# Multichart Schemes for Detecting Changes in Disease Incidence

Authors: Gideon Mensah Engmann; Dong Han
Journal: Computational and Mathematical Methods in Medicine
 (2020)

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## Abstract

Several methods have been proposed in open literatures for detecting changes in disease outbreak or incidence. Most of these methods are likelihood-based as well as the direct application of Shewhart, CUSUM and EWMA schemes. We use CUSUM, EWMA and EWMA-CUSUM multi-chart schemes to detect changes in disease incidence. Multi-chart is a combination of several single charts that detects changes in a process and have been shown to have elegant properties in the sense that they are fast in detecting changes in a process as well as being computationally less expensive. Simulation results show that the multi-CUSUM chart is faster than EWMA and EWMA-CUSUM multi-charts in detecting shifts in the rate parameter. A real illustration with health data is used to demonstrate the efficiency of the schemes.


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## Body

## 1. Introduction

In this era of bioterrorism, outbreak of diseases and surge in disease incidence; statisticians, epidemiologists, informaticians and surveillance scientists are designing algorithms to detect changes in disease occurrence or outbreak in order to avert any possible public health pandemonium. Most of these models or algorithms are modifications of the statistical process control (SPC) schemes, namely Shewhart, CUmulative SUM (CUSUM) and Exponentially Weighted Moving Average (EWMA) statistics.Biosurveillance in the context of human health (health surveillance) is a term for the science and practice of managing health-related data and information for early warning of threats and hazards, early detection of events and rapid characterization of the event so that effective actions can be taken to mitigate adverse health effects [1]. Biosurveillance systems have two main purposes: to support health situational awareness and for early event/outbreak detection. In the past two or three decades, many biosurveillance systems have been developed. Bravata et al. [2] in their review identified 115 health surveillance systems and 9 syndromic surveillance systems. Most of these surveillance systems have been developed and are in use in countries like US, UK, China and Japan among others.Statistical methods or algorithms have been widely applied to solve biosurveillance problems. These statistical methods or algorithms for monitoring bioterrorism, incidence, or outbreak of diseases can be categorized into temporal (see, for example, Reis [3] and Brookmeyer and Stroup [4]), spatio (Waller and Gotway [5] and Lawson and Kleinman [6]), spatio-temporal (Diggle [7] and Fricker [8]), multivariate temporal (see Vial [9]), multivariate spatial monitoring (see, for example, Corberán-Vallet [10]), multivariate spatio-temporal (Quick et al. [11]), and Bayesian (Tzala [12]). Multivariate monitoring methods are extension of the univariate methods.Several methods have been proposed in literatures, for example, Farrington et al. [13] proposed a robust statistical algorithm to process weekly reports of infections received at the Communicable Disease Surveillance Centre. The algorithm calculates suitable thresholds and organisms exceeding their thresholds are then flagged for further investigation. Le Strat and Carrat [14] proposed a hidden markov model to monitor epidemiologic surveillance data. A rule-based method was also proposed by Wong [15] to solve a surveillance classification problem. Many Point Process Models (PPM) were also discussed by Brookmeyer and Stroup [4]. Shmueli et al. [16] proposed a wavelet-based automated algorithm for detecting disease outbreaks in temporal syndromic data. Their method improves upon the Goldenberg et al. [17] algorithm on a diverse set of real syndromic data from multiple data sources and multiple geographical locations. Sebastiani et al. [18] proposed a Bayesian dynamic model to monitor influenza surveillance data. They integrated different data sources into a dynamic model, which identified in-children and infants pediatric emergency departments with respiratory syndromes as an early indicator of impending influenza morbidity and mortality. Their findings show that dynamic Bayesian networks could be suitable modeling tools for developing epidemic surveillance systems. Forsberg et al. [19] also proposed the so-called distance-based method where they assessed possible disease clusters based on M-statistic on the distribution of the pairwise distance between cases. Fricker [20] and Joner et al. [21] also considered Directional Multivariate Exponentially Weighted Moving Average (DMEWMA) and Multivariate CUmulative SUM (MCUSUM) schemes. Fricker et al. [22] proposed CUSUM-based methods with adaptive regression. Fricker and Chang [23] considered Repeated Two-sample Rank (RTR)-based methods. Their proposed method is spatio-temporal and can subsequently be used to track the spread of an outbreak. Lu et al. [24] proposed the Markov switching model to detect disease outbreak. Cowling et al. [25] developed a statistical algorithm using sentinel surveillance data for early detection of the annual influenza peak season in Hong Kong. Bédubourg and Le Strat [26] compared and evaluated using the simulation of several statistical methods for early temporal detection of outbreaks. Even though many methods have been developed and proposed in literature, there is still the need for more concerted efforts to develop and improve methods that will detect changes in disease incidence or outbreak and monitor bioterrorism.Many researchers have used a Poisson process to monitor changes in disease outbreak or incidence. Rossi et al. [27] used CUSUM charts to monitor changes in disease occurrence after the transformation of the Poisson data into approximately normal random variables. Mei et al. [28] proposed a weighted CUSUM chart placing more weight on recent observations and compared their method with other common CUSUM techniques. Jiang et al. [29] compared the performance of several CUSUM methods subject to Poisson distribution. Richards et al. [30] proposed an invariant Poisson control charting scheme and applied it to monitor the number of emergency arrivals observed at the Baltimore Veterans Affairs Medical Center. Most of these models or schemes monitor one variable or disease at a time. Even in cases where two or more diseases are monitored, MCUSUM and MEWMA schemes have been used widely. Multi-charts have been shown in literature to be very powerful to detect changes in random events. They are different from MCUSUM (see Crosier [31], Golosnoy [32], and Raji et al. [33]), MEWMA (see Lowry et al. [34], Hussain et al. [35] and Ajadi and Riaz [36]), and multi-hypothesis testing (see Baum and Veeravalli [37] and Lai [38]) in terms of methodology. For example, multi-chart schemes can tell which of the charts triggered detection, a property that falls short of multivariate charts (MCUSUM and MEWMA). Multi-chart consists of several single charts with different reference values that are used simultaneously to detect and monitor process changes. CUSUM multi-chart scheme has been shown to be more efficient than the EWMA multi-chart scheme in detecting changes in a random process (Han et al. [39]). Multi-chart schemes have elegant properties in the sense that they are fast in detecting changes in a process and computationally less expensive than sister charts like Generalized Exponential Weighted Moving Average (GEWMA) by Han and Tsung [40] and the CUSUM-like control chart by Siegmund and Venkatraman [41].Multi-chart schemes have rarely been used in the field of biosurveillance and health monitoring. A wealth of research is ongoing in disease surveillance and these methods are implemented in health surveillance systems to detect abnormal changes in disease occurrences. The ability to detect abnormal changes in disease occurrence is of uttermost concern to the public health workers for them to trigger public awareness and education. It is in this light that we applied the methodology of multi-chart schemes to detect changes in disease incidence and also evaluate the efficiency of the methods. Many researchers have proposed charting performance indices (for example, Overall Charting Performance Index (OCPI), Relative Mean Index (RMI) among others) to evaluate the performance of CUSUM and EWMA schemes. In the computation of these indices, we need the optimalARL (ARL∗) which is found subject to normal distribution (continuous distribution). The CUSUM ARLs with reference values as charting statistic subject to Poisson distribution are not optimal as we are dealing with discrete distribution; hence, we also propose new measures (called Expectation of the Time for Detecting mean shifts (ETD) and Expectation of the Time for Detecting mean shifts with Equal weights of shifts (ETDE)) to evaluate the efficiency of the schemes.Basically, the objectives of this study are to monitor tuberculosis disease based on multi-chart schemes and also evaluate the efficiency of the methods using a new performance index. Generally, we only know the possible postchange region but rarely know the exact magnitude of mean shift of a process before it is detected; we therefore use a range of known shifts in the rate parameter. The main contribution of our paper is as follows: we present a new performance index measure to evaluate the performance of the charts.The article is organized as follows: materials and methods are presented in section2, subsection 2.1, presents the multi-chart schemes subject to Poisson distribution for detecting changes in disease incidence. Subsection 2.2 presents the performance index measures, while subsection 2.3 gives a theoretical performance comparison of the multi-chart schemes with that of single charts and subsection 2.4 gives the procedural description of multi-chart schemes. Results and discussion are presented in section 3, where these theoretical results are compared by numerical simulations in subsections 3.1, 3.2, 3.3, and 3.4, while subsection 3.5 gives a real example based on tuberculosis data from Ghana. Section 4 concludes and gives remarks.


## 2. Materials and Methods

### 2.1. Multi-chart Schemes

Generally in health care monitoring, the observations are counts and let's assume they follow the Poisson distribution. The Poisson distribution is usually used to describe the number of events that occurred in a unit time interval or within a unit space.Let's assumeXi~Poissonλ, where λ is the average count of a disease occurring in a week or in a month. Usually, at some time period ν, the probability distribution of Xi changes from Poissonλ0 to Poissonλν. We generally refer to ν as a change point. In general, ν=1,2,3,⋯,∞, but in this article, we assume ν=1, which means the first time there is a change in distribution. Intuitively, the mean of Xi undergoes a shift of size λi(λ0), where λ0 is known and assumed to be 1. In biosurveillance problems or health surveillance, we normally monitor for upward change in distribution, since the increment in disease counts pose challenges to the public health workers. For the Poisson distribution, the mean is equal to the variance; hence, developing a chart to monitor the mean jointly monitors both the mean and the variance simultaneously.Mathematically, the prechange distribution with mean (λ0=1) is given by
(1)X1,X2,⋯,Xν−1~Pλ0Xk=λ0xke−λ0xk!And also the postchange distribution with mean (λ≠1) is given by
(2)Xν,Xν+1,⋯,Xn~PλXk=λxke−λxk!The log-likelihood ratioγk for ν≤k≤n is given by
(3)γk=lnPλXkPλ0Xk=Xklnλλ0+λ0−λSo we define a single upward CUSUM chart as(4)Tμi=minn:max1≤k≤n∑j=n−k+1nXklnμiλ0+λ0−μi>diwhere di is the width of the control limit and μi are some reference values satisfying λ0=a<μ1<μ2<,⋯,<μm≤b. We assumed that the possible range of the rate parameter shifts is a,b.LetΔm=μi:1≤i≤m and Wm=wi:1≤i≤m be a set of numbers (known reference values) where μi>1, m≥2, and 0<wi≤1. Also, let Dm=di:1≤i≤m and Hm=hi:1≤i≤m be a set of numbers (width of control limit) where di>0 which usually depends on μi and hi>0 also depends on wi.We define a single upward exponential weighted moving average (EWMA) chart as(5)TEiwi,hi=minn:∑k=0n−1wi1−wikXn−k>hiLet us define the one-sided CUSUM and EWMA multicharts asTCMΔm,Dm and TEMWm,Hm, respectively, where
(6)TCMΔm,Dm=minμi∈ΔmTμiμi,di(7)TEMWm,Hm=min1≤i≤mTEiwi,hiWe also define the one-sided EWMA-CUSUM mixed charts asTEC where
(8)TEC=min1≤i≤mTEiwi,hiorTμiμi,di


### 2.2. Charting Performance Index

The most widely used measure to determine which control chart performs better is the average run length (ARL). Ultimately, we force all the charts to have the same in-control average run lengthARL0 then for a desired shift in the parameter of interest, the chart with the lowest out-of-control average run length (ARLλ) has the greatest ability to determine the prespecified shift. The ARL used in evaluating chart performance is weak due to the fact that its performance will deteriorate if the actual size of a mean shift is significantly different from the assumed size. To help address this problem, a number of novel charting performance indices have been proposed in the literature. For example, Han et al. [39] proposed the Overall Charting Performance Index (OCPI). Other charting performance measures include but not limited to Relative Mean Index (RMI) [42], Charting Performance Index (CPI) [43], etc. In the computation of these indices, we need the optimal ARL (ARL∗) which is found subject to normal distribution (continuous distribution). The CUSUM ARLs with reference values as charting statistic subject to Poisson distribution are not optimal as we are dealing with discrete distribution; hence, we also propose new measures (called Expectation of the Time for Detecting mean shifts (ETD) and Expectation of the Time for Detecting mean shifts with Equal weights of shifts (ETDE)) as performance index measures to evaluate the efficiency of the schemes.We define theETD of a chart T for a range of shifts in the rate parameter a,b by
(9)ETDT=∑i=1lwiARLλiTwhere wi=λi/∑j=1lλj, λ1<λ2<⋯<λl are real rate parameters and l is the number of shifts considered in the study.Whenwi=1/l, we consider
(10)ETDET=1l∑i=1lARLλiTwhere ETDET is the expectation of the time for detecting mean shifts when the ARLλi are assigned equal weights of the inverse of the number of shifts considered in the study. Different forms of the weights can be studied, but here, we restrict it to these two scenarios. The chart with the smallest ETD and ETDE performs better.


### 2.3. Comparison of the Multi-chart Schemes with Its Constituent Charts

Without loss of generality, letP0· and E0· represent the probability and expectation that there is no change in the rate parameter, respectively. Let Pλ· and Eλ· represent the probability and expectation when there is a change in the true rate parameter λ at change point ν=1, respectively. Normally for a stopping time T, we use out-control average run length ARLλ to judge which chart is performing better. All the charts were designed with a common ARL0 and for a shift in the rate parameter; we adjudge a chart with smaller ARLλ to be the best performing. Intuitively, we define ARL0T=E0T and ARLλT=EλT. Let's also assume that the rate parameter λ≥1 and we choose some reference values satisfying λ0=a<μ1<μ2<⋯<μm≤b, where m is the number of charts. Let d1,d2,⋯,dm be the width of the individual control limits. We take the multi-chart control limits; d1′,d2′,⋯,dm′ such that di′>di, for 1≤i≤m.We can compare the multi-chartTCM=TCMd1′,d2′,⋯,dm′ with its constituent charts Tμidi. If we choose d1′,d2′,⋯,dm′ according to the restrictions
(11)L0=ARL0Tμ1,d1≈ARL0Tμ2,d2≈⋯≈ARL0Tμm,dm.That is if we force the in-control average run lengths of all the single CUSUM charts to be approximately equal. Similarly, to construct EWMA multi-chart, we force the in-control average run lengths of all the single EWMA charts to be approximately equal.Preposition 1.
Under the condition (11) and for large L0, we have
(12)ETDTCM≤ETDTμifor1≤i≤m(13)ETDETCM≤ETDETμifor1≤i≤mBy inequalities (12) and (13), CUSUM multi-chart has better detection performance than single CUSUM charts. The proofs of these prepositions are in the Appendix.Usually, it is difficult to predetermine the exact size of the mean shift before it is detected. Instead, a range of shift sizes of interest could be considered. We can compare the performances of these single charts with the average of these charts. We define the average CUSUM chart, average EWMA chart and average EWMA-CUSUM chart respectively as(14)AverageCUSUMChart=1m∑i=1mARLλiTμiμi,di(15)AverageEWMAChart=1m∑i=1mARLλiTEiwi,hi(16)AverageEWMA−CUSUMChart=1m∑i=1mARLλiTEC


### 2.4. Procedural Description of Multi-chart Schemes

This section provides a detailed description of the simulation procedure used for the computation of ARL at each shift (λ), computation of ETD and ETDE for the comparison of the charts. We used Monte Carlo simulations for the computation of the ARLs. Simulation analyses were carried for a 10,000-repetition experiment. We generally set the in-control rate parameter λ=1.


#### 2.4.1. Computation of the CUSUM Multi-chart Statistic

(1)
Determine the number of charts to be used for the CUSUM multichart. Sparks [44] suggested that three or more single charts are needed to achieve an efficient multi-chart scheme(2)
Determine the reference parametersμ1,μ2,⋯,μk(3)
Generate a random sample of size 1 at each step (denoted byXk) from the Poisson distribution with the specified reference value(4)
Determine the in-controlARL0 of the single charts say ARL0≈200 or 500 and use Monte Carlo simulations to find the control limits (d1,d2,⋯,dk) of the single charts using equation (4)(5)
Normally to arrive at an in-control ARL of CUSUM multi-chartTCM of approximately 200 or 500, we had to choose the single charts to have approximately equal in-control ARL0=L0. Set L0 and use step (4) to determine the control limits; (d1′,d2′,⋯,dk′). Adjust di′ until the in-control ARL0 of CUSUM multi-chart is arrived at(6)
Compute theARLs of the single charts and CUSUM multi-chart using charting statistic (4) and (6), respectively. Compute the ARLs of the average CUSUM chart by equation (14)(7)
Compute theETD and ETDE of the CUSUM charts, average CUSUM, and CUSUM multi-chart using equations (9) and (10)


#### 2.4.2. Computation of the EWMA Multi-chart Statistic

(1)
Determine the number of charts to be used for the EWMA multi-chart. Generally, for the sake of comparison, we use the same number of charts as in the CUSUM setting(2)
Determine the smoothing parametersw1,w2,⋯,wk(3)
Generate a random sample of size 1 at each step from the Poisson distribution(4)
Determine the in-controlARL0 of the single charts say ARL0≈200or500 and use Monte Carlo simulations to find the control limits (h1,h2,⋯,hk) of the single charts using equation (5)(5)
Normally to arrive at an in-control ARL of EWMA multi-chartTEM of approximately 200 or 500, we had to choose the single charts to have approximately equal in-control ARL0=L0. Set L0 and use step (4) to determine the control limits; (h1′,h2′,⋯,hk′). Adjust the hi′ until the in-control ARL0 of EWMA multi-chart is arrived at(6)
Compute theARLs of the single charts and EWMA multi-chart using charting statistics (5) and (7), respectively. Compute the ARLs of the average EWMA chart by equation (15)(7)
Compute theETD and ETDE of the single EWMA charts, average EWMA chart and EWMA multi-chart using equations (9) and (10)


#### 2.4.3. Computation of the EWMA-CUSUM Multi-chart Statistic

(1)
Determine the number of charts to be used for the EWMA-CUSUM multi-chart. Generally, for the sake of comparison, we use the same number of charts as in the EWMA and CUSUM setting(2)
Determine the smoothing parametersw1,w2,⋯,wk and reference values μ1,μ2,⋯,μk(3)
Generate a random sample of size 1 at each step from the Poisson distribution(4)
Determine the in-controlARL0 of the single charts say ARL0≈200or500and use Monte Carlo simulations to find the control limits (h1,h2,⋯,hk;d1,d2,⋯,dk) of the single charts using charting statistics (5) and (4).(5)
Normally to arrive at an in-control ARL of EWMA-CUSUM multi-chartTEC of approximately 200 or 500, we had to choose the single charts to have approximately equal in-control ARL0=L0. Set L0 and use step (4) to determine the control limits; (h1′,h2′,⋯,hk′;d1′,d2′,⋯,dk′). Adjust the hi′ and di′ until the in-control ARL0 of EWMA-CUSUM multi-chart is arrived at(6)
Compute theARLs of the single charts and EWMA-CUSUM multi-chart using charting statistic (5) and (8), respectively. Compute the ARLs of the average EWMA-CUSUM chart by equation (16).(7)
Compute theETD and ETDE of the single EWMA-CUSUM charts and EWMA-CUSUM multi-chart using equations (9) and (10).


## 2.1. Multi-chart Schemes

Generally in health care monitoring, the observations are counts and let's assume they follow the Poisson distribution. The Poisson distribution is usually used to describe the number of events that occurred in a unit time interval or within a unit space.Let's assumeXi~Poissonλ, where λ is the average count of a disease occurring in a week or in a month. Usually, at some time period ν, the probability distribution of Xi changes from Poissonλ0 to Poissonλν. We generally refer to ν as a change point. In general, ν=1,2,3,⋯,∞, but in this article, we assume ν=1, which means the first time there is a change in distribution. Intuitively, the mean of Xi undergoes a shift of size λi(λ0), where λ0 is known and assumed to be 1. In biosurveillance problems or health surveillance, we normally monitor for upward change in distribution, since the increment in disease counts pose challenges to the public health workers. For the Poisson distribution, the mean is equal to the variance; hence, developing a chart to monitor the mean jointly monitors both the mean and the variance simultaneously.Mathematically, the prechange distribution with mean (λ0=1) is given by
(1)X1,X2,⋯,Xν−1~Pλ0Xk=λ0xke−λ0xk!And also the postchange distribution with mean (λ≠1) is given by
(2)Xν,Xν+1,⋯,Xn~PλXk=λxke−λxk!The log-likelihood ratioγk for ν≤k≤n is given by
(3)γk=lnPλXkPλ0Xk=Xklnλλ0+λ0−λSo we define a single upward CUSUM chart as(4)Tμi=minn:max1≤k≤n∑j=n−k+1nXklnμiλ0+λ0−μi>diwhere di is the width of the control limit and μi are some reference values satisfying λ0=a<μ1<μ2<,⋯,<μm≤b. We assumed that the possible range of the rate parameter shifts is a,b.LetΔm=μi:1≤i≤m and Wm=wi:1≤i≤m be a set of numbers (known reference values) where μi>1, m≥2, and 0<wi≤1. Also, let Dm=di:1≤i≤m and Hm=hi:1≤i≤m be a set of numbers (width of control limit) where di>0 which usually depends on μi and hi>0 also depends on wi.We define a single upward exponential weighted moving average (EWMA) chart as(5)TEiwi,hi=minn:∑k=0n−1wi1−wikXn−k>hiLet us define the one-sided CUSUM and EWMA multicharts asTCMΔm,Dm and TEMWm,Hm, respectively, where
(6)TCMΔm,Dm=minμi∈ΔmTμiμi,di(7)TEMWm,Hm=min1≤i≤mTEiwi,hiWe also define the one-sided EWMA-CUSUM mixed charts asTEC where
(8)TEC=min1≤i≤mTEiwi,hiorTμiμi,di


## 2.2. Charting Performance Index

The most widely used measure to determine which control chart performs better is the average run length (ARL). Ultimately, we force all the charts to have the same in-control average run lengthARL0 then for a desired shift in the parameter of interest, the chart with the lowest out-of-control average run length (ARLλ) has the greatest ability to determine the prespecified shift. The ARL used in evaluating chart performance is weak due to the fact that its performance will deteriorate if the actual size of a mean shift is significantly different from the assumed size. To help address this problem, a number of novel charting performance indices have been proposed in the literature. For example, Han et al. [39] proposed the Overall Charting Performance Index (OCPI). Other charting performance measures include but not limited to Relative Mean Index (RMI) [42], Charting Performance Index (CPI) [43], etc. In the computation of these indices, we need the optimal ARL (ARL∗) which is found subject to normal distribution (continuous distribution). The CUSUM ARLs with reference values as charting statistic subject to Poisson distribution are not optimal as we are dealing with discrete distribution; hence, we also propose new measures (called Expectation of the Time for Detecting mean shifts (ETD) and Expectation of the Time for Detecting mean shifts with Equal weights of shifts (ETDE)) as performance index measures to evaluate the efficiency of the schemes.We define theETD of a chart T for a range of shifts in the rate parameter a,b by
(9)ETDT=∑i=1lwiARLλiTwhere wi=λi/∑j=1lλj, λ1<λ2<⋯<λl are real rate parameters and l is the number of shifts considered in the study.Whenwi=1/l, we consider
(10)ETDET=1l∑i=1lARLλiTwhere ETDET is the expectation of the time for detecting mean shifts when the ARLλi are assigned equal weights of the inverse of the number of shifts considered in the study. Different forms of the weights can be studied, but here, we restrict it to these two scenarios. The chart with the smallest ETD and ETDE performs better.


## 2.3. Comparison of the Multi-chart Schemes with Its Constituent Charts

Without loss of generality, letP0· and E0· represent the probability and expectation that there is no change in the rate parameter, respectively. Let Pλ· and Eλ· represent the probability and expectation when there is a change in the true rate parameter λ at change point ν=1, respectively. Normally for a stopping time T, we use out-control average run length ARLλ to judge which chart is performing better. All the charts were designed with a common ARL0 and for a shift in the rate parameter; we adjudge a chart with smaller ARLλ to be the best performing. Intuitively, we define ARL0T=E0T and ARLλT=EλT. Let's also assume that the rate parameter λ≥1 and we choose some reference values satisfying λ0=a<μ1<μ2<⋯<μm≤b, where m is the number of charts. Let d1,d2,⋯,dm be the width of the individual control limits. We take the multi-chart control limits; d1′,d2′,⋯,dm′ such that di′>di, for 1≤i≤m.We can compare the multi-chartTCM=TCMd1′,d2′,⋯,dm′ with its constituent charts Tμidi. If we choose d1′,d2′,⋯,dm′ according to the restrictions
(11)L0=ARL0Tμ1,d1≈ARL0Tμ2,d2≈⋯≈ARL0Tμm,dm.That is if we force the in-control average run lengths of all the single CUSUM charts to be approximately equal. Similarly, to construct EWMA multi-chart, we force the in-control average run lengths of all the single EWMA charts to be approximately equal.Preposition 1.
Under the condition (11) and for large L0, we have
(12)ETDTCM≤ETDTμifor1≤i≤m(13)ETDETCM≤ETDETμifor1≤i≤mBy inequalities (12) and (13), CUSUM multi-chart has better detection performance than single CUSUM charts. The proofs of these prepositions are in the Appendix.Usually, it is difficult to predetermine the exact size of the mean shift before it is detected. Instead, a range of shift sizes of interest could be considered. We can compare the performances of these single charts with the average of these charts. We define the average CUSUM chart, average EWMA chart and average EWMA-CUSUM chart respectively as(14)AverageCUSUMChart=1m∑i=1mARLλiTμiμi,di(15)AverageEWMAChart=1m∑i=1mARLλiTEiwi,hi(16)AverageEWMA−CUSUMChart=1m∑i=1mARLλiTEC


## 2.4. Procedural Description of Multi-chart Schemes

This section provides a detailed description of the simulation procedure used for the computation of ARL at each shift (λ), computation of ETD and ETDE for the comparison of the charts. We used Monte Carlo simulations for the computation of the ARLs. Simulation analyses were carried for a 10,000-repetition experiment. We generally set the in-control rate parameter λ=1.


### 2.4.1. Computation of the CUSUM Multi-chart Statistic

(1)
Determine the number of charts to be used for the CUSUM multichart. Sparks [44] suggested that three or more single charts are needed to achieve an efficient multi-chart scheme(2)
Determine the reference parametersμ1,μ2,⋯,μk(3)
Generate a random sample of size 1 at each step (denoted byXk) from the Poisson distribution with the specified reference value(4)
Determine the in-controlARL0 of the single charts say ARL0≈200 or 500 and use Monte Carlo simulations to find the control limits (d1,d2,⋯,dk) of the single charts using equation (4)(5)
Normally to arrive at an in-control ARL of CUSUM multi-chartTCM of approximately 200 or 500, we had to choose the single charts to have approximately equal in-control ARL0=L0. Set L0 and use step (4) to determine the control limits; (d1′,d2′,⋯,dk′). Adjust di′ until the in-control ARL0 of CUSUM multi-chart is arrived at(6)
Compute theARLs of the single charts and CUSUM multi-chart using charting statistic (4) and (6), respectively. Compute the ARLs of the average CUSUM chart by equation (14)(7)
Compute theETD and ETDE of the CUSUM charts, average CUSUM, and CUSUM multi-chart using equations (9) and (10)


### 2.4.2. Computation of the EWMA Multi-chart Statistic

(1)
Determine the number of charts to be used for the EWMA multi-chart. Generally, for the sake of comparison, we use the same number of charts as in the CUSUM setting(2)
Determine the smoothing parametersw1,w2,⋯,wk(3)
Generate a random sample of size 1 at each step from the Poisson distribution(4)
Determine the in-controlARL0 of the single charts say ARL0≈200or500 and use Monte Carlo simulations to find the control limits (h1,h2,⋯,hk) of the single charts using equation (5)(5)
Normally to arrive at an in-control ARL of EWMA multi-chartTEM of approximately 200 or 500, we had to choose the single charts to have approximately equal in-control ARL0=L0. Set L0 and use step (4) to determine the control limits; (h1′,h2′,⋯,hk′). Adjust the hi′ until the in-control ARL0 of EWMA multi-chart is arrived at(6)
Compute theARLs of the single charts and EWMA multi-chart using charting statistics (5) and (7), respectively. Compute the ARLs of the average EWMA chart by equation (15)(7)
Compute theETD and ETDE of the single EWMA charts, average EWMA chart and EWMA multi-chart using equations (9) and (10)


### 2.4.3. Computation of the EWMA-CUSUM Multi-chart Statistic

(1)
Determine the number of charts to be used for the EWMA-CUSUM multi-chart. Generally, for the sake of comparison, we use the same number of charts as in the EWMA and CUSUM setting(2)
Determine the smoothing parametersw1,w2,⋯,wk and reference values μ1,μ2,⋯,μk(3)
Generate a random sample of size 1 at each step from the Poisson distribution(4)
Determine the in-controlARL0 of the single charts say ARL0≈200or500and use Monte Carlo simulations to find the control limits (h1,h2,⋯,hk;d1,d2,⋯,dk) of the single charts using charting statistics (5) and (4).(5)
Normally to arrive at an in-control ARL of EWMA-CUSUM multi-chartTEC of approximately 200 or 500, we had to choose the single charts to have approximately equal in-control ARL0=L0. Set L0 and use step (4) to determine the control limits; (h1′,h2′,⋯,hk′;d1′,d2′,⋯,dk′). Adjust the hi′ and di′ until the in-control ARL0 of EWMA-CUSUM multi-chart is arrived at(6)
Compute theARLs of the single charts and EWMA-CUSUM multi-chart using charting statistic (5) and (8), respectively. Compute the ARLs of the average EWMA-CUSUM chart by equation (16).(7)
Compute theETD and ETDE of the single EWMA-CUSUM charts and EWMA-CUSUM multi-chart using equations (9) and (10).


## 2.4.1. Computation of the CUSUM Multi-chart Statistic

(1)
Determine the number of charts to be used for the CUSUM multichart. Sparks [44] suggested that three or more single charts are needed to achieve an efficient multi-chart scheme(2)
Determine the reference parametersμ1,μ2,⋯,μk(3)
Generate a random sample of size 1 at each step (denoted byXk) from the Poisson distribution with the specified reference value(4)
Determine the in-controlARL0 of the single charts say ARL0≈200 or 500 and use Monte Carlo simulations to find the control limits (d1,d2,⋯,dk) of the single charts using equation (4)(5)
Normally to arrive at an in-control ARL of CUSUM multi-chartTCM of approximately 200 or 500, we had to choose the single charts to have approximately equal in-control ARL0=L0. Set L0 and use step (4) to determine the control limits; (d1′,d2′,⋯,dk′). Adjust di′ until the in-control ARL0 of CUSUM multi-chart is arrived at(6)
Compute theARLs of the single charts and CUSUM multi-chart using charting statistic (4) and (6), respectively. Compute the ARLs of the average CUSUM chart by equation (14)(7)
Compute theETD and ETDE of the CUSUM charts, average CUSUM, and CUSUM multi-chart using equations (9) and (10)


## 2.4.2. Computation of the EWMA Multi-chart Statistic

(1)
Determine the number of charts to be used for the EWMA multi-chart. Generally, for the sake of comparison, we use the same number of charts as in the CUSUM setting(2)
Determine the smoothing parametersw1,w2,⋯,wk(3)
Generate a random sample of size 1 at each step from the Poisson distribution(4)
Determine the in-controlARL0 of the single charts say ARL0≈200or500 and use Monte Carlo simulations to find the control limits (h1,h2,⋯,hk) of the single charts using equation (5)(5)
Normally to arrive at an in-control ARL of EWMA multi-chartTEM of approximately 200 or 500, we had to choose the single charts to have approximately equal in-control ARL0=L0. Set L0 and use step (4) to determine the control limits; (h1′,h2′,⋯,hk′). Adjust the hi′ until the in-control ARL0 of EWMA multi-chart is arrived at(6)
Compute theARLs of the single charts and EWMA multi-chart using charting statistics (5) and (7), respectively. Compute the ARLs of the average EWMA chart by equation (15)(7)
Compute theETD and ETDE of the single EWMA charts, average EWMA chart and EWMA multi-chart using equations (9) and (10)


## 2.4.3. Computation of the EWMA-CUSUM Multi-chart Statistic

(1)
Determine the number of charts to be used for the EWMA-CUSUM multi-chart. Generally, for the sake of comparison, we use the same number of charts as in the EWMA and CUSUM setting(2)
Determine the smoothing parametersw1,w2,⋯,wk and reference values μ1,μ2,⋯,μk(3)
Generate a random sample of size 1 at each step from the Poisson distribution(4)
Determine the in-controlARL0 of the single charts say ARL0≈200or500and use Monte Carlo simulations to find the control limits (h1,h2,⋯,hk;d1,d2,⋯,dk) of the single charts using charting statistics (5) and (4).(5)
Normally to arrive at an in-control ARL of EWMA-CUSUM multi-chartTEC of approximately 200 or 500, we had to choose the single charts to have approximately equal in-control ARL0=L0. Set L0 and use step (4) to determine the control limits; (h1′,h2′,⋯,hk′;d1′,d2′,⋯,dk′). Adjust the hi′ and di′ until the in-control ARL0 of EWMA-CUSUM multi-chart is arrived at(6)
Compute theARLs of the single charts and EWMA-CUSUM multi-chart using charting statistic (5) and (8), respectively. Compute the ARLs of the average EWMA-CUSUM chart by equation (16).(7)
Compute theETD and ETDE of the single EWMA-CUSUM charts and EWMA-CUSUM multi-chart using equations (9) and (10).


## 3. Simulation Results and Discussion

In this section, we shall present and discuss the numerical results of the CUSUM and multi-CUSUM chart in subsection1, present and discuss results for EWMA and EWMA multi-chart in subsection 2, discuss results for EWMA-CUSUM multi-chart in subsection 3, and compare results in subsection 4.


### 3.1. Simulation Results of CUSUM and Multi-CUSUM Chart

Simulation analyses were carried out for a10,000-repetition experiment. We analyzed the simulation results for ten mean shifts in the rate parameter (λ1=1.25,λ2=1.50, λ3=1.75, λ4=2.00, λ5=2.25, λ6=2.50, λ7=2.75, λ8=3.00, λ9=3.25, and λ10=3.50) with change point ν=1 that is the first time there is signal or change. For comparison sake, the in-control ARLARL0 of all the charts were assumed to be equal and was taken to be 200 and 500, respectively. The reference values μi were chosen to be μ1=1.5, μ2=2.0, and μ3=2.5, where μ1 is termed as a small mean shift in the rate parameter, μ2 is a medium mean shift, and μ3 is a large mean shift in the rate parameter, respectively. The simulation results for the out-control average run length ARLλ of the Poisson CUSUM charts with parameter μ1,μ2, μ3, average CUSUM chart and multi-chart TCM were listed in column two, column three, column four, column five, and column six, respectively. The parameter di and di′ are the width of the control limits for the single CUSUM charts and CUSUM multi-chart, respectively. We chose three separate CUSUM charts because as suggested by Sparks [40], three or more single charts are needed to achieve an efficient multi-chart scheme. The control limits were obtained using Monte Carlo simulations. To arrive at an in-control ARL of CUSUM multi-chart TCM of approximately 200, we had to choose ARL0T1μ1=279.86 with d1′=2.914062, ARL0T2μ2=280.86 with d2′=3.59375, and ARL0T3μ3=280.62 with d3′=3.749023. Similarly, to arrive at an in-control ARL of CUSUM multi-chart TCM of approximately 500, we had to choose ARL0T1μ1=740.57 with d1′=3.794189, ARL0T2μ2=740.86 with d2′=4.47998, and ARL0T3μ3=741.11 with d3′=4.744361. In other words, we force all the ARL0 of the single charts to be approximately equal to guarantee an ARL0 of multi-chart to be approximately 200 and 500, respectively.Tables1 and 2 show that each of the schemes has its merits and demerits over a range, and perhaps, it is conflicting to compare the charts in relation to the average run length ARL. Ultimately, the ETD and ETDE enable us to compare the charts over the whole range of shifts. CUSUM multi-chart TCM has the smallest ETD and ETDE followed by CUSUM chart Tμ2d2, Tμ1d1, and Tμ3d3, respectively, for ARL0=200. Also, CUSUM multi-chartTCM has the smallest ETD and ETDE for the range of shifts followed by chart Tμ1d1, Tμ2d2, and Tμ3d3, respectively, for ARL0=500.Table 1
ARLs with their SDRL (in bracket) for the CUSUM and multi-CUSUM chart withARL0≈200.
Shifts inλT1μ1T2μ2T3μ3Average CUSUMTCMμ1,μ2,μ3d1=2.609375d2=3.238342d3=3.4531251.00202.53 (192.35)204.21 (203.3)203.86 (202.29)203.53 (199.31)200.91 (194.15)1.2545.44 (37.36)51.29 (46.61)58.45 (56.17)51.73 (46.71)48.11 (41.63)1.5020.64 (14.18)21.77 (18.01)24.25 (21.58)22.22 (17.92)20.90 (15.07)1.7512.79 (8.96)12.67 (8.70)13.50 (10.94)12.99 (9.53)12.61 (7.75)2.009.17 (6.71)8.71 (4.92)8.95 (5.62)8.94 (5.75)8.85 (5.44)2.257.26 (4.05)6.59 (3.89)6.71 (3.65)6.85 (3.86)6.74 (3.98)2.506.03 (3.08)5.41 (2.98)5.25 (2.91)5.56 (2.99)5.47 (3.01)2.755.15 (2.55)4.51 (2.37)4.29 (2.31)4.65 (2.41)4.60 (2.50)3.004.47 (2.12)3.93 (2.06)3.72 (1.94)4.04 (2.04)3.91 (1.98)3.254.01 (1.77)3.51 (1.73)3.27 (1.64)3.60 (1.71)3.45 (1.69)3.503.59 (1.49)3.15 (1.44)2.92 (1.41)3.22 (1.45)3.07 (1.42)ETD8.9718.8989.3899.0868.692ETDE11.85512.15413.13112.38011.771Table 2
ARLs with their SDRL (in bracket) for the CUSUM and multi-CUSUM chart withARL0≈500.
Shifts inλT1μ1T2μ2T3μ3Average CUSUMTCMμ1,μ2,μ3d1=3.430471d2=4.090408d3=4.3310551.00500.54 (490.24)501.72 (492.04)501.17 (495.91)501.14 (492.73)500.17 (485.63)1.2570.77 (58.68)88.97 (83.20)102.39 (99.86)87.38 (80.58)75.69 (63.93)1.5027.66 (18.25)30.87 (24.72)36.28 (32.23)31.60 (25.07)28.81 (19.96)1.7516.54 (9.14)16.75 (11.80)18.06 (14.48)17.12 (11.81)16.41 (10.24)2.0011.79 (5.67)10.99 (6.84)11.45 (8.25)11.41 (6.92)11.23 (6.53)2.259.28 (4.13)8.13 (4.45)8.02 (5.12)8.48 (4.57)8.51 (4.80)2.507.55 (3.25)6.48 (3.38)6.30 (3.79)6.78 (3.47)6.68 (3.65)2.756.48 (2.65)5.42 (2.70)5.14 (2.79)5.68 (2.71)5.51 (2.86)3.005.65 (2.23)4.71 (2.23)4.39 (2.24)4.92 (2.23)4.69 (2.40)3.255.03 (1.90)4.10 (1.89)3.89 (1.90)4.34 (1.90)4.05 (2.02)3.504.52 (1.65)3.65 (1.61)3.46 (1.61)3.88 (1.62)3.60 (1.75)ETD12.17612.56613.59012.77911.783ETDE16.52718.00719.93818.15916.518Each of the single CUSUM charts has its main strength. For example,T1μ1 is tuned to detect small shifts of the rate parameter, and it is the fastest for detecting (λ=1.25 and 1.50). Chart T2μ2 is the fastest for detecting medium shifts in the rate parameter (λ=2.00 and 2.25) while chart T3μ3 is the fastest for detecting large shifts in the rate parameter (λ=2.50,2.75,3.00,3.25 and 3.50). The CUSUM multi-chart is also faster in detecting shifts in the mean than the average of the three single CUSUM charts.


### 3.2. Simulation Results of EWMA and EWMA Multi-chart

Tables3 and 4 show the simulation analyses for EWMA single charts and EWMA multi-chart for an ARL0 of 200 and 500, respectively. The simulation analyses for EWMA and EWMA multi-charts were carried out for a 10,000-repetition experiment. We analyzed the simulation results for ten mean shifts in the rate parameter (λ1=1.25, λ2=1.50, λ3=1.75, λ4=2.00, λ5=2.25, λ6=2.50, λ7=2.75, λ8=3.00, λ9=3.25, and λ10=3.50) with change point ν=1 that is the first time there is signal or change. We chose values of the smoothing parameter wi to be w1=0.1, w2=0.5, and w3=0.9. The simulation results for the out-control average run length ARLλ of the EWMA and EWMA multi-charts were listed on Tables 3 and 4, respectively. The parameter hi and hi′ are the width of the control limits for the single EWMA charts and EWMA multi-chart, respectively. We chose three separate EWMA charts similar to the CUSUM simulation setting.Table 3
ARLs with their SDRL (in bracket) for the EWMA chart and EWMA multichart withARL0≈200.
Shifts inλTE10.1TE20.5TE30.9Average EWMATEMh1=1.517578h2=2.815918h3=3.8064451.00201.51 (180.30)200.89 (198.31)199.37 (194.44)200.59 (191.02)201.30 (191.34)1.2551.92 (35.82)66.89 (63.94)76.04 (76.97)64.95 (58.91)54.83 (44.98)1.5027.19 (14.17)30.43 (27.86)35.04 (34.94)30.89 (25.66)25.93 (18.08)1.7518.47 (8.01)17.20 (14.83)19.79 (18.99)18.49 (13.94)16.22 (10.57)2.0014.05 (5.34)11.10 (8.85)12.25 (11.33)12.47 (8.51)11.30 (7.25)2.2511.39 (3.98)7.94 (5.98)8.47 (7.62)9.27 (5.86)8.57 (5.49)2.509.69 (3.15)6.14 (4.36)6.34 (5.45)7.39 (4.32)6.74 (4.36)2.758.41 (2.63)4.94 (3.15)4.90 (4.04)6.08 (3.27)5.49 (3.53)3.007.48 (2.23)4.13 (2.58)3.97 (3.17)5.19 (2.66)4.59 (2.93)3.256.74 (1.96)3.58 (2.16)3.31 (2.48)4.54 (2.20)3.97 (2.48)3.506.13 (1.75)3.15 (1.81)2.88 (2.05)4.05 (1.87)3.42 (2.08)ETD12.83711.09112.12112.01610.454ETDE16.14715.55017.29916.33214.106Table 4
ARLs with their SDRL (in bracket) for the EWMA charts and EWMA multichart withARL0≈500.
Shifts inλTE10.1TE20.5TE30.9Average EWMATEMh1=1.641053h2=3.102122h3=4.5991331.00500.94 (484.69)501.15 (500.08)501.23 (496.67)501.11 (493.81)501.97 (492.12)1.2582.02 (61.84)138.95 (136.88)163.51 (162.92)128.16 (120.55)95.57 (82.52)1.5035.52 (20.29)54.14 (53.46)72.77 (71.59)54.14 (48.45)37.43 (25.96)1.7522.22 (10.15)27.86 (25.25)37.67 (36.63)29.25 (24.01)21.23 (13.16)2.0016.54 (6.44)16.56 (14.17)22.99 (21.50)18.70 (14.04)14.37 (8.78)2.2513.04 (4.67)11.19 (8.98)14.90 (13.57)13.04 (9.07)10.47 (6.36)2.5010.82 (3.54)8.14 (6.07)10.61 (9.44)9.86 (6.35)8.17 (5.14)2.759.44 (2.96)6.31 (4.46)7.99 (6.95)7.91 (4.79)6.52 (4.19)3.008.27 (2.48)5.19 (3.40)6.11 (5.02)6.52 (3.63)5.37 (3.44)3.257.47 (2.19)4.36 (2.75)5.13 (4.02)5.65 (2.99)4.54 (2.90)3.506.76 (1.91)3.78 (2.23)4.20 (3.15)4.91 (2.43)3.92 (2.48)ETD16.12118.63723.45819.40414.653ETDE21.21027.64834.58827.81420.759The control limits were obtained using Monte Carlo simulations. To arrive at an in-control ARL of EWMA multi-chartTEM of approximately 200, we had to choose ARL0TE1w1=360.65 with h1′=1.59916, ARL0TE2w2=360.27 with h2′=3.00625, and ARL0TE3w3=359.26 with h3′=4.51543. Similarly, to arrive at an in-control ARL of EWMA multi-chart TEM of approximately 500, we had to choose ARL0TE1w1=899.40 with h1′=1.711293, ARL0TE2w2=900.53 with h2′=3.00625, and ARL0TE3w3=900.22 with h3′=4.689937. In other words, we force all the ARL0 of the single charts to be approximately equal to guarantee an ARL0 of multi-chart to be approximately 200 and 500, respectively.Tables3 and 4 show that each of the schemes has its merits and demerits over a range, and perhaps, it is conflicting to compare the charts in relation to the average run length ARL. We use the ETD and ETDE to compare the charts over the whole range of shifts. EWMA multi-chart TEM has the smallest ETD and ETDE followed by EWMA chart TE20.5, TE30.9, and TE10.1, respectively, for ARL0=200. Also, EWMA multi-chart TEM has the smallest ETD and ETDE followed by chart TE10.1, TE20.5 and TE30.9, respectively, for ARL0=500. The EWMA multi-chart is also faster in detecting shifts in the mean than the average of the three single EWMA charts.


### 3.3. Simulation Results of CUSUM-EWMA-CUSUM Charts

Tables5–8 show the simulation analyses for EWMA-CUSUM single charts and EWMA-CUSUM multi-chart for an ARL0 of 200 and 500, respectively. The simulation analyses were carried out for a 10,000-repetition experiment. We analyzed the simulation results for ten mean shifts in the rate parameter (λ1=1.25, λ2=1.50, λ3=1.75, λ4=2.00, λ5=2.25, λ6=2.50, λ7=2.75, λ8=3.00, λ9=3.25, and λ10=3.50) with change point ν=1 that is the first time there is signal or change. For comparison sake, the in-control ARLARL0 of all the charts were assumed to be equal and was taken to be 200 and 500, respectively. We considered one EWMA chart and two CUSUM charts. We chose values of the smoothing parameter wi, to be w1=0.1 and reference parametersμ1=1.5 and μ2=2.5 for TEC1 and w1=0.1, μ1=1.5 and μ2=2 for TEC2. The simulation results for the out-control average run length ARLλ of the EWMA-CUSUM multi-charts were listed on Tables 5–8, respectively.Table 5
ARLs with their SDRL (in bracket) for the CUSUM-EWMA-CUSUM charts withARL0≈200.
Shifts inλTE10.1T1μ1=1.5T2μ2=2.5Average EWMA-CUSUMTEC1h1=1.517578d1=2.609375d2=3.4531251.00201.51 (180.30)202.53 (192.35)203.86 (202.29)202.63 (191.65)200.20 (190.23)1.2551.92 (35.82)45.44 (37.36)58.45 (56.17)51.94 (43.12)47.14 (39.18)1.5027.19 (14.17)20.64 (14.18)24.25 (21.58)24.03 (16.64)21.66 (15.37)1.7518.47 (8.01)12.79 (8.96)13.50 (10.94)14.92 (9.30)13.15 (8.58)2.0014.05 (5.34)9.17 (6.71)8.95 (5.62)10.72 (5.89)9.19 (5.66)2.2511.39 (3.98)7.26 (4.05)6.71 (3.65)8.45 (3.89)7.08 (4.21)2.509.69 (3.15)6.03 (3.08)5.25 (2.91)6.99 (3.05)5.57 (3.22)2.758.41 (2.63)5.15 (2.55)4.29 (2.31)5.95 (2.50)4.66 (2.65)3.007.48 (2.23)4.47 (2.12)3.72 (1.94)5.22 (2.10)3.96 (2.19)3.256.74 (1.96)4.01 (1.77)3.27 (1.64)4.67 (1.79)3.46 (1.90)3.506.13 (1.75)3.59 (1.49)2.92 (1.41)4.21 (1.55)3.11 (1.62)ETD12.8378.9719.38910.3988.821ETDE16.14711.85513.13113.71011.898Table 6
ARLs with their SDRL (in bracket) for the CUSUM-EWMA-CUSUM charts withARL0≈500.
Shifts inλTE10.1T1μ1=1.5T2μ2=2.5Average EWMA-CUSUMTEC1h1=1.641053d1=3.794189d2=4.7443611.00500.94 (484.69)500.54 (490.24)501.17 (495.91)500.88 (490.28)501.78 (486.69)1.2582.02 (61.84)70.77 (58.68)102.39 (99.86)85.06 (73.46)76.35 (62.88)1.5035.52 (20.29)27.66 (18.25)36.28 (32.23)33.15 (23.59)29.13 (19.82)1.7522.22 (10.15)16.54 (9.14)18.06 (14.48)18.94 (11.26)17.05 (10.40)2.0016.54 (6.44)11.79 (5.67)11.45 (8.25)13.26 (6.79)11.52 (6.74)2.2513.04 (4.67)9.28 (4.13)8.02 (5.12)10.11 (4.64)8.67 (5.04)2.5010.82 (3.54)7.55 (3.25)6.30 (3.79)8.22 (3.53)6.75 (3.78)2.759.44 (2.96)6.48 (2.65)5.14 (2.79)7.02 (2.80)5.54 (2.99)3.008.27 (2.48)5.65 (2.23)4.39 (2.24)6.10 (2.32)4.71 (2.44)3.257.47 (2.19)5.03 (1.90)3.89 (1.90)5.46 (2.00)4.12 (2.10)3.506.76 (1.91)4.52 (1.65)3.46 (1.61)4.91 (1.72)3.65 (1.77)ETD16.12112.17613.59013.96011.955ETDE21.2116.52719.93819.22316.749Table 7
ARLs with their SDRL (in bracket) for the CUSUM-EWMA-CUSUM charts withARL0≈200.
Shifts inλTE10.1T1μ1=1.5T2μ2=2.0Average EWMA-CUSUMTEC2h1=1.517578d1=2.609375d2=3.4531251.00201.51 (180.30)202.53 (192.35)204.21 (203.3)202.75 (191.98)199.58 (192.24)1.2551.92 (35.82)45.44 (37.36)51.29 (46.61)49.55 (39.93)46.76 (38.31)1.5027.19 (14.17)20.64 (14.18)21.77 (18.01)23.20 (15.45)21.45 (15.34)1.7518.47 (8.01)12.79 (8.96)12.67 (8.70)14.64 (8.56)12.80 (8.03)2.0014.05 (5.34)9.17 (6.71)8.71 (4.92)10.64 (5.66)9.18 (5.46)2.2511.39 (3.98)7.26 (4.05)6.59 (3.89)8.41 (3.97)7.00 (3.86)2.509.69 (3.15)6.03 (3.08)5.41 (2.98)7.04 (3.07)5.76 (3.10)2.758.41 (2.63)5.15 (2.55)4.51 (2.37)6.02 (2.52)4.84 (2.41)3.007.48 (2.23)4.47 (2.12)3.93 (2.06)5.29 (2.14)4.20 (2.02)3.256.74 (1.96)4.01 (1.77)3.51 (1.73)4.75 (1.82)3.70 (1.68)3.506.13 (1.75)3.59 (1.49)3.15 (1.44)4.29 (1.56)3.35 (1.47)ETD12.8378.9718.89810.2338.892ETDE16.14711.85512.15413.38311.904Table 8
ARLs with their SDRL (in bracket) for the CUSUM-EWMA-CUSUM charts withARL0≈500.
Shifts inλTE10.1T1μ1=1.5T2μ2=2.0Average EWMA-CUSUMTEC2h1=1.641053d1=3.430471d2=4.0904081.00500.94 (484.69)500.54 (490.24)501.72 (492.04)501.07 (488.99)501.25 (492.97)1.2582.02 (61.84)70.77 (58.68)88.97 (83.2)80.59 (67.91)74.70 (62.12)1.5035.52 (20.29)27.66 (18.25)30.87 (24.72)31.35 (21.09)29.24 (19.67)1.7522.22 (10.15)16.54 (9.14)16.75 (11.80)18.50 (10.36)16.84 (10.09)2.0016.54 (6.44)11.79 (5.67)10.99 (6.84)13.11 (6.32)11.65 (6.54)2.2513.04 (4.67)9.28 (4.13)8.13 (4.45)10.15 (4.42)8.79 (4.68)2.5010.82 (3.54)7.55 (3.25)6.48 (3.38)8.28 (3.39)7.02 (3.50)2.759.44 (2.96)6.48 (2.65)5.42 (2.70)7.11 (2.77)5.86 (2.83)3.008.27 (2.48)5.65 (2.23)4.71 (2.23)6.21 (2.31)5.04 (2.35)3.257.47 (2.19)5.03 (1.90)4.10 (1.89)5.53 (1.99)4.44 (2.00)3.506.76 (1.91)4.52 (1.65)3.65 (1.61)4.98 (1.72)3.98 (1.72)ETD16.12112.17612.56613.62012.081ETDE21.21016.52718.00718.58116.756The control limits were obtained using Monte Carlo simulations. To arrive at an in-control ARL of EWMA-CUSUM multi-chartTEC1 of approximately 200, we had to choose ARL0TE1w1=304.92 with h1′=1.575862, ARL0T1μ1=304.31 with d1′=3.828079, and ARL0T2μ2=305.31 with d2′=4.825815. Similarly, to arrive at an in-control ARL of EWMA-CUSUM multi-chart TEC1 of approximately 500, we had to choose ARL0TE1w1=819.18 with h1′=1.699961, ARL0T1μ1=819.22 with d1′=3.900391, and ARL0T2μ2=820.04 with d2′=4.825815.Also, to arrive at an in-control ARL of EWMA-CUSUM multi-chartTEC2 of approximately 200, we had to choose ARL0TE1w1=270.8328 with h1′=1.559829, ARL0T1μ1=269.995 with d1′=2.87793, and ARL0T2μ2=270.5916 with d2′=3.545212. Similarly, to arrive at an in-control ARL of EWMA-CUSUM multi-chart TEC1 of approximately 500, we had to choose ARL0TE1w1=740.0857 with h1′=1.687119, ARL0T1μ1=739.1194 with d1′=3.803393, and ARL0T2μ2=740.4778 with d2′=4.482666.In other words, we force all theARL0 of the single charts to be approximately equal to guarantee an ARL0 of multi-chart to be approximately 200 and 500, respectively.Tables5–8 show that each of the schemes has its merits and demerits over a range, and perhaps, it is conflicting to compare the charts in relation to the average run length ARL. We use the ETD and ETDE to compare the charts over the whole range of shifts. EWMA-CUSUM multi-chart TEC1 has the smallest ETD and ETDE and hence better detection performance than TEC2.


### 3.4. Comparison of Results

The CUSUM multi-chart is better on the whole in detecting various mean shifts in the rate parameter than EWMA multi-chart and EWMA-CUSUM multi-chart. Furthermore, the EWMA multi-chart is better on the whole in detecting various mean shifts than EWMA-CUSUM multi-chart. We subsequently used CUSUM multi-chart to monitor the real data. Also, the simulation results support the theoretical analysis.


### 3.5. An Illustration with Health Surveillance Data

We use monthly tuberculosis (TB) data (seeSupplementary Materials) from the northern regional health directorate of the Ghana Health Service, spanning the period of 2010 to 2017 to illustrate the implementation of a multi-CUSUM scheme for monitoring health data. The tuberculosis data consists of mainly monthly cases of three types of tuberculosis, namely tuberculosis arthritis (TB arthritis), tuberculosis meningitis (TB meningitis), and tuberculosis miliary (TB miliary). Tuberculosis is basically an infectious disease caused by a bacterial microorganism called mycobacterium tuberculosis. The disease mostly affects the lungs but can affect or spread to other parts of the body as well. TB is contagious and normally spreads into the air through sneezing, talking, and coughing of a person with TB of the lungs or throat. Symptoms of TB in the lungs may include bad cough that lasts three weeks or longer, weight loss, loss of appetite, coughing up blood or mucus, weakness or fatigue, fever, and night sweats. TB can be deadly if it is not treated well. Normally patients can take antibiotics like rifampicin through the supervision of a medical doctor [45].Tuberculosis (TB) is one of the top ten causes of death worldwide [46]. In 2017, there were more than 10 million cases of active TB which resulted in 1.6 million deaths including 0.3 million among people with HIV. New infections occur in about 1% of the population each year and about 25% of the world’s population is thought to be infected with TB [42]. More than 95% of deaths occurred in developing countries, and more than 50% in India, China, Indonesia, Pakistan, and the Philippines [46]. In Ghana, the total cases of notified tuberculosis in 2017 were about 14,550 [47].Tuberculosis arthritis is a joint inflammation caused by the invasion of the joint by tuberculosis bacilli that have migrated from a primary infection, usually in the chest. The most common joints affected include the wrists, ankles, knees, hips, and spine [45].Tuberculosis meningitis is a disease that affects the tissues covering the brain and spinal cord. Tuberculosis meningitis is caused by mycobacterium tuberculosis. The bacterial spreads to the brain and spine from other parts of the body usually the lungs [45].Miliary tuberculosis is another form of tuberculosis where the disease or infection spreads through the entire body. This type of tuberculosis is normally associated with people whose immune system has already been compromised. This is also caused by mycobacterium tuberculosis [45].Figure1 shows the box plot of the count of TB arthritis, TB meningitis, and TB miliary between the years of 2010 to 2017. The average count of TB Arthritis seems to be greater than the average of TB miliary and the average of TB meningitis. Also, the annual average incidence of the diseases varied from year to year as shown in (Figure 2). The means of the diseases seem to be dynamic, so we seek to detect changes in the average counts of the diseases. Many researchers have developed statistical methods for detecting changes in disease incidence or rates (see Mei et al. [28], Jiang et al. [29] and Richards et al. [30]). We proposed the multi-CUSUM chart for detecting changes in disease incidence. We consider the disease incidence as an i.i.d. random sequence, and we monitor the three tuberculosis diseases, namely tuberculosis arthritis, tuberculosis meningitis, and tuberculosis miliary. We applied the chi-square goodness-of-fit test to ascertain whether the data; Yi:i=1,2,⋯,96 is indeed coming from the Poisson distribution. The hypothesis of interest is H0: The form of the distribution for the data is Poisson; verses H1: The form of the distribution for the data is not Poisson. We control the test at a significance level of α=0.05. If the p value of the goodness-of-fit test is greater than the specified significance level α, we fail to reject H0 and conclude that the data is indeed Poisson i.i.d. We performed the chi-square goodness-of-fit test for the three diseases counts.Figure 1
Box plot of count of TB arthritis, TB meningitis, and TB miliary between the years of 2010 to 2017.Figure 2
Annual average incidence of the diseases.The p-values for the chi-square goodness-of-fit test arep value (TB arthritis) = 2.977376×10−22, p value (TB miliary) = 1.393786×10−13, and p value(TB meningitis) = 2.534261×1009. We therefore reject the assertion that the diseases are Poisson i.i.d.′s since the p values are less than the specified significance level. We consequently transformed the data by {Yij/λi:i=1,⋯,m and j=1,2,⋯,96}.To detect changes in the tuberculosis diseases using the multi-CUSUM chart, we obtain estimates of the reference values of the diseases using data before 2013 as phase I data to estimate the rate parameters. The in-control reference values areμ0i=μ01,μ02,μ03, where say μ01= median or mean of the in-control data, μ02= third quarter of the in-control data, and μ03= max of the in-control data; thus, we choose the reference values such that 1<μ01<μ02<μ03. We then purpose to determine the detection capability of the multi-CUSUM chart to detect changes in the diseases starting from 2013.We briefly expound the procedural steps for implementing the multi-CUSUM scheme for detecting changes in the tuberculosis disease incidence:(1)
Standardized your observations {Yij/λi:i=1,⋯,m and j=1,2,⋯,n}(2)
Determine the in-control values of the reference parametersμ01,μ02,μ03, Normally the in-control values are unknown; hence, we estimate them from the phase I historical data (e.g., estimate from data before 2013, say μ01 = median or mean of in-control data, μ02 = third quarter of in-control data, and μ03 = max of in-control data)(3)
Determine the in-controlARL0 of the individual charts say ARL0≈50 and use Monte Carlo simulations to find the control limits of single charts(4)
Assume some in-controlARL0 of the multi-CUSUM charts say ARL0≈50, then use Monte Carlo simulations to arrive at the control limits of the multi-CUSUM charts(5)
Establish the multi-CUSUM chart then monitor new observations over timeTable9 presents numerical results for monitoring TB arthritis, TB meningitis, and TB miliary starting from the 37th month (January 2013). According to Table 9, the in-control rate parameter λ0=1. We use the same range of shifts in the rate parameter a,b and number of shifts l as in the simulation setting. The in-control ARL0≈50 means the expected number of false alarms for monitoring in phase II.Table 9
CUSUM and Multi-CUSUM chart results for monitoring TB arthritis, TB meningitis, and TB miliary withARL0≈50.
Control chartT1μ1T2μ2T3μ3TCMμ1,μ2,μ3TB arthritisControl limitd1=0.72d2=1.54d3=1.1919d′=0.8367187,1.75,2.28125Reference valuesμ01=1.1609μ02=1.5142μ03=2.1199ETD5.5125.1195.1225.071ETDE6.8326.4436.5566.405TB meningitisControl limitd1=1.046875d2=1.657812d3=1.964062d′=1.1375,1.817188,2.179652Reference valuesμ01=1.2632μ02=1.6168μ03=2.0211ETD5.2545.1755.0855.041ETDE6.5426.5376.4786.372TB miliaryControl limitd1=0.7148438d2=1.504687d3=2.354688d′=0.9222656,1.889893,2.591463Reference valuesμ01=1.1597μ02=1.5126μ03=3.0252ETD5.5055.1054.9544.878ETDE6.8056.4206.3706.228CUSUM multi-chartTCM was the quickest to detect changes in TB arthritis, since it has the smallest ETD and ETDE for detecting shifts in a range. Chart Tμ2d2 also outperformed chart Tμ3d3 and chart Tμ1d1 in that order, respectively.For TB meningitis and TB miliary disease, CUSUM multi-chartTCM was the quickest to detect changes, since it has the smallest ETD and ETDE for detecting shifts in a range followed by chart Tμ3d3, Tμ2d2, and Tμ1d1 in that respective order. In general, CUSUM multi-chart is faster in detecting changes in the diseases (TB arthritis, TB meningitis, and TB miliary) than the single charts.


## 3.1. Simulation Results of CUSUM and Multi-CUSUM Chart

Simulation analyses were carried out for a10,000-repetition experiment. We analyzed the simulation results for ten mean shifts in the rate parameter (λ1=1.25,λ2=1.50, λ3=1.75, λ4=2.00, λ5=2.25, λ6=2.50, λ7=2.75, λ8=3.00, λ9=3.25, and λ10=3.50) with change point ν=1 that is the first time there is signal or change. For comparison sake, the in-control ARLARL0 of all the charts were assumed to be equal and was taken to be 200 and 500, respectively. The reference values μi were chosen to be μ1=1.5, μ2=2.0, and μ3=2.5, where μ1 is termed as a small mean shift in the rate parameter, μ2 is a medium mean shift, and μ3 is a large mean shift in the rate parameter, respectively. The simulation results for the out-control average run length ARLλ of the Poisson CUSUM charts with parameter μ1,μ2, μ3, average CUSUM chart and multi-chart TCM were listed in column two, column three, column four, column five, and column six, respectively. The parameter di and di′ are the width of the control limits for the single CUSUM charts and CUSUM multi-chart, respectively. We chose three separate CUSUM charts because as suggested by Sparks [40], three or more single charts are needed to achieve an efficient multi-chart scheme. The control limits were obtained using Monte Carlo simulations. To arrive at an in-control ARL of CUSUM multi-chart TCM of approximately 200, we had to choose ARL0T1μ1=279.86 with d1′=2.914062, ARL0T2μ2=280.86 with d2′=3.59375, and ARL0T3μ3=280.62 with d3′=3.749023. Similarly, to arrive at an in-control ARL of CUSUM multi-chart TCM of approximately 500, we had to choose ARL0T1μ1=740.57 with d1′=3.794189, ARL0T2μ2=740.86 with d2′=4.47998, and ARL0T3μ3=741.11 with d3′=4.744361. In other words, we force all the ARL0 of the single charts to be approximately equal to guarantee an ARL0 of multi-chart to be approximately 200 and 500, respectively.Tables1 and 2 show that each of the schemes has its merits and demerits over a range, and perhaps, it is conflicting to compare the charts in relation to the average run length ARL. Ultimately, the ETD and ETDE enable us to compare the charts over the whole range of shifts. CUSUM multi-chart TCM has the smallest ETD and ETDE followed by CUSUM chart Tμ2d2, Tμ1d1, and Tμ3d3, respectively, for ARL0=200. Also, CUSUM multi-chartTCM has the smallest ETD and ETDE for the range of shifts followed by chart Tμ1d1, Tμ2d2, and Tμ3d3, respectively, for ARL0=500.Table 1
ARLs with their SDRL (in bracket) for the CUSUM and multi-CUSUM chart withARL0≈200.
Shifts inλT1μ1T2μ2T3μ3Average CUSUMTCMμ1,μ2,μ3d1=2.609375d2=3.238342d3=3.4531251.00202.53 (192.35)204.21 (203.3)203.86 (202.29)203.53 (199.31)200.91 (194.15)1.2545.44 (37.36)51.29 (46.61)58.45 (56.17)51.73 (46.71)48.11 (41.63)1.5020.64 (14.18)21.77 (18.01)24.25 (21.58)22.22 (17.92)20.90 (15.07)1.7512.79 (8.96)12.67 (8.70)13.50 (10.94)12.99 (9.53)12.61 (7.75)2.009.17 (6.71)8.71 (4.92)8.95 (5.62)8.94 (5.75)8.85 (5.44)2.257.26 (4.05)6.59 (3.89)6.71 (3.65)6.85 (3.86)6.74 (3.98)2.506.03 (3.08)5.41 (2.98)5.25 (2.91)5.56 (2.99)5.47 (3.01)2.755.15 (2.55)4.51 (2.37)4.29 (2.31)4.65 (2.41)4.60 (2.50)3.004.47 (2.12)3.93 (2.06)3.72 (1.94)4.04 (2.04)3.91 (1.98)3.254.01 (1.77)3.51 (1.73)3.27 (1.64)3.60 (1.71)3.45 (1.69)3.503.59 (1.49)3.15 (1.44)2.92 (1.41)3.22 (1.45)3.07 (1.42)ETD8.9718.8989.3899.0868.692ETDE11.85512.15413.13112.38011.771Table 2
ARLs with their SDRL (in bracket) for the CUSUM and multi-CUSUM chart withARL0≈500.
Shifts inλT1μ1T2μ2T3μ3Average CUSUMTCMμ1,μ2,μ3d1=3.430471d2=4.090408d3=4.3310551.00500.54 (490.24)501.72 (492.04)501.17 (495.91)501.14 (492.73)500.17 (485.63)1.2570.77 (58.68)88.97 (83.20)102.39 (99.86)87.38 (80.58)75.69 (63.93)1.5027.66 (18.25)30.87 (24.72)36.28 (32.23)31.60 (25.07)28.81 (19.96)1.7516.54 (9.14)16.75 (11.80)18.06 (14.48)17.12 (11.81)16.41 (10.24)2.0011.79 (5.67)10.99 (6.84)11.45 (8.25)11.41 (6.92)11.23 (6.53)2.259.28 (4.13)8.13 (4.45)8.02 (5.12)8.48 (4.57)8.51 (4.80)2.507.55 (3.25)6.48 (3.38)6.30 (3.79)6.78 (3.47)6.68 (3.65)2.756.48 (2.65)5.42 (2.70)5.14 (2.79)5.68 (2.71)5.51 (2.86)3.005.65 (2.23)4.71 (2.23)4.39 (2.24)4.92 (2.23)4.69 (2.40)3.255.03 (1.90)4.10 (1.89)3.89 (1.90)4.34 (1.90)4.05 (2.02)3.504.52 (1.65)3.65 (1.61)3.46 (1.61)3.88 (1.62)3.60 (1.75)ETD12.17612.56613.59012.77911.783ETDE16.52718.00719.93818.15916.518Each of the single CUSUM charts has its main strength. For example,T1μ1 is tuned to detect small shifts of the rate parameter, and it is the fastest for detecting (λ=1.25 and 1.50). Chart T2μ2 is the fastest for detecting medium shifts in the rate parameter (λ=2.00 and 2.25) while chart T3μ3 is the fastest for detecting large shifts in the rate parameter (λ=2.50,2.75,3.00,3.25 and 3.50). The CUSUM multi-chart is also faster in detecting shifts in the mean than the average of the three single CUSUM charts.


## 3.2. Simulation Results of EWMA and EWMA Multi-chart

Tables3 and 4 show the simulation analyses for EWMA single charts and EWMA multi-chart for an ARL0 of 200 and 500, respectively. The simulation analyses for EWMA and EWMA multi-charts were carried out for a 10,000-repetition experiment. We analyzed the simulation results for ten mean shifts in the rate parameter (λ1=1.25, λ2=1.50, λ3=1.75, λ4=2.00, λ5=2.25, λ6=2.50, λ7=2.75, λ8=3.00, λ9=3.25, and λ10=3.50) with change point ν=1 that is the first time there is signal or change. We chose values of the smoothing parameter wi to be w1=0.1, w2=0.5, and w3=0.9. The simulation results for the out-control average run length ARLλ of the EWMA and EWMA multi-charts were listed on Tables 3 and 4, respectively. The parameter hi and hi′ are the width of the control limits for the single EWMA charts and EWMA multi-chart, respectively. We chose three separate EWMA charts similar to the CUSUM simulation setting.Table 3
ARLs with their SDRL (in bracket) for the EWMA chart and EWMA multichart withARL0≈200.
Shifts inλTE10.1TE20.5TE30.9Average EWMATEMh1=1.517578h2=2.815918h3=3.8064451.00201.51 (180.30)200.89 (198.31)199.37 (194.44)200.59 (191.02)201.30 (191.34)1.2551.92 (35.82)66.89 (63.94)76.04 (76.97)64.95 (58.91)54.83 (44.98)1.5027.19 (14.17)30.43 (27.86)35.04 (34.94)30.89 (25.66)25.93 (18.08)1.7518.47 (8.01)17.20 (14.83)19.79 (18.99)18.49 (13.94)16.22 (10.57)2.0014.05 (5.34)11.10 (8.85)12.25 (11.33)12.47 (8.51)11.30 (7.25)2.2511.39 (3.98)7.94 (5.98)8.47 (7.62)9.27 (5.86)8.57 (5.49)2.509.69 (3.15)6.14 (4.36)6.34 (5.45)7.39 (4.32)6.74 (4.36)2.758.41 (2.63)4.94 (3.15)4.90 (4.04)6.08 (3.27)5.49 (3.53)3.007.48 (2.23)4.13 (2.58)3.97 (3.17)5.19 (2.66)4.59 (2.93)3.256.74 (1.96)3.58 (2.16)3.31 (2.48)4.54 (2.20)3.97 (2.48)3.506.13 (1.75)3.15 (1.81)2.88 (2.05)4.05 (1.87)3.42 (2.08)ETD12.83711.09112.12112.01610.454ETDE16.14715.55017.29916.33214.106Table 4
ARLs with their SDRL (in bracket) for the EWMA charts and EWMA multichart withARL0≈500.
Shifts inλTE10.1TE20.5TE30.9Average EWMATEMh1=1.641053h2=3.102122h3=4.5991331.00500.94 (484.69)501.15 (500.08)501.23 (496.67)501.11 (493.81)501.97 (492.12)1.2582.02 (61.84)138.95 (136.88)163.51 (162.92)128.16 (120.55)95.57 (82.52)1.5035.52 (20.29)54.14 (53.46)72.77 (71.59)54.14 (48.45)37.43 (25.96)1.7522.22 (10.15)27.86 (25.25)37.67 (36.63)29.25 (24.01)21.23 (13.16)2.0016.54 (6.44)16.56 (14.17)22.99 (21.50)18.70 (14.04)14.37 (8.78)2.2513.04 (4.67)11.19 (8.98)14.90 (13.57)13.04 (9.07)10.47 (6.36)2.5010.82 (3.54)8.14 (6.07)10.61 (9.44)9.86 (6.35)8.17 (5.14)2.759.44 (2.96)6.31 (4.46)7.99 (6.95)7.91 (4.79)6.52 (4.19)3.008.27 (2.48)5.19 (3.40)6.11 (5.02)6.52 (3.63)5.37 (3.44)3.257.47 (2.19)4.36 (2.75)5.13 (4.02)5.65 (2.99)4.54 (2.90)3.506.76 (1.91)3.78 (2.23)4.20 (3.15)4.91 (2.43)3.92 (2.48)ETD16.12118.63723.45819.40414.653ETDE21.21027.64834.58827.81420.759The control limits were obtained using Monte Carlo simulations. To arrive at an in-control ARL of EWMA multi-chartTEM of approximately 200, we had to choose ARL0TE1w1=360.65 with h1′=1.59916, ARL0TE2w2=360.27 with h2′=3.00625, and ARL0TE3w3=359.26 with h3′=4.51543. Similarly, to arrive at an in-control ARL of EWMA multi-chart TEM of approximately 500, we had to choose ARL0TE1w1=899.40 with h1′=1.711293, ARL0TE2w2=900.53 with h2′=3.00625, and ARL0TE3w3=900.22 with h3′=4.689937. In other words, we force all the ARL0 of the single charts to be approximately equal to guarantee an ARL0 of multi-chart to be approximately 200 and 500, respectively.Tables3 and 4 show that each of the schemes has its merits and demerits over a range, and perhaps, it is conflicting to compare the charts in relation to the average run length ARL. We use the ETD and ETDE to compare the charts over the whole range of shifts. EWMA multi-chart TEM has the smallest ETD and ETDE followed by EWMA chart TE20.5, TE30.9, and TE10.1, respectively, for ARL0=200. Also, EWMA multi-chart TEM has the smallest ETD and ETDE followed by chart TE10.1, TE20.5 and TE30.9, respectively, for ARL0=500. The EWMA multi-chart is also faster in detecting shifts in the mean than the average of the three single EWMA charts.


## 3.3. Simulation Results of CUSUM-EWMA-CUSUM Charts

Tables5–8 show the simulation analyses for EWMA-CUSUM single charts and EWMA-CUSUM multi-chart for an ARL0 of 200 and 500, respectively. The simulation analyses were carried out for a 10,000-repetition experiment. We analyzed the simulation results for ten mean shifts in the rate parameter (λ1=1.25, λ2=1.50, λ3=1.75, λ4=2.00, λ5=2.25, λ6=2.50, λ7=2.75, λ8=3.00, λ9=3.25, and λ10=3.50) with change point ν=1 that is the first time there is signal or change. For comparison sake, the in-control ARLARL0 of all the charts were assumed to be equal and was taken to be 200 and 500, respectively. We considered one EWMA chart and two CUSUM charts. We chose values of the smoothing parameter wi, to be w1=0.1 and reference parametersμ1=1.5 and μ2=2.5 for TEC1 and w1=0.1, μ1=1.5 and μ2=2 for TEC2. The simulation results for the out-control average run length ARLλ of the EWMA-CUSUM multi-charts were listed on Tables 5–8, respectively.Table 5
ARLs with their SDRL (in bracket) for the CUSUM-EWMA-CUSUM charts withARL0≈200.
Shifts inλTE10.1T1μ1=1.5T2μ2=2.5Average EWMA-CUSUMTEC1h1=1.517578d1=2.609375d2=3.4531251.00201.51 (180.30)202.53 (192.35)203.86 (202.29)202.63 (191.65)200.20 (190.23)1.2551.92 (35.82)45.44 (37.36)58.45 (56.17)51.94 (43.12)47.14 (39.18)1.5027.19 (14.17)20.64 (14.18)24.25 (21.58)24.03 (16.64)21.66 (15.37)1.7518.47 (8.01)12.79 (8.96)13.50 (10.94)14.92 (9.30)13.15 (8.58)2.0014.05 (5.34)9.17 (6.71)8.95 (5.62)10.72 (5.89)9.19 (5.66)2.2511.39 (3.98)7.26 (4.05)6.71 (3.65)8.45 (3.89)7.08 (4.21)2.509.69 (3.15)6.03 (3.08)5.25 (2.91)6.99 (3.05)5.57 (3.22)2.758.41 (2.63)5.15 (2.55)4.29 (2.31)5.95 (2.50)4.66 (2.65)3.007.48 (2.23)4.47 (2.12)3.72 (1.94)5.22 (2.10)3.96 (2.19)3.256.74 (1.96)4.01 (1.77)3.27 (1.64)4.67 (1.79)3.46 (1.90)3.506.13 (1.75)3.59 (1.49)2.92 (1.41)4.21 (1.55)3.11 (1.62)ETD12.8378.9719.38910.3988.821ETDE16.14711.85513.13113.71011.898Table 6
ARLs with their SDRL (in bracket) for the CUSUM-EWMA-CUSUM charts withARL0≈500.
Shifts inλTE10.1T1μ1=1.5T2μ2=2.5Average EWMA-CUSUMTEC1h1=1.641053d1=3.794189d2=4.7443611.00500.94 (484.69)500.54 (490.24)501.17 (495.91)500.88 (490.28)501.78 (486.69)1.2582.02 (61.84)70.77 (58.68)102.39 (99.86)85.06 (73.46)76.35 (62.88)1.5035.52 (20.29)27.66 (18.25)36.28 (32.23)33.15 (23.59)29.13 (19.82)1.7522.22 (10.15)16.54 (9.14)18.06 (14.48)18.94 (11.26)17.05 (10.40)2.0016.54 (6.44)11.79 (5.67)11.45 (8.25)13.26 (6.79)11.52 (6.74)2.2513.04 (4.67)9.28 (4.13)8.02 (5.12)10.11 (4.64)8.67 (5.04)2.5010.82 (3.54)7.55 (3.25)6.30 (3.79)8.22 (3.53)6.75 (3.78)2.759.44 (2.96)6.48 (2.65)5.14 (2.79)7.02 (2.80)5.54 (2.99)3.008.27 (2.48)5.65 (2.23)4.39 (2.24)6.10 (2.32)4.71 (2.44)3.257.47 (2.19)5.03 (1.90)3.89 (1.90)5.46 (2.00)4.12 (2.10)3.506.76 (1.91)4.52 (1.65)3.46 (1.61)4.91 (1.72)3.65 (1.77)ETD16.12112.17613.59013.96011.955ETDE21.2116.52719.93819.22316.749Table 7
ARLs with their SDRL (in bracket) for the CUSUM-EWMA-CUSUM charts withARL0≈200.
Shifts inλTE10.1T1μ1=1.5T2μ2=2.0Average EWMA-CUSUMTEC2h1=1.517578d1=2.609375d2=3.4531251.00201.51 (180.30)202.53 (192.35)204.21 (203.3)202.75 (191.98)199.58 (192.24)1.2551.92 (35.82)45.44 (37.36)51.29 (46.61)49.55 (39.93)46.76 (38.31)1.5027.19 (14.17)20.64 (14.18)21.77 (18.01)23.20 (15.45)21.45 (15.34)1.7518.47 (8.01)12.79 (8.96)12.67 (8.70)14.64 (8.56)12.80 (8.03)2.0014.05 (5.34)9.17 (6.71)8.71 (4.92)10.64 (5.66)9.18 (5.46)2.2511.39 (3.98)7.26 (4.05)6.59 (3.89)8.41 (3.97)7.00 (3.86)2.509.69 (3.15)6.03 (3.08)5.41 (2.98)7.04 (3.07)5.76 (3.10)2.758.41 (2.63)5.15 (2.55)4.51 (2.37)6.02 (2.52)4.84 (2.41)3.007.48 (2.23)4.47 (2.12)3.93 (2.06)5.29 (2.14)4.20 (2.02)3.256.74 (1.96)4.01 (1.77)3.51 (1.73)4.75 (1.82)3.70 (1.68)3.506.13 (1.75)3.59 (1.49)3.15 (1.44)4.29 (1.56)3.35 (1.47)ETD12.8378.9718.89810.2338.892ETDE16.14711.85512.15413.38311.904Table 8
ARLs with their SDRL (in bracket) for the CUSUM-EWMA-CUSUM charts withARL0≈500.
Shifts inλTE10.1T1μ1=1.5T2μ2=2.0Average EWMA-CUSUMTEC2h1=1.641053d1=3.430471d2=4.0904081.00500.94 (484.69)500.54 (490.24)501.72 (492.04)501.07 (488.99)501.25 (492.97)1.2582.02 (61.84)70.77 (58.68)88.97 (83.2)80.59 (67.91)74.70 (62.12)1.5035.52 (20.29)27.66 (18.25)30.87 (24.72)31.35 (21.09)29.24 (19.67)1.7522.22 (10.15)16.54 (9.14)16.75 (11.80)18.50 (10.36)16.84 (10.09)2.0016.54 (6.44)11.79 (5.67)10.99 (6.84)13.11 (6.32)11.65 (6.54)2.2513.04 (4.67)9.28 (4.13)8.13 (4.45)10.15 (4.42)8.79 (4.68)2.5010.82 (3.54)7.55 (3.25)6.48 (3.38)8.28 (3.39)7.02 (3.50)2.759.44 (2.96)6.48 (2.65)5.42 (2.70)7.11 (2.77)5.86 (2.83)3.008.27 (2.48)5.65 (2.23)4.71 (2.23)6.21 (2.31)5.04 (2.35)3.257.47 (2.19)5.03 (1.90)4.10 (1.89)5.53 (1.99)4.44 (2.00)3.506.76 (1.91)4.52 (1.65)3.65 (1.61)4.98 (1.72)3.98 (1.72)ETD16.12112.17612.56613.62012.081ETDE21.21016.52718.00718.58116.756The control limits were obtained using Monte Carlo simulations. To arrive at an in-control ARL of EWMA-CUSUM multi-chartTEC1 of approximately 200, we had to choose ARL0TE1w1=304.92 with h1′=1.575862, ARL0T1μ1=304.31 with d1′=3.828079, and ARL0T2μ2=305.31 with d2′=4.825815. Similarly, to arrive at an in-control ARL of EWMA-CUSUM multi-chart TEC1 of approximately 500, we had to choose ARL0TE1w1=819.18 with h1′=1.699961, ARL0T1μ1=819.22 with d1′=3.900391, and ARL0T2μ2=820.04 with d2′=4.825815.Also, to arrive at an in-control ARL of EWMA-CUSUM multi-chartTEC2 of approximately 200, we had to choose ARL0TE1w1=270.8328 with h1′=1.559829, ARL0T1μ1=269.995 with d1′=2.87793, and ARL0T2μ2=270.5916 with d2′=3.545212. Similarly, to arrive at an in-control ARL of EWMA-CUSUM multi-chart TEC1 of approximately 500, we had to choose ARL0TE1w1=740.0857 with h1′=1.687119, ARL0T1μ1=739.1194 with d1′=3.803393, and ARL0T2μ2=740.4778 with d2′=4.482666.In other words, we force all theARL0 of the single charts to be approximately equal to guarantee an ARL0 of multi-chart to be approximately 200 and 500, respectively.Tables5–8 show that each of the schemes has its merits and demerits over a range, and perhaps, it is conflicting to compare the charts in relation to the average run length ARL. We use the ETD and ETDE to compare the charts over the whole range of shifts. EWMA-CUSUM multi-chart TEC1 has the smallest ETD and ETDE and hence better detection performance than TEC2.


## 3.4. Comparison of Results

The CUSUM multi-chart is better on the whole in detecting various mean shifts in the rate parameter than EWMA multi-chart and EWMA-CUSUM multi-chart. Furthermore, the EWMA multi-chart is better on the whole in detecting various mean shifts than EWMA-CUSUM multi-chart. We subsequently used CUSUM multi-chart to monitor the real data. Also, the simulation results support the theoretical analysis.


## 3.5. An Illustration with Health Surveillance Data

We use monthly tuberculosis (TB) data (seeSupplementary Materials) from the northern regional health directorate of the Ghana Health Service, spanning the period of 2010 to 2017 to illustrate the implementation of a multi-CUSUM scheme for monitoring health data. The tuberculosis data consists of mainly monthly cases of three types of tuberculosis, namely tuberculosis arthritis (TB arthritis), tuberculosis meningitis (TB meningitis), and tuberculosis miliary (TB miliary). Tuberculosis is basically an infectious disease caused by a bacterial microorganism called mycobacterium tuberculosis. The disease mostly affects the lungs but can affect or spread to other parts of the body as well. TB is contagious and normally spreads into the air through sneezing, talking, and coughing of a person with TB of the lungs or throat. Symptoms of TB in the lungs may include bad cough that lasts three weeks or longer, weight loss, loss of appetite, coughing up blood or mucus, weakness or fatigue, fever, and night sweats. TB can be deadly if it is not treated well. Normally patients can take antibiotics like rifampicin through the supervision of a medical doctor [45].Tuberculosis (TB) is one of the top ten causes of death worldwide [46]. In 2017, there were more than 10 million cases of active TB which resulted in 1.6 million deaths including 0.3 million among people with HIV. New infections occur in about 1% of the population each year and about 25% of the world’s population is thought to be infected with TB [42]. More than 95% of deaths occurred in developing countries, and more than 50% in India, China, Indonesia, Pakistan, and the Philippines [46]. In Ghana, the total cases of notified tuberculosis in 2017 were about 14,550 [47].Tuberculosis arthritis is a joint inflammation caused by the invasion of the joint by tuberculosis bacilli that have migrated from a primary infection, usually in the chest. The most common joints affected include the wrists, ankles, knees, hips, and spine [45].Tuberculosis meningitis is a disease that affects the tissues covering the brain and spinal cord. Tuberculosis meningitis is caused by mycobacterium tuberculosis. The bacterial spreads to the brain and spine from other parts of the body usually the lungs [45].Miliary tuberculosis is another form of tuberculosis where the disease or infection spreads through the entire body. This type of tuberculosis is normally associated with people whose immune system has already been compromised. This is also caused by mycobacterium tuberculosis [45].Figure1 shows the box plot of the count of TB arthritis, TB meningitis, and TB miliary between the years of 2010 to 2017. The average count of TB Arthritis seems to be greater than the average of TB miliary and the average of TB meningitis. Also, the annual average incidence of the diseases varied from year to year as shown in (Figure 2). The means of the diseases seem to be dynamic, so we seek to detect changes in the average counts of the diseases. Many researchers have developed statistical methods for detecting changes in disease incidence or rates (see Mei et al. [28], Jiang et al. [29] and Richards et al. [30]). We proposed the multi-CUSUM chart for detecting changes in disease incidence. We consider the disease incidence as an i.i.d. random sequence, and we monitor the three tuberculosis diseases, namely tuberculosis arthritis, tuberculosis meningitis, and tuberculosis miliary. We applied the chi-square goodness-of-fit test to ascertain whether the data; Yi:i=1,2,⋯,96 is indeed coming from the Poisson distribution. The hypothesis of interest is H0: The form of the distribution for the data is Poisson; verses H1: The form of the distribution for the data is not Poisson. We control the test at a significance level of α=0.05. If the p value of the goodness-of-fit test is greater than the specified significance level α, we fail to reject H0 and conclude that the data is indeed Poisson i.i.d. We performed the chi-square goodness-of-fit test for the three diseases counts.Figure 1
Box plot of count of TB arthritis, TB meningitis, and TB miliary between the years of 2010 to 2017.Figure 2
Annual average incidence of the diseases.The p-values for the chi-square goodness-of-fit test arep value (TB arthritis) = 2.977376×10−22, p value (TB miliary) = 1.393786×10−13, and p value(TB meningitis) = 2.534261×1009. We therefore reject the assertion that the diseases are Poisson i.i.d.′s since the p values are less than the specified significance level. We consequently transformed the data by {Yij/λi:i=1,⋯,m and j=1,2,⋯,96}.To detect changes in the tuberculosis diseases using the multi-CUSUM chart, we obtain estimates of the reference values of the diseases using data before 2013 as phase I data to estimate the rate parameters. The in-control reference values areμ0i=μ01,μ02,μ03, where say μ01= median or mean of the in-control data, μ02= third quarter of the in-control data, and μ03= max of the in-control data; thus, we choose the reference values such that 1<μ01<μ02<μ03. We then purpose to determine the detection capability of the multi-CUSUM chart to detect changes in the diseases starting from 2013.We briefly expound the procedural steps for implementing the multi-CUSUM scheme for detecting changes in the tuberculosis disease incidence:(1)
Standardized your observations {Yij/λi:i=1,⋯,m and j=1,2,⋯,n}(2)
Determine the in-control values of the reference parametersμ01,μ02,μ03, Normally the in-control values are unknown; hence, we estimate them from the phase I historical data (e.g., estimate from data before 2013, say μ01 = median or mean of in-control data, μ02 = third quarter of in-control data, and μ03 = max of in-control data)(3)
Determine the in-controlARL0 of the individual charts say ARL0≈50 and use Monte Carlo simulations to find the control limits of single charts(4)
Assume some in-controlARL0 of the multi-CUSUM charts say ARL0≈50, then use Monte Carlo simulations to arrive at the control limits of the multi-CUSUM charts(5)
Establish the multi-CUSUM chart then monitor new observations over timeTable9 presents numerical results for monitoring TB arthritis, TB meningitis, and TB miliary starting from the 37th month (January 2013). According to Table 9, the in-control rate parameter λ0=1. We use the same range of shifts in the rate parameter a,b and number of shifts l as in the simulation setting. The in-control ARL0≈50 means the expected number of false alarms for monitoring in phase II.Table 9
CUSUM and Multi-CUSUM chart results for monitoring TB arthritis, TB meningitis, and TB miliary withARL0≈50.
Control chartT1μ1T2μ2T3μ3TCMμ1,μ2,μ3TB arthritisControl limitd1=0.72d2=1.54d3=1.1919d′=0.8367187,1.75,2.28125Reference valuesμ01=1.1609μ02=1.5142μ03=2.1199ETD5.5125.1195.1225.071ETDE6.8326.4436.5566.405TB meningitisControl limitd1=1.046875d2=1.657812d3=1.964062d′=1.1375,1.817188,2.179652Reference valuesμ01=1.2632μ02=1.6168μ03=2.0211ETD5.2545.1755.0855.041ETDE6.5426.5376.4786.372TB miliaryControl limitd1=0.7148438d2=1.504687d3=2.354688d′=0.9222656,1.889893,2.591463Reference valuesμ01=1.1597μ02=1.5126μ03=3.0252ETD5.5055.1054.9544.878ETDE6.8056.4206.3706.228CUSUM multi-chartTCM was the quickest to detect changes in TB arthritis, since it has the smallest ETD and ETDE for detecting shifts in a range. Chart Tμ2d2 also outperformed chart Tμ3d3 and chart Tμ1d1 in that order, respectively.For TB meningitis and TB miliary disease, CUSUM multi-chartTCM was the quickest to detect changes, since it has the smallest ETD and ETDE for detecting shifts in a range followed by chart Tμ3d3, Tμ2d2, and Tμ1d1 in that respective order. In general, CUSUM multi-chart is faster in detecting changes in the diseases (TB arthritis, TB meningitis, and TB miliary) than the single charts.


## 4. Conclusion

Basically, this study seeks to monitor tuberculosis diseases based on multi-chart schemes and also evaluate the efficiency of the proposed scheme. To achieve this purpose we carried out a simulation study for CUSUM multi-chart, EWMA multi-chart and EWMA-CUSUM multi-chart subject to the Poisson distribution. The chart with the smallest expectation of the time for detecting mean shifts (ETD) and the smallest expectation of the time for detecting mean shifts when theARLλi are assigned equal weights (ETDE) is better. The simulation results show that CUSUM multi-chart had the smallest ETD and ETDE; hence, CUSUM multi-chart has better detection performance than EWMA multi-chart and EWMA-CUSUM multi-chart. Also, the average of the CUSUM charts performed less better than CUSUM multi-chart likewise the average of the EWMA charts performed less better than EWMA multi-chart.We subsequently used CUSUM multi-chart to monitor tuberculosis (TB) disease from the northern region of Ghana spanning the period of 2010 to 2017. The tuberculosis diseases were TB arthritis, TB meningitis, and TB miliary. We used the data before 2013 as phase 1 historical data for estimating reference values then starting from the 37th month (January 2013) we seek to monitor changes in disease incidence. CUSUM multi-chartTCM was the quickest to detect changes followed by chart Tμ2d2, Tμ3d3, and Tμ1d1 in that order, respectively, for TB arthritis disease. For TB meningitis and TB miliary disease, CUSUM multi-chart TCM was the quickest to detect changes followed by chart Tμ3d3, Tμ2d2, and Tμ1d1 in that respective order. Apparently, the size of shift in TB arthritis disease was medium, while the size of the shift was pretty large for TB meningitis and TB miliary.Early detection of upward abrupt changes in the diseases could send warning signals to public health workers to trigger public awareness, education, and general control of tuberculosis in the northern region as well as other regions of Ghana. Further research can consider how the procedures considered in the article may be modified or adapted using nonparametric monitoring methods and also methods that will account for dependence among the diseases in case there are strong correlations among the diseases. Also, other distributions like the negative binomial can be considered to account for the possible effect of overdispersion.



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