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ESTIMATING CHANGE POINT AND PROCESS MEAN IN CONTROL CHARTS
CHEN YAN
(M.Sc.)
NATIONAL UNIVERSITY OF SINGAPORE
2005
ESTIMATING CHANGE POINT AND PROCESS MEAN IN CONTROL CHARTS
CHEN YAN
(M.Sc. Huazhong University of Science & Technology)
A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF SCIENCE
DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY
NATIONAL UNIVERSITY OF SINGAPORE
2005
Acknowledgements
I would like to take this opportunity to express my sincere gratitude to my supervisor Assoc. Prof. Gan Fah Fatt for his patience, advice, and continuous support
throughout my study at National University of Singapore. I am really grateful to
him for his generous help and valuable suggestions to this thesis.
I wish to contribute the completion of this thesis to my dearest parents and
my husband who have always been supporting me with their encouragement and
understanding. Special thanks to all the staff in my department and all my friends
for their concern and inspiration in the two years.
3
Summary
Statistical process control (SPC) charts are used to monitor for process changes
by distinguishing the assignable causes of variation from the common causes of variation. When a control chart does signal that a process change has occurred, engineers
must initiate a search for the assignable cause and make a suitable adjustment. But
most of the research work on control charting procedures has concentrated on the
detection and signaling of process shifts. If a signal is due to an assignable cause,
then the magnitude of the shift and the time when the shift occurred are useful
information which can help engineers to narrow the search window and hence less
possible downtime. A signal issued could also be due to the naturally randomness
of a process and if evidence can be gathered in support of a false signal, then unnecessary and often expensive search effort can be avoided. All the published methods
of estimations of process mean associated with a signal are grossly biased and often
the bias depends on the magnitude of the shift.
In this thesis, we propose a new and effective method of estimating a process
i
mean following a signal and change point estimators based on this process mean
estimator when a step change occurs in a normal process mean. Then we discuss the performance of proposed and usually used change point and process mean
estimators when used with control charts. The simulation results show that the
proposed estimators provides a useful and much better alternative to the usually
used estimators. The new process mean estimator is able to remove the bias that
is inherent in existing methods. The performance of change point estimators will
increase based on this more accurate process mean estimator.
ii
Chapter 1
Introduction
Quality has always been an integral and important part of manufacturing and
service industries. It was not until 1924 that formal statistical methods for quality control were introduced. Shewhart (1924) developed a simple control charting
procedure for process monitoring and quality improvement. Subgroup mean or
individual observation of a quality characteristic is plotted against the time and
any point that is beyond the control limits provides evidence that a change in the
process mean might have occurred. When a process mean shifts, a signal from the
chart would allow quality control engineers to determine possible assignable causes.
A signal due to an assignable cause does suggest that a process change has occurred,
but it does not indicate what the cause is, nor does it indicate when the change
actually occurred.
1
Most of the research published in the literature has concentrated on the detection of process shifts. However, when a process shift has actually occurred, the
magnitude of the shift and the time when the shift occurred (change point) are
valuable information which can help quality control engineers to determine quickly
possible assignable causes and hence less process downtime. Note that the time
when a signal is issued is usually not the same as the change point. If the change
point could be determined, process engineers would have a smaller search window
within which to look for possible assignable causes. Consequently, the assignable
causes can be identified more quickly, and appropriate actions needed to improve
quality can be implemented sooner. Although estimations of change point and
process shift are important and related, joint research in these two areas is still
lacking.
A step change is a common type of change in industrial processes that often
results from tool breakage, the introduction of a new material or other abrupt and
sudden changes. This thesis will be based on the step change model. We also
assume that samples of size n are taken independently from a normal distribution
with mean µ and variance σ0 . We let τ denote the last sample number from the
¯1, X
¯ 2, · · · , X
¯ τ are the sample
in-control process. In other words, we assume that X
¯ τ +1 , X
¯ τ +2 , · · · , X
¯ T are from
means from the in-control process with mean µ0 , while X
2
the out-of-control process with mean µ1 , that is,
µ0 + εt , t ≤ τ
¯
Xt =
,
µ1 + εt , t > τ
(1.1)
where t = 1, 2, · · · , T and εt ∼ N(0, σ02 /n) and a signal is given at sample number
T . We will assume without loss of generality that µ0 = 0, σ0 = 1 and n = 1. The
main emphasis of this research is the estimation of the change point τ and shifted
process mean µ1 .
Barnard (1959) suggested that control charts should not only be used as a
monitoring tool. They should also be used to estimate the change point τ . For
a process shift δ, the upper and lower cumulative sums (CUSUMs)
+
St+ = max{0, Zt − k + St−1
},
(1.2)
−
St− = max{0, −k − Zt + St−1
},
(1.3)
and
¯ t −µ0 )/σX¯ ,
are compared to the control limits h+ and h− respectively, where Zt = (X
√
σX¯ = σ0 / n, µ0 and σ0 are the target or in-control process mean and standard
deviation respectively. The starting values are usually taken to be S0+ = 0 and
S0− = 0, and the reference k is positive. The CUSUM provides estimates of both
change point and magnitude of the process shift. Page (1954) proposed to use the
first point when a CUSUM chart became last active as a change point estimator,
that is,
τˆCU SU M = max{t : St+ = 0, 0 ≤ t < T },
3
(1.4)
for ST+ > h+ or
τˆCU SU M = max{t : St− = 0, 0 ≤ t < T },
(1.5)
for ST− > h− , where T in ST+ > h+ or ST− > h− is often referred to as the run
length and expectation of T , E(T ) is the average run length (ARL). Srivastava
(1993) discussed this estimator and concluded that it was biased. Hinkley (1971)
and Nishina (1992) studied the distribution of this estimator.
Pignatiello, Samuel and Calvin (1998) developed a maximum likelihood estimator (MLE) of the change point τ for a step change in a normal process mean. Let
¯ =
X
T,t
T
i=t+1
¯ i /(T − t) be an estimate of the shifted process mean µ1 based on the
X
last T − t subgroups, which is denoted as µ
ˆ1 . Their proposed MLE of the change
¯ − µ )2 over 0 ≤ t < T , that is
point is the value of t which maximizes (T − t)(X
T,t
0
¯ − µ )2 , 0 ≤ t < T }.
τˆM LE = arg max{(T − t)(X
T,t
0
t
(1.6)
Pignatiello and Samuel (2001) showed that this estimator is much less biased than
Page’s estimator τˆCU SU M . Siegmund (1986) also derived a confidence set for this
estimator. Furthermore, this estimator can be applied when a signal is given by
any control chart.
Nishina (1992) proposed a change point estimator based on a sequence of points
that plot continuously on one side of the center line of an exponentially weighted
moving average (EWMA) chart. The EWMA chart was introduced by Roberts
4
(1959) and it was obtained by plotting
Qt = λZt + (1 − λ)Qt−1 ,
(1.7)
against the sample number t, where 0 < λ ≤ 1 is a constant and the starting value
Q0 is usually chosen to be Q0 = µ0 . If Qt is larger than the upper control limit h+
or smaller than the lower control limit h− at time T . The control limits h+ and
h− can be taken as two constants or can be take as h+ = L
and h− = −L
λ
[1 − (1 − λ)2t ]
2−λ
λ
[1 − (1 − λ)2t ], where L is a suitably chosen positive constant.
2−λ
Nishina proposed the following change point estimators of τ
τˆEW M A = max{t : Qt ≤ 0, t = 1, 2, · · · , T − 1},
(1.8)
τˆEW M A = max{t : Qt ≥ 0, t = 1, 2, · · · , T − 1},
(1.9)
for QT > h+ and
for QT < h− .
Pignatiello and Samuel (2001) compared τˆM LE , τˆCU SU M and τˆEW M A based on
a simulation study. They concluded that τˆM LE is more accurate than τˆCU SU M and
τˆEW M A when there is an abrupt change in the process mean.
As for the estimation of shifted mean, Taguchi (1985) used the last observed in¯ T −1 based on a signal
control data point to estimate the shifted mean, that is, µ
ˆT = X
¯ chart. This estimator always overestimates the shifted mean
T of a Shewhart X
because the last data point is associated with an out-of-control signal. Wiklund
5
(1992) found that Taguchi’s method is biased for small to moderate shifts, and that
his MLE estimator performs comparatively better, but it still may be inefficient
especially for large shifts. Adams and Woodall (1989) also showed that the optimal
control parameters and loss functions given by Taguchi are severely misleading in
many situations. In Montgomery (2005, page 394), an estimator based on the
CUSUM chart is given as
µ
ˆM
S+
k + T , if ST+ > h+
N+
=
,
−
S
−
−
T
k−
, if ST > h
N−
(1.10)
where N + and N − is given as T − τˆCU SU M depending on whether ST+ > h+ or
ST− > h− respectively.
Since the EWMA chart was introduced by Roberts (1958), the chart was studied
by many researchers including Lucas and Saccucci (1990), Crowder (1987), Gan
(1991), MacGregor (1988), Box and Kramer (1992), Ingolfsson and Sachs (1993),
and also Yashchin (1995). EWMA is a popular and well-known statistic used for
smoothing and forecasting time series and as a process mean estimator, due to its
simplicity and ability to capture non-stationarity. The estimator of the current
process mean using EWMA is defined as:
¯ t + (1 − λ)ˆ
µ
ˆE,t = λX
µE,t−1,
6
(1.11)
which is equivalent to
¯ t + (1 − λ)X
¯ t−1 + (1 − λ)2 X
¯ t−2 + · · ·
µ
ˆ E,t = λ X
¯ 1 + (1 − λ)t X
¯0.
+ (1 − λ)t−1 X
(1.12)
Yashchin (1995) showed that µ
ˆE,t has optimality properties within the class of
linear estimators for estimating the current process mean of a process subject to
a step change. It is also well known that the EWMA has some optimal prediction properties and it is thus frequently used as a forecasting tool. The EWMA
is usually used as a one-step ahead forecast of the process mean. The EWMA estimator is optimal when the process mean follows a first-order integrated moving
average IMA(1,1) model (Box and Jenkins, 1970). In fact, the EWMA estimator
can be implemented as a widely used proportional-integral-derivative (PID) controller (˚
Astr¨om and H¨agglund, 1995). Although the Estimator performs well for
various other processes, few studies have analytically shown the estimator’s wide
applicability.
Wiklund (1992) proposed a MLE estimator of the process mean based on a
truncated normal probability density function. His estimation of the process mean
¯ chart. In his study, he concluded
is based on a signal point from a Shewhart X
that Taguchi’s method is biased for small to moderate shifts, MacGregor’s EWMA
estimator is not a sensitive estimator of the mean shift, and that his MLE estimator
performs comparatively better, but it may still be inefficient especially for large
7
shifts.
In order to understand change point and process mean estimation better, we
will first present Table 1.1. This will highlight problems associated with well-known
and established methods. In Table 1.1, we consider an out-of-control mean µ1 =
√
√
µ0 + δσ0 / n = δ for µ0 = 0 and σ0 / n = 1, where δ =0.00, 0.25, 0.50, 0.75 1.00,
1.50, 2.00, 3.00 at change point τ = 100. For each setting, repeated runs were
simulated, τˆCU SU M (Page, 1954), µ
ˆM (Montgomery, 2005) and E(T ) were found.
And a CUSUM chart with reference k = 0.5, the control limits h+ = 4.77 and
h− = 4.77 is considered in Table 1.1. The CUSUM chart with these parameters has
¯ chart. The expectation
an in-control ARL of 370, the same as the 3-σ Shewhart X
of T in Table 1.1 E(T ) is equal to 100 plus in-control ARL of CUSUM chart for
different δ. These in-control ARL values for different magnitudes of change are
computed using an integral equation computer program developed by Gan (1993).
Table 1.1: Estimation of Change Point Using Page’s Method and Estimation of
Shifted Mean Using Montgomery’s Method Based on 10,000 Simulation Runs
µ1
µ
ˆM
τ
τˆCU SU M E(T )
0.00 0.57 100 362.66 370.37
0.25 0.78 100 213.06 221.31
0.50 0.95 100 121.78 135.28
0.75 0.97 100 107.90 116.17
1.00 1.05 100 100.07 109.93
1.50 1.25 100
98.70
105.52
2.00 1.48 100
98.50
103.86
3.00 2.68 100
98.47
102.49
8
Table 1.1 reveals that for µ1 close to 1.00, µ
ˆM is nearly unbiased. It overestimates
µ1 for µ1 < 1.00, but underestimates µ1 for µ1 > 1.00. As the actual change point
was at time 100, τˆCU SU M should be close to 100 except for the case µ1 = 0.00.
In Table 1.1, τˆCU SU M based on a CUSUM chart’s signal is close to 100 for µ1
close to 1.00. τˆCU SU M overestimates the change point for small values of µ1 but
underestimates change point for large values of µ1 . For an upward shift in the
process mean, µ
ˆM is given as
µ
ˆM
ST+
= k + +,
N
where N + equals to T − τˆCU SU M and ST+ is given as
ST+ = ST+−1 + XT − k
= (ST+−2 + XT −1 − k) + XT − k
= (ST+−3 + xt−2 − k) + XT + XT −1 − 2k
..
.
= Sτˆ+CU SU M + XT + · · · + XτˆCU SU M +1 − (T − τˆCU SU M )k
= XT + · · · + XτˆCU SU M +1 − (T − τˆCU SU M )k.
Hence, Montgomery’s process mean estimator µ
ˆM is actually the sample average
from sample τˆCU SU M +1 to sample T . Thus µ
ˆM is based on a sample associated with
a signal which means that µ
ˆM must be biased. For small values of µ1 , although
the value of N + is comparatively large, the samples come from the right tail of
the distribution will cause that St+ signals after the steady state. So τˆCU SU M will
9
overestimate the process mean for small values of µ1 . For µ1 = 1.00, the estimation
of change point is almost nearly unbiased. N + samples after τˆCU SU M come from the
shifted process and N + is also a suitable sample number for estimation. Therefore,
the performance µ
ˆM is also nearly unbiased. For large values of µ1 , the value of N +
is quite small, and furthermore, we take both samples in control and samples out
of control to estimate process mean. So µ
ˆM will underestimate process mean for
large values of µ1 . The results here show that the inadequacy of well-known and
established methods and thus there is a need to find more accurate estimators of
change point and shifted process mean.
In Chapter 2, we will propose new change point and process mean estimators.
In Chapter 3, we present a comparison of the performance of the proposed and
commonly used change point and process mean estimators based on simulation
studies. Chapter 4 is a numerical example based on piston rings data so as to
provide a good understanding of the estimator discussed in previous chapters. This
is followed by a summary of the research and recommendations for future work in
Chapter 5.
10
Chapter 2
Estimating the Change Point and
Current Process Mean
In this chapter, we will introduce new change point and process mean estimators. In Section 2.1, we use adjusted MLE methods τˆM LE,µ∗1 ,S , τˆM LE,µ∗1 ,CU SU M and
τˆM LE,µ∗1 ,EW M A to estimate change point. In Section 2.2, new process mean estimators µ∗1,N =5 , µ∗1,N =10 , µ∗1,N =20 and µ∗1,N =50 associated with the CUSUM chart are
introduced.
11
2.1
Estimating the Change Point
¯ chart after a
It is well known that a signal could be issued by the Shewhart X
substantial amount of time from a change point. Estimating a change point using
the time at which a control chart signals would lead to a biased and, therefore,
possibly misleading estimate of the change point. This bias is due to the potentially
large delay in issuing a signal using a control chart. Thus, it is not suitable to use
the signal point T to estimate the change point τ .
Pignatiello, Samuel and Calvin (1998) considered the use of a MLE of the process
change point τ and investigated its performance based on a signal from a Shewhart
¯ chart. Their proposed estimator of the change point τ for a step change in a
X
normal process mean is given in equation 1.6. For this MLE estimator, T would
also be a signal point of other charts, such as a CUSUM or an EWMA chart.
We will proceed to derive the MLE of τ using a signal point T from the CUSUM
¯1, X
¯2, · · · , X
¯ T , the MLE of τ is
and EWMA charts. Given the subgroup averages X
the value of τ that maximizes the likelihood function or, equivalently, its logarithm.
The logarithm of the likelihood function can be derived as
¯ = − n
log L(τ, µ1 |X)
2σ02
n
= − 2
2σ0
τ
i=1
T
¯ i − µ 0 )2 +
(X
T
i=1
i=τ +1
¯ i − µ 1 )2
(X
τ
¯ 2 − 2µ0
X
i
i=1
T
¯ i + τ µ2 − 2µ1
X
0
i=τ +1
¯ i + (T − τ )µ2
X
1
.
(2.1)
12
¯ =
If the change point were known, the MLE of µ1 would be µ
ˆ1 = X
T,τ
1
T −τ
T
¯i,
X
i=τ +1
the average of the T − τ most recent samples. It can be shown that equation 2.1 is
equivalent to
T
¯ = − n
log L(τ, µ1 |X)
2σ02
i=1
T
¯ 2 − 2µ0
X
i
¯ i + T µ2
X
0
i=1
¯ − µ )2 .
−(T − t)(X
T,t
0
(2.2)
Since µ0 is assumed to be 0 in Chapter 1, if we assume µ1 to be known (µ0 = µ1 )
but τ to be unknown, the MLE of τ can be obtained as the value of t which
¯ 2 , over 0 ≤ t < T , that is
maximizes (T − t)X
T,t
¯2 , 0 ≤ t < T .
τˆM LE,ˆµ1 = arg max (T − t)X
T,t
(2.3)
t
But µ1 is not likely to be known and we can use
µ∗1
1
=
N
T +N
¯i,
X
i=T +1
that is, the average of the next N observations after the signal point as an estimator
of µ1 into equation 2.1, which is equivalent to
¯ = −
log L(τ, µ1 |X)
n
2σ02
T
i=1
¯ i − µ 0 )2
(X
¯ − 1 µ∗2 + 1 µ2
−2(T − t) (µ∗1 − µ0 )X
T,t
2 1 2 0
,
(2.4)
then τˆM LE,µ∗1 is given as
¯ µ∗ − 1 µ∗ 2 ), 0 ≤ t < T
τˆM LE,µ∗1 = arg max (T − t)(X
T,t 1
t
2 1
13
.
(2.5)
We propose that these MLE change point estimators, τˆM LE,ˆµ1 and τˆM LE,µ∗1 , can
¯ chart, a CUSUM chart or an EWMA
be used with a signal from a Shewhart X
chart.
2.2
Estimating the Shifted Process Mean
In traditional statistical process control (SPC) it is frequently assumed that
an initially in-control process is subjected to random shocks, which may shift the
process mean to an off-target value. Then a control chart is employed to detect such
a shift in mean. The estimation of the current process mean provides opportunities
for quality monitoring and fault diagnosis. In many cases when the resulting process
output deviation can be adjusted to bring the process output to the target value, a
good estimator will certainly provide a more accurate evaluation on how much the
adjustment should be made.
Having seen that using the last in-control sample as the sole basis for estimating
a process mean always overestimates the process mean (Taguchi, 1985) and a well
published method given in Montgomery (2005, page 394) that is biased for nearly
every situation, we will proceed to find better estimators. We first examine the case
where the change point is known based on an out-of-control signal from a Shewhart
¯ chart. Given the change point information, the process mean estimator can be
X
derived easily. Suppose that the current observation is T and the most recent mean
14
change occurred at observation τ . A naive estimate of the process mean is the
sample mean based on samples τ + 1, τ + 2, · · · , T,
T
µ
ˆ1 =
¯i
X
i=τ +1
T −τ
.
(2.6)
Since the shift occurred after τ , all samples from τ + 1 onwards have the same
mean. However, this estimate has large variance especially when T − τ is small
which is the case that the shift is large. The shifted mean by
T
µ
ˆ1 =
¯i
X
i=ˆ
τ +1
T − τˆ
,
(2.7)
using estimators derived earlier, where τˆ can be replaced by different point estimators to obtain different estimators. Montgomery’s process mean estimator µ
ˆM is
actually a special case of µ
ˆ1 in equation 2.7, that is, µ
ˆM is a sample average from
τˆCU SU M + 1 to T based on a CUSUM chart as discussed in Chapter 1.
The estimators µ
ˆM and µ
ˆ 1 are always biased because the samples are associated
with a signal from a chart. To get unbiased estimators after a signal, the process
is allowed to continue without adjustment until N additional subgroup means have
been observed. This aims at collecting information on the magnitude of shift. It
is certainly not necessary to maintain the same subgroup size. Taking immediate
samples after a signal will also allow the signal to be checked to see if it is a genuine
out-of-control signal or it is a signal due to randomness; it is advisable to alter sampling frequency and sample size as before the signal during this period to minimize
15
¯ T +1 , X
¯ T +2 · · · , X
¯ T +N denote the N subgroup means
defective production. Let X
collected following the out-of-control signal; note that since there is no condition on
these values being in or outside the control limits, these constitute random samples
from the process distribution. Hence, an unbiased estimator of µ1 is given as
µ∗1
1
=
N
T +N
¯i.
X
(2.8)
i=T +1
The signal time T in equation 2.8 can be from a Shewhart chart, a CUSUM chart
or an EWMA chart.
The use of exponential smoothing for forecasting was first arrived at empirically
on the grounds that it was a weighted average with the sensible property of giving
most weight to the last observation and less to the next-but-last and so on. Thus,
the general idea is that, given data up to and including time t, which is then called
the forecast origin, we can use the EWMA Qt or µ
ˆ E,t to provide an estimate of the
next value Qt+1 . For µ
ˆE,t , if the value of Qt remains large, the estimator becomes
oversensitive. We should make some adjustment on this estimator to overcome the
trade-off between large and small Qt and to design a more effective estimator for
processes subject to sudden shifts. It is effective in many applications.
In order to swiftly compensate for the sudden shift, the value of λ should be
set larger instantly after the change point to capture the shift. However, the step
change occurs only once and the process mean remains unchanged after τ . If the
value of λ remains large, the estimator becomes oversensitive to the white noises.
16
A novel dynamic-tuning EWMA estimator was proposed by Guo (2002) that has
the capability of adjusting the control parameter dynamically in response to the
underlying process random shifts. The current process mean is estimated using
the EWMA equation and the newly adjusted control parameter. The proposed
estimator is very easy to implement and effective under many disturbance situations.
17
Chapter 3
Comparison of Change Point and
Process Mean Estimators
In this chapter, we compare the performance of all the change point estimators
and process mean estimators developed in the previous chapter with the performance of the commonly used estimators. The parameters of the charts are chosen
such that each chart has an in-control ARL of 370. We consider a CUSUM chart
with parameters k = 0.5, h+ = 4.77 and h− = 4.77, and an EWMA chart with
control parameters λ = 0.14, h+ = 0.7628 and h− = −0.7628. The CUSUM and
EWMA charts with these parameters have the same in-control ARL as a 3-σ Shewhart chart. Moreover, the EWMA chart with these parameters is also optimal
in detecting µ1 = 1, the same as the CUSUM chart. The ARL profiles of these
18
σ
σ
control charts is presented in Table 3.1 for µ1 = µ0 + δ √ , where µ0 = 0, √ = 1,
n
n
and δ = 0.00, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 3.00. We can see that an EWMA chart
with constant control limits has similar ARL performance to an EWMA chart with
varying control limits. Hence, it’s sufficient to use the EWMA chart with λ = 0.14,
h+ = 0.7628 and h− = −0.7628 in our simulation study.
Table 3.1. ARL Profiles of 3-σ Shewhart chart, CUSUM chart
(k = 0.5, h± = 4.77), EWMA chart (λ = 0.14, h± = ±0.7628) and
EWMA chart (λ = 0.14, L = 2.785)
µ1 Shewhart CUSUM EW MA1 EW MA2
0.00
370
370
370
370
0.25
279.58
121.31
102.39
103.76
0.50
155.86
35.21
30.98
30.94
0.75
81.68
16.17
15.14
15.24
1.00
44.09
9.92
9.58
9.61
1.50
15.06
5.52
5.46
5.47
2.00
6.31
3.86
3.86
3.87
3.00
1.99
2.48
2.48
2.52
We used simulation to study the performance of the change point estimators.
The change point was fixed at sample τ = 10, 50, 100. Observations were randomly
generated from a standard normal distribution for samples 1, 2, · · · , τ , where none
of them issues a signal. Then starting after sample τ , observations were randomly
generated from a normal distribution with mean µ1 = δ and standard deviation
1 until a signal is issued. The simulation was repeated for each of the values of
δ studied. The number of simulation runs for each case was selected such that
the standard error of the estimate is less than 1% of the average. The results are
19
displayed in Tables 3.2-3.4.
Consider the case µ1 = 0.25 and τ = 10 in Table 3.2, τˆM LE,ˆµ1 ,S is 60.82 (closer
to τ = 10) and τˆM LE,µ∗1,N=5 ,S is 173.22 although µ∗1,N =5 has an average of 0.2438
which is much closer to µ1 = 0.25 as compared to µ
ˆ1 which has an average value of
0.9017. According to the minimazation function
τ
τˆ =
τ : max
i=1
T
¯ i − µ 0 )2 +
(X
¯ i − µ 1 )2
(X
i=τ +1
the accuracy of a change point estimate depends on the accuracy of the process
mean estimator. Although µ∗1,N =5 = 0.2438 is much closer to µ1 = 0.25, this value
is an average of all the simulation runs, and for N = 5, the variance of the process
mean estimator is large. If we examine the median of the change point estimators,
we can see than the two are much closer, suggesting that τˆM LE,µ∗1,N=5 ,S is far more
right-skewed than τˆM LE,ˆµ1 ,S .
ˆ samples are used to estimate µ
For τˆM LE,ˆµ1 ,S , an average of N
ˆ1 as compared to
τˆM LE,µ∗1,N=5 ,S , where only 5 samples are used to estimate µ∗1,N =5 . This explains why
τˆM LE,ˆµ1 ,S is much more accurate than τˆM LE,µ∗1,N=5 ,S . As N increases, change point,
process mean estimators improves significantly. For τ = 10, the accuracy of both
estimators improves as µ1 increases. As τ increases from 10 to 100, the accuracy of
change point estimators increases. The reason is that the accuracy also depends on
the number of samples used before and after the change point τ . If more samples
20
are available to compute
τ
i=1
T
¯ i − µ 0 )2 +
(X
¯ i − µ 1 )2 ,
(X
i=τ +1
then this should results in more accurate change point and process mean estimators.
In general, MLE change point estimators based on CUSUM chart have a similar
performance to those based on Shewhart chart. The overall performance of the
two is similar. Except for the case µ1 = 0.25 and τ = 10, the difference between
τˆM LE,ˆµ1 ,S and τˆM LE,µ∗1,N=5 ,S is much larger than the difference between τˆM LE,ˆµ1 ,C
and τˆM LE,µ∗1,N=5 ,C . The reason is that T is much smaller and hence smaller number
of samples are used in the minimization function for the CUSUM chart.
21
22
0
10
10
10
10
10
10
10
50
50
50
50
50
50
50
100
100
100
100
100
100
100
0
0.25
0.50
0.75
1.00
1.50
2.00
3.00
0.25
0.50
0.75
1.00
1.50
2.00
3.00
0.25
0.50
0.75
1.00
1.50
2.00
3.00
137.83
103.91
101.10
100.40
99.67
99.65
99.69
91.83
55.45
51.51
50.45
49.95
49.71
49.76
367.42
60.82
18.56
12.72
11.15
10.21
9.99
9.87
τ τˆM LE,ˆµ1 ,S
µ1
118
102
100
100
100
100
100
69
50
50
50
50
50
50
259
34
12
11
10
10
10
10
0.8524
0.9536
1.1633
1.4008
1.9235
2.4624
3.2663
0.8678
0.9625
1.1611
1.4190
1.9964
2.4655
3.2800
0.0016
0.9017
0.9923
1.1856
1.4305
1.9926
2.4982
3.3246
246
154
82
46
16
8
4
237
152
79
45
16
8
3
4
231
149
79
45
15
8
3
254.86
136.03
106.10
100.64
99.86
99.76
100.02
207.16
91.06
57.94
51.56
49.76
49.93
49.98
327.71
173.22
56.23
20.54
13.01
10.04
9.97
10.00
164
104
100
100
100
100
100
112
54
50
50
50
50
50
227
73
15
10
10
10
10
10
0.2503
0.4850
0.7478
1.0132
1.5299
1.9740
3.0070
0.2459
0.5002
0.7548
1.0199
1.5164
2.0344
2.9985
0.0015
0.2438
0.5020
0.7477
1.0059
1.4922
2.0040
2.9811
215.99
116.05
100.86
99.99
99.98
99.95
100.01
169.45
69.46
52.15
50.14
50.02
44.99
49.99
321.43
135.31
34.67
14.36
10.99
9.97
9.97
10.01
129
101
100
100
100
100
100
80
51
50
50
50
50
50
211
44
12
10
10
10
10
10
0.2476
0.4979
0.7517
1.0059
1.5154
1.9984
2.9867
0.2448
0.4993
0.7503
1.0102
1.5104
2.0086
3.0093
0.0014
0.2428
0.5032
0.7507
1.0092
1.4959
1.9898
3.0005
107.24
100.53
100.05
100.05
100.06
100.02
99.99
64.78
50.65
50.03
50.14
49.96
49.93
49.98
246.91
34.76
14.58
11.14
10.30
9.99
10.00
10.04
100
100
100
100
100
100
100
51
50
50
50
50
50
50
0.2497
0.2497
0.7496
1.0010
1.4968
1.9974
2.9941
0.2502
0.5002
0.7495
1.0007
1.4983
1.9995
2.9997
138 -0.0010
15 0.2498
10 0.5009
10 0.7495
10 1.0004
10 1.5007
10 1.9999
10 2.9987
Table 3.2. MLE Change Point and Process Mean Estimators Based on a Shewhart Chart†
ˆ τˆM LE,µ∗
m1,S µ
ˆ1
N
m2,S µ∗1,N =5 τˆM LE,µ∗1,N =10 ,S m3,S µ∗1,N =10 τˆM LE,µ∗1,N =200 ,S m4,S µ∗1,N =200
1,N =5 ,S
†
¯ (taking information of N
ˆ
τˆM LE,ˆµ1 ,S MLE change point estimator with µ
ˆ1 = X
T,τ
¯ chart
samples) based on a signal from the 3-σ Shewhart X
m1,S median of τˆM LE,ˆµ1 ,S
τˆM LE,µ∗1,N=5 ,S MLE change point estimator with µ∗1,N =5 based on a signal from the
¯ chart
3-σ Shewhart X
τˆM LE,µ∗1,N=10 ,S MLE change point estimator with µ∗1,N =10 based on a signal from the
¯ chart
3-σ Shewhart X
τˆM LE,µ∗1,N=200 ,S MLE change point estimator with µ∗1,N =200 based on a signal from
¯ chart
the 3-σ Shewhart X
µ∗1,N =5 , µ∗1,N =10 and µ∗1,N =200 are MLE process mean estimators using N = 5, 10, 200
samples after a signal based on a 3-σ Shewhart chart
m2,S , m3,S and m4,S medians of τˆM LE,µ∗1,N=5 ,S , τˆM LE,µ∗1,N=10 ,S and τˆM LE,µ∗1,N=200 ,S respectively
23
24
145.19
106.08
100.93
99.21
99.43
99.53
99.80
126
103
101
100
100
100
100
0.95
1.0634
1.2379
1.3980
1.7789
2.2291
3.1449
0.9523
1.0644
1.2467
1.4131
1.8044
2.1828
3.1177
72
28
14
10
6
4
2
70
27
14
9
6
4
2
164.32
105.08
97.68
97.97
99.83
99.95
99.98
118.96
58.63
49.84
48.88
49.82
49.94
50.02
132
102
100
100
100
100
100
84
53
50
50
50
50
50
0.2475
0.4974
0.7460
0.9933
1.5157
2.00
2.9831
0.2462
0.4988
0.7469
0.9951
1.4949
1.9921
2.9759
0.0027
0.2506
0.4952
0.7545
1.0206
1.4880
2.0004
2.9997
149.70
100.66
98.59
99.43
99.96
100.07
100.00
105.33
54.39
49.55
49.59
49.95
49.99
50.06
325.68
71.15
18.32
11.86
10.20
9.94
9.99
10.00
119
101
100
100
100
100
100
71
51
50
50
50
50
50
218
32
11
10
10
10
10
10
0.2512
0.4997
0.7495
1.0038
1.4882
1.9976
2.9975
0.2475
0.5036
0.7518
0.9999
1.5183
2.0055
2.9937
0.0033
0.2508
0.4988
0.7539
0.9983
1.4882
1.9989
3.0056
108.13
100.75
100.32
100.32
100.02
100.04
100.02
64.34
51.05
50.15
49.93
50.02
50.02
50.01
263.51
32.45
14.07
11.04
10.41
10.04
9.98
9.99
101
100
100
100
100
100
100
52
50
50
50
50
50
50
0.2498
0.5010
0.7501
1.0005
1.4993
1.9985
2.9997
0.2503
0.5002
0.7492
1.0003
1.5003
2.0010
2.9976
146 -0.0004
15 0.2500
10 0.4998
10 0.7492
10 1.0017
10 1.4998
10 1.9989
10 2.9982
213.06
121.78
107.90
100.07
98.70
98.50
98.47
128.61
71.79
60.12
50.14
48.77
48.54
48.42
362.66
71.25
31.73
18.74
10.31
8.92
8.64
8.63
0.51
0.72
0.87
0.96
1.20
1.65
2.84
0.64
0.86
0.95
1.05
1.25
1.49
2.68
0.58
0.87
0.97
1.07
1.20
1.37
1.68
2.43
100
100
100
100
100
100
100
78
53
51
50
50
50
50
229
44
13
10
10
10
10
10
0.25
0.50
0.75
1.00
1.50
2.00
3.00
98.18
57.16
51.55
49.74
49.59
49.15
49.73
336.06
83.93
22.23
12.58
10.44
9.93
9.98
9.9970
50
50
50
50
50
50
50
10
65
24
12
8
5
4
3
0.25
0.50
0.75
1.00
1.50
2.00
3.00
0.0059
0.9645
1.0928
1.2582
1.4539
1.8485
2.2644
3.1514
0
0.25
0.50
0.75
1.00
1.50
2.00
3.00
228
40
14
11
10
10
10
10
0
10
10
10
10
10
10
10
µ1
367.02
63.00
19.65
12.95
10.96
10.03
9.80
9.86
Table 3.3. MLE Change Point and Process Mean Estimators Based on a CUSUM Chart (k = 0.5, h± = 4.77)
ˆ τˆM LE,µ∗
τ τˆM LE,ˆµ1 ,C m1,C µ
ˆ1 N
m2,C µ∗1,N =5 τˆM LE,µ∗1,N =10 ,C m3,C µ∗1,N =10 τˆM LE,µ∗1,N =200 ,C m4,C µ∗1,N =200 τˆCU SU M µ
ˆM
1,N =5 ,C
¯
ˆ
τˆM LE,ˆµ1 ,C MLE change point estimator with µ
ˆ1 = X
T,τ (taking information of N
samples) based on a signal from the CUSUM chart (k = 0.5 and h± = 4.77)
m1,C median of τˆM LE,ˆµ1 ,C
τˆM LE,µ∗1,N=5 ,C MLE change point estimator with µ∗1,N =5 based on a signal from the
CUSUM chart (k = 0.5 and h± = 4.77)
τˆM LE,µ∗1,N=10 ,C MLE change point estimator with µ∗1,N =10 based on a signal from the
CUSUM chart (k = 0.5 and h± = 4.77)
τˆM LE,µ∗1,N=200 ,C MLE change point estimator with µ∗1,N =200 based on a signal from
the CUSUM chart (k = 0.5 and h± = 4.77)
µ∗1,N =5 , µ∗1,N =10 and µ∗1,N =200 are MLE process mean estimators using N = 5, 10, 200
samples after a signal based on a CUSUM chart (k = 0.5 and h± = 4.77)
m2,C , m3,C and m4,C medians of τˆM LE,µ∗1,N=5 ,C , τˆM LE,µ∗1,N=10 ,C and τˆM LE,µ∗1,N=200 ,C
respectively
τˆCU SU M Page’s change point estimator based on the CUSUM chart (k = 0.5 and
h± = 4.77)
µ
ˆM =
T
i=ˆ
τCU SU M +1
¯i
X
T − τˆCU SU M
is a process mean estimator based on a CUSUM chart
(k = 0.5 and h± = 4.77), which is a special case of µ
ˆ1
25
26
144.10
106.70
100.61
99.43
99.29
99.15
99.58
127
104
101
100
100
100
100
1.0235
1.1240
1.2563
1.4136
1.8166
2.2016
3.0772
1.0237
1.1329
1.2681
1.4298
1.8031
2.1838
3.1117
59
24
14
10
6
5
3
56
23
13
9
6
4
3
154.93
103.75
98.01
98.13
99.65
99.92
100.04
109.49
56.94
49.75
49.27
49.75
49.97
50.01
130
102
100
100
100
100
100
80
52
50
50
50
50
50
0.2538
0.5009
0..7478
0.9902
1.4843
2.0059
2.9846
0.2509
0.4945
0.7470
1.0068
1.4807
2.0126
2.9875
141.95
100.65
98.30
99.13
100.08
99.93
100.03
97.14
53.47
49.31
49.78
49.92
50.07
50.01
332.74
63.01
17.50
11.80
10.46
10.21
9.93
10.00
120
101
100
100
100
100
100
68
51
50
50
50
50
50
215
31
11
10
10
10
10
10
0.2548
0.5014
0.7492
1.0109
1.5047
1.9896
2.9989
0.2527
0.4992
0.7476
1.0037
1.4981
2.0199
2.9968
0.0002
0.2556
0.5004
0.7520
1.0087
1.5176
2.0061
3.0067
107.62
100.59
100.29
99.96
99.97
10.06
99.99
62.60
57.18
50.36
50.14
50.15
50.0150
49.99
266.79
32.21
13.89
10.96
10.25
10.04
9.99
9,97
101
100
100
100
100
100
100
51
50
50
50
50
50
50
151
15
10
10
10
10
10
10
0.2506
0.5004
0.7502
1.0015
1.4991
1.9990
3.0010
0.2502
0.5009
0.7504
1.0013
1.4957
1.9996
2.9988
0.0000
0.2505
0.4998
0.7496
1.0009
1.4977
1.9998
2.9979
175.14
104.18
102.78
96.50
95.75
95.74
95.43
164.68
53.73
49.59
46.68
45.86
45.62
45.63
340.26
69.01
15.21
11.35
8.26
7.66
7.53
7.32
0.90
1.05
1.12
1.21
1.27
1.35
1.40
1.02
1.13
1.21
1.26
1.29
1.31
1.44
0.86
1.12
1.20
1.28
1.32
1.38
1.42
1.44
100
100
100
100
100
100
100
78
54
51
50
50
50
50
41 -0.0030
13 0.2526
10 0.4975
10 0.7570
10 1.0053
10 1.5032
10 2.0020
10
2.99
0.25
0.50
0.75
1.00
1.50
2.00
3.00
95.54
57.66
51.66
49.66
49.46
49.19
49.74
344.69
73.43
20.88
12.48
10.33
10.11
9.96
10.01
50
50
50
50
50
50
50
11
50
21
12
8
6
4
3
0.25
0.50
0.75
1.00
1.50
2.00
3.00
-0.0138
1.0515
1.1469
1.2956
1.4707
1.8328
2.2059
3.1102
0
0.25
0.50
0.75
1.00
1.50
2.00
3.00
146
41
15
11
10
10
10
10
0
10
10
10
10
10
10
10
µ1
360.18
60.63
19.72
12.96
10.91
9.85
9.67
9.84
Table 3.4. MLE Change Point and Process Mean Estimators Based on an EWMA Chart (λ = 0.14, h± = ±4.77)
ˆ τˆM LE,µ∗
τ τˆM LE,ˆµ1 ,E m1,E µ
ˆ1 N
m2,E µ∗1,N =5 τˆM LE,µ∗1,N =10 ,E m3,E µ∗1,N =10 τˆM LE,µ∗1,N =200 ,E m4,E µ∗1,N =200 τˆEW M A µ
ˆE
1,N =5 ,E
¯
ˆ
τˆM LE,ˆµ1 ,E MLE change point estimator with µ
ˆ1 = X
T,τ (taking information of N
samples) based on a signal from the EWMA chart (λ = 0.14 and h± =
±0.7628)
m1,E median of τˆM LE,ˆµ1 ,E
τˆM LE,µ∗1,N=5 ,EW M A MLE change point estimator with µ∗1,N =5 based on a signal from
the EWMA chart (λ = 0.14 and h± = ±0.7628)
τˆM LE,µ∗1,N=10 ,EW M A MLE change point estimator with µ∗1,N =10 based on a signal from
the EWMA chart (λ = 0.14 and h± = ±0.7628)
τˆM LE,µ∗1,N=200 ,EW M A MLE change point estimator with µ∗1,N =200 based on a signal
from the EWMA chart (λ = 0.14 and h± = ±0.7628)
µ∗1,N =5 , µ∗1,N =10 and µ∗1,N =200 are MLE process mean estimators using N = 5, 10, 200
samples after a signal based on an EWMA chart (λ = 0.14 and h± = ±0.7628)
m2,E , m3,E and m4,E medians of τˆM LE,µ∗1,N=5 ,E , τˆM LE,µ∗1,N=10 ,E and τˆM LE,µ∗1,N=200 ,E
respectively
τˆEW M A Nishina’s change point estimator based on an the EWMA chart (λ = 0.14
and h± = ±0.7628)
µ
ˆE = QT is a process mean estimator based on an EWMA chart (λ = 0.14 and
h± = ±0.7628)
27
Page’s estimator τˆCU SU M and Nishina’s estimator τˆEW M A are the worst compared with the other MLE estimators. They underestimate τ for large shifts in
the mean. They overestimate τ for small shifts. It is clear from Figure 3.1 that
for a small shift δ = 0.5, a CUSUM chart might quite likely become inactive and
then active again after τ = 100. This explains the overestimation of τ by τˆCU SU M
for small shifts in the mean. From this figure, we can see that the signal point is
T = 117 and the change point estimator is τˆCU SU M = 103. For a large shift δ = 2.0
after the change point, a CUSUM chart is likely to be active before τ = 100 and
quickly issues a signal. For δ = 2.0, τˆCU SU M = 97 and T = 103. Figure 3.2 helps
to explain as to why τˆCU SU M underestimates τ = 100 for large shifts. τˆEW M A and
τˆCU SU M are similar in the way they estimate the change point, thus τˆEW M A also
underestimates τ for large shifts like δ = 2.00 and overestimates τ for small shifts
like µ1 = 0.25.
28
3
2
0
1
Upper Cumulative sums
4
5
h+=4.77
0
20
40
60
80
100
120
Observation Number
4
h+=4.77
0
2
Upper Cumulative sums
6
8
Figure 3.1: A Typical CUSUM Chart with a Shift of δ = 0.5 at τ = 100
0
20
40
60
80
100
Observation Number
Figure 3.2: A Typical CUSUM Chart with a Shift of δ = 2.0 at τ = 100
29
In general, MLE estimators based on CUSUM and EWMA charts have similar
performance. This is because the quantity T is similar for the CUSUM and EWMA
charts. They are almost spot on for µ1 ≥ 1.00. Overestimation of τ becomes
more severe for smaller shifts in the mean. For small shifts in the mean, the MLE
estimators based on EWMA charts have slightly better performance than the MLE
estimators based on CUSUM charts. Estimation of change point is appreciably
affected by the accuracy of the estimators of the process mean. That is, MLE
estimators based on µ∗1 perform better for accurate estimation of process mean
based on a larger sample size N.
¯ chart, τˆM LE,µ∗ ,S is much closer to the actual change point
For the Shewhart X
1
than τˆM LE,ˆµ1 ,S regardless of the magnitude of the change. For large step changes
in the process mean, the chances of identifying correctly the time of the change
increase for these estimators.
¯ chart issues a signal is
The reason is that the value of T at which a Shewhart X
larger than the value of T at which a CUSUM or an EWMA chart issues a signal.
¯ chart takes more information than the CUSUM and EWMA
Hence the Shewhart X
chart in estimating the change point to get a more accurate estimate.
Tables 3.2-3.4 also contain the simulation results for process mean. EWMA
estimator µ
ˆ E is not a sensitive estimator of the shifts. The CUSUM estimator µ
ˆM
performs comparatively better. They are both biased when the shift is too small or
30
too large, because µ
ˆ M and µ
ˆ E are associated with signal.
In Table 3.2, µ∗1,N =5 , µ∗1,N =10 and µ∗1,N =200 are sample averages of 5, 10 and 200
observations after a signal from a Shewhart chart. The performance of µ∗1,N =5 ,
µ∗1,N =10 and µ∗1,N =200 is much better than that of all the other estimators and also
they are better than that of µ
ˆ1 . These four estimator are unbiased. From the
performance of these four estimators, we can see that even for sample size N as
small as 5 it is still sufficient for us to get an unbiased process mean estimator. The
estimates obtained for µ∗1,N =5 , µ∗1,N =10 and µ∗1,N =200 are similar for various charts.
31
Chapter 4
Applications
In this chapter, we will give one numerical example based on a data set so as to
provide a good understanding of the change point and process mean estimators. We
use the piston rings data set in Montgomery (2005). Piston rings for an automotive
engine are produced by a forging process. We will estimate change point and the
process mean of the inside diameter (mm) of the rings manufactured by this process
with control charts. Thirty-seven samples, each of size five, have been taken from
the process. The first 25 samples are from an in-control process with an in-control
mean µ
ˆ0 = 74.0012 and a standard deviation σ
ˆ0 = 0.0047, and the last 12 samples
are from an out-of-control mean µ
ˆ1 = 74.0049 and a standard deviation σ
ˆ0 = 0.0047.
The Shewhart chart, CUSUM charts and EWMA charts constructed using this data
set are displayed in Figures 4.1-4.5.
32
74.03
74.02
xbar
73.99
74.00
74.01
UCL=74.014
LCL=73.9869
0
5
10
15
20
25
30
35
sample number
0.010
0.015
0.020
H+=0.0226
0.000
0.005
Upper Cumulative sums
0.025
0.030
¯ Chart for the Piston Rings Data
Figure 4.1: Shewhart X
0
5
10
15
20
25
30
35
sample number
Figure 4.2: Upper CUSUM Chart (k = 0.5 and h+ = 4.77) for the Piston Rings
Data
33
0.015
0.010
0.000
0.005
Lower Cumulative sums
0.020
H−=0.226
0
5
10
15
20
25
30
35
sample number
Figure 4.3: Lower CUSUM Chart (k = 0.5 and h− = 4.77) for the Piston Rings
74.006
Data
74.002
73.998
74.000
EWMA
74.004
H+=74.0048
H−=73.99756
0
5
10
15
20
25
30
35
sample number
Figure 4.4: EWMA Chart (λ = 0.14 and h± = ±0.7628) for the Piston Rings Data
34
74.006
74.004
74.002
73.998
74.000
EWMA
UCL
LCL
0
5
10
15
20
25
30
35
sample number
Figure 4.5: EWMA Chart (λ = 0.14 and L = 2.785) for the Piston Rings Data
All the charts issued a signal at sample number 37. In order to demonstrate our
new methods, we simulated two additional runs of data after signal. We simulated
20 samples from a normal distribution with µ1 = 74.0049 (the same as the mean of
the last 12 samples) and σ
ˆ0 = 0.0047. In other words, we assume the out-of-control
mean to be 74.0049. We also simulated 20 samples from a normal distribution with
µ1 = 74.0012 (the same as in-control mean) and σ
ˆ0 = 0.0047.
The estimated change points using various methods discussed in earlier chapters
are displayed in Table 4.1. τˆCU SU M = 33 and τˆEW M A = 30 can be determined easily
from Figure 4.2 and Figure 4.4. For the change point estimators based on the
MLE method using µ∗1 , the estimate process mean µ∗1 was calculated using N = 20
samples. The Shewhart, CUSUM and EWMA charts all issued a signal at the same
35
sample number 37, thus τˆM LE,ˆµ1 ,S , τˆM LE,ˆµ1 ,C and τˆM LE,ˆµ1 ,E are the same. This is
because the estimation of τ depends only on the sample number at which a control
chart signals. For the change point estimators based on the MLE method using
µ∗1,N =20 , τˆM LE,µ∗1,N=20 ,S , τˆM LE,µ∗1 ,C and τˆM LE,µ∗1,N=20 ,E are also the same because the
method depend only on the sample number at which a signal is issued and the
value of µ∗1,N =20 . From Table 4.1, one could deduce that the change point occurred
at sample number 33 to 34. The change point estimate τˆEW M A is probably too
small. The EWMA charts in Figures 4.4-4.5 provide some evidence in support of
this claim.
Table 4.1: Estimation of Change Point Based on Piston Rings Data
τˆM LE,ˆµ1 ,S τˆM LE,µ∗1,N=20 ,S
τˆM LE,ˆµ1 ,C
τˆM LE,µ∗1,N=20 ,C
34
33
34
33
τˆCU SU M τˆM LE,ˆµ1 ,EW M A τˆM LE,µ∗1,N=20 ,EW M A
τˆEW M A
33
34
33
30
Table 4.2: Estimation of Process Mean Based on
µ1
µ
ˆT
µ
ˆM
µ
ˆE
µ∗1,N =5
74.0049 74.0040† 74.0096 74.0060 74.0044
(0.20)
(1.00)
(0.23)
(0.11)
74.0012 74.0040 74.0096 74.0060 73.9983
(0.60)
(1.79)
(1.02)
(0.61)
†
Piston Rings Data
µ∗1,N =10 µ∗1,N =20
74.0058 74.0048
(0.19)
(0.02)
73.9993 74.0011
(0.40)
(0.02)
74.0040 is (0.20) standard deviations away from µ1
The estimated process means are displayed in Table 4.2. The quantity µ
ˆE is for
an EWMA chart with constant control limits. For the EWMA chart with varying
36
control limits, the estimate obtained is similar. If we assume that µ1 = 74.0012,
note that the performance of µ
ˆT , µ
ˆM and µ
ˆE does not change because they are
calculated based on samples associated with the signal. This explains why these
estimators are biased. For this case of false signal, they give misleading estimates.
Our new estimators are able to distinguish between a genuine signal (µ1 = 74.0049)
and a false signal (µ1 = 74.0012).
37
Chapter 5
Conclusions
Control charts are usually used to detect process shifts. When a control chart
does signal that a process shift has occurred, process engineers will usually initiate a
search for possible assignable causes and make a suitable adjustment if such a cause
can be found. However, given a signal from a control chart, process engineers generally do not know what caused the process to change, when the process changed or
how much the process changed. Knowing the change point and the current process
mean would aid the search for possible assignable causes. This would increase their
chances of identifying correctly the assignable causes quickly and allow them to
take the appropriate actions immediately to improve quality.
Several new change point estimators and process mean estimators for detecting
abrupt step changes in the mean of a normal process were developed in Chapter 2.
38
These estimators provide process engineers with a tool to identify the assignable
causes more quickly and to collect information to improve the quality of the process
sooner.
In general, MLE estimators perform better than τˆCU SU M and τˆEW M A . For She¯ chart, τˆM LE,µ∗ ,S with N = 200 is fairly closer to the actual change point
whart X
1
than τˆM LE,ˆµ1 ,S regardless of the magnitude of the change. The accuracy of these
estimators increases with the magnitude of the shift size. MLE estimators based
on CUSUM and EWMA charts have similar performance.
τˆM LE,µ∗1 ,S , τˆM LE,µ∗1 ,C and τˆM LE,µ∗1 ,E based on µ∗1 with sample size N = 200 have
better performance than τˆM LE,ˆµ1 ,S , τˆM LE,ˆµ1 ,C and τˆM LE,ˆµ1 ,E based on µ
ˆ1 . Thus
estimation of change point is affected by the accuracy of the estimators of the
process mean. When τ increases, the accuracy of all the change point estimators
increase accordingly.
As for the process mean estimators, µ
ˆT , µ
ˆM and µ
ˆE are biased because these
estimators are associated with a signal. However, µ∗1,N =5 , µ∗1,N =10 and µ∗1,N =200 are
based on samples independent of the control chart’s signal and hence are unbiased
estimates of the shifted mean. The study conducted here further shows that these
estimators are more accurate process mean estimators. These estimators can easily
distinguish a genuine signal from a false signal.
The estimators presented in this thesis have only been applied to abrupt step
39
changes in a normal mean. These estimators could be used for estimating other
types of process shifts. These estimators could be also used to estimate change
point and process mean shift based on autocorrelated samples and for multivariate
control charts.
40
Bibliography
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˚
Astr¨om, K. L. and H¨agglund, T. (1995). PID Controller: Theory, Design and
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B21, pp. 239–267.
Box, G. and Kramer, T. (1992). “Statistical process monitoring and feedback adjustment - a discussion”. Technometrics 34, pp. 251–267.
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control”. Biometrika 57, pp. 1–17.
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ution of an Exponentially Weighted Moving Average Control Chart”. Journal of
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Wiley and Sons, New York.
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43
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44
Appendix
SAS code for change point and process mean estimators in Chapter 3
τˆM LE,ˆµ1 ,S , µˆ1
data;
seed = 727189910;
delta = 0;
tau = 100;
taup1 = tau + 1;
lcl = -3;
ucl = 3;
maxrun = 10000;
maxsam = 5000;
array xxx xxx1-xxx5000;
45
do runno = 1 to maxrun;
signal:
do iii = 1 to tau;
xxx{iii} = rannor(seed);
if (xxx{iii} > ucl or xxx{iii} < lcl) then goto signal;
end;
do iii = taup1 to maxsam;
xxx{iii} = rannor(seed) + delta;
if (xxx{iii} > ucl or xxx{iii} < lcl) then goto endrun;
end;
problem="problem";
output;
stop;
endrun:
ttt = iii;
mu1 = 0;
do iii = 1 to ttt;
mu1 = mu1 + xxx{iii};
end;
mu1 = mu1/ttt;
46
sumx = 0;
do iii = 1 to ttt;
arg = xxx{iii} - mu1;
sumx = sumx + arg*arg;
end;
mint = 0;
muhat = mu1;
tttm1 = ttt - 1;
do tt = 1 to tttm1;
mu1 = 0;
ttp1 = tt + 1;
do iii = ttp1 to ttt;
mu1 = mu1 + xxx{iii};
end;
mu1 = mu1/(ttt-tt);
sumy = 0;
do iii = 1 to tt;
sumy = sumy + xxx{iii}*xxx{iii};
end;
do iii = ttp1 to ttt;
arg = xxx{iii} - mu1;
47
sumy = sumy + arg*arg;
end;
*output;
if (sumy < sumx) then do;
sumx = sumy;
mint = tt;
muhat = mu1;
end;
end;
output;
end;
keep mint muhat ttt;
proc print; run; proc means; var mint muhat ttt; run; proc means
stderr; var mint muhat ttt; proc sort; by mint; proc print; var
mint; run;
τˆM LE,ˆµ1 ,S , µ∗1
data;
48
seed = 727189910;
delta = 0;
tau = 10;
nnn = 5;
taup1 = tau + 1;
lcl = -3;
ucl = 3;
maxrun = 100000;
maxsam = 5000;
array xxx xxx1-xxx5000;
do runno = 1 to maxrun;
signal:
do iii = 1 to tau;
xxx{iii} = rannor(seed);
if (xxx{iii} > ucl or xxx{iii} < lcl) then goto signal;
end;
do iii = taup1 to maxsam;
xxx{iii} = rannor(seed) + delta;
if (xxx{iii} > ucl or xxx{iii} < lcl) then goto endrun;
end;
problem="problem";
49
output;
stop;
endrun:
ttt = iii;
tttp1 = ttt + 1;
tttpn = ttt + nnn;
mu1 = 0;
do iii = tttp1 to tttpn;
tmp = rannor(seed) + delta;
mu1 = mu1 + tmp;
*output;
end;
mu1 = mu1/nnn;
sumx = 0;
*sumx = (xxx{1}-mu1)*(xxx{1}-mu1);
do iii = 1 to ttt;
arg = xxx{iii} - mu1;
sumx = sumx + arg*arg;
end;
mint = 0;
*output;
50
tttm1 = ttt - 1;
do tt = 1 to tttm1;
sumy = 0;
do iii = 1 to tt;
sumy = sumy + xxx{iii}*xxx{iii};
end;
ttp1 = tt + 1;
do iii = ttp1 to ttt;
arg = xxx{iii} - mu1;
sumy = sumy + arg*arg;
end;
*output;
if (sumy < sumx) then do;
sumx = sumy;
mint = tt;
end;
end;
output;
end;
keep mint mu1 ttt;
proc print; run; proc means; var mint mu1 ttt; run; proc means
51
stderr; var mint mu1 ttt; proc sort; by mint; proc print; var mint;
run;
τˆM LE,ˆµ1 ,C , µˆ1
data;
seed = 727189910;
delta = 0;
tau = 10;
taup1 = tau + 1;
kkk = 0.5;
hhh = 4.77;
maxrun = 100000;
maxsam = 5000;
array xxx xxx1-xxx5000;
do runno = 1 to maxrun;
signal:
sss1 = 0;
sss2 = 0;
do iii = 1 to tau;
xxx{iii} = rannor(seed);
sss1 = sss1 + xxx{iii}-kkk;
52
if (sss1 < 0) then do;
sss1 = 0;
end;
sss2 = sss2 - xxx{iii}-kkk;
if (sss2 < 0) then do;
sss2 = 0;
end;
if (sss1 > hhh or sss2 > hhh) then goto signal;
end;
do iii = taup1 to maxsam;
xxx{iii} = rannor(seed) + delta;
sss1 = sss1 + xxx{iii}-kkk;
if (sss1 < 0) then do;
sss1 = 0;
end;
sss2 = sss2 - xxx{iii}-kkk;
if (sss2 < 0) then do;
sss2 = 0;
end;
*output;
if (sss1 > hhh or sss2 > hhh) then goto endrun;
53
end;
problem="problem";
output;
stop;
endrun:
ttt = iii;
mu1 = 0;
do iii = 1 to ttt;
mu1 = mu1 + xxx{iii};
end;
mu1 = mu1/ttt;
sumx = 0;
do iii = 1 to ttt;
arg = xxx{iii} - mu1;
sumx = sumx + arg*arg;
end;
mint = 0;
muhat = mu1;
tttm1 = ttt - 1;
do tt = 1 to tttm1;
mu1 = 0;
54
ttp1 = tt + 1;
do iii = ttp1 to ttt;
mu1 = mu1 + xxx{iii};
end;
mu1 = mu1/(ttt-tt);
sumy = 0;
do iii = 1 to tt;
sumy = sumy + xxx{iii}*xxx{iii};
end;
do iii = ttp1 to ttt;
arg = xxx{iii} - mu1;
sumy = sumy + arg*arg;
end;
if (sumy < sumx) then do;
sumx = sumy;
mint = tt;
muhat = mu1;
end;
end;
output;
end;
55
keep mint muhat ttt; proc print; run; proc means; var mint muhat
ttt; run; proc means stderr; var mint muhat ttt; run;
τˆM LE,µ∗1 ,C , µ∗1
data;
seed = 727189910;
delta = 0;
tau = 10;
nnn = 5;
taup1 = tau + 1;
kkk = 0.5;
hhh = 4.77;
maxrun = 100000;
maxsam = 5000;
array xxx xxx1-xxx5000;
do runno = 1 to maxrun;
signal:
sss1 = 0;
sss2 = 0;
do iii = 1 to tau;
56
xxx{iii} = rannor(seed);
sss1 = sss1 + xxx{iii}-kkk;
if (sss1 < 0) then do;
sss1 = 0;
end;
sss2 = sss2 - xxx{iii}-kkk;
if (sss2 < 0) then do;
sss2 = 0;
end;
if (sss1 > hhh or sss2 > hhh) then goto signal;
end;
do iii = taup1 to maxsam;
xxx{iii} = rannor(seed) + delta;
sss1 = sss1 + xxx{iii}-kkk;
if (sss1 < 0) then do;
sss1 = 0;
end;
sss2 = sss2 - xxx{iii}-kkk;
if (sss2 < 0) then do;
sss2 = 0;
end;
57
if (sss1 > hhh or sss2 > hhh) then goto endrun;
end;
problem="problem";
output;
stop;
endrun:
ttt = iii;
tttp1 = ttt + 1;
tttpn = ttt + nnn;
mu1 = 0;
do iii = tttp1 to tttpn;
tmp = rannor(seed) + delta;
mu1 = mu1 + tmp;
*output;
end;
mu1 = mu1/nnn;
sumx = 0;
do iii = 1 to ttt;
arg = xxx{iii} - mu1;
sumx = sumx + arg*arg;
end;
58
mint = 0;
*output;
tttm1 = ttt - 1;
do tt = 1 to tttm1;
sumy = 0;
do iii = 1 to tt;
sumy = sumy + xxx{iii}*xxx{iii};
end;
ttp1 = tt + 1;
do iii = ttp1 to ttt;
arg = xxx{iii} - mu1;
sumy = sumy + arg*arg;
end;
*output;
if (sumy < sumx) then do;
sumx = sumy;
mint = tt;
end;
end;
output;
end;
59
keep mint mu1 ttt;
proc means;
var mint mu1 ttt;
run;
proc means stderr;
var mint mu1
ttt;
run;
τˆM LE,ˆµ1 ,E , µ
ˆ1
data;
seed = 717189911;
delta = 0;
tau = 10;
taup1 = tau + 1;
lambda = 0.14;
ucl = 0.7628;
lcl = -0.7628;
60
maxrun = 10000;
maxsam = 2000;
array xxx xxx1-xxx2000;
do runno = 1 to maxrun;
signal:
qqq = 0;
do iii = 1 to tau;
xxx{iii} = rannor(seed);
qqq = (1-lambda)*qqq + lambda*xxx{iii};
if (qqq > ucl or qqq < lcl) then goto signal;
end;
do iii = taup1 to maxsam;
xxx{iii} = rannor(seed) + delta;
qqq = (1-lambda)*qqq + lambda*xxx{iii};
if (qqq > ucl or qqq < lcl) then goto endrun;
end;
problem="problem";
output;
stop;
endrun:
ttt = iii;
61
mu1 = 0;
do iii = 1 to ttt;
mu1 = mu1 + xxx{iii};
*output;
end;
mu1 = mu1/ttt;
sumx = 0;
do iii = 1 to ttt;
arg = xxx{iii} - mu1;
sumx = sumx + arg*arg;
end;
mint = 0;
*output;
muhat = mu1;
tttm1 = ttt - 1;
do tt = 1 to tttm1;
mu1 = 0;
ttp1 = tt + 1;
do iii = ttp1 to ttt;
mu1 = mu1 + xxx{iii};
end;
62
mu1 = mu1/(ttt-tt);
sumy = 0;
do iii = 1 to tt;
sumy = sumy + xxx{iii}*xxx{iii};
end;
do iii = ttp1 to ttt;
arg = xxx{iii} - mu1;
sumy = sumy + arg*arg;
end;
*output;
if (sumy < sumx) then do;
sumx = sumy;
mint = tt;
muhat = mu1;
end;
end;
output;
end;
keep mint muhat ttt;
proc means;
var mint muhat ttt;
63
run;
proc means stderr; var mint muhat ttt;
run;
τˆM LE,µ∗1 ,E
data;
seed = 727189910;
delta = 0;
tau = 10;
nnn = 5;
taup1 = tau + 1;
lambda = 0.14;
ucl = 0.7628;
lcl = -0.7628;
maxrun = 100000;
maxsam = 5000;
array xxx xxx1-xxx5000;
do runno = 1 to maxrun;
signal:
qqq = 0;
do iii = 1 to tau;
64
xxx{iii} = rannor(seed);
qqq = (1-lambda)*qqq + lambda*xxx{iii};
if (qqq > ucl or qqq < lcl) then goto signal;
end;
do iii = taup1 to maxsam;
xxx{iii} = rannor(seed) + delta;
qqq = (1-lambda)*qqq + lambda*xxx{iii};
if (qqq > ucl or qqq < lcl) then goto endrun;
end;
problem="problem";
output;
stop;
endrun:
ttt = iii;
tttp1 = ttt + 1;
tttpn = ttt + nnn;
mu1 = 0;
do iii = tttp1 to tttpn;
tmp = rannor(seed) + delta;
mu1 = mu1 + tmp;
*output;
65
end;
mu1 = mu1/nnn;
sumx = 0;
do iii = 1 to ttt;
arg = xxx{iii} - mu1;
sumx = sumx + arg*arg;
end;
mint = 0;
*output;
tttm1 = ttt - 1;
do tt = 1 to tttm1;
sumy = 0;
do iii = 1 to tt;
sumy = sumy + xxx{iii}*xxx{iii};
end;
ttp1 = tt + 1;
do iii = ttp1 to ttt;
arg = xxx{iii} - mu1;
sumy = sumy + arg*arg;
end;
*output;
66
if (sumy < sumx) then do;
sumx = sumy;
mint = tt;
end;
end;
output;
end;
keep mint mu1 ttt;
proc means; var mint mu1 ttt; run; proc means stderr; var mint mu1
ttt; run;
67
[...]... methods and thus there is a need to find more accurate estimators of change point and shifted process mean In Chapter 2, we will propose new change point and process mean estimators In Chapter 3, we present a comparison of the performance of the proposed and commonly used change point and process mean estimators based on simulation studies Chapter 4 is a numerical example based on piston rings data... good understanding of the estimator discussed in previous chapters This is followed by a summary of the research and recommendations for future work in Chapter 5 10 Chapter 2 Estimating the Change Point and Current Process Mean In this chapter, we will introduce new change point and process mean estimators In Section 2.1, we use adjusted MLE methods τˆM LE,µ∗1 ,S , τˆM LE,µ∗1 ,CU SU M and τˆM LE,µ∗1... ,EW M A to estimate change point In Section 2.2, new process mean estimators µ∗1,N =5 , µ∗1,N =10 , µ∗1,N =20 and µ∗1,N =50 associated with the CUSUM chart are introduced 11 2.1 Estimating the Change Point ¯ chart after a It is well known that a signal could be issued by the Shewhart X substantial amount of time from a change point Estimating a change point using the time at which a control chart signals... signals would lead to a biased and, therefore, possibly misleading estimate of the change point This bias is due to the potentially large delay in issuing a signal using a control chart Thus, it is not suitable to use the signal point T to estimate the change point τ Pignatiello, Samuel and Calvin (1998) considered the use of a MLE of the process change point τ and investigated its performance based... underlying process random shifts The current process mean is estimated using the EWMA equation and the newly adjusted control parameter The proposed estimator is very easy to implement and effective under many disturbance situations 17 Chapter 3 Comparison of Change Point and Process Mean Estimators In this chapter, we compare the performance of all the change point estimators and process mean estimators... for estimating a process mean always overestimates the process mean (Taguchi, 1985) and a well published method given in Montgomery (2005, page 394) that is biased for nearly every situation, we will proceed to find better estimators We first examine the case where the change point is known based on an out-of -control signal from a Shewhart ¯ chart Given the change point information, the process mean. .. these MLE change point estimators, τˆM LE,ˆµ1 and τˆM LE,µ∗1 , can ¯ chart, a CUSUM chart or an EWMA be used with a signal from a Shewhart X chart 2.2 Estimating the Shifted Process Mean In traditional statistical process control (SPC) it is frequently assumed that an initially in- control process is subjected to random shocks, which may shift the process mean to an off-target value Then a control chart... explains why τˆM LE,ˆµ1 ,S is much more accurate than τˆM LE,µ∗1,N=5 ,S As N increases, change point, process mean estimators improves significantly For τ = 10, the accuracy of both estimators improves as µ1 increases As τ increases from 10 to 100, the accuracy of change point estimators increases The reason is that the accuracy also depends on the number of samples used before and after the change point. .. signal point is T = 117 and the change point estimator is τˆCU SU M = 103 For a large shift δ = 2.0 after the change point, a CUSUM chart is likely to be active before τ = 100 and quickly issues a signal For δ = 2.0, τˆCU SU M = 97 and T = 103 Figure 3.2 helps to explain as to why τˆCU SU M underestimates τ = 100 for large shifts τˆEW M A and τˆCU SU M are similar in the way they estimate the change point, ... larger instantly after the change point to capture the shift However, the step change occurs only once and the process mean remains unchanged after τ If the value of λ remains large, the estimator becomes oversensitive to the white noises 16 A novel dynamic-tuning EWMA estimator was proposed by Guo (2002) that has the capability of adjusting the control parameter dynamically in response to the underlying ... future work in Chapter 10 Chapter Estimating the Change Point and Current Process Mean In this chapter, we will introduce new change point and process mean estimators In Section 2.1, we use adjusted... 2.1 Estimating the Change Point ¯ chart after a It is well known that a signal could be issued by the Shewhart X substantial amount of time from a change point Estimating a change point using... good understanding of the change point and process mean estimators We use the piston rings data set in Montgomery (2005) Piston rings for an automotive engine are produced by a forging process We