Robust control chart based on mixed-effects modeling framework: A case study in NAND flash memory industry

Abstract In this research, we analyze the real data in the NAND Flash memory industry using a control chart. There are thousands of electrical measures for each NAND Flash memory chip. We monitor these data through a control chart to ensure that the manufacturing process is in control. For better interpretability, we apply a univariate control chart technique to each variable. However, most existing control charts, such as the EWMA chart, do not include between-subgroup variations but only within-subgroup variations. They often obtain too narrow control limits for some variables, which leads too many subgroups to fall outside the control limits. To overcome this issue, we apply a control chart under a mixed-effects modeling framework to include both within-subgroup and between-subgroup variations. Additionally, the EWMA chart assumes that all the items are normally distributed; however, we frequently encounter that a normal assumption is violated. To overcome this limitation, we apply a robust approach based on a nonparametric sign chart. Furthermore, we introduce a p-value combination method to increase the statistical power for the gradual change detection of a statistical process. Our study show that the proposed control chart can efficiently monitor the real data in the NAND Flash memory industry.


Introduction
Control charts are statistical tools frequently used in many manufacturing industries, including the NAND Flash memory industry, to check whether the quality of a manufacturing process is in a state of control.NAND Flash memory is an electronic nonvolatile computer memory storage medium, which is widely used in storage devices such as solid-state drives (SSD) and USB flash drives.In the NAND Flash memory industry, all the NAND Flash memory chips produced are evaluated through many electrical tests to eliminate infant failures.Meanwhile, we obtain thousands of measures for all the NAND Flash memory chips produced.Using these data, we monitor if the manufacturing process is in a state of control through control charts.To the best of our knowledge, this study is the first to apply a control chart in the NAND Flash memory industry.
Control charts, first introduced by Shewhart (1931), are divided into two types, location and dispersion control charts, which are used to monitor changes in the center and variability of a statistical process, respectively.Shewhart control charts are based on the sample mean � x for the process center and range R or standard deviation S for the process variability.As Shewhart control charts use only the information of the current subgroup, they are insensitive to small process shifts.To detect small process shifts, memorytype control charts have been developed: the cumulative sum (CUSUM; Page 1954) and exponentially weighted moving average (EWMA; Roberts 1959) charts.By using the information from both the past and current subgroups, memory-type control charts can efficiently detect small process shifts.Afterward, several control charts based on adaptive approaches have been proposed, such as adaptive cumulative sum (ACUSUM; Jiang, Shu, and Apley 2008) and adaptive exponentially weighted moving average (AEWMA; Capizzi and Masarotto 2003;Haq, Gulzar, and Khoo 2018;Haq and Khoo 2019) charts.Adaptive control charts are useful for efficiently detecting both large and small process shifts when the shift sizes are unknown.
In this study, we focus on univariate location control charts to detect small location shifts in statistical processes.Although we deal with high-dimensional data, we focus on univariate control charts, rather than multivariate control charts because the results of multivariate control charts are hard to interpret.For example, a NAND Flash memory chip contains millions of memory cells, wherein each cell may be charged or discharged with electrons, which indicates 0 or 1 (1 bit), respectively.In this sense, a NAND Flash memory chip can store data with a complex and sophisticated structure in a small size.To check whether a NAND Flash memory chip is normal, we measure and monitor many variables in various areas of each NAND Flash memory chip, for example leakage current in various areas.Excessive leakage current in each region can be caused by different problems, leading to various defects, with different solutions.For engineers to investigate the causes and solutions after detecting alarms from a control chart, a multivariate control chart methodology may be inappropriate because the region of the chip causing a problem may be difficult to recognize.Therefore, we construct a univariate control chart to monitor high-dimensional electrical measures of NAND Flash memory by applying a univariate control chart to each variable.
As we monitor high-dimensional data using a univariate control chart, we detect many false alarms.This is because a control chart can be viewed as a series of hypothesis testing (Montgomery 2013;Woodall and Faltin 2019).However, we have limited number of engineers to analyze statistically detected alarms electrically, that is, engineers cannot investigate all the alarms.Thus, we cannot aggregate the statistical process into a series of subgroups for a statistical monitoring with too short time interval.In this study, we set the time interval for each subgroup to 1 week.For determining the size and frequency of sampling, more frequent samples are needed with small sample sizes, while less frequent samples are sufficient with large sample sizes (Montgomery 2013;Zwetsloot and Woodall 2021).Data is measured on every NAND Flash chip produced, thus, the sample sizes are sufficiently large (n ¼ 10000 À 25000 for a week), and it is appropriate to choose a long time interval, 1 week.Our real data benefits from small practitioner-to-practitioner variations because the measurements are made on every NAND Flash chip produced (Quesenberry 1993;Saleh et al. 2015).
In this study, we apply a univariate control chart based on a mixed-effects modeling framework as in Woodall and Thomas 1995 to include not only within-week, but also between-week variations.In addition, we employ a sign chart by following Graham, Chakraborti, and Human 2011 to monitor the process nonparametrically because the statistical processes are usually non-normal in the NAND Flash industry.Furthermore, the proposed control chart is constructed under a sliding window approach for the setup of Phase I. Furthermore, we introduce a metaanalysis by combining pvalue to efficiently detect the gradual changes in the statistical processes.
The remainder of the paper is organized as follows.Following the four subsections, the motivations of the proposed control chart appropriate for our real data application, we describe a univariate control chart applied to the motivating example data.The next section discusses the application of the proposed control chart to real data in the NAND flash industry.The final section concludes the paper.

Motivation 1: Between-subgroup variation
To monitor our real data, the first motivation is to consider between-subgroup variations, while most widely used control charts such as Shewhart � X-chart or EWMA chart are based on only within-subgroup variations.In many practical applications, including our research, it is not realistic to assume only withinsubgroup variations (Woodall and Thomas 1995).For our real data, there are various sources that can induce a considerable amount of between-subgroup variations.For example, the manufacturing process of NAND Flash memory is highly complicated.Many manufacturing equipment lines are there for mass production, wherein different manufacturing machines may cause additional variations.Furthermore, each subgroup has a large sample size because all measurements are obtained on every NAND Flash chip produced, and the frequency interval, 1 week, is long.This makes it difficult or impossible to remove the sources inducing additional between-subgroup variations.If we apply a control chart based on only the within-subgroup variations for our real data, we often encounter too narrow control limits for some variables, leading to too many false alarms.To deal with this issue, our control chart includes both within-week variations and between-week variations.
where � X t is the sample mean at t-th week and k is the weight of the moving averages.The control limits are given by l6L � r ffi ffi ffi ffi n t p ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where l and r 2 are the statistical process mean and variance, respectively.The control limits for the two electrical measures are plotted with L ¼ 12 as red lines, whereas plotting statistics are plotted as black lines.The sample size at each week, n t is large, that is, there are about 10000 -25000 wafers at each week.
The EWMA chart is based on only the within-week variations: where X it is the i-th wafer produced at t-th week.
Figure 1 indicates that we may encounter too narrow control limits for some variables when using an EWMA chart based on only the within-week variations.Figure 1(a) shows narrow control limits, which leads many subgroups to fall outside the control limits.There was a serious defect associated with the process decrease in the 26 -28th weeks, but the narrow control limits led to the detection of false alarms before the defect occurred.However, Figure 1(b) has quite wide control limits with the same value L ¼ 12, implying that we cannot widen the control limits with bigger L.Moreover, we cannot adjust the control limits manually for all the variables because we have thousands of measures to be monitored.To solve this problem, we need a control chart based on the both within-week and between-week variations.Following Woodall and Thomas 1995, we construct a control chart based on a mixed-effects modeling framework to include both within-week and between-week variations.
To demonstrate the mixed-effects modeling to include between-week variations, we conduct a simple simulation study.Details of the simulation study are explained in the Supplementary Materials.In the simulation study, we found that narrow control limits are often obtained when sample sizes are large for the Shewhart � X-chart.One can avoid too narrow control limits by adding a between-week variation component under a mixed-effects modeling framework in the simulation study.In our real data application, there are many sources inducing between-week variations, and we have a large sample size, n t ¼ 10000 À 25000: Moreover, we should monitor thousands of variables, with various magnitudes of between-week variations among the variables.Thus, we cannot deal with this issue by widening control limits with larger L using a control chart based on only within-week variations.Therefore, we need a control chart including both within-week and between-week variations.For this, we construct a control chart based on a mixed-effects modeling framework.

Motivation 2: Nonparametric approach
The second motivation is to construct a nonparametric control chart.In the NAND Flash memory industry, we frequently encounter the violation of a normal assumption, while many existing control charts are based on the normality.Specifically, we often encounter some measurement errors in the process of electrical tests, which can be considered as outliers.In this case, a mean-based monitoring can be sensitive to these outliers.To overcome this issue, a nonparametric approach can allow for efficient monitoring of a process even if its underlying distribution is non-normal.Therefore, we aim to construct a nonparametric control chart robust to outliers and non-normality.To this end, we construct a control chart based on a sign chart approach following Graham, Chakraborti, and Human 2011.

Motivation 3: Sliding approach for Phase I
The third motivation is that it is difficult to set a fixed period to Phase I for our real data.Generally, a control chart design is divided into two phases (Montgomery 2013;Woodall and Faltin 2019;Atalay et al. 2020).In Phase I, we assume that a historical set of data is in control and analyzed all at once in a retrospective analysis to estimate unknown parameters.As the statistical process may be initially outof-control in Phase I, one should make the process in-control by excluding the points outside control limits or looking for assignable causes for such occurrences.In Phase II, one monitors the process online through a control chart using the parameters estimated at the Phase I.However, there are continuous changes in the manufacturing process of NAND Flash memory, which leads to continuous changes in the distributions of statistical processes of many variables.In addition, we store the data of NAND Flash memory chips for only a limited period.This is because we obtain all the measurements of thousands of variables for numerous NAND Flash memory chips and it is very expensive to store all the data for a long time.Thus, we need to update the Phase I continuously for each variable.For this, we construct a control chart under a sliding window approach for the setup of Phase I.

Motivation 4: Gradual process change
The fourth motivation is to detect gradual changes of a statistical process efficiently.In the NAND Flash memory industry, there are continuous changes in the manufacturing process for reasons such as improving the yield of NAND Flash memory and responding to changes in raw material prices.In these cases, the changes in a statistical process are presented in a form of 'gradual change', not 'step function'.This is because the manufacturing process of NAND Flash memory is highly complex, so the distribution of a statistical process cannot be changed all at once like a step function.Figure 1(a) presents this motivation, where the variable had decreased drastically, but gradually, from 20-th week to 30-th week.However, most existing control charts focus on detecting step-function-like changes, which is not efficient for monitoring our real data.In this sense, we aim to construct a control chart which can efficiently detect gradual changes in a statistical process.For this, we adopt the pvalue combination method.
To demonstrate the usefulness of the pvalue combination method to efficiently detect gradual process changes, we conduct a simple simulation study.Details of the simulation study are explained in the Supplementary Materials.

The proposed control chart
In this section, we describe a univariate control chart applied to our motivating example data.We introduce a nonparametric control chart based on a mixedeffects modeling framework to account for the between-subgroup variation and to increase robustness to outliers.In addition, we explain a sliding window approach to update the Phase I data to reflect systematic changes that do not necessarily represent the out-of-control state.Additionally, we explain a pvalue combination approach to facilitate detecting the gradual process changes as an out-of-control signal with increased statistical power.

Nonparametric control chart based on a mixed-effects modeling framework
First, we construct a mean control chart based on a mixed-effects modeling framework following (Woodall and Thomas 1995).Let X it be the ith observation at t-th week for t ¼ 1, 2, ::: and i ¼ 1, 2, :::, n t : We assume where l t is a week-specific mean, r 2 is a within-week variance, l is a global mean, and r 2 b is a betweenweek variance.Equivalently, Eq. [4] can be represented as where � X t ¼ 1 n t P n t i¼1 X it is a sample mean at week t.Then, the marginal distribution of � X t is as follows.
Based on Eq.
[6], one can construct a control chart by using � X t as a monitoring statistic.Control limits are specified after estimating the parameters, l, r 2 , and r 2 b using the data in Phase I sample.Although the control chart that monitors the sample means is widely used, we often encounter a situation where outliers appear in X it 's particularly when the normal assumption of Eq. [4] is violated.The plotting statistic � X t is sensitive to these outliers and may cause a false alarm indicating an out-of-control state, although the production process is in control.To be robust to outlying observations and non-normality, we adopt a sign chart approach introduced by Graham, Chakraborti, and Human 2011 as follows.Let M be the global median of X it 's and Y it ¼ IðX it > MÞ: We assume IðX it > MÞ � Bðn t , P t Þ for t ¼ 1, 2, ::: where P t represents the proportion of observations bigger than the global median M at week t.Then, provided that n t P t > 5 and n t ð1 À P t Þ > 5, we can apply the Central Limit Theorem and obtain the following result.
Then, letting l t and r 2 t denote P t and P t ð1 À P t Þ, respectively, we assume Marginally and additionally assuming a common variance, that is r 2 ¼ r 2 t for all t, we obtain which is the same as Eq.[6] except that � X t is replaced by � Y t : Now, one can construct a control chart based on Eq. [10] using � Y t as a monitoring statistic.Control limits are specified once the parameters, M, l, r 2 , and r 2 b are estimated by the proportion data in Phase I sample.
The control chart constructed based on Eq. [10] uses only the current week information to decide if an alarm should be made.Such a method is prone to the failure to detect a small shift.To overcome the limitation, we combine the control chart based on Eq. [10] with the exponentially weighted moving average (EWMA) approach proposed by Roberts (1959) as follows.
[11], we obtain the following result where Now, using W t as a monitoring statistic, a control chart can be constructed based on Eqs.
The proposed control chart gives the following control limits with a pre-specified value L at week t.
where l and V T are estimated by the data in Phase I sample.The proposed control chart detects an out-ofcontrol signal at week t if W t falls outside the control limits.Hereafter, we refer the control chart represented by Eq. [14] as MECC-EWMA representing a mixed effects modeling based control chart combined with the EWMA approach.
As discussed above, to specify the control limits, we need to estimate the parameters, M, r 2 , l and r 2 b , based on the Phase I sample.We denote the Phase I sample by X � it for i ¼ 1, :::n � t and t ¼ 1, 2, :::, T � : Then, M is estimated as the overall sample median of X � it 's for i ¼ 1, :::n � t and t ¼ 1, 2, :::, T � , and r 2 is estimated by where Y � it ¼ IðX � it > MÞ for i ¼ 1, :::n � t and t ¼ 1, 2, :::, T � : Additionally, l and r 2 b are estimated through the maximum likelihood estimation in the mixedeffects modeling framework.We used R statistical software (version 4.0.3;R Development Core Team) with functions from the package mixmeta.(Gasparrini and Armstrong 2011;Sera et al. 2019).Typically, the Phase I sample is defined as a historical sample that is collected during an in-control state of the production process.However, in the NAND Flash memory production, the process can systematically change even when the state is not out-of-control.Moreover, the data of produced NAND Flash memory chips are stored for only a limited period.Consequently, it is practically difficult to define a Phase I sample as a certain historical data and to use it constantly.In this study, we update the Phase I sample using a sliding window approach.In other words, when we evaluate if the process is out-of-control at time t ¼ T, the Phase I data X � it 's are defined as X it 's for i ¼ 1, :::, n t and t ¼ T À t W , :::, T À 1 with a window of length t W .

pvalue combinations for detecting gradual process changes effectively
In the NAND Flash memory industry, we usually detect out-of-control alarms in a statistical process in the form of either 'gradual change' or 'step-functionlike change'.For the gradual change detection, one may need to increase a statistical power, for which we use a pvalue combination method.As a control chart can be viewed as a series of hypothesis testing (Montgomery 2013;Woodall and Faltin 2019), we can express the MECC-EWMA in terms of pvalue.At t-th week, a decision on whether the process is out-of-control can be equivalently represented as a hypothesis testing for the null and alternative hypotheses specified below.
where m is the mean of Z t ¼ ðW t À lÞ=V t , which is a Z-score following a standard normal distribution.Then, the pvalue, p t , can be calculated as where Uð�Þ is the cumulative distribution function (cdf) of the standard normal distribution.Using the p t , outof-control signal is alarmed at week t when the pvalue p t is lower than the significance level a ¼ 2UðÀ LÞ: When a statistical process increases or decreases gradually, at a particular week t, p t may not be lower than the significance level a, but rather be moderately low.Specifically, our proposed control chart may fail to detect these gradual process changes because the upper control limit (or lower control limit, respectively) increases (or decreases) with increasing (or decreasing) statistical processes gradually under a sliding window approach to define the Phase I sample.Moreover, we have thousands of measures to be monitored, and engineers cannot investigate all the numerous alarms electrically.Thus, we set a large L to detect only the alarms with certain and irregular changes in the statistical processes.Under this circumstance, we need to detect gradual process changes more efficiently.For this, we combine pvalue through a meta-analysis framework; we can then obtain a very low pvalue representing a plausible evidence of the gradual process change.
The pvalue combination is conducted as follows.To obtain the combined pvalue at t ¼ T, we consider a window of length t C .Then, the joint null hypothesis for the pvalue in the window is Among various options for the pvalue combination method, we use the Stouffer's weighted Z-test (Stouffer, DeVinney, and Suchmen 1949;Zaykin 2011).The combined pvalue at T-th week is given by where ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi and w t is the weight for the t-th subgroup.Intuitively, it is reasonable to impose more weight for recent w t 's.
For this, we impose exponential weights with the rate 0 < c < 1, that is, w t ¼ c TÀ t : Lower rate imposes more weight on the recent Z t 's.For this study, we choose c ¼ 0:8: Through the above meta-analysis framework, we can combine out-of-control evidences for t ¼ T À t C þ 1, :::, T, and the combined pvalue, q t 's, with exponentially decreasing weights show the combined evidence of both small shifts and gradual changes of the statistical process.Then, the proposed control chart detects an out-of-control signal at t ¼ T when jQ T j > L q with the upper control limit L q .
Hereafter, the MECC-EWMA combined with pvalue combination is referred as MECCp-EWMA.

Real data application
In this section, we applied the proposed control chart to real data in the NAND Flash industry.The NAND Flash architecture is highly sophisticated and has thousands of measures, such as voltages, currents, and operation times, for various regions in the structure.These measures need to be monitored to ensure a reliable quality.In this application, we analyzed nine variables among thousands of measures for 48 wk from January 1, 2021, to December 3, 2021, using the proposed control chart.Variables 1 -7 are seven measures of voltages and leakage currents for various regions in the NAND Flash memory, where each observation indicates the mean of all the chips in a wafer.For Variables 8 and 9, each observation indicates the proportion of passed chips in a wafer for two electrical tests.As most produced chips pass the electrical tests, Variables 8 and 9 are zero-inflated.
Because of the information security policy of the company, we normalized the original data X it with the global sample mean and standard deviation, and then conducted the analysis.Each subgroup indicates a week and sample sizes for each subgroup ranges from 10000 to 25000.We monitored both directions for only Variable 1, 2, and 5 and only the upper direction for other variables.In other words, the null hypothesis in the Eq.[16] is H 1 : m 6 ¼ 0 for Variables 1, 2, and 5 and H 1 : m > 0 for others.We applied the proposed control chart under a sliding window framework for Variables 1 -9, where the window length for the sliding window is 24 wk.This value was selected by taking into account the domain knowledge.Because there are continual changes in the manufacturing process of NAND flash memory, we select a sufficiently long window length to obtain wide control limits for accommodating such changes.However, too long window length can include samples with drastically different chip properties in Phase I. Considering these two aspects, we set the length of the sliding window to 24 wk.We represent the result regarding whether the process is incontrol or not in terms of both W t 's (MECC-EWMA) and Q t 's (MECCp-EWMA) after the 25th week.For comparison, we also applied the EWMA chart under a sliding window framework with 25 wk of window length.The control limits of the EWMA chart are calculated with L ¼ 12 after the 15th week.For the EWMA chart, we truncated outliers with 9 sigma limits to reduce measurement errors.We set the weight of the EWMA trend, k ¼ 0:2 for the two control charts.For the proposed control chart, we set c ¼ 0:8 and t C ¼ 10.We also conducted the sensitivity analysis for c 2 f0:2, 0:4, 0:6, 0:8g and t C 2 f7, 10, 14g and found that there is no significant variations (Figures S2-S12 in the Supplementary material).We set L ¼ 3 and L q ¼ 5:5 because pvalue tend to be lowered after pvalue combinations.
Figure 2 (a)-(i) show the result of MECC-EWMA and MECCp-EWMA for Variables 1 -9, respectively.The upper plot shows the proportion bigger than the global median of all values in the sliding window, in which green lines indicate the control limits without pvalue combinations where L ¼ 3.Meanwhile, the lower plot shows jQ t j's with pvalue combinations, where the red horizontal line indicates the control limit L q ¼ 5:5: For Variable 1, the trend in the upper plot drastically decreased since the 27-th week and gets alarms since the 28-th week.Indeed, there was a serious defect at that point, which was detected by both MECC-EWMA and MECCp-EWMA.For Variable 2, the process had been in a certain level of quality until the 35-th week but drastically decreased after that and both MECC-EWMA and MECCp-EWMA detected alarms for the drastic decrease.Although MECC-EWMA detected alarms with L ¼ 3 at the points of the gradual increase, we may use wider control limits to reduce false alarms.This is like the Bonferroni correction method for multiple testing problem as we are monitoring thousands of variables.In this case, we may not be able to detect it without pvalue combinations.Additionally, MECC-EWMA did not detect alarms near the end of the period, even though the decreasing trend seemed to continue.After pvalue combinations, we can detect such gradual changes with much higher scores than the control limit L q .Next, the trend of Variable 3 was in a certain quality level for the whole period; thus, MECC-EWMA and MECCp-EWMA did not detect any alarm for Variable 3. Furthermore, the trends of Variables 4 and 5 were in a certain level of quality.MECC-EWMA and MECCp-EWMA detected alarms for the points of the drastic increase and decrease near around the 45-th week for Variables 4 and 5, respectively.Subsequently, the trends of Variables 6 and 7 were in a certain level of quality for the whole study period; thus, MECC-EWMA and MECCp-EWMA did not detect any alarms.For Variables 8 and 9, we applied MECC-EWMA and MECCp-EWMA after jittering the data as they were zero-inflated.The trends of Variables 8 and 9 were in a certain level of quality for the whole period; thus, MECC-EWMA and MECCp-EWMA did not detect any alarm.
Figure 3 (a)-(i) show the result of the EWMA chart for Variables 1 -9, respectively, where black and red lines indicate the EWMA trend and control limits with L ¼ 12, respectively.Compared to MECC-EWMA and MECCp-EWMA, the EWMA chart shows much narrower control limits for Variables 1, 2, 4 and 5. First, the control limits for Variable 1 are very narrow, which leads to too many alarms, even at the points before the defect occurred.For Variable 2, the EWMA chart detected the drastic decrease after the 35-th week like MECC-EWMA and MECCp-EWMA; however, it detected other alarms around the 28-th week for the process decrease, while MECC-EWMA and MECCp-EWMA approached to the control limits but no alarm was detected.For Variable 3, the EWMA chart showed much wider control limits compared with those for Variables 1 and 2, which leads to a similar conclusion as in MECC-EWMA and MECCp-EWMA.Meanwhile, Variables 4 and 5 showed similar results with Variable 2. Figures 3(d) and (e) show that the EWMA chart detected other alarms because of the narrow control limits, especially for Variable 4. Since we could not find any electrical defect issue for Variable 4 at that time, engineers would not want any alarm for the trend of Variable 4.
For Variables 6 and 7, there was a big difference between the trends based on the sample mean-based and nonparametric approaches.The trends of EWMA chart for Variables 6 and 7 had rapidly increased near the end of the study period; thus, the EWMA chart detected alarms for it, while those for MECC-EWMA and MECCp-EWMA charts of Variables 6 and 7 were in a certain level of quality and no alarm was detected.For Variables 8 and 9, the trends based on the sample mean show jumps unlike MECC-EWMA and MECCp-EWMA, implying that there was no wafers containing failed chips after the truncation before jumps, and a couple of wafers contained failed chips at the jumps.The EWMA chart detects alarms for all the jumps, which may lead to unstable results and false alarms since jumping points may vary depending on the truncation limits.Figures S13 and  S14 show the result of sensitivity analysis for the outlier truncation (6, 9, 12 sigma limits) in the Supplementary material.For Variable 8, the starting points for the alarm detection of the EWMA chart vary depending on truncation limits, that is, 41st, 42nd, 43rd week for 6, 9, 12 sigma limits, respectively.For Variable 9, the trend based on the sample mean with the 6-sigma truncation limit shows no variation, but there was a drastic increase near the 30th week with the 9-and 12-sigma truncation limit.However, MECC-EWMA and MECCp-EWMA did not detect any alarm because the trends were in a certain level of quality for the whole period.
In summary, the result implies that control limits of the EWMA chart for some variables are very narrow, which may lead to unreasonable monitoring results.As it is difficult to choose appropriate control limits manually for each variable in a high-dimensional data setting, the proposed control chart can be an effective alternative to overcome the issue.This is because the proposed control chart is based on a mixed-effects modeling framework that can include both within-week and between-week variations.In addition, we can see that MECCp-EWMA can detect gradual process changes effectively.

Conclusion
In this study, we analyze the real data of electronic measures in NAND Flash memory industry using a control chart.To the best of our knowledge, this is the first research to apply a control chart to the real data in the NAND Flash memory industry.To reflect the properties of the real data in NAND Flash memory industry, we applied a nonparametric control chart based on a mixed-effects modeling framework to include both within-subgroup and between- subgroup variations.Specifically, as there are sources that can induce between-subgroup variations in the NAND Flash memory industry, we have found very narrow control limits for some variables if a control chart is applied based on only the within-subgorup variations.Our study show that the proposed control chart under the mixed-effects modeling framework can efficiently monitor high-dimensional statistical processes when there are between-subgroup variations.This should be useful for monitoring statistical processes in various manufacturing industries, with highly complicated processes and thus a considerable amount of between-subgroup variations.
One limitation of the proposed control chart is that a subgroup for which an out-of-signal alarm is detected is also included in the Phase I sample for the subsequent monitoring.This is led by the sliding window approach of the proposed control chart; thus, we may obtain wider control limits because of the out-of-signal subgroups in the Phase I sample.Therefore, one needs to investigate the first out-of-signal alarm immediately and rigorously.Otherwise, wider control limits may prevent an out-of-signal alarm from being detected.
Although the proposed control chart is formulated based on only the sample global median, it can be applied to other percentiles.One may consider a tail proportion, which represents the proportion lower than 100p-th or bigger than 100ð1 À pÞ-th percentiles with a small p such as p ¼ 0.01 or 0.001.This concept is similar to dispersion control charts based on the sample range R or standard deviation S. We aim to combine the proposed control charts based on both the sample global median and tail proportion in our future study.We plan to design a methodology for combining the concepts of both location and dispersion control charts based on the proposed control charts using both the sample global median and tail proportion by extending the pvalue combination method.

Figure 1
Figure 1 is our motivating example, which shows the result of the EWMA chart for two electrical measures of a NAND Flash memory product.The EWMA chart is based on the following plotting statistics

Figure 1 .
Figure 1.The result of the EWMA chart with control limits L ¼ 12 of two electrical measures for a NAND Flash memory product.Each black points indicates the mean of each subgroup.The black and red lines indicate the EWMA trend and control limits, respectively.Each subgroup indicates a week and the first 15 wk are considered as Phase I.

Figure 2 .
Figure 2. The result of MECC-EWMA and MECCp-EWMA for the real data application.(a)-(i) indicates the results for Variables 1-9, respectively.Green lines in the upper plots and blue lines in the lower plots are plotted with L ¼ 3 of MECC-EWMA and L q ¼ 5:5 of MECCp-EWMA, respectively.

Figure 3 .
Figure 3.The result of the EWMA chart with L ¼ 12 for the real data application.(a)-(i) indicate the results for Variables 1-9, respectively.