Multivariate Mixed EWMA-CUSUM Control Chart for Monitoring the Process Variance-Covariance Matrix

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I. INTRODUCTION
Control charts are widely used to detect changes in a process location and/or dispersion parameter.These charts are categorized as memory and memoryless charts.Shewhart [1] initiated the idea of a control chart named by Shewhart chart, which is a memoryless control chart; it identifies large shifts in a process and uses only the current information.Memory type control charts are efficient in identifying small changes in the process parameter(s).The most common examples include Cumulative sum (CUSUM) control chart proposed by Page [2] and Exponentially Weighted Moving Average (EWMA) control chart by Roberts [3].The afore-mentioned charts are univariate charts that monitor a single quality characteristic of interest.
Sometimes, we are interested in the monitoring of more than one correlated quality characteristics like the hardness and tensile strength of steel; thus multivariate control charts are employed.Hotelling [4] introduced a chart that monitors two or more correlated quality characteristics and named it as Chi-squared control chart.Shewhart

control chart
The associate editor coordinating the review of this manuscript and approving it for publication was Bora Onat.
(location) in the univariate set-up is an analog of Chi-squared control chart (mean vector).Pignatiello Jr and Runger [5] and Crosier [6] proposed memory type multivariate control charts.They offered Multivariate CUSUM (MCUSUM) control charts that monitor the mean vector.Lowry et al. [7] developed a Multivariate EWMA (MEWMA) control chart; this chart follows a direct analog of univariate EWMA.Multivariate memory-type control charts are efficient to identify small changes in the process mean vector.
Alt [8] proposed a multivariate control chart that monitored the variance-covariance matrix and named it as generalized variance chart.This chart is not effective to detect small shifts in the process variance-covariance matrix.Djauhari et al. [9] introduced vector variance control chart, which can be employed when the variance-covariance matrix is singular.This chart monitors both rational subgroups and individual observations.It was also combined with the generalized variance chart to produce an effective detecting ability of the variance-covariance matrix chart.Memar and Niaki [10] proposed multivariate charts used to monitor the variance-covariance matrix with individual observations.Healy [11] developed two charts that monitor process mean vector and process variance-covariance matrix by using MCUSUM statistics.Also, Chen et al. [12] developed a MEWMA (Max-MEWMA) chart that monitors shift in both process parameters such as location and dispersion simultaneously.Recently, Adegoke et al. [13] proposed a multivariate version of Homogeneously Exponentially Weighted Moving Average (HEWMA) control chart for the monitoring of process mean vector.
Abbas et al. [14], [15] combined the structure of EWMA and CUSUM charts to gain sensitive scheme for the monitoring of process parameter(s).Ajadi et al. [16] extended this idea by raising the sensitivity of mixed EWMA-CUSUM (MEC) chart in the univariate set-up.Later, Ajadi and Riaz [17] introduced a multivariate MEC chart for the monitoring the process mean vector.Following the same inspiration, we intend to design, in this article, a multivariate MEC control chart for the monitoring of process variance-covariance matrix.The study proposal will serve the purpose for different kind of processes such as carbon fiber tubing process, material flow controlling process and bayer process.
The rest of this study is organized as: Section II presents the information of the existing multivariate control charts for monitoring the process variance-covariance matrix, along with the newly proposed control chart.Section III offers the performance evaluations and comparison of the proposed chart and its counterparts.Section IV provides a real application to validate the superiority of the proposed scheme to its counterparts.Finally, Section V gives the summary, conclusions and recommendation of this study.

II. CONTROL CHARTS FOR THE PROCESS VARIANCE-COVARIANCE MATRIX
This section discusses some useful control charts used to monitor process variance-covariance matrix, such as generalized variance chart, multivariate EWMA and CUSUM control charts for monitoring the process variance-covariance matrix.The design structures of these charts will be given, and it will be discussed how the process is declared in-control (IC) or out-of-control (OOC).

A. PRELIMINARIES
Let X be a p dimensional vector (X p×1 ) following a multivariate normal distribution with mean vector µ and variance-covariance matrix .Symbolically, we may write it as: X ∼N p (µ, ) , where µ is p dimensional mean vector µ p×1 and is p dimensional variance-covariance matrix ( p×p ).The mean and variance-covariance matrix are defined as follows: For our study purposes, we will use µ 0 and 0 as the known mean vector and variance-covariance matrix, respectively.Let X i be the i th sample matrix consisting of the x ijk as the i th (i = 1, 2, . . ., n) observation of the j th (j = 1, 2, • • • , p) quality characteristic on the k th (k = 1, 2, . . ., m) sample.Let Xi and S i are p dimensional i th sample mean vector and sample variance-covariance matrix ( Xp×1 and S p×p ) respectively, defined as: Based on these terminologies, we outline brief details of some commonly used multivariate control charts for dispersion and propose a new control chart in the following subsections.

B. GENERALIZED VARIANCE CONTROL CHART
Generalized variance (GenVar) chart, proposed by Alt [8], was developed for monitoring the determinant of the sample variance-covariance matrix |S|.The decision limits including upper control limit (UCL), center line (CL) and lower control limit (LCL) for this chart are given as where and L 1 is the width of the control limit.In most of the time when the actual value of 0 is unknown then, it is estimated by where ˆ is the Phase I estimate for the variance-covariance matrix.The plotting statistic is taken as |S i | which is compared against the above-mentioned control limits.If |S i | falls outside UCL or LCL, then the process is declared as OOC, otherwise IC.

C. MULTIVARIATE EWMA CONTROL CHART
Chen et al. [12] proposed a multivariate chart based on EWMA statistic (MEWMAD) for the simultaneous monitoring of process mean vector and variance-covariance matrix.In this study, we are only interested in the process dispersion, and the variability statistic of the MEWMAD control chart is given as: where H(.;p (n − 1)) represents the Chi-squared distribution with p (n − 1) degrees of freedom, λ is the smoothing constant which always lies between zero and one, and −1 (.) is the normal inverse cumulative distribution function.The other notations are defined in Section II(A).
Based on Y i , we may define a new statistics V i as (that will be used as plotting statistic): The plotting statistic |V i | is compared against the control limit (h 1 ).If |V i | exceeds h 1 , the process is declared OOC, otherwise IC.

D. MULTIVARIATE CUSUM CONTROL CHART
The MCUSUM control chart for monitoring the process variability is named by MCUSUMD control chart, proposed by Healy [11] and defined as: where k 1 = pn δ δ − 1 log δ and δ refers to the amount of shift (see section III(A)).
According to Cheng and Thaga [18], the statistic was standardized by using the following expression Therefore, the plotting statistic is defined as The process is stated as the IC state as long as the S i is below the control limit h 2 , otherwise, it is considered as OOC.It is to be mentioned that for our study purposes, we have fixed the value of k 1 = 0.5, in order to make the chart more sensitive for the smaller shift.

E. THE PROPOSED MULTIVARIATE MIXED EWMA-CUSUM CONTROL CHART
In this section, we propose a new multivariate dispersion chart by integrating the effects of MEWMAD and MCUSUMD control charts into a single structure.This idea was initially developed in the univariate setup by Abbas et al. [14], [15].Later, Ajadi et al. [16] and Ajadi and Riaz [17] made further developments on it.This study follows their inspirations and develops a new multivariate dispersion chart, namely Multivariate Mixed EWMA-CUSUM (MMECD) control chart.The methodological details of the proposed MMECD chart are as follows: Firstly, we compute the W i statistic given in (4) and convert it into Chi-squared value with p(n − 1) degrees of freedom.
After this, we applied normal inverse cumulative distribution function to obtained the standardized statistic such as: Next, M i is transformed into the MEWMA statistic as given below: We can integrate U i into MCUSUM dispersion statistics as: where k * 2 is chosen equal to half of the shift in terms of standard deviation.The statistic MMECD i is compared with the control limit (h 3 ) and the process is declared OOC when MMECD i is greater than h 3 .

III. PERFORMANCE EVALUATION AND COMPARISONS
This section will serve the following purposes: discuss the performance measures used to evaluate the performance of the charts under investigation; describe the construction of the control limits of various charts of this study; outline the algorithm of run length, and design of control charting constants; provide a detail comparison between the proposed multivariate variance-covariance matrix chart (MMECD) and its various counterparts.

A. PERFORMANCE MEASURES
In this study, we use various run length (RL) properties to assess the performance of the control charts under discussion, by considering different amounts of shifts (δ) in a process.Following Chen et al. [12], the shift in the variance-covariance matrix is defined as follows: where δ = 1 refers to an IC state, otherwise OOC.For the sake of simplicity and a fair comparison with existing charts, we have used ρ = 0.2 and the case of equal variances.However, one may expect similar findings for the other choices of ρ and variances.For OOC, we have considered the case of an increase in variability (i.e.δ > 1).
The measures covered in this study include average run length (ARL), standard deviation run length (SDRL), median run length (MDRL), extra quadratic loss (EQL), and sequential extra quadratic loss (SEQL), and some useful percentiles/quantiles (Q i s) of the run length distribution.These measures are briefly described as: A series of points in an IC state until an OOC signal is received referred to a run.The number of points in a run is termed as run length.ARL represents the average number of sample points awaited until the first OOC signal is received.It is classified into two types, ARL 0 (i.e.IC state) and ARL 1 (i.e.OOC state) [19].SDRL is another useful measure used to assess the spread of the run length distribution.MDRL refers to the midpoint of run length distribution (i.e. the point that covers 50% of the area).EQL is defined as the weighted ARL with respect to the range of shift (δ min to δ max ) by considering the square of shift (δ 2 ) as weight.Mathematically, it is defined as: A discrete form of the EQL measure may be defined as: where q refers to the number of shifts covered in the performance evaluation.SEQL is the cumulative measure that refers to the EQL up to a certain shift (say δ i ), mathematically defined as: A discrete form of the SEQL measure may also be defined as: For more details on these performance measures, one may be seen in [20]- [24] and the references therein.

B. ALGORITHM FOR CHOOSING THE CONTROL LIMITS OF MMECD CHART
Step 1 Algorithm for Run Length: (i) Generate a sample from the multivariate normal distribution and calculate the sample statistic (W i ) and its inverse normal using ( 4) and ( 7) respectively.(ii) Calculate Z i and substitute its value in U i using (8) and ( 9) respectively; then substitute U i in (10).(iii) Evaluate statistic MMECD i as given in (10) and plot it against the control limit h 3 .If MMECD i is plotted beyond the control limit, then the process is declared OOC and the corresponding sample number (which is one in this case) is the run length.On the other hand, we proceed to (iv) if MMECD i is plotted inside the control limit h 3 .(iv) We generate another sample from the multivariate normal distribution.Compute the plotting statistic and compare it with the control limit, as we did in (ii) and (iii) above.If the process is declared OOC, then stop at this stage and report 2 as run length, otherwise continue this method for several iterations.
Step 2 Iterative Procedure: Repeat step 1 iteratively to get a large number of RL values (say 10,000 run lengths), and calculate the average of these RL values, producing ARL.If the process in the IC state, then the resulting ARL will be ARL 0 and for OOC state the resulting ARL will be ARL 1 .

C. DESIGN STRUCTURE OF CHARTING CONSTANT AND LIMITS
The design structures of the proposed chart and its counterparts depend on the sample size (n) and the number of correlated quality characteristics to be monitored simultaneously (p).We have evaluated the performance of the charts as a function of n and p.For our study purposes, we have evaluated the results for n = 5 and p = 2, 3, 4 for the proposed MMECD control chart.For the comparison purpose with the existing counterparts, we have covered the case of p = 2.Some selective results for control limits are given below for different charts of this study.

D. COMPARATIVE ANALYSIS
The run length profile (i.e., ARL, SDRL, and percentiles values) with respect to various shifts in the process variance-covariance matrix (1 ≤ δ ≤ 2) for all charts under considerations are provided in Tables 1-5.The results of the generalized variance control chart are reported in Table 1.The run length profile of the MEWMAD control chart with respect to different choices of smoothing parameters (λ = 0.10, 0.20, 0.30 and 0.50) is given in Table 2.The findings of MCUSUMD control chart are provided in Table 3.The run length profile of the proposed MMECD control chart with respect to different choices of λ and quality characteristics (p = 2,3 and 4) is given in Tables 4 and 5.The findings of the GenVar control chart revealed that 20% increase in the dispersion parameter might cause 71.46% decrease in the ARL 1 (cf.Table 1).The run length profile of MEWMAD control chart with respect to different choices of smoothing parameters (λ = 0.10, 0.20, 0.30 and 0.50) depicted that 86.77% decrease reported in the ARL 1 of MEWMAD chart due to 20% shift in the dispersion parameter (i.e.δ = 1.2)(cf.Table 2).The findings of the MCUSUMD control chart exhibited that there is 86.56% decrease in the ARL 1 with the 20% increase of dispersion parameter (i.e., δ = 1.2)(cf.Table 2).The run length profile of the MMECD control chart with respect to different choices of λ and quality characteristics (p = 2,3 and 4) is revealed that 88.34% decrease is reported in ARL 1 of MMECD chart due to 20% shift in the dispersion parameter.
In addition, the run length curves, along with the box plots, are presented in Figure 1.When the process is in a stable state or IC state (i.e.δ = 1), all charts have similar performance.For a shifted environment in dispersion parameter (e.g.δ = 1.5), then the proposed MMECD chart outperforms the other competitive charts under study, as may be seen in Figure 1.
Further, other performance measures, such as EQL and SEQL are reported in Table 6.The SEQL is employed to check the performance of a chart over a specific range of shifts in the dispersion parameter.It is to be mentioned that EQL is based on all the shifts in the process dispersion (that is the last column of Table 6).The findings depict that MMECD chart having smoothing parameter (λ = 0.5) offers the minimum SEQL and EQL as compared to all the competitive charts under consideration.
Moreover, the prime findings of this study are summarized as follows: The proposed MMECD scheme is better than the Gen-Var chart for small and moderate shifts in the process dispersion for all values of λ.
The proposed MMECD chart outperforms MCUSUMD chart for the monitoring of process variance-covariance matrix when the shifts in the process are very small.It holds true for all values of λ.The proposed structure is better than the MEWMAD chart at small shifts when λ < 0.5.When λ ≥ 0.5 (for the sake of brevity, we did not include results in Tables), the proposed MMECD is better than MEWMAD chart for small and moderate shifts of the process dispersion.The sensitivity of the proposed chart to detect small and moderate shifts in the variance-covariance matrix increases as p increases.

IV. AN APPLICATION
In this section, we provide an application of the proposed chart for the manufacturing process.We outline brief details of the process related to carbon fiber tubes, followed by the implementation of the proposed and some existing charts.

A. CARBON FIBER TUBING PROCESS B. A QUALITY CONTROL APPLICATION
In this Section, we apply a dataset related to the industrial manufacturing of a carbon fiber tubing process.In an application, it was observed that the quality of carbon fiber tubes might depend on three variables, including inner diameter, thickness and length [25], as shown in Figure 2. The dataset is categorized as carbon1 and carbon2; where carbon1 and carbon2 datasets contain 30 and 25 multivariate samples, respectively, of the three correlated quality characteristics, each of size 8.This data can be obtained by installing the MSQC package in R program.The implementation procedure used for this dataset is outlined below.First, we used carbon1 to get the Phase I estimates for the required parameters and employed carbon2 for Phase II monitoring of the multivariate variability of the process.We noticed that all the data points are in-control in Phase II for each chart.We have considered the GenVar, MEWMAD, MCUSMD, MMECD control charts.Thus, we combined both carbon1 and carbon2 datasets, computed the sample variance-covariance matrix of the combined dataset as provided in (11)    Based on this estimate, we generated 50 tri-variate samples (referring to the three correlated quality characteristics), each of size 8.The first 20 samples are kept in-control, while we shifted the sample variance-covariance matrix of the last 30 samples by rescaling by 1.2.The resulting Phase II dataset is presented in Table 7.
For this dataset, we constructed all the control charts under investigation in this study.For all the charts, control limits coefficients are set such that ARL 0 = 250.The graphical displays of the proposed chart and its counterparts are shown in Figures 3-6.The result of our findings is outlined as follows: The GenVar chart detected 3 out-of-control signals, at sample numbers 35, 48 and 49 (cf. Figure 3).The MEWMAD chart for the process dispersion alarmed 4 out-of-control signals, at sample numbers 26,48, 49 and 50 (cf.Figure 4).The MCUSUMD chart captured 15 out-of-control points, at sample numbers 31 and 37-50 (cf. Figure 5).The proposed MMECD chart detected 27 out-of-control signals at sample numbers 23-50 (cf. Figure 6).From the analysis above, it is evident that the proposed chart is very effective in detecting small shifts (such as δ = 1.2) in process variance-covariance matrix.The potential reasons for such OOC might include a special cause(s) in one of the important steps in the carbon fiber tubing process such as spinning, stabilizing, carbonizing etc. and these un-natural variations may be caused by the variables   that might include temperature, gas flow, and chemical composition.It is important to timely fix any issues with these stages/variables as it may lead to serious health issues such as skin allergy, lungs infections etc.

V. SUMMARY, CONCLUSION AND RECOMMENDATIONS
A new multivariate control chart, namely MMECD chart is proposed to monitor changes in the process dispersion in a multivariate setup.It is designed by mixing the features of multivariate EWMA and CUSUM charts for dispersion matrix of a process.The performance of the proposed scheme is evaluated in terms of several useful measures such as ARL, SDRL, MDRL, EQL and SEQL.The performance of the MMECD control chart is compared with the other competing charts, including GenVar control chart, MEWMAD and MCUSUMD control charts.The comparisons revealed that the proposed scheme has better performance than its counterparts for detecting the small shifts in the process dispersion in a multivariate environment.The scope of this study may be extended to investigate the Multivariate Mixed EWMA-CUSUM under contaminated normal and non-normal processes.

TABLE 1 .
Run length profile of the generalized variance chart at fixed p = 2, when ARL 0 = 250.

FIGURE 1 .
FIGURE 1. RL curves and box plots of the proposed MMECD control chart and its counterparts.

FIGURE 2 .
FIGURE 2. An image of a carbon fiber tube from a tubing process.
to obtain a new Phase I estimate:

TABLE 5 .
Run length profile of the MMECD control chart for p =3 and 4 at fixed ARL 0 = 250.

TABLE 6 .
SEQL of the proposed MMECD control chart and its Counterparts for p = 2 at fixed ARL 0 = 250.

TABLE 7 .
Specific carbon fiber tubing phase II dataset.