Phase II monitoring of multivariate simple linear profiles with estimated parameters
 514 Downloads
Abstract
In some applications of statistical process monitoring, a quality characteristic can be characterized by linear regression relationships between several response variables and one explanatory variable, which is referred to as a “multivariate simple linear profile.” It is usually assumed that the process parameters are known in Phase II. However, in most applications, this assumption is violated; the parameters are unknown and should be estimated based on historical data sets in Phase I. This study aims to compare the effect of parameter estimation on the performance of three Phase II approaches for monitoring multivariate simple linear profiles, designated as MEWMA, MEWMA_3 and \({\text{MEWMA}}/\chi^{2}\). Three metrics are used to accomplish this objective: AARL, SDARL and CVARL. The superior method may be different in terms of the AARL and SDARL metrics. Using the CVARL metric helps practitioners make reliable decisions. The comparisons are carried out under both incontrol and outofcontrol conditions for all competing approaches. The corrected limits are also obtained by a Monte Carlo simulation in order to decrease the required number of Phase I samples for parameter estimation. The results reveal that parameter estimation strongly affects the incontrol and outofcontrol performance of monitoring approaches, and a large number of Phase I samples are needed to achieve a parameter estimation that is close to the known parameters. The simulation results show that the MEWMA and \({\text{MEWMA}}/\chi^{2}\) methods perform better than the MEWMA_3 method in terms of the CVARL metric. However, the superior approach is different in terms of AARL and SDARL.
Keywords
Profile monitoring Multivariate simple linear profiles Estimation effect Average run length Statistical process monitoring Phase II analysisIntroduction
In most statistical process monitoring (SPM) applications, it is assumed that the quality of a process or product is characterized by the statistical distribution of a single quality characteristic or a vector of several quality characteristics. However, in some cases, process quality could be characterized by a functional relationship between a response variable and one or more explanatory variables, which is referred to as a “profile.” Similar to other process monitoring approaches, profile monitoring is carried out by sampling. Several sets of data points are collected to represent their relationship with a curve (profile). The main objective of profile monitoring is to monitor the stability of this curve over time.
There are different types of profiles. If a linear regression model can represent the relationship between one response variable and one explanatory variable that is a simple linear profile. As an application of simple linear profiles, Kang and Albin (2000) addressed a calibration problem in semiconductor manufacturing and monitored the relationship between measured pressure (Y) and the amount of flow (X) using profiles. Monitoring and change point estimation of simple linear profiles have been considered by some researchers, including Stover and Brill (1998), Mestek et al. (1994), Mahmoud and Woodall (2004), Wang and Tsung (2005), Gupta et al. (2006), Mahmoud et al. (2007), Zou et al. (2007), Noorossana et al. (2008), Jensen et al. (2008), Narvand et al. (2013), Khedmati and Niaki (2015, 2016) and Kalaei et al. (2018).
There are other more complicated types of profiles such as multiple, polynomial and nonlinear profiles that have been studied by many researchers. Several approaches to monitor multiple linear profiles have been developed by some researchers, including Mahmoud (2008), Jensen et al. (2008), Zou et al. (2007), Parker and Finley (2007) and Amiri et al. (2012). In addition, Kazemzadeh et al. (2008, 2009) proposed several methods for monitoring polynomial profiles. Nonlinear profile monitoring through parametric and nonparametric methods and mixed linear profiles has been studied by some researchers, including Ding et al. (2006), Jeong et al. (2006), Williams et al. (2007), Moguerza et al. (2007) and Vaghefi et al. (2009).
In the case of multivariate simple linear (MSL) profiles, which are the main focus of the current paper, the term “simple” refers to the uniqueness of the explanatory variable, and the term “multivariate” refers to the multiplicity of the response variable. In such profiles, there are several response variables, each of which has a linear regression relationship with one explanatory variable. Some researchers, such as Noorossana et al. (2010), Zou et al. (2012), Ayoubi et al. (2014) and Adibi et al. (2014), have studied different aspects of multivariate profile monitoring.
Noorossana et al. (2010) were the first researchers who proposed three methods, designated as MEWMA_3, \({\text{MEWMA}}/\chi^{2}\) and MEWMA, for monitoring MSL profiles in Phase II. They evaluated the performance of these methods in detecting shifts in profile parameters under incontrol and outofcontrol conditions in terms of average run length (ARL). Their results showed that the MEWMA and \({\text{MEWMA}}/\chi^{2}\) methods outperform MEWMA_3. They also used these three methods in a real case study.
Based on the literature, profile monitoring approaches are divided into two phases: Phase I and Phase II. These phases are distinguished by the difference in their goals. In Phase I, a set of historical data points is available (m samples, each based on n observations). The main goals of Phase I are evaluating process stability, recognizing and eliminating assignable causes and estimating process parameters from incontrol samples. The objective of Phase I is to recognize assignable causes with high probability. The main interest in Phase II is quick detection of shifts based on the estimated parameters obtained in Phase I. Existing monitoring approaches in these two phases are dissimilar and use different evaluation metrics. In Phase I, the probability of a signal under outofcontrol conditions is used to describe the ability of a control chart to detect shifts in parameters. The ARL metric and its statistical properties are applied to evaluate control chart performance in Phase II. To find out more about profile monitoring approaches in Phases I and II, see the review paper by Woodall et al. (2004) and Woodall 2007) and the book edited by Noorossana et al. (2011).

Does profile parameter estimation really affect the performance of Phase II control chart schemes?

What is the proper metric for measuring the effect of parameter estimation?

Which method of profile monitoring is less affected by parameter estimation?
There are many studies in which the effect of parameter estimation on the performance of control charts is estimated for quality characteristics except profiles, including Burroughs et al. (1993), Chen (1997), Chakraborti (2000), Shishebori and Zeinal Hamadani (2009), Shishebori et al. (2015), Jones et al. (2001, 2004), Shu et al. (2004), Jones (2002), Zhang and Chen (2002), Zwetsloot and Woodall (2017), Khoo (2005), Castagliola et al. (2016), Saleh et al. (2015). For information on more research on this area, see the following review papers: Jensen et al. (2006) and Psarakis et al. (2014).
While a substantial number of studies have been done on evaluating the effect of parameter estimation on the performance of control charts for nonprofile characteristics, there are only a few studies on this topic for profile characteristics. Woodall and Montgomery (2014) discussed the field of statistical process control, stating that “There is also work needed on the effect of parameter estimation error on the Phase II performance of profile monitoring methods.”
Mahmoud (2012) was the first author to compare the incontrol and outofcontrol performance of three wellknown simple linear profile monitoring approaches that had first been proposed by Kang and Albin (2000), Kim et al. (2003) and Mahmoud et al. (2010), for use when parameters are estimated. In this study, the ARL and standard deviation of run length (SDRL) metrics are used to compare the performance of simple linear profile monitoring approaches. The results showed that using estimated profile parameters instead of known values in Phase II strongly affects the performance of all three methods under incontrol and outofcontrol conditions in terms of both the ARL and SDRL metrics. This author also used simulated corrected limits to investigate the outofcontrol performance of the monitoring approaches. Corrected limits are usually wider than control limits based on known parameters and reflect variability added to the process by parameters estimation. A smaller number of Phase I samples are needed for proper estimation when the corrected limits are applied. The simulation results showed that the method described by Mahmoud et al. (2010) has better outofcontrol run length performance than other competing methods.
Mahmoud (2012) used only the ARL and SDRL metrics to compare control chart performance based on estimated parameters. However, the standard deviation of average run length (SDARL) is also important in comparison with control chart performance. When different practitioners take samples in Phase I, they may estimate different values for process parameters. Consequently, they may obtain different values for incontrol ARL, adding a new source of variability to the process: practitionertopractitioner variability. In other words, when it is assumed that the process parameters are unknown in Phase II and should be estimated from the Phase I data set, the ARL is no longer a parameter, but becomes a random variable. The ARL curve has almost a rightskewed distribution (Jensen et al. 2006). Zhang et al. (2014) maintained that practitionertopractitioner variation is inevitable because different Phase I data sets are used. The SDARL is a very useful metric for measuring this variation. Researchers have suggested that the SDARL should be within 5–10% of the desired incontrol ARL value. It is obvious that when the process parameters are known, the SDARL is equal to zero. Aly et al. (2015), similar to Mahmoud (2012), compared the performance of three methods of simple linear profile monitoring [the method proposed by Kang and Albin (2000), Kim et al. (2003) and Mahmoud et al. (2010)] in terms of incontrol AARL and SDARL, but they did not investigate the outofcontrol performance of the mentioned control charts. The results of their study showed that when the parameters are estimated, the Kim et al. (2003) method generally shows better incontrol performance compared to the other competing methods in terms of the SDARL. They also illustrated that if a control chart shows a particular performance in terms of the ARL, it may not show the same performance in terms of the SDARL.
To the best of the authors’ knowledge, there is no study in the literature that evaluates the effect of parameter estimation on the performance of control charts for monitoring MSL profiles. This paper can be an useful source for quality control engineers in choosing the best control chart for monitoring MSL profiles. According to the simulation results, using parameter estimates with upper control limits designed based on the known parameters can result in a significant deterioration of the chart performance. Because these estimators add extra variability in the chart control limit(s). Therefore, applying a control chart scheme which is less affected by this effect would be very constructive and leads to a huge cost saving in terms of time or expenses for any manufacturing systems. Measuring the estimation effect can be carried out by computing some metrics. In this paper, for the first time, the \(AARL\), \(SDARL\) and \(CVARL\) are used to evaluate the incontrol and outofcontrol performance of control chart schemes. Note that, using different metrics such as the \(AARL\) or SDARL may lead to different results, which can lead to ambiguity in choosing the best method. In this situation, using the \(CVARL\), which considers both the \(AARL\) and \(SDARL\) metrics, can help achieve a reliable decision and choose the best method.
According to the literature, it is obvious that the large Phase I sample leads to more accurate estimates and, consequently, better Phase II performance. However, there might be situations in which collecting a large number of samples is not possible. Hence, using corrected control limits is suggested; in this procedure, a large number of m Phase I data are not required to achieve the desired incontrol \(ARL\). Using wider control chart limits to reflect the variability of parameters estimation is the main idea of using corrected limits. In previous studies such as Mahmoud (2012), corrected limits have been obtained in order to achieve an incontrol value of ARL = 200. However, in the current study, corrected limits are established in order to achieve an incontrol value of \(AARL = 200\) through simulation runs to consider practitionertopractitioner variability and reduce the number of Phase I samples for parameters estimation. Note that the outofcontrol performance of monitoring approaches is also evaluated based on simulated corrected limits.
The rest of this paper is organized as follows. “The multivariate simple linear regression model” section contains a brief explanation of the multivariate simple linear regression model. In “Phase II monitoring methods for multivariate simple linear profiles” section, three MSL profile monitoring approaches in Phase II are presented. “The proposed approach for measuring the effect of parameter estimation on the performance of Phase II control charts” section presents a description of the procedure for evaluating the effect of parameter estimation on the performance of monitoring approaches. Then, “Comparison of control chart performance under incontrol conditions” section presents the incontrol performance of competing approaches in terms of the \(AARL\), \(SDARL\) and \(CVARL\) metrics. “The proposed approach for establishing corrected limits” section presents the simulated corrected limits to achieve the desired incontrol \(AARL\). In “Comparison of control chart performance under outofcontrol conditions” section, the detection performance of the control chart schemes is compared under different types of shifts. Finally, the conclusions and suggestion for further research are provided in “Conclusion and suggestions for future research” section.
The multivariate simple linear regression model
Phase II monitoring methods for multivariate simple linear profiles
Noorossana et al. (2010) proposed three control chart approaches for monitoring MSL profiles in Phase II. Then, they evaluated the performance of the proposed methods in terms of the ARL metric under incontrol and outofcontrol conditions. These three approaches are briefly discussed below.
MEWMA control chart
In this method, the chart alarms when \(T_{{\varvec{z}_{k} }}^{2} > h_{\beta }\). The value of \(h_{\beta }\). is determined through the simulation to achieve a specified incontrol \(ARL\).
\({\mathbf{MEWMA}}/{\varvec{\upchi}}^{2}\) control chart
This method is an extension of the second approach proposed by Kang and Albin (2000). The vector of the average error for the kth sample is denoted by \(\bar{\varvec{e}}_{k} = \left( {\bar{e}_{1k} ,\bar{e}_{2k} , \ldots ,\bar{e}_{pk} } \right)^{\text{T}}\), in which \(\bar{e}_{jk} = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {e_{ijk} }\). It can be shown that under incontrol conditions, \(\bar{\varvec{e}}_{k}\) follows a pvariate normal distribution with a mean vector of zero and a covariance matrix \({\varvec{\Sigma}}_{{\bar{\varvec{e}}}} = n^{  1} {\varvec{\Sigma}}\).
The chart signals as soon as \(T_{{\varvec{z}_{k,e} }}^{2} > h_{e}\). The value of \(h_{e}\) is determined through the simulation to achieve a specified incontrol \(ARL\).
Noorossana et al. (2010) developed the method proposed by Noorossana et al. (2004) for monitoring process variability and proposed a new Chisquare statistic, \(\chi_{ik}^{2} = \varvec{e}_{ik} {\varvec{\Sigma}}^{  1} \varvec{e}_{ik}^{\text{T}} ,\) which follows a Chisquare distribution with p degrees of freedom; consequently, \(\chi_{k}^{2} = \sum\nolimits_{i = 1}^{n} {\chi_{ik}^{2} }\) follows a Chisquare distribution with \(np\) degrees of freedom. Hence, \(\chi_{np,\alpha }^{2}\) which denotes the \(1  \alpha\) percentile of a Chisquare distribution with \(np\) degrees of freedom can be used as an upper control limit for this statistic.
MEWMA_3 control chart
The chart signals when \(T_{Ik}^{2} > h_{I}\) where \(h_{I}\) is determined to achieve a specified incontrol \(ARL\).
The chart signals when \(T_{Sk}^{2} > h_{S}\) and \(h_{S}\) is determined such that a specified incontrol \(ARL\) is obtained.
The proposed approach for measuring the effect of parameter estimation on the performance of Phase II control charts
Based on the literature, it is usually assumed that the parameters of a profile (matrix B) and the covariance matrix of the response variables (\({\varvec{\Sigma}}\)) are known in Phase II. In the current study, it is assumed that matrix B is unknown and should be estimated through incontrol Phase I samples. Now the question is whether the B estimation affects the performance of the MSL profile monitoring approaches (\({\text{MEWMA}}\), \({\text{MEWMA}}\_3\) and \({\text{MEWMA}}/\chi^{2}\) methods) that have been developed based on known parameters, and which method is more robust to the effects of parameter estimation.
To answer these questions, a Monte Carlo simulation algorithm is proposed to evaluate the effect of parameter estimation on the performance of control chart schemes.
 1.Assume that B and \({\varvec{\Sigma}}\) are known. Using 10,000 simulation runs under incontrol conditions, calculate the upper control limits for each of the three methods to achieve \(ARL_{0} = 200\) when the parameters are known. The upper control limits for different values of \(\theta\) are shown in Table 1. Note that Noorossana et al. (2010) obtained the upper control limits for each of the three methods just for \(\theta = 0.2\). In the current study, the control limits were obtained for \(\theta = 0.05, 0.1\) to evaluate the effect of \(\theta\) on the control limit values. Table 1 shows that the control limit values increase when \(\theta\) increases. It should be noted that since the \(\chi^{2}\) statistic is not dependent on \(\theta\), this parameter does not affect the upper limit of the \(\chi^{2}\) control chart.Table 1
Incontrol upper control limits of three control charts approaches
\(\theta\)
Control chart
\({\text{MEWMA}}\)
\({\text{MEWMA}}/\chi^{2}\)
MEWMA_3
0.05
11.2
\({\text{MEWMA}}\)
9.07
\({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)
10.68
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)
10.68
\(\chi^{2}\)
23.77
\({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)
2.208
0.1
12.7
\({\text{MEWMA}}\)
10.25
\({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)
11.82
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)
11.82
\(\chi^{2}\)
23.77
\({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)
2.298
0.2
13.9
\({\text{MEWMA}}\)
11.1
\({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)
12.55
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)
12.55
\(\chi^{2}\)
23.77
\({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)
2.43
 2.
Generate m incontrol MSL profiles based on known parameters using a multivariate normal distribution with a mean vector of zero and a covariance matrix \({\varvec{\Sigma}}\).
 3.
Calculate \({\hat{\mathbf{B}}}_{k}\) where \(k = 1,2, \ldots ,m\) for each of the profiles generated in step 2, then calculate \({\mathbf{\hat{\bar{B}}}} = \frac{{\mathop \sum \nolimits_{k = 1}^{m} {\hat{\mathbf{B}}}_{k} }}{m}\).
 4.
Generate an incontrol profile based on known parameters and estimate the matrix of profile parameters (B) using Eq. 5, and calculate the chart statistic. Note that \({\mathbf{\hat{\bar{B}}}}\) obtained in step 3 should be considered as the estimation of profile parameters, instead of \({\mathbf{B}}\) which contains known parameters. Then, set \(RL = 1\).
 5.
Compare the calculated chart statistic with the corresponding upper control limit (UCL) (Table 1). If the chart statistic is larger than the \(UCL\), go to step 6; otherwise, set \(RL = RL + 1\) and go to step 4.
 6.
Record the \(RL\) value and go to step 4.
 7.
Repeat steps 4–6, 10,000 times and calculate the \(ARL\) by averaging the \(RL\) values. Then, go to step 2.
 8.
Repeat steps 2–7, 10,000 times to achieve 10,000 different values of the \(ARL\). Then, calculate the \(AARL = Mean\left( {ARL} \right)\), \(SDARL = STD\left( {ARL} \right)\) and \(CVARL\) using Eq. 1.
Comparison of control chart performance under incontrol conditions
InControl \(AARL\), \(SDARL\) and \(CVARL\) comparisons of \({\text{MEWMA}}\), \({\text{MEWMA}}/\chi^{2}\) and \({\text{MEWMA}}\_3\) methods when m samples of Phase I are used to estimate the unknown parameters
m  \(AARL\)  \(SDARL\)  \(CVARL \left( \% \right)\)  

\({\text{MEWMA}}\)  \({\text{MEWMA}}/\chi^{2}\)  \({\text{MEWMA}}\_3\)  \({\text{MEWMA}}\)  \({\text{MEWMA}}/\chi^{2}\)  \({\text{MEWMA}}\_3\)  \({\text{MEWMA}}\)  \({\text{MEWMA}}/\chi^{2}\)  \({\text{MEWMA}}\_3\)  
\(\theta\) = 0.2  
10  43.8  85.08  57.9  42.64  47.86  39.85  97.35  56.24  68.82 
70  132.56  155.2  142.96  34.93  32.93  29.25  26.35  21.22  20.44 
200  166.53  180.57  179.35  20.69  18.24  15.31  12.42  10.1  8.5 
500  185.53  194.99  191.81  12.43  8.6  9.3  6.70  4.42  4.8 
1000  194.79  197.95  198.05  7.32  6.04  7.78  3.76  3.05  3.9 
2000  200.07  201.19  199.68  6.2  5.9  6.63  3.10  2.9  3.32 
\(\theta\)= 0.1  
10  35.72  70.27  44.83  37.654  46.33  38.65  105.41  65.93  86.21 
70  105.36  139.58  128.54  34.54  40.22  36.9  32.78  28.82  28.71 
200  149.22  174.57  179.13  25.75  25.33  19.48  17.26  14.51  10.87 
500  175.19  195.51  195.82  16.22  11.5  13.04  9.26  5.88  6.66 
1000  186.89  197.63  206  9.48  10.24  8.15  5.07  5.18  3.96 
2000  191.36  202.06  201.33  7.46  6.65  7.75  3.90  3.29  3.85 
\(\theta\)= 0.05  
10  32.35  54.84  39.83  36.47  48.64  39.11  112.74  88.69  98.19 
70  90.72  131.03  116.51  32.32  42.26  35.89  35.63  32.25  30.80 
200  143.39  168  161.81  30.14  27.42  26.28  21.02  16.32  16.24 
500  169.68  188.31  186.77  15.76  16.04  15.98  9.29  8.52  8.56 
1000  181.53  198.32  200.45  12.33  10.51  10.24  6.79  5.30  5.11 
2000  193.68  203.82  205.63  6.6  8.05  8.16  3.41  3.95  3.97 
Table 1 shows that the \(AARL\) value increases when the value of the smoothing parameter (\(\theta\)) increases. This could be very useful in practical environments. In some industries, such as military or hightech industries, a high number of false alarms lead to high expenses; therefore, choosing the method that has a larger incontrol \(AARL\) is suggested in order to reduce the rate of false alarms.
The proposed approach for establishing corrected limits
As observed in “Comparison of control chart performance under incontrol conditions” section, in the \({\text{MEWMA}}/\chi^{2}\) method, we need at least 200 incontrol Phase I samples for parameters estimation so that the \(AARL\) value is at least 90% of the specified incontrol \(ARL_{0} = 200\). Therefore, the practitioner should wait too long to collect samples. Hence, in these situations, using corrected control limits is strongly suggested. By using corrected limits which are usually wider than original ones due to the extra variability added to the process by estimators, the problem of large Phase I samples requirement for estimation can be solved. Corrected limits can also be established based on the small size of historical data set. However, it should be noted that very wide corrected limits due to very small Phase I samples can deteriorate the detection performance of Phase II control charts. To learn more about corrected limits please see Quesenberry (1993), Jones (2002), Champ et al. (2005) and Mahmoud and Maravelakis (2010).
Corrected limits for three control charts in order to achieve \(AARL = 200\) for different values of m and \(\theta\)
m  \({\text{MEWMA}}\)  \({\text{MEWMA}}/\chi^{2}\)  MEWMA_3  

\(\theta\) = 0.2  
30  16.9  \({\text{MEWMA}}\)  12.76  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  14.13 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  14.1  
\(\chi^{2}\)  27.62  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  4.12  
50  15.5  \({\text{MEWMA}}\)  12.12  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  13.71 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  13.64  
\(\chi^{2}\)  26.86M  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  3.82  
100  14.9  \({\text{MEWMA}}\)  12.21  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  13.33 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  13.2  
\(\chi^{2}\)  24.91  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  3.22  
300  14.2  \({\text{MEWMA}}\)  12.24  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  13.19 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  12.94  
\(\chi^{2}\)  24.35  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  2.9  
600  14.05  \({\text{MEWMA}}\)  11.73  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  12.91 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  12.83  
\(\chi^{2}\)  24.1  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  2.68  
\(\theta\) = 0.1  
30  16.52  \({\text{MEWMA}}\)  12.6  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  13.45 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  13.81  
\(\chi^{2}\)  27.62  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  3.515  
50  15.3  \({\text{MEWMA}}\)  11.85  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  13.18 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  13.11  
\(\chi^{2}\)  26.86  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  3.322  
100  14.1  \({\text{MEWMA}}\)  11.5  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  12.83 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  12.75  
\(\chi^{2}\)  24.91  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  3.032  
300  13.4  \({\text{MEWMA}}\)  10.75  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  12.32 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  12.44  
\(\chi^{2}\)  24.35  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  2.882  
600  13.05  \({\text{MEWMA}}\)  10.55  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  12.18 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  12.11  
\(\chi^{2}\)  24.1  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  2.544  
\(\theta\) = 0.05  
30  16.25  \({\text{MEWMA}}\)  11.44  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  13.42 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  13.88  
\(\chi^{2}\)  27.62  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  3.201  
50  15.1  \({\text{MEWMA}}\)  10.93  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  13.19 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  13.46  
\(\chi^{2}\)  26.86  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  2.881  
100  13.4  \({\text{MEWMA}}\)  10.54  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  12.87 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  12.65  
\(\chi^{2}\)  24.91  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  2.621  
300  12.1  \({\text{MEWMA}}\)  9.92  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  11.83 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  11.92  
\(\chi^{2}\)  24.35  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  2.411  
600  11.7  \({\text{MEWMA}}\)  9.47  \({\text{MEWMA}}_{{\left( {\text{I}} \right)}}\)  11.48 
\({\text{MEWMA}}_{{\left( {\text{S}} \right)}}\)  11.34  
\(\chi^{2}\)  24.1  \({\text{MEWMA}}_{{\left( {\text{E}} \right)}}\)  2.369 
Corrected limits could be used in evaluating the outofcontrol performance of control charts. Assume that when parameters are estimated, a shift has occurred in one of the profile parameters and the control chart has alarmed. In this situation, we cannot determine whether this alarm is due to the effect of parameter estimation or the effect of the shift in the incontrol parameter. Hence, using corrected limits ensures that chart alarming occurs only because of the shift in profile parameters. Consequently, evaluating the performance of the control chart schemes under outofcontrol conditions was carried out using corrected limits; the results are shown in Table 3.
Comparison of control chart performance under outofcontrol conditions
Several types of shifts are considered to evaluate the outofcontrol performance of control chart schemes using parameter estimation, including: (1) a shift in the intercept of the first profile (\(\beta_{01}\)); (2) a shift in the slope of the first profile (\(\beta_{11}\)). To apply these shifts, we should generate an outofcontrol profile based on the shifted parameters in step 4 of the simulation algorithm presented in “The proposed approach for measuring the effect of parameter estimation on the performance of Phase II control charts” section.
Note that the applied shifts in this paper are the same as the shifts applied in a study by Noorossana et al. (2010), which was based on known parameters, in order to compare the results of our study (when parameters are estimated) with the results of Noorossana et al. (2010) (when parameters are known). In addition, the corrected limits (Table 3) are used to compare the outofcontrol performance of the control chart schemes.
\(AARL\), \(SDARL\) and \(CVARL\) comparison when \(\beta_{01}\) has shifted to \(\beta_{01} + \tau_{0} \sigma_{1}\)
m  Method  Metric  \(\tau_{0}\)  

0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8  2  
30  \({\text{MEWMA}}\)  \(AARL\)  18.43  5.31  3.26  2.54  1.98  1.91  1.54  1.20  1.02  1.00 
\(SDARL\)  7.43  0.49  0.28  0.09  0.07  0.04  0.05  0.03  0.01  0.00  
\(CVARL\)  40.31  9.23  8.59  3.54  3.54  2.09  3.25  2.50  0.98  0.00  
\({\text{MEWMA}}\_3\)  \(AARL\)  21.36  6.11  3.40  2.57  2.03  1.93  1.54  1.23  1.02  1.08  
\(SDARL\)  5.42  0.63  0.28  0.19  0.13  0.07  0.05  0.03  0.01  0.01  
\(CVARL\)  25.37  10.31  8.24  7.39  6.40  3.63  3.25  2.44  0.98  0.93  
\({\text{MEWMA}}/\chi^{2}\)  \(AARL\)  17.64  5.12  2.93  1.98  1.55  1.42  1.21  1.07  1.01  1.00  
\(SDARL\)  9.24  0.62  0.33  0.11  0.09  0.05  0.05  0.03  0.01  0.00  
\(CVARL\)  52.38  12.11  11.26  5.56  5.81  3.52  4.13  2.80  0.99  0.00  
50  \({\text{MEWMA}}\)  \(AARL\)  17.53  5.26  3.21  2.43  1.97  1.91  1.52  1.18  1.08  1.00 
\(SDARL\)  4.70  0.48  0.18  0.07  0.07  0.04  0.04  0.03  0.01  0.00  
\(CVARL\)  26.81  9.13  5.61  2.88  3.55  2.09  2.63  2.54  0.93  0.00  
\({\text{MEWMA}}\_3\)  \(AARL\)  21.73  5.41  3.36  2.52  2.16  1.86  1.50  1.20  1.12  1.06  
\(SDARL\)  5.41  0.53  0.26  0.11  0.08  0.08  0.06  0.05  0.03  0.01  
\(CVARL\)  24.90  9.80  7.74  4.37  3.70  4.30  4.00  4.17  2.68  0.94  
\({\text{MEWMA}}/\chi^{2}\)  \(AARL\)  16.11  5.03  2.87  1.98  1.54  1.41  1.18  1.07  1.01  1.00  
\(SDARL\)  5.34  0.51  0.20  0.08  0.06  0.05  0.05  0.03  0.01  0.00  
\(CVARL\)  33.15  10.14  6.97  4.04  3.90  3.55  4.24  2.80  0.99  0.00  
100  \({\text{MEWMA}}\)  \(AARL\)  17.41  5.25  3.19  2.41  1.94  1.87  1.49  1.16  1.08  1.00 
\(SDARL\)  2.73  0.27  0.13  0.04  0.04  0.03  0.03  0.02  0.01  0.00  
\(CVARL\)  15.68  5.14  4.08  1.66  2.06  1.60  2.01  1.72  0.93  0.00  
\({\text{MEWMA}}\_3\)  \(AARL\)  21.44  5.34  3.34  2.51  2.16  1.86  1.51  1.18  1.10  1.03  
\(SDARL\)  2.84  0.34  0.19  0.14  0.07  0.06  0.04  0.04  0.03  0.01  
\(CVARL\)  13.25  6.37  5.69  5.58  3.24  3.23  2.65  3.39  2.73  0.97  
\({\text{MEWMA}}/\chi^{2}\)  \(AARL\)  15.34  4.88  2.43  1.94  1.48  1.36  1.15  1.05  1.00  1.00  
\(SDARL\)  2.76  0.36  0.16  0.08  0.05  0.05  0.04  0.02  0.00  0.00  
\(CVARL\)  17.99  7.38  6.58  4.12  3.38  3.68  3.48  1.90  0.00  0.00  
300  \({\text{MEWMA}}\)  \(AARL\)  17.46  5.23  3.17  2.37  1.97  1.83  1.43  1.16  1.06  1.00 
\(SDARL\)  2.59  0.25  0.11  0.03  0.02  0.02  0.01  0.02  0.01  0.00  
\(CVARL\)  14.85  4.70  3.47  1.47  0.83  1.09  0.70  1.72  0.94  0.00  
\({\text{MEWMA}}\_3\)  \(AARL\)  19.76  5.34  3.31  2.46  2.14  1.86  1.47  1.16  1.08  1.00  
\(SDARL\)  1.83  0.34  0.18  0.10  0.07  0.05  0.05  0.04  0.02  0.00  
\(CVARL\)  9.26  6.37  5.44  4.07  3.27  2.69  3.40  3.45  1.85  0.00  
\({\text{MEWMA}}/\chi^{2}\)  \(AARL\)  15.37  4.86  2.39  1.92  1.43  1.31  1.15  1.03  1.00  1.00  
\(SDARL\)  2.63  0.27  0.15  0.08  0.04  0.04  0.03  0.00  0.00  0.00  
\(CVARL\)  17.11  5.56  6.28  4.17  2.80  3.05  2.61  0.00  0.00  0.00  
600  \({\text{MEWMA}}\)  \(AARL\)  17.24  5.14  3.23  2.37  1.99  1.73  1.40  1.12  1.02  1.00 
\(SDARL\)  1.31  0.19  0.04  0.02  0.02  0.02  0.02  0.02  0.01  0.00  
\(CVARL\)  7.60  3.70  1.13  0.80  0.81  1.21  1.82  1.65  0.69  0.17  
\({\text{MEWMA}}\_3\)  \(AARL\)  19.58  5.16  3.25  2.41  2.03  1.76  1.41  1.16  1.00  1.00  
\(SDARL\)  1.81  0.26  0.16  0.05  0.05  0.04  0.02  0.02  0.00  0.00  
\(CVARL\)  9.24  5.04  4.92  2.07  2.46  2.27  1.42  1.72  0.00  0.00  
\({\text{MEWMA}}/\chi^{2}\)  \(AARL\)  15.34  4.75  2.35  1.92  1.39  1.24  1.15  1.00  1.00  1.00  
\(SDARL\)  1.36  0.22  0.06  0.03  0.03  0.03  0.02  0.00  0.00  0.00  
\(CVARL\)  8.87  4.62  2.55  1.56  2.16  2.42  1.74  0.00  0.00  0.00  
\(\infty\) (Noorossana et al. 2010)  \({\text{MEWMA}}\)  \(ARL\)  14.80  4.90  3.00  2.20  1.90  1.60  1.30  1.00  1.00  1.00 
\({\text{MEWMA}}\_3\)  \(ARL\)  16.10  5.10  3.00  2.20  1.90  1.60  1.30  1.10  1.00  1.00  
\({\text{MEWMA}}/\chi^{2}\)  \(ARL\)  13.70  4.50  2.60  1.80  1.30  1.10  1.00  1.00  1.00  1.00 
\({\text{MEWMA}}/\chi^{2}\) method is better than the \({\text{MEWMA}}\) method for small shifts; for the large shifts, both methods perform similarly. On the other hand, the \({\text{MEWMA}}\) method has a smaller \(SDARL\) value than the other competing methods. However, the \(SDARL\) performance of \({\text{MEWMA}}\) and \({\text{MEWMA}}/\chi^{2}\) methods are almost similar.
Finally, considering the \(CVARL\) metric, showed better performance of the \({\text{MEWMA}}\) method compared with the other competing methods. However, Noorossana et al. (2010) found that the \({\text{MEWMA}}/\chi^{2}\) method performs better compared to the other competing methods in terms of outofcontrol \(ARL\) when the parameters are known.
\(AARL\), \(SDARL\) and \(CVARL\) comparison when \(\beta_{11}\) has shifted to \(\beta_{11} + \tau_{1} \sigma_{1}\)
m  Method  Metric  \(\tau_{1}\)  

0.025  0.05  0.075  0.1  0.125  0.15  0.175  0.2  0.225  0.25  
30  \({\text{MEWMA}}\)  \(AARL\)  42.08  9.64  5.22  3.60  2.79  2.45  2.11  1.94  1.72  1.57 
\(SDARL\)  20.14  1.49  0.57  0.24  0.11  0.09  0.08  0.06  0.04  0.01  
\(CVARL\)  47.86  15.46  10.92  6.67  3.94  3.67  3.79  3.25  2.33  0.64  
\({\text{MEWMA}}\_3\)  \(AARL\)  69.12  17.43  7.76  6.17  4.67  2.93  2.31  2.08  1.93  1.74  
\(SDARL\)  23.87  1.73  0.84  0.45  0.21  0.10  0.09  0.07  0.04  0.02  
\(CVARL\)  34.53  9.93  10.82  7.29  4.50  3.41  3.90  3.37  2.07  1.15  
\({\text{MEWMA}}/\chi^{2}\)  \(AARL\)  49.23  9.86  5.86  3.84  2.71  2.18  1.75  1.68  1.59  1.42  
\(SDARL\)  20.29  2.82  0.66  0.27  0.14  0.12  0.06  0.05  0.03  0.01  
\(CVARL\)  41.21  28.60  11.26  7.03  5.17  5.50  3.43  2.98  1.89  0.70  
50  \({\text{MEWMA}}\)  \(AARL\)  41.98  9.62  5.21  3.59  2.76  2.42  2.09  1.92  1.72  1.56 
\(SDARL\)  20.27  1.32  0.38  0.18  0.08  0.08  0.06  0.05  0.02  0.01  
\(CVARL\)  48.28  13.72  7.29  5.01  2.90  3.31  2.87  2.60  1.16  0.64  
\({\text{MEWMA}}\_3\)  \(AARL\)  67.15  17.36  7.64  5.91  4.41  2.87  2.29  1.97  1.89  1.73  
\(SDARL\)  21.68  1.49  0.52  0.23  0.15  0.09  0.08  0.07  0.03  0.02  
\(CVARL\)  32.29  8.58  6.81  3.89  3.40  3.14  3.49  3.55  1.59  1.33  
\({\text{MEWMA}}/\chi^{2}\)  \(AARL\)  48.25  9.76  5.61  3.79  2.66  2.16  1.72  1.63  1.59  1.38  
\(SDARL\)  19.56  1.38  0.42  0.21  0.09  0.08  0.07  0.05  0.02  0.01  
\(CVARL\)  40.54  14.14  7.49  5.54  3.38  3.70  4.07  3.07  1.26  0.72  
100  \({\text{MEWMA}}\)  \(AARL\)  39.36  9.57  5.18  3.54  2.75  2.37  2.08  1.90  1.72  1.52 
\(SDARL\)  14.07  1.01  0.37  0.16  0.08  0.05  0.03  0.02  0.01  0.01  
\(CVARL\)  35.75  10.55  7.14  4.52  2.91  2.11  1.44  1.05  0.58  0.66  
\({\text{MEWMA}}\_3\)  \(AARL\)  59.45  17.11  7.62  5.82  4.38  2.86  2.27  1.97  1.88  1.61  
\(SDARL\)  20.52  1.42  0.48  0.19  0.12  0.08  0.07  0.04  0.03  0.02  
\(CVARL\)  34.51  8.30  6.30  3.26  2.74  2.80  3.08  2.03  1.60  1.24  
\({\text{MEWMA}}/\chi^{2}\)  \(AARL\)  46.21  9.65  5.53  3.78  2.64  2.13  1.67  1.45  1.58  1.36  
\(SDARL\)  16.54  1.31  0.42  0.18  0.08  0.06  0.05  0.05  0.03  0.01  
\(CVARL\)  35.79  13.58  7.59  4.76  3.03  2.82  2.99  3.45  1.90  0.74  
300  \({\text{MEWMA}}\)  \(AARL\)  39.20  9.31  5.11  3.53  2.74  2.33  2.05  1.89  1.71  1.52 
\(SDARL\)  4.81  0.70  0.24  0.12  0.08  0.04  0.03  0.02  0.01  0.01  
\(CVARL\)  12.27  7.52  4.70  3.40  2.92  1.72  1.46  1.06  0.58  0.66  
\({\text{MEWMA}}\_3\)  \(AARL\)  54.12  15.24  7.65  5.80  4.36  2.84  2.23  1.97  1.88  1.59  
\(SDARL\)  5.73  1.08  0.35  0.15  0.09  0.09  0.08  0.05  0.02  0.01  
\(CVARL\)  10.59  7.09  4.58  2.59  2.06  3.17  3.59  2.53  1.06  0.63  
\({\text{MEWMA}}/\chi^{2}\)  \(AARL\)  43.01  9.62  5.44  3.72  2.64  2.12  1.59  1.43  1.57  1.35  
\(SDARL\)  4.62  0.84  0.26  0.18  0.09  0.08  0.06  0.06  0.04  0.01  
\(CVARL\)  10.74  8.73  4.78  4.84  3.41  3.77  3.77  4.20  2.55  0.74  
600  \({\text{MEWMA}}\)  \(AARL\)  37.34  9.22  5.00  3.51  2.70  2.28  2.03  1.88  1.69  1.46 
\(SDARL\)  3.24  0.53  0.15  0.08  0.04  0.03  0.02  0.02  0.01  0.01  
\(CVARL\)  8.68  5.75  3.00  2.28  1.48  1.18  0.99  1.06  0.59  0.68  
\({\text{MEWMA}}\_3\)  \(AARL\)  48.23  13.14  6.85  4.11  3.22  2.60  2.16  1.92  1.86  1.49  
\(SDARL\)  4.12  1.03  0.35  0.12  0.09  0.08  0.08  0.04  0.02  0.01  
\(CVARL\)  8.54  7.84  5.04  2.92  2.80  3.08  3.70  2.08  1.08  0.67  
\({\text{MEWMA}}/\chi^{2}\)  \(AARL\)  39.12  9.46  5.31  3.53  2.56  1.86  1.45  1.26  1.16  1.23  
\(SDARL\)  3.89  0.73  0.21  0.16  0.09  0.07  0.06  0.03  0.01  0.01  
\(CVARL\)  9.94  7.72  3.95  4.53  3.52  3.76  4.14  2.38  0.86  0.81  
\(\infty\) (Noorossana et al. 2010)  \({\text{MEWMA}}\)  \(ARL\)  30.20  8.60  4.80  3.30  2.60  2.20  2.00  1.80  1.60  1.40 
\({\text{MEWMA}}\_3\)  \(ARL\)  40.70  10.20  5.30  3.70  2.80  2.30  2.10  1.90  1.70  1.40  
\({\text{MEWMA}}/\chi^{2}\)  \(ARL\)  34.40  9.10  4.90  3.30  2.40  1.80  1.40  1.20  1.10  1.00 
Based on the results obtained, it is possible to conclude that control chart performance under outofcontrol conditions is strongly affected by the parameter estimation, similar to incontrol conditions. When estimated parameters are used for calculating the control chart statistic, a large number of Phase I samples are needed in order to achieve the same incontrol performance as when the parameters are known. On the other hand, taking more Phase I samples for the parameter estimation leads to better detection performance.
Therefore, we can say with confidence not only that the incontrol and outofcontrol performance of control chart schemes for monitoring MSL profiles is seriously affected by parameter estimation, but also that ignoring the variability added to the process by parameter estimation can affect choosing the superior monitoring scheme.
This could have many economic and competitive advantages in manufacturing and nonmanufacturing industries because of decreasing the rate of false alarms.
Conclusion and suggestions for future research
The current study was an investigation of the effect of parameter estimation on three approaches to MSL profile monitoring. The approaches are: (1) a \({\text{MEWMA}}\) control chart for monitoring profile parameters; (2) a combination of a \({\text{MEWMA}}\) control chart based on the vector of residual means and a \(\chi^{2}\) chart for monitoring process variability (\({\text{MEWMA}}/\chi^{2}\)); and (3) a combination of three separate \({\text{MEWMA}}\) control charts (\({\text{MEWMA}}\_3\)) for monitoring the vector of intercepts, the vector of slopes and the process variability. Since the \(ARL\) is no longer a parameter when it is assumed that the process parameters are estimated, we used three metrics, the \(AARL\), \(SDARL\) and \(CVARL\), which are based on the statistical properties of the \(ARL\) distribution. Our goal is to compare the methods and choose the best one, which is more robust to the effect of parameter estimation. The simulation results showed that the \({\text{MEWMA}}/\chi^{2}\) method performs better in terms of the incontrol \(AARL\) metric and needs fewer Phase I samples to achieve 90% of the desired incontrol \(ARL = 200\). Although it is confidently inferred that the \({\text{MEWMA}}\_3\) method performs better than the other competing methods in terms of the \(SDARL\) metric when \(\theta = 0.2\), the results are not the same for other values of \(\theta\). We recommended using the \(CVARL\) metric as the basis of the comparison in order to simultaneously consider the AARL and \(SDARL\). The results showed that the \({\text{MEWMA}}\_3\)method performs uniformly better than other approaches in terms of the \(CVARL\), although the \({\text{MEWMA}}/\chi^{2}\) method performs better than the \({\text{MEWMA}}\_3\) method for small values of m. The performance of the \({\text{MEWMA}}\_3\) method improves by increasing m. After obtaining simulated corrected limits in order to reflect the variability added to the process by parameter estimation, we investigated the outofcontrol performance of the control chart schemes in terms of all the metrics, considering different outofcontrol scenarios. The simulation results showed that the \({\text{MEWMA}}\) and \({\text{MEWMA}}/\chi^{2}\) charts have similar outofcontrol performance in most cases. However, it can be declared that the \({\text{MEWMA}}\) method generally performs better than other competing methods in terms of the \(AARL\), \(SDARL\) and \(CVARL\) for all outofcontrol scenarios.
Future research in this area could involve investigation of the parameter estimation effect on the incontrol and outofcontrol performance of different monitoring approaches for other types of profiles, such as nonlinear or polynomial profiles. Also, new metrics could be used in order to better measure the effect of parameter estimation, which is very useful in deciding on superior methods.
Notes
Acknowledgments
The authors thank to the anonymous referee for detailed and thoughtful recommendations which led to significant improvement in the paper. This research is partially supported by a grant from Iran National Science Foundation (INSF).
References
 Adibi A, Montgomery DC, Borror CM (2014) A Pvalue approach for phase II monitoring of multivariate profiles. Int J Qual Eng Technol 4(2):133–143CrossRefGoogle Scholar
 Aly AA, Mahmoud MA, Woodall WH (2015) A comparison of the performance of phase II simple linear profile control charts when parameters are estimated. Commun Stat Simul Comput 44(6):1432–1440MathSciNetzbMATHCrossRefGoogle Scholar
 Aly AA, Mahmoud MA, Hamed R (2016) The performance of the multivariate adaptive exponentially weighted moving average control chart with estimated parameters. Qual Reliab Eng Int 32(3):957–967CrossRefGoogle Scholar
 Amiri A, Eyvazian M, Zou C, Noorossana R (2012) A parameters reduction method for monitoring multiple linear regression profiles. Int J Adv Manuf Technol 58(5–8):621–629CrossRefGoogle Scholar
 Ayoubi M, Kazemzadeh RB, Noorossana R (2014) Estimating multivariate linear profiles change point with a monotonic change in the mean of response variables. Int J Adv Manuf Technol 75(9–12):1537–1556CrossRefGoogle Scholar
 Burroughs TE, Rigdon SE, Champ CW (1993) An analysis of Shewhart charts with runs rules when no standards are given. In: Proceedings of the quality and productivity section of the American statistical association, pp 8–12Google Scholar
 Castagliola P, Maravelakis PE, Figueiredo FO (2016) The EWMA median chart with estimated parameters. IIE Trans 48(1):66–74CrossRefGoogle Scholar
 Chakraborti S (2000) Run length, average run length and false alarm rate of Shewhart Xbar chart: exact derivations by conditioning. Commun Stat Simul Comput 29(1):61–81MathSciNetzbMATHCrossRefGoogle Scholar
 Champ CW, JonesFarmer LA, Rigdon SE (2005) Properties of the T^{2} control chart when parameters are estimated. Technometrics 47(4):437–445MathSciNetCrossRefGoogle Scholar
 Chen G (1997) The mean and standard deviation of the run length distribution of X charts when control limits are estimated. Stat Sin 7(3):789–798zbMATHGoogle Scholar
 Crowder S, Hamilton M (1992) An EWMA for monitoring a process standard deviation. J Qual Technol 24(1):12–21CrossRefGoogle Scholar
 Ding Y, Zeng L, Zhou S (2006) Phase I analysis for monitoring nonlinear profiles in manufacturing processes. J Qual Technol 38(3):199–216CrossRefGoogle Scholar
 Gupta S, Montgomery DC, Woodall WH (2006) Performance evaluation of two methods for online monitoring of linear calibration profiles. Int J Prod Res 44(10):1927–1942CrossRefGoogle Scholar
 Jensen WA, JonesFarmer LA, Champ CW, Woodall WH (2006) Effects of parameter estimation on control chart properties: a literature review. J Qual Technol 38(4):349–364CrossRefGoogle Scholar
 Jensen WA, Birch JB, Woodall WH (2008) Monitoring correlation within linear profiles using mixed models. J Qual Technol 40(2):167–183CrossRefGoogle Scholar
 Jeong MK, Lu JC, Wang N (2006) Waveletbased SPC procedure for complicated functional data. Int J Prod Res 44(4):729–744zbMATHCrossRefGoogle Scholar
 Jones LA (2002) The statistical design of EWMA control charts with estimated parameters. J Qual Technol 34(3):277–288CrossRefGoogle Scholar
 Jones LA, Champ CW, Rigdon SE (2001) The performance of exponentially weighted moving average charts with estimated parameters. Technometrics 43(2):156–167MathSciNetCrossRefGoogle Scholar
 Jones LA, Champ CW, Rigdon SE (2004) The run length distribution of the CUSUM with estimated parameters. J Qual Technol 36(1):95–108CrossRefGoogle Scholar
 Kalaei M, Soleimani P, Niaki STA, Atashgar K (2018) PhaseI monitoring of standard deviations in multistage linear profiles. J Ind Eng Int 14(1):133–142CrossRefGoogle Scholar
 Kang L, Albin SL (2000) Online monitoring when the process yields a linear profile. J Qual Technol 32(4):418–426CrossRefGoogle Scholar
 Kazemzadeh RB, Noorossana R, Amiri A (2008) Phase I monitoring of polynomial profiles. Commun Stat Theory Methods 37(10):1671–1686MathSciNetzbMATHCrossRefGoogle Scholar
 Kazemzadeh RB, Noorossana R, Amiri A (2009) Monitoring polynomial profiles in quality control applications. Int J Adv Manuf Technol 42(7):703–712CrossRefGoogle Scholar
 Khedmati M, Niaki STA (2015) Identifying the time of a step change in AR (1) autocorrelated simple linear profiles. J Ind Eng Int 11(4):473–484CrossRefGoogle Scholar
 Khedmati M, Niaki STA (2016) Monitoring simple linear profiles in multistage processes by a MaxEWMA control chart. Comput Ind Eng 98:125–143CrossRefGoogle Scholar
 Khoo MBC (2005) A control chart based on sample median for the detection of a permanent shift in the process mean. Qual Eng 17(2):243–257CrossRefGoogle Scholar
 Kim K, Mahmoud MA, Woodall WH (2003) On the monitoring of linear profiles. J Qual Technol 35(3):317–328CrossRefGoogle Scholar
 Lowry CA, Woodall WH, Champ CW, Rigdon SE (1992) A multivariate exponentially weighted moving average control chart. Technometrics 34(1):46–53zbMATHCrossRefGoogle Scholar
 Mahmoud MA (2008) Phase I analysis of multiple linear regression profiles. Commun Stat Simul Comput 37(10):2106–2130MathSciNetzbMATHCrossRefGoogle Scholar
 Mahmoud MA (2012) The performance of phase II simple linear profile approaches when parameters are estimated. Commun Stat Simul Comput 41(10):1816–1833MathSciNetzbMATHCrossRefGoogle Scholar
 Mahmoud MA, Maravelakis PE (2010) The performance of the MEWMA control chart when parameters are estimated. Commun Stat Simul Comput 39(9):1803–1817MathSciNetzbMATHCrossRefGoogle Scholar
 Mahmoud MA, Woodall WH (2004) Phase I analysis of linear profiles with calibration applications. Technometrics 46(4):380–391MathSciNetCrossRefGoogle Scholar
 Mahmoud MA, Parker PA, Woodall WH, Hawkins DM (2007) A change point method for linear profile data. Qual Reliab Eng Int 23(2):247–268CrossRefGoogle Scholar
 Mahmoud MA, Morgan JP, Woodall WH (2010) The monitoring of simple linear regression profiles with two observations per sample. J Appl Stat 37(8):1249–1263MathSciNetCrossRefGoogle Scholar
 Mestek O, Pavlík J, Suchánek M (1994) Multivariate control charts: control charts for calibration curves. Fresenius’ J Anal Chem 350(6):344–351CrossRefGoogle Scholar
 Moguerza JM, Muñoz A, Psarakis S (2007) Monitoring nonlinear profiles using support vector machines. In: Iberoamerican congress on pattern recognitionGoogle Scholar
 Narvand A, Soleimani P, Raissi S (2013) Phase II monitoring of autocorrelated linear profiles using linear mixed model. J Ind Eng Int 9(1):1–12CrossRefGoogle Scholar
 Noorossana R, Amiri A, Vaghefi A, Roghanian E (2004) Monitoring quality characteristics using linear profile. In: Proceeding of the 3rd international industrial engineering conference, Tehran, IranGoogle Scholar
 Noorossana R, Amiri A, Soleimani P (2008) On the monitoring of autocorrelated linear profiles. Commun Stat Theory Methods 37(3):425–442MathSciNetzbMATHCrossRefGoogle Scholar
 Noorossana R, Eyvazian M, Vaghefi A (2010) Phase II monitoring of multivariate simple linear profiles. Comput Ind Eng 58(4):563–570CrossRefGoogle Scholar
 Noorossana R, Saghaei A, Amiri A (2011) Statistical analysis of profile monitoring, vol 865. Wiley, New YorkCrossRefGoogle Scholar
 Parker PA, Finley TD (2007) Advancements in aircraft model force and attitude instrumentation by integrating statistical methods. J Aircr 44(2):436–443CrossRefGoogle Scholar
 Psarakis S, Vyniou AK, Castagliola P (2014) Some recent developments on the effects of parameter estimation on control charts. Qual Reliab Eng Int 30(8):1113–1129CrossRefGoogle Scholar
 Quesenberry CP (1993) The effect of sample size on estimated limits for X control charts. J Qual Technol 25(4):237–247CrossRefGoogle Scholar
 Saleh NA, Mahmoud MA, JonesFarmer LA, Zwetsloot I, Woodall WH (2015) Another look at the EWMA control chart with estimated parameters. J Qual Technol 47(4):363CrossRefGoogle Scholar
 Shishebori D, Zeinal Hamadani A (2009) The effect of gauge measurement capability on MC p and its statistical properties. Int J Qual Reliab Manag 26(6):564–582CrossRefGoogle Scholar
 Shishebori D, Akhgari MJ, Noorossana R, Khaleghi GH (2015) An efficient integrated approach to reduce scraps of industrial manufacturing processes: a case study from gauge measurement tool production firm. Int J Adv Manuf Technol 76(5–8):831–855CrossRefGoogle Scholar
 Shu L, Tsung F, KwokLeung T (2004) Runlength performance of regression control charts with estimated parameters. J Qual Technol 36(3):280–292CrossRefGoogle Scholar
 Stover FS, Brill RV (1998) Statistical quality control applied to ion chromatography calibrations. J Chromatogr 804(1–2):37–43CrossRefGoogle Scholar
 Vaghefi A, Tajbakhsh SD, Noorossana R (2009) Phase II monitoring of nonlinear profiles. Commun Stat Theory Methods 38(11):1834–1851MathSciNetzbMATHCrossRefGoogle Scholar
 Wang K, Tsung F (2005) Using profile monitoring techniques for a datarich environment with huge sample size. Qual Reliab Eng Int 21(7):677–688CrossRefGoogle Scholar
 Williams JD, Woodall WH, Birch JB (2007) Statistical monitoring of nonlinear product and process quality profiles. Qual Reliab Eng Int 23(8):925–941CrossRefGoogle Scholar
 Woodall WH (2007) Current research on profile monitoring. Production 17(3):420–425MathSciNetCrossRefGoogle Scholar
 Woodall WH, Montgomery DC (2014) Some current directions in the theory and application of statistical process monitoring. J Qual Technol 46(1):78–94CrossRefGoogle Scholar
 Woodall WH, Spitzner DJ, Montgomery DC, Gupta S (2004) Using control charts to monitor process and product quality profiles. J Qual Technol 36(3):309–320CrossRefGoogle Scholar
 Zhang L, Chen G (2002) A note on EWMA charts for monitoring mean changes in normal processes. Commun Stat Theory Methods 31(4):649–661MathSciNetzbMATHCrossRefGoogle Scholar
 Zhang M, Megahed FM, Woodall WH (2014) Exponential CUSUM charts with estimated control limits. Qual Reliab Eng Int 30(2):275–286CrossRefGoogle Scholar
 Zou C, Tsung F, Wang Z (2007) Monitoring general linear profiles using multivariate exponentially weighted moving average schemes. Technometrics 49(4):395–408MathSciNetCrossRefGoogle Scholar
 Zou C, Ning X, Tsung F (2012) LASSObased multivariate linear profile monitoring. Ann Oper Res 192(1):3–19MathSciNetzbMATHCrossRefGoogle Scholar
 Zwetsloot IM, Woodall WH (2017) A headtohead comparative study of the conditional performance of control charts based on estimated parameters. Qual Eng 29(2):244–253CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.