A modified exponentially weighted moving average (EWMA) scheme is one of the quality control charts such that this control chart can quickly detect a small shift. The average run length (ARL) is frequently used for the performance evaluation on control charts. This paper proposes the explicit formula for evaluating the average run length on a two-sided modified exponentially weighted moving average chart under the observations of a first-order autoregressive process, referred to as AR(1) process, with an exponential white noise. The performance comparison of the explicit formula and the numerical integral technique is carried out using the absolute relative change for checking the correct formula and the CPU time for testing speed of calculation. The results show that the ARL of the explicit formula and the numerical integral equation method are hardly different, but this explicit formula is much faster for calculating the ARL and offered accurate values. Furthermore, the cumulative sum, the classical EWMA and the modified EWMA control charts are compared and the results show that the latter is better for small and intermediate shift sizes. In addition, the explicit formula is successfully applied to real-world data in the health field as COVID-19 data in Thailand and Singapore.
Explicit formulaaverage run lengthmodified EWMA chartAR(1) processexponential white noiseIntroduction
Statistical analysis is important for confirming consistency and has received interest from many business areas requiring increased reliability. In the manufacturing industry, statistical process control is used extensively and a control chart is one of the quality control tools and has been applied in many fields such as finance [1], health [2] and medicine [3]. In 1924, Shewhart created the first control chart which can detect a large-sized process shift quickly [4]. Next, the exponentially weighted moving average (EWMA) chart [5] and the cumulative sum (CUSUM) chart [6] were developed and found to be faster for small shift detection. In addition, other new control charts have been developed, one of which is the modified EWMA control chart. Patel et al. [7] expanded the modified EWMA control chart from a fusion of the characteristics of the Shewhart and EWMA control charts that could detect a small size shift quickly and was more effective for autocorrelated data. The modified EWMA statistic is based on the classical EWMA statistic by using past observations with additional consideration of changes in the recent past for pairs of observations in the process. Next, Khan et al. [8] increased the performance of the modified EWMA scheme by maintaining a constant r in the last term of the modified EWMA statistic.
The average run length (ARL) [9] is the average number of in-control observations before an out-of-control signal occurs and is classified based on two stages: ARL at the initial mean (ARL0) and ARL when the process mean has shifted (ARL1). It is now a popular performance measure for a control chart and has been frequently used for research evaluation with many methods, such as Monte Carlo simulation, a Markov Chain approach, a Martingale approach, a numerical integral equation (NIE), and an explicit formula. All of these methods can be used to approximate the ARL but only the last one can be calculated using the least time and obtained the accuracy ARL. For research examples, Flury et al. [10] simulated the ARL to evaluate the multivariate EWMA control chart performance with highly asymmetric gamma distributions. The Markov Chain approach was used by Chananet et al. [11] for the ARL evaluation of EWMA and CUSUM control charts based on the zero-inflated negative binomial (ZINB) model. The Martingale approach was predicated by Sukparungsee [12] to approximate the ARL with optimal parameters of one and two-sided EWMA control chart.
There are many examples in the literature of using the explicit formula and the NIE method for the ARL on control charts. First, Suriyakat et al. [13] presented an explicit formula and a numerical method to solve the ARL on an EWMA control chart for a first-order autoregressive (AR(1)) process with an exponential white noise. Meanwhile, Busaba et al. [14] proposed numerical approximations of ARL on a CUSUM chart for an AR(1) model with exponential white noise. Next, Pecharat et al. [15] derived an explicit formula and a numerical integration scheme for the ARL on a CUSUM control chart for a moving average process of order q (MA(q)) with an exponential white noise. Later, Peerajit et al. [16] solved the NIEs for the ARL for a long memory process with non-seasonal and seasonal autoregressive fractionally integrated moving average (ARFIMA) models on a CUSUM control chart. After that, Sukparungsee et al. [17] analyzed explicit formulas of the ARL for an EWMA control chart using an autoregressive model. Moreover, Sunthornwat et al. [18] reported an explicit formula and a numerical technique to find the ARL on an EWMA control chart for a long memory ARFIMA process, and also presented the evaluation of ARL of an EWMA control chart for a long memory ARFIMA process with optimal parameters [19]. Recently, Peerajit et al. [20] introduced the ARL evaluation for CUSUM chart on a seasonal autoregressive fractionally integrated moving average (SARFIMA) process with an exponential white noise by using explicit analytical solutions.
Autoregressive models are often used on control charts in the recent literature [21–23]. The order of an autoregressive model is the number of immediate previous values used to predict the present value. A first-order autoregressive model considered in this research is an appropriate process for numerous data sources in real life such as environmental [24], physical [25] and industrial [26] data. Moreover, the modified EWMA control chart can be effectively used with an autocorrelated data.
In this paper, we present the explicit formula to evaluate the ARL on a two-sided modified EWMA control chart for the AR(1) process with an exponential white noise, which is the error term with an exponential distribution that has been studied recently [27–29]. In the next section, the AR(1) process features are shown and applied to a modified EWMA scheme. Following this, the ARL is calculated using the explicit formula and the NIE method, and after that, the performance of the schemes to detect shifts between the CUSUM, the classical EWMA and the modified EWMA control charts with simulated and real-world data are reported and compared.
Control Chart and ProcessClassical and Modified EWMA Control Charts
The modified exponentially weighted moving average (EWMA) control chart introduced by Patel et al. [7] and Khan et al. [8] were adjustments of the classical EWMA scheme. Let {Xt} for t=1,2,3,… be a sequence of random variables on the AR(1) process with mean μ and variance σ2, and λ is an exponential smoothing parameter, where 0<λ<1. The modified EWMA statistic with adjusted constant r is expressed as:
Zt=(1-λ)Zt-1+λXt+r(Xt-Xt-1);t=1,2,3,…
and the asymptotic variance of Zt is (λ+2λr+2r22-λ)σ2. From Eq. (1), we can show that
If r = 0, the classical EWMA statistic coincides with Zt=(1-λ)Zt-1+λXt.
If r = 1, the primal modified EWMA statistic coincides with Zt=(1-λ)Zt-1+λXt+(Xt-Xt-1).
The upper and lower control limits of the classical EWMA control chart are:
μ±Lσλ2-λ
and the bound control limits of the modified EWMA control chart are:
μ±Lmσλ+2λr+2r22-λ
where μ is the process mean, σ is the process standard deviation and L and Lm are suitable control width limits.
CUSUM Control Chart
The cumulative sum (CUSUM) chart has been widely used to detect small shifts similar to the EWMA chart on the control process and was initially proposed by Page [6]. Let {Xt} for t=1,2,3,… be a sequence of random variables on the AR(1) process with mean μ and variance σ2. The upper CUSUM statistic can be expressed by:
Zt= max{0,Zt-1+Xt-q};t=1,2,3,…
where q is usually called a reference value or a constant of CUSUM chart, Z0 = u is an initial value with u∈[0,b] and b is an upper control limit (UCL).
Modified EWMA Chart for the AR(1) Process
The equation of observations for the AR(1) process in the case of an exponential white noise is defined as:
Xt=η+ϕXt-1+εt
where Xt(t=1,2,3,…) is a sequence of random variables, η is a suitable constant, ϕ is an autoregressive coefficient (|ϕ|<1), and εt denotes white noise sequences of the exponential distribution (εt~Exp(β)).
The modified EWMA statistic under the upper bound assumption for the AR(1) process substituted from Eq. (5) into Eq. (1) can be arranged by recursion:
The ARL of the modified EWMA control chart with the AR(1) model is given by:
ARL=Eθ(τa,b)
where θ is the change-point time, Eθ(⋅) is the expectation under the assumption that the change-point occurs at time θ.
Integral Equation Method for Solving ARL
For analytical solutions, the ARL on the modified EWMA control chart with the AR(1) model is developed under the condition of a unique integral equation. L(u) is defined for an initial ARL and L(u)=E∞(τa,b). In accordance with the method of Champ et al. [30], L(u) can be found as:
L(u)=1+∫ a-(1-λ)u-(λϕ+rϕ-r)Xt-1(r+λ)-ηb-(1-λ)u-(λϕ+rϕ-r)Xt-1(r+λ)-ηL[(1-λ)u+(λϕ+rϕ-r)Xt-1+(r+λ)y+(r+λ)η]f(y)dy.
If k=(1-λ)u+(λϕ+rϕ-r)Xt-1+(r+λ)y+(r+λ)η is defined for changing the variable of integration, then L(u) is transformed as:
From Eq. (11), the existence and uniqueness of the solutions for the integral equation are proved by the mathematical theory. The integral equation for solving the ARL on the modified EWMA chart with the AR(1) process is proved that can be existed and has a unique solution by using Banach’s fixed point theory [31].
Theorem 1: Let X be a complete metric space (V,∥⋅∥). If Λ is a contraction mapping whereby Λ:X→X, then Λ has a unique fixed point x.
Theorem 2: Let X be a complete metric space (V,∥⋅∥). Λ:X→X is a contraction mapping if L is any vector in X and Λ has a Lipschitz constant c∈[0,1) such that L= limn→∞Λ(Ln) and
∥Λ(Lg)-Λ(Lh)∥≤c∥Lg-Lh∥for all Lg,Lh∈X.
Proof: For a first part, let Λ be a contraction mapping in the complete metric space (V,∥⋅∥) when L0∈V, and define a sequence {Ln}n∈N given by Ln+1=Λ(Ln) for n≥0. Show that L= limn→∞Λ(Ln) such that {Ln}n∈N is supposed to be Cauchy sequence, so
∥Ln+1-Ln∥=∥Λ(Ln)-Λ(Ln-1)∥≤c∥Ln-Ln-1∥≤cn∥L1-L0∥
where ∥L1-L0∥ is a finite number. After that, using this solution and the triangle inequality for all m>n≥N can be solved as:
∥Ln+m-Ln∥≤∑i=0m-1cn+i∥L1-L0∥≤cn1-c∥L1-L0∥<ε,
where ε>0 and c∈[0,1). Therefore, the sequence {Ln}n∈N is a Cauchy such that:
Λ(L)=Λ(limn→∞Ln)= limn→∞Λ(Ln)= limn→∞Ln+1=L.
Thus, L is called a fixed point of Λ.
Next part has to show that this fixed point is unique. Given Lg,Lh∈V in the complete metric space (V,∥⋅∥) for all u∈[a,b], then Eqs. (11) and (12) are considered as:
‖Λ(Lg)−Λ(Lh)‖∞=supu∈[a,b]|Lg(u)−Lh(u)|‖Λ(Lg)−Λ(Lh)‖∞≤supu∈[a,b]{‖Lg−Lh‖∞e(1−λ)uβ(r+λ)⋅e(λϕ+rϕ−r)Xt−1β(r+λ)⋅eηββ(r+λ)|∫abe−kβ(r+λ)dk|}‖Λ(Lg)−Λ(Lh)‖∞=supu∈[a,b]{e(1−λ)bβ(r+λ)⋅e(λϕ+rϕ−r)Xt−1β(r+λ)⋅eηβ(e−aβ(r+λ)−e−bβ(r+λ))}‖Lg−Lh‖∞‖Λ(Lg)−Λ(Lh)‖∞≤c‖Lg−Lh‖∞.
Hence, ∥Λ(Lg)-Λ(Lh)∥∞≤c∥Lg-Lh∥∞, where positive constant c∈[0,1) and Λ is the contraction mapping on a complete metric space (V,∥⋅∥). Therefore, the integral equation for solving of ARL on the modified EWMA chart with the AR(1) process has a unique solution.
Explicit Formula for ARL
The explicit analytical solutions of the ARL on the modified EWMA control chart with the AR(1) model constructed after checking for unique solutions are:
Proof: In the first step, the solution of Eq. (11) as: L(u)=1+e(1-λ)uβ(r+λ)⋅e(λϕ+rϕ-r)Xt-1β(r+λ)⋅eηββ(r+λ)∫ abL(k)⋅e-kβ(r+λ)dk is determined as: D=∫ abL(k)⋅e-kβ(r+λ)dk which is a constant and C(u)=e(1-λ)uβ(r+λ)⋅e(λϕ+rϕ-r)Xt-1β(r+λ)⋅eηβ; a≤u≤b such that it can be rewritten as:
L(u)=1+C(u)β(r+λ)⋅D.
Consider D and take turns L(k) with Eq. (14), then
Finally, by substituting constant D from Eq. (15) into Eq. (14), then L(u) can be found:
L(u)=1+C(u)β(r+λ)⋅DL(u)=1+e(1−λ)uβ(r+λ)⋅e(λϕ+rϕ−r)Xt−1β(r+λ)⋅eηββ(r+λ)(−β(r+λ)[e−bβ(r+λ)−e−aβ(r+λ)]1+1λ⋅e(λϕ+rϕ−r)Xt−1β(r+λ)⋅eηβ[e−λbβ(r+λ)−e−λaβ(r+λ)])L(u)=1−λe(1−λ)uβ(r+λ)⋅e(λϕ+rϕ−r)Xt−1β(r+λ)⋅eηβ[e−bβ(r+λ)−e−aβ(r+λ)]λ+e(λϕ+rϕ−r)Xt−1β(r+λ)⋅eηβ[e−λbβ(r+λ)−e−λaβ(r+λ)]L(u)=1−λe(1−λ)uβ(r+λ)[e−bβ(r+λ)−e−aβ(r+λ)]λe−(λϕ+rϕ−r)Xt−1β(r+λ)⋅e−ηβ+[e−λbβ(r+λ)−e−λaβ(r+λ)]L(u)=1−λe(1−λ)uβ(r+λ)(e−bβ(r+λ)−e−aβ(r+λ))λe−(λϕ+rϕ−r)Xt−1+(r+λ)ηβ(r+λ)+e−λbβ(r+λ)−e−λaβ(r+λ)
Therefore, the solution to Eq. (13) is obtained.
For an in-control process, the exponential parameter is set to β=β0 and the explicit formula with β0 called ARL0 can be written as:
On the other hand, the exponential parameter for an out-of-control process is defined as β1=(1+δ)β0, where β1>β0 and δ is the shift size such that the explicit formula of β1 can be described as:
Numerical Method for Solving the Integral Equation
The NIE method is used to solve the ARL for the AR(1) process of the two-sided modified EWMA control chart in Eq. (10). The ARL solution or LNIE(u) is approximated with the m linear equation systems over the interval [a, b] by using the composite midpoint quadrature rule [32]. This NIE method is proposed for the two-sided modified EWMA control chart by using the length of m equal divided intervals wj=b-am and the middle point of the jth interval xj=(j-12)wj+a. Therefore, the ARL solution of the NIE method can be rewritten as:
In this section, the performances of the explicit formula and the NIE method are compared by using the ARL solutions such that the NIE method determines the number of division points m = 1,000. ARL0 of 370 is used for the experiments with the classical and modified EWMA control charts and the AR(1) process; a lower ARL value signifies better effective detection. For the in-control process, the initial parameter values β0=1, u = 1, X0 = 1 as this process mean are determined. In addition, the process mean is tested with shift sizes (δ) of 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.08, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.80 and 1.00. The optimization procedure in Fig. 1 can be summarized as follows:
The summarized processing diagram
Step 1: Specify λ, ϕ, η, r and ARL0.
Step 2: Determine the initial values of the process mean on an exponential distribution as: β0, u, X0.
Step 3: Compute b when a is known by using Eq. (13) for the explicit formula or Eq. (18) for the NIE method.
Step 4: Compute ARL1 from the b solution in Step 3 by shifting mean (δ) where β1=(1+δ)β0.
The efficiency comparison of the ARL between the explicit formula and the NIE method is measured using the absolute relative change (ARC) [33] which can be calculated as:
ARC(%)=|L(u)-LNIE(u)L(u)|×100%.
In addition, the speed test results are computed by the CPU time (PC System: Windows 10 Education, i7-6500U CPU@2.50 GHz Processor, 8.00 GB RAM, 64-bit Operating System) in seconds.
The performance comparison of the explicit formula and the NIE method is explained with the ARL, the ARC and the CPU time (Tab. 1). The first results indicate that the ARL values of the explicit formula are similar to those of the NIE method according to the ARC criterion such that ARC solutions are very low and converge to 0. For the CPU time for calculating the ARL, the explicit formula is faster than the NIE method by around 9 s.
Comparison of ARL values on the modified EWMA control chart with the AR(1) model using explicit formulas against the NIE method given <inline-formula id="ieqn-89"><!--<alternatives><inline-graphic xlink:href="ieqn-89.png"/><tex-math id="tex-ieqn-89"><![CDATA[$ \lambda = 0.05$]]></tex-math>--><mml:math id="mml-ieqn-89"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>05</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-90"><!--<alternatives><inline-graphic xlink:href="ieqn-90.png"/><tex-math id="tex-ieqn-90"><![CDATA[$ \eta = 2$]]></tex-math>--><mml:math id="mml-ieqn-90"><mml:mi>η</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <italic>r</italic> = 1, <italic>a</italic> = 0 at <inline-formula id="ieqn-91"><!--<alternatives><inline-graphic xlink:href="ieqn-91.png"/><tex-math id="tex-ieqn-91"><![CDATA[$ \phi = 0.1$]]></tex-math>--><mml:math id="mml-ieqn-91"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:math><!--</alternatives>--></inline-formula> (<italic>b</italic> = 0.333987011), <inline-formula id="ieqn-92"><!--<alternatives><inline-graphic xlink:href="ieqn-92.png"/><tex-math id="tex-ieqn-92"><![CDATA[$ \phi = - 0.1$]]></tex-math>--><mml:math id="mml-ieqn-92"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>1</mml:mn></mml:math><!--</alternatives>--></inline-formula> (<italic>b</italic> = 0.408730497) for <inline-formula id="ieqn-93"><!--<alternatives><inline-graphic xlink:href="ieqn-93.png"/><tex-math id="tex-ieqn-93"><![CDATA[$\mathrm{ARL}_{0}= 370$]]></tex-math>--><mml:math id="mml-ieqn-93"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>A</mml:mi><mml:mi>R</mml:mi><mml:mi>L</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>370</mml:mn></mml:math><!--</alternatives>--></inline-formula>
Shift size (δ)
ϕ=0.1
ϕ=-0.1
Explicit (CPU time)
NIE (CPU time)
ARC (%)
Explicit (CPU time)
NIE (CPU time)
ARC (%)
0.00
370.00008812
370.00008589
6.03×10-07
370.00004893
370.00004557
9.08×10-07
(< 0.001)
(9.281)
(< 0.001)
(9.359)
0.01
78.37858370
78.37858335
4.11×10-07
82.65057512
82.65057456
6.78×10-07
(< 0.001)
(8.969)
(< 0.001)
(9.312)
0.02
43.82837788
43.82837770
4.11×10-07
46.53183718
46.53183689
6.23×10-07
(< 0.001)
(10.313)
(< 0.001)
(10.453)
0.03
30.42557907
30.42557895
3.94×10-07
32.39371610
32.39371591
5.87×10-07
(< 0.001)
(9.281)
(< 0.001)
(9.328)
0.04
23.30847758
23.30847749
3.86×10-07
24.85646420
24.85646406
5.63×10-07
(< 0.001)
(10.468)
(< 0.001)
(10.328)
0.05
18.89793321
18.89793314
3.70×10-07
20.17481918
20.17481907
5.45×10-07
(< 0.001)
(9.344)
(< 0.001)
(9.547)
0.06
15.89855292
15.89855286
3.77×10-07
16.986143197
16.986143105
5.42×10-07
(< 0.001)
(10.438)
(< 0.001)
(10.156)
0.08
12.08490680
12.08490676
3.31×10-07
12.925534561
12.925534495
5.11×10-07
(< 0.001)
(9.406)
(< 0.001)
(9.500)
0.10
9.765566083
9.765566052
3.17×10-07
10.452061817
10.452061767
4.78×10-07
(< 0.001)
(9.562)
(< 0.001)
(9.468)
0.20
5.090970157
5.090970145
2.36×10-07
5.453366997
5.453366978
3.48×10-07
(< 0.001)
(10.392)
(< 0.001)
(10.515)
0.30
3.554052055
3.554052049
0.69×10-07
3.802097396
3.802097385
2.89×10-07
(< 0.001)
(9.546)
(< 0.001)
(9.500)
0.40
2.807071681
2.807071677
1.42×10-07
2.996034011
2.996034005
2.00×10-07
(< 0.001)
(10.063)
(< 0.001)
(10.422)
0.50
2.373151393
2.373151390
1.26×10-07
2.525809691
2.525809687
1.58×10-07
(< 0.001)
(9.515)
(< 0.001)
(9.391)
0.60
2.093319852
2.093319850
9.55×10-08
2.221328911
2.221328908
1.35×10-07
(< 0.001)
(10.219)
(< 0.001)
(10.109)
0.80
1.759313098
1.759313097
5.68×10-08
1.855915275
1.855915274
5.39×10-08
(< 0.001)
(10.500)
(< 0.001)
(10.079)
1.00
1.570797672
1.570797671
6.37×10-08
1.648205459
1.648205458
6.07×10-08
(< 0.001)
(9.657)
(< 0.001)
(9.484)
The performance of the modified EWMA control chart on varying scales of constant r, different bound control limits [a, b] and various λ are further tested by using the relative mean index (RMI) [34] which can be written as:
RMI=1n∑i=1n[ARLi(c)-ARLi(s)ARLi(s)]
where ARLi(c) is the ARL of the control chart for the shift size of row i, ARLi(s) is the smallest ARL of all of the control charts on row i and the RMI value is smaller, then the control chart has more performance for detecting shifts.
The CUSUM, the classical EWMA (r = 0), the classical modified EWMA (r = 1) and the modified EWMA control charts with adjusted r = 0.5 and r = 2 measured a capability by using the ARL and the RMI at ϕ=0.2, −0.2 for ARL0=370 are compared and reported in Tab. 2 and Fig. 2. When the process mean is shifted, the ARL results of the modified EWMA control charts have abrupt decrease and lower values in small and intermediate shifts. For large shifts, the classical EWMA chart is obtained the least ARL. Therefore, the performance of modified EWMA control charts is better than the classical EWMA scheme for small and intermediate shifts. Moreover, the results show that the CUSUM chart has lower performance than modified EWMA control charts for all levels of shifts. From ARL and RMI comparisons, the modified EWMA control charts with higher r values are more effective.
Comparison of the ARL values between the CUSUM, the classical EWMA and modified EWMA control charts given <inline-formula id="ieqn-129"><!--<alternatives><inline-graphic xlink:href="ieqn-129.png"/><tex-math id="tex-ieqn-129"><![CDATA[$ \lambda = 0.05$]]></tex-math>--><mml:math id="mml-ieqn-129"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>05</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-130"><!--<alternatives><inline-graphic xlink:href="ieqn-130.png"/><tex-math id="tex-ieqn-130"><![CDATA[$ \eta = 2$]]></tex-math>--><mml:math id="mml-ieqn-130"><mml:mi>η</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <italic>a</italic> = 0 at <inline-formula id="ieqn-131"><!--<alternatives><inline-graphic xlink:href="ieqn-131.png"/><tex-math id="tex-ieqn-131"><![CDATA[$\mathrm{ARL}_{0}= 370$]]></tex-math>--><mml:math id="mml-ieqn-131"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>A</mml:mi><mml:mi>R</mml:mi><mml:mi>L</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>370</mml:mn></mml:math><!--</alternatives>--></inline-formula>
ϕ
Shift size (δ)
CUSUM (q = 4)
EWMA (r = 0)
Modified EWMA
Adjusted r = 0.5
Basic r = 1
Adjusted r = 2
(b = 5.45278)
(b=1.145388×10-8)
(b = 0.150278601)
(b = 0.301950105)
(b = 0.604752895)
0.2
0.00
370.000
370.000
370.000
370.000
370.000
0.01
338.746
297.174
134.052
76.388
53.985
0.02
310.682
239.724
80.967
42.581
29.372
0.03
285.440
194.207
57.568
29.521
20.296
0.04
262.698
157.991
44.414
22.599
15.576
0.05
242.174
129.058
35.997
18.313
12.684
0.06
223.623
105.850
30.157
15.401
10.731
0.08
191.601
72.056
22.602
11.701
8.262
0.10
165.199
49.824
17.945
9.452
6.768
0.20
85.904
9.981
8.493
4.926
3.761
0.30
50.819
3.130
5.435
3.442
2.765
0.40
33.462
1.617
3.992
2.722
2.275
0.50
24.017
1.210
3.180
2.304
1.987
0.60
18.436
1.081
2.673
2.036
1.799
0.80
12.507
1.017
2.091
1.716
1.570
1.00
9.553
1.005
1.780
1.536
1.437
RMI
15.058
3.955
1.411
0.481
0.168
(b = 4.150138)
(b=1.70872×10-8)
(b = 0.225127154)
(b = 0.452229145)
(b = 0.905706536)
−0.2
0.00
370.000
370.000
370.000
370.000
370.000
0.01
345.684
298.349
145.780
84.954
60.565
0.02
323.398
241.604
89.916
48.004
33.268
0.03
302.945
196.471
64.581
33.470
23.068
0.04
284.146
160.426
50.138
25.705
17.735
0.05
266.843
131.521
40.817
20.876
14.456
0.06
250.895
108.251
34.313
17.584
12.238
0.08
222.579
74.193
25.848
13.388
9.427
0.10
198.341
51.632
20.599
10.830
7.722
0.20
118.297
10.600
9.850
5.654
4.277
0.30
76.717
3.336
6.325
3.940
3.128
0.40
53.181
1.692
4.643
3.101
2.558
0.50
38.902
1.240
3.687
2.611
2.220
0.60
29.728
1.095
3.084
2.293
1.998
0.80
19.212
1.020
2.385
1.910
1.725
1.00
13.717
1.006
2.005
1.692
1.565
RMI
18.278
3.487
1.498
0.539
0.211
Plot of the CUSUM, the classical EWMA and modified EWMA control charts given (a) <inline-formula id="ieqn-139"><!--<alternatives><inline-graphic xlink:href="ieqn-139.png"/><tex-math id="tex-ieqn-139"><![CDATA[$ \phi = 0.2$]]></tex-math>--><mml:math id="mml-ieqn-139"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>2</mml:mn></mml:math><!--</alternatives>--></inline-formula> and (b) <inline-formula id="ieqn-140"><!--<alternatives><inline-graphic xlink:href="ieqn-140.png"/><tex-math id="tex-ieqn-140"><![CDATA[$ \phi = - 0.2$]]></tex-math>--><mml:math id="mml-ieqn-140"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>2</mml:mn></mml:math><!--</alternatives>--></inline-formula>
For Tab. 3, the ARL results with ARL0=370 show the performance of the modified EWMA control chart on various bound control limits [a, b] for ϕ=0.3, −0.3 and a = 0, 0.1, 0.3, 0.4. The RMI values of the lower bound a = 0.4 are 0. When the lower bound (a) is higher, then the modified EWMA control chart can detect better shifts when comparing the ARL and the RMI values in one direction.
Comparison of the ARL values on the modified EWMA control chart with difference control bounds given <inline-formula id="ieqn-143"><!--<alternatives><inline-graphic xlink:href="ieqn-143.png"/><tex-math id="tex-ieqn-143"><![CDATA[$ \lambda = 0.05$]]></tex-math>--><mml:math id="mml-ieqn-143"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>05</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-144"><!--<alternatives><inline-graphic xlink:href="ieqn-144.png"/><tex-math id="tex-ieqn-144"><![CDATA[$ \eta = 2$]]></tex-math>--><mml:math id="mml-ieqn-144"><mml:mi>η</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <italic>r</italic> = 1 at <inline-formula id="ieqn-145"><!--<alternatives><inline-graphic xlink:href="ieqn-145.png"/><tex-math id="tex-ieqn-145"><![CDATA[$\mathrm{ARL}_{0}= 370$]]></tex-math>--><mml:math id="mml-ieqn-145"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>A</mml:mi><mml:mi>R</mml:mi><mml:mi>L</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>370</mml:mn></mml:math><!--</alternatives>--></inline-formula>
ϕ
Shift size (δ)
a = 0
a = 0.1
a = 0.3
a = 0.4
(b = 0.273008016)
(b = 0.374461655)
(b = 0.57735612)
(b = 0.67879871)
0.3
0.00
370.000
370.000
370.000
370.000
0.01
74.484
69.483
60.267
56.043
0.02
41.394
38.391
32.967
30.528
0.03
28.663
26.553
22.770
21.079
0.04
21.926
20.317
17.440
16.158
0.05
17.760
16.469
14.165
13.140
0.06
14.930
13.860
11.951
11.101
0.08
11.338
10.552
9.148
8.523
0.10
9.157
8.544
7.450
6.962
0.20
4.771
4.509
4.037
3.825
0.30
3.336
3.186
2.914
2.790
0.40
2.642
2.544
2.365
2.283
0.50
2.240
2.171
2.045
1.986
0.60
1.982
1.931
1.837
1.793
0.80
1.676
1.645
1.587
1.560
1.00
1.504
1.484
1.445
1.427
RMI
0.232
0.168
0.052
0
(b = 0.500416482)
(b = 0.603072843)
(b = 0.80836715)
(b = 0.911008116)
−0.3
0.00
370.000
370.000
370.000
370.000
0.01
87.389
81.769
71.317
66.487
0.02
49.572
46.055
39.668
36.781
0.03
34.619
32.105
27.577
25.546
0.04
26.612
24.674
21.196
19.641
0.05
21.626
20.059
17.254
16.001
0.06
18.224
16.916
14.577
13.534
0.08
13.884
12.913
11.178
10.404
0.10
11.236
10.474
9.110
8.501
0.20
5.869
5.533
4.927
4.654
0.30
4.087
3.890
3.532
3.370
0.40
3.214
3.083
2.843
2.733
0.50
2.702
2.608
2.436
2.356
0.60
2.370
2.299
2.168
2.108
0.80
1.968
1.924
1.842
1.804
1.00
1.739
1.709
1.652
1.626
RMI
0.238
0.173
0.054
0
Moreover, the modified EWMA control chart is compared for various λ = 0.01, 0.05, 0.10 and 0.20 at ARL0=370, a = 0.4 and ϕ=0.3, −0.3 (Tab. 4). The RMI values of λ=0.20 are 0. The ARL and the RMI results show that more λ values affect for increasing a detected performance of the modified EWMA control chart.
Comparison of the ARL values on the modified EWMA control chart with various <inline-formula id="ieqn-150"><!--<alternatives><inline-graphic xlink:href="ieqn-150.png"/><tex-math id="tex-ieqn-150"><![CDATA[$ \lambda $]]></tex-math>--><mml:math id="mml-ieqn-150"><mml:mi>λ</mml:mi></mml:math><!--</alternatives>--></inline-formula> given <inline-formula id="ieqn-151"><!--<alternatives><inline-graphic xlink:href="ieqn-151.png"/><tex-math id="tex-ieqn-151"><![CDATA[$ \eta = 2$]]></tex-math>--><mml:math id="mml-ieqn-151"><mml:mi>η</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <italic>r</italic> = 1, <italic>a</italic> = 0.4 at <inline-formula id="ieqn-152"><!--<alternatives><inline-graphic xlink:href="ieqn-152.png"/><tex-math id="tex-ieqn-152"><![CDATA[$\mathrm{ARL}_{0}= 370$]]></tex-math>--><mml:math id="mml-ieqn-152"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>A</mml:mi><mml:mi>R</mml:mi><mml:mi>L</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>370</mml:mn></mml:math><!--</alternatives>--></inline-formula>
ϕ
Shift size (δ)
λ=0.01
λ=0.05
λ=0.10
λ=0.20
(b = 0.672828274)
(b = 0.67879871)
(b = 0.686452008)
(b = 0.702326332)
0.3
0.00
370.000
370.000
370.000
370.000
0.01
58.170
56.043
53.700
49.853
0.02
31.757
30.528
29.182
26.993
0.03
21.934
21.079
20.146
18.632
0.04
16.808
16.158
15.448
14.299
0.05
13.662
13.140
12.571
11.649
0.06
11.535
11.101
10.627
9.861
0.08
8.844
8.523
8.172
7.605
0.10
7.215
6.962
6.686
6.240
0.20
3.940
3.825
3.700
3.498
0.30
2.859
2.790
2.715
2.592
0.40
2.331
2.283
2.231
2.147
0.50
2.022
1.986
1.948
1.885
0.60
1.821
1.793
1.763
1.714
0.80
1.578
1.560
1.540
1.508
1.00
1.440
1.427
1.412
1.388
RMI
0.118
0.088
0.0542
0
(b = 0.897887577)
(b = 0.911008116)
(b = 0.92785533)
(b = 0.962983493)
−0.3
0.00
370.000
370.000
370.000
370.000
0.01
68.657
66.487
64.109
60.251
0.02
38.078
36.781
35.370
33.100
0.03
26.460
25.546
24.553
22.962
0.04
20.343
19.641
18.880
17.662
0.05
16.568
16.001
15.387
14.404
0.06
14.008
13.534
13.021
12.200
0.08
10.758
10.404
10.020
9.407
0.10
8.782
8.501
8.198
7.713
0.20
4.784
4.654
4.513
4.288
0.30
3.450
3.370
3.283
3.143
0.40
2.789
2.733
2.672
2.575
0.50
2.398
2.356
2.310
2.237
0.60
2.141
2.108
2.072
2.014
0.80
1.826
1.804
1.779
1.740
1.00
1.642
1.626
1.608
1.579
RMI
0.106
0.078
0.048
0
Application to Real Data
As an example, the modified EWMA control chart is applied to COVID-19 data in Thailand and Singapore [35,36] such that the 100 days of newly infected cases are studied when the more summation of 100 cases. These data are checked to a suitable AR(1) process with an exponential white noise at parameters of COVID-19 data in Thailand β0=34.78, η=10.06, ϕ=0.664 and Singapore β0=159.47, η=48.99, ϕ=0.863. Moreover, their error term have be tested an exponential distribution and plotted graphs in Figs. 3 and 4, respectively.
Plot of the noise residual of COVID-19 data in Thailand
Plot of the noise residual of COVID-19 data in Singapore
From Tabs. 5 and 6, the results of ARL and the RMI are used to compare the CUSUM, the classical EWMA and modified EWMA control charts with real-world data of COVID-19 data in Thailand and Singapore. The results are in accordance with the simulation data in Tab. 2 and show that the modified EWMA control chart adjusted for high r performs well for small and intermediate level shifts.
Comparison of ARL values between the CUSUM, the classical EWMA and modified EWMA control charts for COVID-19 data in Thailand with <inline-formula id="ieqn-170"><!--<alternatives><inline-graphic xlink:href="ieqn-170.png"/><tex-math id="tex-ieqn-170"><![CDATA[$ \beta_{0}= 34.78$]]></tex-math>--><mml:math id="mml-ieqn-170"><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>34</mml:mn><mml:mo>.</mml:mo><mml:mn>78</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-171"><!--<alternatives><inline-graphic xlink:href="ieqn-171.png"/><tex-math id="tex-ieqn-171"><![CDATA[$ \eta = 10.06$]]></tex-math>--><mml:math id="mml-ieqn-171"><mml:mi>η</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn><mml:mo>.</mml:mo><mml:mn>06</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-172"><!--<alternatives><inline-graphic xlink:href="ieqn-172.png"/><tex-math id="tex-ieqn-172"><![CDATA[$ \phi = 0.664$]]></tex-math>--><mml:math id="mml-ieqn-172"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>664</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <italic>a</italic> = 0 at <inline-formula id="ieqn-173"><!--<alternatives><inline-graphic xlink:href="ieqn-173.png"/><tex-math id="tex-ieqn-173"><![CDATA[$ \lambda = 0.05$]]></tex-math>--><mml:math id="mml-ieqn-173"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>05</mml:mn></mml:math><!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-174"><!--<alternatives><inline-graphic xlink:href="ieqn-174.png"/><tex-math id="tex-ieqn-174"><![CDATA[$\mathrm{ARL}_{0}= 370$]]></tex-math>--><mml:math id="mml-ieqn-174"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>A</mml:mi><mml:mi>R</mml:mi><mml:mi>L</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>370</mml:mn></mml:math><!--</alternatives>--></inline-formula>
Shift size (δ)
CUSUM (q = 100)
EWMA (r = 0)
Modified EWMA
Adjusted r = 0.5
Basic r = 1
Adjusted r = 2
(b = 167.3)
(b=1.386×10-6)
(b = 18.5648)
(b = 37.2657)
(b = 74.6315)
0.00
370
370
370
370
370
0.01
343.476
300.868
180.287
113.091
83.291
0.02
319.132
245.645
118.387
66.819
47.287
0.03
296.959
201.362
87.706
47.459
33.189
0.04
276.733
165.712
69.400
36.826
25.669
0.05
258.253
136.898
57.248
30.105
20.993
0.06
241.345
113.522
48.602
25.475
17.805
0.08
211.636
78.935
37.134
19.512
13.738
0.10
186.554
55.686
29.889
15.841
11.253
0.20
106.523
12.056
14.660
8.301
6.181
0.30
67.202
3.840
9.494
5.751
4.462
0.40
45.918
1.881
6.971
4.483
3.598
0.50
33.445
1.318
5.507
3.729
3.078
0.60
25.638
1.130
4.566
3.233
2.732
0.80
16.897
1.029
3.452
2.624
2.298
1.00
12.397
1.009
2.828
2.267
2.039
RMI
13.085
2.367
1.797
0.752
0.380
Comparison of ARL values between the CUSUM, the classical EWMA and modified EWMA control charts for COVID-19 data in Singapore with <inline-formula id="ieqn-177"><!--<alternatives><inline-graphic xlink:href="ieqn-177.png"/><tex-math id="tex-ieqn-177"><![CDATA[$ \beta_{0}= 159.47$]]></tex-math>--><mml:math id="mml-ieqn-177"><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>159</mml:mn><mml:mo>.</mml:mo><mml:mn>47</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-178"><!--<alternatives><inline-graphic xlink:href="ieqn-178.png"/><tex-math id="tex-ieqn-178"><![CDATA[$ \eta = 48.99$]]></tex-math>--><mml:math id="mml-ieqn-178"><mml:mi>η</mml:mi><mml:mo>=</mml:mo><mml:mn>48</mml:mn><mml:mo>.</mml:mo><mml:mn>99</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-179"><!--<alternatives><inline-graphic xlink:href="ieqn-179.png"/><tex-math id="tex-ieqn-179"><![CDATA[$ \phi = 0.863$]]></tex-math>--><mml:math id="mml-ieqn-179"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>863</mml:mn></mml:math><!--</alternatives>--></inline-formula>, <italic>a</italic> = 0 at <inline-formula id="ieqn-180"><!--<alternatives><inline-graphic xlink:href="ieqn-180.png"/><tex-math id="tex-ieqn-180"><![CDATA[$ \lambda = 0.05$]]></tex-math>--><mml:math id="mml-ieqn-180"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo><mml:mn>05</mml:mn></mml:math><!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-181"><!--<alternatives><inline-graphic xlink:href="ieqn-181.png"/><tex-math id="tex-ieqn-181"><![CDATA[$\mathrm{ARL}_{0}= 370$]]></tex-math>--><mml:math id="mml-ieqn-181"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>A</mml:mi><mml:mi>R</mml:mi><mml:mi>L</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>370</mml:mn></mml:math><!--</alternatives>--></inline-formula>
Shift size (δ)
CUSUM (q = 500)
EWMA (r = 0)
Modified EWMA
Adjusted r = 0.5
Basic r = 1
Adjusted r = 2
(b = 745.43)
(b=5.115×10-6)
(b = 68.1366)
(b = 136.8)
(b = 273.966)
0.00
370
370
370
370
370
0.01
343.887
300.195
169.332
103.606
75.401
0.02
320.094
244.582
108.987
60.300
42.324
0.03
298.381
200.081
79.937
42.565
29.584
0.04
278.536
164.327
62.867
32.917
22.836
0.05
260.371
135.488
51.644
26.854
18.657
0.06
243.720
112.138
43.711
22.693
15.816
0.08
214.385
77.685
33.255
17.353
12.200
0.10
189.532
54.613
26.691
14.075
9.995
0.20
109.528
11.662
13.006
7.375
5.509
0.30
69.646
3.701
8.408
5.122
3.995
0.40
47.802
1.828
6.175
4.005
3.236
0.50
34.882
1.296
4.886
3.344
2.781
0.60
26.737
1.120
4.062
2.910
2.478
0.80
17.558
1.026
3.089
2.379
2.101
1.00
12.812
1.008
2.549
2.071
1.876
RMI
14.423
2.689
1.697
0.678
0.320
In Fig. 5, the modified EWMA control chart with r = 2 and the classical EWMA control chart are plotted by calculating Zt of COVID-19 data in Thailand and Singapore at the exponential smoothing parameter λ=0.05 with the optimal control width limit of EWMA chart L = 2.615 [37] and the calculated control width limit of the modified EWMA chart Lm = 0.204 at bound control limits of Thailand [10.92, 49.84] and bound control limits of Singapore [519.16, 237]. The results show that Zt values of the modified EWMA control chart with r = 2 exceed the bound since the 4th observation, while Zt values of the classical EWMA control chart are the out-of-control limit at the 9th observation for Thailand. In Singapore, Zt values of the modified EWMA control chart with r = 2 exceed the bound since the first observation, while the classical EWMA control chart signals an alarm in the 10th observation. Therefore, the modified EWMA control chart can detect shifts more quickly than the classical EWMA control chart.
Plot of the classical EWMA and modified EWMA control chart with <italic>r</italic> = 2 for COVID-19 data in (a) Thailand and (b) Singapore
Discussion and Conclusions
The performance of a control chart can be evaluated by using the ARL. In this paper, the explicit formula and the numerical integral equation (NIE) method of ARL solutions are established on a two-sided modified EWMA control chart for an AR(1) process with an exponential white noise. The ARL results of both methods are computed and their performances compared via the ARC and the CPU time. The explicit formula shows the actual values of the ARL and is faster calculation than the NIE approach. Moreover, the ARL and RMI results are compared between the CUSUM, the classical and modified EWMA control charts with various r, for which the latter provides better detection for small and intermediate shifts. Next, the performance of the modified EWMA control chart is tested on various bound control limits and is found to be better for higher upper bound values. In addition, this model is applied to the real-world data (COVID-19 data in Thailand and Singapore), with which it obtains similar results as with the simulated data; this result supports the excellent performance of the modified EWMA control chart. In future research, we will establish the optimal bound control limits of the modified EWMA control chart and hope to extend our approach to many new control charts currently under development for different processes.
Funding Statement: The research was supported by King Mongkut’s University of Technology North Bangkok Contract No. KMUTNB-62-KNOW-018.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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