In-control Phase Research Articles

Adaptive CUSUM chart with cautious parameter learning

AbstractThe classic CUSUM chart assumes that the in‐control (IC) mean and variance are known. In practice, these parameters are usually estimated from an IC Phase I sample. Recently, Capizzi and Masarotto proposed a cautious parameter learning scheme to incorporate Phase II IC observations in the estimation of the IC mean and variance to reduce the variation of conditional average run lengths (ARLs). In this paper, we develop a new cautious parameter learning scheme that can distinguish IC observations from out‐of‐control (OC) observations in the Phase II sample more effectively than Capizzi and Masarotto's scheme. As a result, our cautious parameter learning scheme can provide better estimation of the IC mean and variance. Combining the new cautious parameter learning scheme with an adaptive CUSUM chart, our proposed monitoring procedure is easy to implement, and is shown to have less variability in conditional ARLs and better overall performance for detecting different mean shifts than existing methods.

The performance of the hypergeometric [formula omitted] chart with estimated parameter

Although it is well known that the performance of attribute control charts decreases significantly when the assumption of known process parameters is invalid, this assumption is prevalent in the pertinent literature. However, in most practical applications, the process parameters have to be estimated from a finite in-control Phase I sample, and therefore the performance of attribute control charts should be evaluated from the perspective of estimated process parameters. In this paper, we compare the run length properties of the hypergeometric np chart in both the known and estimated parameter cases. In particular, we investigate the required number of Phase I samples and new specific chart parameters that allow the hypergeometric np chart with estimated parameter p to have approximately the same in-control performance as in the known parameters case. Moreover, we perform a comprehensive in-control and out-of-control comparison of the hypergeometric np chart with its binomial counterpart. In order to achieve these objectives, we also present a new approach to effectively compute the probability distribution of the sum of independent and identically hypergeometric-distributed random variables. The proposed approximation reduces the computational effort to a few seconds while keeping a remarkable high accuracy with only negligible deviations compared to the exact distribution obtained via convolution.

Parameter Estimation Effect of the Homogeneously Weighted Moving Average Chart to Monitor the Mean of Autocorrelated Observations With Measurement Errors

In statistical process monitoring, the usual assumption when designing monitoring schemes is that process parameters are known and have perfect measurements with independent and identically distributed observations. However, in real-life situation, these assumptions rarely hold. Hence, in this paper, the Phase II performance of the homogenously weighted moving average (HWMA) $\bar {X}$ monitoring scheme under the combined effect of autocorrelation and measurement errors is investigated when the unknown process parameters are estimated from an in-control Phase I dataset. Two models are considered, i.e. the first-order autoregressive model for within-sample autocorrelation and the linear covariate model for (constant and linearly increasing variance) measurement system error. Sampling strategies based on skipping some observations as well as mixing different subgroup samples and taking multiple measurements are implemented to reduce the negative effect of autocorrelation and measurement errors. Since the latter sampling strategies incur costs, as an alternative, increasing the slope coefficient of the linear covariate model compensate the negative effect of measurement errors. The new HWMA $\bar {X}$ scheme is shown to have some interesting detection abilities as compared to its competitors. A real-life example is used to illustrate the implementation of the proposed monitoring scheme.

Phase II monitoring of multivariate profiles with estimated parameters and optimal phase I subgroups

In this paper, we investigate the effect of parameter estimation from in-control Phase I samples on the in-control and out-of-control performance of the four Phase II control charts used for monitoring multivariate multiple linear profiles. These methods are evaluated in terms of out-of-control performance using corrected limits. The monitoring approaches investigated are then compared using the statistical properties of the ARL distribution. Furthermore, two optimization models are proposed for finding the optimum number of Phase I samples needed to achieve proper parameter estimations of desired accuracy and solved by using a novel hybrid simulation algorithm named HBSO.

Addressing the effect of parameter estimation on phase II monitoring of multivariate multiple linear profiles via a new cluster-based approach

Profile monitoring is applied when the quality of a product or a process can be determined by the relationship between a response variable and one or more independent variables. In most Phase II monitoring approaches, it is assumed that the process parameters are known. However, it is obvious that this assumption is not valid in many real-world applications. In fact, the process parameters should be estimated based on the in-control Phase I samples. In this study, the effect of parameter estimation on the performance of four Phase II control charts for monitoring multivariate multiple linear profiles is evaluated. In addition, since the accuracy of the parameter estimation has a significant impact on the performance of Phase II control charts, a new cluster-based approach is developed to address this effect. Moreover, we evaluate and compare the performance of the proposed approach with a previous approach in terms of two metrics, average of average run length and its standard deviation, which are used for considering practitioner-to-practitioner variability. In this approach, it is not necessary to know the distribution of the chart statistic. Therefore, in addition to ease of use, the proposed approach can be applied to other type of profiles. The superior performance of the proposed method compared to the competing one is shown in terms of all metrics. Based on the results obtained, our method yields less bias with small-variance Phase I estimates compared to the competing approach.

Robustification of CUSUM control structure for monitoring location shift of skewed distributions based on modified one-step M-estimator

Including three existing charts, a new approach employing a modified one-step M-estimator (MOM) with Cumulative Sum (CUSUM) control structure were evaluated and compared for their Phase II performances based on the average run length (ARL) under various skewed distributions. The primary focus was on the robustness of the CUSUM charts in two separate cases: (i) when the process parameters are known and (ii) when the process mean is unknown and estimated from an in-control Phase I sample. The simulation and real data analysis showed the proposed technique is comparable or sometimes better than the existing charts.

Self-starting X-bar control chart based on Six Sigma quality and sometimes pooling procedure

ABSTRACTSelf-starting control charts have been proposed in the literature to allow process monitoring when only a small amount of relevant data is available. In fact, self-starting charts are useful in monitoring a process quickly, without having to collect a sizable Phase I sample for estimating the in-control process parameters. In this paper, a new self-starting control charting procedure is proposed in which first an effective initial sample is chosen from the perspective of Six Sigma quality, then the successive sample means are either pooled or not pooled (sometimes pooling procedure) for computing next Q-statistics depending upon its signal. It is observed that the sample statistics obtained so from this in-control Phase I situation can serve as more efficient estimators of unknown parameters for Phase II monitoring. An example is considered to illustrate the construction of the proposed chart and to compare its performance with the existing ones.

Optimal design of the side‐sensitive modified group runs (SSMGR) chart when process parameters are estimated

AbstractA side‐sensitive modified group runs (SSMGR) chart is commonly examined under the assumption of known process parameters (denoted as Case‐K). However, in practical situations, the process parameters are seldom known, and it is essential to estimate them from in‐control (IC) phase I samples. From the study of the minimum amount of phase I samples, m that is required by the SSMGR chart when the process parameters are estimated (represented as Case‐U), to achieve practically the same performance as that of the Case‐K, it was found that a huge number of IC phase I samples are needed. Due to time and cost factors, accumulating a substantial amount of IC phase I samples is difficult. To circumvent this problem, we study the optimal designs of the Case‐U SSMGR chart, based on both the zero and steady states' average number of observations to signal (ANOS) and expected ANOS (EANOS). Tables containing optimal charting parameters are provided to ease the implementation of the Case‐U SSMGR chart. By using these optimal parameters specifically designed for the Case‐U SSMGR chart, we can obtain similar zero and steady states' ANOS and EANOS performances to that of the Case‐K. The use of the optimal Case‐U SSMGR chart is demonstrated with real data taken from a hard‐bake process in a manufacturing company.

An EWMA chart for monitoring the covariance matrix of a multivariate process based on dissimilarity index

AbstractIn this article, we propose an exponentially weighted moving average (EWMA) control chart for monitoring the covariance matrix of a multivariate process based on the dissimilarity index of 2 matrices. The proposed control chart essentially monitors the covariance matrix by comparing the individual eigenvalues of the estimated EWMA covariance matrix with those of the estimated covariance matrix from the in‐control (IC) phase I data. It is different from the conventional EWMA charts for monitoring the covariance matrix, which are either based on comparing the sum or product or both of the eigenvalues of the estimated EWMA covariance matrix with those of the IC covariance matrix. We compare the performance of the proposed chart with that of the best existing chart under the multivariate normal process. Furthermore, to prevent the control limit of the proposed EWMA chart developed using the limited IC phase I data from having extensively excessive false alarms, we use a bootstrap resampling method to adjust the control limit to guarantee that the proposed chart has the actual IC ARL(average run length) not less than the nominal level with a certain probability. Finally, we use an example to demonstrate the applicability and implementation of the proposed EWMA chart.

자기상관 데이터 모니터링에서 일단계 모수 추정이 이단계 관리한계선에 미치는 영향 연구

1920년대에 소개되었던 Shewhart 관리도는 관측치가 서로 독립임을 가정했다. 오늘날은 데이터 측정과 자료수집 기술이 발전하면서 자기상관 공정 데이터가 많이 발생하고 있으며, 이것은 통계적 공정 관리의 성능에 부정적인 영향을 끼치게 된다. 자기상관이 존재하는 데이터에 대하여 가장 쉽게 접근할 수 있는 관리도는 먼저 자기상관구조를 모형화할 수 있는 적절한 시계열 모형을 가정한 다음 잔차를 구하여, 그 잔차에 기반한 Shewhart 관리도를 적용하는 것이다. 실제 문제에서 시계열 모형의 참 모수값은 알려져 있지 않으므로, 이 값은 일단계 표본(과거의 관리상태 표본)으로부터 추정된다. 본 논문에서는 이러한 모수추정이 이단계 표본을 모니터링하는데 어떠한 영향이 있는지 살펴보았다. Traditional Shewhart control charts assume that the observations are independent over time. Current progress in measurement and data collection technology lead to the presence of autocorrelated process data that may affect poor performance in statistical process control. One of the most popular charts for autocorrelated data is to model a correlative structure with an appropriate time series model and apply control chart to the sequence of residuals. Model parameters are estimated by an in-control Phase I reference sample since they are usually unknown in practice. This paper deals with the effects of parameter estimation on Phase II control limits to monitor autocorrelated data.

Run rules based phase II c and np charts when process parameters are unknown

The performance of attributes control charts is usually evaluated under the assumption of known process parameters (i.e., the nominal proportion of non conforming units or the nominal average number of nonconformities). However, in practice, these process parameters are rarely known and have to be estimated from an in-control Phase I data set. The major contributions of this paper are (a) the derivation of the run length properties of the Run Rules Phase II c and np charts with estimated parameters, particularly focusing on the ARL, SDRL, and 0.05, 0.5, and 0.95 quantiles of the run length distribution; (b) the investigation of the number m of Phase I samples that is needed by these charts in order to obtain similar in-control ARLs to the known parameters case; and (c) the proposition of new specific chart parameters that allow these charts to have approximately the same in-control ARLs as the ones obtained in the known parameters case.

Properties of the Markov-Dependent Attribute Control Chart with Estimated Parameters

The performance of attribute control charts that monitor Markov-dependent data is usually evaluated under the assumption of known process parameters, that is, known values of a the probability an item is nonconforming given the previous item is conforming and b the probability an item is conforming given the previous item is nonconforming. In practice, these parameters are usually not known and are calculated from an in-control Phase I-data set. In this paper, a comparison of the in-control ARL (average run length) properties of the attribute chart for Markov-dependent data with known and estimated parameters is presented. The probability distribution of the estimators is developed and used to calculate the in-control ARL and standard deviation of the run length of the chart with estimated parameters. For particular values of a and b, the in-control ARL values of the charts with estimated parameters may be very different than those with known parameters. The size of the Phase-I data set needed for charts with estimated parameters to exhibit the same in-control ARL properties as those with known parameters may vary widely depending on the parameters of the process, but in general, large samples are needed to obtain accurate estimates. As the Phase-I sample size increases, the in-control ARL values of the charts with estimated parameters approach that of the known parameter case but not in a monotonic fashion as in the case of the X-bar chart. Copyright © 2015 John Wiley & Sons, Ltd.

Synthetic Double Sampling X̄ Chart with Estimated Process Parameters

The synthetic double sampling (SDS) X̄ chart comprises the double sampling (DS) X̄ and conforming run length (CRL) sub-charts. The SDS X̄ chart has been studied under the assumption of known process parameters in the literature. Nevertheless, in practice, process parameters are usually unknown and are estimated from an in-control Phase I dataset. This paper investigates the performance of the SDS X̄ chart with estimated process parameters, in terms of the average run length (ARL), average number of observations to signal (ANOS) and standard deviation of the run length (SDRL). The performance of the SDS X̄ chart with estimated process parameters by minimizing the out-of-control ARL and the out-of-control ANOS is compared with the corresponding chart’s performance with known process parameters. In addition, the minimum number of Phase I samples required by the SDS X̄ chart with estimated process parameters so that it has approximately the same in-control ARL and ANOS performances as the chart with known process parameters is studied. The ARL, ANOS and SDRL properties of the SDS X̄ chart with estimated process parameters differ significantly from that of the chart with known process parameters. Therefore, suitable optimal charting parameters are introduced so that the SDS X̄ chart with estimated process parameters has an adequate performance as its known process parameters counterpart without having to use large number of Phase I samples and sample size.

The Performance of Phase II Simple Linear Profile Approaches when Parameters Are Estimated

Previous studies of statistical performance of Phase II simple linear profile approaches were reported only for the case of known profile parameters assumption. The main objective of this article is to evaluate and compare the performance of these approaches when the profile parameters are estimated from an in-control Phase I profile data set. Simulations establish that the performance of these approaches is strongly affected when the parameters are estimated compared to the known parameters case. The in-control performance of the competing approaches significantly deteriorates if estimated parameters are used with control limits intended for known parameters, especially when only a few Phase I samples are used to estimate the parameters. The results show also that some profile monitoring approaches need much larger number of Phase I profiles than other approaches to achieve the expected statistical performance. They also show that the profile monitoring approach proposed by Mahmoud et al. (2010) has generally better out-of-control run length performance than the competing approaches when the estimated parameters are used in the charts design.

Robustness of the EWMA control chart for individual observations

The traditional exponentially weighted moving average (EWMA) chart is one of the most popular control charts used in practice today. The in-control robustness is the key to the proper design and implementation of any control chart, lack of which can render its out-of-control shift detection capability almost meaningless. To this end, Borror et al. [5] studied the performance of the traditional EWMA chart for the mean for i.i.d. data. We use a more extensive simulation study to further investigate the in-control robustness (to non-normality) of the three different EWMA designs studied by Borror et al. [5]. Our study includes a much wider collection of non-normal distributions including light- and heavy-tailed and symmetric and asymmetric bi-modal as well as the contaminated normal, which is particularly useful to study the effects of outliers. Also, we consider two separate cases: (i) when the process mean and standard deviation are both known and (ii) when they are both unknown and estimated from an in-control Phase I sample. In addition, unlike in the study done by Borror et al. [5], the average run-length (ARL) is not used as the sole performance measure in our study, we consider the standard deviation of the run-length (SDRL), the median run-length (MDRL), and the first and the third quartiles as well as the first and the 99th percentiles of the in-control run-length distribution for a better overall assessment of the traditional EWMA chart's in-control performance. Our findings sound a cautionary note to the (over) use of the EWMA chart in practice, at least with some types of non-normal data. A summary and recommendations are provided.

The synthetic [Xbar] chart with estimated parameters

A synthetic [Xbar] chart consists of an integration of a Shewhart [Xbar] chart and a conforming run length chart. This type of chart has been extensively used to detect a process mean shift under the assumption of known process parameters. However, in practice, the process parameters are rarely known and are usually estimated from an in-control Phase I data set. The goals of this article are to (i) evaluate (using a Markov chain model) the performances of the synthetic [Xbar] chart when the process parameters are estimated; (ii) compare it with the case where the process parameters are assumed known to demonstrate that these performances are quite different when the number of samples used during Phase I is small; and (iii) suggest guidelines concerning the choice of the number of Phase I samples and to provide new optimal constants, especially dedicated to the number of samples used in practice.

A CUSUM control chart for monitoring the variance when parameters are estimated

CUSUM control chart has been widely used for monitoring the process variance. It is usually used assuming that the nominal process variance is known. However, several researchers have shown that the ability of control charts to signal when a process is out of control is seriously affected unless process parameters are estimated from a large in-control Phase I data set. In this paper we derive the run length properties of a CUSUM chart for monitoring dispersion with estimated process variance and we evaluate the performance of this chart by comparing it with the same chart but with assumed known process parameters.

RUN RULES $\bar{X}$ CHARTS WHEN PROCESS PARAMETERS ARE UNKNOWN

Supplementary Run Rules are applied to increase the power of the standard Shewhart [Formula: see text] chart for detecting small shift. The performance of this chart is usually evaluated under the assumption of known parameters. However, in practice, the process parameters are rarely known and, usually estimated from an in-control Phase I data set. In this paper, we evaluate (using a Markov Chain model) the performances of the Run Rules [Formula: see text] chart when the process parameters are estimated, and compare them when the process parameters are assumed known. We draw the conclusion that these performances are quite different when the number of samples used during the phase I is small and we suggest new chart parameters in order to overcome this problem.

An EWMA chart for monitoring the process standard deviation when parameters are estimated

The EWMA chart for the standard deviation is a useful tool for monitoring the variability of a process quality characteristic. The performance of this chart is usually evaluated under the assumption of known parameters. However, in practice, process parameters are estimated from an in-control Phase I data set. A modified EWMA control chart is proposed for monitoring the standard deviation when the parameters are estimated. The Run Length properties of this chart are studied and its performance is evaluated by comparing it with the same chart but with process parameters assumed known.

Effects of Parameter Estimation on Control Chart Properties: A Literature Review

Control charts are powerful tools used to monitor the quality of processes. In practice, control chart limits are often calculated using parameter estimates from an in-control Phase I reference sample. In Phase II of the monitoring scheme, statistics based on new samples are compared with the estimated control limits to monitor for departures from the in-control state. Many studies that evaluate control chart performance in Phase II rely on the assumption that the in-control parameters are known. Although the additional variability introduced into the monitoring scheme through parameter estimation is known to affect the chart performance, many studies do not consider the effect of estimation on the performance of the chart. This paper contains a review of the literature that explicitly considers the effect of parameter estimation on control chart properties. Some recommendations are made and future research ideas in this area are provided.