# Average Run Length on CUSUM Control Chart for Seasonal and Non-Seasonal Moving Average Processes with Exogenous Variables

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}) and the out-of-control state (ARL

_{1}) are evaluated as the efficiency criteria; a large ARL

_{0}means that the control chart can be applied to efficiently control the process, whereas a small ARL

_{1}demonstrates that the control chart can detect a change in the process quickly. The Numerical Integral Equation (NIE) method is widely used for the continuous distribution of observations in real-world applications. Furthermore, Banach’s fixed-point theory has been adopted to prove the existence and uniqueness of the ARL in the following researches. Sunthornwat et al. investigated an explicit formula for the ARL on an EWMA control chart for the Autoregressive Fractionally Integrated Moving Average (ARFIMA) model with exponential white noise [4], while Mititelu et al. [22] solved one representing the ARL on a CUSUM control chart based on observations in a hyper-exponential distribution. Their findings show that the explicit formula ARL was more quickly evaluated than the NIE ARL. Petcharat et al. derived an explicit formula ARL, while using an integral equation method on a CUSUM control chart for an MA model with exponential white noise [23], with its existence and uniqueness being proved via Banach’s fixed-point theory. However, the optimal parameters could not be obtained in an MA model.

## 2. Preliminaries

#### 2.1. The MAX(q,r) and SMAX(Q,r)_{L} Processes

**Definition**

**1.**

_{L}process given by the expression

- ${\epsilon}_{t}$is a exponential white noise process,
- $\mu $is a process mean,
- $(1-{\theta}_{1}\mathrm{B}-{\theta}_{2}{\mathrm{B}}^{2}-\dots -{\theta}_{q}{\mathrm{B}}^{q})$is the moving average polynomials in$\mathrm{B}$of order$q$.
- $(1-{\theta}_{L}{\mathrm{B}}^{L}-{\theta}_{2L}{\mathrm{B}}^{2L}-\dots -{\theta}_{QL}{\mathrm{B}}^{QL})$is the seasonal moving average polynomials in$\mathrm{B}$of order$Q;$$L$is a natural number,
- $B$is the backward shift operator, i.e.,${B}^{q}{\epsilon}_{t}={\epsilon}_{t-q},$and
- ${X}_{it}$is exogenous variable and${\beta}_{i}$is a coefficient of${X}_{it}.$

#### 2.2. Fixed Point and Metric Space

**Definition**

**2.**

**Definition**

**3.**

- (1)
- $\rho (x,y)\ge 0$ i.e., $\rho $ is finite and non-negative real valued function.
- (2)
- $\rho (x,y)=0$ if, and only if, $x=y.$
- (3)
- $\rho (x,y)=\rho (y,x),$ (Symmetric property)
- (4)
- $\rho (x,z)\le \rho (x,y)+\rho (y,z)$, (Triangular inequality).

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Theorem**

**1.**

#### 2.3. CUSUM and EWMA Control Charts for the MAX(q,r) and SMAX(Q,r)_{L} Processes

_{L}processes. ${Y}_{t}$ can be expressed by the recursive formula, as

_{L}processes with exponential white noise, the starting value $\text{\hspace{0.17em}}{Y}_{0}=s$ is an initial value; $s\in [0,h]$, where $h$ is a control limit and $a$ is usually called the reference value of CUSUM chart. The CUSUM stopping time (${\tau}_{h}$) with predetermined threshold $h$ is defined as

_{L}processes with exponential white noise.

- Upper control limit: $UC{L}_{t}=\mu +\kappa \sigma \sqrt{\frac{\eta}{(2-\eta )}\left[1-{(1-\eta )}^{2t}\right]}$
- Center Line: $CL=\mu $
- Lower control limit: $LC{L}_{t}=\mu -\kappa \sigma \sqrt{\frac{\eta}{(2-\eta )}\left[1-{(1-\eta )}^{2t}\right]}$

#### 2.4. Characteristics of Average Run Length

_{0,}or the state of no change $(\lambda ={\lambda}_{0})$. The expectation of the run length ${\tau}_{h}$ in the -control state can be defined as

_{1}, which can be defined as

## 3. The Explicit Formulas for Average Run Lengths with MAX(q,r) and SMAX(Q,r)_{L} Processes

_{L}processes with exponential white noise from the integral equations, after checking the existence and uniqueness of the solutions for the ARL, are presented, as follows:

**Theorem**

**2.**

**Proof.**

_{0}can be written as

_{1}can be written as

**Theorem**

**3.**

_{L}process is

**Proof.**

_{0}can be written as

_{1}can be written as

## 4. Explicit Formulas and Numerical Integral Equation Method for Average Run Length

#### 4.1. Existence and Uniqueness of the Explicit Formulas for Average Run Lengths

**Theorem**

**4.**

**Proof**

**(Existence)**

**Proof**

**(Uniqueness)**

#### 4.2. The Numerical Integral Equation Method

#### 4.3. The Numerical Procedure for Obtaining Optimal Parameters for MAX Designs

## 5. Computational Results and Real Application

#### 5.1. Numeric Results

_{12,}are reported in Table 1 and Table 2 for $AR{L}_{0}=370,500$, with which they are in good agreement. Notice that the absolute percentage relative error is small. In Table 3 and Table 4, we use Equations (8) and (9) to show $AR{L}_{0}$ and $AR{L}_{1}$ for the MAX (2,1) process with parameter $a=3$ and the coefficient parameters of the process ${\theta}_{1}=0.1,{\theta}_{2}=0.2$, and ${\beta}_{1}=0.5.$ For Equations (12) and (13), the parameters $a=3,\text{\hspace{0.17em}}{\Theta}_{1}=0.1,$ ${\Theta}_{2}=0.2,\text{\hspace{0.17em}}$ ${\Theta}_{3}=0.3$, and ${\beta}_{1}=0.5.$ were used for the SMAX (3,1)

_{12}process. The parameter value ${\lambda}_{0}=1$ was applied to the in-control process. Meanwhile, for the out-of-control process (${\lambda}_{1}>{\lambda}_{0}$), parameter values ${\lambda}_{1}={\lambda}_{0}(1+\delta )$ were used, with shift sizes of 0.01, 0.03, 0.05, 0.10, 0.30, 0.50, 1.0, 1.5, and 2.0. The first row in both tables shows that the results of ARL

_{0}with the explicit formula were close to the NIE method, when ARL

_{0}approached 370 and 500. The values in parentheses are the CPU times of the ARL from both of the methods. The ARL values of the explicit formula and the NIE method were similar and tended to decrease when the level of the shift size increased. Note that the absolute percentage relative error was very small and the CPU time with the explicit formula was just a fraction of a second, while the NIE method took around 11–13 min.

_{12}processes for $AR{L}_{0}=370$ are shown. For example, if we want to detect a parameter change from ${\lambda}_{0}=1$ to ${\lambda}_{1}>{\lambda}_{0}$ and the $AR{L}_{0}$ is 370, then the optimality procedure given above will give the optimal parameter values $a$ = 1.66999118582 and $h$ = 5.9737144930593 and $AR{L}_{1}{}^{*}$ value = 13.552. The suggested explicit formulas are useful to practitioners, especially when finding the optimal parameters of the MAX and SMAX processes for the CUSUM chart.

#### 5.2. Real-World Application

_{1}than the EWMA control chart when the shift size was small, but the EWMA control chart performed better than the CUSUM control chart when the shift size was $\delta \ge 0.015$.

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Table 1.**ARL values for in control process for MAX (2,1) using explicit formula against numerical integral equation given ${\beta}_{1}=0.5$ for $AR{L}_{0}=370$.

$\mathit{a}$ | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | $\mathit{h}$ | Explicit | NIE | APRE (%) |
---|---|---|---|---|---|---|

1.5 | 0.1 | 0.2 | 4.889 | 370.485 | 369.123 (12.821) | 0.365197 |

0.1 | −0.2 | 5.442 | 370.211 | 368.657 (12.885) | 0.317387 | |

2.5 | 0.1 | 0.2 | 3.971 | 370.432 | 369.123 (12.897) | 0.353371 |

0.1 | −0.2 | 4.887 | 370.008 | 368.657 (12.777) | 0.365127 | |

3 | 0.1 | 0.2 | 3.265 | 370.225 | 369.087 (12.818) | 0.307381 |

0.1 | −0.2 | 3.811 | 370.154 | 368.879 (12.843) | 0.344451 |

**Table 2.**ARL values for in control process for SMAX (3,1)

_{12}using explicit formula against numerical integral equation given ${\beta}_{1}=0.5$ for $AR{L}_{0}=370$.

$\mathit{a}$ | ${\mathbf{\Theta}}_{1}$ | ${\mathbf{\Theta}}_{2}$ | ${\mathbf{\Theta}}_{3}$ | $\mathit{h}$ | Explicit | NIE | APRE (%) |
---|---|---|---|---|---|---|---|

1.5 | 0.1 | 0.2 | 0.3 | 3.525 | 370.411 | 369.202 (13.159) | 0.326394 |

0.1 | −0.2 | 0.3 | 4.145 | 370.071 | 369.652 (13.258) | 0.396805 | |

2.5 | 0.1 | 0.2 | 0.3 | 3.525 | 370.411 | 369.202 (13.810) | 0.326394 |

0.1 | −0.2 | 0.3 | 4.145 | 370.132 | 368.797 (13.574) | 0.360682 | |

3 | 0.1 | 0.2 | 0.3 | 2.906 | 370.008 | 368.979 (13.763) | 0.278102 |

0.1 | −0.2 | 0.3 | 3.392 | 370.202 | 369.029 (13.564) | 0.316854 |

**Table 3.**ARL values for MAX (2,1) process using explicit formula against numerical integral equation given $a=3$, ${\theta}_{1}=0.1,{\theta}_{2}=0.2,$ ${\beta}_{1}=0.5$, $h=3.265$ for $AR{L}_{0}=370$ and $h=3.588$ for $AR{L}_{0}=500$.

δ | $\mathit{A}\mathit{R}{\mathit{L}}_{0}=370$ | APRE (%) | $\mathit{A}\mathit{R}{\mathit{L}}_{0}=500$ | APRE (%) | ||
---|---|---|---|---|---|---|

(Shift Size) | Explicit | NIE | Explicit | NIE | ||

0.00 | 370.225 | 369.087 (12.235) | 0.307381 | 500.080 | 498.380 (11.088) | 0.339946 |

0.01 | 347.839 | 346.783 12.684) | 0.303589 | 468.139 | 466.570 (11.137) | 0.335157 |

0.03 | 308.154 | 307.243 (12.486) | 0.295631 | 411.811 | 410.466 (11.199) | 0.326606 |

0.05 | 274.253 | 273.462 (12.264) | 0.28842 | 364.017 | 362.859 (10.497) | 0.318117 |

0.07 | 245.143 | 244.453 (12.994) | 0.281468 | 323.248 | 322.245 (10.706) | 0.310288 |

0.1 | 208.758 | 208.191 (11.092) | 0.271606 | 272.684 | 271.869 (11.218) | 0.298881 |

0.3 | 86.578 | 86.391 (11.165) | 0.21599 | 107.354 | 107.100 (11.242) | 0.23660 |

0.5 | 45.641 | 45.561 (11.441) | 0.175281 | 54.493 | 54.389 (11.282) | 0.19085 |

1.0 | 16.512 | 16.494 (11.559) | 0.109012 | 18.611 | 18.588 (11.670) | 0.123583 |

1.5 | 9.183 | 9.176 (11.412) | 0.076228 | 10.045 | 10.037 (11.363) | 0.079642 |

2.0 | 6.288 | 6.256 (11.012) | 0.38610 | 6.761 | 6.758 (11.360) | 0.044372 |

**Table 4.**ARL values using explicit formula against numerical integral equation for SMAX (3,1)

_{12}given $a=3$, ${\Theta}_{1}=0.1,{\Theta}_{2}=0.2,{\Theta}_{3}=0.3,$ ${\beta}_{1}=0.5,$ $h=2.906$ for $AR{L}_{0}=370$ and $h=3.223$ for $AR{L}_{0}=500$.

δ | $\mathit{A}\mathit{R}{\mathit{L}}_{0}=370$ | APRE (%) | $\mathit{A}\mathit{R}{\mathit{L}}_{0}=500$ | APRE (%) | ||
---|---|---|---|---|---|---|

(Shift Size) | Explicit | NIE | Explicit | NIE | ||

0.00 | 370.008 | 368.979 (13.753) | 0.278102 | 500.438 | 498.887 (13.859) | 0.309929 |

0.01 | 348.077 | 347.120 (13.961) | 0.274939 | 469.150 | 467.714 (14.024) | 0.306085 |

0.03 | 309.124 | 308.294 (13.439) | 0.268501 | 413.854 | 412.618 (14.150) | 0.298656 |

0.05 | 275.763 | 275.040 (11.606) | 0.262182 | 366.803 | 365.733 (14.016) | 0.29171 |

0.07 | 247.047 | 246.414 (11.660) | 0.256227 | 326.556 | 325.625 (14.204) | 0.285097 |

0.1 | 211.048 | 210.525 (13.293) | 0.247811 | 276.476 | 275.714 (13.411) | 0.275612 |

0.3 | 88.943 | 88.764 (13.893) | 0.201252 | 110.871 | 110.625 (11.886) | 0.221879 |

0.5 | 47.311 | 47.233 (13.971) | 0.164867 | 56.868 | 56.764 (11.878) | 0.18288 |

1.0 | 17.208 | 17.190 (14.135) | 0.104603 | 19.542 | 19.518 (12.244) | 0.122812 |

1.5 | 9.530 | 9.523 (13.920) | 0.073452 | 10.493 | 10.484 (11.859) | 0.085771 |

2.0 | 6.486 | 6.482 (14.323) | 0.061671 | 7.011 | 7.006 (12.324) | 0.071317 |

**Table 5.**Optimal design parameters and $AR{L}_{1}^{*}$ of MAX (1,2) and SMAX (2,2)

_{12}processes for $AR{L}_{0}=370$.

δ | MAX (1,2) | $\mathit{A}\mathit{R}{\mathit{L}}_{1}^{*}$ | SMAX (2,2)_{12} | $\mathit{A}\mathit{R}{\mathit{L}}_{1}^{*}$ | ||
---|---|---|---|---|---|---|

(Shift Size) | $\mathit{a}$ | $\mathit{h}$ | $\mathit{a}$ | $\mathit{h}$ | ||

0.01 | 1.66999118602 | 5.9737144875716 | 315.357 | 2.354154802 | 5.4156807185 | 165.571 |

005 | 1.66999118587 | 5.973714491718 | 207.994 | 2.360429504 | 5.28394587940 | 126.065 |

0.1 | 1.66999118585 | 5.9737144920653 | 128.53 | 2.361119789 | 5.28742459511 | 87.092 |

0.3 | 1.66999118582 | 5.9737144930593 | 29.355 | 2.361410035 | 5.2778673182 | 29.659 |

0.5 | 1.66999118582 | 5.9737144930593 | 13.552 | 2.361410035 | 5.2778673147 | 16.122 |

1.0 | 1.66999118585 | 5.9737144920654 | 7.619 | 2.361410035 | 5.2778673174 | 8.158 |

2.0 | 1.66999118611 | 5.973714485385 | 4.936 | 2.354154797 | 5.415680824 | 4.641 |

**Table 6.**Comparison of ARL values using explicit formula against numerical integral equation for MAX (1,2) when given a = 550, ${\theta}_{1}=0.746083$, ${\beta}_{1}=-11.773,$ ${\beta}_{2}=-61.94972$, $h=22.48$ for $AR{L}_{0}=370$ and $h=24.71$ for $AR{L}_{0}=500$ ${\lambda}_{0}=6.9094$.

δ | $\mathit{A}\mathit{R}{\mathit{L}}_{0}=370$ | APRE (%) | $\mathit{A}\mathit{R}{\mathit{L}}_{0}=500$ | APRE (%) | ||
---|---|---|---|---|---|---|

(Shift Size) | Explicit | NIE | Explicit | NIE | ||

0.00 | 370.437 | 369.302 (10.642) | 0.306394 | 500.389 | 498.693 (10.630) | 0.338936 |

0.01 | 367.082 | 365.959 (11.017) | 0.305926 | 495.596 | 493.919 (11.0590 | 0.33838 |

0.05 | 354.056 | 352.981 (11.106) | 0.303624 | 477.007 | 475.406 (11.122) | 0.335634 |

0.1 | 338.617 | 337.598 (11.103) | 0.300929 | 455.083 | 453.511 (11.586) | 0.345431 |

0.3 | 285.049 | 284.222 (10.352) | 0.290125 | 379.214 | 378.000 (11.111) | 0.320136 |

0.5 | 242.186 | 241.508 (10.965) | 0.279950 | 319.135 | 318.150 (10.628) | 0.308647 |

0.7 | 207.537 | 206.976 (11.193) | 0.270313 | 271.011 | 270.204 (10.755) | 0.297774 |

1.0 | 167.067 | 166.638 (11.902) | 0.256783 | 215.395 | 214.787 (11.016) | 0.282272 |

2.0 | 90.192 | 89.995 (11.758) | 0.218422 | 112.120 | 111.851 (11.116) | 0.239922 |

3.0 | 55.293 | 55.189 (11.212) | 0.188088 | 66.776 | 66.639 (11.467) | 0.205164 |

**Table 7.**Performance comparison of cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts using explicit formula (Explicit) for MAX (1,2) when ${\lambda}_{0}=6.9094$ for $AR{L}_{0}=370$ and $AR{L}_{0}=500$.

δ | $\mathit{A}\mathit{R}{\mathit{L}}_{0}=370$ | $\mathit{A}\mathit{R}{\mathit{L}}_{0}=500$ | ||
---|---|---|---|---|

(Shift Size) | CUSUM | EWMA | CUSUM | EWMA |

$\mathit{h}=0.5333$ | $\mathit{h}=0.8791$ | |||

0.000 | 370.437 | 370.423 (16.215) | 500.389 | 500.433 (16.036) |

0.005 | 368.755 | 369.083 (16.202) | 497.985 | 498.303 (16.265) |

0.010 | 367.082 | 368.191 (16.112) | 495.596 | 495.817 (16.085) |

0.015 | 365.420 | 365.311 (15.489) | 493.221 | 490.892 (16.878) |

0.050 | 354.056 | 348.928 (16.156) | 477.007 | 471.804 (16.966) |

0.1 | 338.617 | 335.141 (15.005) | 455.083 | 453.656 (16.255) |

0.3 | 285.049 | 275.716 (16.485) | 379.214 | 375.205 (16.255) |

0.5 | 242.186 | 229.117 (15.152) | 319.135 | 313.385 (16.008) |

1.0 | 167.067 | 149.136 (15.156) | 215.395 | 207.585 (16.026) |

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## Share and Cite

**MDPI and ACS Style**

Sunthornwat, R.; Areepong, Y.
Average Run Length on CUSUM Control Chart for Seasonal and Non-Seasonal Moving Average Processes with Exogenous Variables. *Symmetry* **2020**, *12*, 173.
https://doi.org/10.3390/sym12010173

**AMA Style**

Sunthornwat R, Areepong Y.
Average Run Length on CUSUM Control Chart for Seasonal and Non-Seasonal Moving Average Processes with Exogenous Variables. *Symmetry*. 2020; 12(1):173.
https://doi.org/10.3390/sym12010173

**Chicago/Turabian Style**

Sunthornwat, Rapin, and Yupaporn Areepong.
2020. "Average Run Length on CUSUM Control Chart for Seasonal and Non-Seasonal Moving Average Processes with Exogenous Variables" *Symmetry* 12, no. 1: 173.
https://doi.org/10.3390/sym12010173