The Parameter Selection and Average Run Length Computation
for EWMA Control Charts
Sheng Shu Cheng
1
, Fong-Jung Yu
2
, Shih-Ting Yang
3
and Jiang-Liang Hou
4
1
Department of Business Administration, Yu-Da University of Science and Technology, Miaoli, Taiwan
2
Department of Industrial Engineering and Management, Da-Yeh University, Changhua, Taiwan
3
Department of Information Management, Nanhua University, Chiayi, Taiwan
4
Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan
Keywords: Statistical Process Control, Exponentially Weighted Moving Average, Smoothing Parameter Selection,
Determination of the Control Limits.
Abstract: In the Statistical Process Control (SPC) field, an Exponentially Weighted Moving Average for Stationary
processes (EWMAST) chart with proper control limits has been proposed to monitor the process mean of a
stationary autocorrelated process. There are two issues of note when using the EWMAST charts. These are
the smoothing parameter selections for the process mean shifts, and the determination of the control limits
to meet the required average run length (ARL). In this paper, a guideline for selecting the smoothing
parameter is discussed. These results can be used to select the optimal smoothing parameter in the
EWMAST chart. Also, a numerical procedure using an integration approach is presented for the ARL
computation with the specified control limits. The proposed approach is easy to implement and provides a
good approximation to the average run length of EWMAST charts.
1 INTRODUCTION
The majority of traditional control charts are based
on an assumption that processed data is statistically
independent; however, this assumption does not hold
in certain production environments. It is well known
that using traditional charts for monitoring
autocorrelated processes usually results in an
unnecessarily high occurrence of false positives. A
common approach to handling autocorrelated data is
to apply traditional control charts on the stream of
residuals after the process data have been fitted to a
time-series model. Several residual control charts
have been proposed in recent years. Such as: Alwan
and Roberts (1998) proposed a special cause chart
(SCC) which uses a time-series model to obtain and
monitor the residuals. Montgomery and Mastrangelo
(1991) provided a procedure of plotting one-step-
ahead EWMA prediction errors on a control chart
(M-M chart). English et al (1991) and Wincek
(1990) suggested Kalman filtering to obtain the
residuals. The Proportional-Integral- Derivative
(PID) chart by Jiang et al (2002) and the Dynamic
2
T
chart by Tsung and Apley (2002) have been
proposed for monitoring of processes with a
feedback controller.
There are some problems in the above control
charts. Zhang (1998) proposed an EWMAST control
chart to monitor the original autocorrelated data. The
EWMAST chart is very similar to the traditional
EWMA chart, except that it is designed to be applied
to the monitoring of stationary, autocorrelated data.
Both the EWMA and EWMAST charts are used for
charting the same statistic, but the EWMAST chart
is used in conjunction with modified control limits
to account for the additional variation within an
autocorrelated process. The major advantages in
using an EWMAST chart are: there is no
requirement for an operator to use time-series
techniques; and this method has a comparable ability
to the residual-based charts for detecting large mean
shifts. Specifically, when we set the smoothing
parameter,
=1, when using an EWMAST chart,
then it is equivalent to the traditional Shewhart chart.
Therefore, the EWMAST chart can be more flexible
than the Shewhart chart as its smoothing parameter
can be set according to the magnitude of the mean
shift in the stationary autocorrelated process.
According to the simulation results in Zhang (1998),
the EWMAST chart is more sensitive than residual-
294
Cheng S., Yu F., Yang S. and Hou J..
The Parameter Selection and Average Run Length Computation for EWMA Control Charts.
DOI: 10.5220/0005146902940299
In Proceedings of the International Conference on Neural Computation Theory and Applications (NCTA-2014), pages 294-299
ISBN: 978-989-758-054-3
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
based charts for positive autocorrelated data. For
negative autocorrelated data, the performance of the
EWMAST chart is still superior, but is not as
significant as the positively correlated cases.
Regarding the selection of an optimal
smoothing parameter for the EWMAST chart,
Zhang’s (2000) suggestion was to set the smoothing
parameter
equal 0.2 for most applications, and
provided the ARLs for many autocorrelated
processes. In general, the simulation technique is
rather costly, especially in the case of on-target
analysis. Therefore, it is not a suitable approach
when investigating the ARL performance in
practical applications.
In order to set the parameters of EWMAST charts
more easily, a computing algorithm is presented and
tabulated parameters, which yield the shortest out-
of-control ARL for EWMAST charts, are provided
in this article.
2 DESCRIPTION OF THE
EWMAST CHART
To illustrate the autocorrelated process, we consider
the important case of monitoring the process mean
0
of a stationary AR(1) process and defined as:
ttt
eXX
)(
010
, ,...2,1t (1)
where
t
X represents the process output, 1
is a
constant representing the stationary process, and
}{
t
e is a normally distributed white noise with finite
variance,
2
e
. The EWMAST chart is constructed by
charting the EWMA control statistic under the
stationary process. The chart statistic
t
Z is defined
as:
,...2,1 ,)1(
1
tXZZ
ttt
, (2)
where
00
Z is the on-target process mean, and
is a smoothing parameter within the range
)10(
.
Assume the process
}{
t
X
with stationary variance
2
X
undergoes a single assignable cause that shifts
the process mean to
Xt
0
at time t. To
monitor
}{
t
, a plot of
t
Z is made by selecting
suitable values of the smoothing parameter
and
the width of the control limits
t
Z
L
0
, where
0L . The parameter L serves as a width adjustment
for control limits to meet the required in-control
ARL. Zhang (1998) indicates that the control
statistic
t
Z
has variance:
1
1
)(2
2
22
])1(1[)1)((2
)1(1
2
t
k
ktk
t
x
z
k
t
(3)
where )(k
is the autocorrelation function of }{
t
X
at time lag
k , and )(k
can simplify to
k
for the
AR(1) processes. The limiting value of the variance
of
t
Z in equation (3) as t increases to infinity is
given in (4):
1
22
)1)((21
2
k
k
xz
k
(4)
Hence, the control limits are taken to be:
.
0
0
z
z
LUCL
LLCL
(5)
3 DESIGN OF THE EWMAST
CHART
Constructing an EWMAST chart requires the
specification of
and the constant L. In general, the
choice of parameters (
, L) is based on a
compromise for certain statistic or economic
constraints. It is well known that small values of
are better for detecting small shifts in the mean and
large values of
are better for detecting large
shifts. Although the suggestion for designing the
EWMAST smoothing parameter (
2.0
) by Zhang
(1998) is given in practical terms, there remain
multiple issues that have not been clearly specified.
The
2.0
can only be viewed as a heuristic
suggestion when the magnitude of the shift is
unknown. Another concern is that Zhang’s
suggestion does not consider the in-control ARL.
To select optimal parameters
and L for an
EWMAST chart, an extensive simulation, with
10,000 runs per parameter setting, was implemented.
Thus, the standard deviation of the ARL estimation
error is less than 1% of the actual ARL. For each
simulation run, the
}{
t
e are generated as an
independent sequence of random numbers, based
upon the standard normal distribution utilized in the
IMSL
®
software (1989). The value of }{
t
X is
generated from equation (1). The charting statistic is
calculated via equation (2), with
0
Z initialized at 0
and terminated when
t
Z
exceeds the control limits.
TheParameterSelectionandAverageRunLengthComputationforEWMAControlCharts
295
Table 1: Optimal EWMAST control schemes.
=0.5
=1.5
0
ARL
L
min
ARL
0
ARL
L
min
ARL
0.25 100 0.05 1.815 23.36 0.25 100 0.3 2.388 5.28
370 0.05 2.432 37.99 370 0.2 2.800 7.21
500 0.05 2.555 41.90 500 0.2 2.903 7.68
1000 0.05 2.828 52.42 1000 0.2 3.133 8.83
0.50 100 0.05 1.730 32.40 0.50 100 0.2 2.185 7.42
370 0.05 2.356 58.17 370 0.2 2.714 10.93
500 0.05 2.483 65.72 500 0.1 2.670 11.73
1000 0.05 2.759 87.24 1000 0.1 2.925 13.59
0.75 100 0.05 1.585 49.05 0.75 100 0.2 2.013 11.92
370 0.05 2.230 104.07 370 0.1 2.410 19.14
500 0.05 2.358 121.86 500 0.1 2.532 21.01
1000 0.05 2.641 177.68 1000 0.05 2.641 25.44
0.90 100 0.05 1.372 68.04 0.90 100 1.0 2.185 17.24
370 0.05 2.041 176.47 370 0.05 2.041 35.88
500 0.05 2.175 217.04 500 0.05 2.175 40.07
1000 0.05 2.470 349.61 1000 0.05 2.470 51.46
=2.5
=3.0
0
ARL
L
min
ARL
0
ARL
L
min
ARL
0.25 100 0.8 2.552 2.25 0.25 100 1.00 2.570 1.60
370 0.5 2.954 3.04 370 0.70 2.976 2.12
500 0.5 3.041 3.22 500 0.70 3.071 2.25
1000 0.4 3.230 3.63 1000 0.70 3.284 2.59
0.50 100 1.0 2.538 2.67 0.50 100 1.00 2.538 1.72
370 0.5 2.887 4.13 370 1.00 2.980 2.52
500 0.5 2.985 4.45 500 1.00 3.071 2.76
1000 0.4 3.173 5.18 1000 0.80 3.259 3.41
0.75 100 1.0 2.430 3.30 0.75 100 1.00 2.430 1.95
370 1.0 2.903 5.97 370 1.00 2.903 3.19
500 1.0 3.000 6.79 500 1.00 3.000 3.59
1000 0.3 3.008 8.92 1000 1.00 3.218 4.68
0.90 100 1.0 2.185 4.02 0.90 100 1.00 2.185 2.16
370 1.0 2.711 8.54 370 1.00 2.711 4.18
500 1.0 2.821 10.03 500 1.00 2.821 4.89
1000 1.0 3.064 14.46 1000 1.00 3.064 6.88
The same program was used for an out-of-control
ARL with a mean shift
X
added to }{
t
X at time
t =1. The design of an EWMAST chart consists of
the selection of charting parameters
),( L
that
satisfy certain
,
, and in-control ARL. For most
practical purposes, Table I show the particular
values of
and
L
on the EWMAST chart for
certain
(
0.25,0.5,0.75 and 0.9) to provide the
required in-control ARLs (
0
ARL =100, 370, 500 and
1000).
Table I shows that, as we expected, when
=0.25 and
0
ARL
=100 the optimal
value is 0.3
for detecting a 1.5-
X
shift; and
=0.8 is optimal
NCTA2014-InternationalConferenceonNeuralComputationTheoryandApplications
296
for detecting a 2.5-
X
shift, but in the case of
=0.25 and
0
ARL =500 the optimal
value is 0.2
for detecting a 1.5-
X
shift and
=0.5 for a 2.5-
X
shift. This phenomenon is the same as which
occurs in EWMA charts that have been studied by
Lucas and Saccucci (1990). Another test, with
specified
0
ARL =370 and a three-
X
shift
determined that the optimal value of
is 0.7 for
=0.25, but the optimal value of
is 1.0 for
=0.75. Therefore, the optimal
of an EWMAST
chart may be affected by the different values of
,
the magnitude of the mean shift (
X
) and the in-
control average run length (
0
ARL ).
The optimal
values should be dependent on
the magnitude of the mean shift, the autoregressive
parameter, and the in-control average run length.
This is helpful for the operator to set an appropriate
value of
when an EWMAST chart is chosen to
monitor a stationary process. The following steps are
recommended as guidelines for designing an optimal
EWMAST chart.
Step 1. First, specify the desired in-control
0
ARL ,
the autocorrelation coefficient
, and decide
upon the smallest process mean shift, in
terms of
X
, that must be rapidly
detected..
Step 2. Next, select the optimal parameters
),( L
from Table I.
Step 3. Finally, evaluate the entire ARL performance
for this EWMAST chart to determine
whether the chart provides suitable
protection against other shifts.
4 THE NUMERICAL
PROCEDURE FOR FINDING
THE ARL OF THE EWMAST
CHART
In this section, a numerical procedure is presented
for the investigation of the ARL of EWMAST
charts. Knowledge of the run length is important and
permits us to illustrate the performance of a chart in
terms of average run length. In general, since the run
length of a chart is a nonnegative, random, integer
variable, chart performance can be evaluated by
averaging:
0
)RL()RL(ARL
k
kpkE (6)
In the iid case, a traditional Shewhart chart can be
easily calculated. The
)RL( kp
are evaluated
directly and summed to obtain the ARL. However,
in charting procedures that use recursive charting
statistics, such as EWMA charts and CUSUM
charts, equation (6) is not easily evaluated. In this
situation, equation (6) can be rewritten as:
00
)RL(ARL
k
k
k
Pkp
. (7)
Let the random variable
N
be the time of the
first passage of the process
t
Z
exceeding the
control limits given by equation (5). The probability
of an in-control process at time
k
, given the initial
condition
0
Z ( UCLZLCL
0
), is written as:
,)RL(
1 kkk
rPkpP
(8)
where
0.1
0
P
, and
k
r
is the shrink ratio of
k
P
relative to
1k
P . By using equation (8) to establish a
recursive formulation for the probability of a ruined
problem, the average run lengths can be found by
directly summing the
k
P terms. These computations
demonstrate the behavior of run length distributions
over various AR(1) parameters.
To investigate the behavior of
k
P and
k
r , our
analysis shows that there is a linear relationship
between
k
P and
k
r with increasing k .
The following steps constitute the ARL
computation method for an EWMAST chart applied
to the AR(1) process.
Step 1. Given
and L values, and letting k be the
time index, set
k =1 as the beginning of
the process;
Step 2. Calculate the
2
k
z
using equation (3) for
each k, if
2k , and also calculate
),(Cov
ji
ZZ for kji ,1 using the
equation (A.3) in Zhang’s (1998) studies ;
Step 3. Use Alan’s (1998) algorithm to find the
probability of
)RL( kp ;
Step 4. Find the shrink ratio of
)1RL(
)RL(
kp
kp
r
k
;
Step 5. Collect
k
rrr ,...,,
21
, if }{
k
r is converging
to a constant
p
or
5
10)RL(
kp ; then,
set
Nk
and go to Step 6; otherwise, set
1
kk and go to Step 2;
Step 6. Compute
p
P
iP
N
N
i
1
)RL(ARL
1
0
.
TheParameterSelectionandAverageRunLengthComputationforEWMAControlCharts
297
Table 2: ARLs for the EWMAST chart applied to an AR(1) process with
0
.
(
, L)
shift
r.d.
(1)
a
(2)
b
=[(1)-(2)]/(2)*100%
=0.25, L=2.80
0.0 373.41 379.49 -1.60%
0.5 56.31 56.57 -0.46%
1.0 14.72 14.74 -0.14%
2.0 4.676 4.68 -0.09%
3.0 2.797 2.73 2.45%
=0.50, L=2.72
0.0 378.15 372.43 1.54%
0.5 88.97 86.28 3.12%
1.0 24.29 23.22 4.61%
2.0 6.580 6.33 3.95%
3.0 3.571 3.50 2.03%
=0.75, L=2.59
0.0 390.41 378.90 3.04%
0.5 149.71 146.07 2.49%
1.0 47.40 45.45 4.29%
2.0 11.279 10.80 4.44%
3.0 5.129 5.03 1.97%
=0.90, L=2.43
0.0 423.87 377.46 12.30%
0.5 228.02 201.73 13.03%
1.0 89.14 82.62 7.89%
2.0 20.791 19.26 7.95%
3.0 7.615 7.09 7.40%
a
Computational results.
b
Zhang’s (2000) simulation
results
.
Table 3: ARLs for the EWMAST chart applied to an AR(1) process with 0
.
(
, L)
shift
r.d.
(1)
a
(2)
b
=[(1)-(2)]/(2)*100%
=-0.25, L=2.92
0.0 384.96 374.52 2.79%
0.5 23.31 22.35 4.30%
1.0 6.892 6.72 2.56%
2.0 2.862 2.85 0.42%
3.0 1.935 1.94 -0.26%
=-0.50, L=2.94
0.0 366.15 374.90 -2.33%
0.5 14.056 13.97 0.62%
1.0 4.840 4.64 4.31%
2.0 2.239 2.27 -1.37%
3.0 1.590 1.62 -1.85%
=-0.75, L=2.94
0.0 347.80 375.88 -7.47%
0.5 8.010 8.25 -2.91%
1.0 3.319 3.38 -1.80%
2.0 1.704 1.78 -4.27%
3.0 1.291 1.24 4.11%
=-0.90, L=2.90
c
0.0 376.05 377.17 -0.30%
0.5 5.336 5.37 -0.63%
1.0 2.480 2.53 -1.98%
2.0 1.480 1.45 2.07%
3.0 1.147 1.07 7.20%
a
Computational results.
b
Zhang’s (2000) simulation results.
c
Result from Zhang’s (2000) use of L=2.65.
NCTA2014-InternationalConferenceonNeuralComputationTheoryandApplications
298
5 COMPUTATIONAL RESULTS
Zhang (2000) estimated these ARLs at
0, 0.5, 1,
2 and 3.0 in units of
X
on simulations utilizing at
least 4,000 realizations from the AR(1) processes
with
= 0.25, 0.5, 0.75 and 0.9. In contrast
to the proposed methodology, we do the same
parameter combinations in Zhang’s studies. The
results are listed in Table II for
>0 and in Table III
for
<0.
As indicated in Table II, it is clear that when the
process is positively autocorrelated, the ARLs of our
computational results are in agreement with those
obtained by Zhang’s simulation results. Let the
relative difference (r.d.) represent the difference
between Zhang’s results and those obtained by the
proposed method. Table II also shows that when
mean shifts are small and
is large, the simulation
results deviate more from the computational results.
This phenomenon indicates that due to the inflation
of
2
t
z
, the larger the
, the more simulation runs
are required.
As indicated in Table III, it is clear that when the
process is negatively autocorrelated, the ARLs are
also in agreement with Zhang’s results. Table III
also shows an interesting phenomenon: when
becomes increasingly negative and large, the
EWMAST chart becomes more sensitive. This
property is completely opposite to a positive
autocorrelated process. As for the r.d. index, we can
also observe that the results of a simulation with few
realizations results in an unstable estimate of ARL,
especially in the case of an in-control situation with
highly correlated data.
6 CONCLUSIONS
In this research, the performance of an EWMAST
chart has been investigated for various parameter
settings when the AR(1) process is utilized. These
results demonstrate guidelines for parameter (
, L)
selection when the in-control ARL and the
autogressive parameter are specified. A numerically
analytical expression was also used to evaluate the
ARLs of the EWMAST chart, in the important
special case of an autocorrelated process.
Importantly, this method enables the assessment of
the run-length distribution of an EWMAST chart
using underlying data from an AR(1) process. For an
application of the results, the ARL algorithm can be
extended to calculate run-length distribution and
ARLs for other stationary-process data with
determined parameters. Although these results are
relatively narrow in scope when compared to the
results in Lucas and Saccucci (1990), they are still
helpful to the operator for setting parameters when
using an EWMAST chart. As for the other
requirements of in-control ARLs and the different
values listed in Table I, a larger table covering a
wider range of ARL values is available from the
authors on request.
ACKNOWLEDGEMENTS
The authors wish to express their appreciation for
the financial support by National Sciences Council
of the Republic of China, Grant No NSC 102-2221-
E-212-018, MOST 103-2221-E-343 -002, NSC 101-
2221-E-007-045-MY3 and Da-Yeh University,
Taiwan,
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