Exponentially Weighted Moving Average Control Charts for Monitoring Increases in Poisson Rate

Transcription

1 Exponentially Weighted Moving Average Control Charts for Monitoring Increases in Poisson Rate Lianjie SHU 1,, Wei JIANG 2, and Zhang WU 3 EndAName 1 Faculty of Business Administration University of Macau Taipa, Macau * Corresponding Author ( ljshu@umac.mo) 2 Department of Industrial Engineering & Logistics Management The Hong Kong University of Science and Technology Clearwater Bay, Hong Kong 3 Division of Systems and Engineering Management School of Mechanical and Aerospace Engineering Nanyang Technological University, Singapore Abstract The exponentially weighted moving average (EWMA) control chart has been widely studied as a tool for monitoring normal processes due to its simplicity and efficiency. However, relatively little attention has been paid to EWMA charts for monitoring Poisson processes. This paper extends EWMA charts to Poisson processes with emphasis on quick detection of increases in Poisson rate. Both cases with and without normalzing transformation for Poisson data are considered. A Markov chain model is established to analyze and design the proposed chart. The comparison results indicate that the EWMA chart based on normalized data is nearly optimal. Keywords: Poisson Distribution; Average Run Length; Statistical Process Control; Markov Chain 1

2 1 Introduction Besides monitoring continuous data, there are many applications where it is important to monitor a sequence of discrete counts, such as quality control and productivity improvement in manufacturing processes and health surveillance in health care management (Montgomery 2005; Woodall 2006; Tsui et al. 2008). Count data per unit time or area is often described by a Poisson model. For example, the number of nonconforming items in an inspection unit, the number of reported failures per week, and the monthly number of visits to a certain clinic are often assumed to follow a Poisson distribution. The Poisson distribution has only one parameter, rate of count per unit, which represents both the mean and variance of the distribution. A simple charting procedure for monitoring changes of rate of a Poisson distribution is the Shewhart c-chart or u-chart. The c-chart is used when the count data is based on one unit while the u-chart is applicable if it is based on n units (n > 1). Both the Shewhart c- and u-charts make use of only the most recent observations to determine the statistical status of the process. To make use of the past observations, the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts have been discussed. A sample of research demonstrating the use of CUSUM chart for monitoring the mean of a Poisson process includes Brook and Evans (1972), Lucas (1985), White and Keats (1996), White et al. (1997), Rossi et al. (1999), and Chan et al. (2007). In contrast, the work on the EWMA chart for Poisson data is relatively less. Few exceptions include Gan (1990), Martz and Kvam (1996), and Borror et al. (1998). A comprehensive review on attribute control charts was given by Woodall (1997). A control chart is often designed in one-sided or two-sided form, depending on the objective of detecting one directional changes (either upward or downward) or changes in both directions. When there is an increase in Poisson rate, the process is likely to produce more nonconforming items, which in turn incurs heavier loss and higher cost for companies. On the other hand, a decrease in Poisson mean may imply an improvement in the process. Recently, there is a growing interest on detecting increases of disease incidence rate in health surveillance (Tsui et al. 2008). Therefore, in order to prevent the underlying process from producing too many nonconforming items, efficient control charts being able to early signal upward shifts in the Poisson rate would be greatly desired. The one-sided Poisson CUSUM chart, which originates from sequential likelihood ratio test theory, has been widely investigated as it serves the basis for the analysis of overall performance of a two-sided CUSUM chart. However, little attention 2

3 has been paid to one-sided Poisson EWMA chart, while most of the existing literature mainly focuses on two-sided Poisson EWMA charts, see, for example, Gan (1990) and Borror et al. (1998). To fill the gap in part, this paper aims to develop some one-sided Poisson EWMA control charts with primary interests on the detection of increases in the Poisson rate. Unlike a one-sided Poisson CUSUM chart, one-sided EWMA chart may serve as a baseline model to capture the dynamics of process mean levels when observations are counts. Note that the Poisson data is discrete and nonnormal. It has long been an intuitive idea in Statistical Process Control (SPC) to use transformed data for monitoring when the data is nonnormal. For example, Crowder and Hamilton (1992) considered using logarithmic transformation of the sample variance for monitoring changes in variability. Quesenberry (1991a,b,c) considered using the inverse transformation of a cumulative distribution function for monitoring Binomial and Poisson data. The performance of EWMA charts based on the normalizing transformation of Poisson data has not been previously investigated. The question whether the normalizing transformation can lead to better chart performance arises in both theory and practice. Therefore, it is of great interest to investigate the performance of Poisson EWMA control charts in the case when the Poisson data are normalized. The rest of this paper is organized as follows. First, several one-sided Poisson EWMA control charts are developed based on transformations. Next, the performance of various upper Poisson EWMA control charts is analyzed, followed by some design guidelines. An example of male thyroid cancer surveillance in New Mexico is presented to demonstrate the use of the proposed chart. Lastly, some concluding remarks are discussed. The appendices provide the Markov chain model for the upper Poisson EWMA control charts. 2 One-Sided Poisson EWMA Charts Assume that the process observations, {X t, t = 1, 2, }, are independent following a Poisson distribution with mean µ. The process is said to be in control when µ = µ 0 and out of control when the mean changes to some other values, say, µ = µ 1. The traditional Poisson EWMA control chart is defined by Q t = X t + (1 )Q t 1, (1) where is a smoothing constant such that 0 < 1. The starting value is often set to the target, Q 0 = µ 0 while other head start values can be chosen for Q 0 for fast initial response (FIR) 3

4 features (Lucas and Crosier 1982). The asymptotic control limits are often written as follows, i.e., ( ) LCL = max 0, µ 0 L (2 ) µ 0, UCL = µ 0 + L (2 ) µ 0. This chart is two-sided, which can detect both increases and decreases in Poisson rate. For the purpose of quick detection of either upward or downward changes only, the onesided EWMA chart is desirable. The conventional way to modify a two-sided EWMA chart into the one-sided upper/lower form is to reset the EWMA statistic to the target whenever it is less/greater than the target. See, for example, Champ et al. (1991), Crowder and Hamilton (1992) and Gan (1998). Taking the upper EWMA chart as an example, by resetting the EWMA statistic in Equation (1) to µ 0 whenever it is less than µ 0, an upper Poisson EWMA chart can be obtain as E X t = max[µ 0, X t + (1 )E X t 1], (2) where E X 0 = µ 0. We refer this chart as EX chart in this paper. It signals when E X t exceeds the upper control limit h = µ 0 + L 2 µ 0. Note that the Poisson distribution is asymmetric and is not normal. normality, one may consider the linear transformation To achieve better Y t = X t µ 0 µ0. (3) It is often assumed in practice that Y t is asymptotically normal when the in-control mean µ 0 is sufficiently large. However, it is important to note that this linear transformation just changes the location to zero and standardizes the spread of the Poisson distribution to one. It actually does not affect the shape of the Poisson distribution. To obtain a distribution that is close to normal, a nonlinear transformation based on square root is often suggested, see, for example, Anscombe (1947), Freeman and Tukey (1950), and Ryan and Schwertman (1997). The square root transformation M t = X t + c is approximately normal with mean and variance approximated by (Anscombe 1947) µ M = µ + c 1 8 µ, 4

5 and σ 2 M = 1 4 ( c ), 8µ respectively. Note that when c = 3/8, the variance of M t is nearly independent of the Poisson parameter µ. In this case, the variance of M t is stabilized at σm 2 = 0.25, and any change in the Poisson parameter will be transformed into a change in the mean level of M t. For this reason, we use c = 3/8 in this paper while other choices of c values can be similarly analyzed. Assuming that M t is approximately normal, the statistic X t Z t = µ M µ=µ 0 ( = 2 X t + 3 σ M µ=µ0 8 µ ) 8 µ 0 will be close to the standard normal distribution when the process is in control. When X t in Equation (2) is replaced by Y t or Z t, we obtain another upper Poisson EWMA scheme as E Y 0 = 0 (4) E Y t = max[0, Y t + (1 )E Y t 1], t = 1, 2,..., or E Z 0 = 0 (5) E Z t = max[0, Z t + (1 )E Z t 1], t = 1, 2,..., For simplicity, denote the EWMA schemes in Equations (4) and (5) as EY and EZ charts, respectively. The EY/EZ chart signals when EY/EZ exceeds the upper control limit h E = L 2, which is expressed in the same form as the limit for the EX chart. The values of L are generally different from EY and EZ charts. It is easy to see that both EX and EY charts have the same performance since (E X t µ 0 )/ µ 0 = E Y t. For this reason, we will discuss EY chart only but not the EX chart to keep simplicity throughout the remaining of this paper. Recently, Shu et al. (2007) and Shu and Jiang (2008) considered another resetting rule in the one-sided EWMA scheme, particularly for nonnormal data. They suggested to reset the current observation or normalized observation to the target but not the EWMA statistic. This scheme has been shown to have better performance than the direct reset of the EWMA statistic. Take 5

6 the chart based on Z t as an example. The revised procedure first winsorizes Z t and then applies a conventional EWMA scheme to the winsorized data. That is, where Z + t defined as R t = Z + t + (1 )R t 1, (6) = max[0, Z t ] and R 0 = E(Z + t µ = µ 0). A lower-sided EWMA chart can be similarly R t = Z t + (1 )R t 1, where Z t = min[0, Z t ] and R 0 = E(Z t µ = µ 0). Note that if Z t N(0, 1), the mean and variance of the winsorized normal variable, Z + t 1999) E(Z t + ) = 1 2π σ 2 = 1 Z t π. = max[0, Z t ], are given by (Barr and Sherrill Therefore, the asymptotic mean of R t is not equal to zero but 1/ 2π in the in-control situation. To shift the mean of R t to zero, it is convenient to rewrite the EWMA recursion in Equation (6) as and R Z 0 variance as Z + t Z + t with Y + t = 0. Analogously, define Y + t R Z t = (Z + t 1 2π ) + (1 )R Z t 1, (7) = max[0, Y t ], which has approximately the same mean and since Y t follows approximately a standard normal distribution. Now replacing in Equation (7) leads to the control statistic R Y t = (Y + t 1 2π ) + (1 )R Y t 1 (8) with R Y 0 = 0. For simplicity, denote the revised upper EWMA charts in Equations (7) and (8) as RZ and RY charts, respectively. The RZ or RY chart declares an alarm when R Z t exceeds the upper control limit h R = L 2 σ Z + t = L 2 The values of L are generally different from RY and RZ charts. ( ). 2π or R Y t 3 Performance Analysis In this section, we first compare the effect of two types of resetting schemes in the EWMA on the performance of Poisson EWMA charts. Then, the effect of using normalizing transformation on 6

7 the Poisson data is investigated. All the ARL results are obtained using Markov chain approach, details of which are given in Appendices A and B. The discussions will be based on two cases: the case with a relatively small in-control mean of µ 0 = 4 and the case with a relatively large value of µ 0 = 16. The other cases with different values of µ 0 would provide qualitatively the same conclusions, which are not presented here for the sake of simplicity. 3.1 Comparisons between EY and RY charts For illustration, Table 1 compares the zero-state ARL values of EY and RY charts when µ 0 = 4 and 16 and = 0.05 and The zero-state in-control ARL values are maintained approximately 1,000 for both charts. The shift size, δ, is measured in the unit of the in-control standard deviation of the Poisson process, namely, δ = µ µ 0 µ0. From Table 1, it can be observed that for a fixed value of, the RY chart performs closely to the EY chart at small shifts but has much better performance at large shifts. For example, when µ 0 = 16 and = 0.05, the percentage decrease in the out-of-control (OC) ARL of the RY chart at δ = 0.5 relative to the EY chart is 1.52%(( )/32.96) but the corresponding percentage decrease at δ = 3 is 23.97% (( )/3.88). Note that when = 1, the EY chart reduces to E Y t = max[0, Y t ] = Y + t, and the RY chart reduces to R Y t = (Y + t 1 2π ). Clearly, both charts are essentially the same upper-sided Shewhart chart of Y t. Therefore, they would perform the same when = 1. This implies that when increases, the improvement of the RY chart over the EY chart at large shifts tends to diminish. To illustrate this point, Figure 1 further plots the percentage decrease in the OC ARL of the RY chart relative to the EY chart for different values of when µ 0 = 16. From Figure 1, it is clear that the relative OC ARL percentage decrease reduces from about 20%-25% to 2%-5% for detecting shifts of δ 2 as increases from 0.05 to 0.5. Note that the same value of is used for both EY and RY charts in the above comparison, which might not be optimal for either chart at the same mean shift. For this reason, Table 2 7

8 Table 1: Zero-State ARL Comparisons between the EY and RY Charts µ 0 = 4 µ 0 = 16 = 0.05 = 0.15 = 0.05 = 0.15 EY RY EY RY EY RY EY RY δ L= further presents the optimal performance of both charts aimed at detecting shifts of sizes of δ opt = 0.5 and δ opt = 1.5. The optimal values can be obtained from the design guidelines presented in the next section. To provide a more complete comparison, the steady-state ARL is also included. The steady-state ARL refers to the ARL obtained assuming that the control statistic has reached steady state before changes occur, which was computed in a way similar to that of Lucas and Saccucci (1990). Details of the computation are given in Appendix A. From Table 2, it is interesting to note that the optimal values of the RY chart are always smaller than those of the EY chart for detecting the same size of shifts. The RY chart always performs better than the EY chart at the optimized shifts in all cases considered here. In fact, the RY chart performs uniformly better than the EY chart at all shift sizes. The steady-state results provide qualitatively the same conclusion. This observation is clearly an evidence of superior performance by resetting the observation/transformed observation but not the EWMA statistic in detecting increases in the Poisson rate. The observation is consistent with that in the case of monitoring normal process mean (Shu et al. 2007). 8

9 Table 2: Optimal ARL Comparisons between the EY and RY Charts µ 0 = 4 µ 0 = 16 δ opt = 0.5 δ opt = 1.5 δ opt = 0.5 δ opt = 1.5 EY RY EY RY EY RY EY RY = δ L= Zero-State Steady-State

10 Percentage Decrease in OC ARL =0.05 =0.15 =0.25 = δ Figure 1: Percentage Decrease of Out-of-Control ARL of the RY Chart Relative to the EY Chart When µ 0 = The case with normalizing transformation We now compare the run length performance of the upper Poisson EWMA charts based on data with and without normalization transformation. Tables 3 and 4 provide the comparison results for µ 0 = 4 and µ 0 = 16, respectively, with optimally chosen for certain shift sizes. It can be observed that the optimal performance of the upper Poisson EWMA chart based on normalized data is nearly the same as the chart based on un-normalized data. For example, when µ 0 = 4 and δ opt = 0.5, the EY and EZ charts have the respective ARL values of and at δ = 0.5, while the RY and RZ charts have the ARL values of and 31.23, respectively. Clearly, the differences in these optimal ARL values are negligible. This indicates that the normalization transformation of the data merely makes data more normal but not results in any performance improvement of detection. This observation is similar to that commented in Gan and Fan (1995). They showed that the EWMA chart based on the logarithmic transformation of the sample variance is nearly optimal, compared to the EWMA chart based on the sample variance. However, it is surprising to note that the normalization transformation tends to deteriorate the performance for detecting other shifts in most cases. 10

11 Table 3: Zero-State ARL Performance of the Upper Poisson EWMA Chart in Cases with and without Normalizing Transformation When µ 0 = 4 δ opt = 0.5 δ opt = 1.5 EY EZ RY RZ EY EZ RY RZ = δ L= Table 4: Zero-State ARL Performance of the Upper Poisson EWMA Chart in Cases with and without Normalizing Transformation When µ 0 = 16 δ opt = 0.5 δ opt = 1.5 EY EZ RY RZ EY EZ RY RZ = δ L=

12 4 Design of Upper EWMA Poisson Charts In the above comparisons, the RY chart demonstrates some advantages for practical use. It is shown that the RY chart performs better than the EY chart and has performance similar or superior to the RZ chart. To facilitate the design of the RY chart for practitioners, this section provides some design guidelines. The design of the RY chart is a two-degree-of-freedom problem, which involves determination of the values of and L. Similar to the design approach of Lucas and Saccucci (1990), one can select the value of to optimize the detection performance for a specific change in the Poisson mean and then choose L to provide a desired in-control ARL. There is no simple guidelines for finding the optimal parameters of RY charts. Extensive computation was thus performed to numerically search the nearly optimal values of and L for RY charts. Tables 5 to 8 present a list of optimal parameters of RY charts when in-control ARL values are set as 370, 500, 1000, and 1500, respectively. These parameters are adequate for most practical purposes, which are obtained using a numerical search method through the Markov chain approximation. For example, suppose we want to find the optimal parameters of the RY chart for the target shift of size δ opt = 1 with in-control zero-state ARL of 370 and µ 0 = 4. The L value can be paired with a wide range values of = 0.01, 0.02,,(stepsize=0.01) to produce the desired in-control zero-state ARL of 370. The out-of-control ARL at δ = 1 is then evaluated for each of the (, L) combinations. The chart with the combination ( = 0.09, L = 3.044) gives the minimum ARL of 9.0, and thus these parameters are selected to be optimal. The same approach is used to determine the optimal parameters of RY charts for other in-control ARL values and different values of µ 0 and δ opt. From Tables 5-8, it can be seen that small values of are more sensitive to small shifts while large values of are more sensitive to shifts of relatively large sizes, as expected. For example, when the in-control ARL is 370 and µ 0 = 2, from Table 5, the optimal values of increases from = 0.02 to = 0.19 as δ opt increases from δ opt = 0.25 to δ opt = 2.0. Furthermore, it can be observed from Tables 5-8 that the in-control mean value of the Poisson distribution has impact on the choice of optimal parameters for Poisson charts for given target sizes of shifts, especially when µ 0 is small and the target shift size is large. This observation is different from the case of monitoring normal process means. However, this effect tends to be negligible when µ 0 is sufficiently large, say µ From Tables 5-8, it can be seen that the optimal values of at a particular shift are nearly the same for different values of µ For 12

13 example, when the in-control ARL is 370, the optimal choices of aimed at detecting a shift of size δ opt = 1 is always given by = 0.10 for µ 0 16, as shown in Table 5. This is mainly due to the property that the Poisson distribution approaches a normal distribution when µ 0 is large. Meanwhile small values of also make the approximation of the EWMA statistics based on nonnormal data to be close to normal (Borror et al. 1998). 5 An Example of Male Thyroid Cancer Surveillance in New Mexico To illustrate the use of RY chart, the data on the incidence of male thyroid cancer in New Mexico was used here. The data was collected by at the National Cancer Institute and was available through the Surveillance, Epidemiology, and End Results (SEER) Program. This example was also studied in Mei et al. (2010). The annual incidence of male thyroid cancer per 100,000 persons over was presented in Table 9. The main objective of this application is to detect an increase in the incidence of male thyroid cancer. It is well known that the incidence is stable during the period from year 1973 to year 1988 and increases since year For this reason, the first 16 observations during the period were considered to be in control and the remaining observations are out of control. The incidence rate based on the first 16 observations can be estimated as 2 per 100,000 persons. For the sake of simplicity, we assume that the estimate is the true value of µ 0. However, it is important note that estimation error often exists in practice, which would result in negative effects on control chart performance. A detailed review on effects of parameter estimation on control chart properties was given by Jensen et al. (2006). Suppose that we are interested in quickly detecting an increase of size δ opt = 0.5 in the incidence, and that the desired in-control ARL is 500. According to Table 6, the optimal parameters for the RY chart are chosen as: = 0.02 and L = The control limit for the RY chart is h R = L 2 ( π ) = The charting statistic of the RY chart at each time period is obtained according to Equation (8), which is given in Table 9. Figures 2(a) and (b) plot the sample data, X t, and the charting statistic, R Y t, respectively. From Figure 2(a), it can be observed that the observations during years tend to be 13

14 Table 5: Optimal Parameters for the RY Chart When In-Control ARL is 370 δ opt (, L) ( 0.02, 1.809) ( 0.02, 1.832) ( 0.02, 1.846) ( 0.02, 1.828) ( 0.02, 1.834) ( 0.02, 1.828) ( 0.02, 1.820) ARL min (, L) ( 0.02, 1.809) ( 0.02, 1.832) ( 0.02, 1.846) ( 0.02, 1.828) ( 0.02, 1.834) ( 0.02, 1.828) ( 0.02, 1.820) ARL min (, L) ( 0.02, 1.809) ( 0.04, 2.383) ( 0.06, 2.656) ( 0.06, 2.621) ( 0.06, 2.605) ( 0.06, 2.586) ( 0.06, 2.574) ARL min (, L) ( 0.07, 2.895) ( 0.09, 3.044) ( 0.10, 3.044) ( 0.10, 2.984) ( 0.10, 2.965) ( 0.10, 2.938) ( 0.10, 2.922) ARL min (, L) ( 0.10, 3.223) ( 0.13, 3.340) ( 0.13, 3.246) ( 0.14, 3.226) ( 0.14, 3.208) ( 0.15, 3.223) ( 0.15, 3.205) ARL min (, L) ( 0.14, 3.536) ( 0.16, 3.513) ( 0.18, 3.497) ( 0.20, 3.486) ( 0.20, 3.463) ( 0.20, 3.422) ( 0.20, 3.400) ARL min (, L) ( 0.17, 3.716) ( 0.19, 3.660) ( 0.20, 3.574) ( 0.23, 3.594) ( 0.23, 3.568) ( 0.23, 3.526) ( 0.23, 3.501) ARL min (, L) ( 0.19, 3.820) ( 0.21, 3.750) ( 0.22, 3.658) ( 0.25, 3.656) ( 0.25, 3.630) ( 0.25, 3.583) ( 0.25, 3.560) ARL min µ 0 14

15 Table 6: Optimal Parameters for the RY Chart When In-Control ARL is 500 δ opt (, L) ( 0.02, 2.047) ( 0.02, 2.063) ( 0.02, 2.074) ( 0.02, 2.055) ( 0.02, 2.047) ( 0.02, 2.039) ( 0.02, 2.031) ARL min (, L) ( 0.02, 2.047) ( 0.02, 2.063) ( 0.03, 2.363) ( 0.03, 2.336) ( 0.03, 2.328) ( 0.03, 2.324) ( 0.03, 2.303) ARL min (, L) ( 0.03, 2.391) ( 0.06, 2.926) ( 0.06, 2.860) ( 0.06, 2.816) ( 0.06, 2.801) ( 0.07, 2.883) ( 0.07, 2.865) ARL min (, L) ( 0.07, 3.117) ( 0.09, 3.249) ( 0.09, 3.164) ( 0.09, 3.100) ( 0.10, 3.153) ( 0.10, 3.129) ( 0.11, 3.173) ARL min (, L) ( 0.10, 3.606) ( 0.12, 3.480) ( 0.12, 3.381) ( 0.12, 3.302) ( 0.14, 3.393) ( 0.15, 3.404) ( 0.15, 3.386) ARL min (, L) ( 0.12, 3.606) ( 0.16, 3.718) ( 0.17, 3.647) ( 0.18, 3.600) ( 0.19, 3.614) ( 0.19, 3.575) ( 0.20, 3.583) ARL min (, L) ( 0.16, 3.880) ( 0.18, 3.826) ( 0.19, 3.739) ( 0.20, 3.674) ( 0.23, 3.755) ( 0.23, 3.709) ( 0.23, 3.683) ARL min (, L) ( 0.19, 4.044) ( 0.22, 3.999) ( 0.22, 3.856) ( 0.23, 3.782) ( 0.24, 3.785) ( 0.24, 3.740) ( 0.24, 3.714) ARL min µ 0 15

16 Table 7: Optimal Parameters for the RY Chart When In-Control ARL is 1,000 δ opt (, L) ( 0.02, 2.548) ( 0.02, 2.548) ( 0.02, 2.543) ( 0.02, 2.510) ( 0.02, 2.503) ( 0.02, 2.492) ( 0.02, 2.480) ARL min (, L) ( 0.02, 2.547) ( 0.02, 2.548) ( 0.03, 2.814) ( 0.03, 2.773) ( 0.03, 2.764) ( 0.03, 2.744) ( 0.04, 2.902) ARL min (, L) ( 0.05, 3.308) ( 0.05, 3.236) ( 0.06, 3.295) ( 0.06, 3.233) ( 0.06, 3.215) ( 0.06, 3.188) ( 0.06, 3.171) ARL min (, L) ( 0.08, 3.719) ( 0.08, 3.601) ( 0.09, 3.592) ( 0.09, 3.511) ( 0.09, 3.492) ( 0.10, 3.528) ( 0.10, 3.507) ARL min (, L) ( 0.10, 3.923) ( 0.11, 3.857) ( 0.11, 3.743) ( 0.13, 3.773) ( 0.13, 3.750) ( 0.12, 3.656) ( 0.13, 3.687) ARL min (, L) ( 0.13, 4.167) ( 0.14, 4.056) ( 0.14, 3.926) ( 0.16, 3.926) ( 0.16, 3.895) ( 0.17, 3.895) ( 0.17, 3.869) ARL min (, L) ( 0.16, 4.360) ( 0.19, 4.328) ( 0.19, 4.162) ( 0.20, 4.087) ( 0.20, 4.058) ( 0.21, 4.049) ( 0.21, 4.108) ARL min (, L) ( 0.19, 4.532) ( 0.23, 4.498) ( 0.24, 4.357) ( 0.24, 4.230) ( 0.24, 4.196) ( 0.24, 4.142) ( 0.25, 4.141) ARL min µ 0 16

17 Table 8: Optimal Parameters for the RY Chart When In-Control ARL is 1,500 δ opt (, L) ( 0.02, 2.810) ( 0.02, 2.814) ( 0.02, 2.784) ( 0.02, 2.758) ( 0.02, 2.748) ( 0.02, 2.727) ( 0.02, 2.719) ARL min (, L) ( 0.02, 2.810) ( 0.02, 2.814) ( 0.03, 3.056) ( 0.03, 3.010) ( 0.03, 2.996) ( 0.03, 2.978) ( 0.03, 2.961) ARL min (, L) ( 0.05, 3.562) ( 0.05, 3.481) ( 0.05, 3.401) ( 0.05, 3.337) ( 0.06, 3.442) ( 0.06, 3.412) ( 0.06, 3.391) ARL min (, L) ( 0.07, 3.858) ( 0.08, 3.848) ( 0.08, 3.738) ( 0.09, 3.736) ( 0.09, 3.714) ( 0.09, 3.677) ( 0.09, 3.656) ARL min (, L) ( 0.09, 4.088) ( 0.10, 4.025) ( 0.12, 4.040) ( 0.12, 3.940) ( 0.12, 3.915) ( 0.12, 3.871) ( 0.12, 3.846) ARL min (, L) ( 0.13, 4.429) ( 0.14, 4.305) ( 0.14, 4.159) ( 0.16, 4.153) ( 0.16, 4.122) ( 0.16, 4.075) ( 0.16, 4.047) ARL min (, L) ( 0.15, 4.568) ( 0.17, 4.480) ( 0.19, 4.409) ( 0.19, 4.282) ( 0.19, 4.209) ( 0.20, 4.230) ( 0.20, 4.200) ARL min (, L) ( 0.17, 4.692) ( 0.19, 4.579) ( 0.21, 4.489) ( 0.22, 4.393) ( 0.22, 4.358) ( 0.23, 4.335) ( 0.23, 4.300) ARL min µ 0 17

18 Table 9: An Illustrative Example for the RY Chart Year X t Y t Rt Y Year X t Y t Rt Y

19 8 0.4 Cancer Incidence Per 100, Year Year (a) (b) Figure 2: The Annual Incidence Data (a) and the RY Chart (b) larger than the first 16 observations. This exhibits the pattern of an increase in the incidence. From Figure 2(b), it is clear that the RY chart triggers an out-of-control signal in year The RY charting statistic remains above the control limit for the rest of observations and starts the increase since year Therefore, the search for assignable cause can be traced back to year Concluding Remarks This paper extends one-sided EWMA charts from the case of monitoring normal process means to the case of monitoring Poisson data. A Markov chain model is established to analyze the ARL performance for these charts. Both cases with and without normalizing transformation applied to the Poisson data are investigated. The results show that resetting of the current observation or the transformed observation would improve the performance of detecting increases in the Poisson rate, compared to the conventional way of resetting the EWMA statistic to the target. Although the normalization transformation makes the Poisson data much closer to normal, it results in slightly worse performance than the Poisson EWMA chart without transformation. Therefore we suggest using the original observations in the new EWMA Poisson chart. When designing the one-sided Poisson EWMA procedure in our paper, the in-control mean, µ 0, is assumed to be known. In practice; however, it would be more common that the incontrol process parameter of the Poisson distribution is unknown and needs to be estimated. 19

20 The estimation error may have considerable effects on the performance of control charts (Jensen et al. 2006). To make a control chart to perform like the case with known parameter(s), a sufficiently large sample size is often required during a Phase I analysis. In parallel with the work of Jensen et al. (2006), one can determine the appropriate sample size required for establishing the one-sided Poisson EWMA chart. This will be further discussed in another paper. Although the discussions in this paper aim at quick detection of upward changes in Poisson rate, one can easily extend the suggested chart to the case for detecting downward changes. The lower Poisson EWMA chart can be obtained based on the same logic by truncating the observation to the target whenever it is larger than the target. Lucas and Saccucci (1990) have suggested some enhancements for the two-sided EWMA charts, including a combined Shewhart- EWMA and the FIR feature. Although these enhancements are not presented in this paper, it is straightforward to extend them to the one-sided Poisson EWMA charts. Finally, due to the increasing interests in health surveillance, it would be interesting to extend the proposed one-sided EWMA scheme to monitor other attribute data such as Binomial distributions in the future research. Appendix A: Approximation of ARLs for the EZ and EY Charts Using Markov Chain Suppose the in-control region [0, h E ] of the EZ charting statistic is divided into m subintervals, which are labelled as i = 1, 2,, m. The width of each interval is ω = 2h E /(2m 1) except the first is ω/2. Let P (i, j) be the transition probability of Et Z from state i to state j. Define d 1 = 1 [j 1.5 (1 )(i 1)]ω + µ µ 0 d 2 = 1 [j 0.5 (1 )(i 1)]ω + µ µ 0 Then for i = 1, 2,..., m and j 1, P (i, j) = Pr{ E Z t in state j E Z t 1 in state i} = Pr{(j 1)ω 0.5ω < Z t + (1 )(i 1)ω (j 1)ω + 0.5ω [j 1.5 (1 )(i 1)]ω = Pr{ = Pr{d 1 < X t + 3/8 d 2 } < Z t [j 0.5 (1 )(i 1)]ω } 20

21 = 0, d 2 < F (d , µ), F (d , µ) F (d , µ), d 2 d and d 1 < 3 8 and d , where F (, µ) is the cumulative distribution function of a Poisson random variable with mean µ. For j = 1, P (i, j) = Pr{ X t d 2} = F (d , µ). Although the Poisson random variable takes only nonnegative integer values, the left and right sides of the above inequalities in general will not be integers. Denote T the run length of the EZ chart. Then, the probabilities that Et Z goes from one transient state to another state in t steps are given by the matrix R t. Hence, the probability mass function (PMF) of the run length follows Pr(T = t E0 Z ) = p T ini(r t 1 R t ) 1, (9) where E0 Z is an initial value, p ini is the probability vector corresponding to E0 Z, and 1 is a column vector of ones. The zero-state or initial-state ARL is computed by ARL = p T ini (I R) 1 1. (10) For the EZ chart, E0 Z = 0. To compute the steady-state ARL of a control chart using Markov chain approach, one needs to compute the steady-state probability vector. However, the exact stationary probability vector does not exist because the transition probability matrix is not ergodic. To overcome this problem, Lucas and Saccucci (1990) suggested using a cyclical steady-state probability vector as an alternative. It is obtained by adjusting the transition probability matrix so that the control statistic was reset to the initial state whenever it goes into the out-of-control state, namely, P = R (I R) Let p ss be the cyclical steady-state probability vector. Then, p ss is the solution to the system p = P T p subject to 1 T p =1. The cyclical steady-state ARL is given by ARL = p T ss(i R)

22 Analogously, this method was employed to compute the steady-state ARL for the EZ chart in this paper. The zero-state and steady-state ARLs for other control controls can be obtained similarly except that the transition probability matrices need to be redefined. The number of states m may usually be chosen as 30 to evaluate the run length performance, as recommended by Brook and Evans (1972). However, we prefer a much larger m value because this may improve the overall accuracy while affecting the total execution time only marginally, given modern computing power available today. In this paper, we have used the value of m up to m = 300, and the execution time is in the order of a few seconds. Define F (, µ) and ω as above. The transition probability matrix for the EY chart can be obtained as follows: for j 1, P (i, j) = Pr{ E Y t in state j E Y t 1 in state i} = Pr{(j 1)ω 0.5ω < Y t + (1 )(i 1)ω (j 1)ω + 0.5ω [j 1.5 (1 )(i 1)]ω = Pr{ = F (c 2, µ) F (c 1, µ) < Y t [j 0.5 (1 )(i 1)]ω } where c 1 = c 2 = [j 1.5 (1 )(i 1)]ω µ0 + µ 0 [j 0.5 (1 )(i 1)]ω µ0 + µ 0. For j = 1, P (i, j) = Pr{Y t c 2 } = F (c 2, µ). Appendix B: The Markov Chain Model for the RZ and RY Charts Note that the charting statistic of the RZ chart can be written as Since Z + t easy to show that t 1 Rt Z = (1 ) j (Z t j + 1 ) + (1 ) t R0 Z. 2π j=0 0 and t 1 j=0 (1 )j = 1 (1 ) t approaches one as t + for 0 < 1, it is R Z t 1 2π [1 (1 ) t ] 1 2π. 22

23 Thus, the in-control region of Rt Z can be approximated by [ 1/ 2π, h R ], and the interval (h R, + ) is considered to be an absorbing state. The interval [ 1/ 2π, h R ] was partitioned into m sub-intervals, each of width = (h R + 1/ 2π)/m. The mid value for these sub-intervals is 1/ 2π+ (i 0.5) for i = 1, 2,, m. For i = 1, 2,..., m and j = 1, 2,..., m, the transition probability for the charting statistic Rt Z from state i to state j is P (i, j) = Pr{ R Z t in state j R Z t 1 in state i} = Pr{ 1 2π + (j 1) < (Z + t 1 2π ) + (1 )( 1 2π + (i 0.5) ) 1 2π + j } [j 1 (1 )(i 0.5)] = Pr{ < Z t + [j (1 )(i 0.5)] }. Define and then a 1 = a 2 = [j 1 (1 )(i 0.5)], [j (1 )(i 0.5)], P (i, j) = Pr{a 1 < Z t + a 2 }, 0, a 2 < 0 = F (b 2, µ), a 2 0 and a 1 < 0 F (b 2, µ) F (b 1, µ), a 2 0 and a 1 0, where b 1 = b 2 = ( ) 2 0.5a 1 + µ µ 0 8, ( ) 2 0.5a 2 + µ µ 0 8. The transition probability for the RY chart can be calculated without modifications except that values of b 1 and b 2 are redefined as b 1 = a 1 µ0 + µ 0, b 2 = a 2 µ0 + µ 0. 23

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