Extended Exponentially Weighted Moving Average (Extended EWMA or EEWMA) control chart is one of the control charts which can quickly detect a small shift. The average run length (ARL) measures the performance of control chart. Due to the derivation of the explicit formulas for ARL on the EEWMA control chart for the autoregressive AR(p) process has not previously been reported. The aim of the article is to derive explicit formulas of ARL using a Fredholm integral equation of the second kind on EEWMA control chart for Autoregressive process, as AR(2) and AR(3) processes with exponential white noise. The accuracy of the solution obtained with the EEWMA control chart was compared to the numerical integral equation (NIE) method and extended to compare performance with Cumulative Sum (CUSUM) and Exponentially Weighted Moving Average (EWMA) control charts. The results show that the ARL obtained by the explicit formula and the NIE method are hardly different but ARL of explicit formula is less the computational (CPU) time than ARL of NIE method. The performance of EEWMA control chart is better than the CUSUM and EWMA control charts for all situations except when the large shift sizes the EEWMA control chart performed as well as the EWMA control chart for AR(2) and AR(3) processes. And then, the EEWMA control chart is also extended to compare efficiency of EEWMA control chart with various λ. An exponential smoothing parameter of 0.05 is recommended. In addition, the simulation study, and efficacy illustration with real data on new COVID-19 cases in Thailand and Vietnam provided similar results.
Extended EWMA control chartautoregressive processaverage run lengthexplicit formulaIntroduction
Currently, the statistical process control (SPC) is very important in the manufacturing industry for monitoring, controlling, and improving processes. Control charts are one of the efficient tools of SPC and have been applied in many fields such as finance [1], health [2], and medicine [3]. The Shewhart control chart was the first to be reported and is widely used for detecting large changes in a process mean [4]. Subsequently, the Cumulative Sum (CUSUM) chart [5] and the Exponentially Weighted Moving Average (EWMA) chart [6] have been widely employed to monitor a process mean due to their excellent performance in detecting small to moderate mean shifts. In addition, Patel et al. [7]-proposed the Modified Exponentially Weighted Moving Average (Modified EWMA) chart that is effective at detecting small size shift quickly for observations both autocorrelation and independently normally distribution. Later, Neveed et al. [8] proposed the Extended Exponentially Weighted Moving Average (EEWMA) chart that performed better than other control charts for detecting small shifts in the mean of a monitored process.
The performance of the chart is measured by Average Run Length (ARL). The ARL0 denote the average number of observations before an in-control process is taken to signal to be out of control and should be large whereas the ARL1 denote the average number of observations taken from out of control and should be as small as possible.
Many methods for evaluating ARL for control charts have been studied. For example, Monte Carlo simulations (MC), Markov Chain approach (MCA), Martingale approach (MA) and Numerical Integral Equation approach (NIE) and explicit formulas. Mastrangelo et al. [9] evaluated ARL of the traditional EWMA chart for serially correlated processes by using the Monte Carlo simulation method. Zhang et al. [10] proposed the ARL of the multivariate exponentially weighted moving average (MEWMA) chart and the combined control chart were evaluated with Monte Carlo simulation. Sukparungsee [11] approximated the ARL with optimal parameters of one and two-sided EWMA control chart using by Martingale approach. Chananet et al. [12] evaluated the ARL of EWMA and CUSUM control charts with Markov Chain approach based on the zero-inflated negative binomial (ZINB) model.
Many literatures for evaluating the ARL using NIE method and explicit formula have been studied. Areepong et al. [13] proposed the ARL using the numerical integral equation approach of the EWMA chart and compared the results with the Monte Carlo simulation method. Khoo et al. [14] presented a Markov chain approach for computing the ARL of EWMA charts. Moreover, Phanyaem et al. [15] derived the ARL for ARMA processes via explicit formula and numerical integral equation (NIE) method of EWMA chart. Petcharat et al. [16] investigated the derivation of the ARL for moving average order q process with exponential white noise by explicit formula. After that, Peerajit et al. [17] studied the NIE method of ARL on CUSUM chart. Supharakonsakun et al. [18] evaluated the ARL by NIE method on modified EWMA and compared efficiency with EWMA control chart. Sunthornwat et al. [19] derived explicit formulas of ARL on CUSUM chart for seasonal and non-seasonal moving average processes with exogenous variables and evaluated against the NIE method. Later, Anwar et al. [20] proposed modified-mxEWMA chart that performs very well for the monitoring of small to moderate shifts in the process and show the implementation of the wood industry. Saghir et al. [21] proposed modified EWMA chart and the performance is evaluated by ARL. Aslam et al. [22] proposed new Bayesian Modified-EWMA chart and its applications in mechanical and sport industry. Karoon et al. [23] developed the numerical integral equation (NIE) methods for evaluating the ARL on Extended EWMA chart for AR(p) process. Supharakonsakun et al. [24] presented the exact average run length based on explicit formula the observations are from moving average process with exponential white noise for modified EWMA chart. Phanthuna et al. [25] proposed the explicit formula for evaluating the ARL on a two-sided modified EWMA chart under the observations of AR(1) process. Recently, Phanthuna et al. [26] presented explicit formula of ARL for modified EWMA chart with autoregressive model involving exponential white noise.
However, the derivation of the explicit formulas for ARL on the EEWMA chart for autoregressive AR(p) process has not previously been reported. Therefore, the aim of this study is to derive explicit formulas of the ARL on the EEWMA control chart for AR(p) process, as AR(2) and AR(3) processes with exponential white noise. The explicit formulas for ARL were compared with the (NIE) method as the benchmark. Besides, the performance of the explicit formulas for deriving the ARL on the EEWMA chart was compared with those on the CUSUM and EWMA charts for both simulated data and real-world data reported.
Materials and MethodsCumulative Sum (CUSUM) Control Chart
The CUSUM control chart was originally introduced by Page [5] in quality control to detect small changes in process mean, as an extension of Shewhart control chart. The CUSUM control chart can be expressed by the recursive equation below.Ct=max(0,Ct−1+Xt−a),t=1,2,…where a is non-zero constant, C0 is the initial value of CUSUM statistics with u ∈ [0, b], C0 = u.
The stopping time of the CUSUM control chart is given byτb=inf{t>0;Ct>b},b>uwhere τb is the stopping time, b is upper control limit (UCL).
Exponentially Weighted Moving Average (EWMA) Control Chart
The EWMA control chart was initially proposed by Robert [6]. It is usually used to monitor and detect small changes in process mean. The EWMA control chart can be expressed by the recursive equation below.Zt=(1−λ)Zt−1+λXt,t=1,2,…
where Xt is a process with mean, λ is an exponential smoothing parameter with 0 < λ < 1 and Z0 is the initial value of EWMA statistics, Z0 = u. The upper control limit (UCL) and Lower control limit (LCL) of EWMA control charts are given byUCL=μ0+Qσλ2−λ,LCL=μ0−Qσλ2−λ,where μ0 is the target mean, σ is the process standard deviation, and Q is suitable control limit width.
The stopping time of the EWMA control chart is given byτh=inf{t≥0:Zt>h},h>uwhere τh is the stopping time, h is UCL.
Extended Exponentially Weighted Moving Average (Extended EWMA or EEWMA) Control Chart
The EEWMA control chart was proposed by Neveed et al. [8]. It is developed from the EWMA control chart. This is effective to monitored and detected small changes in process mean. The EWMA control chart can be expressed by the recursive equation below.
Et=λ1Xt−λ2Xt−1+(1−λ1+λ2)Et−1,t=1,2,…,
where λ1 and λ2 are exponential smoothing parameters with (0 < λ1 ≤ 1) and (0 ≤ λ2 < λ1) and the initial value is a constant, E0 = u. The upper control limit (UCL) and Lower control limit (LCL) of the EEWMA control charts are given by
UCL=μ0+Lσλ12+λ22−2λ1λ2(1−λ1+λ2)2(λ1−λ2)−(λ1−λ2)2,
LCL=μ0−Lσλ12+λ22−2λ1λ2(1−λ1+λ2)2(λ1−λ2)−(λ1−λ2)2,
where μ0 is the target mean, σ is the process standard deviation, and L is suitable control limit width.
The stopping time of the EEWMA control chart is given byτh′=inf{t≥0:Et>h′},h′>uwhere τh′ is the stopping time, h′ is UCL.
Explicit Formulas of ARL on the EEWMA Control Chart for AR(p) Processes
Let L(u) denote the ARL for the autoregressive process, to define function L(u) asARL=L(u)=Eθ(τb)≥Twhere Eθ(⋅) is the expectation under the assumption that the change point occurs at time θ and θ is the change point time.
The equation of observations for autoregressive (AR(p)) process in the case of an exponential while noise denoted can be described byXt=η+ϕ1Xt−1+ϕ2Xt−2+…+ϕpXt−p+ϵtwhere Xt (t = 1, 2, 3, …) is a sequence of random variables, η is a suitable constant, Φ is an autoregressive coefficient (− 1 ≤ Φ ≤ 1), and ɛt is white noise sequence of exponential (ɛt ∼ Exp(α)). The probability density function of ɛt is given by f(x)=1αe−xα where x ≥ 0.
Let L(u) denote ARL for AR(p) process, the EEWMA statistics Et can be written as:Et=(1−λ1+λ2)Zt−1+(λ1ϕ1−λ2)Xt−1+λ1ϕ2Xt−2+λ1ϕ3Xt−3+…+λ1ϕpXt−p+λ1η+λ1ϵtwhere (0 < λ1 ≤ 1), (0 ≤ λ2 < λ1) and the initial value E0 = u, and Xt−1, Xt−2, …, Xt−p.
Consequently, the EEWMA statistics Et can be written asEt=(1−λ1+λ2)u+(λ1ϕ1−λ2)Xt−1+λ1ϕ2Xt−2+λ1ϕ3Xt−3+…+λ1ϕpXt−p+λ1η+λ1ϵt
If ɛt = 0 LCL = 0 and UCL = h′, respectively. Then0≤Et≤h′0≤(1−λ1+λ2)u+(λ1ϕ1−λ2)Xt−1+λ1ϕ2Xt−2+λ1ϕ3Xt−3+…+λ1ϕpXt−p+λ1η+λ1ϵt≤h′0−(1−λ1+λ2)u−(λ1ϕ1−λ2)Xt−1−λ1ϕ2Xt−2−λ1ϕpXt−pλ1−η≤εt≤h'−(1−λ1+λ2)u−(λ1ϕ1−λ2)Xt−1−λ1ϕ2Xt−2−λ1ϕpXt−pλ1−η
Let L(u) denote the ARL on the EEWMA control chart. The function L(u) can be derived by Fredholm integral equation of the second kind, L(u) is defined as follows:
L(u)=1+∫L(E1)f(ϵ1)dϵ1
Therefore, the function L(u) is obtained as follows:L(u)=1+∫0−(1−λ1+λ2)u−(λ1ϕ1−λ2)Xt−1−λ1ϕ2Xt−2−λ1ϕpXt−pλ1−ηh′−(1−λ1+λ2)u−(λ1ϕ1−λ2)Xt−1−λ1ϕ2Xt−2−λ1ϕpXt−pλ1−ηL((1−λ1+λ2)u+(λ1ϕ1−λ2)Xt−1+λ1ϕ2Xt−2+…+λ1ϕpXt−p+λ1η+λ1y)f(y)dy
If k = (1 − λ1 + λ2)u + (λ1Φ1 − λ2)Xt−1 + λ1Φ2Xt−2 + … + λ1ΦpXt−p + λ1η + λ1y is defined for changing the integration variable, the function L(u) is given byL(u)=1+1λ1∫0h′L(k)f(k−(1−λ1+λ2)u−(λ1ϕ1−λ2)Xt−1λ1−ϕ2Xt−2−…−ϕpXt−p−η)dk
The L(u) is Fredholm integral equation of the second kind. If ɛt ∼ Exp(α), then
Finally, substituting constant F form Eq. (17) into Eq. (16), then L(u) can be written asL(u)=1−(λ1−λ2)e(1−λ1+λ2)uλ1α⋅(e−h′λ1α−1)(λ1−λ2)e−{(λ1ϕ1−λ2)Xt−1λ1α+ϕ2Xt−2+…+ϕpXt−p+ηα}+(e−(λ1−λ2)h′λ1α−1).
The process is “in-control” with the exponential parameter α = α0, the explicit formula of the ARL0 for AR(p) process on the EEWMA control chart can be written as follows:ARL0=1−(λ1−λ2)e(1−λ1+λ2)uλ1α0⋅(e−h′λ1α0−1)(λ1−λ2)e−{(λ1ϕ1−λ2)Xt−1λ1α0+ϕ2Xt−2+…+ϕpXt−p+ηα0}+(e−(λ1−λ2)h′λ1α0−1).
Meanwhile, the process is “out-of-control” with the exponential parameter α = α1 and then α1 = (1 + δ)α0, where α1 > α0 and δ is the shift size, the explicit formula of ARL1 for AR(p) process on the EEWMA control chart can be written as follows:ARL1=1−(λ1−λ2)e(1−λ1+λ2)uλ1α1⋅(e−h′λ1α1−1)(λ1−λ2)e−{(λ1ϕ1−λ2)Xt−1λ1α1+ϕ2Xt−2+…+ϕpXt−p+ηα1}+(e−(λ1−λ2)h′λ1α1−1).
while (− 1 ≤ Φ ≤ 1) is the autoregressive coefficient, (0 < λ1 ≤ 1), (0 ≤ λ2 < λ1) are the smoothing parameters, the initial value E0 = u, and Xt−1, Xt−2, …, Xt−p and h′ is the upper control limit.
Numerical Integral Equation Method of ARL on the EEWMA Control Chart for AR(p) Processes
The NIE method is used to solve the ARL for the AR(p) process on the EEWMA control chart in Eq. (14). The ARL solution or L~(u) is approximated with the m linear equation systems over the interval [0, h′]. A quadrature rule is used to approximate the integral by a finite sum of areas of rectangles with base h′/m and heights chosen as the values of f(aj) at the midpoints of intervals of length beginning at zero with a set of constant weights wj=h′m;j=1,2,…,m and aj=h′m(j−12)(see [24]).
Therefore, the approximating NIE method for the ARL on the EEWMA control chart is evaluated as follows:∫0h′L(k)f(k)dk≈∑j=1mwjf(aj)
The system of m linear equation is showed as:
Lm×1 = 1m×1 + Rm×mLm×1 or (Im − Rm×m)Lm×1 = 1m×1 or Lm×1 = (Im − Rm×m)−11m×1
Lm×1 = (Im − Rm×m)−11m×1 where Lm×1=[L~(a1),L~(a2),…,L~(am)]T, Im = diag(1, 1, …, 1) and 1m×1 = [1, 1, …, 1] T.
Let Rm×m be a matrix, the definition of the m to mth element of the matrix R is given by[Rij]≈1λ1wjf(aj−(1−λ1+λ2)ai−(λ1ϕ1−λ2)Xt−1λ1−ϕ2Xt−2−…−ϕpXt−p−η)
Finally, the numerical approximation for the function L~(u) is as follows:L~(u)=1+1λ1∑j=1mwjL(aj)f(aj−(1−λ1+λ2)u−(λ1ϕ1−λ2)Xt−1λ1−ϕ2Xt−2−…−ϕpXt−p−η)
Existence and Uniqueness of ARL
The solution of ARL shows that there uniquely exists the integral equation for explicit formulas by the Banach’s Fixed-point Theorem. In this study, let T be an operation in the class of all continuous functions defined byT(L(u))=1+1λ1∫0h′L(k)f(k−(1−λ1+λ2)u−(λ1ϕ1−λ2)Xt−1λ1−ϕ2Xt−2−…−ϕpXt−p−η)dk
According to Banach’s Fixed-point Theorem, if an operator T is a contraction, and then the fixed-point equation T(L(u)) = L(u) has a unique solution. To show that Eq. (23) exists and has a unique solution, theorem can be used as follows below.
Theorem 1 Banach’s Fixed-point Theorem: Let (X, d) be a complete metric space and T: X → X be a contraction mapping with contraction constant 0 ≤ r < 1 such that ‖T(L1) − T(L2)‖ ≤ r‖L1 − L2‖, ∀L1,L2∈X. Then there exists a unique L( ⋅ ) ∈ X such that T(L(u)) = L(u), i.e., a unique fixed-point in X.
Proof: Let T defined in Eq. (23) is a contraction mapping for L1, L2 ∈ G[0, h′], such that ‖T(L1) − T(L2)‖ ≤ r‖L1 − L2‖, ∀L1,L2∈G[0,h′] with 0 ≤ r < 1 under the norm ‖L‖∞=supu∈[0,h′]|L(u)|, so
The absolute percentage relative error (APRE) to measure the accuracy of the ARL is defined asAPRE(%)=|L(u)−L˜(u)|L(u)×100
where L(u) is the explicit formulas of the ARL on the EEWMA control chart for AR(p) process shows that Eq. (18) , which ARL0 and ARL1 are Eqs. (19) and (20), respectively, and L~(u) in Eq. (22) is the NIE method of the ARL using the Gauss-Legendre quadrature rule on the EEWMA control chart for AR(p) with the number of division points m =500 nodes. The numerical results were computed by MATHEMATICA. The initial parameter values are studied at ARL0 = 370 on the EEWMA control chart for AR(p) process, referred to as AR(2) and AR(3) processes with exponential white noise and given λ1 = 0.05, 0.10, λ2 = 0.01, 0.02, 0.03. The ‘in-control’ process had parameter value as α = α0 with shift size (δ = 0). On the other hand, the ‘out-of-control’ process was presented with parameter values as α1 = (1 + δ)α0 with shift sizes (δ) equals 0.001, 0.003, 0.005, 0.010, 0.030, 0.050, 0.100, 0.500 and 1.000 were determined. Furthermore, the coefficient parameters of the process Φ1 = 0.2, Φ2 = 0.2, − 0.2 were used for the AR(2) process, and Φ1 = Φ2 = 0.2, Φ3 = 0.2, − 0.2 were used for the AR(3) process. In addition, the speed test results were computed by the CPU time (PC System: windows10, 64-bit, Intel® Core™ i5-8250U 1.60 GHz 1.80 GHz, RAM 4 GB) in seconds.
In Tabs. 1 and 2, the ARL results by using the explicit formula (Eqs. (19) and (20)) and the NIE method (Eq. (22) then the absolute percentage relative error named APRE(%) show that Eq. (24). It showed that the ARL values derived from the explicit formulas give results close to those from the NIE method both AR(2) and AR(3) process. AR(2) process, Tab. 1 showed that the analytical results agree with NIE approximations with APRE(%) less than 0.000239% and CPU time of approximately 2.7–3.5 s whereas the CPU time of the explicit formulas is not much. AR(3) process in Tab. 2 showed that the analytical results agree with NIE approximations with APRE(%) less than 0.000216% and CPU time of approximately 2.8–3.5 s whereas the CPU time of the explicit formulas is not much. The entries inside the parentheses are the CPU time in seconds.
Comparing ARL values on the EEWMA control chart for the AR(2) process using explicit formulas against the NIE method given λ1 = 0.05, 0.10, λ2 = 0.01, η = 0 for ARL0 = 370
λ1
Shift size (δ)
Φ1 = Φ2 = 0.2*
Φ1 = 0.2, Φ2 = −0.2**
Explicit
NIE (CPU time)
APRE (%)
Explicit
NIE (CPU time)
APRE (%)
0.05
0.000
370.321304
370.321192 (2.891)
0.000030
370.388734
370.388600 (3.266)
0.000036
0.001
234.777706
234.777647 (3.047)
0.000025
239.110276
239.110204 (3.001)
0.000030
0.003
135.885390
135.885362 (3.406)
0.000021
140.253416
140.253381 (3.079)
0.000025
0.005
95.8318800
95.8318615 (2.906)
0.000019
99.4478549
99.4478320 (3.032)
0.000023
0.010
55.4896135
55.4896039 (3.047)
0.000017
57.8903579
57.8903459 (2.875)
0.000021
0.030
21.2886283
21.2886251 (3.125)
0.000015
22.2934984
22.2934945 (2.907)
0.000018
0.050
13.5344720
13.5344701 (2.921)
0.000014
14.1755969
14.1755945 (3.187)
0.000016
0.100
7.48364935
7.48364848 (3.172)
0.000012
7.82854173
7.82854066 (3.078)
0.000014
0.500
2.45093356
2.45093345 (2.999)
0.000004
2.53934226
2.53934213 (2.719)
0.000005
1.000
1.77981293
1.77981290 (3.016)
0.000002
1.83156186
1.83156182 (3.016)
0.000002
0.10
0.000
370.042850
370.042130 (2.812)
0.000195
370.069839
370.068954 (3.235)
0.000239
0.001
258.305964
258.305590 (2.844)
0.000145
265.398874
265.398393 (3.157)
0.000181
0.003
161.332490
161.332326 (3.093)
0.000101
169.772828
169.772610 (2.967)
0.000128
0.005
117.496757
117.496661 (2.890)
0.000082
124.992900
124.992771 (3.203)
0.000103
0.010
70.2760299
70.2759877 (3.062)
0.000060
75.6205876
75.6205304 (3.063)
0.000076
0.030
27.5853972
27.5853864 (3.203)
0.000039
29.9372209
29.9372064 (3.204)
0.000048
0.050
17.5425526
17.5425468 (3.000)
0.000033
19.0425702
19.0425625 (2.891)
0.000041
0.100
9.60903193
9.60902943 (2.985)
0.000026
10.3968364
10.3968331 (2.875)
0.000032
0.500
2.94823486
2.94823458 (3.172)
0.000009
3.11870875
3.11870839 (3.171)
0.000012
1.000
2.05578182
2.05578173 (3.109)
0.000004
2.14628399
2.14628388 (3.156)
0.000005
Notes: *h′ = 0.0488991 for λ1 = 0.05 and h′ = 0.1376787 for λ1 = 0.10. **h′ = 0.0530625 for λ1 = 0.05 and h′ = 0.1499641 for λ1 = 0.10.
Comparing ARL values on the EEWMA control chart for the AR(3) process using explicit formulas against the NIE method given λ1 = 0.05, 0.10, λ2 = 0.01, η = 0 for ARL0 = 370
λ1
Shift size(δ)
Φ1 = Φ2 = Φ3 = 0.2*
Φ1 = Φ2 = 0.2, Φ3 = −0.2**
Explicit
NIE (CPU time)
APRE (%)
Explicit
NIE (CPU time)
APRE (%)
0.05
0.000
370.152690
370.152587 (3.250)
0.000028
370.369025
370.368902 (3.187)
0.000033
0.001
232.684141
232.684088 (3.063)
0.000023
236.904912
236.904847 (3.298)
0.000027
0.003
133.850860
133.850835 (2.953)
0.000019
138.011528
138.011496 (3.219)
0.000023
0.005
94.1665824
94.1665658 (3.095)
0.000018
97.5864564
97.5864358 (3.126)
0.000021
0.010
54.3952320
54.3952233 (3.063)
0.000016
56.6510616
56.6510508 (3.079)
0.000019
0.030
20.8336361
20.8336333 (3.125)
0.000014
21.7738180
21.7738144 (3.015)
0.000016
0.050
13.2442206
13.2442189 (2.907)
0.000013
13.8440442
13.8440422 (3.203)
0.000015
0.100
7.32712291
7.32712213 (3.030)
0.000011
7.65034375
7.65034278 (2.827)
0.000013
0.500
2.41014802
2.41014793 (3.125)
0.000004
2.49392222
2.49392210 (3.234)
0.000005
1.000
1.75574743
1.75574740 (3.093)
0.000002
1.80505114
1.80505111 (3.484)
0.000002
0.10
0.000
370.144195
370.143544 (3.265)
0.000176
370.111076
370.110277 (3.015)
0.000216
0.001
255.145929
255.145598 (3.172)
0.000130
261.759999
261.759576 (3.093)
0.000162
0.003
157.658228
157.658085 (3.140)
0.000090
165.370557
165.370369 (3.015)
0.000114
0.005
114.276687
114.276603 (3.171)
0.000073
121.058937
121.058826 (3.328)
0.000092
0.010
68.0136145
68.0135780 (3.265)
0.000054
72.7984472
72.7983982 (3.141)
0.000067
0.030
26.6006234
26.6006140 (3.281)
0.000035
28.6897880
28.6897755 (3.063)
0.000044
0.050
16.9144058
16.9144007 (3.141)
0.000030
18.2468150
18.2468084 (3.140)
0.000037
0.100
9.27732544
9.27732323 (3.312)
0.000024
9.97959567
9.97959280 (3.218)
0.000029
0.500
370.144195
370.143544 (3.265)
0.000176
370.111076
370.110277 (3.015)
0.000216
1.000
255.145929
255.145598 (3.172)
0.000130
261.759999
261.759576 (3.093)
0.000162
Notes: *h′ = 0.0469439 for λ1 = 0.05 and h′ = 0.1319420 for λ1 = 0.10. **h′ = 0.0509374 for λ1 = 0.05 and h′ = 0.1436815 for λ1 = 0.10.
Performance Comparing the ARL Results
For Tabs. 3 and 4, the EEWMA control chart is compared for various λ1 = 0.05, 0.10 and λ2 = 0.01, 0.02, 0.03 at ARL0 = 370, η = 0, Φ1 = Φ2 = 0.2 (as an AR(2) process) and Φ1 = Φ2 = Φ3 = 0.2 (as an AR(3) process). The ARL values are indicated that the ARL1 on the EEWMA (λ2 = 0.03) or EEWMA_03 control chart was reduced more sensitively than on the EEWMA with either λ2 = 0.01 (EEWMA_01) or λ2 = 0.02 (EEWMA_02) for all magnitudes of changes both AR(2) and AR(3) processes. Moreover, the ARL1 on the EEWMA control chart with λ1 = 0.05 was reduced more sensitively than on the EEWMA control chart with λ1 = 0.10 for all situations running AR(2) and AR(3) processes. The exponential smoothing parameter 0.05 is recommended. Tab. 5 showed that the comparison of ARL values for the AR(2) process on CUSUM, EWMA and EEWMA control charts, the results presented that the ARL1 on the EEWMA control chart with λ2 = 0.03 was reduced the ARL1 more than the CUSUM, EWMA, EEWMA with either λ2 = 0.01 or λ2 = 0.02 control charts for all shift sizes and all exponential smoothing parameter values. Similarly, the ARL results of AR(3) process in Tab. 6, the results presented that the ARL1 on the EEWMA control chart with λ2 = 0.03 was reduced the ARL1 more than the CUSUM, EWMA, EEWMA with either λ2 = 0.01 or λ2 = 0.02 control charts for all shift sizes and all exponential smoothing parameter values as same as the ARL results of AR(2) process. Therefore, the performance of the EEWMA control chart with λ2 = 0.03 is more efficient than the performance of the CUSUM, EWMA, EEWMA with either λ2 = 0.01 or λ2 = 0.02 control charts for all situations except when the large shift sizes (δ ≥ 0.5), the EEWMA control chart with λ2 = 0.03 was reduced as well as the EWMA, EEWMA with either λ2 = 0.01 or λ2 = 0.02 control charts.
Comparing ARL on the EEWMA control chart for AR(2) process with various λ when given η = 0, Φ1 = Φ2 = 0.2 for ARL0 = 370
Shift size(δ)
λ1 = 0.05
λ1 = 0.10
λ2=0.01h′=0.0488991
λ2=0.02h′=0.0265188
λ2=0.03h′=0.0144770
λ2=0.01h′=0.1376787
λ2=0.02h′=0.0997870
λ2=0.03h′=0.0728639
0.000
370.3213
370.6220
370.1861
370.0429
370.1648
370.2258
0.001
234.7777
209.5365
190.2497
258.3060
237.4070
222.4548
0.003
135.8854
112.4374
96.80430
161.3325
138.5722
124.0426
0.005
95.83188
77.04483
65.11853
117.4968
98.05656
86.21334
0.010
55.48961
43.40275
36.07534
70.27603
56.96144
49.21775
0.030
21.28863
16.33608
13.43369
27.58540
21.89414
18.68865
0.050
13.53447
10.37563
8.532907
17.54255
13.91433
11.87089
0.100
7.483649
5.770465
4.769831
9.609032
7.680046
6.578097
0.500
2.450934
1.989591
1.716927
2.948235
2.491119
2.203938
1.000
1.779813
1.504181
1.342614
2.055782
1.799900
1.630912
Comparing ARL on the EEWMA control chart for AR(3) process with various λ when given η = 0, Φ1 = Φ2 = Φ3 = 0.2 for ARL0 = 370
Shift size(δ)
λ1 = 0.05
λ1 = 0.10
λ2=0.01h′=0.0469439
λ2=0.02h′=0.0254704
λ2=0.03h′=0.0139076
λ2=0.01h′=0.1319420
λ2=0.02h′=0.0957177
λ2=0.03h′=0.0699340
0.000
370.1527
370.0058
370.2755
370.1442
370.0809
370.0634
0.001
232.6841
207.9606
189.1197
255.1459
235.2129
220.7210
0.003
133.8509
111.2001
95.92305
157.6582
136.3804
122.4781
0.005
94.16658
76.10072
64.45710
114.2767
96.24664
84.96887
0.010
54.39523
42.82084
35.67538
68.01361
55.76243
48.42211
0.030
20.83364
16.10460
13.27734
26.60062
21.39311
18.36408
0.050
13.24422
10.22846
8.433957
16.91441
13.59477
11.66399
0.100
7.327123
5.690395
4.716288
9.277325
7.508189
6.465857
0.500
2.410148
1.967488
1.702598
2.873554
2.447154
2.173623
1.000
1.755747
1.490908
1.334281
2.015192
1.774239
1.612790
Comparing ARL values for the AR(2) process on CUSUM, EWMA and EEWMA control charts when given λ1 = 0.05, λ2 = 0.01, 0.02, 0.03, η = 0 for ARL0 = 370
Φ1
Φ2
Shift size(δ)
CUSUM(a = 2)
EWMA(λ2 = 0)
EEWMA
EEWMA_01(λ2 = 0.01)
EEWMA_02(λ2 = 0.02)
EEWMA_03(λ2 = 0.03)
0.2
0.2
b = 3.579
h = 0.0914794
h′ = 0.0488991
h′ = 0.0265188
h′ = 0.0144770
0.000
370.1644
370.0520
370.3213
370.6220
370.1861
0.001
367.7875
284.1654
234.7777
209.5365
190.2497
0.003
363.0933
194.2544
135.8854
112.4374
96.80430
0.005
358.4773
147.7005
95.83188
77.04483
65.11853
0.010
347.2692
92.59445
55.48961
43.40275
36.07534
0.030
306.7755
37.68188
21.28863
16.33608
13.43369
0.050
272.2869
23.98603
13.53447
10.37563
8.532907
0.100
205.9789
12.95413
7.483649
5.770465
4.769831
0.500
43.83369
3.616823
2.450934
1.989591
1.716927
1.000
15.80483
2.397395
1.779813
1.504181
1.342614
−0.2
b = 3.471
h = 0.0994968
h′ = 0.0530625
h′ = 0.0287478
h′ = 0.0156870
0.000
370.2999
370.2746
370.3887
370.5370
370.6751
0.001
367.9426
296.0073
239.1103
212.3404
192.7199
0.003
363.2867
211.3611
140.2534
114.8813
98.65484
0.005
358.7077
164.4419
99.44785
78.95453
66.49796
0.010
347.5872
105.9012
57.89036
44.60220
36.90566
0.030
307.3817
44.05912
22.29350
16.81948
13.75746
0.050
273.0998
28.06060
14.17560
10.68373
8.737752
0.100
207.0696
15.01612
7.828542
5.938363
4.880706
0.500
44.50081
3.955546
2.539342
2.036011
1.746727
1.000
16.06022
2.550454
1.831562
1.532108
1.360015
Comparing ARL values for the AR(3) process on CUSUM, EWMA and EEWMA control charts when given λ1 = 0.05, λ2 = 0.01, 0.02, 0.03, η = 0 for ARL0 = 370
Φ1 = Φ2
Φ3
Shift size(δ)
CUSUM(a = 2)
EWMA(λ2 = 0)
EEWMA
EEWMA_01(λ2 = 0.01)
EEWMA_02(λ2 = 0.02)
EEWMA_03(λ2 = 0.03)
0.2
0.2
b = 3.635
h = 0.0877273
h′ = 0.0469439
h′ = 0.0254704
h′ = 0.0139076
0.000
370.1955
370.1327
370.1527
370.0058
370.2755
0.001
367.8070
279.1179
232.6841
207.9606
189.1197
0.003
363.0902
187.3070
133.8509
111.2001
95.92305
0.005
358.4522
141.0997
94.16658
76.10072
64.45710
0.010
347.1919
87.53147
54.39523
42.82084
35.67538
0.030
306.5263
35.32701
20.83364
16.10460
13.27734
0.050
271.9131
22.48317
13.24422
10.22846
8.433957
0.100
205.4330
12.18450
7.327123
5.690395
4.716288
0.500
43.48171
3.478614
2.410148
1.967488
1.702598
1.000
15.67118
2.331644
1.755747
1.490908
1.334281
0.2
−0.2
b = 3.524
h = 0.0953998
h′ = 0.0509374
h′ = 0.0276106
h′ = 0.0150698
0.000
370.0741
370.0517
370.3690
370.4431
370.2123
0.001
367.7086
289.7383
236.9049
210.8827
191.4205
0.003
363.0365
202.1826
138.0115
113.6314
97.70710
0.005
358.4420
155.3759
97.58646
77.98044
65.79521
0.010
347.2848
98.61321
56.65106
43.99142
36.48428
0.030
306.9602
40.53274
21.77382
16.57357
13.59355
0.050
272.5951
25.80669
13.84404
10.52704
8.634091
0.100
206.4616
13.87987
7.650344
5.852996
4.824608
0.500
44.16430
3.774240
2.493922
2.012414
1.731631
1.000
15.93231
2.469918
1.805051
1.517905
1.351188
Application to Real Data
The performance of the ARL constructed using explicit formulas for the EEWMA control chart for λ1 = 0.05 and various λ2 = 0.01, 0.02, 0.03 was compared with those of CUSUM and EWMA (λ2 = 0) control charts using data on new COVID-19 cases in Thailand and in Vietnam from March 30th to July 7th, 2021. Lately, Areepong et al. [27] investigated monitoring COVID-19 outbreaks in Thailand, Singapore, Vietnam, and Hong Kong using by the EWMA control chart. There are 100 observations of daily. This data is a stationary time series. By looking at the autocorrelation function (ACF) and partial autocorrelation function (PACF). Both countries are the worst affected in Southeast Asia. For λ1 = 0.05 and λ2 = 0.01, 0.02, or 0.03, the settings for the Thailand dataset are that it is an AR(2) process with ARL0= 370; the significance of the mean and standard deviation are 2.774663 and 1.663941, respectively; process coefficients Φ1 = 0.343110, Φ2 = 0.527991; the error is exponential white noise (α0 = 0.665927) whereas the settings for the Vietnam dataset are it is an AR(3) process with ARL0 = 370; the significance of the mean and standard deviation are 0.214103 and 0.259871, respectively; process coefficients Φ1 = 0.269717, Φ2 = 0.572229, Φ3 = 0.219039; the error is exponential white noise (α0 = 0.129397).
The results for the ARL of CUSUM, EWMA and EEWMA with various λ2 = 0.01, 0.02, 0.03 control charts on AR(2) process for the Thailand dataset in Tab. 7 are agreement to the simulation results in Tab. 5. Similarly, These control charts on AR(3) process for the Vietnam dataset in Tab. 8 are agreement to the simulation results in Tab. 6. ARL1 on an EEWMA control chart with λ2 = 0.03 was reduced more sensitively than CUSUM, EWMA, EEWMA with λ2 = 0.01 and EEWMA with λ2 = 0.02 control charts for all magnitudes of changes except when the large shift sizes (δ ≥ 0.5), the EEWMA control chart with λ2 = 0.03 was reduced as well as the EWMA, EEWMA with λ2 = 0.01 and EEWMA with λ2 = 0.02 control charts both AR(2) and AR(3) processes. The results indicate that the performances of the control charts were, in ascending order, EEWMA for λ2 = 0.03, EEWMA for λ2 = 0.02, EEWMA for λ2 = 0.01, EWMA, and CUSUM, as illustrated in Figs. 1 and 2.
Comparing ARL values for the AR(2) process on CUSUM, EWMA and EEWMA control charts with COVID-19 data in Thailand when given ARL0 = 370, α0 = 0.665927, η = 3.445847, Φ1 = 0.343110, Φ2 = 0.527991, λ1 = 0.05 and λ2 = 0.01, 0.02, 0.03
Shift size(δ)
CUSUM(a = 1.5)
EWMA(λ2 = 0)
EEWMA
EEWMA_01(λ2 = 0.01)
EEWMA_02(λ2 = 0.02)
EEWMA_03(λ2 = 0.03)
b = 2.8223
h = 0.0307285
h′ = 0.00149023
h′ = 0.0000738797
h′ = 0.00000366589
0.000
370.0535
370.0669
370.0996
370.1763
370.2019
0.001
367.5050
232.2021
145.5442
102.3868
78.57510
0.003
362.4756
133.3967
66.08133
42.17770
30.84456
0.005
357.5350
93.79691
42.92652
26.73477
19.36131
0.010
345.5633
54.15912
23.10964
14.16717
10.22973
0.030
302.5634
20.73229
8.508357
5.271144
3.884294
0.050
266.2953
13.17821
5.444146
3.446960
2.600501
0.100
197.6295
7.289428
3.120606
2.083703
1.655609
0.500
38.98113
2.397793
1.304992
1.094978
1.032944
1.000
14.08608
1.747635
1.116857
1.024074
1.005267
Comparing ARL values for the AR(3) process on CUSUM, EWMA and EEWMA control charts with COVID-19 data in Vietnam when given ARL0 = 370 α0=0.129397,η=0,ϕ1=0.269717,ϕ2=0.572229,ϕ3=0.219039, λ1 = 0.05 and λ2 = 0.01, 0.02, 0.03
Shift size(δ)
CUSUM(a = 0.2)
EWMA(λ2 = 0)
EEWMA
EEWMA_01(λ2 = 0.01)
EEWMA_02(λ2 = 0.02)
EEWMA_03(λ2 = 0.03)
b = 0.7101
h = 0.0085715
h′ = 0.0017763
h′ = 0.000376829
h′ = 0.0000802665
0.000
370.0149
370.0735
370.3624
370.1065
370.0699
0.001
366.7198
253.6347
189.3055
151.1184
123.8997
0.003
360.2121
155.7430
95.87017
69.39629
53.37970
0.005
353.8458
112.5812
64.38790
45.22358
34.19629
0.010
338.5372
66.84116
35.63176
24.40944
18.23388
0.030
284.9166
26.10418
13.25846
8.989864
6.728492
0.050
241.4673
16.60412
8.421038
5.743410
4.342770
0.100
164.3741
9.121001
4.708184
3.278241
2.545492
0.500
24.28932
2.848698
1.699288
1.341181
1.181372
1.000
10.04217
2.005967
1.332079
1.135106
1.058714
ARL values of the AR(2) process on CUSUM, EWMA and EEWMA control charts with new cases COVID-19 data in Thailand when given ARL0 = 370
ARL values of the AR(3) process on CUSUM, EWMA and EEWMA control charts with new cases COVID-19 data in Vietnam when given ARL0 = 370
As mentioned above, the EEWMA (λ2 = 0.03) and EWMA (λ2 = 0) control charts are plotted by calculating Et and Zt for the two datasets for λ1 = 0.05. The detecting the process with real data of the new cases COVID-19 data in Thailand (as an AR(2) process) and Vietnam (as an AR(3) process) are shown in Figs. 3 and 4, respectively. In Fig. 3, the ARL of the AR(2) process for the Thailand COVID-19 data on the EEWMA control chart for λ2 = 0.03 indicates that the process was signaled as out-of-control at the 6th observation whereas on the EWMA control chart, it was detected at the 11th observation. In Fig. 4 for the EEWMA (λ2 = 0.03) control chart, the ARL of AR(3) process for the Vietnam COVID-19 data on the EEWMA control chart for λ2 = 0.03 was signaled as out-of-control process at the 9th observation whereas on the EWMA control chart, it was detected as out-of-control at the 20th observation. Therefore, the EEWMA control chart can detect shift more quickly than the EWMA control chart.
The detecting the AR(2) process with the Thailand COVID-19 data when given ARL0 = 370; (a) EWMA control chart and (b) EEWMA control chart at λ2 = 0.03
The detecting the AR(3) process with the Vietnam COVID-19 data when given ARL0 = 370; (a) EWMA control chart and (b) EEWMA control chart at λ2 = 0.03
Discussions and Conclusions
In the study, the performances of control charts were evaluated by using ARL. The explicit formulas comprise a good alternative to the NIE method for constructing the ARL. The analytical results agree with the NIE approximations for AR(2) and AR(3) processes with absolute percentage relative errors of less than 0.000239% and 0.000216%, respectively. The CPU time to calculate the ARL by using the NIE methods were approximately 2.7–3.5 and 2.8–3.5 s for the AR(2) and AR(3) processes, respectively, whereas they were almost instantaneous when using the explicit formulas. The performance comparison of the ARL using explicit formulas on the EEWMA with various λ performed better than on the CUSUM and EWMA control charts running AR(2) or AR(3) processes for most cases except for large shift sizes (δ ≥ 0.5) when even then, the EEWMA control chart with various λ performed as well as the EWMA control chart. The EEWMA control chart with λ2 = 0.03 performed better than the EEWMA with either λ2 = 0.01 or λ2 = 0.02, CUSUM, or EWMA control charts for most magnitudes of changes except for a large shift sizes (δ ≥ 0.5) when it performed at least as well as the others for AR(2) and AR(3) processes. Besides, an exponential smoothing parameter value of 0.05 is recommended. In addition, the simulation study, and the efficacy illustration with real data of new COVID-19 cases in Thailand and Vietnam provided similar results.
Funding Statement: Thailand Science Research and Innovation Fund, and King Mongkut’s University of Technology North Bangkok Contract no. KMUTNB-FF-65–45.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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