3. Compute the control limits with the centerline at MR:
Recall that or = d^R/d>■ and d-> and d3 depend on a.
6.1.4 Exponentially Weighted Moving-Average Chart
The exponentially weighted moving-average (EWMA) Zi is defined as
where 0 < w < 1 is a constant weight, x. is the mean of sample i of size n, and the starting value at i = 1 is zq = i. EWMA attaches a higher weight to more recent data and has a fading memory where old data are discarded from the average. Since the EWMA is a weighted average of several consecutive observations, it is insensitive to nonnormality in the distribution of the data. It is a very useful chart for plotting individual observations (n = 1). If £Ci arc independent random variables with variance a2/n, the variance of 2, is
The last term (in brackets) in Eq. 6.37 quickly approaches 1 as i increases and the variance reaches a limiting value. Often the asymptotic expression for the variance is used for computing the control limits. The weight constant w determines the memory of EWMA, the rate of decay of past sample information. For w = 1, the chart becomes a Shewhart chart. As w —> 0 EWMA approaches a CUSUM. A good value for most cases is in the range 0.2 < w < 0.3. A more appropriate value of w for a specific application can be computed by considering the ARL for detecting a specific magnitude of level shift or by searching w which minimizes the prediction error for a historical data set by an iterative least squares procedure. 50 or more observations should be utilized in such procedures. EWMA is also known as geometric moving average, exponential smoothing, or first order pole filter.
Upper and the lower control limits are calculated as
Example Develop an EWMA chart to detect a shift in the mean by using the first column of the example data set in Figure 6.6 and w = 0.25. Compute the variance of z by using asymptotic version of Eq. 6.37 and the values of z% from Eq. 6.36. The resulting charts where the first twenty samples are used to develop the charts are shown in Figure 6.7. From the EWMA control chart (Figure 6.7) signal for observation 23, we conclude that the process is out of control at that point.
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