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Adaptive CUSUM charts for monitoring linear drifts in Poisson rates
Int. J. Production Economics 148 (2014) 14–20
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Adaptive CUSUM charts for monitoring linear drifts in Poisson rates Feng He a, Lianjie Shu b,n, Kwok-Leung Tsui c a b c
Department of Statistics and Actuarial Science, University of Waterloo, Canada Faculty of Business Administration, University of Macau, Macau Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong
art ic l e i nf o
a b s t r a c t
Article history: Received 30 August 2012 Accepted 4 November 2013 Available online 15 November 2013
There has been relatively less attention paid to the monitoring of linear drifts in Poisson rates. Although the conventional CUSUM charts can be used for monitoring linear drifts in Poisson rates, they often rely on the assumption of a known parameter. To get rid of this assumption, this paper develops adaptive CUSUM (ACUSUM) charts for monitoring drifts in Poisson rates. The basic idea is to first estimate the current Poisson mean level and then to dynamically update the likelihood ratio in the CUSUM chart based on the estimated mean. Three different mean estimators based on the exponentially weighted moving average (EWMA) schemes are discussed. The comparison results are shown to favor the ACUSUM chart, especially at small drifts. & 2013 Elsevier B.V. All rights reserved.
Keywords: Change-point detection Linear drift Poisson distribution Statistical process control
1. Introduction In manufacturing and business processes as well as health surveillance, it is often important to detect changes in count data in addition to monitoring variable data (Woodall, 2006; Faltin, 2008; Tsui et al., 2008; Montgomery, 2009). The Poisson distribution is usually used to model count data over some interval unit based on time, distance, area or some similar unit. For example, the number of nonconformities per unit of product in the industrial setting and the number of new infections in a hospital per day in healthcare are often modeled by the Poisson distribution. The Poisson distribution has only one parameter, rate of count per unit, which characterizes both mean and variance. Various approaches have been proposed for monitoring changes of rate of a Poisson distribution. The Shewhart c-chart or u-chart is the simplest procedure to monitor Poisson processes. However, one drawback of it is that it is not effective for the quick detection of small shifts in the Poisson mean parameter as it only uses the information in the current sample. To improve the detection ability at small shifts, the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts, which incorporate information in both current and past samples, are often utilized. The pioneering work on the use of the CUSUM chart for monitoring Poisson processes is given by Brook and Evans (1972). They introduced the Markov chain approach for evaluating the run length of the one-sided Poisson CUSUM chart under a step shift. Lucas (1985) discussed the design and implementation
n
Corresponding author. Tel.: +853-8397-4741; fax: +853-2883-8320. E-mail address: [email protected] (L. Shu).
0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2013.11.004
procedures for both one- and two-sided Poisson CUSUM charts and enhancements such as the fast initial response (FIR) feature. Some recent studies on the Poisson CUSUM chart include White and Keats (1996), White et al. (1997), Hawkins and Olwell (1998), Chan et al. (2007), Han et al. (2010), Ryan and Woodall (2010), Aebtarm and Bouguila (2011), and Jiang et al. (2011). For an early review on attribute control charts, interested readers can be referred to Woodall (1997). However, the above cited papers mainly focus on the detection of step shifts in the Poisson rate parameter. In practice, there are often settings where a linear drift needs to be considered. For example, tool wear may cause the mean count rate of nonconformities of a machined part to gradually increase per unit sampled. There is relatively less attention devoted to the detection of linear drifts in the Poisson mean rate. Some exceptions include Perry et al. (2006, 2007). They compared the performance of change point estimators under a linear trend and monotonic changes in the Poisson rate. However, they did not address the monitoring issues under a linear trend in the Poisson rate. Although the conventional Poisson CUSUM chart designed for step changes can be utilized for monitoring linear drift in the Poisson rate, it may be inefficient due to the mis-specification of the shift form. Another drawback of the traditional Poisson CUSUM charts is that it requires the prior knowledge of the shift size, which also implies that it may not work well for detecting linear drifts. Motivated by this, this paper suggests an adaptive CUSUM (ACUSUM) chart for monitoring linear drifts in the Poisson rate. The rest of this paper is organized as follows. In Section 2, the ACUSUM charts with different estimators of Poisson rate are introduced. In Section 3, the effects of mean estimators as well as effects of the smoothing parameter on the ACUSUM charts are
F. He et al. / Int. J. Production Economics 148 (2014) 14–20
discussed. In Section 4, the performance between CUSUM and ACUSUM charts under linear drifts is compared. In Section 5, an example is given to illustrate the implementation of the proposed chart. Concluding remarks are given in Section 6.
2. ACUSUM charts Let X 1 ; X 2 ; …, be a sequence of observations collected from a Poisson distribution with mean λt at fixed intervals of time. When the process is in the state of statistical control, observations are assumed to be independently distributed from a Poisson distribution with a known mean λ0. After an unknown time point τ, the process mean is subject offfiffiffiffiffiin-control pffiffiffiffiffito a linear drift in the unitp standard deviation ( λ0 ). The amount of drift is θ λ0 per unit time, where θ is unknown. In other words, the process mean at time t can be represented as ( λ0 t o τ; pffiffiffiffiffi λt ¼ t ¼ 1; 2; …; λ0 þ ðt τ þ 1Þθ λ0 t Z τ; For simplicity, we focus on the one-sided upper control chart aimed at detecting an increase of θ from zero, which is common in industrial quality control. The CUSUM chart proposed by Page (1954) can be formulated as a sequential hypothesis testing procedure for the change point from a known in-control distribution to another known alternative distribution. It has the basic form C t ¼ maxf0; C t 1 þLt g;
t ¼ 1; 2; …
ð1Þ
where C 0 ¼ 0 and the increment Lt ¼ log ½f 1 ðX t Þ=f 0 ðX t Þ is the loglikelihood ratio to contrast the alternative distribution f1 with the null distribution f0. At time t, the observed count Xt follows a Poisson distribution with mean λt. Thus, the log-likelihood ratio statistic for the tth observation follows Lt ¼ log f 1 ðX t Þ log f 0 ðX t Þ ¼ X t ðlog λt log λ0 Þ ðλt λ0 Þ:
ð2Þ
When λt is known, the CUSUM chart can be easily established by substituting Eq. (2) into Eq. (1). However, the drift coefficient θ is rarely unknown in real practice. To deal with this issue, one natural idea is to replace θ by its estimate. This is the key idea of the adaptive CUSUM chart that has been widely discussed for monitoring process changes under normal distributions. See, for example, Sparks (2000), Shu and Jiang (2006), and Jiang et al. (2008). The EWMA scheme has been widely used as an effective tool for parameter estimation due to its simplicity and efficiency (Roberts, 1959). Analogously, it can also be used to estimate the Poisson rate. The EWMA estimate of λt is based on Y tð1Þ ¼ ð1 γ ÞY ð1Þ t 1 þ γXt ;
ð3Þ
¼ λ0 and γ (0 o γ o 1) is a smoothing where the initial value is parameter. Note that when γ ¼ 1, the EWMA estimate reduces to Xt. This could cause the log-likelihood ratio in Eq. (2) to be undefined as log X t is not well defined when X t ¼ 0. Therefore, we restrict γ on the range 0 o γ o1 in this case. Note that the Poisson rate keeps increasing over time once a shift occurred in the process. In order to improve the estimation efficiency of the EWMA scheme, one can reset the EWMA estimate to the nominal rate whenever it is less than the nominal rate. This leads to the second type of EWMA estimation scheme: Y 0ð1Þ
Y tð2Þ ¼ maxfλ0 ; ð1 γ ÞY ð2Þ t 1 þ γ X t g;
ð4Þ
where Y ð2Þ 0 ¼ λ0 . This estimation scheme has been discussed by Sparks (2000) and Jiang et al. (2008) for estimating the mean level of a normal process.
15
Note that the second EWMA estimation scheme may not be able to accumulate the information of historical observations in a smooth way due to the reset of EWMA statistic to the in-control rate when Y ð2Þ t o λ0 . To overcome this drawback, we suggest a third estimation scheme based on the combined use of the above two estimators. In particular, the EWMA scheme in Eq. (3) is used to smooth the observed data as usual. When the EWMA statistic is less than λ0, the process is considered to be in control. It is thus reasonable to assume the current mean stays at the level of λ0. Otherwise when EWMA statistic is larger than λ0, we may have reasons to doubt that the process is out of control. In this case, the EWMA statistic can be served as an online estimate of the current Poisson mean level. To sum up, n o ð1Þ Y ð3Þ ; ð5Þ t ¼ max λ0 ; Y t where Y tð1Þ is defined by Eq. (3). Compared to the second estimation scheme, this estimation scheme does not directly reset the EWMA statistic but resets the estimate to λ0 when the EWMA statistic Y tð1Þ is smaller λ0. The ACUSUM chart can be established by replacing Lt in Eq. (1) with an estimate of the Poisson rate. That is, the ACUSUM charting statistics are based on At ¼ maxf0; At 1 þ X t ðlog λ^ t log λ0 Þ ðλ^ t λ0 Þg;
t ¼ 1; 2; …;
where λ^ t denotes an estimate of λt. For the sake of simplicity, the resulting ACUSUM charts based on the mean estimates in (Eqs. (3), 4), and (5) will be referred to as ACUSUM-I, ACUSUM-II, and ACUSUM-III charts, respectively. An out-of-control signal is triggered when the ACUSUM charts exceed the threshold values of h1, h2, and h3, respectively.
3. Performance analysis of ACUSUM charts The average run length (ARL), which describes the average number of samples until the first out-of-control signal on a control chart, has been widely used for chart performance comparisons. The out-of-control ARL (ARL1) refers to the average number of samples collected before a shift is detected, while the in-control ARL (ARL0) is the expected number of samples until a false alarm. The ideal chart is expected to have small ARL1 but large ARL0. In this section, we first compare the ARL performance among the ACUSUM charts based on the above estimation schemes. Then the choice of the smoothing parameter γ for designing ACUSUM charts is discussed. The measured ARL values can be either in the zero state or steady state. The zero-state ARL refers to the ARL value obtained assuming that the change occurs at the initial startup of a control chart, while the steady-state ARL is based on a delayed shift which occurs after a period of monitoring. To provide a complete comparison, both the zero-state and steady-state ARL results have been analyzed based on Monte Carlo simulations. The simulation results were based on 40,000 replicates. The zero-state ARL results were obtained based on τ ¼ 1. The steady-state ARL results were simulated based on the fact that the control chart has run in the in-control state for 50 samples before the drift in the rate parameter takes place. 3.1. Comparison among adaptive CUSUM charts Table 1 presents both the zero-state ARL and steady-state ARL values of the three ACUSUM charts under drifts in the Poisson rate when λ0 ¼ 4. Three different smoothing parameter values of γ ¼ 0:005, 0.01, and 0.05 were considered. The in-control ARL values of all the charts are maintained around 200. The same
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F. He et al. / Int. J. Production Economics 148 (2014) 14–20
Table 1 Comparison of ARL values among ACUSUM charts when λ0 ¼ 4 and ARL0 200. Limit
γ ¼ 0:005
γ ¼ 0:01
γ ¼ 0:05
ACUSUM-I 0.8873
II 0.7991
III 0.1843
ACUSUM-I 1.6976
II 1.5075
III 0.5190
ACUSUM-I 7.5961
II 6.1395
III 3.5006
θ 0.0000 0.0005 0.0010 0.0050 0.0100 0.0500 0.1000 0.5000 1.0000 2.0000 3.0000 4.0000
Zero-state ARL 199.57 182.01 156.17 81.99 59.25 26.96 19.03 8.38 5.87 4.08 3.28 2.88
200.13 156.44 131.85 72.94 53.64 25.34 18.10 8.11 5.69 3.98 3.19 2.82
199.51 109.21 87.41 48.11 36.10 17.30 12.37 5.52 3.85 2.70 2.17 1.93
199.70 184.25 159.64 83.57 60.17 27.11 19.06 8.36 5.84 4.06 3.25 2.87
200.20 160.48 136.19 74.67 54.57 25.37 18.06 8.05 5.64 3.94 3.16 2.78
199.65 124.21 100.25 55.32 41.25 19.30 13.74 6.07 4.24 2.96 2.37 2.04
199.52 192.21 177.10 100.34 70.97 29.85 20.53 8.66 5.99 4.14 3.31 2.90
199.89 173.49 154.63 88.99 63.70 27.36 18.89 8.09 5.62 3.90 3.12 2.73
200.18 152.41 129.01 72.33 52.35 23.19 16.14 6.93 4.80 3.32 2.69 2.25
0.0005 0.0010 0.0050 0.0100 0.0500 0.1000 0.5000 1.0000 2.0000 3.0000 4.0000
Steady-state ARL 143.33 128.56 73.36 53.99 25.14 17.78 7.90 5.50 3.83 3.10 2.67
124.49 107.20 60.61 45.01 21.09 15.05 6.67 4.69 3.29 2.67 2.30
107.60 86.74 47.10 35.48 17.03 12.21 5.48 3.84 2.69 2.18 1.88
144.82 130.59 74.68 54.58 25.12 17.74 7.85 5.46 3.80 3.07 2.65
127.55 110.63 62.88 46.27 21.41 15.18 6.70 4.71 3.28 2.67 2.29
111.11 91.83 50.58 37.98 18.16 13.03 5.80 4.07 2.84 2.30 1.99
146.47 138.96 86.84 62.65 27.13 18.77 7.97 5.51 3.81 3.07 2.65
134.21 122.32 75.35 55.20 24.15 16.65 7.07 4.90 3.38 2.74 2.33
122.13 105.46 61.40 45.52 20.77 14.60 6.31 4.39 3.05 2.46 2.11
control limit as the zero-state analysis was used to analyze the steady-state results for each chart. A pronounced observation from Table 1 is that the ACUSUM-III chart has shorter zero-state out-ofcontrol ARL values than the ACUSUM-II chart, and than the ACUSUM-I chart for all the cases considered here. Similar observations can be made from the steady-state results. This indicates that the ACUSUM-III chart performs uniformly better than the ACUSUM-II chart, and than the ACUSUM-I chart for monitoring linear drifts in Poisson rates. The above observation demonstrates that the reset of EWMA estimate in the ACUSUM chart improves over the chart without reset. The resetting operation sets the EWMA statistic to a lower bound when it is small. This enables the EWMA statistic to quickly catch up with the current Poisson mean level when there is a shift, which in turn speeds up the detection performance of the ACUSUM chart. Therefore, it is expected that the ACUSUM-II chart is more sensitive than the ACUSUM-I chart. Instead of resetting the EWMA statistic directly, the third mean estimation scheme, ð1Þ Y ð3Þ t ¼ max fλ0 ; Y t g; resets the current estimate to λ0 when Y ð1Þ o λ . By comparing Y ð3Þ and Y tð2Þ , it can be observed that both 0 t t methods can provide a fast reaction to changes of mean levels in the out-of-control case as both methods involve the use of maximum operation. However, in the in-control case, Y ð3Þ tends to provide a t more accurate estimate of Poisson mean than Y ðt2Þ . This is because the observed count is always nonnegative over time. Therefore, Y ð2Þ t would be very likely to exceed λ0 in the next few steps after it was reset to the lower bound, even when the process is still in the status of in-control. In contrast, Y ð3Þ is likely to maintain at λ0 t conditioned that the EWMA statistic, Y ð1Þ t , is not too close to λ0. This advantage of the third estimator leads to better detection performance of the ACUSUM-III chart as compared to the ACUSUM-II chart. 3.2. Effects of
γ
Due to the superior performance of the ACUSUM-III chart, we will only use it for discussion throughout the remaining of the
paper. The design of ACUSUM-III chart involves the choices of γ and h3. To provide some guidelines on the choice of γ, we first investigate the effect of γ on the performance of ACUSUM-III chart. Fig. 1 shows the ARL values of the ACUSUM-III chart when γ varies over the interval ½0:1; 1. It can be observed that the ACUSUM-III chart with smaller values of γ has consistently smaller ARL1 values at both small and large drifts. For example, when γ decreases from 1 to 0.1, the zero-state ARL1 value of the ACUSUM-III chart at θ ¼ 0:01 decreases from 92.46 to 60.45. The zero-state ARL1 value of the same chart decreases slightly from 3.24 to 2.37 at a large drift of θ ¼ 4. This observation also holds in the steady-state analysis. For a full evaluation of the effect of γ, it is also necessary to consider the ARL changes for values of γ smaller than 0.1. For this purpose, Fig. 2 further plots the zero-state and steady-state ARL1 curves of the ACUSUM-III chart for different values of γ that vary over the small interval ½0:001; 0:1. Three drift coefficients of θ ¼ 0:001, 0.01, and 0.5 are considered, which correspond to small, medium, and large magnitudes. From Fig. 2, it is surprising to observe that the zero-state ARL1 curve increases with γ, regardless of whether the drift coefficient θ is small or large. The optimal γ for minimizing the zero-state ARL1 approaches zero. This observation is similar to the observation that the optimal smoothing parameter of the EWMA chart for monitoring normal means should approach zero in order to give the zerostate ARL optimality, as discussed by Frisén (2003). In this case, nearly equal weight is given to all observations, and the EWMA chart gives the minimal ARL1 for fixed ARL0. This has also been confirmed by simulations in Chan and Zhang (2000) in the study of design issues about EWMA charts. Different from the zero-state results, the steady-state ARL1 curves are no longer an increasing function of γ. It is clear from Fig. 2 that the steady-state ARL1 value achieves the minimum approximately at γ ¼ 0.002 under the three cases of θ ¼0.001, 0.01, and 0.5. Note that the observed optimal value of γ ¼ 0:002 is much smaller than the values (0:05 r γ r 0:25) commonly suggested based on the rule of thumb.
F. He et al. / Int. J. Production Economics 148 (2014) 14–20
12
Zero-state ARL
Zero-state ARL
200 150 100
10 8 6 4
50 θ= 0 θ= 5e-04
0 0.2
θ= 0.001 θ= 0.005
0.4
θ= 0.01 θ= 0.05
0.6
2
θ= 0.1
0.8
θ= 0.5 θ= 1
0.2
1.0
θ= 2 θ= 3
0.4
γ
θ= 4
0.6
0.8
1.0
0.8
1.0
γ
150
10
Steady-state ARL
Steady-state ARL
17
100
50
θ= 0.001 θ= 0.005
θ= 0 θ= 5e-04
0 0.2
0.4
θ= 0.01 θ= 0.05
0.6
2 4 6
θ= 0.1
0.8
8
1.0
θ= 0.5 θ= 1
0.2
θ= 2 θ= 3
0.4
γ
θ= 4
0.6
γ
Fig. 1. Effects of γ on ARL values of the ACUSUM-III chart when λ0 ¼ 4 and ARL0 200.
From Fig. 1, the steady-state ARL values tend to be smaller than the corresponding zero-state ARL values when the same control limit is used for analyzing both zero-state and steady-state results. However, if γ is too small, it was found within our investigation that the steady-state ARL can be much larger than the zero-state ARL. In line with Table 1, Table 2 presents both zero-state and steady-state ARL values of the ACUSUM-III chart when γ takes extremely small values, say 0:001 r γ r0:01. The values in bold represent the minimum ARL1 value at a particular drift coefficient. It is clear from Table 2 that when γ r0:003, the steady-state ARL values of the ACUSUM-III chart are larger than the zero-state ARL values. This observation is opposite to that observed from Table 1.
specified for early detection. Based on the assumption that a step shift occurs at the initial time step with magnitude of λ1, the corresponding log-likelihood ratio at time t is given by
4. Comparison with CUSUM charts under linear drifts
k¼
It is of interest to compare with the ACUSUM chart with some CUSUM charts under linear drift in the Poisson rate, including the CUSUM chart developed for monitoring linear drifts and the CUSUM chart developed for monitoring step mean shifts. However, both CUSUM charts require the assumption of a known outof-control parameter, details of which are summarized below. Let θ1 denote the out-of-control drift coefficient (in the unit of pffiffiffiffiffi λ0 ) pre-specified for early detection. Assuming a linear drift occurs at the rate of θ1 at the initial time, the process mean at time t is expected to be pffiffiffiffiffi λnt ¼ λ0 þ t θ1 λ0 :
is often called the reference value of the Poisson CUSUM chart. For the sake of simplicity, the above two charts in Eqs. (6) and (7) designed for detecting linear drifts and step shifts will be denoted as CUSUMD and CUSUMS charts, respectively. In the simulation studies, the parameters for the ACUSUM-III chart were set to γ ¼ 0:002 and h3 ¼ 0:0378 for providing an approximate in-control ARL of 200. Two values of θ1 ¼ 0:001 and 0.0025 were considered for the CUSUMD chart. For each value of λ0, two values of λ1 were considered for the CUSUMS chart. The combinations of λ0 and λ1 used in the simulation studies are summarized in Table 3. Table 4 compares the ARL values among the ACUSUM-III, CUSUMD , CUSUMS charts when λ0 ¼ 4. In the zero-state analysis, the ACUSUM-III chart has substantially short ARL1 values than the other charts, especially when the drift coefficient is small. Based on the steady-state results, the ACUSUM-III chart outperforms the CUSUMS chart at small drift coefficients and performs slightly worse than the CUSUMS chart with λ1 ¼ 6 at relatively large drift coefficients. Both the ACUSUM-III and CUSUMS charts perform better than the CUSUMD chart. A pronounced observation from Table 4 is that the CUSUMD chart performs poorly at the pre-specified drift coefficient, although the CUSUMD chart is especially designed for detecting linear drifts in the Poisson rate. For example, the CUSUMD chart designed with θ1 ¼ 0:001 has the zero-state ARL1 value of 145.56 at θ ¼ 0:001, which is substantially larger than the respective
Based on the assumption of known
θ1, it follows that
Lt ¼ log f 1 ðX t Þ log f 0 ðX t Þ ¼ X t ðlog λt log λ0 Þ ðλt λ0 Þ: n
n
Substituting this log-likelihood ratio into Eq. (1) leads to the following CUSUM charting statistics: n n Dt ¼ maxf0; Dt 1 þX t log λt log λ0 λt λ0 g; t ¼ 1; 2; …; ð6Þ which signals when Dt 4 hD . The conventional CUSUM chart designed for detecting step mean shifts can also be used for monitoring linear drifts. It is designed aimed at quickly detecting a pre-specified mean shift. Let λ1 be the out-of-control mean in the underlying Poisson process
Lt ¼ log f 1 ðX t Þ log f 0 ðX t Þ ¼ X t ðlog λ1 log λ0 Þ ðλ1 λ0 Þ: Likewise, substituting it into Eq. (1) leads to the following CUSUM charting statistics:
λ1 λ1 λ0 Xt ; ð7Þ St ¼ max 0; St 1 þ log λ0 log λ1 log λ0 which alarms when St 4 hS . This CUSUM chart is equivalent to the conventional Poisson CUSUM chart discussed by Lucas (1985) except a constant multiplier log ðλ1 =λ0 Þ on the term (X t kÞ, where
λ1 λ0 log λ1 log λ0
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F. He et al. / Int. J. Production Economics 148 (2014) 14–20
Steady state ARL
Zero state ARL
140 120 100 80
105
95 90
60 85 0.001
0.03
0.06
0.09
0.001
0.03
0.06
0.09
0.001
0.03
0.06
0.09
0.001
0.03
0.06
0.09
60 Steady state ARL
Zero state ARL
50 50 40 30
45
40
35 0.001
0.03
0.06
0.09
6.6 Steady state ARL
Zero state ARL
7.0
6.0
5.0
6.2
5.8
5.4
4.0 0.001
0.03
0.06
0.09
Fig. 2. Plots of out-of-control ARL values for the ACUSUM-III chart when λ0 ¼ 4 and ARL0 200. (a) θ ¼ 0:001, (b) θ ¼ 0:01, (c) θ ¼ 0:5.
values of 115.24 and 67.58 for the CUSUMS (λ1 ¼ 6) and ACUSUMIII charts. Although the CUSUMD chart tries to optimize the performance for detecting the shift of a particular size at each step, it cannot provide an overall good detection performance as the Poisson mean varies over time. Clearly, the optimality property of the conventional CUSUM chart for detecting step shifts cannot apply to the case under linear drifts. The comparison is also conducted for larger values of λ0 ¼ 6, 10, 20 but the results are omitted in this paper. Similar observations to the case with λ0 ¼ 4 can be made. The above comparisons indicate a good property of the ACUSUM-III chart. In particular, the ACUSUM-III chart can always provide the shortest ARL1 values at small drift coefficients and the ARL1 values close to or slightly larger than the shortest at relatively large drift coefficients.
Moreover, compared to the CUSUMD and CUSUMS charts, the ACUSUM chart has the advantage of being free from the requirement of a known shift size.
5. A simulation example In this section, we use a simulation example from Perry et al. (2007) to demonstrate the implementation of the ACUSUM-III chart. As done in Perry et al. (2007), the data were simulated from a Poisson distribution with a rate of 5 for the first 25 observations and from a Poisson distribution with rate λi ¼ λ0 þ 0:15ði 25Þ starting at observation 26. The in-control rate was assumed to be λ0 ¼ 5. Clearly, the simulated process maintains in control for the
F. He et al. / Int. J. Production Economics 148 (2014) 14–20
19
Table 2 Comparisons of ARL values of the ACUSUM-III chart when λ0 ¼ 4 and ARL0 200. γ
Drift (θ)
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.010
202.04 67.41 54.30 29.23 22.35 11.44 8.35 3.87 2.74 1.95 1.59 1.35
200.17 86.21 67.58 37.37 28.15 13.99 10.15 4.58 3.23 2.25 1.83 1.58
199.90 96.87 76.52 42.13 31.69 15.51 11.16 5.01 3.52 2.46 2.01 1.78
200.68 104.44 82.88 45.50 34.30 16.54 11.88 5.31 3.71 2.60 2.10 1.86
199.70 109.57 87.34 47.96 36.07 17.30 12.37 5.52 3.85 2.70 2.17 1.92
199.52 113.68 90.99 49.98 37.51 17.90 12.75 5.69 3.97 2.76 2.21 1.94
201.04 117.51 94.32 51.87 38.72 18.40 13.08 5.81 4.07 2.84 2.28 1.99
199.61 119.93 96.64 53.16 39.63 18.72 13.33 5.92 4.14 2.89 2.32 2.00
199.42 122.20 98.61 54.43 40.52 19.06 13.55 6.02 4.19 2.94 2.35 2.04
199.26 124.23 100.35 55.32 41.24 19.30 13.75 6.08 4.25 2.95 2.37 2.04
Steady-state ARL 0.0005 122.96 0.0010 93.80 0.0050 47.87 0.0100 35.07 0.0500 16.56 0.1000 11.91 0.5000 5.29 1.0000 3.71 2.0000 2.60 3.0000 2.12 4.0000 1.84
112.51 87.76 46.14 34.33 16.33 11.71 5.26 3.69 2.57 2.10 1.82
108.98 85.77 45.92 34.39 16.47 11.86 5.31 3.73 2.60 2.11 1.82
107.46 85.94 46.63 34.88 16.77 12.00 5.40 3.79 2.65 2.15 1.85
107.54 86.84 47.21 35.41 17.08 12.23 5.48 3.85 2.69 2.19 1.88
108.62 87.91 48.10 35.84 17.42 12.45 5.57 3.89 2.73 2.21 1.91
109.51 89.12 48.88 36.42 17.65 12.60 5.66 3.95 2.77 2.24 1.93
109.80 90.07 49.57 37.01 17.85 12.73 5.71 3.99 2.79 2.27 1.95
110.89 91.04 50.23 37.59 18.03 12.87 5.79 4.04 2.82 2.29 1.97
111.35 91.60 50.76 37.95 18.17 13.03 5.81 4.07 2.84 2.30 1.99
Zero-state ARL 0.0000 0.0005 0.0010 0.0050 0.0100 0.0500 0.1000 0.5000 1.0000 2.0000 3.0000 4.0000
Table 3 Combinations of λ0 and λ1 studied through simulation.
Table 4 Comparison of ARL values for λ0 ¼ 4 with ARL0 200. hS ¼ 3:3530 CUSUMS λ1 ¼ 6
hD ¼ 1:2007 CUSUMD θ1 ¼ 0:001
hD ¼ 2:1245 CUSUMD θ1 ¼ 0:0025
h3 ¼ 0:0378 ACUSUM-III γ ¼ 0:002
Zero-state ARL 0.0000 200.76 0.0005 132.95 0.0010 107.43 0.0050 56.38 0.0100 40.09 0.0500 17.30 0.1000 12.06 0.5000 5.32 1.0000 3.74 2.0000 2.67 3.0000 2.16 4.0000 1.93
204.22 141.43 115.24 59.66 41.85 17.14 11.56 4.85 3.36 2.34 1.93 1.69
200.54 166.19 145.56 90.35 71.73 41.86 33.16 19.36 15.35 12.12 10.75 9.83
200.36 164.61 142.08 83.97 65.59 37.61 29.70 17.28 13.73 10.94 9.49 8.76
200.17 86.21 67.58 37.37 28.15 13.99 10.15 4.58 3.23 2.25 1.83 1.58
Steady-state ARL 0.0005 127.25 0.0010 104.35 0.0050 53.78 0.0100 38.07 0.0500 16.36 0.1000 11.34 0.5000 4.87 1.0000 3.41 2.0000 2.40 3.0000 1.97 4.0000 1.74
137.15 113.70 58.28 41.00 16.76 11.24 4.60 3.17 2.20 1.80 1.57
126.67 111.87 67.01 51.05 25.65 18.79 8.86 6.37 4.56 3.77 3.22
125.32 109.48 62.90 46.79 22.47 16.29 7.54 5.40 3.86 3.15 2.80
112.51 87.76 46.14 34.33 16.33 11.71 5.26 3.69 2.57 2.10 1.82
Drift (θ) λ0
λ1
4 4 6 6 10 10 20 20
5 6 8 9 12 14 25 30
first 25 observations and a drift occurs at the rate of 0.15 following time step 25. Suppose the desired in-control ARL of the ACUSUM-III chart is 400. The parameter γ of the ACUSUM-III chart is set to be γ ¼ 0:002 based on the previous study. Then simulations can be performed to determine the threshold for providing the desired in-control ARL, leading to h3 ¼ 0:1319. For comparison, the CUSUMS chart is also considered here. The CUSUMS chart designed is aimed at detecting a sudden step increase of 25% in-control Poisson rate, i.e. λ1 ¼ λ0 þ 0:25λ0 ¼ 6:25. The threshold for the CUSUMS chart was determined as hS ¼ 3:4717 for providing an approximated in-control ARL of 400. Table 5 presents the simulated observations, CUSUMS and ACUSUM-III charting statistics computed at each time step. Fig. 3 plots the charting patterns of both the ACUSUM-III and CUSUMS charts. From Table 5, the ACUSUM-III chart signals an out-ofcontrol situation at observation 33 while the CUSUMS chart signals at observation 37. Clearly, the ACUSUM-III chart signals 4 steps earlier than the CUSUMS chart. This example again illustrates that the conventional CUSUM chart optimally designed for detecting step shifts of a particular size is less efficient when linear drifts actually occurred. In contrast, the ACUSUM chart that dynamically updates the likelihood ratio based on an estimate of the Poisson
hS ¼ 2:7100 CUSUMS λ1 ¼ 5
rate can provide a relatively good detection performance under linear drifts.
6. Concluding remarks When linear drifts are introduced to a Poisson process, the mean of the underlying process is no longer constant but changes over time. Although the conventional CUSUM charts developed
20
F. He et al. / Int. J. Production Economics 148 (2014) 14–20
Table 5 A simulation example. t
Obs
CUSUMS
ACUSUM-III
t
Obs
CUSUMS
ACUSUM-III
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3 3 8 8 6 6 7 4 6 4 1 6 7 8 6 3 11 4 5 2
0.0000 0.0000 0.5351 1.0703 1.1592 1.2480 1.5600 1.2026 1.2915 0.9340 0.0000 0.0889 0.4009 0.9360 1.0249 0.4443 1.6489 1.2915 1.1572 0.3535
0.0000 0.0000 0.0000 0.0024 0.0036 0.0052 0.0100 0.0080 0.0104 0.0084 0.0068 0.0076 0.0108 0.0191 0.0222 0.0175 0.0459 0.0415 0.0415 0.0321
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
12 6 4 4 5 4 2 6 7 9 6 8 7 6 7 7 9 7 7 10
1.7812 1.8700 1.5126 1.1552 1.0209 0.6635 0.0000 0.0889 0.4009 1.1592 1.2480 1.7832 2.0952 2.1840 2.4960 2.8080 3.5663 3.8783 4.1903 5.1718
0.0733 0.0795 0.0735 0.0680 0.0679 0.0628 0.0512 0.0554 0.0654 0.0917 0.0986 0.1229 0.1406 0.1497 0.1697 0.1911 0.2405 0.2664 0.2939 0.3727
The simulation results show that the proposed ACUSUM chart outperforms the CUSUMS chart designed for detecting step mean shifts at small drift coefficient and performs only sightly worse than the latter at relatively large drift coefficients. Both the ACUSUM and CUSUMS charts perform better than the CUSUMD chart. Although the CUSUMD chart aims at optimizing the performance for detecting the shift of a particular size at each step, it cannot provide an overall good detection performance under linear drifts where the Poisson mean is time-varying.
Acknowledgments The authors would like to thank the referees for their valuable suggestions. Dr. L. J. Shu's work was supported in part by the Research Committee under the Grant MYRG096(Y2-L2)-FBA12-SLJ and FDCT/002/2013/A. Professor K.-L. Tsui's work was supported in part by the Research Committee under the Grant HK RGC GRF 121410 and Food and Health Bureau RFCID Fund 11101262.
References
0
CUSUM_S ACUSUM III
Changing Statistic
1 2 3 4 5 0
10 20 30 Observation Number
40
Fig. 3. A simulation example.
based on the likelihood ratio can be used for monitoring linear drifts, they often require the assumption of a known out-ofcontrol parameter. This parameter is often unknown in practice. To relax this limitation, this paper develops ACUSUM charts for monitoring drifts in the Poisson rate. Compared to the conventional CUSUM charts, the ACUSUM chart can dynamically update the likelihood ratio based on an estimate of the current Poisson mean level. Three different EWMA-type schemes for estimating Poisson means are considered, including the EWMA estimator without reset, the EWMA estimator with reset, and a combined estimator. The second estimator with reset enables the EWMA estimator to quickly catch up with the change in the Poisson mean level once a shift occurs. However, it tends to overestimate the mean level in the in-control situation due to reset at the same time. As compared to the second mean estimator, the third estimator can provide a more accurate estimate of the Poisson rate in the incontrol case while maintaining the same estimation efficiency in the out-of-control case. Therefore, the ACUSUM-III chart is expected to perform better than the other two ACUSUM charts.
Aebtarm, S., Bouguila, N., 2011. An empirical evaluation of attribute control charts for monitoring defects. Expert Syst. Appl. 38 (6), 7869–7880. Brook, D., Evans, D.A., 1972. An approach to the probability distribution of CUSUM run length. Biometrika 59 (3), 539–549. Chan, L.K., Zhang, J., 2000. Some issues in the design of EWMA charts. Commun. Stat.: Simul. Comput. 29 (1), 207–217. Chan, L.-Y., Ouyang, J., Lau, H.Y.-K., 2007. A two-stage cumulative quantity control chart for monitoring poisson processes. J. Qual. Technol. 39 (3), 203–223. Faltin, F.W., 2008. Control charts, overview. In: Encyclopedia of Statistics in Quality and Reliability. John Wiley & Sons, Ltd. Frisén, M., 2003. Statistical surveillance, optimality and methods. Int. Stat. Rev. 71 (2), 403–434. Han, S.W., Tsui, K.-L., Ariyajunya, B., Kim, S.B., 2010. A comparison of CUSUM, EWMA, and temporal scan statistics for detection of increases in Poisson rates. Qual. Reliab. Eng. Int. 26 (3), 279–289. Hawkins, D.M., Olwell, D.H., 1998. Cumulative Sum Charts and Charting for Quality Improvement. Springer, New York. Jiang, W., Shu, L., Apley, D.W., 2008. Adaptive CUSUM procedures with EWMAbased shift estimators. IIE Trans. 40 (10), 992–1003. Jiang, W., Shu, L., Tsui, K.-L., 2011. Weighted CUSUM control charts for monitoring Poisson processes with varying sample sizes. J. Qual. Technol. 43 (4), 346–362. Lucas, J.M., 1985. Counted data CUSUM's. Technometrics 27 (2), 129–144. Montgomery, D.C., 2009. Introduction to Statistical Quality Control. John Wiley & Sons, Inc., Hoboken, NJ. Page, E.S., 1954. Continuous inspection schemes. Biometrika 41 (1/2), 100–115. Perry, M.B., Pignatiello, J.J.J., Simpson, J.R., 2006. Estimating the change point of a Poisson rate parameter with a linear trend disturbance. Qual. Reliab. Eng. Int. 22 (4), 371–384. Perry, M.B., Pignatiello, J.J.J., Simpson, J.R., 2007. Change point estimation for monotonically changing Poisson rates in SPC. Int. J. Product. Res. 45 (8), 1791–1813. Roberts, S.W., 1959. Control chart tests based on geometric moving averages. Technometrics 1 (3), 239–250. Ryan, A.G., Woodall, W.H., 2010. Control charts for Poisson count data with varying sample sizes. J. Qual. Technol. 42 (3), 260–275. Sparks, R.S., 2000. CUSUM charts for signalling varying location shifts. J. Qual. Technol. 32 (2), 157–171. Shu, L., Jiang, W., 2006. A Markov chain model for the adaptive CUSUM control chart. J. Qual. Technol. 38 (2), 135–147. Tsui, K.-L., Chiu, W., Gierlich, P., Goldsman, D., Liu, X., Maschek, T., 2008. A review of healthcare, public health, and syndromic surveillance. Qual. Eng. 20 (4), 435–450. White, C.H., Keats, J.B., 1996. ARLs and higher-order run-length moments for the Poisson CUSUM. J. Qual. Technol. 28 (3), 363–369. White, C.H., Keats, J.B., Stanley, J., 1997. Poisson CUSUM versus c chart for defect data. Qual. Eng. 9 (4), 673–679. Woodall, W.H., 1997. Control charts based on attribute data: bibliography and review. J. Qual. Technol. 29 (2), 172–183. Woodall, W.H., 2006. The use of control charts in health-care and public-health surveillance. J. Qual. Technol. 38 (2), 89–104.
Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Adaptive CUSUM charts for monitoring linear drifts in Poisson rates Feng He a, Lianjie Shu b,n, Kwok-Leung Tsui c a b c
Department of Statistics and Actuarial Science, University of Waterloo, Canada Faculty of Business Administration, University of Macau, Macau Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong
art ic l e i nf o
a b s t r a c t
Article history: Received 30 August 2012 Accepted 4 November 2013 Available online 15 November 2013
There has been relatively less attention paid to the monitoring of linear drifts in Poisson rates. Although the conventional CUSUM charts can be used for monitoring linear drifts in Poisson rates, they often rely on the assumption of a known parameter. To get rid of this assumption, this paper develops adaptive CUSUM (ACUSUM) charts for monitoring drifts in Poisson rates. The basic idea is to first estimate the current Poisson mean level and then to dynamically update the likelihood ratio in the CUSUM chart based on the estimated mean. Three different mean estimators based on the exponentially weighted moving average (EWMA) schemes are discussed. The comparison results are shown to favor the ACUSUM chart, especially at small drifts. & 2013 Elsevier B.V. All rights reserved.
Keywords: Change-point detection Linear drift Poisson distribution Statistical process control
1. Introduction In manufacturing and business processes as well as health surveillance, it is often important to detect changes in count data in addition to monitoring variable data (Woodall, 2006; Faltin, 2008; Tsui et al., 2008; Montgomery, 2009). The Poisson distribution is usually used to model count data over some interval unit based on time, distance, area or some similar unit. For example, the number of nonconformities per unit of product in the industrial setting and the number of new infections in a hospital per day in healthcare are often modeled by the Poisson distribution. The Poisson distribution has only one parameter, rate of count per unit, which characterizes both mean and variance. Various approaches have been proposed for monitoring changes of rate of a Poisson distribution. The Shewhart c-chart or u-chart is the simplest procedure to monitor Poisson processes. However, one drawback of it is that it is not effective for the quick detection of small shifts in the Poisson mean parameter as it only uses the information in the current sample. To improve the detection ability at small shifts, the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts, which incorporate information in both current and past samples, are often utilized. The pioneering work on the use of the CUSUM chart for monitoring Poisson processes is given by Brook and Evans (1972). They introduced the Markov chain approach for evaluating the run length of the one-sided Poisson CUSUM chart under a step shift. Lucas (1985) discussed the design and implementation
n
Corresponding author. Tel.: +853-8397-4741; fax: +853-2883-8320. E-mail address: [email protected] (L. Shu).
0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2013.11.004
procedures for both one- and two-sided Poisson CUSUM charts and enhancements such as the fast initial response (FIR) feature. Some recent studies on the Poisson CUSUM chart include White and Keats (1996), White et al. (1997), Hawkins and Olwell (1998), Chan et al. (2007), Han et al. (2010), Ryan and Woodall (2010), Aebtarm and Bouguila (2011), and Jiang et al. (2011). For an early review on attribute control charts, interested readers can be referred to Woodall (1997). However, the above cited papers mainly focus on the detection of step shifts in the Poisson rate parameter. In practice, there are often settings where a linear drift needs to be considered. For example, tool wear may cause the mean count rate of nonconformities of a machined part to gradually increase per unit sampled. There is relatively less attention devoted to the detection of linear drifts in the Poisson mean rate. Some exceptions include Perry et al. (2006, 2007). They compared the performance of change point estimators under a linear trend and monotonic changes in the Poisson rate. However, they did not address the monitoring issues under a linear trend in the Poisson rate. Although the conventional Poisson CUSUM chart designed for step changes can be utilized for monitoring linear drift in the Poisson rate, it may be inefficient due to the mis-specification of the shift form. Another drawback of the traditional Poisson CUSUM charts is that it requires the prior knowledge of the shift size, which also implies that it may not work well for detecting linear drifts. Motivated by this, this paper suggests an adaptive CUSUM (ACUSUM) chart for monitoring linear drifts in the Poisson rate. The rest of this paper is organized as follows. In Section 2, the ACUSUM charts with different estimators of Poisson rate are introduced. In Section 3, the effects of mean estimators as well as effects of the smoothing parameter on the ACUSUM charts are
F. He et al. / Int. J. Production Economics 148 (2014) 14–20
discussed. In Section 4, the performance between CUSUM and ACUSUM charts under linear drifts is compared. In Section 5, an example is given to illustrate the implementation of the proposed chart. Concluding remarks are given in Section 6.
2. ACUSUM charts Let X 1 ; X 2 ; …, be a sequence of observations collected from a Poisson distribution with mean λt at fixed intervals of time. When the process is in the state of statistical control, observations are assumed to be independently distributed from a Poisson distribution with a known mean λ0. After an unknown time point τ, the process mean is subject offfiffiffiffiffiin-control pffiffiffiffiffito a linear drift in the unitp standard deviation ( λ0 ). The amount of drift is θ λ0 per unit time, where θ is unknown. In other words, the process mean at time t can be represented as ( λ0 t o τ; pffiffiffiffiffi λt ¼ t ¼ 1; 2; …; λ0 þ ðt τ þ 1Þθ λ0 t Z τ; For simplicity, we focus on the one-sided upper control chart aimed at detecting an increase of θ from zero, which is common in industrial quality control. The CUSUM chart proposed by Page (1954) can be formulated as a sequential hypothesis testing procedure for the change point from a known in-control distribution to another known alternative distribution. It has the basic form C t ¼ maxf0; C t 1 þLt g;
t ¼ 1; 2; …
ð1Þ
where C 0 ¼ 0 and the increment Lt ¼ log ½f 1 ðX t Þ=f 0 ðX t Þ is the loglikelihood ratio to contrast the alternative distribution f1 with the null distribution f0. At time t, the observed count Xt follows a Poisson distribution with mean λt. Thus, the log-likelihood ratio statistic for the tth observation follows Lt ¼ log f 1 ðX t Þ log f 0 ðX t Þ ¼ X t ðlog λt log λ0 Þ ðλt λ0 Þ:
ð2Þ
When λt is known, the CUSUM chart can be easily established by substituting Eq. (2) into Eq. (1). However, the drift coefficient θ is rarely unknown in real practice. To deal with this issue, one natural idea is to replace θ by its estimate. This is the key idea of the adaptive CUSUM chart that has been widely discussed for monitoring process changes under normal distributions. See, for example, Sparks (2000), Shu and Jiang (2006), and Jiang et al. (2008). The EWMA scheme has been widely used as an effective tool for parameter estimation due to its simplicity and efficiency (Roberts, 1959). Analogously, it can also be used to estimate the Poisson rate. The EWMA estimate of λt is based on Y tð1Þ ¼ ð1 γ ÞY ð1Þ t 1 þ γXt ;
ð3Þ
¼ λ0 and γ (0 o γ o 1) is a smoothing where the initial value is parameter. Note that when γ ¼ 1, the EWMA estimate reduces to Xt. This could cause the log-likelihood ratio in Eq. (2) to be undefined as log X t is not well defined when X t ¼ 0. Therefore, we restrict γ on the range 0 o γ o1 in this case. Note that the Poisson rate keeps increasing over time once a shift occurred in the process. In order to improve the estimation efficiency of the EWMA scheme, one can reset the EWMA estimate to the nominal rate whenever it is less than the nominal rate. This leads to the second type of EWMA estimation scheme: Y 0ð1Þ
Y tð2Þ ¼ maxfλ0 ; ð1 γ ÞY ð2Þ t 1 þ γ X t g;
ð4Þ
where Y ð2Þ 0 ¼ λ0 . This estimation scheme has been discussed by Sparks (2000) and Jiang et al. (2008) for estimating the mean level of a normal process.
15
Note that the second EWMA estimation scheme may not be able to accumulate the information of historical observations in a smooth way due to the reset of EWMA statistic to the in-control rate when Y ð2Þ t o λ0 . To overcome this drawback, we suggest a third estimation scheme based on the combined use of the above two estimators. In particular, the EWMA scheme in Eq. (3) is used to smooth the observed data as usual. When the EWMA statistic is less than λ0, the process is considered to be in control. It is thus reasonable to assume the current mean stays at the level of λ0. Otherwise when EWMA statistic is larger than λ0, we may have reasons to doubt that the process is out of control. In this case, the EWMA statistic can be served as an online estimate of the current Poisson mean level. To sum up, n o ð1Þ Y ð3Þ ; ð5Þ t ¼ max λ0 ; Y t where Y tð1Þ is defined by Eq. (3). Compared to the second estimation scheme, this estimation scheme does not directly reset the EWMA statistic but resets the estimate to λ0 when the EWMA statistic Y tð1Þ is smaller λ0. The ACUSUM chart can be established by replacing Lt in Eq. (1) with an estimate of the Poisson rate. That is, the ACUSUM charting statistics are based on At ¼ maxf0; At 1 þ X t ðlog λ^ t log λ0 Þ ðλ^ t λ0 Þg;
t ¼ 1; 2; …;
where λ^ t denotes an estimate of λt. For the sake of simplicity, the resulting ACUSUM charts based on the mean estimates in (Eqs. (3), 4), and (5) will be referred to as ACUSUM-I, ACUSUM-II, and ACUSUM-III charts, respectively. An out-of-control signal is triggered when the ACUSUM charts exceed the threshold values of h1, h2, and h3, respectively.
3. Performance analysis of ACUSUM charts The average run length (ARL), which describes the average number of samples until the first out-of-control signal on a control chart, has been widely used for chart performance comparisons. The out-of-control ARL (ARL1) refers to the average number of samples collected before a shift is detected, while the in-control ARL (ARL0) is the expected number of samples until a false alarm. The ideal chart is expected to have small ARL1 but large ARL0. In this section, we first compare the ARL performance among the ACUSUM charts based on the above estimation schemes. Then the choice of the smoothing parameter γ for designing ACUSUM charts is discussed. The measured ARL values can be either in the zero state or steady state. The zero-state ARL refers to the ARL value obtained assuming that the change occurs at the initial startup of a control chart, while the steady-state ARL is based on a delayed shift which occurs after a period of monitoring. To provide a complete comparison, both the zero-state and steady-state ARL results have been analyzed based on Monte Carlo simulations. The simulation results were based on 40,000 replicates. The zero-state ARL results were obtained based on τ ¼ 1. The steady-state ARL results were simulated based on the fact that the control chart has run in the in-control state for 50 samples before the drift in the rate parameter takes place. 3.1. Comparison among adaptive CUSUM charts Table 1 presents both the zero-state ARL and steady-state ARL values of the three ACUSUM charts under drifts in the Poisson rate when λ0 ¼ 4. Three different smoothing parameter values of γ ¼ 0:005, 0.01, and 0.05 were considered. The in-control ARL values of all the charts are maintained around 200. The same
16
F. He et al. / Int. J. Production Economics 148 (2014) 14–20
Table 1 Comparison of ARL values among ACUSUM charts when λ0 ¼ 4 and ARL0 200. Limit
γ ¼ 0:005
γ ¼ 0:01
γ ¼ 0:05
ACUSUM-I 0.8873
II 0.7991
III 0.1843
ACUSUM-I 1.6976
II 1.5075
III 0.5190
ACUSUM-I 7.5961
II 6.1395
III 3.5006
θ 0.0000 0.0005 0.0010 0.0050 0.0100 0.0500 0.1000 0.5000 1.0000 2.0000 3.0000 4.0000
Zero-state ARL 199.57 182.01 156.17 81.99 59.25 26.96 19.03 8.38 5.87 4.08 3.28 2.88
200.13 156.44 131.85 72.94 53.64 25.34 18.10 8.11 5.69 3.98 3.19 2.82
199.51 109.21 87.41 48.11 36.10 17.30 12.37 5.52 3.85 2.70 2.17 1.93
199.70 184.25 159.64 83.57 60.17 27.11 19.06 8.36 5.84 4.06 3.25 2.87
200.20 160.48 136.19 74.67 54.57 25.37 18.06 8.05 5.64 3.94 3.16 2.78
199.65 124.21 100.25 55.32 41.25 19.30 13.74 6.07 4.24 2.96 2.37 2.04
199.52 192.21 177.10 100.34 70.97 29.85 20.53 8.66 5.99 4.14 3.31 2.90
199.89 173.49 154.63 88.99 63.70 27.36 18.89 8.09 5.62 3.90 3.12 2.73
200.18 152.41 129.01 72.33 52.35 23.19 16.14 6.93 4.80 3.32 2.69 2.25
0.0005 0.0010 0.0050 0.0100 0.0500 0.1000 0.5000 1.0000 2.0000 3.0000 4.0000
Steady-state ARL 143.33 128.56 73.36 53.99 25.14 17.78 7.90 5.50 3.83 3.10 2.67
124.49 107.20 60.61 45.01 21.09 15.05 6.67 4.69 3.29 2.67 2.30
107.60 86.74 47.10 35.48 17.03 12.21 5.48 3.84 2.69 2.18 1.88
144.82 130.59 74.68 54.58 25.12 17.74 7.85 5.46 3.80 3.07 2.65
127.55 110.63 62.88 46.27 21.41 15.18 6.70 4.71 3.28 2.67 2.29
111.11 91.83 50.58 37.98 18.16 13.03 5.80 4.07 2.84 2.30 1.99
146.47 138.96 86.84 62.65 27.13 18.77 7.97 5.51 3.81 3.07 2.65
134.21 122.32 75.35 55.20 24.15 16.65 7.07 4.90 3.38 2.74 2.33
122.13 105.46 61.40 45.52 20.77 14.60 6.31 4.39 3.05 2.46 2.11
control limit as the zero-state analysis was used to analyze the steady-state results for each chart. A pronounced observation from Table 1 is that the ACUSUM-III chart has shorter zero-state out-ofcontrol ARL values than the ACUSUM-II chart, and than the ACUSUM-I chart for all the cases considered here. Similar observations can be made from the steady-state results. This indicates that the ACUSUM-III chart performs uniformly better than the ACUSUM-II chart, and than the ACUSUM-I chart for monitoring linear drifts in Poisson rates. The above observation demonstrates that the reset of EWMA estimate in the ACUSUM chart improves over the chart without reset. The resetting operation sets the EWMA statistic to a lower bound when it is small. This enables the EWMA statistic to quickly catch up with the current Poisson mean level when there is a shift, which in turn speeds up the detection performance of the ACUSUM chart. Therefore, it is expected that the ACUSUM-II chart is more sensitive than the ACUSUM-I chart. Instead of resetting the EWMA statistic directly, the third mean estimation scheme, ð1Þ Y ð3Þ t ¼ max fλ0 ; Y t g; resets the current estimate to λ0 when Y ð1Þ o λ . By comparing Y ð3Þ and Y tð2Þ , it can be observed that both 0 t t methods can provide a fast reaction to changes of mean levels in the out-of-control case as both methods involve the use of maximum operation. However, in the in-control case, Y ð3Þ tends to provide a t more accurate estimate of Poisson mean than Y ðt2Þ . This is because the observed count is always nonnegative over time. Therefore, Y ð2Þ t would be very likely to exceed λ0 in the next few steps after it was reset to the lower bound, even when the process is still in the status of in-control. In contrast, Y ð3Þ is likely to maintain at λ0 t conditioned that the EWMA statistic, Y ð1Þ t , is not too close to λ0. This advantage of the third estimator leads to better detection performance of the ACUSUM-III chart as compared to the ACUSUM-II chart. 3.2. Effects of
γ
Due to the superior performance of the ACUSUM-III chart, we will only use it for discussion throughout the remaining of the
paper. The design of ACUSUM-III chart involves the choices of γ and h3. To provide some guidelines on the choice of γ, we first investigate the effect of γ on the performance of ACUSUM-III chart. Fig. 1 shows the ARL values of the ACUSUM-III chart when γ varies over the interval ½0:1; 1. It can be observed that the ACUSUM-III chart with smaller values of γ has consistently smaller ARL1 values at both small and large drifts. For example, when γ decreases from 1 to 0.1, the zero-state ARL1 value of the ACUSUM-III chart at θ ¼ 0:01 decreases from 92.46 to 60.45. The zero-state ARL1 value of the same chart decreases slightly from 3.24 to 2.37 at a large drift of θ ¼ 4. This observation also holds in the steady-state analysis. For a full evaluation of the effect of γ, it is also necessary to consider the ARL changes for values of γ smaller than 0.1. For this purpose, Fig. 2 further plots the zero-state and steady-state ARL1 curves of the ACUSUM-III chart for different values of γ that vary over the small interval ½0:001; 0:1. Three drift coefficients of θ ¼ 0:001, 0.01, and 0.5 are considered, which correspond to small, medium, and large magnitudes. From Fig. 2, it is surprising to observe that the zero-state ARL1 curve increases with γ, regardless of whether the drift coefficient θ is small or large. The optimal γ for minimizing the zero-state ARL1 approaches zero. This observation is similar to the observation that the optimal smoothing parameter of the EWMA chart for monitoring normal means should approach zero in order to give the zerostate ARL optimality, as discussed by Frisén (2003). In this case, nearly equal weight is given to all observations, and the EWMA chart gives the minimal ARL1 for fixed ARL0. This has also been confirmed by simulations in Chan and Zhang (2000) in the study of design issues about EWMA charts. Different from the zero-state results, the steady-state ARL1 curves are no longer an increasing function of γ. It is clear from Fig. 2 that the steady-state ARL1 value achieves the minimum approximately at γ ¼ 0.002 under the three cases of θ ¼0.001, 0.01, and 0.5. Note that the observed optimal value of γ ¼ 0:002 is much smaller than the values (0:05 r γ r 0:25) commonly suggested based on the rule of thumb.
F. He et al. / Int. J. Production Economics 148 (2014) 14–20
12
Zero-state ARL
Zero-state ARL
200 150 100
10 8 6 4
50 θ= 0 θ= 5e-04
0 0.2
θ= 0.001 θ= 0.005
0.4
θ= 0.01 θ= 0.05
0.6
2
θ= 0.1
0.8
θ= 0.5 θ= 1
0.2
1.0
θ= 2 θ= 3
0.4
γ
θ= 4
0.6
0.8
1.0
0.8
1.0
γ
150
10
Steady-state ARL
Steady-state ARL
17
100
50
θ= 0.001 θ= 0.005
θ= 0 θ= 5e-04
0 0.2
0.4
θ= 0.01 θ= 0.05
0.6
2 4 6
θ= 0.1
0.8
8
1.0
θ= 0.5 θ= 1
0.2
θ= 2 θ= 3
0.4
γ
θ= 4
0.6
γ
Fig. 1. Effects of γ on ARL values of the ACUSUM-III chart when λ0 ¼ 4 and ARL0 200.
From Fig. 1, the steady-state ARL values tend to be smaller than the corresponding zero-state ARL values when the same control limit is used for analyzing both zero-state and steady-state results. However, if γ is too small, it was found within our investigation that the steady-state ARL can be much larger than the zero-state ARL. In line with Table 1, Table 2 presents both zero-state and steady-state ARL values of the ACUSUM-III chart when γ takes extremely small values, say 0:001 r γ r0:01. The values in bold represent the minimum ARL1 value at a particular drift coefficient. It is clear from Table 2 that when γ r0:003, the steady-state ARL values of the ACUSUM-III chart are larger than the zero-state ARL values. This observation is opposite to that observed from Table 1.
specified for early detection. Based on the assumption that a step shift occurs at the initial time step with magnitude of λ1, the corresponding log-likelihood ratio at time t is given by
4. Comparison with CUSUM charts under linear drifts
k¼
It is of interest to compare with the ACUSUM chart with some CUSUM charts under linear drift in the Poisson rate, including the CUSUM chart developed for monitoring linear drifts and the CUSUM chart developed for monitoring step mean shifts. However, both CUSUM charts require the assumption of a known outof-control parameter, details of which are summarized below. Let θ1 denote the out-of-control drift coefficient (in the unit of pffiffiffiffiffi λ0 ) pre-specified for early detection. Assuming a linear drift occurs at the rate of θ1 at the initial time, the process mean at time t is expected to be pffiffiffiffiffi λnt ¼ λ0 þ t θ1 λ0 :
is often called the reference value of the Poisson CUSUM chart. For the sake of simplicity, the above two charts in Eqs. (6) and (7) designed for detecting linear drifts and step shifts will be denoted as CUSUMD and CUSUMS charts, respectively. In the simulation studies, the parameters for the ACUSUM-III chart were set to γ ¼ 0:002 and h3 ¼ 0:0378 for providing an approximate in-control ARL of 200. Two values of θ1 ¼ 0:001 and 0.0025 were considered for the CUSUMD chart. For each value of λ0, two values of λ1 were considered for the CUSUMS chart. The combinations of λ0 and λ1 used in the simulation studies are summarized in Table 3. Table 4 compares the ARL values among the ACUSUM-III, CUSUMD , CUSUMS charts when λ0 ¼ 4. In the zero-state analysis, the ACUSUM-III chart has substantially short ARL1 values than the other charts, especially when the drift coefficient is small. Based on the steady-state results, the ACUSUM-III chart outperforms the CUSUMS chart at small drift coefficients and performs slightly worse than the CUSUMS chart with λ1 ¼ 6 at relatively large drift coefficients. Both the ACUSUM-III and CUSUMS charts perform better than the CUSUMD chart. A pronounced observation from Table 4 is that the CUSUMD chart performs poorly at the pre-specified drift coefficient, although the CUSUMD chart is especially designed for detecting linear drifts in the Poisson rate. For example, the CUSUMD chart designed with θ1 ¼ 0:001 has the zero-state ARL1 value of 145.56 at θ ¼ 0:001, which is substantially larger than the respective
Based on the assumption of known
θ1, it follows that
Lt ¼ log f 1 ðX t Þ log f 0 ðX t Þ ¼ X t ðlog λt log λ0 Þ ðλt λ0 Þ: n
n
Substituting this log-likelihood ratio into Eq. (1) leads to the following CUSUM charting statistics: n n Dt ¼ maxf0; Dt 1 þX t log λt log λ0 λt λ0 g; t ¼ 1; 2; …; ð6Þ which signals when Dt 4 hD . The conventional CUSUM chart designed for detecting step mean shifts can also be used for monitoring linear drifts. It is designed aimed at quickly detecting a pre-specified mean shift. Let λ1 be the out-of-control mean in the underlying Poisson process
Lt ¼ log f 1 ðX t Þ log f 0 ðX t Þ ¼ X t ðlog λ1 log λ0 Þ ðλ1 λ0 Þ: Likewise, substituting it into Eq. (1) leads to the following CUSUM charting statistics:
λ1 λ1 λ0 Xt ; ð7Þ St ¼ max 0; St 1 þ log λ0 log λ1 log λ0 which alarms when St 4 hS . This CUSUM chart is equivalent to the conventional Poisson CUSUM chart discussed by Lucas (1985) except a constant multiplier log ðλ1 =λ0 Þ on the term (X t kÞ, where
λ1 λ0 log λ1 log λ0
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F. He et al. / Int. J. Production Economics 148 (2014) 14–20
Steady state ARL
Zero state ARL
140 120 100 80
105
95 90
60 85 0.001
0.03
0.06
0.09
0.001
0.03
0.06
0.09
0.001
0.03
0.06
0.09
0.001
0.03
0.06
0.09
60 Steady state ARL
Zero state ARL
50 50 40 30
45
40
35 0.001
0.03
0.06
0.09
6.6 Steady state ARL
Zero state ARL
7.0
6.0
5.0
6.2
5.8
5.4
4.0 0.001
0.03
0.06
0.09
Fig. 2. Plots of out-of-control ARL values for the ACUSUM-III chart when λ0 ¼ 4 and ARL0 200. (a) θ ¼ 0:001, (b) θ ¼ 0:01, (c) θ ¼ 0:5.
values of 115.24 and 67.58 for the CUSUMS (λ1 ¼ 6) and ACUSUMIII charts. Although the CUSUMD chart tries to optimize the performance for detecting the shift of a particular size at each step, it cannot provide an overall good detection performance as the Poisson mean varies over time. Clearly, the optimality property of the conventional CUSUM chart for detecting step shifts cannot apply to the case under linear drifts. The comparison is also conducted for larger values of λ0 ¼ 6, 10, 20 but the results are omitted in this paper. Similar observations to the case with λ0 ¼ 4 can be made. The above comparisons indicate a good property of the ACUSUM-III chart. In particular, the ACUSUM-III chart can always provide the shortest ARL1 values at small drift coefficients and the ARL1 values close to or slightly larger than the shortest at relatively large drift coefficients.
Moreover, compared to the CUSUMD and CUSUMS charts, the ACUSUM chart has the advantage of being free from the requirement of a known shift size.
5. A simulation example In this section, we use a simulation example from Perry et al. (2007) to demonstrate the implementation of the ACUSUM-III chart. As done in Perry et al. (2007), the data were simulated from a Poisson distribution with a rate of 5 for the first 25 observations and from a Poisson distribution with rate λi ¼ λ0 þ 0:15ði 25Þ starting at observation 26. The in-control rate was assumed to be λ0 ¼ 5. Clearly, the simulated process maintains in control for the
F. He et al. / Int. J. Production Economics 148 (2014) 14–20
19
Table 2 Comparisons of ARL values of the ACUSUM-III chart when λ0 ¼ 4 and ARL0 200. γ
Drift (θ)
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.010
202.04 67.41 54.30 29.23 22.35 11.44 8.35 3.87 2.74 1.95 1.59 1.35
200.17 86.21 67.58 37.37 28.15 13.99 10.15 4.58 3.23 2.25 1.83 1.58
199.90 96.87 76.52 42.13 31.69 15.51 11.16 5.01 3.52 2.46 2.01 1.78
200.68 104.44 82.88 45.50 34.30 16.54 11.88 5.31 3.71 2.60 2.10 1.86
199.70 109.57 87.34 47.96 36.07 17.30 12.37 5.52 3.85 2.70 2.17 1.92
199.52 113.68 90.99 49.98 37.51 17.90 12.75 5.69 3.97 2.76 2.21 1.94
201.04 117.51 94.32 51.87 38.72 18.40 13.08 5.81 4.07 2.84 2.28 1.99
199.61 119.93 96.64 53.16 39.63 18.72 13.33 5.92 4.14 2.89 2.32 2.00
199.42 122.20 98.61 54.43 40.52 19.06 13.55 6.02 4.19 2.94 2.35 2.04
199.26 124.23 100.35 55.32 41.24 19.30 13.75 6.08 4.25 2.95 2.37 2.04
Steady-state ARL 0.0005 122.96 0.0010 93.80 0.0050 47.87 0.0100 35.07 0.0500 16.56 0.1000 11.91 0.5000 5.29 1.0000 3.71 2.0000 2.60 3.0000 2.12 4.0000 1.84
112.51 87.76 46.14 34.33 16.33 11.71 5.26 3.69 2.57 2.10 1.82
108.98 85.77 45.92 34.39 16.47 11.86 5.31 3.73 2.60 2.11 1.82
107.46 85.94 46.63 34.88 16.77 12.00 5.40 3.79 2.65 2.15 1.85
107.54 86.84 47.21 35.41 17.08 12.23 5.48 3.85 2.69 2.19 1.88
108.62 87.91 48.10 35.84 17.42 12.45 5.57 3.89 2.73 2.21 1.91
109.51 89.12 48.88 36.42 17.65 12.60 5.66 3.95 2.77 2.24 1.93
109.80 90.07 49.57 37.01 17.85 12.73 5.71 3.99 2.79 2.27 1.95
110.89 91.04 50.23 37.59 18.03 12.87 5.79 4.04 2.82 2.29 1.97
111.35 91.60 50.76 37.95 18.17 13.03 5.81 4.07 2.84 2.30 1.99
Zero-state ARL 0.0000 0.0005 0.0010 0.0050 0.0100 0.0500 0.1000 0.5000 1.0000 2.0000 3.0000 4.0000
Table 3 Combinations of λ0 and λ1 studied through simulation.
Table 4 Comparison of ARL values for λ0 ¼ 4 with ARL0 200. hS ¼ 3:3530 CUSUMS λ1 ¼ 6
hD ¼ 1:2007 CUSUMD θ1 ¼ 0:001
hD ¼ 2:1245 CUSUMD θ1 ¼ 0:0025
h3 ¼ 0:0378 ACUSUM-III γ ¼ 0:002
Zero-state ARL 0.0000 200.76 0.0005 132.95 0.0010 107.43 0.0050 56.38 0.0100 40.09 0.0500 17.30 0.1000 12.06 0.5000 5.32 1.0000 3.74 2.0000 2.67 3.0000 2.16 4.0000 1.93
204.22 141.43 115.24 59.66 41.85 17.14 11.56 4.85 3.36 2.34 1.93 1.69
200.54 166.19 145.56 90.35 71.73 41.86 33.16 19.36 15.35 12.12 10.75 9.83
200.36 164.61 142.08 83.97 65.59 37.61 29.70 17.28 13.73 10.94 9.49 8.76
200.17 86.21 67.58 37.37 28.15 13.99 10.15 4.58 3.23 2.25 1.83 1.58
Steady-state ARL 0.0005 127.25 0.0010 104.35 0.0050 53.78 0.0100 38.07 0.0500 16.36 0.1000 11.34 0.5000 4.87 1.0000 3.41 2.0000 2.40 3.0000 1.97 4.0000 1.74
137.15 113.70 58.28 41.00 16.76 11.24 4.60 3.17 2.20 1.80 1.57
126.67 111.87 67.01 51.05 25.65 18.79 8.86 6.37 4.56 3.77 3.22
125.32 109.48 62.90 46.79 22.47 16.29 7.54 5.40 3.86 3.15 2.80
112.51 87.76 46.14 34.33 16.33 11.71 5.26 3.69 2.57 2.10 1.82
Drift (θ) λ0
λ1
4 4 6 6 10 10 20 20
5 6 8 9 12 14 25 30
first 25 observations and a drift occurs at the rate of 0.15 following time step 25. Suppose the desired in-control ARL of the ACUSUM-III chart is 400. The parameter γ of the ACUSUM-III chart is set to be γ ¼ 0:002 based on the previous study. Then simulations can be performed to determine the threshold for providing the desired in-control ARL, leading to h3 ¼ 0:1319. For comparison, the CUSUMS chart is also considered here. The CUSUMS chart designed is aimed at detecting a sudden step increase of 25% in-control Poisson rate, i.e. λ1 ¼ λ0 þ 0:25λ0 ¼ 6:25. The threshold for the CUSUMS chart was determined as hS ¼ 3:4717 for providing an approximated in-control ARL of 400. Table 5 presents the simulated observations, CUSUMS and ACUSUM-III charting statistics computed at each time step. Fig. 3 plots the charting patterns of both the ACUSUM-III and CUSUMS charts. From Table 5, the ACUSUM-III chart signals an out-ofcontrol situation at observation 33 while the CUSUMS chart signals at observation 37. Clearly, the ACUSUM-III chart signals 4 steps earlier than the CUSUMS chart. This example again illustrates that the conventional CUSUM chart optimally designed for detecting step shifts of a particular size is less efficient when linear drifts actually occurred. In contrast, the ACUSUM chart that dynamically updates the likelihood ratio based on an estimate of the Poisson
hS ¼ 2:7100 CUSUMS λ1 ¼ 5
rate can provide a relatively good detection performance under linear drifts.
6. Concluding remarks When linear drifts are introduced to a Poisson process, the mean of the underlying process is no longer constant but changes over time. Although the conventional CUSUM charts developed
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F. He et al. / Int. J. Production Economics 148 (2014) 14–20
Table 5 A simulation example. t
Obs
CUSUMS
ACUSUM-III
t
Obs
CUSUMS
ACUSUM-III
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
3 3 8 8 6 6 7 4 6 4 1 6 7 8 6 3 11 4 5 2
0.0000 0.0000 0.5351 1.0703 1.1592 1.2480 1.5600 1.2026 1.2915 0.9340 0.0000 0.0889 0.4009 0.9360 1.0249 0.4443 1.6489 1.2915 1.1572 0.3535
0.0000 0.0000 0.0000 0.0024 0.0036 0.0052 0.0100 0.0080 0.0104 0.0084 0.0068 0.0076 0.0108 0.0191 0.0222 0.0175 0.0459 0.0415 0.0415 0.0321
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
12 6 4 4 5 4 2 6 7 9 6 8 7 6 7 7 9 7 7 10
1.7812 1.8700 1.5126 1.1552 1.0209 0.6635 0.0000 0.0889 0.4009 1.1592 1.2480 1.7832 2.0952 2.1840 2.4960 2.8080 3.5663 3.8783 4.1903 5.1718
0.0733 0.0795 0.0735 0.0680 0.0679 0.0628 0.0512 0.0554 0.0654 0.0917 0.0986 0.1229 0.1406 0.1497 0.1697 0.1911 0.2405 0.2664 0.2939 0.3727
The simulation results show that the proposed ACUSUM chart outperforms the CUSUMS chart designed for detecting step mean shifts at small drift coefficient and performs only sightly worse than the latter at relatively large drift coefficients. Both the ACUSUM and CUSUMS charts perform better than the CUSUMD chart. Although the CUSUMD chart aims at optimizing the performance for detecting the shift of a particular size at each step, it cannot provide an overall good detection performance under linear drifts where the Poisson mean is time-varying.
Acknowledgments The authors would like to thank the referees for their valuable suggestions. Dr. L. J. Shu's work was supported in part by the Research Committee under the Grant MYRG096(Y2-L2)-FBA12-SLJ and FDCT/002/2013/A. Professor K.-L. Tsui's work was supported in part by the Research Committee under the Grant HK RGC GRF 121410 and Food and Health Bureau RFCID Fund 11101262.
References
0
CUSUM_S ACUSUM III
Changing Statistic
1 2 3 4 5 0
10 20 30 Observation Number
40
Fig. 3. A simulation example.
based on the likelihood ratio can be used for monitoring linear drifts, they often require the assumption of a known out-ofcontrol parameter. This parameter is often unknown in practice. To relax this limitation, this paper develops ACUSUM charts for monitoring drifts in the Poisson rate. Compared to the conventional CUSUM charts, the ACUSUM chart can dynamically update the likelihood ratio based on an estimate of the current Poisson mean level. Three different EWMA-type schemes for estimating Poisson means are considered, including the EWMA estimator without reset, the EWMA estimator with reset, and a combined estimator. The second estimator with reset enables the EWMA estimator to quickly catch up with the change in the Poisson mean level once a shift occurs. However, it tends to overestimate the mean level in the in-control situation due to reset at the same time. As compared to the second mean estimator, the third estimator can provide a more accurate estimate of the Poisson rate in the incontrol case while maintaining the same estimation efficiency in the out-of-control case. Therefore, the ACUSUM-III chart is expected to perform better than the other two ACUSUM charts.
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