Statistical Monitoring of Covariance Matrix in Multivariate Processes: A Literature Review

02/14/2020
by   Mohsen Ebadi, et al.
0

Monitoring several correlated quality characteristics of a process is common in modern manufacturing and service processes. For this purpose, control charts have been developed to detect assignable causes before producing nonconforming products. Although a lot of attention has been paid to monitoring the multivariate process mean, not many control charts are available for monitoring the covariance matrix. This paper presents a comprehensive overview of the literature on control charts for monitoring covariance matrix in multivariate statistical process monitoring (MSPM) framework. It classifies the research that have previously appeared in the literature. We highlight the challenging areas for research and provide some directions for future research.

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1 Introduction

Statistical process monitoring (SPM) is an effective tool for quality improvement of manufacturing and service products. It has been widely used in industrial and service sectors for process monitoring. Among the tools of SPM, control charts are the most powerful techniques. In many situations, the interests lies in monitoring many inter-related characteristics of a process simultaneously. For example, Huwang et al. [2007] provided an application of wafer manufacturing in the chemical mechanical planarization process from semiconductor industries and Maboudou-Tchao and Agboto [2013] explained an example of service industry in which it is desired to monitor the vital signs of a patient, such as systolic blood pressure, heart rate, and body temperature in an intensive care unit (ICU). This type of examples has motivated development of multivariate statistical process monitoring (MSPM) and as a result, many multivariate control charts have been introduced in the literature. In practical applications, one might be interested in monitoring different features of the underlying multivariate processes such as mean vector, covariance matrix, shape matrix, and etc.

Since the publication of the pioneering paper Hotelling [1947] on multivariate control charts, numerous techniques have been introduced. Jackson [1985], Lowry and Montgomery [1995], and Bersimis et al. [2007] have provided excellent reviews of the literature up to those dates, but their focus is mainly on the mean shift problem. Monitoring covariance structures of multivariate processes is of a great importance too. However, this problem has received much less attention in the literature. In addition, common methods of multivariate quality monitoring like that of Hotelling, which is based on quadratic forms, confound mean shifts with scale shifts. This implies that upon receiving a signal from a control chart, an extensive analysis is needed to determine the nature of the shift.

Monitoring covariance or dispersion matrices is much more challenging problem for several reasons. It is not clear what functionals of the covariance matrices summarize variance-covariance entries in a more effective way. In addition, the number of unknown parameters to be estimated is much larger than those in the mean vector. These types of considerations have manifested in methodologies that are inherently more diverse than those for the mean shift monitoring and therefore not easily accessible for practitioners. It is of great interest to summarize and discuss the diverse literature on the topic in a structured and easily accessible manner. Yeh et al. [2006] reviewed control chart methodologies for covariance matrix monitoring developed between 1990 and 2005. However, since then there have been numerous advancement on the topic in several fronts. The purpose of this paper is to update the existing review literature of this area by focusing on developments occurred since 2006 up to 2019. To do this, perhaps in more systematic way, we classify the research in this area into certain distinctive categories. The key idea in the current review paper is to highlight critical issues and provide guidelines for selecting suitable control charts in every specific settings.

This review mainly focuses on Phase II control charts designed for multivariate normal processes with independent subgroups of observations, but the related topics such as diagnostic procedures, Phase I analysis methods, robust control charts, high dimensional problems and etc are also going to be discussed. The remainder of this paper is as follows. Section 2 concerns with definitions, notations, and problem setup and describes the classification approach that we adopt for grouping relevant publications. Section 3 discusses the methods available in the literature for monitoring covariance matrix when the sample size is greater than or equal to the number of process variables, while Section 4 presents the relevant works on monitoring covariance matrix when sample size is less than the number of variables. Section 5 reviews the control charts for simultaneous monitoring of mean vector and covariance matrix. In Section 6, other developments related to covariance monitoring such as Phase I analysis and interpretation of an out-of-control signal are given. Section 7 provides recommendations for future research in covariance monitoring. Finally, concluding remarks are presented in Section 8.

2 Problem definition and a literature classification scheme

Statistical process monitoring contains two main phases: Phase I and Phase II. In Phase I, historical observations are analyzed in order to determine whether or not a process is in control and then, to estimate the process parameters of the in-control process. The emphasis here is robustness. The Phase II data analysis, however, consists of using the estimates obtained in Phase I and certain control limits to examine whether or not the process remains in control when future observations are drawn. The emphasis here is sensitivity. Typically, a lot of process understandings and improvements is needed in the transition from Phase I to Phase II ( Woodall 2000).

Consider a multivariate process with quality characteristics

to be monitored simultaneously. Under an in-control situation, we assume that the process follows a multivariate normal distribution with mean vector

and covariance matrix . Let

denote the vector of in-control standard deviations of these

variables obtained from the diagonal elements of . Independent sample sets , each with cardinality in the form of

, are taken periodically from this process. In the sampling epoch

, we denote the th observation of the th quality characteristic variable by where , and . Suppose in Phase I, we have collected such sample sets and estimate the parameters and by the sample mean vector and sample covariance matrix of the combined observations, respectively, which are defined as:

(1)

where . The existing literature on control charts for monitoring covariance matrices are mainly motivated by testing two-sided or one-sided hypotheses problems such as:

(2)
(3)

While the methodology for simultaneous monitoring of a mean vector and a covariance matrix is often inspired by a test of hypothesis of the form:

(4)

By surveying some existing multivariate methodologies for two-sided hypothesis test of covariance matrices, Alt (1985) introduced a number of control charts for monitoring covariance matrices of multivariate processes. However, unlike monitoring of a mean shift, in the framework of monitoring dispersion change, the direction of the shift is also important. It is usually much more important to detect a dispersion increase than a decrease, which means that the methodologies based on a one-sided test would be more appropriate than its two-sided counterpart for declaring an out-of-control state.

As mentioned above, Phase II monitoring of process dispersion has been the main concern of existing articles in this research area. Some of the existing methodologies are direct extensions of the univariate control charts to the multivariate framework, while other are based on different approaches. In this review paper, we divide the existing literature into four categories. Given below, we provide some explanations on how these categories are formed:

  • Monitoring covariance matrix alone, when :
    This case has been explored the most among all four categories. The literature assumes that the rational group size is large enough that the sample covariance matrices have full ranks.

  • Monitoring covariance matrix alone, when :
    When the number of variables is large, collecting samples with observations more than variables is sometimes not practical. As an extreme case, there are some industry instances that due to the expenses and epoch of sampling, only individual observations can be sampled for subgroups and therefore the usual methodologies based on the sample covariance matrix are not applicable here.

  • Simultaneous monitoring of mean and covariance:

    In multivariate setting, generally change may happen in many ways. For example, one variable or characteristic may have shift in the mean while another variable may have a change in its variability or in other forms. It is also possible that change may happen in product moments of some variables. This case has received little attention in the literature due to its complexity. Recent papers have mainly focused on procedures of simultaneous monitoring of mean vector and the covariance matrix.

  • Other related topics to covariance monitoring:
    This last subsection briefly reviews the other topics related to monitoring of dispersion in multivariate processes such as diagnostic procedures, robust approaches and Phase I analysis.

In what follows, we shall note that some methods reviewed in this paper can potentially belong to more than one category. For example, the methods used for the case can sometimes be extended to the case . In the next four sections, the main idea of related papers to each category will be discussed.

3 Monitoring covariance matrix alone, when

As previously mentioned, the literature for the case of is abundant. Generally, in almost all methods, one summarizes the information in a multivariate data by means of a univariate statistic. Shewhart-type control charts for monitoring covariance matrices are often constructed based on either the determinant of the sample covariance matrix, known as the sample generalized variance and denoted by , or the trace of the sample covariance matrix, which is the sum of the variances of the variables and denoted by . However, both of and fail to explain certain important features of data or processes captured in the covariance matrix. Since the determinant

is the product of the eigenvalues, it is incapable in detecting covariance matrix changes when some aspects of the variability increase while others decrease. On the other hand,

is the sum of the marginal variances and does not take into account the covariation among different variables.

Alt [1985] used both of these aforementioned quantities and proposed a few methods for Phase II monitoring of covariance matrices. He adapted a multivariate control chart based on the likelihood ratio test (LRT) statistic for the one-sample test of covariance matrix in (2) and defined the charting statistic of the form

(5)

where is the sample variance-covariance matrix of the rational subgroup with size . Asymptotic upper control limits (UCL) for this Shewhart-type chart are given by the upper percentiles of

distribution with degrees of freedom

for large . This method can detect departures in any direction from the in-control (IC) setting. Another method proposed by Alt [1985] is to monitor the square root of , where the control limits were only given for the case of obtained from the identity

(6)

The exact distribution of for is not one of the known distributions. Hence, for higher dimensions, Alt [1985] proposed an alternative method to monitor

by assuming that the most of the probability distribution mass of

is contained in the interval . Since then, several publications have discussed extensions and properties of chart. Two recent publications are Dogu and Kocakoc [2011], and Lee and Khoo [2018]. Ghute and Shirke [2008] proposed a synthetic charting approach for monitoring covariance matrix by combining chart of Alt [1985] and the Conforming Run Length (CRL) chart. This approach is a two-step procedure. For a given rational subgroup, if chart triggers an out of control alarm, that rational subgroup is declared as nonconforming, otherwise it is declared as conforming. CRL is the number of conforming rational subgroups between two consecutive nonconforming rational subgroups. Let be the th () CRL above. The procedure declares the process as out of control if is less than or equal to a predefined value. In order to decrease the amount of false alarms in CRL charts, Gadre [2014] proposed group runs control charts as an extension of the methodology in Ghute and Shirke [2008]. The group runs chart declares a process as out of control, if or for some and for the first time for some predefined value . More works on improving the performance of synthetic control charts for covariance matrix can be found in Lee and Khoo [2015, 2017a, and 2017b].

Costa and Machado [2008, 2009] used the statistic VMAX, the maximum of the marginal sample variances of standardized variables from rational subgroups where standardization is done using the marginal in-control mean and standard deviations. They argued that VMAX has the advantage of faster detection of covariance shifts and is better in identifying the individual out of control variables in comparison to the methods introduced in Alt [1985]. However, this method is sensitive to linear dependency among the variables, as the out of control ARL (

) of the method increases when the correlations among the variables increase. Machado et al. [2009b], Quinino et al. [2012], Costa and Neto [2017], Gadre and Kakade [2018], and Machado et al. [2018] proposed some variants of the VMAX and improved it in bivariate and trivariate cases. In addition to the aforementioned bivariate charts, Cheng and Cheng [2011] proposed an artificial neural network based approach which showed better performance than traditional generalized variance chart in detecting variance shifts of a bivariate process.

Although most of the techniques developed in the literature are centred on detecting changes of any kind in the covariance matrix, in most practical situations it is more critical to detect the process dispersion increase than decrease. Costa and Machado [2011a] introduced the RMAX statistic based on the marginal sample ranges of the quality characteristics. Let denote the sample range of th quality characteristic for the th rational subgroup. Thus for sample is used as a monitoring statisitic. This RMAX chart signals an increase in the process variance. Yen and Shiau [2010] proposed a control chart for detecting dispersion increase, i.e. is positive semidefinite, based on the LRT statistic of a one-sided test. For the rational subgroup of , define

(7)

where are the roots of the equation , and is the number of . The LRT statistic is and whenever , the

th quantile of the distribution of

, the process is considered to be out of control. It is important to notice that detection of dispersion decreases also have some important information about the reasons of process improvement. See Yen et al. [2012] in which the authors used the corresponding one-sided LRT statistic for detecting decrease in multivariate dispersion.

The aforementioned control charts are designed for detecting shifts in any number of elements in the covariance . However, in some real-world applications, one may know a priori that the changes can only occur in a small number of enteries in the covariance matrix (the sparsity feature) asking for more efficient methods of estimation. Recall the in-control process and note that for the lower triangular (inverse-Cholesky root) matrix with property , we have , where is the identity matrix. Thus, without loss of generality, one can assume that the in-control distribution of is , and if the process goes out of control, the distribution changes to with only a few entries of are different from the entries of the identity matrix . To incorporate this information into the design of control charts for monitoring the process variability, Li et al. [2013] proposed to use a penalized likelihood estimation approach. They used an penalty function on the precision matrix in order to force the unchanged off-diagonal elements of the precision matrix to zero. So, for the sample epoch , the respective estimate of the precision matrix can be defined as the solution to the following penalized likelihood function

(8)

where is the matrix norm and is the penalization or tuning parameter that can be tuned to achieve different levels of sparsity for the estimated . Having obtained , the Penalized Likelihood Ratio (PLR) charting statistic is defined as:

(9)

The PLR chart signals an out-of-control alarm when where depends on the selected and the desirable in-control ARL (), which can be obtained via a Monte Carlo simulation.

4 Monitoring covariance matrix alone, when

Gathering observations with desirable rational subgroups is not always plausible and there are certain processes in which only individual observations (i.e. ) can be collected. For this reason, there has been an emphasis in the literature on developing EWMA-type control charts for monitoring covariance matrix of a process with individual observations. Notice that EWMA-type charts belong to the class of accumulative methods and work better with individual observations than with rational groups. When the determinant of the covariance-matrix estimator as a generalized variance measure is always zero. However, in these situations, the trace of the matrix could be employed as an alternative measure of generalized variance but it is more natural to use to monitor the process variability when the matrix is diagonal. Huwang et al. [2007] used this idea and proposed a multivariate exponentially weighted mean square (MEWMS) control chart for individual observations. They used the transformation , whose in-control parameters are and . At sampling point , one can define a multivariate exponentially weighted moving average statistic for individual observations as:

(10)

where is the smoothing parameter and . It can be shown that E( so that can be used as an estimate of . An MEWMS control chart for covariance monitoring can be defined with time-varying control limits of the form

(11)

Here the value of depends on , , and and the formulas for and are given in Huwang et al. [2007]. Notice that this procedure is sensitive to misidentification of the mean vector, meaning that if the mean vector of the process changes in the observation period whilst the covariance matrix remains unchanged, the procedure tends to spuriously signal a covariance change. To overcome this drawback, Huwang et al. [2007] proposed another procedure, called the multivariate exponentially weighted moving variance (MEWMV) chart which is insensitive to shifts in the process mean. Their proposal is to replace in (10) with its deviation from , in form of

(12)

where is an estimate of the possible mean shift at time and it is obtained from the recursion , where and , the matrix is defined by and . The control limits of the MEWMV chart are given by

(13)

Again, the quantities and are given in Huwang et al. [2007].

Gunaratne et al. [2017] proposed a computationally efficient algorithm based on Parallelised Monte Carlo simulation to obtain the optimal values and improve the ability of MEWMS and MEWMV charts in monitoring high-dimensional correlated quality variables. Hawkins and Maboudou-Tchao [2008] proposed yet another multivariate exponentially weighted moving covariance matrix (MEWMC) control chart similar to the MEWMS chart. To be more specific, suppose is an inverse-Cholesky root matrix of , that is a matrix satisfying . Consider the transformation to , which follows the distribution under the in-control setting. Hawkins and Maboudou-Tchao [2008] defined as:

(14)

where is the smoothing constant and

. The test statistic will be then defined as:

(15)

The control chart signalling a loss of control if , where is the upper control limit and is chosen based on the desired . There are two major differences between the methods in Huwang et al. [2007] and Hawkins and Maboudou-Tchao [2008]. Firstly, they have different initialization of EWMA. Secondly, unlike Huwang et al. [2007], Hawkins and Maboudou-Tchao [2008] uses an Alt [1985]-type likelihood ratio statistic to develop their charting statistic. MEWMC chart can also be modified by applying the concept of dissimilarity between two matrices. Huwang et al. [2017] used this idea and developed a multivariate exponentially weighted moving dissimilarity (MEWMD) chart that outperforms MEWMC chart of Hawkins and Maboudou-Tchao [2008] in certain settings.

Another variation of the method in Huwang et al. [2007] was proposed in Memar and Niaki [2009] by employing -norm and -norm based distances between diagonal elements of the matrices in (10) from their expected values instead of using the trace in the control charts (11) proposed by Huwang et al. [2007]. Following the same path, Memar and Niaki [2011] considered the general case of rational subgroups with . Later, Hwang [2017] noticed the large number of false alarms are produced when using charts of Memar and Niaki [2009, 2011] and therefore proposed a chi-square quantile-based monitoring statistic to overcome this problem.

Mason et al. [2009] introduced a charting approach that is in a form of the well known Wilks’ statistic. When , the charting statistic can be expressed as a function of the ratio of the determinant of two separate estimates of the underlying covariance matrix. The numerator is the determinant of the sample covariance matrix based on the Phase I historical data and the denominator is obtained from the historical data plus the most recent observed subgroup of size in Phase II. When the test shows that a new sample does not increase the volume of the space occupied by the historical data set, then it means that the new sample is very similar to the original sample and otherwise, the control chart signals an out of control situation.

Recently, in a line of research similar to the case of discussed before, several authors have proposed methods for monitoring sparse changes in covariance matrices when . Yeh et al. [2012] pointed out that the method proposed by Li et al. [2013], which has been explained in Section 3, requires the subgroup size much larger than the dimensionality , and thus not applicable for individual observations. For this reason, they modified the penalty function in (8) to shrink the sample precision matrix toward the in-control one rather than to and defined as:

(16)

where can be obtained from (10) and is a data-dependent tuning parameter. Using this estimate in MEWMC chart of Hawkins and Maboudou-Tchao [2008], Yeh et al. [2012] introduced the LASSO-MEWMC (LMEWMC) chart for monitoring possible sparse changes in covariance matrices. Maboudou-Tchao and Agboto [2013] proposed a Shewhart-type control chart similar to that of Li et al. [2013] specific to the cases of . At the sampling epoch and subgroup perform the transformation , where is the respective observation vector. Let be an matrix whose th row is given by the row vector . The method of Maboudou-Tchao and Agboto [2013] uses the penalization approach given in (8) while is replaced by , and produces as an estimate of . The statistic is given by

(17)

which is exactly the same as (15) if is replaced by . See also Maboudou-Tchao and Diawara [2013].

Shen et al. [2014] observed that the performances of the proposed methods by Yeh et al. [2012] and Li et al. [2013] strongly depend on their tuning parameters, and proposed a method that overcomes this drawback. Their approach is as follows. Consider defined in (14) for any . Let , and , where () is an entry in the matrix . Letting and , Shen et al. [2014] introduced the MaxNorm charting statistic in the form of

(18)

where , , , and are estimated via a Monte Carlo simulation when is large. In a most recent work, Abdella et al. [2019] discussed shortcomings of the penalized likelihood based methods in a broader fashion and argued two major flaws. Firstly, detection performance of the penalized likelihood-based methods strongly depends on the shift patterns associated with the pre-specified tuning parameter. Secondly, the sparsity assumptions that being used as a building block in these methods could be violated after the data transformation step in such a way that changes in a small number of entries of the original covariance matrix may result in changes to a large number entries in the transformed covariance matrix. Motivated by the adaptive LASSO-thresholding of Cai and Liu [2011], Abdella et al. [2019] proposed a control chart for monitoring of the covariance matrix of a high dimensional process called ALT-norm chart. Via Monte Carlo simulations, they concluded that the ALT-norm chart performs very well, in terms of out-of-control ARL, in detecting several types of shift patterns and a wide range of shift magnitudes. Kim et al. [2019] also addressed the issue of losing sparsity after data transformation and proposed a ridge () type penalized likelihood ratio method. This method does not rely on transform of the data and can detect changes in the covariance matrix without sparsity assumption when the sample size is small. However, it is sensitive to the misspecification of the tuning parameter.

In another attempt to remove the effect of tuning parameter, Li and Tsung [2019] used the two-sample test statistic of Ledoit and Wolf [2004] for high-dimensional covariance matrices (). Their proposed control chart integrated the powerful two sample test of Ledoit and Wolf [2004] with EWMA procedure for multivariate process with individual observation. They called the proposed MEWMV chart, the variability change with a large (MVP) chart. Li and Tsung [2019] showed that the MVP chart is less affected by the high dimensionality than the charts proposed by Yeh et al. [2012], Li et al. [2013] and Shen et al. [2014] and more effective if the process shifts cause changes in variance components.

5 Simultaneous monitoring of mean and covariance

The aforementioned control charts discussed in the previous sections for monitoring the covariance matrix assume that the mean vector is constant over the monitoring period of the covariance matrix. However, in practice, the mean vector rarely remains the same. Therefore, it is important to take into account the effect of a mean shift during the monitoring period, and simultaneously investigate changes in both the mean vector and the covariance matrix. There are several research in the literature that have considered this scenario. For a bivariate process, Niaki and Memar [2009] developed a bivariate method which is based on the maximum of the usual EWMA charts for the mean vector and the covariance matrix to jointly monitor them. Machado et al. [2009a] proposed two charting methods for bivariate processes: (1) the MVMAX chart that only requires the marginal sample means and variances (2) use of the non-central chi-square statistic to monitor the marginal means and the variances of processes simultaneously. These ideas were extended to the general multivariate case with . See Costa and Machado [2011b] and Costa and Machado [2013].

For simultaneous monitoring of the mean vector and covariance matrix, Reynolds and Cho [2006] proposed two control charts for monitoring the covariance matrix, and combined them with the standard MEWMA chart designed for monitoring the mean vector. Specifically, let denote the standardized observations, and be the covariance matrix of the random vector . Denote the respective in-control covariance matrix with . For any and , define , where is the sample mean for variable at the sampling epoch . The EWMA statistics of the sample mean and squared standard deviation of the standardized observations for the variable are defined by

(19)
(20)

respectively, where , are the initial values, and is the tuning parameter. Assuming is given, the MEWMA control chart for monitoring mean vector is given by the quadratic form

(21)

where . In addition the two MEWMA-type statistics of Reynolds and Cho [2006] for monitoring are

where represents the inverse of a matrix whose entries are the squared of the corresponding entries of the matrix . The chart is used with a UCL, while is used with both an LCL and a UCL and is more effective in detecting decreases in variability. Reynolds and Cho [2006] studied the performance of the aforementioned MEWMA control charts and compared to some other existing EWMA charts in the literature. See also Reynolds and Stoumbos [2008, 2010], Reynolds and Kim [2007], and Reynolds and Cho [2011]. It is worthwhile to mention that, despite their effectiveness, the hybrid approaches similar to those in Reynolds and Cho [2006] suffer from the deficiency of losing ability to provide detailed diagnostic information when a process is out-of-control. For example, in simultaneous application of the Hotelling (or MEWMA) and control charts, the former may reacts to mean shifts, dispersion changes, and changes of correlations altogether, which will mask the nature of shift. This is mainly because of the fact that those charts assume that the covariance matrix (or its estimate) remains unchanged during the monitoring process. Zhang and Chang (2008) proposed a combined charting scheme for monitoring individual observations which differentiates mean shifts from variance shifts. Their proposal is composed of two control charts. The first control chart is a dynamic MEWMA (DMEWMA) that sequentially updates the estimate of the covariance matrix only as new data become available. Hence, this chart is sensitive to mean shifts only. The second chart is a multivariate exponentially weighted moving deviation (MEWMD) control chart which monitors the difference between the current observation and the current process mean estimated by a moving average of the certain number of most recent observations. Note that this MEWMD chart detects variance shifts only.

Instead of implementing two separate control charts, Zhang et al. [2010] proposed a single control chart based on integrating EWMA procedure with the generalized LRT. This new chart is similar to MEWMC chart proposed by Hawkins and Maboudou-Tchao (2008) but is designed to monitor the process mean vector and covariance matrix simultaneously. Using EWMA procedure makes the proposal of Zhang et al. [2010] more effective in detecting small or moderate shifts. This control chart can be also used for the case of individual observations. However, it can not determine which parameter or parameters have shifted after a signal occurs. Wang et al. [2014] used the advantage of the penalized likelihood estimation in producing sparse estimates when only a small set of the mean or variance/covariance components contributes to changes in the process. They proposed two control charts based on the penalized likelihood estimate for multivariate process in which only individual observations are available. Their idea extends the previously discussed work of Li et al. [2013] and Yeh et al. [2012] to simultaneously estimate both of the process mean vector and the covariance matrix via the penalized likelihood method and construct the relevant charting statistic. Zamba and Hawkins [2009] developed a multivariate change-point analysis through generalized likelihood ratio statistics applied sequentially for detecting changes in the mean and/or covariance matrix as well as their locations. Their formulation has the benefit of not needing a large Phase I datasets.

6 Other Relevant Topics

In this subsection, we briefly discuss some selected works on other topics relevant to the covariance matrix monitoring. This includes diagnostic approaches, Phase I analysis and robust methods among others.

  • Interpretation of out-of-control signals

    Once a control chart has signalled an alarm, it is important to identify the source of cause by performing diagnostic analysis on the magnitude of the shift and the time of the change. Since many unknown parameters are contained in the covariance matrix, the diagnostic step in covariance matrix monitoring is a much more involved task than that for a mean vector. The bivariate case is relatively simple and detection method such as those of Costa and Machado [2008] and Niaki and Memar [2009] can be easily used to find the out of control variables. However, for a higher dimensional settings, finding the variables responsible for signals is not an easy task. Motivated by the representation of based on squared multiple correlation coefficients, Mason et al. [2010] proposed a decomposition of Wilks’ ratio statistic, introduced in Section 4, to identify the process variables contributing to a signal. Sullivan et al. [2007] extended the step-down diagnostic procedure of the mean vector monitoring to covariance matrix. Step-down approach partitions the parameter set into two subsets, in which one subset contains those parameters that their estimates are statistically different on the two sides of change point and the other subset includes parameters for which their estimates are not statistically different. The procedure searches for the largest subset that have the feature of the second group and then selects the remaining parameters as shifted parameters. The method proposed by Sullivan et al. [2007] is also applicable for non-normal multivariate data. Mingoti and Pinto [2018] showed that the method of Sullivan et al. (2007) has better detection power than both VMAX and VMIX control charts under the bivariate normal assumption. Zou et al. [2011] considered the sparsity feature in high-dimensional processes and argued that a fault diagnosis problem is similar to the variable or model selection problem in this setting. If , of size , is the full set of the indices of the parameters and is a candidate subset of that identifies all the indices corresponding to the changed parameters, the objective is to select an optimal so that

    could be as close as possible to the true fault isolation model. For this purpose, Zou et al. [2011] reduced fault isolation to two-sample variable selection problem to provide a unified diagnosis framework based on Bayesian information criterion (BIC). Zou et al. [2011] then proposed combining BIC with the adaptive LASSO variable selection method (LEB method) to obtain a LASSO-based diagnostic procedure which includes diagnosis in both the mean vector and the covariance matrix. Cheng and Cheng [2008] formulated identifying the source of variance shifts in the multivariate process as a classification problem and proposed two classifiers based on neural networks (NN) and support vector machines (SVM) to identify which variable is responsible for the variance shift. See also Salehi et al. [2012] and Cheng and Lee [2016] in this regard. Hung and Chen [2012] applied the Bartlett’ s decomposition and Cholesky’ s decomposition to classify changes of covariance matrix and to determine the source of fault.

  • Non-Normality of data and Phase I analysis

    The multivariate control charts are generally constructed based on the normality assumption of the underlying process. It is generally well understood that these procedures lose their efficiency when the process distribution deviates from multivariate normality. Therefore, robust and nonparametric control charts are preferable. Riaz and Does [2008] used the sample Gini mean differences based matrix, denoted by , as an estimate of the covariance matrix and proposed a robust control chart for Phase II monitoring of process variability in a bivariate process when departures from bivariate normality is suspected. But as pointed out by Saghir [2015], the design structures of both and charts are based on multivariate normal distribution. The authors considered a variety of non-normal distributions and proposed asymmetrical probability limits for and the charts for each underlying distributions in Phase II analysis of bivariate non-normal processes. Osei-aning et al. [2016] concentrated on the use of maximum of marginal sample standardized dispersion estimates such as the standard deviation, interquartile range, average absolute deviation from median and the median absolute deviation and developed Phase II control charts based on these statistics for normal and non-normal bivariate processes. Haq and Khoo (2018) proposed a non-parametric MEWMA sign chart which is robust to the violation of the normality assumption in simultaneous monitoring of mean and dispersion. Liang et al. [2019] applied spatial sign covariance matrix and maximum norm to the EWMA scheme to propose a robust multivariate control chart which is distribution-free over the elliptical distribution family and more sensitive for various covariance matrix shifts under heavy-tailed distributions.

    As mentioned, the Phase I data are used to gain understanding about the process, assessing process stability, and parameters estimates. This step is difficult and critical in process monitoring since the additional variability introduced from the estimation step affects the Phase II control chart’s performance. The success of Phase I methods depends on their ability to distinguish the outlying observations correctly and providing reliable estimates of the process parameters which is related to the correct specification of the underlying IC model. Saghir et al. [2016] proposed control limits of Phase I and charts for a bivariate normal process based on the false alarm probability (FAP), while Saghir et al. [2017] developed general design structure of Phase I and charts based on FAP when observations are drawn from a bivariate normal distribution and provided the necessary control charting constants. Vargas and Lagos [2007] used the minimum volume ellipsoid estimator (MVE), as a robust estimator of the covariance matrix in developing control charts for the Phase II monitoring of dispersion. The MVE finds the ellipsoid with minimum volume that covers a subset of at least half of the observations. Similarly, Variyath, and Vattathoor (2014) used two robust estimators including re-weighted minimum covariance determinant (RMCD) and the re-weighted minimum volume ellipsoid (RMVE) and proposed RMCD / RMVE-based MEWMS / MEWMV control charts for Phase I monitoring of dispersion in multivariate processes with individual observations.

    Generally, obtaining good estimates in the multivariate setting is not an easy task and it requires a large Phase I exercise. The unknown-parameter self-starting formulation rectifies the need for a large Phase I data set. Maboudou-Tchao and Hawkins [2011] combined the self-starting monitoring scheme for the mean vector and covariance matrix to develop a self-starting chart for joint monitoring of mean and covariance. Self-starting control charts use successive observations for recalculating the parameter estimates each time a new sample is taken and checking process stability. It has the benefits that can blur the differences between Phase I and Phase II and handle sequential monitoring and estimation simultaneously. Zhang et al. [2016] pointed out that self-starting charts assume that process variables follow a multivariate normal distribution. Moreover, although nonparametric control charts robustly detect changes across different types of data distributions, they are not distribution-free, that is they can not obtain the specified IC run-length distribution without knowing the exact IC distribution or requiring a sufficiently large size of IC samples. Zhang et al. [2016] proposed a distribution-free multivariate control chart based on a multivariate goodness of fitness (GoF) test to detect general distributional changes (including changes in the mean vector, the covariance matrix, and the distribution shapes). They showed that it has a satisfying performance even with an unknown IC distribution or limited reference samples. Their chart employs data-dependent control limits and is also applicable for high dimensional observations. Li et al. [2014] applied the concept of data depth and the Mann-Whitney test and proposed a nonparametric multivariate Phase I control chart for detecting shifts in both of the mean vector and the covariance matrix of processes with individual observations with limited information about the underlying process.

7 Future research directions

In this section, some unaddressed issues in the literature will be presented to provide new directions for interested researchers. In what follows, we list some of the interesting future research problems relevant to the aforementioned work discussed in this paper.

  • Misleading signals are common in simultaneous schemes for the process mean vector and the covariance matrix. For example when the process mean increases, the signal might be alarmed by the chart for the covariance or on the negative side of the chart for the mean. Ramos et al. [2013] and Ramos et al. [2016] investigated the probabilities of misleading signals (PMS) for a few control chart methods such as Hotelling- simultaneous scheme or simultaneous EWMA scheme chart. However, this has not been studied for other types of control charts. More research is needed on better understanding of PMS.

  • A common problem in quality control charts is the low power of detecting small changes. The problem is often addressed by using of CUSUM and EWMA control charts (see Woodall 1986 and Lucas and Saccucci 1990), but a more recent approach is using adaptive sample sizes. As an example, Haq (2018) proposed an adaptive EWMA (AEWMA) dispersion chart to achieve better performance over a range of dispersion shifts. More research is still needed, especially for those control charts that their adaptive versions have not been yet considered. Also, the sequential probability ratio test (SPRT) chart showed its optimality in producing a lower expected sampling size for the univariate case. With the exception of Reynolds and Cho [2011], there have very little work has been done on the effectiveness of the SPRT for multivariate charts.

  • The sparsity feature in monitoring covariance matrices has been considered by some authors by employing a penalized likelihood function. The performance of these methods highly depend on the tuning parameters, but there is not much being done on how to choose this parameter effectively. In addition, as discussed in Shen et al. [2014], the joint monitoring of mean vector and covariance matrix under sparsity has not received much attention so far.

  • Generally, monitoring procedures that are capable of distinguishing between mean shifts and covariance shifts are highly preferable. Obviously, using separate charts is a solution, but there still remain problems to be solved when constructing these separate charts. Interpreting an out-of-control signal which includes the identification of variable responsible for shift and also diagnosing whether mean or covariance matrix has shifted should receive greater attention.

  • Effects of parameter estimation, as the main task of Phase I, on control chart properties is an important research subject in SPC area (Woodall and Montgomery 1999). As stated by Jones-Farmer et al. [2014], only a few studies have investigated the effect of estimation on the performance of control charts for covariance matrix. Addressing questions such as ”how well might a chart perform if designed with estimates instead of exact parameters?” or ”what sample size is required in Phase I to ensure proper performance in Phase II?” is more challenging for covariance matrix rather than mean vector due to the large number of unknown components.

  • Using robust estimators instead of classical estimators can increase the performance of Phase I control charts under contaminated data. The literature is lacking on robust and easy-to-use multivariate control charts for Phase I monitoring, especially for the case of individual observations.

  • The ”curse of dimensionality” is increasingly encountered in industries such as biology, stock market analysis in finance, and wireless communication networks. Within the past 20 years, there has been a surge in developing statistical methods for high-dimensional data. See for example, Jiang et al. [2012], and Srivastava et al. [2014]. Using such techniques seems to be very promising for SPM. Furthermore, in high-dimensional applications, the OC conditions typically involve a small number of variables and consequently, combining a multivariate monitoring scheme with effective variable selection based methods would be yet another alternative for high-dimensional process monitoring. Interested readers are referred to Peres and Fogliatto [2018].

  • In many of the methodologies presented in this paper, it is assumed that the underlying process follows a multivariate normal distribution for the generated data. A little work has been done on the dispersion charts with multivariate observations departing from normality and there is an important gap in the literature that needs to be filled. To alleviate the effect of departure from multivariate normality, some distribution-free control schemes have been introduced in the recent for monitoring covariance matrix in multivariate processes. Nonparametric control charts enjoy greater robustness over parametric control schemes. However, the existing work for monitoring covariance matrix, to the best of authors knowledge, is mainly concentrated on observations from the bivariate process. There is a great need for nonparametric control charts for monitoring covariance matrices when two or more quality characteristics are present in the process. A similar gap exists for Phase I analysis, since developing nonparametric methods for Phase I is much more appropriate than Phase II monitoring (Woodall 2017).

  • Monitoring covariance matrices of multivariate processes when the data are auto-correlated is another topic which needs more attention. Chang and Zhang (2007) proposed a novel multivariate dynamic linear model (DLM) instead of the classical time series such as autoregressive integrated moving average (ARIMA) models to filter the autocorrelation in monitoring the covariance matrix of multivariate autocorrelated observations. More is needed to be done on this topic.

For convenience of access, a summary the methods for monitoring covariance matrix are provided in Table 1.

Problem Idea/chart References Process Type Sparsity feature Only individual observations
Synthetic control chart(combining and CRL charts) Ghute & Shirke [2008] Multivariate                  
Group runs chart Gadre [2014] Multivariate                    
Synthetic control charts Lee & Khoo [2015,2017a,b] Multivariate                    
VMAX statistic (based on the marginal sample variances) Costa & Machado [2008] Bivariate                    
VMAX statistic Costa & Machado [2009] Multivariate                   
A variant of VMAX (VMAX Group Runs) Gadre & Kakade [2018] Bivariate                    
A synthetic chart based on the VMAX Machado et al. [2009b] Bivariate                    
Covariance monitoring Use of 3 attribute control charts Machado et al. [2018] Bivariate                   
when (Section 3) An statistic based on the sample variances Costa & Neto [2017] 2 & 3-variate                    
VMIX statistic (based on stacked sample variance) Quinino et al. [2012] Bivariate                    
Artificial Neural Network Cheng & Cheng [2011] Bivariate                    
RMAX statistic for detecting increase in variances (based on the marginal sample ranges) Costa and Machado [2011a] Multivariate                 
LRT statistic of a one-sided test (increase in variances) Yen & Shiau [2010] Multivariate                 
LRT statistic of a one-sided test (decrease in variances) Yen et al. [2012] Multivariate                 
Penalized Likelihood Ratio (PLR) statistic Li et al. [2013] Multivariate        ✓          
Two exponentially weighted chart (MEWMS and MEWMV) Huwang et al. [2007] Multivariate                  ✓
An exponentially weighted chart (MEWMC) Hawkins & Maboudou-Tchao [2008] Multivariate                  ✓
An exponentially weighted chart (MEWMD) Huwang et al. [2017] Multivariate                    ✓
Modification of MEWMS and MEWMV Memar & Niaki [2009] Multivariate                    ✓
Modification of MEWMS and MEWMV Memar & Niaki [2011] Multivariate                    
Chi-square quantile-based monitoring statistic Hwang [2017] Multivariate                    ✓
Covariance monitoring Wilks’ statistic Mason et al. [2009] Multivariate                    
when (Section 4) LASSO-MEWMC (LMEWMC) chart Yeh et al. [2012] Multivariate        ✓            ✓
Modification to PLR chart (Li et al. 2013) Maboudou-Tchao & Agboto [2013] Multivariate        ✓            
MaxNorm charting statistic Shen et al. [2014] Multivariate        ✓            
Adaptive LASSO-Thresholding (ALT) Norm chart Abdella et al. [2019] Multivariate        ✓            
Ridge () type penalized likelihood ratio method Kim et al. [2019] Multivariate        ✓            
Integrating a two-sample test with EWMA procedure Li & Tsung [2019] Multivariate        ✓            
Based on the maximum of the usual EWMA charts Niaki & Memar [2009] Bivariate                    
Two charts based on marginal sample means and variances Machado et al. [2009a] Bivariate                    
Extensions of Machado et al. [2009a] chart to multivariate case Costa & Machado [2011b , 2013] Multivariate                    
Combination of two MEWMA-type control charts Reynolds & Cho [2006] Multivariate                    
Simultaneous monitoring Modifications to Reynolds & Cho [2006] Reynolds & Stoumbos [2008, 2010] Multivariate                    
of mean and covariance Modifications to Reynolds & Cho [2006] Reynolds & Kim [2007] Multivariate                    
(Section 5) Modifications to Reynolds & Cho [2006] Reynolds & Cho [2011] Multivariate                    
Combination of two exponentially weighted control charts (DWEMA and MEWMD) Zhang and Chang (2008) Multivariate                    
A single control chart based on integrating EWMA procedure with the generalized LRT Zhang et al. [2010] Multivariate                    
Two control charts based on the penalized likelihood estimate Wang et al. [2014] Multivariate        ✓            
Multivariate change-point analysis through generalized likelihood ratio statistics Zamba & Hawkins [2009] Multivariate                    
Table 1: A summary of methods for monitoring covariance matrix

8 Conclusion

In real applications, a process shift may occur in either location or scale. In this paper, we review the existing literature on monitoring covariance matrix of a multivariate process. The existing control charts are classified into four major groups where for each group main ideas and results along with their benefits or drawbacks are briefly discussed. Most of the authors have focuses on monitoring covariance matrices in situations where the number of rational subgroups is larger than the number of variables and a full rank estimate of the covariance matrix is available. However, there are many other situations where the literature on monitoring covariance matrices is limited. Some of these potential research areas on the topic are highlighted in the present work. For example, designing robust control charts as well as nonparametric control charts in the high-dimensional framework both for Phases I and II are interesting areas for further research in the domain of MSPM for covariance matrices.

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