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A robust change point estimator
for binomial CUSUM control
charts
Advisor: Takeshi Emura ( 江 村 剛 志 ) 博士
Presenter : Yi Ting Ho
2014/6/24
1
Outline
 Introduction
 Background
1. Binomial CUSUM control chart
2. Maximum likelihood estimator
3. Page’s last zero estimator
 Method
Combine CUSUM estimator and MLE
2014/6/24
2
Outline
 Simulations
Design closely follows those of Perry and Pignatiello (2005)
 Data analysis
Jewelry manufacturing data by Burr (1979)
 Conclusion
2014/6/24
3
Outline
 Introduction
 Background
1. Binomial CUSUM control chart
2. Maximum likelihood estimator
3. Page’s last zero estimator
 Method
Combine CUSUM estimator and MLE
2014/6/24
4
Introduction
• What is the change point?
• We consider observations come from binomial distribution with
the same fraction nonconforming p .
15
CUSUM Chart
10
h=12.043
5
Cumulative Sum
Out-of-control
0
Change
point
0
2014/6/24
10
20
30
40
Subgroup number
50
60
5
Introduction
In SPC, the np-chart is most famous charts used to monitor the
number of nonconforming items for industrial manufactures.
defectives
6
8
upper control limit
4
When sample size is large and defective
rate is not too small, the np-chart works
well.
10
np-chart
2
center line
0
lower control limit
2014/6/24
0
5
10
15
Subgroup number
20
25
30
6
Introduction
Page (1954, 1955) first suggested the CUSUM chart to estimate
the change point.
The binomial CUSUM control chart is a good alternative when
small changes are important.
2014/6/24
7
Introduction
Samuel and Pignatiello (2001) proposed maximum likelihood
estimator (MLE) for the process change point using the step change
Likelihood function for a binomial random variable.
Perry and Pignatiello (2005) shows that the performance of the
MLE is often better than Page’s last zero estimator.
2014/6/24
8
Introduction
The MLE method outperforms CUSUM method when magnitudes
of change is large.
In order to construct more robust in different parameter setting, this
thesis combines CUSUM estimator and MLE. Furthermore,
compares the new method with two estimators.
2014/6/24
9
Outline
 Introduction
 Background
1. Binomial CUSUM control chart
2. Maximum likelihood estimator
3. Page’s last zero estimator
 Method
Combine CUSUM estimator and MLE
2014/6/24
10
Binomial CUSUM control chart
i .i .d
i .i .d
X 1 , X 2 , , X  ~ Bin ( n, p0 ), X  1 , X   2 , , X T ~ Bin ( n, paTrue )
  { (  , paTrue ) |  { 1, 2, , T }, p0  paTrue  1 } for a fixed value
T { 1, 2,  }.
 n x
n x
if i  1, 2, ,  ,
   p0 ( 1  p0 ) ,
  x
P( X i  x )  
  n ( pTrue ) x ( 1  pTrue ) n  x , if i    1,   2, , T .
a
  x a
 
p0 : in-control fraction nonconforming.
paTrue: out-of-control fraction nonconforming.

: the true change point .
2014/6/24
11
Binomial CUSUM control chart
S0  0 and Si  max{ 0, X i  nk  Si 1 },
where
i  1, 2, 
 1  paas 

 ln 
1  p0 

k 
,
as
 pa ( 1  p0 ) 
ln 

as
p
(
1

p
)
a
 0

k : The reference value.
paas :
out-of-control fraction nonconforming for which to design the
CUSUM chart.
2014/6/24
12
Binomial CUSUM control chart
When S i exceeds decision interval h  0 the chart signals that an
increase in the process fraction nonconforming has occurred.
2014/6/24
13
Maximum likelihood estimator
(MLE)
i .i .d
X 1 , X 2 , , X  ~ Bin ( n, p0 ),
i .i .d
X  1 , X   2 , , X T ~ Bin ( n, paTrue )
  { (  , paTrue ) |  { 1, 2, , T }, p0  paTrue  1 } for fixed value
T { 1, 2,  }.
The likelihood function is given by
L(  , p
True
a
T
 n  xi
n 
n  xi
| p0 ,  )     p0 ( 1  p0 )   ( paTrue ) xi ( 1  paTrue ) n xi
i 1  xi 
i  1 xi 

where   ( x1 , x 2 ,, xT ).
2014/6/24
14
Maximum likelihood estimator
(MLE)
The log-likelihood is
log L(  , p
True
a


i 1
i 1
| p0 ,  )  C  log( p0 ) xi  log( 1  p0 ) ( n  xi )
 log( p
T
True
a
)  xi  log( 1  p
i  1
T
True
a
)  ( n  xi ),
i  1
where C is a constant. We can rewrite log L(  , paTrue | p0 ,  ) as
True

p
log L(  , paTrue | p0 ,  )  C *  log a
 p0

 T
 1  pTrue
a
  xi  log
i  1
 1  p0


T
T
i 1
i 1
 T
  ( n  xi ),
i  1

where C  C  log( p0 ) xi  log( 1  p0 ) ( n  xi ) is a constant.
*
2014/6/24
15
Maximum likelihood estimator
(MLE)
If  is given, the log-likelihood equation becomes
T
T
xi  ( n  xi )


i  1
i  1
log L(  , pTrue
|
p
,

)


,
0
True
True
a
p a
pa
1  pTrue
a
T
x


i
i  1
True
pa
T
T
( nx


 i
i
1
1 pa
True
)

pˆ a (  ) 
True
x


i  1
T
i
n


.
i  1
2014/6/24
16
Maximum likelihood estimator
(MLE)
T

2
True2
p a
T
x


( nx


i
i
)
log L(  , pa | p0 ,  )   i  1 2  i  1 True 2  0.
( 1 pa )
pTrue
a
Putting pˆ (  ) into log L(  , pTrue | p0 ,  ), we have a profile
log-likelihood for  as
True
a
a
log L(  , pˆ a (  ) | p0 ,  )  C  log(
True
2014/6/24
*
pˆ True
( )
a
p0
T
1  pˆ True
( )
a
i  1
1  p0
)  xi  log(
T
)  ( n  xi ).
i  1
17
Maximum likelihood estimator
(MLE)
Therefore, the change point estimator ˆMLE is
ˆMLE
True
True
 

 pˆ a (  ) 
 T
 1  pˆ a (  ) 
 T
 arg max log 
  xi  log 
  ( n  xi
p0
 { 1, 2 , , T } 




i  1
 1  p0
i  1
 
2014/6/24

) .

18
Page’s last zero estimator
Under CUSUM control chart we have cumulative sum
S0  0 and Si  max{ 0, X i  nk  Si 1 }, i  1, 2, 
An estimate of the change point is given by ˆCUSUM  max{ i : Si  0 }.
2014/6/24
19
Outline
 Introduction
 Background
1. Binomial CUSUM control chart
2. Maximum likelihood estimator
3. Page’s last zero estimator
 Method
Combine CUSUM estimator and MLE
2014/6/24
20
Proposed method
We propose a new change point estimator that combines ˆMLE and ˆCUSUM .
ˆNEW ( w )  wˆCUSUM( 1  w )ˆMLE ,




as
w  w( p True
|
p
)


a
a




2014/6/24
paTrue  p0
p as
 p0
a




p as
 p0 
a

True
pa  p0 
 paTrue

 p
0





 paTrue 


 p 
 0 
,
0  w  1,
, if paTrue  p0  paas  p0 ,
if paTrue  p0  p as
 p0 .
a
21
Proposed method
True
First, we estimate unknown pa by MLE.
T
pˆ a ( ˆMLE ) 
x


i
i  ˆMLE 1
T
n


,
i  ˆMLE 1
The estimator of weight function as
wˆ  w( pˆ a ( ˆMLE ) | paas
2014/6/24

 pˆ a ( ˆMLE )  p0

as
p
 p0

a
)

as

p
 p0
a

 pˆ a ( ˆMLE )  p0









 pˆ a ( ˆMLE ) 


p0


 pˆ a ( ˆMLE

p0

)


,
, if pˆ a ( ˆMLE )  p0  paas  p0 ,
if pˆ a ( ˆMLE )  p0  p asa  p0 .
22
Proposed method
Hence, we obtain the proposed estimator of the change point as
ˆNEW ( wˆ )  wˆ ˆCUSUM( 1  wˆ )ˆMLE ,
2014/6/24
ˆ  1.
0w
23
Outline
 Simulations
Design closely follows those of Perry and Pignatiello (2005)
 Data analysis
Jewelry manufacturing data by Burr (1979)
 Conclusion
2014/6/24
24
Simulations
i .i .d
i .i .d
X 1 , X 2 , , X  ~ Bin ( n, p0 ), X  1 , X   2 , , X T ~ Bin ( n, paTrue )
•
We assume the true change point   100.
•
The in-control fraction nonconforming p0  0.1.
•
The out-of-control fraction nonconforming
pTrue
{ 0.11, 0.12, , 0.25, 0.30 }.
a
25
2014/6/24
Simulations
• Consider CUSUM charts designed to detect 30% increase in
by setting paas  1.3  p0  0.13.
• Choose h  6.57 or h  11.42 such that in-control process
average run lengths (ARL) is close to 150 or 370.
2014/6/24
26
p0  0.1, paas  0.13,
Simulations
paTrue  0.1, 0.11, 0.15, 0.20, 0.25, 0.30
150
200
250
300
12
0
20
40
60
80
100
40
60
Cusum Chart
80
100
80
100
6
8
10
h=6.57
2
0
2
0
60
100
4
6
8
Cumulative Sum
10
h=6.57
4
8
6
4
2
40
80
12
Cusum Chart
12
Cusum Chart
Cumulative Sum
10
20
i
i
2014/6/24
10
0
i
0
20
8
120
i
h=6.57
0
6
2
0
2
0
100
12
50
h=6.57
4
6
8
Cumulative Sum
10
h=6.57
4
Cumulative Sum
10
8
6
4
0
2
Cumulative Sum
h=6.57
0
Cumulative Sum
Cusum Chart
12
Cusum Chart
12
Cusum Chart
0
20
40
60
i
80
100
0
20
40
60
i
27
p0  0.1, paas  0.13,
Simulations
paTrue  0.1, 0.11, 0.15, 0.20, 0.25, 0.30
100
150
200
250
300
20
40
60
80
100
0
20
40
60
80
Maximum likelihood estimator
Maximum likelihood estimator
Maximum likelihood estimator
60
80
100
15
10
0
0
40
h=6.57
5
Log - Likelihood
15
10
Log - Likelihood
h=6.57
5
15
10
5
0
20
100
20
i
20
i
20
i
i
2014/6/24
15
20
0
h=6.57
0
10
0
0
50
h=6.57
5
Log - Likelihood
15
10
Log - Likelihood
h=6.57
5
15
10
5
0
Log - Likelihood
h=6.57
0
Log - Likelihood
Maximum likelihood estimator
20
Maximum likelihood estimator
20
Maximum likelihood estimator
0
20
40
60
i
80
100
0
20
40
60
i
80
28
100
p0  0.1, paas  0.13,
Simulations
paTrue  0.1, 0.11, 0.15, 0.20, 0.25, 0.30
150
200
250
300
20
0
50
100
0
60
Cusum Chart
80
100
80
100
10
15
h=11.42
0
0
60
100
5
Cumulative Sum
10
15
h=11.42
5
Cumulative Sum
5
40
80
20
Cusum Chart
20
Cusum Chart
10
15
40
i
i
2014/6/24
20
i
0
20
15
150
i
h=11.42
0
10
0
0
100
20
50
h=11.42
5
Cumulative Sum
10
15
h=11.42
5
Cumulative Sum
15
10
5
0
Cumulative Sum
h=11.42
0
Cumulative Sum
Cusum Chart
20
Cusum Chart
20
Cusum Chart
0
20
40
60
i
80
100
0
20
40
60
i
29
p0  0.1, paas  0.13,
Simulations
paTrue  0.1, 0.11, 0.15, 0.20, 0.25, 0.30
100
150
200
250
300
20
0
20
40
60
80
100
0
60
80
Maximum likelihood estimator
i
80
100
15
5
0
5
0
60
h=11.42
10
Log - Likelihood
15
h=11.42
10
Log - Likelihood
5
0
40
100
20
Maximum likelihood estimator
20
Maximum likelihood estimator
20
i
10
15
40
i
20
2014/6/24
20
i
h=11.42
0
15
5
0
5
0
50
h=11.42
10
Log - Likelihood
15
h=11.42
10
Log - Likelihood
15
10
0
5
Log - Likelihood
h=11.42
0
Log - Likelihood
Maximum likelihood estimator
20
Maximum likelihood estimator
20
Maximum likelihood estimator
0
20
40
60
i
80
100
0
20
40
60
i
80
30
100
Simulations
Table 4 Simulation results for estimating the change point  under n  50, p0  0.1
and paas  0.13 based on 1000 simulation runs. The true change point   100.
2014/6/24
31
Simulations
2014/6/24
32
Simulations
Table 5 Simulation results for estimating the mean squared error the change point 
as
Under n  50, p0  0.1 and pa  0.13 based on 1000 simulation runs. The true
change point   100.
2014/6/24
33
Simulations
2014/6/24
34
Simulations
• In most cases, the ˆNEW ( wˆ ) outperforms other estimators.
• Despite some of the mean squared error of ˆNEW ( wˆ ) in all
cases are not the smallest, ˆNEW ( wˆ ) provides very precise
estimator of change point.
2014/6/24
35
Outline
 Simulations
Design closely follows those of Perry and Pignatiello (2005)
 Data analysis
Jewelry manufacturing data by Burr (1979)
 Conclusion
2014/6/24
36
Data Analysis
Jewelry manufacturing
process
2471 yellow beads
(good pieces)
229 red beads
(defective pieces)
We set the in-control fraction nonconforming p0  229 2700  0.085.
Each subgroup contains n  50 beads so the number of
subgroups is T  2700 50  54.
X i  the number of defective pieces in 50 beads, for i  1, 2, , 54.
2014/6/24
37
2014/6/24
38
Data Analysis
• Center line
 np  50  0.085  4.25.
• Upper control limit  np  3 n  p0  ( 1  p0 )
 4.25  3 50  0.085  0.915  10.165.
• Lower control limit  np  3 n  p0  ( 1  p0 )
 4.25  3 50  0.085  0.915  1.665.
39
2014/6/24
Data Analysis
6
4
defectives
8
10
np-chart
0
2
47th sample
0
10
20
30
40
50
60
Subgroup number
Although np-chart do not detect the process out-of-control, the data
slightly increase after the subgroup 40.
2014/6/24
40
Data Analysis
CUSUM Chart
8
15
10
Maximum likelihood estimator
6
2
5
h=12.043
4
10
Log - Likelihood
^τ=50
0
0
Cumulative Sum
Out-of-control
0
10
2014/6/24
20
30
40
Subgroup number
50
60
0
Change
point
10
20
30
40
50
60
i
41
Data Analysis
ˆ )  47.70983.
• We obtain ˆCUSUM  43, ˆMLE  50 and ˆNEW ( w
• ˆCUSUM always underestimate the true change point while ˆMLE slightly
overestimate  when paTrue  Paas.
• Our proposed method may provide more unbiased estimator.
2014/6/24
42
Outline
 Simulations
Design closely follows those of Perry and Pignatiello (2005)
 Data analysis
Jewelry manufacturing data by Burr (1979)
 Conclusion
2014/6/24
43
Conclusion
• The estimator ˆNEW ( wˆ ) contains advantage of ˆCUSUM and ˆMLE .
• Our proposed method are unbiasedness for estimating the true
change point.
• The estimator ˆNEW ( wˆ ) is more robust than ˆCUSUM and ˆMLE
under different parameter setting.
2014/6/24
44
References
•
•
•
•
•
•
•
•
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