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Chapter 36
Exponentially Weighted Moving Average
(EWMA) and
Engineering Process Control (EPC)
Introduction
• Under the Shewhart model for control charting, it is assumed
that the mean is constant. Also, errors are NID(0, 𝜎2). In
many applications this assumption is not true.
• Exponentially weighted moving average (EWMA) techniques
offer an alternative based on exponential smoothing
(sometimes called geometric smoothing).
• The computation of EWMA as a filter is done by taking the
weighted average of past observations with progressively
smaller weights over time.
• EWMA has flexibility of computation through the selection of
a weight factor and can use this factor to achieve balance
between older data and more recent observations.
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Introduction
• EWMA techniques can be combined with engineering
process control (EPC) to indicate when a process should be
adjusted.
• Application examples for EWMA with EPC include the
monitoring of pans produced by a tool that wears and needs
periodic sharpening, adjustment, or replacement.
• Much of the discussion in this chapter is a summary of the
discussion of Hunter (1995, 1996).
36.1 S4/IEE Application Examples:
Three-way Control Chart
• Manufacturing 30,000-foot-level metric: An S4/IEE project was to
decrease the within- and between-part thickness
capability/performance of manufactured titanium metal sheets. A
KPIV to this process is the acid concentration of a chemical-
etching step (pickling), which removes metal from the sheet. As
part of the control phase of the project, an EWMA with EPC
procedure was established that would identify when additional acid
should be added to the etching tank.
• Manufacturing 30,000-foot-level metric: An S4/IEE project was to
decrease the within- and between-part variability of the diameter of
a metal part, which was ground to dimension. As part of the
control phase of the project, an EWMA with EPC procedure was
established that would identify when adjustments should be made
to the grinding wheel because of wear.
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36.2 Description
• Consider a sequence of observations 𝑌1, 𝑌2, 𝑌3, . . . , 𝑌𝑡. We
could examine these data using any of the following
procedures, with the differences noted:
• Shewhart--no weighting of previous data
• CUSUM--equal weights for previous data
• Moving average—weight, for example, the five most
recent responses equally as an average
• EWMA—weight the most recent reading the highest and
decrease weights exponentially for previous readings
36.2 Description
• A Shewhart, CUSUM, or moving average control chart for
these variables data would all be based on the model
𝑌𝑡 = 𝜂 + 𝑚𝑡
where the expected value of the observations 𝐸(𝑌𝑡) is a
constant 𝜂 and 𝑚𝑡, is NID(0, 𝜎𝑚2).
• For the Shewhart model the mean and variance are both
constant, with independent errors.
• Also with the Shewhart model, the forecast for the next
observation or average of observations is the centerline of
the chart (𝜂0).
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36.2 Description
• An EWMA is retrospective when plotted under Shewhart
model conditions. It smoothes the time trace, thereby
reducing the role of noise which can offer insight into what
the level of the process might have been, which can be
helpful when identifying special causes.
• Mathematically, for 0 < 𝜆 < 1 this can be expressed as
𝐸𝑊𝑀𝐴 = 𝑌 𝑡 = 𝜆𝑌𝑡 + 𝜃𝑌 𝑡−1 𝑤ℎ𝑒𝑟𝑒 𝜃 = 1 − 𝜆
• At time 𝑡 the smoothed value of the response equals the
multiple of lambda times today's observation plus theta
times yesterday’s smoothed value.
36.2 Description
• A more typical plotting expression for this relationship is
𝐸𝑊𝑀𝐴 = 𝑌 𝑡+1 = 𝑌 𝑡 + 𝜆𝑒𝑡 𝑤ℎ𝑒𝑟𝑒 𝑒𝑡 = 𝑌𝑡 − 𝑌 𝑡
• The predicted value for tomorrow equals the predicted value
of today plus a “depth of memory parameter“ (lambda) times
the difference between the observation and the current day's
prediction.
• For plotting convenience, EWMA is often put one unit ahead
of 𝑌𝑡.
• Under certain conditions, as described later, EWMA can be
used as a forecast.
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36.2 Description
• The three sigma limits for an EWMA control chart are
±3𝜎𝐸𝑊𝑀𝐴 = 𝜆/(2 − 𝜆)[±3𝜎𝑆ℎ𝑒𝑤ℎ𝑎𝑟𝑡]
• When there are independent events, an EWMA chart with 𝜆
= 0.4 yields results almost identical to the combination of
Western Electric rules, where the control limits are exactly
half of those from a Shewhart chart (Hunter 1989b).
• The underlying assumptions for a Shewhart model are often
not true in reality.
• An EWMA can be used to model processes that have linear
or low-order time trends, cyclic behavior, and a response
that is a function of an external factor, nonconstant variance,
and autocorrelated patterns.
36.3 Example 36.1:
EWMA with Eng. Process Control
• The data (Wheeler l995b; Hunter 1995) are
the bearing diameters of 50 camshafts
collected over time.
• A traditional 𝑋𝑚𝑅 chart indicates that there
are many out-of-control conditions.
• However, this example illustrates how
underlying assumptions for application of
the 𝑋𝑚𝑅 chart to this data set are probably
violated.
• EWMA and EPC alternatives are then
applied.
Dia.
1 50
2 51
3 50.5
4 49
5 50
6 43
7 42
8 45
9 47
10 49
11 46
12 50
13 52
14 52.5
15 51
16 52
17 50
Dia.
18 49
19 54
20 51
21 52
22 46
23 42
24 43
25 45
26 46
27 42
28 44
29 43
30 46
31 42
32 43
33 42
34 45
Dia.
35 49
36 50
37 51
38 52
39 54
40 51
41 49
42 50
43 49.5
44 51
45 50
46 52
47 50
48 48
49 49.5
50 49
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36.3 Example 36.1:
EWMA with Eng. Process Control
36.3 Example 36.1:
EWMA with Eng. Process Control
• First we will check for non-independence. To do this we will
use time series analysis techniques. If data meander, each
observation tends to be close to the previous observation and
there is no correlation between successive observations. That
is, there is no autocorrelation (i.e., correlation with itself).
• If observations are independent of time, their autocorrelation
should equal zero. A test for autocorrelation involves
regressing the current value on previous values of the time
series to determine if there is correlation.
• The term lag quantifies how far back comparisons are made.
• Independence of data across time can be checked by
estimation of the lag autocorrelation coefficients 𝑝𝑘, where k =
1, 2, . . . .
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36.3 Example 36.1:
EWMA with Eng. Process Control
Dia. Lag1 Lag2 Lag3 Lag4 Lag5 Lag6 Lag7 Lag8 Lag9 Lag10
1 50
2 51 50
3 50.5 51 50
4 49 50.5 51 50
5 50 49 50.5 51 50
6 43 50 49 50.5 51 50
7 42 43 50 49 50.5 51 50
8 45 42 43 50 49 50.5 51 50
9 47 45 42 43 50 49 50.5 51 50
10 49 47 45 42 43 50 49 50.5 51 50
11 46 49 47 45 42 43 50 49 50.5 51 50
12 50 46 49 47 45 42 43 50 49 50.5 51
13 52 50 46 49 47 45 42 43 50 49 50.5
14 52.5 52 50 46 49 47 45 42 43 50 49
15 51 52.5 52 50 46 49 47 45 42 43 50
16 52 51 52.5 52 50 46 49 47 45 42 43
17 50 52 51 52.5 52 50 46 49 47 45 42
18 49 50 52 51 52.5 52 50 46 49 47 45
19 54 49 50 52 51 52.5 52 50 46 49 47
20 51 54 49 50 52 51 52.5 52 50 46 49
21 52 51 54 49 50 52 51 52.5 52 50 46
22 46 52 51 54 49 50 52 51 52.5 52 50
23 42 46 52 51 54 49 50 52 51 52.5 52
24 43 42 46 52 51 54 49 50 52 51 52.5
25 45 43 42 46 52 51 54 49 50 52 51
26 46 45 43 42 46 52 51 54 49 50 52
27 42 46 45 43 42 46 52 51 54 49 50
28 44 42 46 45 43 42 46 52 51 54 49
29 43 44 42 46 45 43 42 46 52 51 54
30 46 43 44 42 46 45 43 42 46 52 51
31 42 46 43 44 42 46 45 43 42 46 52
32 43 42 46 43 44 42 46 45 43 42 46
33 42 43 42 46 43 44 42 46 45 43 42
34 45 42 43 42 46 43 44 42 46 45 43
35 49 45 42 43 42 46 43 44 42 46 45
36 50 49 45 42 43 42 46 43 44 42 46
37 51 50 49 45 42 43 42 46 43 44 42
38 52 51 50 49 45 42 43 42 46 43 44
39 54 52 51 50 49 45 42 43 42 46 43
40 51 54 52 51 50 49 45 42 43 42 46
41 49 51 54 52 51 50 49 45 42 43 42
42 50 49 51 54 52 51 50 49 45 42 43
43 49.5 50 49 51 54 52 51 50 49 45 42
44 51 49.5 50 49 51 54 52 51 50 49 45
45 50 51 49.5 50 49 51 54 52 51 50 49
46 52 50 51 49.5 50 49 51 54 52 51 50
47 50 52 50 51 49.5 50 49 51 54 52 51
48 48 50 52 50 51 49.5 50 49 51 54 52
49 49.5 48 50 52 50 51 49.5 50 49 51 54
50 49 49.5 48 50 52 50 51 49.5 50 49 51
Correl 0.7431 0.54573 0.3426 0.239356 0.131113 -0.02377 -0.09443 -0.22902 -0.22966 -0.34415
36.3 Example 36.1:
EWMA with Eng. Process Control Dia. Lag1 Lag2 Lag3 Lag4 Lag5 Lag6 Lag7 Lag8 Lag9 Lag10
1 50
2 51 50
3 50.5 51 50
4 49 50.5 51 50
5 50 49 50.5 51 50
6 43 50 49 50.5 51 50
7 42 43 50 49 50.5 51 50
8 45 42 43 50 49 50.5 51 50
9 47 45 42 43 50 49 50.5 51 50
10 49 47 45 42 43 50 49 50.5 51 50
11 46 49 47 45 42 43 50 49 50.5 51 50
12 50 46 49 47 45 42 43 50 49 50.5 51
13 52 50 46 49 47 45 42 43 50 49 50.5
14 52.5 52 50 46 49 47 45 42 43 50 49
15 51 52.5 52 50 46 49 47 45 42 43 50 : : : : : : : : : : : : : : : : : : : : : : : :
45 50 51 49.5 50 49 51 54 52 51 50 49
46 52 50 51 49.5 50 49 51 54 52 51 50
47 50 52 50 51 49.5 50 49 51 54 52 51
48 48 50 52 50 51 49.5 50 49 51 54 52
49 49.5 48 50 52 50 51 49.5 50 49 51 54
50 49 49.5 48 50 52 50 51 49.5 50 49 51
Correl 0.7431 0.5457 0.3426 0.2393 0.1311 -0.0237 -0.0944 -0.2290 -0.2296 -0.3441
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36.3 Example 36.1:
EWMA with Eng. Process Control
• The estimate of the autocorrelation coefficients are:
𝑟1 = 0.74 𝑟6 = −0.02
𝑟2 = 0.55 𝑟7 = −0.09
𝑟3 = 0.34 𝑟8 = −0.23
𝑟4 = 0.24 𝑟9 = −0.23
𝑟5 = 0.13 𝑟10 = −0.34
• For the hypothesis that all 𝑝𝑘 = 0 the approximate standard
error of 𝑟𝑘 is 1/ 𝑛, which leads to an approximate 95%
confidence interval for 𝜌1 of 𝑟1 ± 2/ 𝑛, which results in
0.74 ± 0.28. Because zero is not contained within this interval,
we reject the null hypothesis. The implication of correlation is
that the moving-range statistic does not provide a good
estimate of standard deviation to calculate the control limits.
36.3 Example 36.1:
EWMA with Eng. Process Control
• George Box (Hunter 1995; Box and Luceno 1997) suggests
using a variogram to check adequacy of the assumptions of
constant mean, independence, and constant variance. It
checks the assumption that data derive from a stationary
process. A variogram does this by taking pairs of observations
1, 2, or m apart to produce alternative time series. When the
assumptions are valid, there should be no difference in the
expectation of statistics obtained from these differences.
• For the standardized variogram
𝐺𝑚 =𝑉𝑎𝑟(𝑌𝑡+𝑚 − 𝑌𝑡)
𝑉𝑎𝑟(𝑌𝑡+1 − 𝑌𝑡)
the ratio 𝐺𝑚 equals 1 for all values of 𝑚, if the data have a
constant mean, independence, and constant variance.
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36.3 Example 36.1:
EWMA with Eng. Process Control
• For processes that have an ultimate constant variance (i.e.,
stationary process), 𝐺𝑚 will increase at first but soon become
constant.
• When the level and variance of a process grows without limit
(i.e., nonstationary processes), 𝐺𝑚 will continually increase.
• For processes that increase as a straight line, an EWMA gives
a unique model.
• A simple method to compute 𝐺𝑚 is to use the moving-range
computations for standard deviations.
36.3 Example 36.1:
EWMA with Eng. Process Control
Dia. m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10
1 50
2 51 1
3 50.5 0.5 0.5
4 49 1.5 2 1
5 50 1 0.5 1 0
6 43 7 6 7.5 8 7
7 42 1 8 7 8.5 9 8
8 45 3 2 5 4 5.5 6 5
9 47 2 5 4 3 2 3.5 4 3
10 49 2 4 7 6 1 0 1.5 2 1
11 46 3 1 1 4 3 4 3 4.5 5 4
12 50 4 1 3 5 8 7 0 1 0.5 1
13 52 2 6 3 5 7 10 9 2 3 1.5
14 52.5 0.5 2.5 6.5 3.5 5.5 7.5 10.5 9.5 2.5 3.5
15 51 1.5 1 1 5 2 4 6 9 8 1
16 52 1 0.5 0 2 6 3 5 7 10 9
17 50 2 1 2.5 2 0 4 1 3 5 8
18 49 1 3 2 3.5 3 1 3 0 2 4
19 54 5 4 2 3 1.5 2 4 8 5 7
20 51 3 2 1 1 0 1.5 1 1 5 2
21 52 1 2 3 2 0 1 0.5 0 2 6
22 46 6 5 8 3 4 6 5 6.5 6 4
23 42 4 10 9 12 7 8 10 9 10.5 10
24 43 1 3 9 8 11 6 7 9 8 9.5
25 45 2 3 1 7 6 9 4 5 7 6
26 46 1 3 4 0 6 5 8 3 4 6
27 42 4 3 1 0 4 10 9 12 7 8
28 44 2 2 1 1 2 2 8 7 10 5
29 43 1 1 3 2 0 1 3 9 8 11
30 46 3 2 4 0 1 3 4 0 6 5
31 42 4 1 2 0 4 3 1 0 4 10
32 43 1 3 0 1 1 3 2 0 1 3
33 42 1 0 4 1 2 0 4 3 1 0
34 45 3 2 3 1 2 1 3 1 0 2
35 49 4 7 6 7 3 6 5 7 3 4
36 50 1 5 8 7 8 4 7 6 8 4
37 51 1 2 6 9 8 9 5 8 7 9
38 52 1 2 3 7 10 9 10 6 9 8
39 54 2 3 4 5 9 12 11 12 8 11
40 51 3 1 0 1 2 6 9 8 9 5
41 49 2 5 3 2 1 0 4 7 6 7
42 50 1 1 4 2 1 0 1 5 8 7
43 49.5 0.5 0.5 1.5 4.5 2.5 1.5 0.5 0.5 4.5 7.5
44 51 1.5 1 2 0 3 1 0 1 2 6
45 50 1 0.5 0 1 1 4 2 1 0 1
46 52 2 1 2.5 2 3 1 2 0 1 2
47 50 2 0 1 0.5 0 1 1 4 2 1
48 48 2 4 2 3 1.5 2 1 3 6 4
49 49.5 1.5 0.5 2.5 0.5 1.5 0 0.5 0.5 1.5 4.5
50 49 0.5 1 1 3 1 2 0.5 1 0 2
Mean 2.0816 2.59375 3.2553 3.391304 3.688889 4.045455 4.209302 4.392857 4.792683 5.2375
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36.3 Example 36.1:
EWMA with Eng. Process Control Dia. m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10
1 50
2 51 1
3 50.5 0.5 0.5
4 49 1.5 2 1
5 50 1 0.5 1 0
6 43 7 6 7.5 8 7
7 42 1 8 7 8.5 9 8
8 45 3 2 5 4 5.5 6 5
9 47 2 5 4 3 2 3.5 4 3
10 49 2 4 7 6 1 0 1.5 2 1
11 46 3 1 1 4 3 4 3 4.5 5 4
12 50 4 1 3 5 8 7 0 1 0.5 1
13 52 2 6 3 5 7 10 9 2 3 1.5 : : : : : : : : : : : : : : : : : : : : : : : :
45 50 1 0.5 0 1 1 4 2 1 0 1
46 52 2 1 2.5 2 3 1 2 0 1 2
47 50 2 0 1 0.5 0 1 1 4 2 1
48 48 2 4 2 3 1.5 2 1 3 6 4
49 49.5 1.5 0.5 2.5 0.5 1.5 0 0.5 0.5 1.5 4.5
50 49 0.5 1 1 3 1 2 0.5 1 0 2
Mean 2.0816 2.5937 3.2553 3.3913 3.6888 4.0454 4.2093 4.3928 4.7926 5.2375
36.3 Example 36.1:
EWMA with Eng. Process Control
m 𝑀𝑅 𝑑2 𝜎𝑌𝑚 = 𝑀𝑅/𝑑2 𝜎𝑌𝑚
2 𝐺𝑚 =
𝜎𝑌𝑚2
𝜎𝑌12
1 2.08163 1.128 1.8454 3.40557 1.0000
2 2.59375 1.128 2.2994 5.28735 1.5526
3 3.25532 1.128 2.8859 8.32854 2.4456
4 3.39130 1.128 3.0065 9.03889 2.6541
5 3.68889 1.128 3.2703 10.69481 3.1404
6 4.04545 1.128 3.5864 12.86224 3.7768
7 4.20930 1.128 3.7317 13.92522 4.0890
8 4.39286 1.128 3.8944 15.16617 4.4533
9 4.79268 1.128 4.2488 18.05258 5.3009
10 5.23750 1.128 4.6432 21.55906 6.3305
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36.3 Example 36.1:
EWMA with Eng. Process Control
0.0000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
1 2 3 4 5 6 7 8 9 10
G(m)
36.3 Example 36.1:
EWMA with Eng. Process Control
• The plot of 𝐺𝑚 versus the interval 𝑚 is an increasing straight
line, which suggests that an EWMA model is reasonable.
• An estimate for 𝜆 can be obtained from the slope of the line.
Because the line must pass through 𝐺𝑚 = 1 and 𝑚 = 1, the
slope can be obtained from the equation
𝑏 = 𝑥𝑦
𝑥2= 𝑚− 1 [𝐺𝑚 − 1]
[𝑚 − 1]2
=0 0 + 1 .5526 + ⋯+ 9(5.3305)
02 + 12 +⋯+ 92=155.9408
285= 0.54716
• Use the following relationship to solve for 𝜆, 𝜆 = 0.76
𝑏 =𝜆2
1 + (1 − 𝜆)2
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36.3 Example 36.1:
EWMA with Eng. Process Control
• The limits of the EWMA control chart are determined by the
equation
±3𝜎𝐸𝑊𝑀𝐴 = 𝜆/(2 − 𝜆) ±3𝜎𝑆ℎ𝑒𝑤ℎ𝑎𝑟𝑡
=0.76
2 − 0.7653.74− 48.20 = 0.783 × 5.54 = 4.33
• The resulting control limits are then 52.53 and 43.87
(48.204.33).
36.3 Example 36.1:
EWMA with Eng. Process Control
Minitab:
Stat
Control Charts
Time-Weighted
Charts
EWMA
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36.3 Example 36.1:
EWMA with Eng. Process Control
• The limits of the EWMA control chart are determined by the
equation
𝑌 𝑡+1 = 𝑌 𝑡 + 0.76 𝑌𝑡 − 𝑌 𝑡 = 0.76𝑌𝑡 + 0.24𝑌 𝑡
• Let us now consider employing the fitted EWMA model. If we
let the target value for the camshaft diameters be 𝜏 = 50 and let
the first prediction be 𝑌 1 = 50, the fitted EWMA gives the
prediction values and errors in the following table
36.3 Example 36.1:
EWMA with Eng. Process Control
𝑌𝑡 𝑌 𝑡+1 𝑒𝑡 𝑌𝑡 − 𝜏 𝑌𝑡 −𝑌 𝑒𝑡 2
1 50.0 50.000 0.000 0 3.152159 0.000
2 51.0 50.000 1.000 1 7.703023 1.000
3 50.5 50.760 -0.260 0.25 5.177591 0.068
4 49.0 50.562 -1.562 1 0.601295 2.441
5 50.0 49.375 0.625 0 3.152159 0.391
6 43.0 49.850 -6.850 49 27.29611 46.922
7 42.0 44.644 -2.644 64 38.74525 6.991
8 45.0 42.635 2.365 25 10.39784 5.595
9 47.0 44.432 2.568 9 1.499567 6.593
10 49.0 46.384 2.616 1 0.601295 6.845
11 46.0 48.372 -2.372 16 4.948703 5.627
12 50.0 46.569 3.431 0 3.152159 11.770
13 52.0 49.177 2.823 4 14.25389 7.971
14 52.5 51.322 1.178 6.25 18.27932 1.387
15 51.0 52.217 -1.217 1 7.703023 1.482
16 52.0 51.292 0.708 4 14.25389 0.501
17 50.0 51.830 -1.830 0 3.152159 3.349
18 49.0 50.439 -1.439 1 0.601295 2.071
19 54.0 49.345 4.655 16 33.35562 21.665
20 51.0 52.883 -1.883 1 7.703023 3.545
21 52.0 51.452 0.548 4 14.25389 0.300
22 46.0 51.868 -5.868 16 4.948703 34.439
23 42.0 47.408 -5.408 64 38.74525 29.251
24 43.0 43.298 -0.298 49 27.29611 0.089
25 45.0 43.072 1.928 25 10.39784 3.719
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36.3 Example 36.1:
EWMA with Eng. Process Control
𝑌𝑡 𝑌 𝑡+1 𝑒𝑡 𝑌𝑡 − 𝜏 𝑌𝑡 − 𝑌 𝑒𝑡 2
26 46.0 44.537 1.463 16 4.948703 2.140
27 42.0 45.649 -3.649 64 38.74525 13.315
28 44.0 42.876 1.124 36 17.84697 1.264
29 43.0 43.730 -0.730 49 27.29611 0.533
30 46.0 43.175 2.825 16 4.948703 7.979
31 42.0 45.322 -3.322 64 38.74525 11.036
32 43.0 42.797 0.203 49 27.29611 0.041
33 42.0 42.951 -0.951 64 38.74525 0.905
34 45.0 42.228 2.772 25 10.39784 7.682
35 49.0 44.335 4.665 1 0.601295 21.764
36 50.0 47.880 2.120 0 3.152159 4.493
37 51.0 49.491 1.509 1 7.703023 2.276
38 52.0 50.638 1.362 4 14.25389 1.855
39 54.0 51.673 2.327 16 33.35562 5.414
40 51.0 53.442 -2.442 1 7.703023 5.961
41 49.0 51.586 -2.586 1 0.601295 6.687
42 50.0 49.621 0.379 0 3.152159 0.144
43 49.5 49.909 -0.409 0.25 1.626727 0.167
44 51.0 49.598 1.402 1 7.703023 1.965
45 50.0 50.664 -0.664 0 3.152159 0.440
46 52.0 50.159 1.841 4 14.25389 3.388
47 50.0 51.558 -1.558 0 3.152159 2.428
48 48.0 50.374 -2.374 4 0.050431 5.636
49 49.5 48.570 0.930 0.25 1.626727 0.865
50 49.0 49.277 -0.277 1 0.601295 0.077
Ave/Sum 48.225 775 613.0302 312.4701
36.3 Example 36.1:
EWMA with Eng. Process Control
𝑌 𝑡+1 = 48.225
(𝑌𝑡 − 𝜏)2= 775.00 → 𝑠𝜏 =775
50 − 1= 3.98
(𝑌𝑡 − 𝑌 )2= 613.03 → 𝑠𝑌 =613
50 − 1= 3.54
(𝑌𝑡 − 𝑌 𝑡)2= 312.47 → 𝑠𝜏 =
312.47
50 − 1= 2.53
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36.3 Example 36.1:
EWMA with Eng. Process Control
• From this it seems possible that use of the EWMA as a forecast
to control the process could result in a very large reduction in
variability (i.e., sum of squares from 775.00 to 312.47).
• A comparison of the autocorrelation coefficients of the original
observations with the residuals 𝑒𝑡, after fitting the EWMA model
is as follows:
• Original Observations Y, EWMA Residuals e,
36.3 Example 36.1:
EWMA with Eng. Process Control
• The residuals suggest independence and support our use of
EWMA as a reasonable model providing useful forecast
information of process performance.
• Consider now what can be done to take active control. We must
be willing to accept a forecast for where a process will be in the
next instant of time.
• When a forecast falls too distant from a target 𝜏, an operator
can then change some influential external factor 𝑋𝑡 to force the
forecast to equal target 𝜏. This differs from the Shewhart model
discussed above in that the statistical approach is now not
hypothesis testing but instead estimation.
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36.3 Example 36.1:
EWMA with Eng. Process Control
• The application of process controls from an external factor can
be conducted periodically when the response reaches a certain
level relative to the specification. However, for this example we
will consider that adjustments are made after each reading and
that full consequences of taking corrective action can be
accomplished within the next time interval.
• Table 36.5 summarizes the calculations, which can be
explained as follows. Let us consider that 𝑋𝑡 is the current
setting of a control factor, where 𝑌 𝑡+1 is the forecast.
• Also, we can exactly compensate for a discrepancy of
𝑧𝑡 = 𝑌 𝑡+1 − 𝜏 by making the change 𝑥𝑡 = 𝑋𝑡+1 − 𝑋𝑡. When
bringing a process back to its target, we set 𝑔𝑥𝑡 = −𝑧𝑡, where 𝑔
is the adjuster gain.
36.3 Example 36.1:
EWMA with Eng. Process Control
• The controlling factor is initially set to zero and the first forecast
is 50, the target. The table contains the original observations
and new observations, which differ from original observations
by the amount shown. The difference between the new
observation and the target of 50 is shown as 𝑒(𝑡), while
0.76 𝑒(𝑡) represents the new amount of adjustment needed.
EWMA is determined from the equation 𝑌 𝑡+1 = 0.76𝑌𝑡 + 0.24𝑌 𝑡
• We can see that the 𝑋𝑚𝑅 chart of the new observations shown
in Figure 36.4 is now in control/predictable. The estimated
implications of control to the process are as follows:
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36.3 Example 36.1:
EWMA with Eng. Process Control
𝑌𝑡 𝑌 𝑡+1 𝑒𝑡 𝜆𝑒𝑡 Adj. New
Obs. EWMA
1 50.0 50.000 0.000 0.000 0.000 50.000 50.000
2 51.0 50.000 1.000 0.760 0.000 51.000 50.000
3 50.5 50.760 -0.260 -0.198 -0.760 49.740 50.760
4 49.0 50.562 -1.562 -1.187 -0.562 48.438 49.985
5 50.0 49.375 0.625 0.475 0.625 50.625 48.809
6 43.0 49.850 -6.850 -5.206 0.150 43.150 50.189
7 42.0 44.644 -2.644 -2.009 5.356 47.356 44.839
8 45.0 42.635 2.365 1.798 7.365 52.365 46.752
9 47.0 44.432 2.568 1.951 5.568 52.568 51.018
10 49.0 46.384 2.616 1.988 3.616 52.616 52.196
11 46.0 48.372 -2.372 -1.803 1.628 47.628 52.515
12 50.0 46.569 3.431 2.607 3.431 53.431 48.801
13 52.0 49.177 2.823 2.146 0.823 52.823 52.320
14 52.5 51.322 1.178 0.895 -1.322 51.178 52.702
15 51.0 52.217 -1.217 -0.925 -2.217 48.783 51.544
16 52.0 51.292 0.708 0.538 -1.292 50.708 49.445
17 50.0 51.830 -1.830 -1.391 -1.830 48.170 50.405
18 49.0 50.439 -1.439 -1.094 -0.439 48.561 48.706
19 54.0 49.345 4.655 3.537 0.655 54.655 48.596
20 51.0 52.883 -1.883 -1.431 -2.883 48.117 53.200
21 52.0 51.452 0.548 0.417 -1.452 50.548 49.337
22 46.0 51.868 -5.868 -4.460 -1.868 44.132 50.257
23 42.0 47.408 -5.408 -4.110 2.592 44.592 45.602
24 43.0 43.298 -0.298 -0.226 6.702 49.702 44.834
25 45.0 43.072 1.928 1.466 6.928 51.928 48.534
𝑌𝑡 𝑌 𝑡+1 𝑒𝑡 𝜆𝑒𝑡 Adj. New Obs. EWMA
26 46.0 44.537 1.463 1.112 5.463 51.463 51.114
27 42.0 45.649 -3.649 -2.773 4.351 46.351 51.379
28 44.0 42.876 1.124 0.854 7.124 51.124 47.558
29 43.0 43.730 -0.730 -0.555 6.270 49.270 50.268
30 46.0 43.175 2.825 2.147 6.825 52.825 49.509
31 42.0 45.322 -3.322 -2.525 4.678 46.678 52.029
32 43.0 42.797 0.203 0.154 7.203 50.203 47.962
33 42.0 42.951 -0.951 -0.723 7.049 49.049 49.665
34 45.0 42.228 2.772 2.106 7.772 52.772 49.197
35 49.0 44.335 4.665 3.546 5.665 54.665 51.914
36 50.0 47.880 2.120 1.611 2.120 52.120 54.005
37 51.0 49.491 1.509 1.147 0.509 51.509 52.572
38 52.0 50.638 1.362 1.035 -0.638 51.362 51.764
39 54.0 51.673 2.327 1.768 -1.673 52.327 51.459
40 51.0 53.442 -2.442 -1.856 -3.442 47.558 52.118
41 49.0 51.586 -2.586 -1.965 -1.586 47.414 48.653
42 50.0 49.621 0.379 0.288 0.379 50.379 47.711
43 49.5 49.909 -0.409 -0.311 0.091 49.591 49.739
44 51.0 49.598 1.402 1.065 0.402 51.402 49.627
45 50.0 50.664 -0.664 -0.504 -0.664 49.336 50.976
46 52.0 50.159 1.841 1.399 -0.159 51.841 49.730
47 50.0 51.558 -1.558 -1.184 -1.558 48.442 51.334
48 48.0 50.374 -2.374 -1.804 -0.374 47.626 49.136
49 49.5 48.570 0.930 0.707 1.430 50.930 47.988
50 49.0 49.277 -0.277 -0.210 0.723 49.723 50.224
Avg 48.20 49.975
36.3 Example 36.1:
EWMA with Eng. Process Control
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36.3 Example 36.1:
EWMA with Eng. Process Control
• The estimated implications of control to the process are as
follows:
• It should be noted that for this situation a chart could be created
from the EWMA relationship that describes how an operator in
manufacturing should adjust a machine depending on its
current output.
Mean Std. Deviation
No Control 48.20 3.54
Every observation control 49.98 2.53