Lecture 14: CUSUM and EWMA
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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 Weighted Moving Average Charts.
Lecture 14: CUSUM and EWMA
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Shewhart Charts cannot detect small shifts
Fig 6-13 pp 195 Montgomery.
The charts discussed so far are variations of the Shewhartchart: 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:
Lecture 14: CUSUM and EWMA
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Cumulative-Sum ChartIf 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.
CUSUM charts are built on the principle of Maximum Likelihood Estimation (MLE).
Lecture 14: CUSUM and EWMA
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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:
This concept is good for estimation as well as for comparison.
Maximum Likelihood Estimation
max θ
m
Πi=1
pdf( xi,θ)
Lecture 14: CUSUM and EWMA
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Maximum Likelihood Estimation ExampleTo estimate the mean value of a normal distribution, collect the observations x1,x2, ... ,xm and solve the non-linear programming problem:
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21max σ
μ
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Lecture 14: CUSUM and EWMA
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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
Σi=1
logfB(xi)fG(xi)
∑=
==Π
m
iii
m
ipp
11)log()log(
Lecture 14: CUSUM and EWMA
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MLE Control Schemes (cont.)
Sm =m
Σi=1
logfB(xi)fG(xi)
- mink < m
k
Σi=1
logfB(xi)fG(xi)
> L
or
Sm = max (Sm-1+logfB(xm)fG(xm), 0) > L
Note that this counts from the beginning of the process. We choose the best k points as "calibration" and we get:
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!
Lecture 14: CUSUM and EWMA
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The Cumulative Sum chart
Sm =m
Σi=1
(xi - µ0)
AdvantagesThe CUSUM chart is very effective for small shifts and when the subgroup size n=1. DisadvantagesThe CUSUM is relatively slow to respond to large shifts. Also, special patterns are hard to see and analyze.
If θ is a mean value of a normal distribution, is simplified to:
where μ0 is the target mean of the process. This can be monitored with V-shaped or tabular “limits”.
Lecture 14: CUSUM and EWMA
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Example
-15-10-505
1015
20 40 60 80 100 120 140 160
µ0=-0.1
LCL=-13.6
UCL=13.4
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020406080
100120
0 20 40 60 80 100 120 140 160
0
Shew
hart
smal
l shi
ftC
USU
M s
mal
l shi
ft
Lecture 14: CUSUM and EWMA
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The V-Mask CUSUM design for standardized observations yi=(xi-μo)/σ
Figure 7-3 Montgomery pp 227
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).
Lecture 14: CUSUM and EWMA
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The V-Mask CUSUM design for standardized observations yi=(xi-μo)/σ
d = 2δ 2
ln1- β
α
θ = tan -1 δ2A
δ = Δσ x
dθ
δ is the amount of shift (normalized to σ) that we wish to detect with type I error α and type II error β.Α is a scaling factor: it is the horizontal distance between successive points in terms of unit distance on the vertical axis.
Lecture 14: CUSUM and EWMA
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ARL vs. Deviation for V-Mask CUSUM
Lecture 14: CUSUM and EWMA
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CUSUM chart of furnace Temperature difference
I
IIIII
Detect 2Co, σ =1.5Co, (i.e. δ=1.33), α=.0027 β=0.05, A=1
=> θ = 18.43o, d = 6.6
100806040200-10
0
10
20
30
40
50
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Lecture 14: CUSUM and EWMA
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Tabular CUSUMA tabular form is easier to implement in a CAM system
Ci+ = max [ 0, xi - ( μo + k ) + Ci-1
+ ]
Ci- = max [ 0, ( μo - k ) - xi + Ci-1
- ]
C0+ = C0
- = 0
k = (δ/2)/σ
h = dσxtan(θ)
dθ
h
Lecture 14: CUSUM and EWMA
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Tabular CUSUM Example
Lecture 14: CUSUM and EWMA
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Various Tabular CUSUM Representations
Lecture 14: CUSUM and EWMA
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CUSUM Enhancements
Other solutions include the application of Fast Initial Response (FIR) CUSUM, or the use of combined CUSUM-Shewhart charts.
To speed up CUSUM response one can use "modified" V masks:
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5
10
15
Lecture 14: CUSUM and EWMA
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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 covariancesAn 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.
Lecture 14: CUSUM and EWMA
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Control Charts Based on Weighted Averages
The 3-sigma control limits for Mt are:
Mt = xt + xt-1 + ... +xt-w+1w
V(Mt) = σ2n w
UCL = x + 3 σn w
LCL = x - 3 σn w
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:
Limits are wider during start-up and stabilize after the first w groups have been collected.
Lecture 14: CUSUM and EWMA
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Example - Moving average chart
-10
-5
0
5
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0 20 40 60 80 100 120 140 160
sample
w = 10
Lecture 14: CUSUM and EWMA
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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.
zt = λ ( 1 - λ )j xt - j + ( 1 - λ )t z0Σj = 0
t - 1
UCL = x + 3 σ λ( 2 - λ ) n
LCL = x - 3 σ λ( 2 - λ ) n
Lecture 14: CUSUM and EWMA
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Two example Weighting Envelopes
0.00.10.20.30.40.50.60.70.80.91.0
0 10 20 30 40 50
EWMA 0.6EWMA 0.1 -> age of sample
relativeimportance
Lecture 14: CUSUM and EWMA
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EWMA Comparisons
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0 20 40 60 80 100 120 140 160sample
-10
-5
0
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10
15
0 20 40 60 80 100 120 140 160
λ=0.6
λ=0.1
Lecture 14: CUSUM and EWMA
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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.
• 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...
xt = f ( xt - 1, xt - 2, xt - 3 ,... )xt- xt = atUsually:
xt = φixt - iΣi = 1
p+ θjat - jΣ
j = 1
q
Lecture 14: CUSUM and EWMA
SpanosEE290H F05
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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 model-based SPC!
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