Generalized Likelihood Ratio Chart Research Articles

A design parameter-free geometric Kullback-Leibler information control chart for monitoring Bernoulli processes

This paper considers a design parameter-free Kullback-Leibler information (KLI) control chart for monitoring the nonconforming proportion p of Bernoulli processes in Phase II. The chart statistic is derived from the Kullback-Leibler information based on a geometric distribution. Unlike conventional charts for Phase II, the proposed chart does not require design parameters. The users only need to determine a desired in-control average run length. The performance of conventional control charts deteriorates if the prespecified optimal design parameter is not appropriate for the actual size of the shift in the monitored quality characteristic. A design parameter-free approach prevents this situation, and the time needed to find the optimal design parameter can be saved as well. The performance of charts is evaluated by the average number of observations to signal under a specific shift in p, and the penalized relative mean index assesses the overall performance. The proposed chart has the best overall performance for detecting an upward shift in p compared with the cumulative count of conforming (CCC-r), Bernoulli cumulative sum (CUSUM), Bernoulli exponentially weighted moving average (EWMA), Bernoulli generalized likelihood ratio (GLR), geometric CUSUM, and binomial GLR charts. The KLI chart has decent performance for detecting a downward shift in p if the size of the shift is small. For detecting a possible shift in p to both sides, the overall performance of the KLI chart is better than those of the charts.

The Design of GLR Control Chart for Monitoring the Geometric Observations Using Sequential Sampling Scheme

The control charts’ design is focused on system forecasting which is important in mathematics and statistics; these techniques are commonly employed in manufacturing industries. The need for a control chart that can conceptualize and identify the symmetric or asymmetric structure of the monitoring phase with more than one aspect of the standard attribute is a necessity of industries. The generalized likelihood ratio (GLR) chart is a well-known method to track both the decrease and increase in the mechanism effectively. A control chart, termed as a GLR control chart, is established in this article, focusing on a sequential sampling scheme (the SS GLR chart) to evaluate the geometrically distributed process parameter. The SS GLR chart statistic is examined on a window of past samples. In contexts of the steady-state average time to signal, the output of the SS GLR control chart is analyzed and compared with the non-sequential geometric GLR chart and the cumulative sum (CUSUM) charts. In this article, the optimum parameter options are presented, and regression equations are established to calculate the SS GLR chart limits.

Designing a parameter‐free Kullback‐Leibler information control chart for monitoring process mean shift

AbstractThis paper proposes a parameter‐free Kullback‐Leibler information control chart for monitoring sustained shifts in the process mean of a normally distributed process in phase II. Two plotted statistics are provided. One is based on our backward empirical sequential test, the other is based on the maximum log‐likelihood ratio change point method. These two achieve similar performances for the control chart. The performance of the proposed chart is compared with those of the cumulative sum chart, the exponentially weighted moving average chart, and the generalized likelihood ratio (GLR) chart. The results show that our proposed chart and the GLR chart have similar performances. Both can detect a wide range of shifts in the process mean, and neither requires design parameters other than the control limits. The proposed chart outperforms GLR when the size of the shift is below 1.24 standard deviations, while GLR outperforms the proposed chart when the size of the shift is above 1.24 standard deviations.

A Kullback‐Leibler information control chart for linear profiles monitoring

AbstractProfile monitoring is referred to as monitoring the functional relationship between the response and explanatory variables. Traditionally, process control charts monitor the mean and/or the variance of a univariate quality characteristic. For several correlated quality characteristics, multivariate process control charts monitor the mean vector and/or the covariance matrix. However, monitoring the functional relationship between variables is sometimes more beneficial. One example is the power output of a Diesel engine and the amount of fuel injected should be related. In this paper, we propose a Kullback‐Leibler information (KLI) control chart for linear profiles monitoring in Phase II. The plotted statistics of the KLI chart are calculated based on a backward procedure. The functional relationship is described by linear regression. The performance of the proposed KLI control chart is compared with those of other existing control charts, especially the Generalized Likelihood Ratio (GLR) chart for both KLI and GLR charts do not require design parameters. The results show that (1) the KLI control chart is better than the GLR control chart in detecting the shift of variance in terms of Average Time to Signal, and (2) if the shift of the regression coefficient is small, the GLR chart outperforms the KLI chart, but if the size of shift is large, the KLI chart outperforms the GLR chart. The plotted statistics of KLI are compared to that of GLR. Their similarity is discussed.

Monitoring of linear profiles using generalized likelihood ratio control chart with variable sampling interval

AbstractThe usual practice in using a control chart to monitor a profile is to take samples from the process with fixed sampling intervals. This article considered the use of generalized likelihood ratio (GLR) chart with a variable sampling interval (VSI) to monitor simple linear profiles. The performance of the VSI GLR chart was compared with that of the GLR chart with a fixed sampling rate. The results indicated the superiority of the VSI GLR chart over all possible single step changes in the intercept, slope, and standard deviation. Also, an example was provided to explain the implementation of the VSI GLR chart.

Process monitoring using PCA-based GLR methods: A comparative study

Statistical process monitoring is a key requirement for many industrial processes. Many of these processes utilize Principal Component Analysis (PCA) in order to carry out statistical process monitoring due to its computational simplicity. Two fault detection charts that are commonly used with the PCA method are the Hotelling T2 and Q statistics. Although these charts are reasonably able to detect most shifts in the process mean, they may be unable to accurately detect other process faults, such as shifts in the process variance. Hypothesis testing methods such as the Generalized Likelihood Ratio (GLR) chart have been developed in order to detect different types of deviations from normal operating conditions, i.e., process faults. Although, GLR charts have shown superior performance in terms of fault detection compared to other existing techniques, literature has only examined the performance of a PCA-based GLR chart designed to detect shifts in the mean. Through a simulated synthetic example, this work evaluates the performance of PCA-based GLR charts designed to independently detect a shift in the mean, independently detect a shift in the variance, and simultaneously detect shifts in the mean and/or variance. The results show that in order to detect a shift in the mean or a shift in the variance, the GLR charts designed to independently detect either type of fault need to be implemented in parallel as they provide significantly lower missed detection rates, than a GLR chart designed to detect shifts in both the process mean and variance simultaneously. The GLR chart designed to simultaneously detect both a shift in the mean and variance does not perform as well as the other GLR charts, since two parameters (both the mean and variance) need to be estimated for this method while maximizing the GLR statistic, as opposed to just a single parameter for the other GLR charts. The practical applicability of the PCA-based GLR charts is demonstrated through the well-known benchmark Tennessee Eastman process, and through a case where autocorrelation is present in the data. Therefore, in order to detect shifts in the mean and/or variance, we recommend parallel implementation of the GLR charts designed to independently detect shifts in the mean, and independently detect shifts in the variance Parallel implementation of these two GLR charts aids with fault classification as well. This work also discusses the importance of selecting an appropriate window length of previous data to be used when computing the maximum likelihood estimates (MLEs) and maximizing the GLR statistic, keeping all fault detection criteria in mind in order to ensure that the desired fault detection results are obtained.

제1형 우측중도절단된 와이블 수명자료를 모니터링하는 GLR 관리도

와이블 분포는 제품의 물리적 특성을 잘 반영하고 고장률이 증가하거나 감소하는 경우 모두를 나타낼 수 있기 때문에 재품의 수명을 모델링할 때 가장 많이 사용되는 분포이다. 일반적으로 수명시험에서 표본의 모든 제품의 수명이 측정될 때까지 시험을 진행하는 것이 원칙이지만 여러 가지 시간적 또는 비용적인 제약으로 인해 시험을 중도절단 하는 경우가 빈번하게 발생한다. 이 논문에서는 제품의 수명 데이터가 와이블 분포를 따르고 제1형 우측중도절단된 경우 척도모수의 변화를 탐지하는 Weibull generalized likelihood ratio (GLR) 관리도 절차를 제안하였다. 모의실험을 실시하여 제안한 GLR 관리도와 기존에 제안된 두 가지 CUSUM 관리도의 성능을 average run length (ARL)을 이용하여 비교하였다. 그 결과 이 논문에서 제안한 GLR 관리도는 형상모수와 표본의 크기가 크고 중도절단율이 아주 높지 않은 경우 척도모수의 다양한 변화를 탐지하는데 효율적임을 알 수 있었다. Weibull distribution is a popular distribution for modeling lifetimes because it reflects the characteristics of failure adequately and it models either increasing or decreasing failure rates simply. It is a standard method of the lifetimes test to wait until all samples failed; however, censoring can occur due to some realistic limitations. In this paper, we propose a generalized likelihood ratio (GLR) chart to monitor changes in the scale parameter for type I right-censored Weibull lifetime data. We also compare the performance of the proposed GLR chart with two CUSUM charts proposed earlier using average run length (ARL). Simulation results show that the Weibull GLR chart is effective to detect a wide range of shift sizes when the shape parameter and sample size are large and the censoring rate is not too high.

Improved data-based fault detection strategy and application to distillation columns

Abstract Chemical and petrochemical processes require continuous monitoring to detect abnormal events and to sustain normal operations. Furthermore, process monitoring enhances productivity, efficiency, and safety in process industries. Here, we propose an innovative statistical approach that exploits the advantages of multiscale partial least squares (MSPLS) models and generalized likelihood ratio (GLR) tests for fault detection in processes. Specifically, we combine an MSPLS algorithm with wavelet analysis to create our modeling framework. Then, we use GLR hypothesis testing based on the uncorrelated residuals obtained from the MSPLS model to improve fault detection. We use simulated distillation column data to evaluate the MSPLS-based GLR chart. Results show that our MSPLS-based GLR method is more powerful than the PLS-based Q and GLR method and MSPLS-based Q method, especially in early detection of small faults with abrupt or incipient behavior.

A GLR Control Chart for Monitoring the Process Mean with Sequential Sampling

Variable sampling rate (VSR) control charts have been used where the sampling rate changes as a function of the data from the process. In this article we investigate a generalized likelihood ratio (GLR) control chart based on sequential sampling (the SS GLR chart). The objective of process monitoring is assumed to be the effective detection of a wide range of two-sided shifts in the process mean. The performance of the SS GLR chart is evaluated and compared with other control charts. The SS GLR chart has much better performance than that of the fixed sampling rate GLR chart. It is also shown that the overall performance of the SS GLR chart is better than that of the variable sampling interval (VSI) GLR chart and the VSR CUSUM chart. The SS GLR chart has the additional advantage that it requires fewer parameters to be specified than other VSR charts. The optimal parameter choices are given in the article, and regression equations are provided to find the limits for the SS GLR chart.

ZIP 공정을 관리하는 GLR 관리도

단위 영역의 결점수는 일반적으로 Poisson 분포를 가정한다. 이 Poisson 분포의 확장된 형태로 ZIP(zero-inflated Poisson) 분포를 고려할 수 있는데, 이 모형은 데이터에 0이 많이 관측되는 경우 잘 적합된다고 알려져 있다. 이 논문에서는 ZIP 분포를 따르는 공정을 관리하는 GLR(generalized likelihood ratio) 관리도 절차를 제안하고 있다. 또한 제안된 GLR 관리도의 효율을 기존에 제안된 CUSUM 관리도들과 비교하였다. 그 결과 제안된 GLR 관리도는 모수의 다양한 변화에 대해 효율이 좋거나 또는 효율이 크게 떨어지지 않았고, 특히 CUSUM 관리도에서 모수가 미리 설정한 방향과 다르게 변화했을 때 효율이 크게 나빠지는 문제를 해결할 수 있는 대안이라는 결론을 얻을 수 있었다. The number of nonconformities in a unit is commonly modeled by a Poisson distribution. As an extension of a Poisson distribution, a zero-inflated Poisson(ZIP) process can be used to fit count data with an excessive number of zeroes. In this paper, we propose a generalized likelihood ratio(GLR) chart to monitor shifts in the two parameters of the ZIP process. We also compare the proposed GLR chart with the combined cumulative sum(CUSUM) chart and the single CUSUM chart. It is shown that the overall performance of the GLR chart is comparable with CUSUM charts and is significantly better in some cases where the actual directions of the shifts are different from the pre-specified directions in CUSUM charts.

GLR Charts for Simultaneously Monitoring a Sustained Shift and a Linear Drift in the Process Mean

This paper considers the problem of monitoring the mean of a normally distributed process variable when the objective is to effectively detect both a sustained shift and a linear drift. The design and application of a generalized likelihood ratio (GLR) chart for simultaneously monitoring a sustained shift and a linear drift are evaluated. The GLR chart has the advantage that when we design this chart, we do not need to specify the size of the parameter change. The performance of the GLR chart is compared with that of other control charts, such as the standard cumulative sum (CUSUM) charts and the cumulative score (CUSCORE) charts. And we compare the proposed GLR chart with the GLR charts designed for monitoring only a sustained shift and for monitoring only a linear drift. Finally, we also compare the proposed GLR chart with the chart combinations. We show that the proposed GLR chart has better overall performance for a wide range of shift sizes and drift rates relative to other control charts, when a special ause produces a sustained shift and/or a linear drift in the process mean.

An Individuals Generalized Likelihood Ratio Control Chart for Monitoring Linear Profiles

This paper investigates a generalized likelihood ratio (GLR) control chart for detecting sustained changes in the parameters of linear profiles when individual observations are sampled. The control charts usually used for monitoring linear profiles are based on taking a sample of n observations at each sampling time point, where n is large enough that a regression model can be fitted at each sampling point using these n observations. For this sampling scenario, it has been shown that a GLR control chart has many advantages over other control chart schemes in terms of convenience of design, fast detection of process changes, and useful diagnostic aids. However, in many applications, it may not be convenient or possible to take a sample larger than n = 1. Therefore, it is desirable to develop some control chart to monitor profile data with individual observations (n = 1) at each sampling point. In this paper, we consider a GLR control chart based on individual observations and show that it has certain advantages compared with the GLR chart based on groups of observations. An important advantage of GLR control charts is that the only design parameter that needs to be specified in order to use a GLR chart is the control limit, and here, control limits for linear profiles up to eight regression coefficients are provided for convenient use by practitioners. Copyright © 2013 John Wiley & Sons, Ltd.

The Design of the Variable Sampling Interval Generalized Likelihood Ratio Chart for Monitoring the Process Mean

This paper considers the problem of monitoring a normally distributed process variable in which some sustained shift in the mean may occur. A generalized likelihood ratio (GLR) control chart with a variable sampling interval (VSI) scheme is proposed to quickly detect a wide range of such shifts. The performance of the VSI GLR chart is evaluated, and the results show that using the VSI feature can greatly reduce the expected detection time. Design methodology for the VSI GLR chart is provided so that practitioners can easily use this chart in applications. An illustrative example is given to explain how to apply the VSI GLR chart. Copyright © 2013 John Wiley & Sons, Ltd.

A GLR Control Chart for Monitoring the Mean Vector of a Multivariate Normal Process

This paper develops a statistical process control (SPC) chart based on a generalized likelihood ratio (GLR) statistic to monitor the mean vector of a multivariate normal process. The performance of the GLR chart is compared with the performance of the Hotelling χ2 chart, the multivariate exponentially weighted moving average (MEWMA) chart, and a multi-MEWMA combination. Results show that the Hotelling χ2 chart and the MEWMA chart are only effective for a small range of shift sizes in the mean vector, while the GLR chart and some carefully designed multi-MEWMA combinations can give similarly better overall performance in detecting a wide range of shift magnitudes. Unlike most of these other options, the GLR chart does not require specification of tuning-parameter values by the user. The GLR chart also has the advantage in process diagnostics: at the time of a signal, estimates of change-point and out-of-control mean vector are immediately available to the user. All these advantages of the GLR chart make it a favorable option for practitioners. For the design of the GLR chart, a series of easy-to-use equations are provided to users for calculating the control limit to achieve a desired in-control performance.

The Design of GLR Control Charts for Monitoring the Process Mean and Variance

This paper investigates generalized likelihood ratio (GLR) control charts for the problem of monitoring the mean and variance of a normally distributed process variable when the objective is to effectively detect both small and large shifts in these parameters. It is shown that the GLR charts are generally as effective at detecting a wide range of shift sizes as other options, such as the traditional practice of using a combination of a control chart for the mean and a control chart for the variance. Using a combination of control charts requires that multiple control-chart design parameters be specified. Although multiple design parameters allow for the charts to be tuned to be more sensitive to certain shifts that may be of particular interest, they complicate the process of designing these charts. The GLR chart for detecting a change in the mean or an increase in the variance does not have any control-chart parameters that need to be specified other than the control limit. The GLR chart for the case in which decreases in the variance are also of interest has one design parameter in addition to the control limit. Tables of control limits corresponding to specified values of the in-control average number of observations to signal are provided, so it is very easy to design the GLR chart for use in applications.

The Monitoring of Linear Profiles with a GLR Control Chart

In this paper, we consider the problem of monitoring a linear functional relationship between a response variable and one or more explanatory variables (a linear profile). The design and application of a generalized likelihood ratio (GLR) control chart are discussed. The likelihood ratio test of the GLR chart is generalized over the regression coefficients, the variance of the error term, and the possible change point. The performance of the GLR chart is compared with various existing control charts. We show that the overall performance of the GLR chart is much better than other options in detecting a wide range of shift sizes. The existing control charts designed for certain shifts that may be of particular interest have several chart parameters that need to be specified by the user, which makes the design of such control charts more difficult. The GLR chart is very simple to design, as it is invariant to the choice of design matrix and the values of in-control parameters. Therefore, there is only one design parameter (the control limit) that needs to be specified. Another advantage of the GLR chart is its built-in diagnostic aids that provide estimates of both the change point and the values of linear profile parameters.

A Binomial GLR Control Chart for Monitoring a Proportion

This paper considers a control chart for monitoring a process proportion based on a generalized likelihood ratio (GLR) statistic. The objective is to effectively detect a wide range of shift sizes. The GLR statistic is obtained from a moving window of past sample statistics that follow a binomial distribution. The Phase II performance of this chart is evaluated using the steady state average number of observations to signal (SSANOS) for detecting sustained increases in the proportion. Comparisons of the binomial GLR chart to the Shewhart np-chart, the individual binomial cumulative sum (CUSUM) chart, and combinations of multiple binomial CUSUM charts show that the overall performance of the binomial GLR chart is at least as good as these other options. Moreover, unlike the other charts, the binomial GLR chart has an advantage that it does not require users to specify multiple charting parameters that may be difficult to obtain, and this makes it easier for the GLR chart to be designed for practical applications.

A Generalized Likelihood Ratio Chart for Monitoring Bernoulli Processes

This paper considers the problem of monitoring the proportion p of nonconforming items when a continuous stream of Bernoulli observations is available and the objective is to effectively detect a wide range of increases in p. The proposed control chart is based on a generalized likelihood ratio (GLR) statistic obtained from a moving window of past Bernoulli observations. The Phase II performance of this chart in detecting sustained increases in p is evaluated using the steady state average number of observations to signal. Comparisons of the Bernoulli GLR chart to the Shewhart CCC‐r chart, the Bernoulli cumulative sum chart, and the Bernoulli exponentially weighted moving average chart show that the overall performance of the Bernoulli GLR chart is better than its competitors. In addition, methods are provided for designing the Bernoulli GLR chart so that this chart can be easily applied in practice. Copyright © 2012 John Wiley & Sons, Ltd.

The GLR Chart for Poisson Process with Individual Observations

A GLR (generalized likelihood ratio) chart for Poisson distributed process with individual observations is proposed and the design procedure of the GLR chart is discussed. The performance of the GLR charts is compared to the exponentially weighted moving average (EWMA) chart and the GWMA chart. The numerical experiments show that the GLR chart has comparable performance as the other two charts. However, the GLR chart is much easier to design and implement since there are more design parameters in these two charts.

A Generalized Likelihood Ratio Control Chart for Monitoring the Process Mean Subject to Linear Drifts

This article considers the problem of monitoring a normally distributed process variable when a special cause may produce a time‐varying linear drift in the mean. The design and application of a generalized likelihood ratio (GLR) control chart for drift detection are evaluated. The GLR drift chart does not require specification of any tuning parameters by the practitioner and has the advantage that, at the time of the signal, estimates of both the change point and the drift size are immediately available. An equation to accurately approximate the control limit is provided. The performance of the GLR drift chart is compared with that of other control charts such as a standard cumulative sum chart and a cumulative score chart designed for drift detection. We also compare the GLR chart designed for drift detection with the GLR chart designed for sustained shift detection because both of them require only a control limit to be specified. In terms of the expected time for detection and in terms of the bias and mean squared error of the change‐point estimators, the GLR drift chart has better performance for a wide range of drift rates relative to the GLR shift chart when the out‐of‐control process is truly a linear drift. Copyright © 2012 John Wiley & Sons, Ltd.