Estimating change point and process mean in control charts

mean following a signal and change point estimators based on this process meanestimator when a step change occurs in a normal process mean.. The performance of change point estimators wi

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 SUBMITTEDFOR THE DEGREE OF MASTER OF SCIENCEDEPARTMENT OF STATISTICS AND APPLIED PROBABILITY

NATIONAL UNIVERSITY OF SINGAPORE

2005

I would like to take this opportunity to express my sincere gratitude to my pervisor Assoc Prof Gan Fah Fatt for his patience, advice, and continuous supportthroughout my study at National University of Singapore I am really grateful tohim for his generous help and valuable suggestions to this thesis

su-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 andunderstanding Special thanks to all the staff in my department and all my friendsfor their concern and inspiration in the two years

Statistical process control (SPC) charts are used to monitor for process changes

by distinguishing the assignable causes of variation from the common causes of tion When a control chart does signal that a process change has occurred, engineersmust initiate a search for the assignable cause and make a suitable adjustment Butmost of the research work on control charting procedures has concentrated on thedetection 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 usefulinformation which can help engineers to narrow the search window and hence lesspossible downtime A signal issued could also be due to the naturally randomness

varia-of a process and if evidence can be gathered in support varia-of a false signal, then essary and often expensive search effort can be avoided All the published methods

unnec-of estimations unnec-of process mean associated with a signal are grossly biased and unnec-oftenthe bias depends on the magnitude of the shift

In this thesis, we propose a new and effective method of estimating a process

mean following a signal and change point estimators based on this process meanestimator when a step change occurs in a normal process mean Then we dis-cuss the performance of proposed and usually used change point and process meanestimators when used with control charts The simulation results show that theproposed estimators provides a useful and much better alternative to the usuallyused estimators The new process mean estimator is able to remove the bias that

is inherent in existing methods The performance of change point estimators willincrease based on this more accurate process mean estimator

Chapter 1

Introduction

Quality has always been an integral and important part of manufacturing andservice industries It was not until 1924 that formal statistical methods for qual-ity control were introduced Shewhart (1924) developed a simple control chartingprocedure for process monitoring and quality improvement Subgroup mean orindividual observation of a quality characteristic is plotted against the time andany point that is beyond the control limits provides evidence that a change in theprocess mean might have occurred When a process mean shifts, a signal from thechart 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 changeactually occurred

Most of the research published in the literature has concentrated on the tion of process shifts However, when a process shift has actually occurred, themagnitude of the shift and the time when the shift occurred (change point) arevaluable information which can help quality control engineers to determine quicklypossible assignable causes and hence less process downtime Note that the timewhen a signal is issued is usually not the same as the change point If the changepoint could be determined, process engineers would have a smaller search windowwithin which to look for possible assignable causes Consequently, the assignablecauses can be identified more quickly, and appropriate actions needed to improvequality can be implemented sooner Although estimations of change point andprocess shift are important and related, joint research in these two areas is stilllacking.

detec-A step change is a common type of change in industrial processes that oftenresults from tool breakage, the introduction of a new material or other abrupt andsudden changes This thesis will be based on the step change model We alsoassume that samples of size n are taken independently from a normal distributionwith mean µ and variance σ0 We let τ denote the last sample number from thein-control process In other words, we assume that ¯X1, ¯X2, · · · , ¯Xτ are the samplemeans from the in-control process with mean µ0, while ¯Xτ +1, ¯Xτ +2, · · · , ¯XT are from

the out-of-control process with mean µ1, that is,

0/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 Themain emphasis of this research is the estimation of the change point τ and shiftedprocess mean µ1

Barnard (1959) suggested that control charts should not only be used as amonitoring tool They should also be used to estimate the change point τ For

a process shift δ, the upper and lower cumulative sums (CUSUMs)

are compared to the control limits h+

and h−respectively, where Zt= ( ¯Xt−µ0)/σX ¯,

ˆ

τCU SU M = max{t : S+

t = 0, 0 ≤ t < T }, (1.4)

Pignatiello, Samuel and Calvin (1998) developed a maximum likelihood tor (MLE) of the change point τ for a step change in a normal process mean Let

Nishina (1992) proposed a change point estimator based on a sequence of pointsthat plot continuously on one side of the center line of an exponentially weightedmoving average (EWMA) chart The EWMA chart was introduced by Roberts

(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

rλ

2 − λ[1 − (1 − λ)

2t]and h− = −L

rλ

Pignatiello and Samuel (2001) compared ˆτM LE, ˆτCU SU M and ˆτEW M A based on

a simulation study They concluded that ˆτM LE is more accurate than ˆτCU SU M andˆ

τEW 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 control data point to estimate the shifted mean, that is, ˆµT = ¯XT −1based on a signal

in-T of a Shewhart ¯X chart This estimator always overestimates the shifted meanbecause the last data point is associated with an out-of-control signal Wiklund

(1992) found that Taguchi’s method is biased for small to moderate shifts, and thathis MLE estimator performs comparatively better, but it still may be inefficientespecially for large shifts Adams and Woodall (1989) also showed that the optimalcontrol parameters and loss functions given by Taguchi are severely misleading inmany situations In Montgomery (2005, page 394), an estimator based on theCUSUM chart is given as

N+, if S+

T > h+

k − S

− T

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 forsmoothing and forecasting time series and as a process mean estimator, due to itssimplicity and ability to capture non-stationarity The estimator of the currentprocess mean using EWMA is defined as:

ˆ

µE,t = λ ¯Xt+ (1 − λ)ˆµE,t−1, (1.11)

Yashchin (1995) showed that ˆµE,t has optimality properties within the class oflinear 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 tion properties and it is thus frequently used as a forecasting tool The EWMA

predic-is usually used as a one-step ahead forecast of the process mean The EWMA timator is optimal when the process mean follows a first-order integrated movingaverage IMA(1,1) model (Box and Jenkins, 1970) In fact, the EWMA estimatorcan be implemented as a widely used proportional-integral-derivative (PID) con-troller (˚Astr¨om and H¨agglund, 1995) Although the Estimator performs well forvarious other processes, few studies have analytically shown the estimator’s wideapplicability

es-Wiklund (1992) proposed a MLE estimator of the process mean based on atruncated normal probability density function His estimation of the process mean

is based on a signal point from a Shewhart ¯X chart In his study, he concludedthat Taguchi’s method is biased for small to moderate shifts, MacGregor’s EWMAestimator is not a sensitive estimator of the mean shift, and that his MLE estimatorperforms comparatively better, but it may still be inefficient especially for large

In order to understand change point and process mean estimation better, wewill first present Table 1.1 This will highlight problems associated with well-knownand 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 weresimulated, ˆτCU SU M (Page, 1954), ˆµM (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

an in-control ARL of 370, the same as the 3-σ Shewhart ¯X chart The expectation

of T in Table 1.1 E(T ) is equal to 100 plus in-control ARL of CUSUM chart fordifferent δ These in-control ARL values for different magnitudes of change arecomputed using an integral equation computer program developed by Gan (1993)

Table 1.1: Estimation of Change Point Using Page’s Method and Estimation ofShifted Mean Using Montgomery’s Method Based on 10,000 Simulation Runs

0.00 0.57 100 362.66 370.370.25 0.78 100 213.06 221.310.50 0.95 100 121.78 135.280.75 0.97 100 107.90 116.171.00 1.05 100 100.07 109.931.50 1.25 100 98.70 105.522.00 1.48 100 98.50 103.863.00 2.68 100 98.47 102.49

Table 1.1 reveals that for µ1close to 1.00, ˆµM is nearly unbiased It overestimates

µ1 for µ1 < 1.00, but underestimates µ1 for µ1 > 1.00 As the actual change pointwas at time 100, ˆτCU SU M should be close to 100 except for the case µ1 = 0.00

In Table 1.1, ˆτCU SU M based on a CUSUM chart’s signal is close to 100 for µ1

close to 1.00 ˆτCU SU M overestimates the change point for small values of µ1 butunderestimates change point for large values of µ1 For an upward shift in theprocess mean, ˆµM is given as

ˆ

µM = k + S

+ T

N+,where N+

= S+ ˆ τCU SU M + XT + · · · + Xˆ τCU SU M +1− (T − ˆτCU SU M)k

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 ˆτCU SU M come from theshifted process and N+

is also a suitable sample number for estimation Therefore,the performance ˆµM 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 ˆµM will underestimate process mean forlarge values of µ1 The results here show that the inadequacy of well-known andestablished methods and thus there is a need to find more accurate estimators ofchange 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 andcommonly used change point and process mean estimators based on simulationstudies Chapter 4 is a numerical example based on piston rings data so as toprovide a good understanding of the estimator discussed in previous chapters This

is followed by a summary of the research and recommendations for future work inChapter 5

estima-τM LE,µ ∗

1 ,EW M A to estimate change point In Section 2.2, new process mean mators µ∗1,N =5, µ∗1,N =10, µ∗1,N =20 and µ∗1,N =50 associated with the CUSUM chart areintroduced

esti-2.1 Estimating the Change Point

It is well known that a signal could be issued by the Shewhart ¯X chart after asubstantial amount of time from a change point Estimating a change point usingthe 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 potentiallylarge delay in issuing a signal using a control chart Thus, it is not suitable to usethe signal point T to estimate the change point τ

Pignatiello, Samuel and Calvin (1998) considered the use of a MLE of the processchange point τ and investigated its performance based on a signal from a Shewhart

¯

X chart Their proposed estimator of the change point τ for a step change in anormal process mean is given in equation 1.6 For this MLE estimator, T wouldalso 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 CUSUMand EWMA charts Given the subgroup averages ¯X1, ¯X2, · · · , ¯XT, the MLE of τ isthe value of τ that maximizes the likelihood function or, equivalently, its logarithm.The logarithm of the likelihood function can be derived as

If the change point were known, the MLE of µ1would be ˆµ1 = ¯¯XT,τ = 1

T,t, over 0 ≤ t < T , that isˆ

τM LE, ˆ µ1 = arg max

t

n(T − t) ¯¯X2

T,t, 0 ≤ t < To (2.3)

But µ1 is not likely to be known and we can use

µ∗1 = 1N

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

2µ

2 0

We propose that these MLE change point estimators, ˆτM LE, ˆ µ1 and ˆτM LE,µ ∗

1, can

be used with a signal from a Shewhart ¯X chart, a CUSUM chart or an EWMAchart

In traditional statistical process control (SPC) it is frequently assumed that

an initially in-control process is subjected to random shocks, which may shift theprocess 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 opportunitiesfor quality monitoring and fault diagnosis In many cases when the resulting processoutput deviation can be adjusted to bring the process output to the target value, agood estimator will certainly provide a more accurate evaluation on how much theadjustment 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 wellpublished method given in Montgomery (2005, page 394) that is biased for nearlyevery situation, we will proceed to find better estimators We first examine the casewhere the change point is known based on an out-of-control signal from a Shewhart

¯

X chart Given the change point information, the process mean estimator can bederived easily Suppose that the current observation is T and the most recent mean

change occurred at observation τ A naive estimate of the process mean is thesample mean based on samples τ + 1, τ + 2, · · · , T,

estima-τCU SU M + 1 to T based on a CUSUM chart as discussed in Chapter 1

The estimators ˆµM and ˆµ1 are always biased because the samples are associatedwith 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 havebeen observed This aims at collecting information on the magnitude of shift It

is certainly not necessary to maintain the same subgroup size Taking immediatesamples after a signal will also allow the signal to be checked to see if it is a genuineout-of-control signal or it is a signal due to randomness; it is advisable to alter sam-pling frequency and sample size as before the signal during this period to minimize

defective production Let ¯XT +1, ¯XT +2· · · , ¯XT +N denote the N subgroup meanscollected following the out-of-control signal; note that since there is no condition onthese values being in or outside the control limits, these constitute random samplesfrom the process distribution Hence, an unbiased estimator of µ1 is given as

µ∗1 = 1N

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 givingmost 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 calledthe forecast origin, we can use the EWMA Qt or ˆµE,t to provide an estimate of thenext value Qt+1 For ˆµE,t, if the value of Qt remains large, the estimator becomesoversensitive We should make some adjustment on this estimator to overcome thetrade-off between large and small Qt and to design a more effective estimator forprocesses subject to sudden shifts It is effective in many applications

In order to swiftly compensate for the sudden shift, the value of λ should beset larger instantly after the change point to capture the shift However, the stepchange occurs only once and the process mean remains unchanged after τ If thevalue of λ remains large, the estimator becomes oversensitive to the white noises

A novel dynamic-tuning EWMA estimator was proposed by Guo (2002) that hasthe capability of adjusting the control parameter dynamically in response to theunderlying process random shifts The current process mean is estimated usingthe EWMA equation and the newly adjusted control parameter The proposedestimator is very easy to implement and effective under many disturbance situations.

= 4.77 and h− = 4.77, and an EWMA chart withcontrol parameters λ = 0.14, h+

= 0.7628 and h− = −0.7628 The CUSUM andEWMA charts with these parameters have the same in-control ARL as a 3-σ She-whart 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

control charts is presented in Table 3.1 for µ1 = µ0+ δ√σ

n, where µ0 = 0,

σ

√

n = 1,and δ = 0.00, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, 3.00 We can see that an EWMA chartwith constant control limits has similar ARL performance to an EWMA chart withvarying 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

of them issues a signal Then starting after sample τ , observations were randomlygenerated 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 thatthe standard error of the estimate is less than 1% of the average The results are

For ˆτM LE, ˆ µ1,S, an average of ˆN samples are used to estimate ˆµ1 as compared toˆ

τM LE,µ ∗

1 ,N =5 This explains whyˆ

τM LE, ˆ µ1,S is much more accurate than ˆτM LE,µ ∗

1 ,N =5 ,S As N increases, change point,process mean estimators improves significantly For τ = 10, the accuracy of bothestimators improves as µ1 increases As τ increases from 10 to 100, the accuracy ofchange point estimators increases The reason is that the accuracy also depends onthe number of samples used before and after the change point τ If more samples

are available to compute

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 similarperformance to those based on Shewhart chart The overall performance of thetwo is similar Except for the case µ1 = 0.25 and τ = 10, the difference betweenˆ

τM LE, ˆ µ1,S and ˆτM LE,µ ∗

1 ,N =5 ,S is much larger than the difference between ˆτM LE, ˆ µ1,C

and ˆτM LE,µ ∗

of samples are used in the minimization function for the CUSUM chart

† τˆM LE, ˆ µ1,S MLE change point estimator with ˆµ1 = ¯¯XT,τ (taking information of ˆNsamples) based on a signal from the 3-σ Shewhart ¯X chart

m1 ,S median of ˆτM LE, ˆ µ1,S

ˆ

τM LE,µ ∗

1 ,N =5 based on a signal from the3-σ Shewhart ¯X chart

the 3-σ Shewhart ¯X chart

1 ,N =5 based on a signal from theCUSUM chart (k = 0.5 and h±= 4.77)

ˆ

τM LE,µ ∗

CUSUM chart (k = 0.5 and h±= 4.77)

ˆ

τM LE,µ ∗

the CUSUM chart (k = 0.5 and h±= 4.77)

the EWMA chart (λ = 0.14 and h±

= ±0.7628)

ˆ

τM LE,µ ∗

1 ,N =10based on a signal fromthe EWMA chart (λ = 0.14 and h±= ±0.7628)

ˆ

τM LE,µ ∗

from the EWMA chart (λ = 0.14 and h±

Page’s estimator ˆτCU SU M and Nishina’s estimator ˆτEW M A are the worst pared with the other MLE estimators They underestimate τ for large shifts inthe mean They overestimate τ for small shifts It is clear from Figure 3.1 thatfor a small shift δ = 0.5, a CUSUM chart might quite likely become inactive andthen active again after τ = 100 This explains the overestimation of τ by ˆτCU SU M

com-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 ˆτCU SU M = 103 For a large shift δ = 2.0after the change point, a CUSUM chart is likely to be active before τ = 100 andquickly issues a signal For δ = 2.0, ˆτCU SU M = 97 and T = 103 Figure 3.2 helps

to explain as to why ˆτCU SU M underestimates τ = 100 for large shifts ˆτEW M A andˆ

τCU SU M are similar in the way they estimate the change point, thus ˆτEW M A alsounderestimates τ for large shifts like δ = 2.00 and overestimates τ for small shiftslike µ1 = 0.25

Figure 3.1: A Typical CUSUM Chart with a Shift of δ = 0.5 at τ = 100

In general, MLE estimators based on CUSUM and EWMA charts have similarperformance This is because the quantity T is similar for the CUSUM and EWMAcharts They are almost spot on for µ1 ≥ 1.00 Overestimation of τ becomesmore severe for smaller shifts in the mean For small shifts in the mean, the MLEestimators based on EWMA charts have slightly better performance than the MLEestimators based on CUSUM charts Estimation of change point is appreciablyaffected by the accuracy of the estimators of the process mean That is, MLEestimators based on µ∗1 perform better for accurate estimation of process meanbased on a larger sample size N.

For the Shewhart ¯X chart, ˆτM LE,µ ∗ 1,S is much closer to the actual change pointthan ˆτM LE, ˆ µ1,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 changeincrease for these estimators

The reason is that the value of T at which a Shewhart ¯X chart issues a signal islarger than the value of T at which a CUSUM or an EWMA chart issues a signal.Hence the Shewhart ¯X chart takes more information than the CUSUM and EWMAchart in estimating the change point to get a more accurate estimate

Tables 3.2-3.4 also contain the simulation results for process mean EWMAestimator ˆµE is not a sensitive estimator of the shifts The CUSUM estimator ˆµM

performs comparatively better They are both biased when the shift is too small or

too large, because ˆµM and ˆµE 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 200observations after a signal from a Shewhart chart The performance of µ∗