Click SigmaXL > Templates & Calculators > Control
Chart Templates > Average Run Length (ARL)
Calculators >EWMA ARL. This template is also
located at SigmaXL > Control Charts > Control
Chart Templates> Average Run Length (ARL)
Calculators >EWMA ARL.
The default template settings are Specify =
Weight (Lambda) & K, Weight (Lambda) = 0.1,
K (Sigma Multiplier) = 2.7, Type of Limits
= Time-Varying, Method = Markov Chain.
Notes: Parameters to be specified will
be shown in yellow highlight, otherwise they are
hidden. The EWMA parameter Weight (Lambda) is a
value between 0 and 1 and controls the amount of
influence that previous observations have on the
current EWMA statistic. A value near 1 puts almost
all weight on the current observation, making it
resemble a Shewhart chart. For values near 0, a
small weight is applied to almost all of the past
observations, so the EWMA chart performance is
similar to that of a CUSUM chart. The EWMA parameter
K (Sigma Multiplier) is a value typically between 2
and 4. It is also referred to as L, but SigmaXL uses
K to avoid confusion with Lambda.
The EWMA control chart template has hard
coded the Type of Limits as Time-Varying, since they
improve the sensitivity of the EWMA to detect early
changes in the process mean. Fixedis included as an
option here for comparison of ARL results to
published papers. Also, if the process is in control
when the EWMA is started but shifts out of control
after the control limits have stabilized, the more
appropriate ARL for such a case would be Type of
Limits = Fixed.
The Markov Chain approximation is fast
and accurate to compute ARLs. Monte Carlo simulation
allows you to assess robustness to nonnormality and
also produces the table of Run Length Standard
Deviation and Percentiles (scroll right to view).
For further details on the Markov Chain
approximation see Lucas [1] for fixed and Steiner
[3] for time-varying. Monte Carlo simulation uses
the Pearson Family of distributions to match the
specified skewness and kurtosis.
The EWMA ARL is for a two-sided chart
with zero-state, i.e., the shift is assumed to occur
at the start. The mean and standard deviation are
also assumed to be known. This will not likely be
the case in use, but is still useful for determining
parameter settings and comparison of ARL across
chart types.
All ARL calculations for EWMA use a
standardized in-control mean=0 and sigma=1.
Click the Calculate EWMA ARL button to
reproduce the ARL table and chart.
The ARL_{0} (in-control ARL with 0 shift in
mean) for the EWMA chart with these settings is
356.1, which is close to the Shewhart ARL_{0} of 370.4.
The ARL_{1} for a small 1 sigma shift in mean is 7.54,
so is much faster to detect than the ARL_{1} of 43.89
for
Shewhart Individuals and faster to detect
than the Monte Carlo ARL_{1} of 9.7 for the
Shewhart Individuals with 8 tests for special causes.
Now we will compare time-varying to fixed limits.
Select Specify = Weight (Lambda) & K. Enter
Weight (Lambda) = 0.1, K (Sigma Multiplier)
= 2.7, Type of Limits = Fixed, Method
= Markov Chain.
Click the Calculate EWMA ARL button to
produce the ARL table and chart for these settings:
The ARL_{0} for the EWMA chart with these
settings is 369.04, which is close to the Shewhart
ARL_{0} of 370.4. The ARL_{1} for a small 1 sigma shift in
mean with fixed limits is 9.73, which is slower than
the ARL_{1} with time-varying limits of 7.54.
Next, we will specify Weight (Lambda) = 0.1, the
Shewhart ARL_{0} value of 370.4 and solve for the K
(Sigma Multiplier). Enter Specify = Weight
(Lambda) & ARL0, Weight (Lambda) = 0.1,
In-Control Average Run Length (ARL0) = 370.4,
Type of Limits = Time-Varying, Method =
Markov Chain.
Click the Calculate EWMA ARL button to
produce the updated EWMA Parameters, ARL table and
chart for these settings:
The ARL_{0} for the EWMA chart with these
settings is 370.4 as specified. The K (Sigma
Multiplier) solved to obtain this ARL_{0} value is
2.7146. The ARL_{1} for a small 1 sigma shift in mean
is 7.62.
Now we will specify Weight (Lambda) = 0.05, the
Shewhart ARL_{0} value of 370.4 and solve for the K
(Sigma Multiplier). Enter Specify = Weight
(Lambda) & ARL0, Weight (Lambda) = 0.05,
In-Control Average Run Length (ARL0) = 370.4,
Type of Limits = Time-Varying, Method =
Markov Chain.
Click the Calculate EWMA ARL button to
produce the updated EWMA Parameters, ARL table and
chart for these settings:
The ARL_{0} for the EWMA chart with these
settings is 370.4 as specified. The K (Sigma
Multiplier) solved to obtain this ARL_{0} value is
2.523. The ARL1 for a small 1 sigma shift in mean is
6.76.
Next, we will specify an ARL_{0} value of 500 with a
desired optimization to detect a 1 sigma shift in
mean. The calculator will solve for the optimal
Weight (Lambda) and K (Sigma Multiplier). Enter
Specify = ARL0 & Shift, In-Control Average Run
Length (ARL0) = 500, Shift in Mean to Detect
(Multiple of Sigma) = 1, Type of Limits =
Time-Varying, Method = Markov Chain.
Click the Calculate EWMA ARL button to
produce the EWMA Parameters, ARL table and chart for
these settings:
The ARL_{0} for the EWMA chart with these
settings is 500.0 as specified. The ARL_{1} for a small
1 sigma shift in mean is 8.66. The solved parameters
are Weight (Lambda) = 0.1336 and K (Sigma
Multiplier) = 2.889.
Note: Weight (Lambda) is first
solved using fixed limits. This value is then used
to solve for K using time-varying limits.
Now we will use Monte Carlo simulation to obtain
approximate Run Length standard deviation and
percentiles. Enter Specify = Weight (Lambda)
& K, Weight (Lambda) = 0.1,
K (Sigma Multiplier) = 2.7, Type of Limits
= Time-Varying, Method = Monte Carlo,
Number of Replications = 1e4, Skewness =
0, Kurtosis (Normal is 0) = 0.
Click the Calculate EWMA ARL button to
produce the Monte Carlo approximate ARL table, ARL
chart and Run Length Standard Deviation and
Percentiles table. Monte Carlo simulation with
10,000 (1e4) replications will take about a minute
to run.
The additional run length statistics show the
large variation of run length values. The MRL_{0} = 238
(in-control median run length with 0 shift in
process mean).
Note: The results will vary
slightly since this is Monte Carlo simulation.
We will now assess robustness to nonnormality using
Monte Carlo simulation with Weight (Lambda) = 0.1
and a specified ARL_{0} of 370.4 for comparison to
Shewhart. Enter Specify = Weight (Lambda)
& ARL0, Weight (Lambda) = 0.1, In-Control
Average Run Length (ARL0) = 370.4, Type of
Limits = Time-Varying, Method = Monte
Carlo, Number of Replications = 1e4,
Skewness = 2, Kurtosis (Normal is 0) = 6.
Note: The EWMA parameter K (Sigma
Multiplier) will be solved using Markov-Chain
approximation and assume a Normal distribution, so
will match the value previously calculated above
(6).
Click the Calculate EWMA ARL button to
produce the updated EWMA Parameters, Monte Carlo
approximate ARL table, ARL chart and Run Length
Standard Deviation and Percentiles table:
ARL_{0} is approximately 266.4 which is a 1.4 x
increase (370.4/266.4) in false alarms compared to
Normal but is a much better performance than the
ARL_{0} = 55 result for
Shewhart Individuals. MRL_{0} is approx. 183.
Next, we will assess robustness to nonnormality
using Monte Carlo simulation with a lower Weight
(Lambda) = 0.05 and a specified ARL_{0} of 370.4. Enter
Specify = Weight (Lambda) & ARL0, Weight
(Lambda) = 0.05, In-Control Average Run
Length (ARL0) = 370.4, Type of Limits =
Time-Varying, Method = Monte Carlo, Number
of Replications = 1e4, Skewness = 2,
Kurtosis (Normal is 0) = 6.
Note: Montgomery [2] (Table 9.12)and
Borror, Montgomery &Runger [4] point out that the
EWMA becomes more robust with lower values of
Lambda.
Click the Calculate EWMA ARL button to
produce the updated EWMA Parameters, Monte Carlo
approximate ARL table, ARL chart and Run Length
Standard Deviation and Percentiles table:
ARL0 is approximately 352 which is close to
the original specified 370.4. The MRL0 is approx.
239. If robustness to non-normality is a concern
then a Weight (Lambda) = 0.05 is recommended.
Template Notes:
Specify EWMA parameters: Weight (Lambda) & K,
Weight (Lambda) &ARL0 or ARL0& Shift using
the drop-down list. Parameters to be specified will be
shown in yellow highlight, otherwise they are hidden.
If applicable, enter the EWMA parameter Weight
(Lambda). This is a value between 0 and 1 and
controls the amount of influence that previous
observations have on the current EWMA statistic. A value
near 1 puts almost all weight on the current
observation, making it resemble a Shewhart chart. For
values near 0, a small weight is applied to almost all
of the past observations, so the EWMA chart performance
is similar to that of a CUSUM chart.
If applicable, enter the EWMA parameter K (Sigma
Multiplier). This is a value typically between 2 and
4. It is also referred to as L, but SigmaXL uses K to
avoid confusion with Lambda.
If applicable, enter the desired In-Control
Average Run Length (ARL0). This will be the target
ARL for mean shift = 0 and should be a large value to
minimize false alarms, typically 370 to 500. The K
(Sigma Multiplier) will be solved to achieve this ARL0,
given a specified Weight (Lambda) value.
If applicable, enter the desired Shift in Mean to
Detect (Multiple of Sigma). The Weight (Lambda)
value that minimizes ARL for the specified shift will be
solved.
Select Type of Limits: Time-Varying or
Fixed using the drop-down list.
The EWMA control chart template uses time-varying
control limits since they improve the sensitivity of the
EWMA to detect early changes in the process mean.
Published ARL tables typically use fixed limits, so
providing both allows comparison between the two types.
Select Method: Markov Chain or Monte Carlo
using the drop-down list. Markov Chain approximation is
fast and accurate to compute ARLs. Monte Carlo
simulation allows you to assess robustness to
non-normality and also produces the table of Run Length
Standard Deviation and Percentiles (scroll right to
view).
For further details on the Markov Chain
approximation see Lucas [1] for fixed and Steiner [3]
for time-varying. Monte Carlo simulation uses the
Pearson Family of distributions to match the specified
skewness and kurtosis.
If applicable, enter Number of Replications.
1000 (1e3) replications will be fast, approx. 10
seconds, but will have an ARL0 error approx. = +/- 10%;
10,000 (1e4) replications will take about a minute, with
an ARL0 error = +/- 3.2%; 100,000 (1e5) replications
will take about ten minutes, with an ARL0 error = +/-
1%.
If applicable, enter Skewness. Skewness must
be >= 0. Skewness = 0 is symmetric.
If applicable, enter Kurtosis (Normal is 0).
Also known as Excess Kurtosis, it must be >= Skewness^2
- 1.48. This is required to keep the distribution
unimodal. If Skewness=0 and Kurtosis = 0, the
distribution is normal.
Click the Calculate EWMA ARL button to
produce the ARL table and chart. If Monte Carlo was
selected, the table of Run Length Standard Deviation and
Percentiles will also be produced.
The EWMA ARL is for a two-sided chart with
zero-state, i.e., the shift is assumed to occur at the
start. The mean and standard deviation are also assumed
to be known. This will not likely be the case in use,
but is still useful for determining parameter settings
and comparison of ARL across chart types.
Due to the complexity of calculations, SigmaXL must
be loaded and appear on the menu in order for this
template to function. Do not add or delete rows or
columns in this template.
REFERENCES:
Lucas J.M. and Saccucci M.S. (1990), “Exponentially
weighted moving average control schemes: Properties and
enhancements”, Technometrics 32, 1-12.
Steiner, S. H. (1999), "EWMA control charts with
time-varying control limits and fast initial response",
Journal of Quality Technology 31(1), 75-86.
Borror, C. M., Montgomery, D.C. and RungerG. C.
(1999). “Robustness of the EWMA Control Chartto
Nonnormality,” Journal of Quality Technology, 31(3),
309–316.
Web Demos
Our CTO and Co-Founder, John Noguera, regularly hosts free Web Demos featuring SigmaXL and DiscoverSim Click here to view some now!