An Enhanced Multivariate EWMA Approach with Variable Selection and Adaptive Sampling for Efficient Process Monitoring

Abstract

Due to the curse of dimensionality faced in modern industrial processes, high-dimensional Statistical Process Control (SPC) faces significant challenges in detecting small and sparse process shifts. Traditional multivariate control charts often suffer from noise accumulation and fail at timely identification of anomalies that affect only a small subset of variables. To address this issue, this study proposes an enhanced Multivariate Exponentially Weighted Moving Average (MEWMA) approach with variable selection and adaptive sampling for efficient process monitoring. The proposed smart approach works in two ways: first, it automatically focuses on the variables that are most likely to have changed (variable selection); second, it takes samples more frequently when things look uncertain, and less frequently when everything appears stable (variable sampling interval). This combination allows problems to be detected earlier. A Monte Carlo approach is used to calculate the the Average Time to Signal (ATS) values of the proposed scheme, and comparative results show that the proposed scheme outperforms standard charts like the Fixed Sampling Intervals (FSI) VSME, VSI-$T^2$, and VSI-MEWMA schemes in terms of detection speed for small-to-moderate sparse shifts. Finally, a real example from car body manufacturing is provided as an illustration for the implementation of the proposed scheme.

1. Introduction

As a key statistical technique in SPC, Control Charts (CCs) serve as graphical tools for analyzing process stability. By plotting quality characteristic values against predetermined control limits, CCs can promptly issue alerts upon detecting anomalous information, enabling proactive identification of production issues rather than relying on post-production inspection. This proactive capability has established control charts as essential methods for improving product quality and yield, with widespread applications across industrial and service sectors (see, e.g., [1,2,3]).

Based on the number of monitored variables, CCs can be classified into univariate and multivariate types. Univariate CCs originated in the 1920s, with the first quality chart proposed by W.A. Shewhart in the United States. However, Shewhart CCs rely solely on current sample information and exhibit limited sensitivity in detecting small shifts in process parameters. To address this limitation, the Cumulative Sum (CUSUM) and Exponentially Weighted Moving Average (EWMA) CCs were developed to incorporate historical data (see, e.g., [4,5,6,7,8,9,10,11]). On the other hand, as products and manufacturing processes have grown increasingly complex, the need to simultaneously monitor multiple correlated quality characteristics has drawn industrial and academic attention to multivariate CCs. After decades of development, numerous multivariate CCs have emerged, with the three most typical structures being Hotelling’s $T^2$, Multivariate CUSUM (MCUSUM), and Multivariate EWMA (MEWMA) (see, e.g., [12,13,14,15,16]). Subsequent research on CCs has primarily focused on three aspects: adapting to different data distributions (see [17,18,19,20]; updating statistical models (see [21,22,23,24]); and modifying charting structures, (see [25,26,27]).

With the rapid advancement of manufacturing capabilities and inspection technologies, the number of variables requiring monitoring in production processes has increased dramatically. However, as dimensionality grows, traditional multivariate statistical methods become increasingly susceptible to noise accumulation, which severely impairs their ability to detect shifts in high-dimensional data. Through simulation studies, Wang and Jiang [28] demonstrated that when the data dimension is 20 and only one variable undergoes a mean shift, the Hotelling’s $T^2$ chart struggles to detect the change due to interference from noise in the remaining dimensions.

In modern quality monitoring scenarios, simultaneous faults across multiple locations are relatively uncommon. Instead, a single fault source is typically detected by a subset of correlated process sensors, meaning that only a few variables experience distributional shifts. Such sparse shifts are often obscured by noise from other dimensions and fail to be prominently reflected in traditional multivariate statistics, leading to monitoring failures. Consequently, an effective strategy for statistical inference in high-dimensional process monitoring is to incorporate variable selection. Wang and Jiang [28] proposed a variable-selection-based multivariate statistical process control method that uses a forward-selection algorithm to screen faulty variables, achieving simultaneous process monitoring and fault diagnosis. In [29,30], variable selection was further applied to the MEWMA and MCUSUM CCs, respectively, improving detection efficiency for shifts of various magnitudes. In [31], an MEWMA chart was proposed based on the Least Absolute Shrinkage and Selection Operator (LASSO) method, integrating high-dimensional data monitoring with change point detection using goodness-of-fit tests. In [32], the retrieval speed of Out-Of-Control (OOC) variables was enhanced by adopting the Least Angle Regression (LARS) algorithm instead of stepwise regression.

Parallel to these developments in the variable dimension, significant advances have been made in the temporal dimension through adaptive sampling strategies. While conventional control charts employ Fixed Sampling Intervals (FSI) predetermined by production schedules, research has demonstrated that allowing sampling frequency to vary based on process data can substantially enhance detection efficiency. This concept underpins the Variable Sampling Interval (VSI) scheme, in which the time until the next sample is shortened if the current control statistic suggests a potential OOC condition and lengthened if no process disturbance is indicated. The primary advantage of VSI over FSI lies in its ability to detect process changes more rapidly (see, e.g., [33,34,35,36,37]). Previous studies have demonstrated that employing just two sampling intervals is sufficient to achieve effective detection performance across a range of process shifts [38,39,40].

Current research on adaptive EWMA CCs mainly includes two aspects. The first aspect concerns dynamically adjusting the smoothing parameter $\lambda$ to achieve global sensitivity to both small and large shifts. For example, ref. [41] estimated the mean shift using an EWMA-based estimator, then determined $\lambda$ for the plotting EWMA statistic through a continuous function. The authors of [42] replaced the conventional score functions with the Hampel function and proposed a new adaptive EWMA chart, while those of [43] introduced a Support Vector Regression (SVR)-based adaptive EWMA scheme for improved mean monitoring in industrial processes. In [44], a streamlined and parameter-free adaptive EWMA method was introduced for monitoring Poisson processes. In [45], the authors constructed an EWMA scheme by introducing an adaptive method, aiming to improve the efficiency of detecting the existing ratio of two normal random variable charts for both small and large shifts. Finally, ref. [46] proposed an EWMA control chart for inverse Maxwell processes.

The second aspect focuses on adjusting sampling intervals and sample sizes in the temporal dimension. In [47], the authors evaluated the Average Time to Signal (ATS) properties of two-sided VSI EWMA CCs and presented a practical design procedure. In [48], an EWMA scheme with VSI was developed to monitor the process coefficient of variation. In [49,50], the authors investigated the performance of the VSI EWMA scheme with estimated parameters, while [37] introduced a Max EWMA scheme with VSI strategy for simultaneously monitoring both the frequency and amplitude of an event. Finally, ref. [51] developed a VSI MEWMA scheme to monitor the mean vector of the Gumbel’s Bivariate Exponential (GBE) distribution. In addition, several studies have extended the VSI strategy to adaptive EWMA schemes [52,53,54,55,56].

The latest developments in intelligent fault diagnosis have highlighted the importance of effective and efficient monitoring structures that can detect complex and compound faults in contemporary industrial facilities, especially when data availability is limited [57]. The problem of detecting weak and incipient faults in high-dimensional process environments still represents a major issue, as small process variations are often lost in the noise and irrelevant variables; this has led to the creation of methodologies that are more sensitive to small and sparse process variations [58]. Furthermore, real-life industrial data are normally noisy, variable, and subject to changing operational conditions, requiring the development of robust and adaptive monitoring plans [59]. All of these issues underscore the relevance of incorporating adaptive control and efficient selection of variables within multivariate statistical process control systems to achieving enhanced speed and trustworthiness when identifying shifts in intricate manufacturing processes.

However, despite the acknowledged relevance of integrating adaptive control with variable selection, existing multivariate SPC methods still exhibit two critical gaps: first, most VSI-based EWMA charts lack variable selection, meaning that sparse shifts are masked by high-dimensional noise; second, existing variable selection SPC methods only use FSI. More importantly, a novel insight is that the choice of the selected variables directly affects the probability of entering the warning region. Selecting too many noisy variables triggers false short intervals, while selecting too few may miss weak signals. This interaction between variable selection and VSI zone division is entirely new and unexplored. To address this, we propose a new adaptive monitoring approach which we term the Variable Sampling Interval–Variable Selection-based Multivariate EWMA (VSI-VSME) chart. The remainder of this paper is structured as follows: Section 2 provides a detailed exposition of the proposed VSI-VSME framework; in Section 3, the statistical performance of the proposed chart is evaluated and discussed through comprehensive simulation studies; Section 4 offers practical applications based on real datasets to demonstrate the methodology’s utility in operational settings; finally, Section 5 concludes with summary remarks and directions for future research. And the Appendix A lists all symbols and parameters used in this paper.

2. Methodology Design

2.1. The VSME Chart

In modern complex manufacturing processes, multivariate quality control issues are increasingly prominent. Traditional multivariate control charts such as Hotelling’s $T^2$ face significant challenges when handling high-dimensional data. This is particularly the case in the presence of sparse shifts, leading to substantially reduced detection efficiency due to only a small subset of variables exhibiting actual anomalies. To overcome these limitations, ref. [28] introduced a Variable Selection-based Multivariate SPC (VS-MSPC) method. The core idea is to incorporate a penalty term into the likelihood function in order to automatically identify potential OOC variables and estimate the magnitude of their shifts, enabling precise monitoring of high-dimensional process data. Specifically, it considers the following penalized optimization problem:

$$S^2=\underset{\mathit{\mu }}{min}\left({({\mathbf{y}}_t-\mathit{\mu })}^T{\mathbf{\Sigma }}^{-1}({\mathbf{y}}_t-\mathit{\mu })+{\displaystyle \sum _{j=1}^p}{p}_{{ϵ}_j}\left(\right|{\mu }_j\left|\right)\right)$$

(1)

where ${\mathbf{y}}_t$ is a p-dimensional observation collected at time t and following a multivariate normal distribution ${\mathbf{y}}_t\sim N(\mathit{\mu },\mathbf{\Sigma })$. Here, $|{\mu }_j|$ denotes the absolute value of the jth element of $\mu$, representing the true mean of the jth variable. The term $p_{{ϵ}_j}(·)$ is a penalty function that controls model complexity by regulating the number of variables with nonzero means, i.e., those for which $|{\mu }_j|>0$.

In [28], the authors adopted the $L_0$ penalty $p_{ϵ}\left(\right|{\mu }_j\left|\right)=ϵI\left(\right|{\mu }_j|\ne 0)$, which directly constrains the number of nonzero coefficients. This naturally aligns with the sparsity assumption commonly hold in engineering practice that most variables are In-Control (IC), while only a few are Out-Of-Control (OOC). Through Cholesky decomposition of the inverse covariance matrix ${\mathbf{\Sigma }}^{-1}={\mathbf{R}}^T\mathbf{R}$ and by setting ${\mathbf{z}}_t=\mathbf{R}{\mathbf{y}}_t$, the problem in (1) can be transformed into the following constrained least squares problem:

$$\underset{\mathit{\mu }}{min}{({\mathbf{z}}_t-\mathbf{R}\mathit{\mu })}^T({\mathbf{z}}_t-\mathbf{R}\mathit{\mu })\mathrm{subject}\mathrm{to}{\displaystyle \sum _{j=1}^p}I\left(\right|{\mu }_j|\ne 0)\le s$$

(2)

where s is a prescribed integer upper bound on the number of nonzero coefficients. To solve this problem efficiently, a forward variable selection algorithm is used to find the solution ${\mathit{\mu }}_t^{*}$ of Equation (2). The following monitoring statistic is constructed:

$$\mathrm{\Lambda }\left({\mathbf{y}}_t\right)=2{\mathbf{y}}_t^T{\mathbf{\Sigma }}^{-1}{\mathit{\mu }}^{*}-{\mathit{\mu }}^{*T}{\mathbf{\Sigma }}^{-1}{\mathit{\mu }}^{*}.$$

(3)

In [29], it was further shown that monitoring this statistic is equivalent to monitoring

$$\mathrm{\Lambda }\left({\mathbf{y}}_t\right)={\mathbf{y}}_t^T{\mathbf{\Sigma }}^{-1}{\mathit{\mu }}^{*}.$$

(4)

Building upon the VS-MSPC framework introduced above, a Variable Selection-based Multivariate EWMA (denoted as VSME) integrates exponential smoothing into the variable selection process to enhance the robustness and accuracy of shift estimation in dynamic processes. Let ${\mathbf{w}}_t$ be the exponentially weighted vector at time t, defined recursively as

$${\mathbf{w}}_t=(1-\lambda ){\mathbf{w}}_{t-1}+\lambda {\mathbf{y}}_t,$$

(5)

where the initial condition $w_0=\mathbf{0}$ and where $0<\lambda \le 1$ is the smoothing constant controlling the memory of the process. When $t\to \infty$, the solution of Equation (3) can equivalently be performed using the simplified statistic

$$M_t={{\mathbf{w}}_t}^T{\mathbf{\Sigma }}^{-1}{\mathit{\mu }}^{*}.$$

(6)

An OOC signal is triggered whenever $M_t$ exceeds a predetermined Upper Control Limit (UCL). Selection of $\lambda$ and the sparsity level s is critical to the performance of the VSME scheme. These parameters can be chosen based on desired statistical properties or through cross-validation in practical applications.

2.2. The VSI-VSME Chart

The VSI strategy is an adaptive approach to statistical process control that is designed to enhance the efficiency of monitoring schemes. Unlike FSI CCs, where the time between samples is constant, a VSI chart dynamically adjusts the waiting time until the next sample based on the current value of the control statistic. This allows the chart to respond more frequently when the data suggest potential process instability and to respond less frequently when the process appears stable.

The VSI-VSME scheme partitions the IC region into three distinct zones using the UCL and the Upper Warning Limit (UWL). After plotting the $t^{th}$ statistic $M_t$, the operational procedure of the VSI chart is as follows:

• If $M_t\in [0,\mathrm{UWL}]$ (the safe region), the process is considered stable. The next sample will be taken after a long sampling interval $h_L$, where $h_L>1$.
• If $M_t\in (\mathrm{UWL},\mathrm{UCL}]$ (the warning region), the process is still considered IC, but the chart signals a higher level of caution. Consequently, the next sample will be taken after a short sampling interval $h_S$, where $0<h_S<1$.
• If $M_t\in (\mathrm{UCL},+\infty )$ (the OOC region), the process is deemed OOC. An OOC signal is issued and necessary corrective actions must be initiated to find the assignable cause(s).

This dynamic adjustment of sampling intervals enhances the ability of CCs to detect shifts while significantly improving the efficiency of resource utilization in monitoring. The Average Run Length ($\mathrm{ARL}$) is the standard performance measure for FSI CCs. However, $\mathrm{ARL}$ is an insufficient metric for VSI CCs, since it only counts the number of samples until a signal and ignores the varying time intervals between them. Therefore, the $\mathrm{ATS}$, which incorporates the actual time elapsed, is the appropriate criterion for evaluating the performance of VSI CCs. The $\mathrm{ATS}$ is defined as the expected time between the start and a control chart signal:

• For FSI CCs, the $\mathrm{ATS}$ is simply the product of the $\mathrm{ARL}$ and the fixed sampling interval h:

  $${\mathrm{ATS}}^{FSI}={\mathrm{ARL}}^{FSI}\times h_0.$$

  (7)
• For VSI CCs, the $\mathrm{ATS}$ depends on the $\mathrm{ARL}$ and the expected (average) sampling interval $E\left(h\right)$:

  $${\mathrm{ATS}}^{VSI}={\mathrm{ARL}}^{VSI}\times E\left(h\right),$$

  (8)

  where $E\left(h\right)={\rho }_S\times h_S+{\rho }_L\times h_L$ and where ${\rho }_S$ and ${\rho }_L$ represent the proportion of times the short and long sampling intervals are used, respectively.

To ensure a fair comparison between an FSI chart and its VSI counterpart, it is standard practice to set both charts to have the same IC performance. This is achieved by choosing charts with the same IC $\mathrm{ARL}$ value (${\mathrm{ARL}}_0$) and constraining the average sampling interval of the VSI chart to match the fixed interval of the FSI chart (typically $h_0=1$ time unit):

$$E\left(h\right)={\rho }_S\times h_S+{\rho }_L\times h_L=h_0=1.$$

(9)

Under this constraint, the IC average time to signal is also identical for both charts:

$${\mathrm{ATS}}_0^{FSI}={\mathrm{ATS}}_0^{VSI}={\mathrm{ARL}}_0=C.$$

(10)

Consequently, $\mathrm{ATS}=\mathrm{ARL}$ in the FSI scheme, since the sampling interval is fixed at 1. Given the same UCL, our use of the UWL in the proposed VSI scheme does not change the alarm rule. Thus, the $\mathrm{ARL}$ values remain the same as in the FSI case in both IC and OOC situations. However, the VSI scheme achieves a smaller $\mathrm{ATS}$ value because it employs shorter sampling intervals in the safe region. For these reasons, the OOC performance of the two schemes can be compared directly using the OOC $\mathrm{ATS}$ (${\mathrm{ATS}}_1$). The chart with the smaller ${\mathrm{ATS}}_1$ is more efficient, as it detects process shifts faster in real time. Figure 1 outlines the implementation procedure of the proposed VSI-VSME scheme.

3. Performance Analysis

This study simulates a 20-dimensional process with different numbers of shifted variables, specifically:

$$y_t\sim N_p\left({[\delta ,\delta ,...,0]}^T,I\right),\mathrm{Number}\left(\delta \right)=p_0\in \{1,2,\dots ,p\}.$$

In this simulation setup, $\delta$ denotes the non-centrality parameter, representing the magnitude of the mean shift in the variables that undergo a change. Specifically, under the IC state, all variables have a mean of zero. When a shift occurs, the first $p_0$ variables experience a mean shift of $\delta$, while the remaining variables stay at zero. By varying $\delta$, the detection performance under different shift sizes can be evaluated. In the implementation, three sets of sampling interval configurations are considered:

• $h_S=0.1$ and $h_L=1.9$,
• $h_S=0.3$ and $h_L=1.7$,
• $h_S=0.5$ and $h_L=1.5$.

Table 1. UCL and UWL values of the VSI-VSME scheme for different s and $\lambda$ values.

C	Limit	$\mathit{s}=1$	$\mathit{s}=2$	$\mathit{s}=3$	$\mathit{s}=4$	$\mathit{s}=5$
$\lambda =0.1$
200	UCL	0.637	0.925	1.134	1.295	1.424
	UWL	0.228	0.389	0.511	0.605	0.687
370	UCL	0.704	1.007	1.226	1.395	1.532
	UWL	0.232	0.395	0.518	0.615	0.692
500	UCL	0.737	1.046	1.269	1.442	1.580
	UWL	0.233	0.397	0.520	0.618	0.697
$\lambda =0.25$
200	UCL	1.861	2.670	3.263	3.721	4.088
	UWL	0.632	1.080	1.412	1.677	1.892
370	UCL	2.034	2.876	3.487	3.963	4.347
	UWL	0.638	1.087	1.426	1.691	1.901
500	UCL	2.119	2.973	3.594	4.080	4.468
	UWL	0.639	1.089	1.424	1.691	1.909
$\lambda =0.5$
200	UCL	4.400	6.361	7.757	8.840	9.715
	UWL	1.494	2.545	3.330	3.965	4.470
370	UCL	4.841	6.813	8.256	9.378	10.281
	UWL	1.496	2.548	3.338	3.969	4.487
500	UCL	5.029	7.034	8.495	9.634	10.551
	UWL	1.496	2.550	3.339	3.965	4.487

summarizes the $\mathrm{UCL}$ and $\mathrm{UWL}$ values for the proposed VSI-VSME scheme to achieve IC performance levels (${\mathrm{ARL}}_0$ = ${\mathrm{ATS}}_0$ = C) of 200, 370, and 500 when $\lambda \in \{0.1,0.25,0.5\}$, $s\in \{1,2,3,4,5\}$, and $p=20$. All reported results are based on ${10}^5$ Monte Carlo replications. The standard errors of the $\mathrm{ATS}$ estimates are consistently below $0.5\%$ of the estimated value. All simulations were performed on an computer with an Intel Core i5 12600KF CPU with 32 GB RAM, using MATLAB R2024b parallelized over six workers. Moreover, the average computational time per replication for a run is approximately $0.002$ s until signal.

3.1. VSI-VSME vs. FSI-VSME

This part evaluates the OOC performance of the proposed VSI-VSME scheme against the FSI-VSME scheme. Here, we set the parameter s equal to $p_0$, which is the actual number of variables affected by the shift. Cases where $s\ne p_0$ are discussed later.

Table 2. ${\mathrm{ATS}}_1$ comparison of the VSI-VSME and VSME schemes when $\lambda =0.1$, $s=p_0$, and $p=20$.

C	$({\mathit{h}}_{\mathit{S}},{\mathit{h}}_{\mathit{L}})$	Chart	$\mathit{s}={\mathit{p}}_0=1$	$\mathit{s}={\mathit{p}}_0=2$	$\mathit{s}={\mathit{p}}_0=3$	$\mathit{s}={\mathit{p}}_0=4$	$\mathit{s}={\mathit{p}}_0=5$
200		VSME	52.315	32.443	24.388	19.835	16.943
	(0.1, 1.9)	VSI-VSME	42.282	24.126	17.915	14.776	13.036
	(0.3, 1.7)	VSI-VSME	44.055	25.951	19.152	15.703	13.708
	(0.5, 1.5)	VSI-VSME	46.343	27.446	20.395	16.567	14.360
370		VSME	68.703	40.379	28.840	22.913	19.245
	(0.1, 1.9)	VSI-VSME	52.648	27.429	19.395	15.699	13.566
	(0.3, 1.7)	VSI-VSME	56.435	30.312	21.523	17.299	14.816
	(0.5, 1.5)	VSI-VSME	59.871	33.213	23.658	18.918	16.096
500		VSME	78.804	44.560	31.329	23.842	20.405
	(0.1, 1.9)	VSI-VSME	58.700	29.003	19.983	15.899	13.838
	(0.3, 1.7)	VSI-VSME	63.242	32.522	22.502	17.647	15.343
	(0.5, 1.5)	VSI-VSME	67.945	36.004	25.093	19.410	16.733

,

Table 3. ${\mathrm{ATS}}_1$ comparison of the VSI-VSME and VSME schemes when $\lambda =0.25$, $s=p_0$, and $p=20$.

C	$({\mathit{h}}_{\mathit{S}},{\mathit{h}}_{\mathit{L}})$	Chart	$\mathit{s}={\mathit{p}}_0=1$	$\mathit{s}={\mathit{p}}_0=2$	$\mathit{s}={\mathit{p}}_0=3$	$\mathit{s}={\mathit{p}}_0=4$	$\mathit{s}={\mathit{p}}_0=5$
200		VSME	92.263	56.022	38.642	28.475	22.600
	(0.1, 1.9)	VSI-VSME	79.636	42.404	26.013	17.589	13.193
	(0.3, 1.7)	VSI-VSME	82.859	45.058	28.474	20.118	15.209
	(0.5, 1.5)	VSI-VSME	84.685	48.142	31.479	22.596	17.248
370		VSME	147.156	83.588	55.119	38.989	29.569
	(0.1, 1.9)	VSI-VSME	126.326	61.497	35.416	22.546	15.766
	(0.3, 1.7)	VSI-VSME	131.589	67.223	39.535	26.339	18.759
	(0.5, 1.5)	VSI-VSME	135.295	72.400	44.086	29.798	21.711
500		VSME	182.890	103.546	65.846	45.743	33.715
	(0.1, 1.9)	VSI-VSME	156.384	75.236	41.101	25.444	17.386
	(0.3, 1.7)	VSI-VSME	163.079	81.131	46.471	29.912	20.975
	(0.5, 1.5)	VSI-VSME	170.173	87.036	52.192	34.232	24.675

and

Table 4. ${\mathrm{ATS}}_1$ comparison of the VSI-VSME and VSME schemes when $\lambda =0.5$, $s=p_0$, and $p=20$.

C	$({\mathit{h}}_{\mathit{S}},{\mathit{h}}_{\mathit{L}})$	Chart	$\mathit{s}={\mathit{p}}_0=1$	$\mathit{s}={\mathit{p}}_0=2$	$\mathit{s}={\mathit{p}}_0=3$	$\mathit{s}={\mathit{p}}_0=4$	$\mathit{s}={\mathit{p}}_0=5$
200		VSME	140.283	100.207	75.468	57.775	43.053
	(0.1, 1.9)	VSI-VSME	131.538	87.204	60.357	45.720	31.586
	(0.3, 1.7)	VSI-VSME	134.323	90.860	64.104	45.822	34.824
	(0.5, 1.5)	VSI-VSME	135.033	93.634	67.001	45.846	37.976
370		VSME	248.898	170.970	124.936	93.450	68.674
	(0.1, 1.9)	VSI-VSME	233.107	148.074	99.395	71.373	48.179
	(0.3, 1.7)	VSI-VSME	237.120	153.631	105.266	71.359	53.459
	(0.5, 1.5)	VSI-VSME	239.537	158.453	110.945	70.832	58.070
500		VSME	324.355	223.380	160.171	118.063	86.141
	(0.1, 1.9)	VSI-VSME	303.375	193.172	127.059	89.580	60.095
	(0.3, 1.7)	VSI-VSME	306.666	200.866	134.913	93.377	65.675
	(0.5, 1.5)	VSI-VSME	313.140	208.699	142.408	89.034	72.700

provide a comprehensive comparison of the OOC ${\mathrm{ATS}}_1$ between the proposed VSI-VSME scheme and the FSI-VSME benchmark across three smoothing parameters $(\lambda =0.1,0.25,0.5)$, five sparsity levels $(s=p_0=1,\dots ,5)$, three IC performance levels $(C={\mathrm{ARL}}_0={\mathrm{ATS}}_0=200,370,500)$, and three VSI configurations $((h_S,h_L)=(0.1,1.9),(0.3,1.7),(0.5,1.5))$. We can draw the following conclusions:

First, regardless of $\lambda$, C, or s, every VSI-VSME setting yields a strictly smaller ${\mathrm{ATS}}_1$ than the corresponding FSI-VSME benchmark, confirming that the adaptive sampling strategy consistently accelerates the detection of sparse shifts. The improvement is most pronounced for the most aggressive VSI design $(h_S=0.1,h_L=1.9)$. For example, in Table 2 $(\lambda =0.1,C=200,s=p_0=1)$ the FSI-VSME gives ${\mathrm{ATS}}_1=52.315$, whereas the VSI-VSME with $(0.1,1.9)$ reduces it to $42.282$, a relative reduction of about $19.2\%$. As $h_S$ increases and $h_L$ decreases, the advantage gradually shrinks; yet, even the mildest VSI setting $(0.5,1.5)$ still outperforms the FSI scheme $({\mathrm{ATS}}_1=46.343)$ for the same case.

Second, the benefit of the VSI mechanism is highly consistent across different $\lambda$ values. When $\lambda$ is small $\left(0.1\right)$, the ${\mathrm{ATS}}_1$ values are generally lower and the percentage improvement from VSI is larger. For instance, at $\lambda =0.1$, $C=370$, $s=2$, the $(0.1,1.9)$ VSI scheme achieves ${\mathrm{ATS}}_1=27.429$ versus an FSI value of $40.379$, a $32.1\%$ reduction. At $\lambda =0.5$ in Table 4, the same comparison $(C=370,s=2)$ yields ${\mathrm{ATS}}_1=148.074$ versus $170.970$, a reduction of only $13.4\%$. These results indicate that the VSI mechanism is particularly effective when $\lambda$ is small.

Third, as the number of truly shifted variables $(p_0=s)$ increases, both the FSI and VSI charts detect the shift faster, as expected, because a larger signal from more variables shortens the $\mathrm{ARL}$. However, the relative improvement due to VSI tends to be larger for moderate s (e.g., $s=2$ or 3) than for $s=1$ or $s=5$. For example, in Table 3 $(\lambda =0.25,C=200)$ the reduction from FSI to VSI $(0.1,1.9)$ is about $13.7\%$ for $s=1$ $(92.263\to 79.636)$ and about $11.8\%$ for $s=5$ $(22.600\to 13.193)$, but reaches $24.8\%$ for $s=2$ $(56.022\to 42.404)$ and $32.7\%$ for $s=3$ $(38.642\to 26.013)$. This suggests that the VSI strategy is most beneficial when the shift is moderately sparse.

In summary, the evidence from Table 2, Table 3 and Table 4 unequivocally demonstrates that the VSI-VSME scheme provides a clear and robust improvement in detection speed over the FSI-VSME scheme across a wide range of parameters and that the improvement is largest for small $\lambda$.

3.2. VSI-VSME vs. VSI-$T^2$ and VSI-MEWMA

Table 5. ATS₁ comparison of the VSI-VSME, VSI-MEWMA, and VSI-$T^2$ schemes when $\lambda =0.1$, $p_0=2$, and $p=20$.

Method	s	$({\mathit{h}}_{\mathit{s}},{\mathit{h}}_{\mathit{l}})$	$\mathit{\delta }$
			0.2	0.4	0.6	0.8	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0
VSI-VSME	1	(0.1, 1.9)	204.689	50.949	19.495	11.969	8.912	5.270	3.607	2.666	2.123	1.775	1.376	0.916	0.535
		(0.3, 1.7)	209.085	55.240	21.714	13.179	9.544	5.634	3.914	2.941	2.386	2.026	1.637	1.208	0.868
		(0.5, 1.5)	214.049	60.011	23.858	14.322	10.246	6.008	4.209	3.227	2.647	2.265	1.891	1.492	1.204
	2	(0.1, 1.9)	193.731	47.088	18.941	12.020	8.980	5.437	3.754	2.743	2.197	1.890	1.504	0.993	0.547
		(0.3, 1.7)	196.219	51.444	20.793	12.897	9.470	5.661	3.953	2.952	2.387	2.056	1.671	1.224	0.872
		(0.5, 1.5)	202.828	55.488	22.655	13.812	10.023	5.927	4.152	3.153	2.583	2.234	1.845	1.461	1.195
	3	(0.1, 1.9)	186.569	46.030	18.933	12.194	9.186	5.669	3.999	2.968	2.324	2.049	1.808	1.399	0.909
		(0.3, 1.7)	192.671	50.497	20.831	13.077	9.701	5.922	4.194	3.159	2.527	2.222	1.950	1.571	1.154
		(0.5, 1.5)	197.461	54.538	22.777	14.001	10.203	6.154	4.375	3.353	2.726	2.400	2.093	1.731	1.399
	4	(0.1, 1.9)	181.568	46.018	19.113	12.348	9.406	5.884	4.217	3.152	2.452	2.123	1.956	1.679	1.232
		(0.3, 1.7)	187.902	50.528	20.966	13.274	9.926	6.115	4.395	3.344	2.650	2.310	2.099	1.800	1.444
		(0.5, 1.5)	194.975	54.428	22.979	14.197	10.431	6.349	4.571	3.521	2.845	2.491	2.243	1.924	1.591
	5	(0.1, 1.9)	182.201	46.202	19.261	12.551	9.575	6.069	4.363	3.311	2.561	2.176	2.028	1.832	1.477
		(0.3, 1.7)	186.523	50.765	21.276	13.483	10.089	6.274	4.541	3.497	2.759	2.366	2.180	1.948	1.613
		(0.5, 1.5)	193.297	55.016	23.231	14.442	10.665	6.532	4.721	3.668	2.959	2.554	2.337	2.086	1.748
VSI-MEWMA	—	(0.1, 1.9)	181.397	49.267	20.656	13.429	10.330	6.752	4.973	3.909	3.109	2.470	2.163	2.072	1.980
		(0.3, 1.7)	186.828	54.139	23.085	14.649	11.015	7.023	5.164	4.066	3.298	2.664	2.353	2.229	2.080
		(0.5, 1.5)	191.876	59.075	25.445	15.852	11.715	7.286	5.346	4.225	3.449	2.836	2.544	2.388	1.801
VSI-$T^2$	—	(0.1, 1.9)	347.563	293.145	222.336	155.214	99.165	26.302	5.931	1.407	0.420	0.197	0.132	0.110	0.102
		(0.3, 1.7)	347.957	294.619	226.776	159.524	104.359	30.527	8.182	2.531	1.012	0.555	0.391	0.329	0.307
		(0.5, 1.5)	347.962	299.305	232.660	165.278	110.019	34.762	10.561	3.654	1.613	0.912	0.650	0.548	0.512

,

Table 6. ATS₁ comparison of the VSI-VSME, VSI-MEWMA, and VSI-$T^2$ schemes when $\lambda =0.25$, $p_0=2$, and $p=20$.

Method	s	$({\mathit{h}}_{\mathit{s}},{\mathit{h}}_{\mathit{l}})$	$\mathit{\delta }$
			0.2	0.4	0.6	0.8	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0
VSI-VSME	1	(0.1, 1.9)	287.122	123.008	38.930	14.227	7.312	3.262	2.035	1.343	0.838	0.463	0.271	0.194	0.160
		(0.3, 1.7)	287.980	128.516	42.980	16.804	8.862	3.901	2.466	1.722	1.187	0.829	0.642	0.549	0.475
		(0.5, 1.5)	293.448	134.568	47.405	19.362	10.413	4.552	2.900	2.071	1.547	1.195	1.012	0.904	0.788
	2	(0.1, 1.9)	274.505	111.596	34.238	12.913	6.941	3.296	2.131	1.449	0.892	0.490	0.270	0.181	0.138
		(0.3, 1.7)	278.250	117.738	39.032	15.189	8.286	3.812	2.468	1.748	1.211	0.829	0.625	0.504	0.408
		(0.5, 1.5)	281.488	122.658	42.293	17.408	9.605	4.335	2.813	2.037	1.511	1.169	0.976	0.829	0.681
	3	(0.1, 1.9)	269.986	107.456	33.500	12.750	6.965	3.363	2.234	1.580	1.033	0.580	0.321	0.203	0.154
		(0.3, 1.7)	272.547	114.598	37.582	15.074	8.252	3.893	2.569	1.867	1.323	0.915	0.674	0.549	0.453
		(0.5, 1.5)	277.680	118.722	41.292	17.214	9.615	4.414	2.913	2.148	1.613	1.240	1.033	0.895	0.753
	4	(0.1, 1.9)	266.036	106.470	33.368	12.760	7.007	3.441	2.314	1.677	1.154	0.682	0.370	0.229	0.169
		(0.3, 1.7)	271.999	111.967	37.622	15.211	8.378	3.998	2.659	1.966	1.431	0.994	0.725	0.586	0.493
		(0.5, 1.5)	271.010	117.909	41.508	17.509	9.755	4.513	3.006	2.243	1.708	1.306	1.080	0.946	0.813
	5	(0.1, 1.9)	263.672	104.679	33.474	12.927	7.103	3.490	2.381	1.762	1.238	0.758	0.421	0.251	0.181
		(0.3, 1.7)	267.623	110.722	37.781	15.288	8.513	4.069	2.734	2.044	1.508	1.062	0.768	0.618	0.521
		(0.5, 1.5)	270.818	116.920	41.831	17.804	9.939	4.604	3.097	2.319	1.789	1.369	1.119	0.980	0.863
VSI-MEWMA	—	(0.1, 1.9)	259.054	107.510	36.838	14.560	7.864	3.383	2.651	2.047	1.608	1.152	0.713	0.407	0.253
		(0.3, 1.7)	264.025	114.285	41.630	17.401	9.611	4.502	3.045	2.340	1.861	1.421	1.017	0.762	0.629
		(0.5, 1.5)	267.782	119.878	46.461	20.242	11.358	5.177	3.437	2.625	2.120	1.678	1.324	1.115	1.000
VSI-$T^2$	—	(0.1, 1.9)	347.563	293.145	222.336	155.214	99.165	26.302	5.931	1.407	0.420	0.197	0.132	0.110	0.102
		(0.3, 1.7)	347.957	294.619	226.776	159.524	104.359	30.527	8.182	2.531	1.012	0.555	0.391	0.329	0.307
		(0.5, 1.5)	347.962	299.305	232.660	165.278	110.019	34.762	10.561	3.654	1.613	0.912	0.650	0.548	0.512

and

Table 7. ATS₁ comparison of the VSI-VSME, VSI-MEWMA, and VSI-$T^2$ schemes when $\lambda =0.5$, $p_0=2$, and $p=20$.

Method	s	$({\mathit{h}}_{\mathit{s}},{\mathit{h}}_{\mathit{l}})$	$\mathit{\delta }$
			0.2	0.4	0.6	0.8	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0
VSI-VSME	1	(0.1, 1.9)	331.791	225.025	110.853	46.337	18.617	3.124	1.207	0.607	0.323	0.202	0.150	0.122	0.107
		(0.3, 1.7)	335.846	230.182	117.498	50.845	21.541	4.334	1.855	1.074	0.715	0.536	0.435	0.364	0.323
		(0.5, 1.5)	331.822	231.154	122.772	55.203	24.923	5.605	2.516	1.544	1.100	0.872	0.721	0.608	0.538
	2	(0.1, 1.9)	322.488	208.620	99.741	39.815	15.643	2.774	1.159	0.584	0.305	0.185	0.134	0.111	0.103
		(0.3, 1.7)	323.106	212.798	105.281	43.845	18.512	3.842	1.717	1.004	0.667	0.494	0.391	0.332	0.308
		(0.5, 1.5)	327.022	215.840	111.233	48.291	21.229	4.902	2.276	1.421	1.024	0.804	0.647	0.554	0.513
	3	(0.1, 1.9)	320.442	202.557	95.733	38.620	15.310	2.816	1.156	0.599	0.316	0.192	0.138	0.114	0.104
		(0.3, 1.7)	322.767	206.293	101.385	43.021	18.171	3.825	1.733	1.021	0.684	0.508	0.403	0.341	0.311
		(0.5, 1.5)	322.412	212.239	106.558	47.072	21.176	4.913	2.304	1.440	1.049	0.826	0.670	0.566	0.518
	4	(0.1, 1.9)	318.274	199.010	95.651	38.754	15.304	2.868	1.197	0.627	0.335	0.203	0.144	0.117	0.105
		(0.3, 1.7)	321.025	203.564	100.061	43.064	18.392	3.949	1.788	1.051	0.706	0.527	0.421	0.352	0.316
		(0.5, 1.5)	319.730	208.378	106.146	47.363	21.313	5.044	2.356	1.488	1.076	0.856	0.697	0.583	0.527
	5	(0.1, 1.9)	315.952	196.505	93.702	38.900	15.581	2.917	1.233	0.648	0.351	0.209	0.151	0.121	0.107
		(0.3, 1.7)	316.331	202.827	99.124	42.941	18.710	4.066	1.826	1.083	0.730	0.547	0.438	0.362	0.320
		(0.5, 1.5)	319.320	205.822	104.926	47.173	21.699	5.140	2.424	1.530	1.105	0.881	0.723	0.601	0.532
VSI-MEWMA	—	(0.1, 1.9)	309.544	193.262	97.249	42.460	18.320	3.406	1.409	0.778	0.438	0.262	0.180	0.143	0.119
		(0.3, 1.7)	312.683	198.374	102.793	47.870	21.831	4.961	2.127	1.266	0.848	0.632	0.511	0.421	0.356
		(0.5, 1.5)	316.425	205.281	108.637	52.780	25.588	6.235	2.826	1.754	1.263	1.002	0.838	0.701	0.592
VSI-$T^2$	—	(0.1, 1.9)	347.563	293.145	222.336	155.214	99.165	26.302	5.931	1.407	0.420	0.197	0.132	0.110	0.102
		(0.3, 1.7)	347.957	294.619	226.776	159.524	104.359	30.527	8.182	2.531	1.012	0.555	0.391	0.329	0.307
		(0.5, 1.5)	347.962	299.305	232.660	165.278	110.019	34.762	10.561	3.654	1.613	0.912	0.650	0.548	0.512

systematically compare the OOC ${\mathrm{ATS}}_1$ of the proposed VSI-VSME scheme with those of the VSI-$T^2$ and VSI-MEWMA schemes under three smoothing parameters $(\lambda =0.1,0.25,0.5)$, various shift magnitudes $\{0.2,0.4,0.6,0.8,1,1.5,2,3,3.5,4,4.5,5\}$, and different choices of the sparsity parameter s (from 1 to 5) while fixing the true number of shifted variables at $p_0=2$. The results reveal several critical insights.

First, across all settings, the VSI-$T^2$ scheme performs dramatically worse than both VSI-MEWMA and VSI-VSME schemes, especially for small to moderate shifts $(\delta \le 1.5)$. For example, in Table 5 $(\lambda =0.1,\delta =0.8,(h_S,h_L)=(0.1,1.9))$ the VSI-$T^2$ scheme yields an ${\mathrm{ATS}}_1$ of $155.214$, whereas the VSI-MEWMA scehme achieves $13.429$ and the VSI-VSME scheme (with $s=1$) reaches as low as $11.969$. This huge gap is expected because the conventional VSI-$T^2$ statistic accumulates noise from all 20 dimensions, severely diluting the sparse signal.

Second, the comparison between VSI-VSME and VSI-MEWMA schemes is more nuanced and depends critically on $\lambda$, $\delta$, and the choice of s. When $\lambda$ is small ($0.1$, Table 5), the VSI-VSME scheme consistently outperforms VSI-MEWMA across almost all $\delta$ and s values. For example, when $(h_S,h_L)=(0.1,1.9))$, for $\delta =0.6$, the VSI-VSME scheme with $s=2$ gives ${\mathrm{ATS}}_1=18.941$ vs. the VSI-MEWMA scheme’s $20.656$; for $\delta =0.8$, we have ${\mathrm{ATS}}_1=12.020$ vs. $13.429$. This demonstrates that the variable selection mechanism effectively filters out noise and provides a cleaner signal for the EWMA recursion, which is particularly beneficial when the shift is sparse (only 2 out of 20 variables shift). As $\delta$ increases beyond $2.0$, the differences narrow, and for very large shifts $(\delta \ge 4.0)$ the VSI-MEWMA scheme occasionally becomes slightly faster. This is because the shift is no longer “sparse” in effect; the mean shift is so large that even a naive multivariate statistic detects it almost instantly.

Third, the role of the sparsity parameter s is highly informative. In Table 5, when $\delta$ is very small $(0.2-0.4)$, larger s (e.g., $s=4$ or 5) yields lower ${\mathrm{ATS}}_1$, meaning that allowing more variables into the selected set helps to capture the weak signal scattered across the two truly shifted variables plus noise. For moderate $\delta$$(0.6-1.0)$, the optimal s shifts to $s=1$ or 2, i.e., a more parsimonious selection works best. For large $\delta$$(\ge 2.0)$, $s=1$ almost always gives the smallest ${\mathrm{ATS}}_1$. This pattern is consistent with the intuition: under a very weak shift, the penalized selection needs more freedom to avoid missing the signal; under a strong shift, a tight sparsity constraint prevents noise from entering the monitoring statistic.

Fourth, as $\lambda$ increases to $0.25$ (Table 6) and $0.5$ (Table 7), the performance gap between the VSI-VSME and VSI-MEWMA schemes gradually diminishes; for $\lambda =0.5$ with small $\delta$ (e.g., $\delta =0.4$), the VSI-MEWMA scheme actually outperforms the VSI-VSME scheme for all s. This occurs because a large $\lambda$ makes the EWMA statistic nearly equivalent to the current observation, reducing the benefit of exponential smoothing. At the same time, the variable selection step becomes more susceptible to sampling variability, slightly delaying detection. Nevertheless, for $\delta \ge 1.0$, VSI-VSME with $s=1$ or $s=2$ still matches or beats the VSI-MEWMA scheme.

In summary, across all three tables, the VSI-VSME scheme exhibits remarkable robustness; even when s is mis-specified (e.g., $s=5$ while true $p_0=2$), the ${\mathrm{ATS}}_1$ remains far below that of the VSI-$T^2$ scheme and often close to or better than that of the VSI-MEWMA scheme. This indicates that the proposed chart does not require exact knowledge of the number of shifted variables to deliver good performance.

4. Example Analysis

To demonstrate the practical usefulness of the proposed VSI-VSME scheme, we apply it to a real dataset from an automotive sheet metal assembly process. The data, adapted from Table 5.14 in [60], consist of thickness deviation measurements recorded at six critical locations on automobile body panels. These six variables are correlated and their dimensional accuracy directly affects the vehicle’s structural integrity, safety, corrosion resistance, and final assembly fit. In this case study, the first 30 observation vectors are treated as IC samples and are used to estimate the process parameters and to establish control limits; the remaining 20 observation vectors are monitored sequentially to detect potential mean shifts that may arise from tool wear, material variations, or assembly misalignment. For all competing schemes, we set the same VSI parameter configuration of $(h_S,h_L)=(0.5,1.5)$ to ensure a fair comparison and fix the IC ${\mathrm{ATS}}_0$ at 370. The VSI-$T^2$, VSI-MEWMA, and VSI-VSME schemes (with $s=1,2,3,4,5$) are then applied to monitor the 20 test observations.

The monitoring results are displayed in Figure 2. The VSI-$T^2$ scheme issues an OOC alarm at time $t=6$, while the VSI-MEWMA scheme triggers an alarm at $t=4.5$. For the proposed VSI-VSME chart, the detection time depends on the choice of the sparsity parameter s. Specifically, when $s=2,3,4$, the chart signals an OOC condition at $t=4$. When $s=1$ or $s=5$, detection occurs slightly later at $t=4.5$, which is still faster than the VSI-$T^2$ scheme and comparable to the VSI-MEWMA scheme. This pattern indicates that the VSI-VSME scheme is not overly sensitive to the exact choice of s, with even moderately mis-specified s still yieldng good performance. More importantly, the fact that $s=2,3,4$ achieve the fastest detection suggests that the underlying shift likely affects about two to four of the six monitored dimensions. In summary, the VSI-VSME scheme offers a practical and efficient tool for high-dimensional process monitoring in real manufacturing environments where sparse shifts are common and quick detection is critical for minimizing quality loss.

5. Conclusions

This study is motivated by a fundamental challenge in high-dimensional statistical process monitoring, namely, detecting small sparse shifts in the mean vector when only a small subset of variables is actually OOC. Traditional multivariate control charts suffer from severe noise accumulation in high dimensions, while standard adaptive sampling approaches predominantly focus on temporal adjustments (varying sampling intervals or sample sizes) without addressing the variable selection problem. To fill this gap, we propose a novel scheme which integrates two complementary mechanisms: (i) a forward variable selection algorithm that automatically identifies the most likely shifted variables and filters out irrelevant noise, and (ii) a variable sampling interval strategy that shortens the sampling interval when the process enters a warning region and lengthens it when the process appears stable.

Through extensive Monte Carlo simulations, we demonstrate several key findings. First, the simulation results strongly support that conclusion that the proposed VSI-VSME scheme consistently detects sparse shifts faster than the FSI-VSME scheme. This improvement is most pronounced for small smoothing parameters. Second, the proposed scheme exhibits a dramatic advantage over the VSI-$T^2$ and VSI-MEWMA schemes. This advantage is particularly evident for small $\lambda$ and small to moderate shift magnitudes, where the variable selection component is able to successfully isolate the sparse signal from high-dimensional noise. Third, the proposed VSI-VSME scheme is robust to mis-specification of the sparsity parameter s. Even when s deviates from the true number of shifted variables $P_0$, the ${\mathrm{ATS}}_1$ performance rarely deteriorates below that of the VSI-$T^2$ and VSI-MEWMA schemes.

A potential direction for future research would be to explore dynamic selection of s during Phase-II monitoring. Moreover, the current variable selection procedure assumes that the IC covariance matrix is known or can be estimated accurately from a large Phase-I dataset. Developing a VSI-VSME chart that accounts for estimation uncertainty (e.g., through adjusted control limits or bootstrap calibration) would enhance its practical reliability. We believe that the VSI-VSME framework provides a solid foundation for these future innovations.