Chapter Contents |
Previous |
Next |
EWMACHART Statement |
The following notation is used in this section:
E_{i} | exponentially weighted moving average for the i^{ th} subgroup |
r | EWMA weight parameter |
process mean (expected value of the population of measurements) | |
process standard deviation (standard deviation of the population of measurements) | |
x_{ij} | j^{ th} measurement in i^{ th} subgroup, with j =1, 2, 3, ..., n_{i} |
n_{i} | sample size of i^{ th} subgroup |
mean of measurements in i^{ th} subgroup. If n_{i}=1, then the subgroup mean reduces to the single observation in the subgroup | |
weighted average of subgroup means | |
inverse standard normal function |
The preceding equation can be rewritten as
The EWMA for the i^{ th} subgroup can also be written as
The following table presents the formulas for the limits:
Table 20.19: Limits for an EWMA ChartControl Limits |
LCL = lower limit = |
UCL = upper limit = |
Probability Limits |
LCL = lower limit = |
UCL = upper limit = |
These formulas assume that the data are normally distributed. If standard values and are available for and , respectively, replace with and with in Table 20.19. Note that the limits vary with both n_{i} and i.
If the subgroup sample sizes are constant (n_{i}=n), the
formulas
for the control limits
simplify to
Consequently, when the subgroup sample sizes are constant, the width of the control limits increases monotonically with i. For probability limits, replace k with in the previous equations. Refer to Roberts (1959) and Montgomery (1996).
As i becomes large, the upper and lower control limits
approach constant values:
Some authors base the control limits for EWMA charts on the asymptotic expressions in the two previous equations. For asymptotic probability limits, replace k with in these equations. You can display asymptotic limits by specifying the ASYMPTOTIC option.
Uniformly weighted moving average charts and exponentially weighted
moving average charts have similar properties, and their asymptotic
control limits are identical provided that
You can specify parameters for the EWMA limits as follows:
Average run lengths for two-sided EWMA charts are shown in Table 20.20, which is patterned after Table 1 of Crowder (1987a,b). The ARLs were computed using the EWMAARL DATA step function (see "EWMAARL Function" for details on the EWMAARL function). Note that Crowder (1987a,b) uses the notation L in place of k and the notation in place of r.
You can use Table 20.20 to find a combination of k and r that
yields a desired ARL for an in-control process () and for a
specified shift of . Note that is assumed to be
standardized; in other words, if a shift of is to be detected
in the process mean , and if is the process standard
deviation, you should select the table entry with
For example, suppose you want to construct an EWMA scheme with an in-control ARL of 90 and an ARL of 9 for detecting a shift of . Table 20.20 shows that the combination r=0.5 and k=2.5 yields an in-control ARL of 91.17 and an ARL of 8.27 for .
Crowder (1987a,b) cautions that setting the in-control ARL at a desired level does not guarantee that the probability of an early false signal is acceptable. For further details concerning the distribution of the ARL, refer to Crowder (1987a,b).
In addition to using Table 20.20 or the EWMAARL DATA step function to choose a EWMA scheme with desired average run length properties, you can use them to evaluate an existing EWMA scheme. For example, the "Getting Started" section of this chapter contains EWMA schemes with r=0.3 and k=3. The following statements use the EWMAARL function to compute the in-control ARL and the ARLs for shifts of and :
data arlewma; arlin = ewmaarl( 0,0.3,3.0); arl1 = ewmaarl(.25,0.3,3.0); arl2 = ewmaarl(.50,0.3,3.0); run;
The in-control ARL is 465.553, the ARL for is 178.741, and the ARL for is 53.1603. See Example 20.5 for an illustration of how to use the EWMAARL function to compute average run lengths for various EWMA schemes and shifts.
Table 20.20: Average Run Lengths for Two-Sided EWMA Charts
r (weight parameter) | |||||||
k | 0.05 | 0.10 | 0.25 | 0.50 | 0.75 | 1.00 | |
2.0 | 0.00 | 127.53 | 73.28 | 38.56 | 26.45 | 22.88 | 21.98 |
2.0 | 0.25 | 43.94 | 34.49 | 24.83 | 20.12 | 18.86 | 19.13 |
2.0 | 0.50 | 18.97 | 15.53 | 12.74 | 11.89 | 12.34 | 13.70 |
2.0 | 0.75 | 11.64 | 9.36 | 7.62 | 7.29 | 7.86 | 9.21 |
2.0 | 1.00 | 8.38 | 6.62 | 5.24 | 4.91 | 5.26 | 6.25 |
2.0 | 1.25 | 6.56 | 5.13 | 3.96 | 3.59 | 3.76 | 4.40 |
2.0 | 1.50 | 5.41 | 4.20 | 3.19 | 2.80 | 2.84 | 3.24 |
2.0 | 1.75 | 4.62 | 3.57 | 2.68 | 2.29 | 2.26 | 2.49 |
2.0 | 2.00 | 4.04 | 3.12 | 2.32 | 1.95 | 1.88 | 2.00 |
2.0 | 2.25 | 3.61 | 2.78 | 2.06 | 1.70 | 1.61 | 1.67 |
2.0 | 2.50 | 3.26 | 2.52 | 1.85 | 1.51 | 1.42 | 1.45 |
2.0 | 2.75 | 2.99 | 2.32 | 1.69 | 1.37 | 1.29 | 1.29 |
2.0 | 3.00 | 2.76 | 2.16 | 1.55 | 1.26 | 1.19 | 1.19 |
2.0 | 3.25 | 2.56 | 2.03 | 1.43 | 1.18 | 1.13 | 1.12 |
2.0 | 3.50 | 2.39 | 1.93 | 1.32 | 1.12 | 1.08 | 1.07 |
2.0 | 3.75 | 2.26 | 1.83 | 1.24 | 1.08 | 1.05 | 1.04 |
2.0 | 4.00 | 2.15 | 1.73 | 1.17 | 1.05 | 1.03 | 1.02 |
2.5 | 0.00 | 379.09 | 223.35 | 124.18 | 91.17 | 82.49 | 80.52 |
2.5 | 0.25 | 73.98 | 66.59 | 59.66 | 58.33 | 61.07 | 65.77 |
2.5 | 0.50 | 26.63 | 23.63 | 23.28 | 27.16 | 33.26 | 41.49 |
2.5 | 0.75 | 15.41 | 12.95 | 11.96 | 13.96 | 18.05 | 24.61 |
2.5 | 1.00 | 10.79 | 8.75 | 7.52 | 8.27 | 10.57 | 14.92 |
2.5 | 1.25 | 8.31 | 6.60 | 5.39 | 5.52 | 6.75 | 9.46 |
2.5 | 1.50 | 6.78 | 5.31 | 4.18 | 4.03 | 4.65 | 6.30 |
2.5 | 1.75 | 5.75 | 4.46 | 3.43 | 3.14 | 3.43 | 4.41 |
2.5 | 2.00 | 5.00 | 3.86 | 2.92 | 2.57 | 2.67 | 3.24 |
2.5 | 2.25 | 4.43 | 3.42 | 2.56 | 2.18 | 2.17 | 2.49 |
2.5 | 2.50 | 4.00 | 3.07 | 2.29 | 1.90 | 1.83 | 2.00 |
2.5 | 2.75 | 3.64 | 2.80 | 2.08 | 1.69 | 1.59 | 1.67 |
2.5 | 3.00 | 3.36 | 2.57 | 1.91 | 1.52 | 1.41 | 1.45 |
2.5 | 3.25 | 3.12 | 2.39 | 1.77 | 1.39 | 1.29 | 1.29 |
2.5 | 3.50 | 2.92 | 2.24 | 1.64 | 1.28 | 1.19 | 1.19 |
2.5 | 3.75 | 2.74 | 2.13 | 1.52 | 1.20 | 1.13 | 1.12 |
2.5 | 4.00 | 2.58 | 2.04 | 1.42 | 1.13 | 1.08 | 1.07 |
3.0 | 0.00 | 1383.62 | 842.15 | 502.90 | 397.46 | 374.50 | 370.40 |
3.0 | 0.25 | 133.61 | 144.74 | 171.09 | 208.54 | 245.76 | 281.15 |
3.0 | 0.50 | 37.33 | 37.41 | 48.45 | 75.35 | 110.95 | 155.22 |
3.0 | 0.75 | 19.95 | 17.90 | 20.16 | 31.46 | 50.92 | 81.22 |
3.0 | 1.00 | 13.52 | 11.38 | 11.15 | 15.74 | 25.64 | 43.89 |
3.0 | 1.25 | 10.24 | 8.32 | 7.39 | 9.21 | 14.26 | 24.96 |
3.0 | 1.50 | 8.26 | 6.57 | 5.47 | 6.11 | 8.72 | 14.97 |
3.0 | 1.75 | 6.94 | 5.45 | 4.34 | 4.45 | 5.80 | 9.47 |
3.0 | 2.00 | 6.00 | 4.67 | 3.62 | 3.47 | 4.15 | 6.30 |
3.0 | 2.25 | 5.30 | 4.10 | 3.11 | 2.84 | 3.16 | 4.41 |
3.0 | 2.50 | 4.76 | 3.67 | 2.75 | 2.41 | 2.52 | 3.24 |
3.0 | 2.75 | 4.32 | 3.32 | 2.47 | 2.10 | 2.09 | 2.49 |
3.0 | 3.00 | 3.97 | 3.05 | 2.26 | 1.87 | 1.79 | 2.00 |
3.0 | 3.25 | 3.67 | 2.82 | 2.09 | 1.69 | 1.57 | 1.67 |
3.0 | 3.50 | 3.42 | 2.62 | 1.95 | 1.53 | 1.41 | 1.45 |
3.0 | 3.75 | 3.22 | 2.45 | 1.84 | 1.41 | 1.29 | 1.29 |
3.0 | 4.00 | 3.04 | 2.30 | 1.73 | 1.31 | 1.20 | 1.19 |
3.5 | 0.00 | 12851.0 | 4106.4 | 2640.16 | 2227.34 | 2157.99 | 2149.34 |
3.5 | 0.25 | 281.09 | 381.29 | 625.78 | 951.18 | 1245.90 | 1502.76 |
3.5 | 0.50 | 53.58 | 64.72 | 123.43 | 267.36 | 468.68 | 723.81 |
3.5 | 0.75 | 25.62 | 25.33 | 38.68 | 88.70 | 182.12 | 334.40 |
3.5 | 1.00 | 16.65 | 14.79 | 17.71 | 35.97 | 78.05 | 160.95 |
3.5 | 1.25 | 12.36 | 10.37 | 10.48 | 17.64 | 37.15 | 81.80 |
3.5 | 1.50 | 9.86 | 8.00 | 7.25 | 10.19 | 19.63 | 43.96 |
3.5 | 1.75 | 8.22 | 6.54 | 5.52 | 6.70 | 11.46 | 24.96 |
3.5 | 2.00 | 7.07 | 5.55 | 4.47 | 4.86 | 7.33 | 14.97 |
3.5 | 2.25 | 6.21 | 4.83 | 3.77 | 3.78 | 5.08 | 9.47 |
3.5 | 2.50 | 5.55 | 4.29 | 3.28 | 3.10 | 3.76 | 6.30 |
3.5 | 2.75 | 5.03 | 3.87 | 2.91 | 2.63 | 2.94 | 4.41 |
3.5 | 3.00 | 4.60 | 3.54 | 2.63 | 2.30 | 2.40 | 3.24 |
3.5 | 3.25 | 4.25 | 3.26 | 2.41 | 2.05 | 2.03 | 2.49 |
3.5 | 3.50 | 3.95 | 3.03 | 2.23 | 1.85 | 1.76 | 2.00 |
3.5 | 3.75 | 3.70 | 2.84 | 2.10 | 1.69 | 1.56 | 1.67 |
3.5 | 4.00 | 3.47 | 2.66 | 1.99 | 1.55 | 1.40 | 1.45 |
Chapter Contents |
Previous |
Next |
Top |
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.