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 (

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 [

The performance of the chart is measured by Average Run Length (_{0} 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 _{1} denote the average number of observations taken from out of control and should be as small as possible.

Many methods for evaluating

Many literatures for evaluating the

However, the derivation of the explicit formulas for

The CUSUM control chart was originally introduced by Page [_{0} is the initial value of CUSUM statistics with _{0} =

The stopping time of the CUSUM control chart is given by_{b} is the stopping time,

The EWMA control chart was initially proposed by Robert [

where _{t} is a process with mean, _{0} is the initial value of EWMA statistics, _{0} = _{0} is the target mean,

The stopping time of the EWMA control chart is given by_{h} is the stopping time,

The EEWMA control chart was proposed by Neveed et al. [

where _{1} and _{2} are exponential smoothing parameters with (0 < _{1} ≤ 1) and (0 ≤ _{2} < _{1}) and the initial value is a constant, _{0} =

where _{0} is the target mean,

The stopping time of the EEWMA control chart is given by^{′} is

Let

The equation of observations for autoregressive (AR(p)) process in the case of an exponential while noise denoted can be described by_{t} (_{t} is white noise sequence of exponential (ɛ_{t} ∼ _{t} is given by

Let _{t} can be written as:_{1} ≤ 1), (0 ≤ _{2} < _{1}) and the initial value _{0} = _{t−1}, _{t−2}, …, _{t−p}.

Consequently, the EEWMA statistics _{t} can be written as

If ɛ_{t} = 0 ^{′}, respectively. Then

Let

Therefore, the function

If _{1} + _{2})_{1}_{1} − _{2})_{t−1} + _{1}_{2}_{t−2} + … + _{1}_{p}_{t−p} + _{1}_{1}

The _{t} ∼

Consider the constant

Finally, substituting constant

The process is “in-control” with the exponential parameter _{0}, the explicit formula of the _{0} for AR(p) process on the EEWMA control chart can be written as follows:

Meanwhile, the process is “out-of-control” with the exponential parameter _{1} and then _{1} = (1 + _{0}, where _{1} > _{0} and _{1} for AR(p) process on the EEWMA control chart can be written as follows:

while (− 1 ≤ _{1} ≤ 1), (0 ≤ _{2} < _{1}) are the smoothing parameters, the initial value _{0} = _{t−1}, _{t−2}, …, _{t−p} and ^{′} is the upper control limit.

The NIE method is used to solve the ^{′}]. A quadrature rule is used to approximate the integral by a finite sum of areas of rectangles with base ^{′}/_{j}) at the midpoints of intervals of length beginning at zero with a set of constant weights

Therefore, the approximating NIE method for the

The system of m linear equation is showed as:

_{m×1} = 1_{m×1} + _{m×m}_{m×1} or (_{m} − _{m×m})_{m×1} = 1_{m×1} or _{m×1} = (_{m} − _{m×m})^{−1}1_{m×1}

_{m×1} = (_{m} − _{m×m})^{−1}1_{m×1} where _{m} = _{m×1} = [1, 1, …, 1] ^{T}.

Let _{m×m} be a matrix, the definition of the ^{th} element of the matrix

Finally, the numerical approximation for the function

The solution of

According to Banach’s Fixed-point Theorem, if an operator

_{1}) − _{2})‖ ≤ _{1} − _{2}‖,

_{1}, _{2} ∈ ^{′}], such that ‖_{1}) − _{2})‖ ≤ _{1} − _{2}‖,

The absolute percentage relative error (APRE) to measure the accuracy of the

where _{0} and _{1} are _{0} = 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 _{1} = 0.05, 0.10, _{2} = 0.01, 0.02, 0.03. The ‘in-control’ process had parameter value as _{0} with shift size (_{1} = (1 + _{0} with shift sizes (_{1} = 0.2, _{2} = 0.2, − 0.2 were used for the AR(2) process, and _{1} = _{2} = 0.2, _{3} = 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

_{1} |
Shift size ( |
_{1} = _{2} = 0.2* |
_{1} = 0.2, _{2} = −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: *^{′} = 0.0488991 for _{1} = 0.05 and ^{′} = 0.1376787 for _{1} = 0.10. **^{′} = 0.0530625 for _{1} = 0.05 and ^{′} = 0.1499641 for _{1} = 0.10.

_{1} |
Shift size |
_{1} = _{2} = _{3} = 0.2* |
_{1} = _{2} = 0.2, _{3} = −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: *^{′} = 0.0469439 for _{1} = 0.05 and ^{′} = 0.1319420 for _{1} = 0.10. **^{′} = 0.0509374 for _{1} = 0.05 and ^{′} = 0.1436815 for _{1} = 0.10.

For _{1} = 0.05, 0.10 and _{2} = 0.01, 0.02, 0.03 at _{0} = 370, _{1} = _{2} = 0.2 (as an AR(2) process) and _{1} = _{2} = _{3} = 0.2 (as an AR(3) process). The _{1} on the EEWMA (_{2} = 0.03) or EEWMA_03 control chart was reduced more sensitively than on the EEWMA with either _{2} = 0.01 (EEWMA_01) or _{2} = 0.02 (EEWMA_02) for all magnitudes of changes both AR(2) and AR(3) processes. Moreover, the _{1} on the EEWMA control chart with _{1} = 0.05 was reduced more sensitively than on the EEWMA control chart with _{1} = 0.10 for all situations running AR(2) and AR(3) processes. The exponential smoothing parameter 0.05 is recommended. _{1} on the EEWMA control chart with _{2} = 0.03 was reduced the _{1} more than the CUSUM, EWMA, EEWMA with either _{2} = 0.01 or _{2} = 0.02 control charts for all shift sizes and all exponential smoothing parameter values. Similarly, the _{1} on the EEWMA control chart with _{2} = 0.03 was reduced the _{1} more than the CUSUM, EWMA, EEWMA with either _{2} = 0.01 or _{2} = 0.02 control charts for all shift sizes and all exponential smoothing parameter values as same as the _{2} = 0.03 is more efficient than the performance of the CUSUM, EWMA, EEWMA with either _{2} = 0.01 or _{2} = 0.02 control charts for all situations except when the large shift sizes (_{2} = 0.03 was reduced as well as the EWMA, EEWMA with either _{2} = 0.01 or _{2} = 0.02 control charts.

Shift size |
_{1} = 0.05 |
_{1} = 0.10 |
||||
---|---|---|---|---|---|---|

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 |

Shift size |
_{1} = 0.05 |
_{1} = 0.10 |
||||
---|---|---|---|---|---|---|

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 |

_{1} |
_{2} |
Shift size |
CUSUM |
EWMA_{2} = 0) |
EEWMA | ||
---|---|---|---|---|---|---|---|

EEWMA_01_{2} = 0.01) |
EEWMA_02_{2} = 0.02) |
EEWMA_03_{2} = 0.03) |
|||||

0.2 | 0.2 | ||||||

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 | |||||||

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 |

_{1} = _{2} |
_{3} |
Shift size |
CUSUM |
EWMA_{2} = 0) |
EEWMA | ||
---|---|---|---|---|---|---|---|

EEWMA_01_{2} = 0.01) |
EEWMA_02_{2} = 0.02) |
EEWMA_03_{2} = 0.03) |
|||||

0.2 | 0.2 | ||||||

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 | ||||||

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 |

The performance of the _{1} = 0.05 and various _{2} = 0.01, 0.02, 0.03 was compared with those of CUSUM and EWMA (_{2} = 0) control charts using data on new COVID-19 cases in Thailand and in Vietnam from March 30^{th} to July 7^{th}, 2021. Lately, Areepong et al. [_{1} = 0.05 and _{2} = 0.01, 0.02, or 0.03, the settings for the Thailand dataset are that it is an AR(2) process with _{0}= 370; the significance of the mean and standard deviation are 2.774663 and 1.663941, respectively; process coefficients _{1} = 0.343110, _{2} = 0.527991; the error is exponential white noise (_{0} = 0.665927) whereas the settings for the Vietnam dataset are it is an AR(3) process with _{0} = 370; the significance of the mean and standard deviation are 0.214103 and 0.259871, respectively; process coefficients _{1} = 0.269717, _{2} = 0.572229, _{3} = 0.219039; the error is exponential white noise (_{0} = 0.129397).

The results for the _{2} = 0.01, 0.02, 0.03 control charts on AR(2) process for the Thailand dataset in _{1} on an EEWMA control chart with _{2} = 0.03 was reduced more sensitively than CUSUM, EWMA, EEWMA with _{2} = 0.01 and EEWMA with _{2} = 0.02 control charts for all magnitudes of changes except when the large shift sizes (_{2} = 0.03 was reduced as well as the EWMA, EEWMA with _{2} = 0.01 and EEWMA with _{2} = 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 _{2} = 0.03, EEWMA for _{2} = 0.02, EEWMA for _{2} = 0.01, EWMA, and CUSUM, as illustrated in

Shift size |
CUSUM |
EWMA_{2} = 0) |
EEWMA | ||
---|---|---|---|---|---|

EEWMA_01_{2} = 0.01) |
EEWMA_02_{2} = 0.02) |
EEWMA_03_{2} = 0.03) |
|||

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 |

Shift size |
CUSUM |
EWMA_{2} = 0) |
EEWMA | ||
---|---|---|---|---|---|

EEWMA_01_{2} = 0.01) |
EEWMA_02_{2} = 0.02) |
EEWMA_03_{2} = 0.03) |
|||

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 |

As mentioned above, the EEWMA (_{2} = 0.03) and EWMA (_{2} = 0) control charts are plotted by calculating _{t} and _{t} for the two datasets for _{1} = 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 _{2} = 0.03 indicates that the process was signaled as out-of-control at the 6^{th} observation whereas on the EWMA control chart, it was detected at the 11^{th} observation. In _{2} = 0.03) control chart, the _{2} = 0.03 was signaled as out-of-control process at the 9^{th} observation whereas on the EWMA control chart, it was detected as out-of-control at the 20^{th} observation. Therefore, the EEWMA control chart can detect shift more quickly than the EWMA control chart.

In the study, the performances of control charts were evaluated by using _{2} = 0.03 performed better than the EEWMA with either _{2} = 0.01 or _{2} = 0.02, CUSUM, or EWMA control charts for most magnitudes of changes except for a large shift sizes (