High-dimensional control charts with application to surveillance of grease damage in bearings of wind turbines

High-dimensional data, characterized by having more attributes or variables than observations, presents unique challenges in industrial operations surveillance. Traditional multivariate control charts, like Hotelling’s T 2 chart, perform adequately with lower-dimensional data. However, they often fail to detect variations in process means as data dimensionality increases. This research proposes new control charts designed to enhance the detection of mean variations in both high and low-dimensional data. Specifically, Srivastava-Du (SD), Bai-Saranadasa (BS) and Dempster (DS) statistic-based charts are introduced, and their effectiveness is evaluated through simulations and real-life data applications. The performance of these charts is compared under various multivariate normal and non-normal distributions. Results indicate that DS and BS charts perform similarly, with the DS chart outperforming in low-dimensional normal distribution. Conversely, the SD chart outper-formed in high-dimensional non-normal distributions. Additionally, the practical application of these proposed charts is illustrated through the monitoring of grease degradation in wind turbine bearings.


Introduction
Effective process monitoring is crucial for ensuring quality, productivity, and efficiency in today's data-driven decision-making era.Traditional control charts have proven effective for monitoring processes with a limited number of variables.However, they face significant challenges when dealing with high-dimensional datasets.For lowdimensional datasets, Hotelling's T 2 control chart is the most widely used method among multivariate control charts (Hotelling, 1947).This chart performs well when the dimensionality is low, but its effectiveness diminishes as the number of variables increases.
High-dimensional data refers to datasets with a large number of variables compared to the number of observations.Modern datasets often contain thousands to millions of features for each object or individual.Such high-dimensional data can be found in various fields, including industry (Kim et al., 2019;Li et al., 2014), medical science (Fan et al., 2021;Kim et al., 2020;Maboudou-Tchao et al., 2023;Yeganeh, Johannssen, Chukhrova, & Rasouli, 2024), chemical engineering (Sun et al., 2013), and image processing (S.He et al., 2018;Yeganeh, Johannssen, & Chukhrova, 2024).Dealing with high-dimensional data is challenging due to the curse of dimensionality, computational complexity, and overfitting.Techniques such as dimension reduction and variable selection are used to overcome these challenges and extract significant insights.
Dimension reduction methods reduce the number of variables in a dataset while retaining essential information.These methods transform high-dimensional data into a lower-dimensional space, facilitating visualization, computational efficiency, and interpretation.In statistical process control (SPC), dimension reduction methods can be categorized into three main types: component analysis, projection-based techniques, and other methods.Component analysis is the most commonly used approach in SPC studies, with methods such as principal component analysis (PCA) (Shaohui et al., 2022), local and global PCA (Yu, 2012), multiway PCA, kernel PCA (Lee et al., 2004;Sun et al., 2013), and multistate adaptive-dynamic PCA (Odom et al., 2018).Projection-based methods include locality preserving projections (LPP) and multiway LPP (Hu & Yuan, 2008).Other techniques, such as joint decorrelation and stochastic proximity embedding (Fujiwara & Kano, 2017), one-dimension residual convolutional auto-encoder (Yu & Liu, 2022), and stacked auto-encoder (Yeganeh et al., 2023), are also employed for dimension reduction in SPC studies.
Variable selection methods aim to select a subset of important variables from a highdimensional dataset.
Step-down procedures (Sullivan et al., 2007) and Least Absolute Shrinkage and Selection Operator (LASSO)-based procedures (Zou et al., 2011) are commonly used for high-dimensional process monitoring.These methods effectively select relevant variables, and then Hotelling's T2 chart is implemented to monitor highdimensional processes.However, Hotelling's T 2 chart is only effective for detecting large shifts in the parameter of interest.With a substantial number of correlated quality variables, its effectiveness diminishes.Correlation in the data complicates the inversion of the covariance matrix, resulting in unreliable outcomes (Mason & Young, 2002;Seborg et al., 2016).Additionally, an abundance of variables can hinder the detection of process shifts and cause multicollinearity, where variables become highly interdependent (Ku et al., 1995).Although dimension reduction and variable selection techniques help reduce dimensions, their efficiency is affected by multicollinearity.
Some studies have designed high-dimensional control charts using techniques other than dimension reduction or variable selection to monitor shifts in the mean of highdimensional processes.For example, Ahmadi-Javid and Ebadi (2021) proposed a twostep technique involving Hotelling's T 2 chart to monitor the location for a normally distributed high-dimensional multi-stream process.Ullah et al. (2017) proposed a James-Stein shrinkage estimator and created independent T 2 statistics using a leave-one-out resampling procedure.Ahmad and Ahmed (2021) proposed a modified T 2 statistic for high-dimensional data monitoring.
In univariate SPC literature, many studies have focused on monitoring means with two samples.For example, Balakrishnan et al. (2009) proposed nonparametric control charts using runs and Wilcoxon-type rank-sum statistics.Mahmood et al. (2017), Mukherjee and Chakraborti (2012) and Chong et al. (2022) proposed nonparametric control charts for two-sample cases using Cucconi and LePage test statistics.Abbas et al. (2021) proposed a univariate chart using a sign test, and recently, Erem and Mahmood (2023) and Mahmood and Erem (2023) proposed univariate control charts using exceedance statistics.In a two-sample monitoring structure, a two-sample mean test compares the means of both samples, and the test statistic is used as the plotting statistic for the chart.
Following the spirit of two-sample univariate control charts, this study proposes new multivariate memory-less control charts for two-sample mean monitoring.The proposed control charts can be implemented for both low-and high-dimensional data without requiring dimension reduction or variable selection techniques, thus preserving all information from the high-dimensional dataset.
The remaining parts of this article are structured as follows: Section 2 introduces the two-sample Hotelling's T 2 chart and develops the proposed charts.Section 3 describes the design of the simulation study.Section 4 presents the performance of the proposed charts for both low-and high-dimensional data.A case study is provided in Section 5, and concluding remarks are given in Section 6.

Methodology
This section discusses the methodologies of the conventional chart and our proposed charts.Assuming X 1 ; X 2 ; . . .; X p be random variables that follow a multivariate normal distribution with mean vector μ 0 and covariance matrix Σ 0 ; also supposing that the i th observation vector of the process be denoted by X ij , where i ¼ 1; 2; . . .:; n; j ¼ 1; 2; . . .; p, the Hotelling's T 2 statistic is defined as: . The Lower Control Limit (LCL) of Hotelling's T 2 chart is set to be zero while the Upper Control Limit (UCL) under phase I is obtained by , where β α;p=2; nÀ pÀ 1 ð Þ=2 is upper α th percentile of beta distribution with parameters p=2 and n À p À 1 ð Þ=2 (Bersimis et al., 2007;Tracy et al., 1992).Similarly, the UCL for phase II under an assumption of normality is obtained by where F α;p; nÀ p ð Þ is the upper α th percentile of the F distribution with parameters p and n À p ð Þ. See (Montgomery, 2020) for more details.The Hotelling's T 2 chart is commonly used in monitoring lower-dimensional data (Bulut, 2023;Mahmood et al., 2019).However, in several high-dimensional studies, Hotelling's T 2 chart was implemented after applying dimension reduction techniques.
The conventional two-sample Hotelling's T 2 chart and its control limits are briefly explained in Sub-Section 2.1.Then, the details of the proposed Srivastava-Du (SD), Bai-Saranadasa (BS) and Dempster (DS) statistic-based charts are discussed in Sub-Section 2.2.

Two-Sample Hotelling's T 2 chart
For a two-sample case (where k refers to sample number, m represents the sample size of the first sample, n represents the sample size of the second sample, and p denotes the number of dimensions), let x ijk , i ¼ 1; 2; . . .; m or i ¼ 1; 2; . . .; n, j ¼ 1; 2; . . .; p; and k ¼ 1; 2 are the p-dimensional multivariate normal random vectors with mean vector μ ¼ ðμ 1 ; . . .; μ p Þ 0 and unknown common positive covariance matrix Σ.The two-sample Hotelling's T 2 statistic is computed as follows: where S p is the pooled covariance matrix.The UCL is given by: The LCL is zero, and the chart detects a signal when T 2 exceeds the UCL.

Proposed two-sample high-dimensional charts
The two-sample Hotelling's T 2 provided in section 2.1 works well when the data has lower dimensions, such as p � 30.When the data have higher dimensions than the existing two-sample Hotelling's T 2 statistics is not calculated due to the singularity problem of the sample covariance matrix (S p ); therefore, a new chart must be proposed to overcome this issue.In the past, some studies are designed based on the Hotelling's T 2 using dimension reduction and variable selection.However, the proposed charts will not require us to adopt dimension reduction and variable selection techniques; hence, no information will be lost from the high-dimensional dataset.
For a two-sample case (k refers to sample number, m represents the sample size of the first sample, n represents the sample size of the second sample, and p is used to determine dimensions), let x ijk , i ¼ 1; 2; . . .; m or i ¼ 1; 2; . . .; n, j ¼ 1; 2; . . .; p; and k ¼ 1; 2 are the p-dimensional multivariate normal random vectors with mean vector μ ¼ ðμ 1 ; . . .; μ p Þ 0 and unknown common positive covariance matrix Σ.In the twosample case, when the number of variables is more than observations, the two-sample statistic T 1 (Srivastava & Du, 2008) can be written as: where � x 1 is the mean of first sample, � x 2 is the mean of the second sample, the diagonal matrix (D S ) of sample variances (S) is defined by The sample correlation matrix (R) is defined as follows: The adjustment coefficient c p;mþn can be chosen by The LCL and UCL of the statistic T 1 for a given false alarm α, can be obtained from the following probability limits: It is to be noted that the test statistic T 1 was proposed by Srivastava and Du (2008).Therefore, the control chart based on statistic T 1 is named as SD chart.
The DS chart is proposed based on the statistic T 2 (Dempster, 1958(Dempster, , 1960)), which can be expressed as follows: The LCL and UCL of the test statistic T 2 for a given false alarm α, can be obtained from the following probability limits: Further, the BS chart is proposed based on statistic T 3 (Bai & Saranadasa, 1996), which is given by: The LCL and UCL of the BS chart for a given false alarm α, can be obtained from the following probability limits: It is to be noted that all proposed three charts, named SD, DS, and BS charts, signal an out-of-control (OOC) point when T 1 , T 2 , and T 3 falls outside of their control limits, respectively.Otherwise, the process is considered to be in a stable or in-control (IC) situation.

Design of simulation study
Following the methodologies outlined Stoumbos and Sullivan (2002), Najarzadeh (2021), andD. He et al. (2021), the following distributions are considered in the simulation setup: (i) Multivariate normal with Σ 0 consisting of unit variances and covariances equal to 0, denoted as MVN 0 .(ii) Multivariate normal with Σ 0 consisting of unit variances and covariances equal to 0.5, denoted as MVN 1 .(iii) Multivariate t with 5 degrees of freedom, denoted as MVT.
(iv) Multivariate gamma with a shape parameter of 5 and a scale parameter of 1, denoted as MVG.
For both MVT and MVG, Σ 0 consists of unit variances and covariances equal to 0.5.The MVT and MVG distributions are used to check the robustness of the proposed methodologies.
For lower dimensions, such as p ¼10, 30, and 50, an equal sample size (m ¼ 30, n ¼ 30) is considered for both samples.With pre-specified ARL 0 values of 168, 200, 370, and 500, Table 1 presents the control limits for the proposed DR, BS, and SD charts for each of the four cases: MVN 0 , MVN 1 , MVT, and MVG.To assess the performance of the proposed charts in comparison to Hotelling's T 2 chart under different shifts (δ ¼ 0; 0:1; 0:2; 0:3; 0:4; 0:5; 0:75; 1) in the second sample observations, a simulation study with 10,000 iterations, pre-specified ARL 0 ¼ 200, and four different distribution cases is designed.The comparative findings are presented in Table 2.
For higher dimensions, such as p ¼100, 300, and 500, the sample sizes for two samples are considered as (a) A simulation study of 1,000,000 iterations has been performed to provide control limits under the distribution cases (i) -(iv) for the two combinations of sample size (a) -(b).Using the same pre-fixed ARL 0 values as for lower dimensions, the control limits of the proposed charts for higher dimensions under four different distribution cases are presented in Table 3.Finally, the comparative analysis of the proposed charts based on the run length profile with 10,000 simulations, ARL 0 ¼ 200, shifts (δ ¼ 0; 0:1; 0:2; 0:3; 0:4; 0:5; 0:75; 1), sample sizes (a) -(b), and distribution cases (i)-(iv) is provided in Table 4.
The implementation structure of the proposed chart for higher dimensions is illustrated in Figure 1.The control chart setup is designed in two phases: Phase I and Phase II.
Phase I is designed to find the control limits based on the specified p, m, n, ARL 0 , and distribution (i.e.MVN 0 , MVN 1 , MVT, and MVG).Both samples are calculated based on the IC parameters μ 0 ¼ 0 and Σ 0 .The samples are generated 1,000,000 times, and statistics are calculated for each set of samples.Finally, control limits are obtained using Equation 6, 7, 9, 10, 12, 13against pre-specified ARL 0 , which are presented in Table 3.
Phase II is the retrospective stage of the analysis.For the specified p, m, n, ARL 0 , and distribution (i.e., MVN 0 , MVN 1 , MVT, and MVG), the first sample is generated using IC parameters μ 0 and Σ 0 , while the second sample values are generated using OOC Further, statistics are obtained based on both samples; when a sample exceeds the control limits obtained in Phase I analysis, the respective sample number is considered a single run length.This process is repeated 10,000 times, and finally, the average run length (ARL) and standard deviation of run length (SDRL), along with Q 1 (1st quartile), Q 2 (2nd quartile or median), and Q 3 (3rd The first row of each cell shows ARL (SDRL), second row of each cell shows the values of 25th, 50th, and 75th percentiles, respectively.Bold numbers mean the smallest ARL for each shift in different dimension size.
Table 3.Control limits of the proposed charts under higher dimensions.
Sample size p Distribution ARL 0 DR BS SD quartile) of run length, are calculated and reported for performance evaluation in Table 4.It is to be noted that the run length profile of Hotelling's T 2 chart for higher dimensions is not possible due to the singularity problem of the sample covariance matrix (S p ) in Equation 1. Furthermore, Bai and Saranadasa (1996) showed that the T 3 and T 1 statistics have similar performance in testing the means of two-sample highdimensional data.Similarly, we found in initial findings that the BS chart has the same run length profile as the DR chart, as also seen in the case study presented in Section 5. Therefore, for brevity, we have excluded the results of the BS chart.

Performance of proposed charts under lower dimensions
Based on the run length profile, the performance comparison between the proposed DR and SD charts, and the existing two-sample Hotelling's T 2 chart under lower dimensions is described as follows: • The control limits of the proposed charts for ARL 0 ¼ 168, 200, 370, and 500 are presented in Table 1.However, only ARL 0 ¼ 200 will be used to conduct a comparative analysis of the charts.• Hotelling's T 2 chart has shown superior detection ability in Table 2, under the distribution structure MVN 0 , for p ¼ 10, as its ARL 1 values for various shifts are less than those of the proposed DR and SD charts.However, since our proposed DR chart offers the least ARL 1 values for various shifts, it works best for p ¼ 30.For p ¼ 50, the ARL 1 values of Hotelling's T 2 chart are least for minor shifts like δ ¼ 0.1, 0.2, and 0.3; our proposed DR chart has shown the greatest detection performance for large shifts like δ ¼ 0.4, 0.5, 0.75, and 1.  197.7, 198.7, and 201.3, respectively (cf.Table 2).Nonetheless, Hotelling's T 2 chart yields the lowest ARL 1 (193.7)for p ¼ 50, δ ¼ 0:2; for the same dimension with δ ¼ 1, the suggested DR chart yields the smallest ARL 1 (45.7)compared to the other charts (refer to Table 2).In comparison to other charts, our proposed SD chart offers the least ARL 1 values for all other shifts with dimension p ¼ 50.• Hotelling's T 2 chart with the distribution structure MVT only shows small ARL 1 values for p ¼ 10, δ ¼ 0:1, and, p ¼ 30, δ ¼ 0:1 & 0.2 (cf.Table 2).In comparison to the other charts, the proposed DR chart has shown superior detection capabilities for all other shifts and dimensions, including p ¼ 10 and 30.The suggested SD chart performs better than the other charts for the dimension p ¼ 50, as shown by its smaller ARL 1 values (see Table 2).• With the exception of δ ¼ 0:1, p ¼ 30, where the suggested SD chart yields a small ARL 1 value (193.9),our proposed DR chart performs best for p ¼ 10 and 30 under the distribution MVG for all shifts (cf.Table 2).Because it gives the lowest ARL 1 values compared to the other charts, the suggested SD chart also demonstrates the greatest detection performance against all shifts for the dimension p ¼ 50 (cf.Table 2).
In conclusion, the typical Hotelling's T 2 chart was only effective for the MVN 0 distribution when p ¼ 10.However, when p ¼ 50, the SD chart outperformed the DR chart with MVT and MVG distributions, but in other cases with low-dimensional data, the DR chart was superior.

Performance of proposed charts under higher dimensions
As aforementioned, the two-sample Hotelling's T 2 chart cannot be implemented for higher dimensions, for example, p ¼ 100, 300, and 500.Therefore, the run length profile performance is compared between the proposed DR and SD charts under a higher dimension environment.
• The control limits of the proposed charts for ARL 0 ¼ 168, 200, 370, and 500 are presented in Table 3.However, only ARL 0 ¼ 200 is used to conduct a comparative analysis of the charts.For example, for ARL 0 ¼ 200, with p ¼ 100 (m ¼ 30, n ¼ 30), MVN 1 distribution, the LCL and UCL of the DR chart are 0:5796 and 1:6214, respectively.Similarly, with p ¼ 300, MVT distribution, the LCL and UCL of the SD chart are À 1:6393 and 2:0354, respectively.7,189.5,171.3,189.9,and 109.1,respectively (cf. Table 4).• Only for some higher shifts, like δ ¼ 0.75 and 1, the DR chart displays small ARL 1 values for all combinations of sample sizes, distributions, and dimensions p ¼ 100 and 300, with the distribution structure MVT.The SD chart has demonstrated superior detection ability over the DR chart for all other shifts with all combinations of sample sizes and dimensions, p ¼ 100 and 300 (cf.Table 4).• For both combinations of sample sizes, the SD chart performs best, with p ¼ 100 and 300 under the distribution MVG for all shifts with the exception of δ ¼ 0.1, p ¼ 100 (m ¼ 30%, n ¼ 45%), when the DR chart yields a modest ARL 1 value of 204.2 (see Table 4).
Overall, for MVN 0 and MVN 1 distributions, the DR chart can identify shifts faster than the SD chart, but for MVT and MVG distributions, the SD chart performs better in terms of detecting the shifts.

A case study on wind turbine bearings
As a reliable, environmentally friendly energy source, wind power has become the fastestgrowing renewable energy globally, paving the way for a future reliant on clean energy (Amano, 2017).Wind turbines are designed to transform the kinetic energy of wind into mechanical or electrical energy.Two popular designs of wind turbines are shown in Figure 2. The main bearings, gearbox bearings, generator bearings, blade bearings, and yaw bearings are vital components in large-scale wind turbines; these bearings play a critical role in converting wind energy into electrical energy (Liu & Zhang, 2020).Large-scale wind farms typically contain hundreds of turbines, each with its main bearing, while smaller wind farms have only a few turbines.Grease, a semi-solid lubricating substance composed of base oil, thickeners, and additives, is used to provide long-lasting lubrication, reduce friction, and protect against wear and corrosion in various industrial and mechanical applications, including wind turbine bearings.
Grease damage is a significant factor that can cause deterioration in wind turbine bearings.This case study aims to apply the proposed control charts to efficiently detect anomalies in main bearings of wind turbines caused by grease damage.This section elaborates the real-life application of our proposed DR, BS, and SD charts to monitor shifts in the mean of high-dimensional data with two samples.For this purpose, a true grease damage dataset for 6 months obtained by Yucesan and Viana (2020) is used.
bearings.Each row represents the measure of true grease damage recorded every 10 minutes for each of the 123 turbines, resulting in 6 months of observations.
Using the 'EnvCpt' R package version 1.1.3(Killick et al., 2021), the change point of the data is identified at 17,280.The package fits all possible models based on various trend, change-point, and autocorrelation models on the 25,920 row means and finds that the change point occurs after the first 4 months, as shown in Figure 3. Hence, the first 17,280 rows (4 months of observations) are considered as IC data.This IC data is divided into 288 samples, with 60 as the sample size.Each sample is grouped into two subsamples by considering equal sample sizes of m ¼ n ¼ 30.Following the structure of the proposed control charts given in Figure 1, Phase I is designed to find the control limits based on the specified ARL 0 ¼ 200.As the distribution of the IC data is unknown, bootstrap is used to generate 1,000,000 samples from IC data, and plotting statistics are calculated for each set of samples.Further, control limits using Equations 6, 7, 9, 10, 12, and 13 for the proposed DR, BS, and SD charts are obtained as (156.59, 194.3), (111.04, 137.99), and (5.20, 38.23), respectively.Finally, the test statistic values for the three proposed charts are obtained using 288 IC samples, resulting in 288 IC values for each chart, which are plotted in the pink-shaded area in Figure 4.
Similar to the simulation study, the OOC data for Phase II is recorded where the mean shift occurs in the second sample while the first sample remains IC.The first sample of the OOC data consists of 8,640 IC observations taken from rows indexed 8,641-17,280 (2nd and 3rd month observations), while the second sample also consists of 8,640 observations taken from rows indexed 17,281-25,920 (last 2 months observations).Both samples have equal sample sizes, that is, m ¼ n ¼ 30; the total sample size is 60.This arrangement of data provided 288 OOC values to find test statistics for DR, BS, and SD charts.

Summary and concluding remarks
High-dimensional data, where the number of features exceeds the number of observations, is becoming increasingly prevalent in modern manufacturing processes due to advancements in sensor technologies, automation, and data collection capabilities.Much of the existing literature on SPC has focused on one-sample high-dimensional data.However, as manufacturing processes grow more complex and data-intensive, resulting in two-sample high-dimensional data, the analysis becomes more challenging.To date, no research has utilized control charts to study the behavior of two-sample highdimensional data.This paper discusses the application of control charts to both twosample low-dimensional and high-dimensional data.For low-dimensional data, equal sample sizes were considered for both samples.For high-dimensional data, two combinations of sample sizes were used.A simulation study was conducted to determine the control limits of the charts under four distribution combinations (MVN 0 , MVN 1 , MVT, and MVG) based on pre-specified ARL 0 values.Several performance measures were considered to compare the performance of the charts.
The conventional Hotelling's T 2 chart performed well only for the MVN 0 distribution with p ¼ 10.Our proposed DR and SD charts demonstrated superior detection ability for all low-dimensional data cases.Comparing the performance of our proposed charts under the low-dimensional framework revealed that the SD chart performed better for p ¼ 50 under MVT and MVG distributions, while the DR chart excelled in other scenarios of low-dimensional data.In the high-dimensional setup, the DR chart quickly detected anomalies compared to the SD chart for MVN 0 and MVN 1 distributions, while the SD chart outperformed the DR chart for MVT and MVG distributions.It is important to note that only the control limits were provided for another proposed chart, the BS chart.The BS chart was excluded from the run length profile performance comparison as it exhibited the same performance as the DR chart.A case study on two-sample highdimensional data revealed that both the DR and BS charts were equally effective in detecting shifts in the means of the second sample.
Future studies could extend this research by considering memory-type structures, such as EWMA and CUSUM charts, to detect small shifts in the location parameter.While this study focused on location monitoring, future research might explore monitoring the covariance parameter of high-dimensional data or the joint monitoring of location and scale under a high-dimensional setup.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Figure 1 .
Figure 1.Schematic diagram for the implementation of proposed charts.

Figure 3 .
Figure 3. Change point analysis by fitting different models.

Figure 4 .
Figure 4. Graphical displays of plotting statistics for DR, BS, and SD charts corresponding to their control limits.

Table 1 .
Control limits of the proposed charts under lower dimensions with m ¼ 30 and n ¼ 30.

Table 4 .
ARL (SDRL) comparison of DR and SD charts with m ¼
Table 2 displays the lowest values of ARL 1 in bold type.• The detection ability of the proposed DR chart under the distribution structure MVN 1 is superior to the other charts for both p ¼10 and 30, with the exception of shift δ ¼ 0.1, whereas the ARL 1 values of Hotelling's T 2 , DR, and SD are reported as

•
Table4illustrates that for p ¼ 100 (m ¼ n ¼ 30% and m ¼ 30%, n ¼ 45%), under the distribution structure MVN 0 , the DR chart has shown superior detection ability, as evidenced by its ARL 1 values being less than those of the SD charts for various shifts.Similarly, the proposed DR chart worked best for the same distribution structure with p ¼ 300 (m ¼ n ¼ 30% and m ¼ 30%, n ¼ 45%) as it offers the lowest ARL 1 values for various shifts.The DR chart has shown the greatest detection ability for large shifts, even while the ARL 1 values of the SD chart are smaller for small shifts like δ ¼ 0:1 (p ¼ 100, m ¼ 30%, n ¼ 45%; p ¼ 300, m ¼ 30%, n ¼ 45%) and δ ¼ 0:1 and 0.2 (p ¼ 100, m ¼ 30%, n ¼ 45%).For the ease of the reader, the lowest values of ARL 1 are shown in bold font in Table4.• The detection ability of DR chart under the distribution structure MVN 1 is superior to the SD chart for various shift values for both p ¼ 100 and 300, except shift δ ¼ 0.1 (p ¼ 300, m ¼ n ¼ 30%), δ ¼ 0.2 and 0.3 (p ¼ 100, m ¼ 30%, n ¼ 45%), and δ ¼ 0.1 and 0.2 (p ¼ 300, m ¼ 30%, n ¼ 45%) where the ARL 1 values of the SD charts are reported as 186.