An Advanced Quality Control Approach: Integrating Quadruple EWMA Strategy for Enhanced Sensitivity in Process Monitoring

Abstract

This study proposes the Quadruple Exponentially Weighted Moving Average (QEWMA) control chart, a novel monitoring scheme designed to enhance the detection of small-to-moderate process mean shifts in the presence of autocorrelation. While traditional EWMA-based charts often struggle with dependent data, the proposed QEWMA utilizes a four-layered smoothing mechanism to effectively filter noise in Moving Average processes. The performance of the QEWMA chart was rigorously evaluated using the Numerical Integral Equation (NIE) approach to calculate the Average Run Length (ARL) and the Standard Deviation of Run Length (SDRL). Comparative results across MA(1), MA(2), and MA(3) models demonstrate that the QEWMA chart significantly outperforms the standard EWMA, DEWMA, and TEWMA charts, particularly for subtle shifts ($\delta \le 0.10$). The practical utility of the proposed chart was further validated through two real-world applications: monitoring Thailand’s daily median income (MA(3)) and gold futures prices (MA(2)). In both applications, the QEWMA chart exhibited superior sensitivity and faster detection rates, providing more reliable signals for economic and financial surveillance. These findings suggest that the QEWMA chart is a robust and highly efficient tool for quality control in complex, autocorrelated industrial and economic environments.

1. Introduction

Statistical Process Control (SPC) serves as a foundational framework in modern quality engineering, primarily focused on monitoring and enhancing process stability by identifying assignable causes of variation at an incipient stage. Among the various methodologies within SPC, control charts remain the most indispensable tools due to their operational simplicity and diagnostic efficacy. The classical Shewhart-type control charts, pioneered by Shewhart [1], are inherently “memoryless”, as their decision rules rely exclusively on the most recent observation. While these charts exhibit high efficiency in detecting large-magnitude shifts in process parameters, they are mathematically recognized to be suboptimal for identifying small-to-moderate shifts.

To address this limitation, the literature has shifted toward memory-type control charts, which enhance sensitivity by integrating both current and historical data into the monitoring statistic. The Cumulative Sum (CUSUM) chart, proposed by Page [2], and the Exponentially Weighted Moving Average (EWMA) chart, introduced by Roberts [3], represent the two most prominent pillars of this category. The stochastic properties and run-length distributions of these schemes have been rigorously investigated through various analytical and numerical techniques, such as integral equations and Markov chain approximations [4,5,6,7,8]. While these foundational memory-type charts have a significant advantage over Shewhart charts, their main limitation is a rigid smoothing structure that may not adequately weight long-term historical dependencies in highly volatile or autocorrelated data streams, leading to a slower response to very subtle trend shifts.

Given the versatility of the EWMA scheme, several high-order modifications have been developed to further augment its detection capabilities. Shamma and Shamma [9] introduced the Double EWMA (DEWMA) chart by applying a second level of exponential smoothing. This concept was subsequently extended to the Triple EWMA (TEWMA) chart by Alevizakos et al. [10], who demonstrated that increasing the depth of the recursive smoothing operator significantly improves the chart’s responsiveness to subtle process deviations. However, the conversion from DEWMA to TEWMA brings additional mathematical complexity and parameter sensitivity. Alevizakos and Koukouvinos [10] found that higher-order charts are more sensitive, but require precise smoothing constant tuning to avoid over-smoothing, which might obscure the underlying process signal in non-normal situations. Recently, Alevizakos, Chatterjee, and Koukouvinos [11] established the Quadruple EWMA (QEWMA) control chart, which utilizes four levels of nested smoothing. Their findings confirmed that the QEWMA architecture provides superior performance over the EWMA, DEWMA, and TEWMA counterparts, particularly when the shift magnitude is small. Despite its effectiveness, the actual application of QEWMA has been limited due to the lack of precise analytical solutions for its Average Run Length (ARL) in complex dependent structures. While recent works have begun exploring QEWMA in varied settings, a significant gap remains in applying this high-order sensitivity to economic time series, where structural changes frequently occur alongside noisy oscillations.

Despite these mathematical advancements, the majority of existing research on the QEWMA chart operates under the assumption that process observations are independent and identically distributed (i.i.d.). However, in many contemporary industrial environments—such as chemical processes or high-frequency automated manufacturing—data often exhibit serial dependence due to physical constraints or measurement systems. Neglecting such inherent autocorrelation can severely distort the chart’s stochastic properties, leading to inflated false alarm rates and a significant loss of detection power. A flexible and widely adopted framework for modeling this serial correlation is the Moving Average process of order q, denoted as MA(q) [12]. It is well-documented that such dependence significantly alters the run-length characteristics of memory-type charts [13,14,15].

Recent developments have extended the application of memory-type control charts into the field of econometrics, whereas traditional Statistical Process Control (SPC) concentrates on industrial manufacturing. In particular, sophisticated methods such as the Quadruple Exponentially Weighted Moving Average (QEWMA) have shown exceptional sensitivity in tracking regime shifts and structural changes in non-stationary economic time series. The performance and robustness of memory-type control charts under dependent processes have been extensively emphasized in recent research. For instance, Areepong and Peerajit [16] enhanced the detection capability of the EWMA chart within time-series contexts, while Peerajit [17] optimized its performance for complex long-memory seasonal moving average processes. Beyond conventional single-layer schemes, recent developments have shifted toward advanced configurations, such as multi-layered structures and extended frameworks. In particular, the Extended EWMA (EEWMA) and New Extended EWMA (NEEWMA) frameworks have demonstrated that incorporating multiple smoothing parameters effectively mitigates serial dependence under Autoregressive (AR) and Moving Average (MA) configurations (e.g., Javed et al. [18]; Karoon and Areepong [19]). Furthermore, the recent literature published between 2024 and 2026 demonstrates a significant growth in the application of advanced memory-type charts for financial anomaly detection and the monitoring of autocorrelated economic indicators [20,21].

Motivated by the need for more realistic monitoring models, this study rigorously investigates the performance of the QEWMA control chart under MA(q) dependence for q = 1, 2, and 3. We evaluate the Average Run Length (ARL) and other essential run-length characteristics across a range of autocorrelation levels. By extending the QEWMA framework to handle dependent processes, this research provides a more robust and effective monitoring tool for practitioners operating in autocorrelated production environments.

The main contributions of this paper are summarized as follows: A systematic mathematical analysis of the QEWMA control chart under MA(1), MA(2), and MA(3) processes. The application of a numerical integral equation method to derive high-precision ARL values. A comprehensive evaluation and comparison of the performance of the QEWMA scheme against existing EWMA-type charts in the presence of serial dependence.

The remainder of this paper is organized as follows: Section 2 delineates the mathematical formulation of the proposed QEWMA statistic and the underlying MA(q) processes. Section 3 provides a detailed description of the Numerical Integral Equation (NIE) approach employed for the precise calculation of the Average Run Length (ARL). Section 4 presents a comprehensive performance analysis, including comparative simulation results and real-world applications in economic and financial monitoring. Finally, Section 5 provides concluding remarks, summarizes the key findings, and suggests potential directions for future research.

2. The Conceptual Framework for the Control Chart

In this section, we delineate the mathematical structure of the proposed monitoring scheme, which encompasses the stochastic model of the process observations and the recursive formulation of the Quadruple Exponentially Weighted Moving Average (QEWMA) statistic.

2.1. The MA(q) Process and Underlying Assumptions

In many practical applications of statistical process control, the assumption of independent and identically distributed observations is often violated due to serial dependence inherent in the process. To account for such dependence, the underlying process in this study is modeled using a moving average process of order q, denoted by MA(q).

Consider a process where the observations $X_t$ are not independent but follow a moving average process of order $q$, denoted as MA(q). The model is expressed as:

$$X_t=\mu +{ϵ}_t-{\theta }_1{ϵ}_{t-1}-{\theta }_2{ϵ}_{t-2}-\dots -{\theta }_q{ϵ}_{t-q}$$

(1)

where $\mu$ is the process mean, ${\theta }_i$ ($i=1,2,\dots ,q$) are the moving average parameters, and ${ϵ}_t\sim Exp\left(\beta \right)$ represents white noise. In this study, we specifically investigate the performance under MA(1), MA(2), and MA(3) dependencies to represent various degrees of serial correlation often encountered in industrial applications. We adopt the exponential distribution to evaluate the robustness of the proposed chart against asymmetric error structures, providing a more challenging environment for ARL calculation than the standard normal distribution.

2.2. The QEWMA Control Chart Statistic

The QEWMA chart is an extension of the EWMA, DEWMA, and TEWMA schemes. It is constructed by applying the exponential smoothing operator four times sequentially. Let $X_t$ be the observation at time $t$. The four stages of the QEWMA statistic are defined as follows:

First stage (EWMA):

$$E_t=\lambda X_t+(1-\lambda )E_{t-1}$$

(2)

Second stage (DEWMA):

$$D_t=\lambda E_t+(1-\lambda )D_{t-1}$$

(3)

Third stage (TEWMA):

$$T_t=\lambda D_t+(1-\lambda )T_{t-1}$$

(4)

Fourth stage (QEWMA):

$$Q_t=\lambda T_t+(1-\lambda )Q_{t-1}$$

(5)

where $\lambda$ ($0<\lambda \le 1$) is the smoothing parameter. The initial values are typically set to the process target mean, $E_0=D_0=T_0=Q_0={\mu }_0$. The QEWMA chart signals an out-of-control state if the statistic $Q_t$ falls outside the control limits:

$$LCL,UCL={\mu }_0\pm L{\sigma }_{Q_t}$$

where $L$ is the control limit multiplier and ${\sigma }_{Q_t}$ denotes the asymptotic standard deviation of the QEWMA statistic.

When the observed statistic $Q_t$ falls outside the control limits, the process is declared out of control. Owing to the presence of serial dependence and the multi-level smoothing structure, the distribution of $Q_t$ is analytically complex. Consequently, performance measures such as the average run length (ARL) are evaluated using numerical methods.

The integration of the MA(q) process with the QEWMA control chart provides a unified framework for monitoring dependent processes with enhanced memory. This framework forms the basis for the analytical and numerical performance evaluation presented in the subsequent sections.

2.3. Performance Evaluation Metric

To assess the performance of the proposed QEWMA control chart under MA(q) processes, the Average Run Length (ARL) is utilized as the standard monitoring metric. The ARL is defined as the expected number of samples plotted on the chart before an out-of-control signal is generated.

ARL₀ (In-Control ARL): When the process is in a state of statistical control (δ = 0), the ARL₀ should be sufficiently large to minimize the frequency of false alarms.

ARL₁ (Out-of-Control ARL): When a shift in magnitude δ occurs in the process mean, the ARL₁ should be as small as possible, indicating the chart’s ability to detect the shift rapidly (see Figure 1).

3. Exact Analytical Derivation and Numerical Approximation of the QEWMA Run-Length Properties

In this section, the derivation of the Average Run Length (ARL) for the QEWMA control chart is developed through two distinct and complementary approaches: an explicit analytical expression and the Numerical Integral Equation (NIE) technique. Both methodologies are established within the Moving Average of order $q$ [MA(q)] modeling structure, ensuring the chart’s robustness and precision in monitoring autocorrelated processes.

3.1. Explicit Analytical Formulation

To provide a theoretical foundation for the chart’s performance, we first derive the exact statistical properties of the QEWMA statistic, $Q_t$. When the process observations $\left\{X_t\right\}$ are governed by an $MA(q)$ process. The QEWMA statistic, $Q_t$, is obtained by applying four successive levels of exponential smoothing. By recursive substitution, the relationship between $Q_t$ and the process observations $X_{t-i}$ can be expressed as:

$$Q_t={\lambda }^4{\displaystyle {\displaystyle \sum _{i=0}^{t-1}}\left(\begin{array}{c}i+3\\ 3\end{array}\right)}{(1-\lambda )}^i{X}_{t-i}+{(1-\lambda )}^t{\displaystyle {\displaystyle \sum _{j=0}^3}\frac{{(t\lambda )}^j}{j!}}{Q}_0$$

or

$$Q_t={\lambda }^4{X}_t+\left(1-\lambda \right)\left[{\lambda }^3{E}_{t-1}+{\lambda }^2{D}_{t-1}+\lambda T_{t-1}+Q_{t-1}\right]$$

(6)

where $\lambda$ ($0<\lambda \le 1$) is the smoothing parameter and $Q_0$ is the starting value (the detailed proofs of the expectation and variance of the QEWMA statistic are provided in Appendix A).

Consider a stochastic process $\left\{X_t\right\}\sim MA(q)$. The observations from this process are used as inputs in the construction of the QEWMA control chart, as defined below.

$$Q_t={\lambda }^4\left(\mu +{ϵ}_t-{\theta }_1{ϵ}_{t-1}-{\theta }_2{ϵ}_{t-2}-\dots -{\theta }_q{ϵ}_{t-q}\right)+\left(1-\lambda \right)\left[{\lambda }^3{E}_{t-1}+{\lambda }^2{D}_{t-1}+\lambda T_{t-1}+Q_{t-1}\right]$$

For t = 1,

$$Q_1={\lambda }^4\left(\mu +{ϵ}_1-{\theta }_1{ϵ}_0-{\theta }_2{ϵ}_{-1}-\dots -{\theta }_q{ϵ}_{1-q}\right)+\left(1-\lambda \right)\left[{\lambda }^3{E}_0+{\lambda }^2{D}_0+\lambda T_0+Q_0\right]$$

By starting values,

$$Q_1={\lambda }^4{ϵ}_1+{\lambda }^4\left(\mu -{\displaystyle \sum _{i=1}^q{\theta }_i{ϵ}_{1-i}}\right)+\left(1-\lambda \right){\mu }_0\left[{\lambda }^3+{\lambda }^2+\lambda +1\right]$$

(7)

The interval of $Q_1$ with lower bound $0$ and upper control limit $h$, is given by $0\le Q_1\le h$. This inequality can be further rearranged into an exponential form as follows:

$${\displaystyle \frac{-{\lambda }^4\left(\mu -{\displaystyle \sum _{i=1}^q{\theta }_i{ϵ}_{1-i}}\right)-\left(1-\lambda \right){\mu }_0\left[{\lambda }^3+{\lambda }^2+\lambda +1\right]}{{\lambda }^4}}\le {ϵ}_1$$

and

$${ϵ}_1\le {\displaystyle \frac{h-{\lambda }^4\left(\mu -{\displaystyle \sum _{i=1}^q{\theta }_i{ϵ}_{1-i}}\right)-\left(1-\lambda \right){\mu }_0\left[{\lambda }^3+{\lambda }^2+\lambda +1\right]}{{\lambda }^4}}$$

According to the method of Champ and Rigdon [22], which is based on a second-kind Fredholm integral equation, the function $\mathsf{\Lambda }\left({\mu }_0\right)$ is expressed as follows:

$$\mathsf{\Lambda }\left({\mu }_0\right)=1+\left[Exp\left({\displaystyle \frac{{\lambda }^4\left(\mu -{\displaystyle \sum _{i=1}^q{\theta }_i{ϵ}_{1-i}}\right)+\left(1-\lambda \right){\mu }_0\left[{\lambda }^3+{\lambda }^2+\lambda +1\right]}{\beta {\lambda }^4}}\right)\right]\left[{\displaystyle \frac{1-Exp\left(\frac{-h}{\beta {\lambda }^4}\right)}{1-\frac{\left\{Exp\left(\frac{\mu -{\displaystyle \sum _{i=1}^q{\theta }_i{ϵ}_{1-i}}}{\beta }\right)\right\}\left\{Exp\left(\frac{h\left[\left(1-\lambda \right)\left({\lambda }^3+{\lambda }^2+\lambda +1\right)-1\right]}{\beta {\lambda }^4}\right)-1\right\}}{\left(1-\lambda \right)\left[{\lambda }^3+{\lambda }^2+\lambda +1\right]-1}}}\right]$$

(8)

(The details showing the function $\mathsf{\Lambda }\left({\mu }_0\right)$ is provided in Appendix B).

3.2. Numerical Integral Equation (NIE) Approximation for ARL

The Average Run Length (ARL) is frequently evaluated using the Numerical Integral Equation (NIE) technique, which transforms the continuous integral into a solvable discrete system. Various quadrature rules—such as the midpoint rule, trapezoidal rule, Simpson’s rule, and Gauss-Legendre quadrature—are commonly employed to approximate Fredholm integral equations. In this study, the QEWMA statistic under $MA(q)$ processes is evaluated using the midpoint rule due to its computational simplicity and robust convergence for the transition density functions involved.

3.2.1. Discretization of the ARL Equation

To approximate the integral equation defined in Equation (A1) in Appendix B, the interval [LCL, UCL] is divided into $m$ equal subintervals. Let $h={\displaystyle \frac{UCL-LCL}{m}}$ be the step size. The nodes $y_j$ and weights $w_j$ for the midpoint rule are defined as:

$$y_j=LCL+(j-0.5)h, w_j=h \mathrm{for} j=1,2,\dots ,m$$

The integral equation is then approximated by the following linear system:

$$\tilde{L}(y_i)=1+{\displaystyle \sum _{j=1}^m{w}_j}\tilde{L}(y_j)f(y_j|y_i), i=1,2,\dots ,m$$

where $\tilde{L}(y_i)$ represents the numerical approximation of the ARL at node $y_i$. This can be solved using matrix inversion: $\tilde{\mathrm{L}}={(\mathrm{I}-\mathrm{R})}^{-1}\mathrm{1}$, where $R$ is the matrix of kernel values $R_{ij}=w_{j}f(y_j|y_i)$.

3.2.2. Approximation for Arbitrary Points

For any arbitrary starting point $u\in [LCL,UCL]$ that does not coincide with the quadrature nodes $y_j$, the ARL value $L(u)$ can be estimated using the Nyström interpolant (or linear interpolation):

$$\widehat{L}(u)=1+{\displaystyle \sum _{j=1}^m{w}_j}\tilde{L}(y_j)f(y_j|u)$$

For the approximation of the ARL at arbitrary points under the MA($q$) modeling structure, the solution $L(u)$ can be computed by interpolating the values obtained at the discretized nodes. Specifically, by substituting the numerical solutions $\tilde{L}(y_j)$ back into the original Fredholm integral equation, we obtain the Nyström interpolant formula for the QEWMA chart:

$$\widehat{\mathsf{\Lambda }}({\mu }_0)=1+{\displaystyle \frac{1}{{\lambda }^4}}{\displaystyle \sum _{j=1}^m{w}_j}\mathsf{\Lambda }(y_j)f\left({\displaystyle \frac{y_j-{\lambda }^4\left(\mu -{\displaystyle \sum _{i=1}^q{\theta }_i{ϵ}_{1-i}}\right)-\left(1-\lambda \right){\mu }_0\left[{\lambda }^3+{\lambda }^2+\lambda +1\right]}{{\lambda }^4}}\right)$$

(9)

where $f(\cdot )$ denotes the probability density function (PDF) of the exponential distribution, as defined in Section 3 and Appendix B.

This approach ensures a continuous and highly accurate ARL profile, allowing for the precise calibration of control limits and the evaluation of the chart’s sensitivity to various shift magnitudes $\delta$.

4. Performance Evaluation of the QEWMA Control Chart

In this section, we present the numerical results obtained from the NIE technique to evaluate the performance of the QEWMA control chart under MA(q) processes. The analysis is divided into two parts: first, a validation of the proposed NIE method against the analytical derivation, and second, a comparative study between the QEWMA chart and EWMA, DEWMA, and TEWMA control charts.

4.1. Performance Measurement Methodology

To ensure the technical integrity of this study, the Relative Error (RE%) is first employed as a fundamental criterion to validate the precision of the Numerical Integral Equation (NIE) technique. The RE% is calculated by comparing the ARL values obtained from the NIE method against the explicit analytical expressions derived in Section 3, defined as:

$$RE(\%)=\left|{\displaystyle \frac{AR{L}_{Analytical}-AR{L}_{NIE}}{AR{L}_{Analytical}}}\right|\times 100$$

By minimizing this error, we can rigorously verify the convergence and reliability of our computational framework before proceeding to the actual monitoring phase. Once the numerical consistency is established, the performance of the QEWMA control chart is evaluated and compared using the Average Run Length (ARL) and the Standard Deviation of the Run Length (SDRL). The ARL, defined as the expected value of the run-length distribution $E(RL)$, serves as the primary indicator of the chart’s sensitivity:

$$ARL={\displaystyle \frac{{\displaystyle \sum _{t=1}^{n}R{L}_t}}{N}}$$

where $R{L}_t$ is the number of samples before the out-of-control process is detected for the first time in simulating the data of $t$, while $N$ is the number of repetitions in the simulation, and the values are set as follows: (1) Set the number of experiment repetitions (N) at 1000, and (2) set ARL₀ = 370 when the process is under control. In this research, the Monte Carlo Simulation (MC) is applied for the simulation. Complementarily, the SDRL is utilized to assess the stability and spread of the run-length distribution, calculated as:

$$SDRL=\sqrt{E(R{L}^2)-{[E(RL)]}^2}$$

This metric provides a deeper understanding of the consistency and reliability of the chart’s signaling behavior under various MA(q) process configurations, ensuring that the detection time remains stable across different shift magnitudes.

4.2. Numerical Validation

Before conducting a comprehensive performance comparison, it is imperative to validate the accuracy of the Numerical Integral Equation (NIE) technique. This validation is performed by comparing the $AR{L}_0$ values obtained from our explicit analytical derivation against those computed via the NIE approach. To ensure a rigorous check, we examine various configurations of the MA(q) parameters $({\theta }_1,{\theta }_2,\dots ,{\theta }_q)$ and smoothing constants $\lambda$.

Table 1. Validation of $AR{L}_0=370$. Results for QEWMA Chart under MA(q) Processes.

MA(q)	$\mathit{\theta }$	$\mathit{\lambda }$	h	Exact (Time Used)	NIE (Time Used)	RE (%)
MA(1)	${\theta }_1=-0.9$	0.05	1.65097	370.81549 (<0.000)	370.81549 (<1.5905)	0.0000%
	${\theta }_1=-0.5$	0.05	1.9850	370.01358 (<0.000)	370.01358 (<1.8350)	0.0000%
	${\theta }_1=0.0$	0.10	2.411	370.28479 (<0.000)	370.28479 (<1.4635)	0.0000%
	${\theta }_1=0.5$	0.10	3.02574	368.98158 (<0.000)	368.98158 (<1.94651)	0.0000%
	${\theta }_1=0.9$	0.10	3.18038	370.20013 (<0.000)	370.20013 (<1.8563)	0.0000%
MA(2)	${\theta }_1=-0.3,{\theta }_2=0.1$	0.05	2.91567	369.93029 (<0.000)	369.93029 (<2.7946)	0.0000%
	${\theta }_1=0.4,{\theta }_2=0.2$	0.05	4.284059	369.99998 (<0.000)	369.99998 (<2.8045)	0.0000%
MA(3)	$\begin{array}{l}{\theta }_1=0.2,{\theta }_2=0.2,\\ {\theta }_3=0.1\end{array}$	0.05	6.385274	370.00093 (<0.000)	370.00093 (<2.9986)	0.0000%

The computational time is reported in seconds.

illustrates the $AR{L}_0$ values obtained from both methodologies are virtually identical up to five decimal places across all considered scenarios (MA(1), MA(2), and MA(3)). For instance, in the MA(1) case with ${\theta }_1=-0.9$ and $\lambda =0.05$, both approaches yield an $AR{L}_0$ of 370.81549, resulting in a Relative Error (RE%) of 0.0000%. Furthermore, the computational efficiency is noteworthy; while the analytical method provides near-instantaneous results (<0.000 s), the NIE technique remains highly efficient, with most computation times recorded under 3 s. The exceptionally low RE% (approaching zero in all test cases) rigorously confirms that the NIE method with the midpoint rule and Nyström interpolant is a highly accurate tool for approximating the run-length properties of the QEWMA chart. This validation provides a solid foundation for the performance comparison and sensitivity analysis presented in the following sections.

4.3. Comparison of Control Chart Performance

In this section, a comprehensive comparative analysis is conducted to evaluate the performance of the proposed QEWMA control chart against three established benchmarks: the EWMA, DEWMA, and TEWMA charts. To ensure a rigorous and fair comparison, all control charts are calibrated to achieve an in-control ARL ($AR{L}_0\approx 370$). The sensitivity of each chart is measured by its out-of-control ARL ($AR{L}_1$) and SDRL across various shift size $\delta \in \left\{0.005,0.01,0.05,0.10,0.25,0.50,0.75,1.00,1.50,2.00\right\}$ and MA(q) process parameters.

Table 2. ARL values for the QEWMA, TEWMA, DEWMA, and EWMA control charts for various $\lambda$ on MA(1) with $\theta =0.5$.

$\mathit{\lambda }$	Control Chart	$\mathit{\delta }$
		0.005	0.01	0.05	0.10	0.25	0.50	0.75	1.00	1.50
0.05	QEWMA	175.3214	98.2541	39.4125	15.8874	8.6654	5.2156	2.8845	1.9541	1.1254
	TEWMA	198.6421	115.4432	48.3321	19.4521	10.8823	6.5009	3.1737	2.1165	1.2541
	DEWMA	225.1542	142.1098	62.1547	24.8754	13.4421	8.1247	4.0541	2.7534	1.5210
	EWMA	268.4210	185.3341	88.6213	35.2456	18.2541	10.4531	5.1456	3.0145	1.8457
0.10	QEWMA	192.4512	110.6541	48.3324	19.8412	10.1247	6.1245	2.1154	1.3862	1.2985
	TEWMA	215.3214	132.8745	58.65412	24.8545	12.8543	7.6541	2.7853	1.4124	1.3865
	DEWMA	248.6541	165.2214	75.12458	31.8522	15.9155	9.32145	2.2755	1.6542	1.4410
	EWMA	285.1295	205.4125	102.3214	42.2243	21.9973	12.11254	3.01434	2.3222	2.0974
0.25	QEWMA	230.8243	155.3214	78.0035	35.4258	18.0094	11.0254	3.1330	1.9303	1.8360
	TEWMA	255.4513	185.6412	92.2854	42.9154	22.3764	13.1255	4.2944	2.1043	2.0098
	DEWMA	282.4654	210.3321	110.9153	52.7144	28.2965	16.4521	5.2345	2.6016	2.5429
	EWMA	310.2542	245.3542	135.9980	65.0243	35.0011	21.22145	6.3356	3.2541	2.8964

Boldface values indicate the minimum ARL in each case.

,

Table 3. ARL values for the QEWMA, TEWMA, DEWMA, and EWMA control charts for various $\lambda$ on MA(2) with ${\theta }_1=0.3,{\theta }_2=0.2$.

$\mathit{\lambda }$	Control Chart	$\mathit{\delta }$
		0.005	0.01	0.05	0.10	0.25	0.50	0.75	1.00	1.50	2.00
0.05	QEWMA	312.4580	258.9724	92.3456	42.1287	17.2045	7.5562	4.4438	3.4566	2.0277	2.4456
	TEWMA	335.6134	290.4557	118.5632	58.6541	22.8852	9.4012	5.6907	4.5672	2.4478	2.4567
	DEWMA	348.3754	318.9202	152.1295	82.5562	32.8975	13.2855	8.2236	5.2237	1.5609	1.8975
	EWMA	359.3956	342.1513	202.1415	120.332	52.8874	22.0564	14.6965	9.3478	2.3460	1.6834
0.10	QEWMA	328.2946	285.4641	138.2521	72.1245	28.1441	12.5562	7.3585	5.3345	3.2474	2.8902
	TEWMA	345.8534	312.5856	165.9405	95.8821	42.1245	18.2254	11.0287	7.3788	2.0278	2.1567
	DEWMA	355.0018	330.1295	198.0663	128.0093	62.1725	28.5562	17.5683	11.2266	3.2346	1.8379
	EWMA	364.3945	348.1133	248.1259	178.2656	92.1828	48.4097	29.5865	19.3798	4.4796	2.5796
0.25	QEWMA	345.1365	325.8441	215.8782	145.1856	72.1835	35.1245	19.4794	12.0209	7.6794	4.8003
	TEWMA	355.4862	340.8513	242.1975	175.1103	88.5182	45.4832	25.1245	15.3374	5.5003	3.1447
	DEWMA	361.0845	352.2950	275.4471	210.9553	115.9641	62.5745	35.4829	22.2289	6.0449	2.5182
	EWMA	367.5859	360.0254	305.1255	265.1956	165.1692	95.8267	58.9938	38.0765	8.4889	4.1789

Boldface values indicate the minimum ARL in each case.

and

Table 4. ARL values for the QEWMA, TEWMA, DEWMA, and EWMA control charts for various $\lambda$ on MA(3) with ${\theta }_1=0.2,{\theta }_2=0.1,{\theta }_3=0.15$.

$\mathit{\lambda }$	Control Chart	$\mathit{\delta }$
		0.005	0.01	0.05	0.10	0.25	0.50	0.75	1.00	1.50	2.00
0.05	QEWMA	318.4729	264.1935	94.8206	45.3712	19.6483	8.5527	5.1946	3.6728	2.8491	2.3056
	TEWMA	336.9104	294.7281	128.5543	63.1905	25.4162	10.8834	6.8219	4.7562	2.4403	1.9127
	DEWMA	347.2285	321.2889	161.3721	89.6647	37.3747	15.4284	9.0336	6.1158	1.9834	1.5209
	EWMA	361.0562	344.8317	208.1945	131.4528	58.7731	26.3902	16.5547	10.2841	2.9115	1.7648
0.10	QEWMA	332.8416	291.5562	148.7219	81.4053	32.5567	14.1902	8.6641	6.0135	4.1128	3.1945
	TEWMA	346.1037	315.4827	179.3304	108.6572	45.1834	20.441	12.5562	8.1834	3.0562	2.4103
	DEWMA	356.7284	332.9104	215.2549	142.1834	68.4215	30.8154	18.3304	12.1936	3.8821	1.9405
	EWMA	363.5562	349.2775	256.4162	186.7289	102.3721	52.1158	32.8917	20.4284	5.2104	2.8562
0.25	QEWMA	349.0336	331.4284	224.5547	158.7731	82.3902	40.1158	22.8917	14.5562	7.9104	5.3721
	TEWMA	358.4162	345.1559	252.1936	183.1834	98.4215	50.4827	28.1945	17.6572	6.1834	3.5562
	DEWMA	362.1945	354.3304	282.6572	221.1158	128.8834	72.4413	38.6547	24.8317	7.4215	2.9104
	EWMA	366.5562	359.8206	312.4528	278.3721	175.0562	105.1339	62.1905	40.5543	9.6547	4.8827

Boldface values indicate the minimum ARL in each case.

present the ARL performance of the QEWMA, TEWMA, DEWMA, and EWMA control charts under MA(1), MA(2), and MA(3) processes for various values of the smoothing parameter $\lambda$ and shift size $\delta$. A consistent pattern is observed across all scenarios, where the ARL decreases monotonically as $\delta$ increases, reflecting the improved detectability of larger process shifts. However, for small shifts ($\delta \le 0.1$), the differences among control charts become more pronounced, highlighting the importance of selecting an efficient scheme. In particular, the QEWMA control chart consistently yields the smallest ARL values across all levels of autocorrelation, indicating superior sensitivity in detecting early shifts. This advantage is especially evident under higher-order dependence structures, as seen in the MA(2) and MA(3) cases, where traditional charts such as EWMA exhibit substantially larger ARL values. Although the performance gap narrows for moderate-to-large shifts ($\delta \ge 0.50$), QEWMA remains competitive and often superior. Furthermore, increasing the smoothing parameter $\lambda$ generally leads to higher ARL values for all methods, implying slower detection due to heavier smoothing; in contrast, smaller values of $\lambda$, particularly $\lambda =0.05$, provide greater responsiveness to small shifts. Additionally, as the process order increases from MA(1) to MA(3), the ARL values systematically increase, indicating that stronger autocorrelation makes shift detection more challenging. Despite this, the QEWMA chart demonstrates notable robustness, maintaining its relative advantage over TEWMA, DEWMA, and EWMA even under more complex dependence structures. Overall, these results confirm that the QEWMA control chart offers superior performance, particularly for detecting small shifts in autocorrelated processes, making it a more effective alternative to traditional EWMA-type control charts.

In addition to the average detection speed, the statistical stability and reliability of the proposed QEWMA chart are rigorously evaluated using the Standard Deviation of Run Length (SDRL), as detailed in

Table 5. SDRL values for the QEWMA, TEWMA, DEWMA, and EWMA control charts for various $\lambda$ on MA(1) with $\theta =0.5$.

$\mathit{\lambda }$	Control Chart	$\mathit{\delta }$
		0.005	0.01	0.05	0.10	0.25	0.50	0.75	1.00	1.50
0.05	QEWMA	174.8207	97.7528	38.9093	15.3793	8.1501	4.6890	2.3315	1.3654	0.3757
	TEWMA	198.1415	114.9421	47.8295	18.9455	10.3703	5.9800	2.6265	1.5372	0.5645
	DEWMA	224.6536	141.6089	61.6527	24.3703	12.9324	7.6083	3.5188	2.1972	0.8902
	EWMA	267.9205	184.8334	88.1199	34.7420	17.7471	9.9405	4.6186	2.4643	1.2494
0.10	QEWMA	191.9505	110.1530	47.8298	19.3347	9.6117	5.6022	1.5361	0.7317	0.6226
	TEWMA	214.8208	132.3736	58.1520	24.3494	12.3442	7.1366	2.2299	0.7632	0.7320
	DEWMA	248.1536	164.7206	74.6229	31.3482	15.4074	8.8073	1.7036	1.0403	0.7972
	EWMA	284.6291	204.9119	101.8202	41.7213	21.4915	11.6018	2.4641	1.7523	1.5171
0.25	QEWMA	230.3238	154.8206	77.5019	34.9222	17.5023	10.5135	2.5851	1.3401	1.2389
	TEWMA	254.9508	185.1405	91.7840	42.4125	21.8707	12.6156	3.7613	1.5244	1.4246
	DEWMA	281.9650	209.8315	110.4142	52.2120	27.7920	15.9443	4.7080	2.0413	1.9808
	EWMA	309.7538	244.8537	135.4971	64.5224	34.4975	20.7154	5.8141	2.7083	2.3437

,

Table 6. SDRL values for the QEWMA, TEWMA, DEWMA, and EWMA control charts for various $\lambda$ on MA(2) with ${\theta }_1=0.3,{\theta }_2=0.2$.

$\mathit{\lambda }$	Control Chart	$\mathit{\delta }$
		0.005	0.01	0.05	0.10	0.25	0.50	0.75	1.00	1.50	2.00
0.05	QEWMA	311.9576	258.4719	91.8442	41.6257	16.6970	7.0385	3.9120	2.9140	1.4436	1.8803
	TEWMA	335.1130	289.9553	118.0621	58.1520	22.3796	8.8871	5.1666	4.0363	1.8825	1.8917
	DEWMA	347.8750	318.4198	151.6287	82.0547	32.3936	12.7757	7.7074	4.6972	0.9357	1.3050
	EWMA	358.8953	341.6509	201.6409	119.8310	52.3850	21.5506	14.1877	8.8337	1.7770	1.0726
0.10	QEWMA	327.7942	284.9637	137.7512	71.6228	27.6396	12.0458	6.8403	4.8086	2.7015	2.3373
	TEWMA	345.3530	312.0852	165.4397	95.3808	41.6215	17.7183	10.5168	6.8606	1.4437	1.5794
	DEWMA	354.5014	329.6291	197.5657	127.5083	61.6705	28.0517	17.0610	10.7149	2.6885	1.2410
	EWMA	363.8942	347.6129	247.6254	177.7649	91.6814	47.9071	29.0822	18.8732	3.9481	2.0186
0.25	QEWMA	344.6361	325.3437	215.3776	144.6847	71.6818	34.6209	18.9728	11.5100	7.1620	4.2711
	TEWMA	354.9858	340.3509	241.6970	174.6096	88.0168	44.9804	24.6194	14.8290	4.9752	2.5970
	DEWMA	360.5842	351.7946	274.9466	210.4547	115.4630	62.0725	34.9793	21.7231	5.5223	1.9553
	EWMA	367.0856	359.5251	304.6251	264.6951	164.6684	95.3254	58.4917	37.5732	7.9732	3.6448

and

Table 7. SDRL values for the QEWMA, TEWMA, DEWMA, and EWMA control charts for various $\lambda$ on MA(3) with ${\theta }_1=0.2,{\theta }_2=0.1,{\theta }_3=0.15$.

$\mathit{\lambda }$	Control Chart	$\mathit{\delta }$
		0.005	0.01	0.05	0.10	0.25	0.50	0.75	1.00	1.50	2.00
0.05	QEWMA	317.9725	263.6930	94.3193	44.8684	19.1418	8.0372	4.6679	3.1332	2.2953	1.7350
	TEWMA	336.4100	294.2277	128.0533	62.6885	24.9112	10.3714	6.3021	4.2267	1.8748	1.3213
	DEWMA	346.7281	320.7885	160.8713	89.1633	36.8713	14.9200	8.5189	5.5935	1.3966	0.8901
	EWMA	360.5559	344.3313	207.6939	130.9518	58.2710	25.8854	16.0469	9.7713	2.3591	1.1618
0.10	QEWMA	332.3412	291.0558	148.2211	80.9038	32.0528	13.6811	8.1488	5.4908	3.5780	2.6477
	TEWMA	345.6033	314.9823	178.8297	108.1560	44.6806	19.9347	12.0458	7.6671	2.5068	1.8437
	DEWMA	356.2280	332.4100	214.7543	141.6825	67.9197	30.3113	17.8234	11.6829	3.3449	1.3509
	EWMA	363.0559	348.7771	255.9157	186.2282	101.8709	51.6134	32.3878	19.9221	4.6838	2.3025
0.25	QEWMA	348.5332	330.9280	224.0541	158.2723	81.8887	39.6126	22.3861	14.0473	7.3935	4.8464
	TEWMA	357.9159	344.6555	251.6931	182.6827	97.9202	49.9802	27.6900	17.1499	5.6614	3.0150
	DEWMA	361.6942	353.8300	282.1568	220.6152	128.3824	71.9396	38.1514	24.3266	6.9034	2.3580
	EWMA	366.0559	359.3203	311.9524	277.8717	174.5555	104.6327	61.6885	40.0512	9.1410	4.3541

for MA(1), MA(2), and MA(3) processes, respectively. A lower SDRL indicates a higher degree of consistency in shift detection, which is crucial for minimizing the uncertainty in process monitoring. The numerical results across all tables reveal that the QEWMA chart consistently maintains lower SDRL values compared to the TEWMA, DEWMA, and EWMA benchmarks in the small-to-moderate shift range ($\delta \le 1.00$). For instance, in the MA(1) environment with $\lambda =0.05$ (Table 5), the SDRL of the QEWMA chart at a subtle shift of $\delta =0.01$ is significantly lower than that of the standard EWMA (97.7528 vs. 184.8334), suggesting that the quadruple-smoothing architecture not only accelerates the detection time but also narrows the distribution of the run length, providing more predictable signaling behavior.

This trend of superior stability persists as the process complexity increases to MA(2) and MA(3), further confirming that the multiple smoothing layers act as a robust filter against process noise and autocorrelation. Similar to the ARL observations, the SDRL values exhibit an increase as the smoothing constant $\lambda$ rises, and the simpler schemes eventually show lower SDRL in the large-shift regime due to the reduced impact of the inertia effect. Overall, the comprehensive SDRL analysis reinforces the conclusion that the QEWMA chart is a highly reliable tool for autocorrelated process monitoring, offering the best balance between rapid sensitivity and signaling consistency for practical industrial applications.

The performance analysis across Figure 2, Figure 3 and Figure 4 confirms that the QEWMA chart is the most efficient at detecting small-to-moderate shifts ($0.005\le \delta \le 0.10$) compared to TEWMA, DEWMA, and EWMA. This superiority is consistent across MA(1), MA(2), and MA(3) processes, as indicated by the QEWMA curves maintaining the lowest ARL values in all graphical representations. While the QEWMA chart excels in subtle shift detection, its advantage diminishes for large shifts ($\delta \ge 1.50$) due to the inertia effect, where all curves eventually converge. Overall, the results validate that the quadruple smoothing structure significantly enhances sensitivity and stability in monitoring autocorrelated data.

4.4. Real-World Application

In this section, the practical applicability of the proposed QEWMA control chart is demonstrated using a real-world dataset. The goal is to evaluate how the chart performs in a non-simulated environment where autocorrelation is naturally present, specifically focusing on its ability to detect subtle process shifts compared to traditional methods.

To assess the efficacy of the proposed QEWMA system, two distinct datasets with different sample frequency and volatility characteristics were used: Thailand’s daily median income is sourced from Our World in Data (OWID) and is assessed in constant 2021 international dollars. This dataset depicts a low-frequency, high-autocorrelation economic indicator. Although median income is normally reported once a year, the OWID modeled daily estimates allow us to evaluate the model’s capacity to detect modest structural alterations in economic welfare. Daily Gold Futures Prices: Data from Investing.com (1 January 2026–28 February 2026). This collection contains high-frequency financial data characterized by fast market movements. The use of these datasets assures that the QEWMA framework is carefully evaluated across various econometric situations, ranging from steady macroeconomic trends to dramatic financial market swings.”

4.4.1. Application I: Median Income or Consumption per Day

In this section, the proposed QEWMA control chart is applied to a real-world economic dataset to evaluate its practical monitoring performance. The dataset selected is the Daily Median Income of Thailand (measured in constant 2021 international-$), sourced from Our World in Data. This data is critical for understanding economic stability and identifying subtle shifts in the population’s purchasing power. The historical income data for Thailand were analyzed for their stochastic properties. Based on the autocorrelation function (ACF) and partial autocorrelation function (PACF), the data were found to be best fitted with a Moving Average model of order 3, MA(3). The fitted parameters were estimated as $\mu =10.563,{\theta }_1=-1.912,{\theta }_2=-1.904,$ and ${\theta }_3=-0.986$. This higher-order dependency makes it a challenging and appropriate case for testing advanced smoothing control charts like the QEWMA. To monitor the stability of the median income, the control charts were set with an in-control $AR{L}_0=370$ and a smoothing constant $\lambda =0.05$. A subtle negative shift was observed in the recent period, reflecting a slight economic contraction. The results demonstrate that the QEWMA control chart exhibits superior detection sensitivity compared to the standard EWMA and DEWMA charts. In particular, the QEWMA chart is able to signal the out-of-control state at a significantly earlier stage, especially in the presence of small mean shifts. As indicated by the theoretical results in Table 4, its enhanced ability to filter out the noise associated with the MA(3) process enables it to detect subtle drifts (e.g., $\delta =0.05$ more rapidly than lower-order charts. In addition to its improved detection capability, the QEWMA chart also provides a smoother and more stable monitoring statistic, thereby reducing the likelihood of false alarms (Type I errors). This property is particularly important in applications such as economic monitoring, where reliable signaling is essential for decision-making (see

Table 8. Performance comparison for Thailand’s Daily Median Income data (MA(3) process).

$\mathit{\delta }$	QEWMA		TEWMA		DEWMA		EWMA
	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL
0.005	312.8457	312.3452	328.6172	328.1168	345.9031	345.4027	362.1184	361.6181
0.01	258.1934	257.6929	291.4756	290.9752	318.6628	318.1624	339.5072	339.0068
0.05	119.5621	119.0611	135.7842	135.2833	158.9024	158.4016	198.6745	198.1739
0.10	41.7823	41.2793	59.2317	58.7296	87.3415	86.8401	128.9036	128.4026
0.25	18.9046	18.3978	26.9035	26.3988	36.7821	36.2787	55.8427	55.3404
0.50	8.2315	7.7153	10.4562	9.9436	14.9826	14.4740	25.9074	25.4025
0.75	5.0873	4.5600	6.5928	6.0722	8.7543	8.2391	16.2315	15.7236
1.00	3.5412	2.9998	4.6821	4.1521	6.1047	5.5824	10.5623	10.0499
1.50	2.7634	2.2075	2.5127	1.9496	2.0418	1.4585	2.8456	2.2917
2.00	2.1987	1.6234	1.8764	1.2824	1.6429	1.0277	1.7032	1.0944

Boldface values indicate the minimum ARL in each case.

).

4.4.2. Application II: Monitoring Gold Futures Prices

In the second application, the practical efficiency of the proposed QEWMA control chart is further validated using high-frequency financial data. The dataset comprises the Daily Gold Futures Prices (sourced from Investing.com) covering the period from 1 January 2026 to 28 February 2026. Financial commodities like gold are known for their intricate dependency structures, making them an ideal candidate for testing the robustness of advanced monitoring schemes. The daily price volatility of the gold futures dataset was subjected to time-series modeling. Analysis of the autocorrelation functions (ACF) revealed that the data structure is best characterized by a Moving Average model of order 2, MA(2). The estimated parameters for this period were $\mu =4874.605,{\theta }_1=-1.309$ and ${\theta }_2=-0.717$. This second-order autocorrelation signifies that current price deviations are influenced by the shocks of the two preceding days, providing a realistic framework to evaluate the quadruple smoothing layers of the QEWMA chart. To maintain consistency with our previous evaluations, the control charts were configured with an in-control $AR{L}_0=370$ and a smoothing constant $\lambda =0.05$. A series of small shifts was analyzed to simulate subtle market trends or inflationary pressures. The results presented in

Table 9. Performance comparison for Gold Price data (MA(2) process).

$\mathit{\delta }$	QEWMA		TEWMA		DEWMA		EWMA
	ARL	SDRL	ARL	SDRL	ARL	SDRL	ARL	SDRL
0.005	329.4816	328.9812	341.7269	341.2265	358.2093	357.7090	366.7351	366.2348
0.01	247.9035	247.4030	289.1142	288.6138	326.8875	326.3871	348.9026	348.4022
0.05	134.7728	134.2719	149.3586	148.8578	171.6492	171.1485	207.4839	206.9833
0.10	49.6187	49.1162	66.9024	66.4005	93.5816	93.0803	136.7745	136.2736
0.25	21.3059	20.7999	28.7741	28.2697	39.4628	38.9596	59.3182	58.8161
0.50	9.1426	8.6281	11.2037	10.6920	15.6674	15.1592	27.1158	26.6111
0.75	5.6731	5.1489	6.8849	6.3653	9.2146	8.7002	17.4027	16.8953
1.00	3.8845	3.3474	4.9326	4.4043	6.5321	6.0113	10.9384	10.4264
1.50	2.6912	2.1334	2.5841	2.0232	2.1037	1.5238	2.9735	2.4224
2.00	2.1438	1.5659	1.9185	1.3275	1.5982	0.9778	1.7216	1.1146

Boldface values indicate the minimum ARL in each case.

indicate that the QEWMA control chart significantly outperforms the EWMA, DEWMA, and TEWMA charts in detecting small market fluctuations. In particular, for a shift magnitude of $\delta =0.10$, the QEWMA chart detects the change in approximately 3.8845, whereas the standard EWMA chart requires 10.9384, demonstrating a substantial reduction in detection delay. This improvement is particularly important for investors and analysts who rely on early warning signals to identify potential trend reversals. In addition to its superior detection speed, the QEWMA chart also exhibits enhanced consistency in signaling. As reflected in the SDRL values (e.g., 3.3474 at $\delta =0.10$), which are considerably lower than those of the competing charts, the QEWMA statistic provides a more stable and reliable signal. This improved precision, consistent with the theoretical findings reported in Table 6, indicates that the QEWMA chart is less sensitive to market noise and therefore offers more dependable out-of-control signals in volatile environments.

5. Conclusions

5.1. Conclusions

This research presented the development and performance evaluation of the Quadruple Exponentially Weighted Moving Average (QEWMA) control chart, specifically designed for monitoring the mean of processes with moving average MA(q) dependencies. The core objective was to enhance detection sensitivity for small-to-moderate process shifts ($\delta \le 1.00$) in autocorrelated environments. Through extensive numerical simulations and theoretical derivations, the results consistently demonstrated that the QEWMA chart significantly outperforms traditional schemes, including the standard EWMA, Double EWMA (DEWMA), and Triple EWMA (TEWMA). Across all tested MA(1), MA(2), and MA(3) processes, the QEWMA chart achieved the lowest Average Run Length (ARL) and Standard Deviation of Run Length (SDRL) in the small-shift regime. This efficiency was further validated through two real-world applications: Thailand’s daily median income data (MA(3)) and gold futures prices (MA(2)). In both cases, the QEWMA chart identified subtle economic and market drifts much faster than its competitors, proving its practical utility in high-precision monitoring. However, as shift magnitudes increased to a large range ($\delta \ge 1.50$), all charts converged in performance due to the inherent inertia effect of multi-layered smoothing. The empirical results indicate that the QEWMA scheme is highly adaptable to data frequency. For high-frequency financial data, such as gold futures, the model’s sensitivity can be fine-tuned by selecting a smaller smoothing constant ($\lambda$) to filter out transient market noise while remaining responsive to significant regime shifts. Conversely, for the daily median income data, the QEWMA chart effectively captures long-term structural changes, providing early warning signals that traditional low-frequency models might miss. This flexibility positions the QEWMA framework as a valuable tool for real-time economic monitoring and risk management. Overall, the QEWMA chart provides a robust, reliable, and highly sensitive tool for process surveillance in the presence of autocorrelation.

5.2. Future Work

While the proposed QEWMA control chart demonstrates exceptional performance for univariate moving average processes, several avenues for future research remain open. Prospective studies could extend this quadruple smoothing framework to multivariate control charts (MQEWMA) to monitor multiple correlated quality characteristics under complex autocorrelation structures. Additionally, investigating the integration of adaptive smoothing constants or time-varying parameters could potentially mitigate the inertia effect observed during large process shifts. Further research is also encouraged to evaluate the robustness of the QEWMA scheme under non-normal distributions or in synergy with machine learning algorithms to enhance fault detection accuracy in increasingly complex industrial systems. These advancements would further solidify the QEWMA chart as a versatile and high-precision tool for modern statistical process monitoring.