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Efficient Design and Analysis of
Markovian-type Control Charts
Prof. Lianjie Shu
Email: [email protected]
Faculty of Business Administration
University of Macau
1
Outline
•
Markovian-type Control Charts
•
Accurate Evaluation
•
Efficient Design and Sensitivity Analysis
•
Steady-State Analysis
•
The Use of Probability Limits for Control Charts
2
Markovian-type Control Charts
• The charting statistic at time t can be
generalized as
Yt  g (Yt 1 , X t ),
where X t  obeservation at time t
It alarms at
ta  min{t ,Yt  h} or ta  min{t ,| Yt | h}
3
Markovian-type Control Charts
• It represents an important class of control
charts in Statistical Process Control
• It includes many control charts as special
cases
– The standard exponentially weighted moving
average (EWMA) and cumulative sum (CUSUM)
charts
– Recent variations such as adaptive EWMA charts,
adaptive CUSUM charts, and Bayesian control
charts.
4
Example: CUSUM Chart
The upper-sided CUSUM charting statistics:
Yt  max(0, Yt 1  X t  k ),
where k  a reference value with popular choice of k   / 2
  a prespecified mean shift for early detection
h  a decision interval
It alarms at
ta  min{t , Yt  h}
5
Example: EWMA
The EWMA charting statistics
Qt   X t  (1   )Qt 1 ,
where 0<  1 and Q0  0 .
It alarms when
ta  min{t ,| Qt | h}
When   1, the EWMA chart reduces to the Shewhart chart.
6
Example: Adaptive EWMA chart
The adaptive EWMA statistics: (Huber 1981)
Qt  Qt 1  hu ( et ),
where et  X t  Qt 1
 e  (1   ) , e  

and hu ( e)  
 e,
| e | 
 e  (1   ) , e   .

Note:
1. When et  0, Qt  (1  w( et ))Qt 1  w( et ) X t , where w( et ) 
hu ( et )
et
.
2. When hu ( e)   e, the AEWMA estimator reduces to the EWMA estimator.
3. When  =0 or  =1, hu ( e)  e and Qt  X t (Shewhart Statistic).
7
Example: Adaptive CUSUM cha
• The Adaptive CUSUM Chart (Sparks 2000; Shu
and Jiang 2006; Shu et al. 2008)
Z t  max{0, Z t1  ( X t  ˆt / 2) / h(ˆt / 2)},
ˆ  max(  , Q ),
t
min
t
where Qt  (EWMA) estimate of the current process mean

 min
 minimum mean shift specified for early detection
h( k )  operating function that relates the decision interval
of the upper-CUSUM chart to k , given an in-control ARL.
The ACUSUM chart signals when Z t  c, where c is a
constant close to 1.
8
Advantages of Markovian-type
Control Charts
• Simplicity: recursive form
• The average run length can be formulated as a
Markov chain or an integral equation. This allows
us to evaluate the charting performance without
running a large number of simulations.
9
Issue I
Accurate Evaluation of Average
Run Length (ARL)
10
Performance Measure
• Run Length: alarm time/stopping time
• Average run length (ARL): expected value of run length
ARL=E(RL)
• In-control ARL: ARL0=E(RL| δ=0)
• Out-of-control ARL: ARL1=E(RL| δ≠0)
11
Evaluation Methods
– Monte Carlo Simulation: feasible but may not be the most
efficient
– Markov Chain
– Integral Equation
• Often Gaussian-Legendre (GL) quadrature applied to evaluate the integral
equation in SPC
• The GL method often produce fast and accurate ARL results when the
integration kernel is smooth but unreliable results when the integration
kernel is not smooth.
• Non-smooth integration kernels could be due to non-normal distributions
12
More Accurate Methods
• The piecewise collocation (PWC) method
 To approximate a function based on a linear combination
of some Chebyshev polynomials
• The Clenshaw-Curtis (CC) quadrature (Clenshaw
and Curtis 1960)
 Basic idea: it can separate the kernel function into two
parts: continuous part and discontinuous part and then
evaluate the discontinuous one individually.
13
Example 1: ARL of CUSUM under
Gamma Distributions
14
Example 1: ARL of CUSUM under
Gamma Distributions
15
Example 1: ARL of CUSUM under
Gamma Distributions
•
The CUSUM charting statistics
•
The ARL integral equation
16
Example 1: ARL of CUSUM under
Gamma Distributions
17
Example 2: ARL of AEWMA Control
Chart under Normal Distributions
 The AEWMA Statistic:
Gt  Gt 1   ( et )  (1  w( et ))Gt 1  w( et ) X t ,
where et  X t  Gt 1 =forecast error
 The Huber's score function (Huber 1981):
 e  (1   ) R, e   R

hu ( e)  
 e,
| e | R
 e  (1   ) R, e  R.

0.8
Weight
 ( et )=score function
 ( et )
w( et ) 
 weight function
et
R=3.5
R=2.5
1
0.6
0.4
0.2
0
-10
-8
-6
-4
-2
0
e
2
4
6
18
8
10
Example 2: ARL of AEWMA Control
Chart under Normal Distributions
• The ARL integral equation
19
(a) λ=0.01, R=3
(b) λ=0.05, R=3
20
Some Publications
•
•
•
•
•
Su, Y., Shu, L., and Tsui, K.-L., “Adaptive EWMA Procedures for Monitoring
Processes Subject to Linear Drifts”, Computational Statistics and Data
Analysis, 55, 2819-2829, 2011.
Huang, W., Shu, L., and Jiang, W., “Evaluation of Exponentially Weighted
Moving Variance Control Chart Subject to Linear Drifts”, Computational
Statistics and Data Analysis, 56, 4278-4289, 2012.
Shu, L., Huang, W., Su, Y., and Tsui, K.-L., “Computation of Run Length
Percentiles of CUSUM Control Charts under Changes in Variances”, Journal of
Statistical Computation and Simulation, 83, 1238-1251, 2013
Huang, W., Shu, L., Jiang, W., and Tsui, K.-L., “Evaluation of Run-Length
Distribution for CUSUM Charts under Gamma Distributions”, IIE Transactions,
981-994, 2013.
Huang, W., Shu, L., Su, Y. “An Accurate Evaluation of Adaptive Exponentially
Weighted Moving Schemes”, IIE Transactions 46, 457-469, 2014
21
Issue II
Efficient Design and
Sensitivity Analysis
22
Design Criteria
• Statistical Design: To optimize the out-of-control
ARL, given a fixed in-control ARL
• Economical Design: To minimize the cost/penalty
function
• Simulations are used to obtain the optimal
parameters for control charts. Can a control chart be
designed in a more efficient manner?
23
Gradient-based Design
• Motivation
 Note that the ARL of Markovian-type control charts can be
formulated as a Markov chain or an integral equation.
Therefore, it is possible to approximate the ARL gradients
from the ARL integral equation or the Markov chain model.
• Advantages of the gradient-based method
 More accurate
 More efficient
 Facilitates the sensitivity analysis
24
Case 1: A Gradient Approach for Efficient
Design of EWMA Charts
The EWMA charting statistics
Qt   X t  (1   )Qt 1 ,
where 0<  1 and Q0  z. It alarms when
ta  min{t ,| Qt | h}
Define L( , h; z,  )  E (ta |Y0  z,  =
Design Criterion:
min. L( , h; z,  )
 ,h
s.t. L( , h; z,0)  ARL0
25
Problem Formulation
minL( , h; z, )
 ,h
subject to L( , h; z,0) = ARL0
h =  ( )
min g ( ; )  L( , (  ); z,  )

26
Problem Solving:
Conventional Method
•
Grid Search Grid Search Procedure:
(Lucas and Saccucci 1990)
(i) Change λ with a particular step size (say
0.01) over the interval (0,1], starting from
λ=0.01;
(ii) Calculate the corresponding control limits for
each λ in the in-control case;
(iii) Select λ that provides the smallest out-ofcontrol ARL at shift size of δ.
27
Problem Solving: A New Approach
•
The first-order condition at the optimal λ
that minimizes g(λ;δ) is
g’ (λ;δ)=0.
28
Estimate of 𝐿𝜆 (𝜆, ℎ; 𝑧, 𝛿)
[u1, u2 ,..., uN ]: Gauss-Legendre abscissas on [-h,h]
[ w1, w2 ,..., wN ]: Gauss-Legendre weights on [-h,h]
29
Estimate of 𝐿𝜆 (𝜆, ℎ; 𝑧, 𝛿)
30
Estimate of 𝐿ℎ (𝜆, ℎ; 𝑧, 𝛿)
31
32
33
Case 2: A Gradient Approach for Efficient Design
of CUSUM Charts Under Random Shifts
• Design criterion under random shift size
–
–
The ARL based design criterion has limited applicability in
practice;
When the shift size δ is assumed to be random, a more
reasonable performance measure is based on the expected
weighted average run length (EWARL)
EWARL = E [ ( ) L(k , h; z ,  )] =
b
  ( ) L(k , h; z,  ) g ( )d ,
a
where
L(k , h; z ,  )  ARL of the CUSUM chart with design parameters k and h,
conditioned on the initial value Y0  z and shift size of 
 ( )  weight function on the ARL
g ( )  random distribution of  over the range [a, b]
34
Problem Formulation
• Two types of weights
1.Expected quality loss (EQL) in Chen and Chen (2007)
() = 1  
2
2.Expected relative ARL (ERARL) (Ryu et al. 2010; Zhao et
al. 2005; and Han et al. 2007):
() = 1/L opt (/2)
35
Problem Formulation
b
min G(k , h) = a ( ) L(k , h; z,  ) g ( )d
k ,h
subject to L(k , h; z,0) = ARL 0
h = ( k )
min(k ) = G (k , (k )) =
k
b
  ( ) L(k , (k ); z,  ) g ( )d
a
36
Problem Solving
•
Find Θ’(k)
37
Problem Solving
• Derivations of the ARL gradients w.r.t. the design
parameters k and h
38
•The simulation results are obtained from Ryu et al. (2010) in
which k is searched in [0.25,2] or [0.25,1] with grid size 0.01. For
each k, the EWARL is simulated based on 1,000 randomly
generated δs .
39
40
Some Publications
•
•
•
Shu, L., Huang, W., and Jiang, W., “A Novel Gradient
Approach for Optimal Design and Sensitivity Analysis of
EWMA Control Charts”, Naval Research Logistics, 61,
223-237, 2014 .
Huang, W., and Shu, L. “A Gradient Approach for
Efficient Design of CUSUM Charts under Uncertainty”,
under revision with Journal of Quality Technology,
2014.
Huang, W. and Shu, L., “A Gradient Approach to
Efficient Design of Multivariate EWMA Control Charts”,
working paper, 2014.
41
Issue III
Steady-State Analysis of Control
Charts
42
Two Types of Steady-State
Distribution
(i) Cyclical steady-state distribution
– It assumes that the charting statistic was reset to its initial state
whenever a signal occurs, e.g., in industrial quality control
(ii) Conditional steady-state distribution
– It assumes that the charting statistic would reach its stationary
distribution, conditioned that no signal is given before the change
point.
– It is common in medical settings. For example, it is unlikely that a
hospital would close or suspend the treatment of patients even if
there is a deterioration of performance suspected (Gandy and Lau
2013)
43
Two Types of Steady-State
Distribution
(i) Cyclical steady-state distribution
– It assumes that the charting statistic was reset to its initial state
whenever a signal occurs, e.g., in industrial quality control
(ii) Conditional steady-state distribution
– It assumes that the charting statistic would reach its stationary
distribution, conditioned that no signal is given before the change
point.
– It is common in medical settings. For example, it is unlikely that a
hospital would close or suspend the treatment of patients even if
there is a deterioration of performance is suspected (Gandy and
Lau 2013)
44
The Conditional Distribution of
the CUSUM Charting Statistics
•
Assumption: 𝐻0 : 𝑋𝑡 ~𝑁 0,1
•
The one-sided upper CUSUM chart:
𝑉. 𝑆. 𝐻1 : 𝑋𝑡 ~𝑁 𝜇, 1
+
𝑆𝑡+ = max(0, 𝑆𝑡−1
+ 𝑋𝑡 − 𝑘),
where 𝑘 is the reference value.
45
The Conditional Distribution of the
CUSUM Charting Statistics
• Define Φ1𝑡 (𝑢; 𝑧) as the probability that 𝑆𝑡+ ≤ 𝑢 (0 ≤ 𝑢 ≤ ℎ)
conditioned that no signal is triggered before and at time step t and
𝑆0+ = 𝑧, i.e.,
Φ1𝑡 (𝑢; 𝑧) = 𝑃𝑟(𝑆𝑡+ ≤ 𝑢|𝑆0+ = 𝑧, 𝑆𝑗+ ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡),
where 𝐼𝑗 = [0, ℎ] denotes the in-control region at time step j.
• Define 𝜑𝑡1 𝑢; 𝑧 = 𝑑 Φ1𝑡 𝑢; 𝑧 /𝑑𝑢
• Then
𝛽𝑡
=
𝛼𝑡
where 𝛽𝑡 = 𝑃𝑟(𝑆𝑡+ ≤ 𝑢|𝑆0+ = 𝑧, 𝑆𝑗+ ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 − 1) and
𝛼𝑡 = 𝑃𝑟(𝑆𝑡+ ∈ [0, ℎ]|𝑆0+ = 𝑧, 𝑆𝑗+ ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 − 1)
Φ1𝑡 (𝑢; 𝑧)
46
The Conditional Distribution of the
CUSUM Charting Statistics
• When t=1
47
The Conditional Distribution of the
CUSUM Charting Statistics
• When 𝑡 ≥ 2
48
The cyclical Distribution of the CUSUM
Charting Statistics
• Define Φ𝑡2 (𝑢; 𝑧) as the probability that 𝑆𝑡+ ≤ 𝑢 (0 ≤ 𝑢 ≤ ℎ)
conditioned on 𝑆0+ = 𝑧, i.e.,
Φ𝑡2 (𝑢; 𝑧) = 𝑃𝑟(𝑆𝑡+ ≤ 𝑢|𝑆0+ = 𝑧)
• Define 𝜑𝑡2 𝑢; 𝑧 = 𝑑 Φ𝑡2 𝑢; 𝑧 /𝑑𝑢
• When 𝑡 = 1
Φ𝑡2 (𝑢; 𝑧) = 𝑃𝑟(0 ≤ 𝑆1+ ≤ 𝑢|𝑆0+ = 𝑧)
= 𝑃𝑟(𝑋1 + 𝑧 − 𝑘 > ℎ)+𝑃𝑟(𝑋1 + 𝑧 − 𝑘 ≤ 𝑢)
= 1 − 𝐹(ℎ + 𝑘 − 𝑧)+𝐹(𝑢 + 𝑘 − 𝑧)
49
The cyclical Distribution of the CUSUM
Charting Statistics
• When 𝑡 ≥ 2
50
The stationary distribution of the
charting statistic
(i)
The probability distribution of the CUSUM charting statistics
converges as t goes to infinity;
(ii) The stationary distribution of the CUSUM charting statistics is
independent of its initial value.
51
52
Computation of the steady-state ARL
(SSARL)
 The SSARL is a weighted average of the zero-state ARL
LSS   L(u )d (u )
where (u) is the stationary distribution of the charting statistic
or
LSS   P(ui ) L(ui )
53
54
Issue IV
The Use of Probability Limits for
Control Charts
55
Motivation
(i)
The use of probability limits is a very general idea and enables us to
control the false alarm rate at each step at a desired level. However,
the use of constant control limits fail to do this.
(ii) The use of probability limits can facilitate the design of control chart in
the case with varying sample sizes. In this case, the pattern of the
varying sample sizes need to be assumed before simulating the
(fixed) control limits. Instead, the use of probability can relax this
assumption.
56
The conventional CUSUM chart unde
varying sample sizes
•
Assume the observations are sampling from a normal distribution with mean
𝜇 and variance 𝜎 2 .
•
Let 𝑛𝑡 be the sample size at time t, and 𝑋𝑡 be the subgroup mean.
•
The standardized sample mean is
𝑍𝑡 =
•
𝑋𝑡 − 𝜇0
𝜎/ 𝑛𝑡
The log-likelihood ratio for testing the hypotheses: 𝐻0 : 𝜇 = 𝜇0 and 𝐻1 : 𝜇 = 𝜇1
𝐿𝑡 = 𝑙𝑜𝑔
𝑓1 (𝑋𝑡 )
𝑓0 (𝑋𝑡 )
=
𝑛𝑡 (𝜇1 −𝜇0 ) 𝑋𝑡 −𝜇0
(
𝜎
𝜎/ 𝑛𝑡
−
𝜇1 −𝜇0
)
𝜎/ 𝑛𝑡
57
The conventional CUSUM chart unde
varying sample sizes
•
The CUSUM chart based on the standardized observations (SD-CUSUM):
𝑆1,𝑡 = max(0, 𝑆1,𝑡−1 + 𝑍𝑡 − 𝑘𝑡 ),
𝜇 −𝜇0
.
𝑛𝑡
1
where 𝑘𝑡 is the reference value 𝑘𝑡 = 2𝜎/
•
The CUSUM chart based on the log-likelihood ratio (Arnold and Reynolds
2001)
𝑆2,𝑡 = max{0, 𝑆2,𝑡−1 + 𝑛𝑡 𝑍𝑡 − 𝑘𝑡 },
which was referred to as the generalized CUSUM chart, denoted as GLRCUSUM.
58
The conventional CUSUM chart unde
varying sample sizes
• The weighted CUSUM (WCUSUM) based on a geometric weight
(Yashchin 1989):
– The weighted standardized CUSUM (WS1-CUSUM) is based on
𝑆3,𝑡 = max(0, 𝛾𝑆3,𝑡−1 + 𝑍𝑡 − 𝑘𝑡 ),
where 0 < 𝛾 ≤ 1 is a discount factor.
– The weighted GLR-CUSUM chart (WGLR1-CUSUM) is based on
𝑆4,𝑡 = max{0, 𝛾𝑆4,𝑡−1 + 𝑛𝑡 𝑍𝑡 − 𝑘𝑡 },
59
The conventional CUSUM chart unde
varying sample sizes
•
The weighted CUSUM (WCUSUM) based on a robust weight function from
Huber’s score function (Shu et al. 2011)
– The weighted standardized CUSUM (WS2-CUSUM) is based on
𝑆5,𝑡 = max{0, 𝑆5,𝑡−1 + 𝑤ℎ𝑢 (𝑒𝑡 ) 𝑍𝑡 − 𝑘𝑡 },
where
𝑒𝑡 =
𝐿𝑡 −𝐸(𝐿𝑡 |𝜇=𝜇0 )
𝑣𝑎𝑟(𝐿𝑡 |𝜇=𝜇0 )
= 𝑍𝑡 and
– The weighted GLR-CUSUM chart (WGLR2-CUSUM) is based on
𝑆6,𝑡 = max{0, 𝑆6,𝑡−1 + 𝑤ℎ𝑢 (𝑒𝑡 ) 𝑛𝑡 𝑍𝑡 − 𝑘𝑡 },
60
The CUSUM Chart with Probability
Limits
• The use of a fixed control limit will lead to different false alarm rate
at each time step. Instead, the use of probability limits can maintain
a desired conditional false alarm rate at each time step.
61
The CUSUM Chart with Probability
Limits
• Define 𝛼𝑡 as the false alarm rate at time t conditioned that there is
no alarms before time t.
• Similar to Margavio et al. (1995), define 𝐻𝑡 𝛼𝑡 as the threshold of
the GLR-CUSUM chart at time t for controlling the desired false
alarm rate of 𝛼𝑡 . Namely,
𝑃𝑟0 𝑆2,1 > 𝐻1 𝛼1
= 𝛼1
𝑃𝑟0 𝑆2,𝑡 > 𝐻𝑡 𝛼𝑡 |𝑆2,𝑖 ≤ 𝐻𝑖 𝛼𝑖 , 𝑖 = 1,2, … , 𝑡 − 1 = 𝛼𝑡 , 𝑡 = 2,3, … .
• The in-control run length distribution is given by
𝑛
𝑃𝑟0 𝑅𝐿 > 𝑛 =
(1 − 𝛼𝑖 ) 𝑓𝑜𝑟 𝑛 ≥ 1.
𝑖=1
• When 𝛼𝑖 maintains the same over time, the in-control run length
distribution would exactly follow a geometric distribution.
62
The CUSUM Chart with Probability
Limits
• Define Φ𝑖,𝑡 (𝑥) as the in-control (i=0) and out-of-control distribution
(i=1) of the GLR-CUSUM charting statistic, conditioned on no signal
before time t, i.e.,
Φ𝑖,𝑡 𝑥 = 𝑃𝑟𝑖 (𝑆2,𝑡 ≤ 𝑥|𝑆2,𝑗 ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 − 1),
where 𝐼𝑗 = [0, 𝐻𝑗 𝛼𝑗 ] denotes the in-control region at time step j.
• Similarly, define Φ𝑖,𝑡 (𝑥) as the probability distribution of the GLRCUSUM charting statistics, conditioned on the event that there is no
signal at and before time step t, i.e.,
Φ𝑖,𝑡 𝑥 = 𝑃𝑟𝑖 𝑆2,𝑡 ≤ 𝑥 𝑆2,𝑗 ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 =
Φ𝑖,𝑡 𝑥
Φ𝑖,𝑡 𝐻𝑡 𝛼𝑡
for 𝑥 ∈ 𝐼𝑡
63
Determination of the Probability Limi
• Define Φ𝑖,𝑡 (𝑥) as the in-control (i=0) and out-of-control distribution
(i=1) of the GLR-CUSUM charting statistic, conditioned on no signal
before time t, i.e.,
Φ𝑖,𝑡 𝑥 = 𝑃𝑟𝑖 (𝑆2,𝑡 ≤ 𝑥|𝑆2,𝑗 ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 − 1),
where 𝐼𝑗 = [0, 𝐻𝑗 𝛼𝑗 ] denotes the in-control region at time step j.
• Similarly, define Φ𝑖,𝑡 (𝑥) as the probability distribution of the GLRCUSUM charting statistics, conditioned on the event that there is no
signal at and before time step t, i.e.,
Φ𝑖,𝑡 𝑥 = 𝑃𝑟𝑖 𝑆2,𝑡 ≤ 𝑥 𝑆2,𝑗 ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 =
Φ𝑖,𝑡 𝑥
Φ𝑖,𝑡 𝐻𝑡 𝛼𝑡
for 𝑥 ∈ 𝐼𝑡
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Determination of the Probability Limits
• Note that
Φ0,𝑡 𝐻𝑡 𝛼𝑡
= 1 −𝛼𝑡
• To search for 𝐻𝑡 𝛼𝑡 , one can employ Newton’s method
Φ0,𝑡 𝐻𝑡𝑘−1 𝛼𝑡 + 𝛼𝑡 − 1
𝐻𝑡𝑘 𝛼𝑡 = 𝐻𝑡𝑘−1 𝛼𝑡 −
𝜑0,𝑡 𝐻𝑡𝑘−1 𝛼𝑡
where 𝜑0,𝑡 𝑥 = [Φ0,𝑡 𝐻𝑡 𝛼𝑡 ]′ .
65
Determination of the Probability Limits
•
When 𝑡 = 1
Φ0,1 𝑥 = 𝐹0 (𝑘1 +
𝜑0,1 𝑥 =
1
𝑓 (𝑘
𝑛1 0 1
𝑥
)
𝑛1
+
𝑥
),
𝑛1
where F0 (. ) and f0 (. ) represent the in-control CDF and PDF of the
standard normal distribution.
Φ0,1 𝑥 =
Φ0,1 𝑥
1−𝛼1
for 0 ≤ 𝑥 ∈ 𝐼𝑡
and
𝐻1 𝛼1 = 𝑛1 𝐹0−1 1 −𝛼1 − 𝑛1 𝑘1
66
Determination of the Probability Limits
• When 𝑡 ≥ 2
67
Determination of the Probability
Limits
68
Out-of-Control Performance
• Define 𝑝𝑡 as the probability that run length is larger than t, i.e.,
𝑝𝑡 = 𝑃𝑟1 (𝑅𝐿 > 𝑡),
+
• Define Φ1,𝑡
(𝑥) as the probability that the chart does not signal before
time t and the charting statistics less than or equal to x, i.e.,
+
Φ1,𝑡
(𝑥) = 𝑃𝑟1 (𝑆2,𝑡 ≤ 𝑥, 𝑆2,𝑗 ∈ 𝐼𝑗 , 𝑗 = 1, … , 𝑡 − 1),
+
+
• Note that 𝑝𝑡 = Φ1,𝑡
𝐻𝑡 𝛼𝑡 , we need to find Φ1,𝑡
𝐻𝑡 𝛼𝑡
compute 𝑝𝑡 .
in order to
69
Out-of-Control Performance
•
+
How to compute Φ1,𝑡
𝐻𝑡 𝛼𝑡 ?
–
Let 𝑞𝑡 be the probability that the chart does not signal at time t given no signal before time t, i.e.,
𝑞𝑡 = Φ1,𝑡 𝐻𝑡 𝛼𝑡 . Then
𝑝𝑡 = 𝑝𝑡−1 𝑞𝑡 = 𝑞1 𝑞2 … 𝑞𝑡 , for t=1, 2, …
–
In the OC case,
–
+
Based on Φ1,𝑡
𝑥 = Φ1,𝑡 (𝑥) 𝑞1 𝑞2 … 𝑞𝑡−1 , multiplying the above equation by 𝑞1 𝑞2 … 𝑞𝑡−1 leads
to
+
+
Φ1,𝑡
𝑥 = 𝐹𝑡 (𝑘𝑡 + (𝑥 − 𝐻𝑡−1 𝛼𝑡−1 )/ 𝑛𝑡 ) Φ1,𝑡−1
𝐻𝑡−1 𝛼𝑡−1
+
𝐻𝑡−1 𝛼𝑡−1
1
𝑛𝑡 0
+
+
𝑓𝑡 (𝑘𝑡 + (𝑥 − 𝑦)/ 𝑛𝑡 ) Φ1,𝑡−1
𝑦 𝑑𝑦
70
Out-of-Control Performance
• Algorithm for computing the out-of-control ARL
71
Performance Comparisons
• Approximation Accuracy
72
Performance Comparisons
• Comparison under varying sample sizes
73
Performance Comparison
• Comparison with CUSUM charts with fast initial response (FIR)
74
• Comparison with CUSUM charts with fast initial response (FIR)
75
Remarks
• An integral equation approach can be developed to determine the
probability limits of the CUSUM charts under varying sample sizes. It
is more efficient than the Monte Carlos simulation.
• Also, an integral equation approach can be employed to analyze the
out-of-control performance. The control chart with dynamic control
limits is a general idea and can be used to improve the ARL
performance of the control charts with a fixed control limit.
• The approach discussed here can be generalized any Markoviantype control charts with/without varying sample sizes.
76
Thank You!
Prof. Lianjie SHU
Email: [email protected]
University of Macau
Taipa, Macau, China
Tel: (853) 8397-4741
homepage:
http://www.umac.mo/fba/staff/shulianjie.html
77
Postgraduate Program
•
Requirements
(http://www.umac.mo/grs/en/admissions.php)
–
–
GPA 3/4 or above
English
•
•
•
•
Application Procedures: online
–
•
CET 6 or
TOFEL 550 or
IELTS 6
https://isw.umac.mo/naweb_grs/faces/index.jspx
Studentship
–
Mop $14,000/month
78