to the process residuals from time series models was proposed in 1988. Later some authors proposed to use a CUSUM chart or an EWMA chart 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. The residual charts, however, do not have the same properties as the traditional charts. Zhang (1997) showed that sometimes the detection capability of a residual X chart was poor for small shifts in the process mean compared to the detection capability of the X chart and EWMA chart.

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. From Zhang (1998), the approximate variance of the EWMA statistic, $\sigma_z$, when the process is a stationary process is obtained in terms of $\lambda$, $\sigma_x$, and the process autocorrelation. The centerline of the EWMAST chart is at $\mu$ and the L$\sigma$ limits are

\begin{displaymath}\mu\pm L\sigma_z.\end{displaymath}

L usually equals two or three. For residual charts, the residual et from a time series model is defined as

\begin{displaymath}e_t = X_t - \hat{x_t},\end{displaymath}

where Xt and $\hat{x_t}$ are the observation and prediction of the process from the time series model at t. Various residual charts are constructed based on et depending on the traditional charts used. For a residual X chart, the chart is constructed by charting et. The centerline of the chart is at 0 and the control limits are

\begin{displaymath}\pm L \sigma_e,\end{displaymath}

where $\sigma_e$ is the standard deviation of the process residuals. The residual CUSUM chart and residual EWMA chart are constructed by applying CUSUM and EWMA charts respectively on et.

In Zhang (1998), comparisons have been made among EWMAST chart, residual X chart, and X chart. The comparisons were based on the average run length (ARL). The EWMAST chart performs better than the residual X chart and the X chart when the process autocorrelation is not very positively strong and the mean shifts are small to medium. In this article, I compare EWMAST chart with residual CUSUM and residual EWMA charts as well as the residual chart and X chart. For AR(1) processes with positive parameters, $\phi$, the ARL's for the EWMAST chart with $\lambda$ = 0.2 and the residual EWMA 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,$\sigma_x$. For EWMAST chart, X chart, and EWMA residual chart the control limits L$\sigma$ are adjusted to have the in-control ARL close to 370, which corresponds to the in-control ARL of 3-sigma X chart applied to i.i.d. process data with a normal distribution. At least two thousand realizations of each of the AR(1) processes with zero mean and normally distributed white noise were generated.

From the simulation results, it is clear that when a process is weakly autocorrelated such as for an AR(1) process with $\phi$ = 0.25, the EWMAST chart, residual EWMA chart, and residual CUSUM chart perform equally well. When $\phi$ = 0.5, the EWMAST and residual CUSUM charts perform better than the other charts when the mean shift are small to medium. When $\phi$ = 0.75, the EWMAST chart performs much better than the residual X chart and residual EWMA chart when the mean shifts are not large. In this case, X chart performs better than the other charts except the EWMAST chart when the mean shifts are medium to large. When a process is strongly positively autocorrelated, such as an AR(1) process with $\phi$ = 0.9, residual X chart performs better than other charts when the mean shifts are medium and large. In this case, EWMAST chart, however, still perform very well. It performs better than other charts when the mean shifts are small to medium and performs well when the mean shifts are large.

For the AR(1) processes with negative parameters, simulations also have been done for $\phi$ = -0.25, -0.5, -0.75 and -0.9, to compare the EWMAST chart with residual X and residual EWMA charts as well as the X chart. The results show that for all these values of $\phi$, the performance of the X chart is the worst among those charts considered for small to medium mean shifts. For large mean shifts, all charts perform well. For a small mean shift such as 0.5, the EWMAST and residual EWMA charts perform best. Overall, EWMAST chart performs better than the other charts.

The comparisons show that the EWMAST chart performs better than the residual CUSUM and residual EWMA charts. Overall, it also better than the X chart and residual X chart. An obvious advantage of using EWMAST chart is that there is no need to build a time series model and therefore it is easy to implement. Thus, I recommend using the EWMAST chart with 3$\sigma$ control limits and $\lambda$ = 0.2 to monitor the process mean for a very wide range of autocorrelated data.


Zhang, N. F. (1997). "Detection Capability of Residual Chart for Autocorrelated Data," Journal of Applied Statistics, 24, 475-492.

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

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