C. J. Spanos
CUSUM, MA and EWMA Control Charts
Increasing the sensitivity and getting ready for
automated control:
The Cumulative Sum chart, the Moving
Average and the Exponentially
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C. J. Spanos
Shewhart Charts cannot detect small shifts
The charts discussed so far are variations of the Shewhart
chart: each new point depends only on one subgroup.
Shewhart charts are sensitive to large process shifts.
The probability of detecting small shifts fast is rather small:
Fig 6-13 pp 195 Montgomery.
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C. J. Spanos
Cumulative-Sum Chart
If each point on the chart is the cumulative history (integral)
of the process, systematic shifts are easily detected. Large,
abrupt shifts are not detected as fast as in a Shewhart chart.
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CUSUM charts are built on the principle of Maximum
Likelihood Estimation (MLE).
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C. J. Spanos
Maximum Likelihood Estimation
The "correct" choice of probability density function (pdf)
moments maximizes the collective likelihood of the
observations.
If x is distributed with a pdf(x,θ) with unknown θ, then θ can
be estimated by solving the problem:
m
maxθ
Σ pdf(xi,θ)
i=1
This concept is good for estimation as well as for comparison.
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C. J. Spanos
Maximum Likelihood Estimation Example
To estimate the mean value of a normal distribution, collect
the observations x1,x2, ... ,xm and solve the maximization
problem:
m
maxµest
1
Σ 1 e-2
i=1 σ 2π
xi-µest 2
σ
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C. J. Spanos
MLE Control Schemes
If a process can have a "good" or a "bad" state (with the
control variable distributed with a pdf fG or fB respectively).
This statistic will be small when the process is "good" and
large when "bad":
m
f (x )
Σ log f GB(xii)
i=1
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MLE Control Schemes (cont.)
Note that this counts from the beginning of the process. We
choose the best k points as "calibration" and we get:
m
k
f B(x i)
f B(x i)
S m = Σ log
- min k < m Σ log
>L
f G(x i)
f G(x i)
i=1
i=1
or
S m = max (S m-1+log
f B(x m)
, 0) > L
f G(x m)
This way, the statistic Sm keeps a cumulative score of all the
"bad" points. Notice that we need to know what the "bad"
process is!
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C. J. Spanos
The Cumulative Sum chart
If θ is a mean value of a normal distribution, is simplified to:
m
S m = Σ (x i - µ 0)
i=1
where µ0 is the target mean of the process. This can be
monitored with V-shaped limits.
Advantages
The Cusum chart is very effective for small shifts and when
the subgroup size n=1.
Disadvantages
The Cusum is relatively slow to respond to large shifts.
Also, special patterns are hard to see and analyze.
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C. J. Spanos
Example
15
10
5
0
-5
-10
-15
UCL=13.4
µ0=-0.1
0
20
40
60
80
100
120
140
LCL=-13.6
160
0
20
40
60
80
100
120
140
160
120
100
80
60
40
20
0
-20
-40
-60
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C. J. Spanos
The CUSUM design
Need to set L(0) (i.e. the run length when the process is in
control), and L(δ) (i.e. the run-length for a specific deviation).
Figure 7-3 Montgomery pp 227
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C. J. Spanos
The CUSUM design
2 ln 1- β
α
δ2
δ= ∆
θ = tan -1 ∆
σx
2A
d=
d
θ
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ARL vs. Deviation for CUSUM
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C. J. Spanos
CUSUM chart of furnace Temperature difference
Detect 2Co, σ =1.5Co, α=.0027 β=0.05, θ = 18.43o, d = 6.6
50
3
40
2
30
B
20
C
0
A
10
1
-1
-2
0
-3
-10
0
20
40
60
80
100
0
20
40
60
80
100
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C. J. Spanos
Tabular CUSUM
A tabular form is easier to implement in a CAM system
SH( i ) = max [ 0, xi - ( µo + K ) + SH( i - 1 ) ]
SL( i ) = max [ 0, ( µo - K ) - xi + SL( i - 1 ) ]
SL( 0 ) = SH( 0 ) = 0
K = ∆/2
H = 2dσxtan(θ)
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H
d
θ
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C. J. Spanos
Cusum Enhancements
To speed up CUSUM response one can use "modified" V
masks:
15
10
5
0
-5
0
20
40
60
80
100
Other solutions include the application of Fast Initial
Response (FIR) CUSUM, or the use of combined CUSUMShewhart charts.
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C. J. Spanos
General MLE Control Schemes
Since the MLE principle is so general, control
schemes can be built to detect:
• single or multivariate deviation in means
• deviation in variances
• deviation in covariances
An important point to remember is that MLE
schemes need, implicitly or explicitly, a definition of
the "bad" process.
The calculation of the ARL is complex but possible.
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C. J. Spanos
Control Charts Based on Weighted Averages
Small shifts can be detected more easily when multiple
samples are combined.
Consider the average over a "moving window" that contains
w subgroups of size n:
Mt =
xt + xt-1 + ... +xt-w+1
w
2
V(Mt) = nσw
The 3-sigma control limits for Mt are:
UCL = x + 3 σ
nw
3
LCL = x - σ
nw
Limits are wider during start-up and stabilize after the first w
groups have been collected.
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C. J. Spanos
Example - Moving average chart
w = 10
15
10
5
0
-5
-10
0
20
40
60
80
100
120
140
160
sample
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C. J. Spanos
The Exponentially Weighted Moving Average
If the CUSUM chart is the sum of the entire process history,
maybe a weighed sum of the recent history would be more
meaningful:
zt = λxt + (1 - λ)zt -1
0 < λ < 1 z0 = x
It can be shown that the weights decrease geometrically
and that they sum up to unity.
t-1
zt = λ Σ ( 1 - λ )j x t - j + ( 1 - λ )t z0
j=0
UCL = x + 3 σ
LCL = x - 3 σ
λ
(2-λ)n
λ
(2-λ)n
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C. J. Spanos
Two example Weighting Envelopes
relative
importance
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
10
EWMA 0.6
EWMA 0.1
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30
40
50
-> age of sample
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EWMA Comparisons
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λ=0.6
C. J. Spanos
10
5
0
-5
-10
0
20
40
60
80
100
120
140
160
0
20
40
60
80
sample
100
120
140
160
15
10
λ=0.1
5
0
-5
-10
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C. J. Spanos
Another View of the EWMA
•The EWMA value zt is a forecast of the sample at the t+1
period.
• Because of this, EWMA belongs to a general category of
filters that are known as “time series” filters.
xt = f (xt - 1, xt - 2, xt - 3 ,... )
xt- xt = at
Usually:
p
xt = Σ φixt - i +
i=1
q
Σ
θjat - j
j=1
•The proper formulation of these filters can be used for
forecasting and feedback / feed-forward control!
•Also, for quality control purposes, these filters can be used
to translate a non-IIND signal to an IIND residual...
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C. J. Spanos
Summary so far..
While simple control charts are great tools for visualizing
the process,it is possible to look at them from another
perspective:
Control charts are useful “summaries” of the process
statistics.
Charts can be designed to increase sensitivity without
sacrificing type I error.
It is this type of advanced charts that can form the
foundation of the automation control of the (near) future.
Next stop before we get there: multivariate and modelbased SPC!
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