The Extended Exponentially Weighted Moving Average (extended EWMA) control chart is one of the control charts and can be used to quickly detect a small shift. The performance of control charts can be evaluated with the average run length (

The Control chart is one of the statistical process control instruments and has been applied in many fields such as finance, economics, industry, health, and medicine (see [

The average run length (_{0} is the expected number of observations before an in-control process is taken to signal to be out of control and should be large, whereas _{1} is the expected number of observations taken from out of control and should be as small as possible. Previous research has shown that the

There is a body of literature on evaluating the

The EWMA control chart was initially proposed by Roberts [_{t} is a process with mean, _{0} is the initial value of the EWMA statistic, _{0} =

The stopping time is given by

The extended EWMA control chart was proposed by Neveed et al. [_{t} is a process with mean, _{1} and _{2} are exponential smoothing parameters with _{1} ∈ (0, 1) and _{2} ∈ (0, _{1}), _{0} is the initial value of the extended EWMA statistic, _{0} =

The stopping time is given by

The observations equation for the autoregressive with trend or the trend AR(p) model in the case of exponential while noise is defined as_{i} is an autoregressive coefficient at _{p}| < 1 and ɛ_{t} is white noise sequence of exponential (ɛ_{t} ∼ _{t} is given by _{t} can be written as

If _{t} for in-control process, then _{t} <

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

Let _{E}(_{E}(_{E}(

So, the function _{E}(

Next step, _{E}(

If ɛ_{t} ∼

Consequently,

Consider the constant

Substituting constant _{E}(

Finally, the solution of _{0}, whereas the process is out-of-control with the exponential parameter _{1}, and then _{1} = (1 + _{0} where _{1} > _{0} and

The NIE method is one of the techniques that is used to approximate the _{N}(

The

The system of m linear equation is showed as:

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

So, the solution of numerical integral equation can be explained as_{j} is a set of the division point on the interval [_{j} is a weight of the composite midpoint formula

By using the Banach's Fixed-Point Theorem, the

If an operator _{E}(_{E}(

_{1}) − _{2})‖ ≤ _{1} − _{2}‖, _{E}(_{E}(

_{1}, _{2} ∈ _{1}) − _{2})‖ ≤ _{1} − _{2}‖,

According to

The speed test results were computed by the CPU time (PC System: windows10, 64-bit, Intel® Core™ i5-8250U 1.60, 1.80 GHz, RAM 4 GB) in seconds. In addition, the numerical results were computed by MATHEMATICA. The initial parameter value was studied at _{0} = 370 on a two-sided extended EWMA chart for the trend AR(p) model namely the trend AR(1) and trend AR(2) models with exponential white noise and given _{1} = 0.05, _{2} = 0.01. The in-control process was presented a parameter value as _{0} with shift size (_{1} = (1 + _{0} with shift sizes (_{1} = 0.1, − 0.1) and (_{1} = 0.1, _{2} = 0.1, − 0.1) were used for the trend AR(1) model, and the trend AR(2) model, respectively. The process has determined that slope

The performance comparisons of the explicit formula (as

Shift size ( |
_{1} = 0.1 |
|||||
---|---|---|---|---|---|---|

Explicit ( |
NIE (_{NIE}( |
APRE(%) | Explicit ( |
NIE (_{NIE}( |
APRE(%) | |

0.000 | 370.0028282 |
370.0028124 |
0.0000043 | 370.0022007 |
370.0021940 |
0.0000018 |

0.001 | 222.6285267 |
222.6285189 |
0.0000035 | 207.5881058 |
207.5881027 |
0.0000015 |

0.003 | 124.2572117 |
124.2572081 |
0.0000030 | 110.8824977 |
110.8824963 |
0.0000013 |

0.005 | 86.39599653 |
86.39599417 |
0.0000027 | 75.85371836 |
75.85371747 |
0.0000012 |

0.010 | 49.34352289 |
49.34352166 |
0.0000025 | 42.66567897 |
42.66567851 |
0.0000011 |

0.030 | 18.74823467 |
18.74823426 |
0.0000022 | 16.04073087 |
16.04073071 |
0.0000010 |

0.050 | 11.91320517 |
11.91320493 |
0.0000020 | 10.18682847 |
10.18682838 |
0.0000009 |

0.100 | 6.606133419 |
6.606133309 |
0.0000017 | 5.666575969 |
5.666575929 |
0.0000007 |

0.500 | 2.217555232 |
2.217555219 |
0.0000006 | 1.959621512 |
1.959621508 |
0.0000002 |

1.000 | 1.640867750 |
1.640867746 |
0.0000002 | 1.485839866 |
1.485839865 |
0.0000001 |

Note: For _{1} = 0.1 (_{1} = −0.1 (

Shift size ( |
_{1} = _{2} = 0.1 |
|||||
---|---|---|---|---|---|---|

Explicit ( |
NIE (_{NIE}( |
APRE(%) | Explicit ( |
NIE (_{NIE}( |
APRE(%) | |

0.000 | 370.0047719 |
370.0047589 |
0.0000035 | 370.0047133 |
370.0047008 |
0.0000034 |

0.001 | 218.9666453 |
218.966639 |
0.0000029 | 218.1789684 |
218.1789624 |
0.0000027 |

0.003 | 120.8909144 |
120.8909115 |
0.0000024 | 120.1764142 |
120.1764114 |
0.0000023 |

0.005 | 83.71006657 |
83.71006468 |
0.0000023 | 83.14282505 |
83.14282326 |
0.0000022 |

0.010 | 47.62257905 |
47.62257806 |
0.0000021 | 47.26085617 |
47.26085524 |
0.0000020 |

0.030 | 18.04477709 |
18.04477677 |
0.0000018 | 17.89740758 |
17.89740727 |
0.0000017 |

0.050 | 11.46428946 |
11.46428927 |
0.0000017 | 11.37026579 |
11.37026561 |
0.0000016 |

0.100 | 6.362069199 |
6.362069111 |
0.0000014 | 6.310913041 |
6.310912957 |
0.0000013 |

0.500 | 2.150983712 |
2.150983702 |
0.0000005 | 2.136969812 |
2.136969802 |
0.0000004 |

1.000 | 1.600855788 |
1.600855785 |
0.0000002 | 1.592424578 |
1.592424575 |
0.0000002 |

Note: For _{1} = _{2} = 0.1 (_{1} = 0.1, _{2} = −0.1 (

The relative mean index (_{i}(_{i}(

For _{1} = 0.05, _{2} = 0.01, _{0} = 370 show that the performance of a two-sided extended EWMA control chart under various bound control limits [_{1} = 0.2, − 0.2(as the trend AR(1) model) and _{1} = 0.2, _{2} = 0.2, − 0.2 (as the trend AR(2) model). The

_{1} |
Shift size | ||||
---|---|---|---|---|---|

0.2 | 0.000 | ( |
( |
( |
( |

0.001 | 231.650 | 215.150 | 180.897 | 146.963 | |

0.003 | 132.865 | 117.512 | 89.966 | 67.222 | |

0.005 | 93.365 | 81.082 | 60.171 | 43.907 | |

0.010 | 53.871 | 46.024 | 33.302 | 23.919 | |

0.030 | 20.616 | 17.507 | 12.640 | 9.174 | |

0.050 | 13.106 | 11.188 | 8.197 | 6.071 | |

0.100 | 7.252 | 6.292 | 4.786 | 3.706 | |

0.500 | 2.391 | 2.231 | 1.965 | 1.756 | |

1.000 | 1.744 | 1.681 | 1.569 | 1.476 | |

RMI | 0.784 | 0.584 | 0.253 | 0 | |

( |
( |
( |
( |
||

−0.2 | 0.000 | 370 | 370 | 370 | 370 |

0.001 | 200.983 | 183.627 | 149.271 | 117.348 | |

0.003 | 105.355 | 91.864 | 68.568 | 50.183 | |

0.005 | 71.594 | 61.491 | 44.788 | 32.225 | |

0.010 | 40.025 | 33.970 | 24.335 | 17.375 | |

0.030 | 14.987 | 12.722 | 9.207 | 6.727 | |

0.050 | 9.517 | 8.145 | 6.019 | 4.519 | |

0.100 | 5.302 | 4.633 | 3.588 | 2.841 | |

0.500 | 1.860 | 1.760 | 1.594 | 1.464 | |

1.000 | 1.426 | 1.389 | 1.325 | 1.271 | |

RMI | 0.793 | 0.588 | 0.250 | 0 |

_{2} |
Shift size | ||||
---|---|---|---|---|---|

0.2 | 0.000 | ( |
( |
( |
( |

0.001 | 227.892 | 211.209 | 176.806 | 142.998 | |

0.003 | 129.210 | 114.050 | 86.999 | 64.802 | |

0.005 | 90.386 | 78.367 | 57.992 | 42.221 | |

0.010 | 51.924 | 44.313 | 32.010 | 22.963 | |

0.030 | 19.810 | 16.818 | 12.140 | 8.814 | |

0.050 | 12.591 | 10.749 | 7.880 | 5.844 | |

0.100 | 6.974 | 6.054 | 4.613 | 3.580 | |

0.500 | 2.317 | 2.166 | 1.913 | 1.715 | |

1.000 | 1.700 | 1.640 | 1.535 | 1.447 | |

RMI | 0.787 | 0.586 | 0.254 | 0 | |

( |
( |
( |
( |
||

−0.2 | 0.000 | 370 | 370 | 370 | 370 |

0.001 | 226.072 | 209.328 | 174.880 | 141.146 | |

0.003 | 127.489 | 112.431 | 85.623 | 63.687 | |

0.005 | 88.995 | 77.106 | 56.987 | 41.447 | |

0.010 | 51.022 | 43.523 | 31.416 | 22.525 | |

0.030 | 19.438 | 16.500 | 11.911 | 8.650 | |

0.050 | 12.353 | 10.548 | 7.735 | 5.739 | |

0.100 | 6.845 | 5.944 | 4.534 | 3.522 | |

0.500 | 2.282 | 2.135 | 1.889 | 1.696 | |

1.000 | 1.680 | 1.621 | 1.519 | 1.434 | |

RMI | 0.788 | 0.586 | 0.254 | 0 |

Besides, a two-sided extended EWMA control chart under various conditions (_{2} = 0.01, 0.02, 0.04) are compared to the EWMA control chart (_{2} = 0) at _{1} = 0.05, 0.10, _{0} = 370 for _{1} = 0.3 (as the trend AR(1) model) and _{1} = _{2} = 0.3 (as the trend AR(2) model). The lower and upper control limits of the EWMA and extended EWMA control charts for the trend AR(1) and trend AR(2) models are obtained in _{1} = 0.05 is equal to 0. The exponential smoothing parameter of 0.05 is recommended. In addition, the _{2} = 0.04(EEWMA0.04), (_{2} = 0.01(EEWMA0.01) or _{2} = 0.02(EEWMA0.02) and the EWMA(_{2} = 0) control chart for all situations, both the trend AR(1) and the trend AR(2) models.

_{1} |
EWMA (_{2} = 0) |
Extended EWMA | ||||||
---|---|---|---|---|---|---|---|---|

_{2} = 0.01 |
||||||||

0.05 | 0.05 | 0.1406020 | 0.05 | 0.10884124 | 0.05 | 0.08858456 | 0.05 | 0.06682472 |

0.10 | 0.05 | 0.2401663 | 0.05 | 0.20085810 | 0.05 | 0.17053690 | 0.05 | 0.12804540 |

_{1} |
EWMA (_{2} = 0) |
Extended EWMA | ||||||
---|---|---|---|---|---|---|---|---|

_{2} = 0.01 |
||||||||

0.05 | 0.05 | 0.1342092 | 0.05 | 0.1047736 | 0.05 | 0.08594707 | 0.05 | 0.06568533 |

0.10 | 0.05 | 0.2260982 | 0.05 | 0.1899824 | 0.05 | 0.16200730 | 0.05 | 0.12265030 |

_{1} |
Shift size ( |
EWMA (_{2} = 0) |
Extended EWMA | ||
---|---|---|---|---|---|

_{2} = 0.01 |
|||||

0.05 | 0.000 | 370 | 370 | 370 | 370 |

0.001 | 206.213 | 158.863 | 135.905 | 108.374 | |

0.003 | 109.835 | 74.781 | 60.584 | 45.468 | |

0.005 | 75.160 | 49.233 | 39.305 | 29.076 | |

0.010 | 42.422 | 26.969 | 21.320 | 15.653 | |

0.030 | 16.237 | 10.324 | 8.202 | 6.100 | |

0.050 | 10.488 | 6.797 | 5.457 | 4.126 | |

0.100 | 6.049 | 4.103 | 3.368 | 2.628 | |

0.500 | 2.375 | 1.881 | 1.649 | 1.400 | |

1.000 | 1.860 | 1.561 | 1.402 | 1.229 | |

RMI (_{1}) |
0 | 0 | 0 | 0 | |

RMI (_{2}) |
1.133 | 0.504 | 0.257 | 0 | |

0.10 | 0.000 | 370 | 370 | 370 | 370 |

0.001 | 260.799 | 228.889 | 208.561 | 182.362 | |

0.003 | 164.213 | 130.260 | 111.791 | 90.978 | |

0.005 | 120.038 | 91.303 | 76.646 | 60.892 | |

0.010 | 72.088 | 52.625 | 43.301 | 33.700 | |

0.030 | 28.412 | 20.265 | 16.515 | 12.749 | |

0.050 | 18.095 | 12.981 | 10.619 | 8.240 | |

0.100 | 9.939 | 7.309 | 6.061 | 4.778 | |

0.500 | 3.111 | 2.576 | 2.272 | 1.918 | |

1.000 | 2.196 | 1.919 | 1.743 | 1.523 | |

RMI (_{1}) |
0.467 | 0.624 | 0.674 | 0.745 | |

RMI (_{2}) |
0.791 | 0.405 | 0.21 | 0 |

_{1} |
Shift size ( |
EWMA |
Extended EWMA | ||
---|---|---|---|---|---|

_{2} = 0.01 |
|||||

0.05 | 0.000 | 370 | 370 | 370 | 370 |

0.001 | 194.938 | 154.309 | 133.141 | 106.664 | |

0.003 | 100.661 | 71.834 | 58.977 | 44.594 | |

0.005 | 68.169 | 47.146 | 38.200 | 28.495 | |

0.010 | 38.149 | 25.769 | 20.700 | 15.336 | |

0.030 | 14.584 | 9.871 | 7.970 | 5.983 | |

0.050 | 9.462 | 6.511 | 5.310 | 4.051 | |

0.100 | 5.518 | 3.947 | 3.286 | 2.587 | |

0.500 | 2.255 | 1.833 | 1.621 | 1.386 | |

1.000 | 1.793 | 1.528 | 1.382 | 1.219 | |

RMI (_{1}) |
0 | 0 | 0 | 0 | |

RMI (_{2}) |
0.853 | 0.475 | 0.248 | 0 | |

0.10 | 0.000 | 370 | 370 | 370 | 370 |

0.001 | 248.774 | 221.608 | 203.526 | 179.223 | |

0.003 | 150.619 | 123.403 | 107.551 | 88.683 | |

0.005 | 108.242 | 85.787 | 73.372 | 59.201 | |

0.010 | 63.899 | 49.071 | 41.271 | 32.694 | |

0.030 | 24.929 | 18.824 | 15.710 | 12.359 | |

0.050 | 15.911 | 12.074 | 10.111 | 7.992 | |

0.100 | 8.827 | 6.832 | 5.787 | 4.642 | |

0.500 | 2.900 | 2.463 | 2.199 | 1.877 | |

1.000 | 2.091 | 1.855 | 1.698 | 1.497 | |

RMI (_{1}) |
0.448 | 0.593 | 0.649 | 0.729 | |

RMI (_{2}) |
0.672 | 0.364 | 0.194 | 0 |

The results indicate that the performances of the control charts were, in ascending order, the extended EWMA with _{2} = 0.04, extended EWMA with _{2} = 0.02, extended EWMA with _{2} = 0.01 and EWMA control charts, as illustrated in

The _{0} = 370 for _{1} = 0.05 and various _{2} = 0.01, 0.02, 0.04, and its performance was compared with the EWMA (_{2} = 0) control chart using data on the number of COVID-19 patients in hospitals per million people in the United Kingdom and Sweden. The observations were made daily from June 26^{th} to September 10^{th}, 2021 and from January 24^{th} to April 20^{th}, 2021, respectively. This data is a stationary time series by looking at the autocorrelation function (ACF) and partial autocorrelation function (PACF). The dataset for the trend AR(1) model was assigned as the significance of the mean and standard deviation were 79.32026 and 27.69376, respectively. The trend AR(p) model in _{t} = (0.810841)_{t−1} + ɛ_{t} and the error was exponential white noise (_{0} = 1.684939). Meanwhile, the observations of the trend AR(2) model was defined as _{t} = (0.592171)_{t−1} + (0.219693)_{t−2} + ɛ_{t} and the error was exponential white noise (_{0} = 0.544925). The _{2} = 0.04 chart reduced the _{2} = 0.01 or _{2} = 0.02 and the EWMA control chart for all situations, both the trend AR(1) and trend AR(2) models. The results in

Shift size ( |
EWMA (_{2} = 0) |
Extended EWMA | ||
---|---|---|---|---|

_{2} = 0.01 |
||||

( |
||||

0.000 | 370 | 370 | 370 | 370 |

0.001 | 269.483 | 223.565 | 198.024 | 168.576 |

0.003 | 174.783 | 125.164 | 103.107 | 81.180 |

0.005 | 129.487 | 87.194 | 70.000 | 53.760 |

0.010 | 78.829 | 49.985 | 39.232 | 29.525 |

0.030 | 31.274 | 19.215 | 14.951 | 11.191 |

0.050 | 19.846 | 12.335 | 9.658 | 7.284 |

0.100 | 10.775 | 6.990 | 5.578 | 4.292 |

0.500 | 3.238 | 2.537 | 2.192 | 1.823 |

1.000 | 2.265 | 1.915 | 1.714 | 1.479 |

RMI | 1.117 | 0.491 | 0.240 | 0 |

Shift size ( |
EWMA (_{2} = 0) |
Extended EWMA | ||
---|---|---|---|---|

_{2} = 0.01 |
||||

( |
||||

0.000 | 370 | 370 | 370 | 370 |

0.001 | 166.252 | 84.701 | 63.958 | 42.457 |

0.003 | 79.672 | 33.997 | 24.756 | 15.968 |

0.005 | 52.719 | 21.615 | 15.680 | 10.147 |

0.010 | 28.983 | 11.715 | 8.556 | 5.656 |

0.030 | 11.079 | 4.794 | 3.642 | 2.597 |

0.050 | 7.272 | 3.378 | 2.644 | 1.978 |

0.100 | 4.367 | 2.309 | 1.890 | 1.512 |

0.500 | 2.016 | 1.443 | 1.275 | 1.128 |

1.000 | 1.699 | 1.318 | 1.185 | 1.073 |

RMI | 2.443 | 0.691 | 0.334 | 0 |

Hence, the extended EWMA (_{2} = 0.04) and EWMA (_{2} = 0) control charts were plotted by calculating _{t} and _{t} for the two datasets when given _{1} = 0.05. Detecting the process with real data of the number of COVID-19 patients in hospitals per million people in the United Kingdom and Sweden were shown in ^{th} and 13^{th} observations, respectively. In ^{st} and 27^{th} observations, respectively. As a result, a two-sided extended EWMA control chart can detect shifts faster than the EWMA control chart.

The performances of control charts were evaluated by using _{1} conditions is examined, the _{1} = 0.05 is equal to 0. So, the exponential smoothing parameter of 0.05 is recommended. Furthermore, the extended EWMA control chart has a higher efficiency if _{2} is increased. After that, the extended EWMA control can detect shifts faster than the EWMA control chart when the datasets were verified by calculating the control charts. Finally, the simulation study and the performance illustration with real data using data on the number of COVID-19 patients in hospitals per million people in the United Kingdom and Sweden provided similar results.

We are grateful to the referees for their constructive comments and suggestions which helped to improve this research.

_{L}with exponential white noise on EWMA chart