Estimated Control Limits Research Articles

Fault Detection and Isolation of Non-Gaussian and Nonlinear Processes Based on Statistics Pattern Analysis and the k-Nearest Neighbor Method.

Only low-order information of process data (i.e., mean, variance, and covariance) was considered in the principal component analysis (PCA)-based process monitoring method. Consequently, it cannot deal with continuous processes with strong dynamics, nonlinearity, and non-Gaussianity. To this aim, the statistics pattern analysis (SPA)-based process monitoring method achieves better monitoring results by extracting higher-order statistics (HOS) of the process variables. However, the extracted statistics do not strictly follow a Gaussian distribution, making the estimated control limits in Hotelling-T2 and squared prediction error (SPE) charts inaccurate, resulting in unsatisfactory monitoring performance. In order to solve this problem, this paper presents a novel process monitoring method using SPA and the k-nearest neighbor algorithm. In the proposed method, first, the statistics of process variables are calculated through SPA. Then, the k-nearest neighbor (kNN) method is used to monitor the extracted statistics. The kNN method only uses the paired distance of samples to perform fault detection. It has no strict requirements for data distribution. Hence, the proposed method can overcome the problems caused by the non-Gaussianity and nonlinearity of statistics. In addition, the potential of the proposed method in early fault detection or safety alarm and fault isolation is explored. The proposed method can isolate which variable or its statistic is faulty. Finally, the numerical examples and Tennessee Eastman benchmark process illustrate the effectiveness of the proposed method.

Estimating X¯ Statistical Control Limits for Any Arbitrary Probability Distribution Using Re-Expressed Truncated Cumulants

Shewhart X¯ control charts commonly used for monitoring the mean of a process may be inaccurate or perform poorly when the subgroup size is small or the distribution of the process variable is skewed. Truncated saddlepoint distributions can increase the accuracy of estimated control limits by including higher order moments/cumulants in their approximation, yet this distribution may not exist in the lower tail, and thus the lower control limit may not exist. We introduce a novel modification in which some usually truncated higher-order cumulants are re-expressed as functions of lower-order cumulants estimated from data in a manner that ensures the existence of the truncated saddlepoint distribution over the complete domain of the random variable. The accuracy of this approach is tested in cases where the cumulants are assumed either known or estimated from sample data, and demonstrated in a healthcare application.

An Integrated Topology of Three-Port DC-DC Converter for PV-Battery Power Systems

This paper introduces an integrated topology of isolated three-port dc-dc converter (TPC) to interface Photovoltaic (PV) and battery for standalone system. To guarantee zero circulating current flow between different ports while permitting bidirectional power flow at the battery port, auxiliary switches are utilized with the active bridges of the proposed TPC topology. In addition, a simple switching scheme based on Pulse Width Modulation (PWM) is proposed for the input ports of the TPC. This action eliminates the need for the phase shift modulation used for conventional isolated TPCs. While the PV port is controlled to track the maximum power point, the battery port is managed to regulate the load voltage. The paper presents the detailed mathematical analysis of the different operation modes of the proposed integrated TPC topology and the estimated control limits during charging and discharging of the battery. Consequently, dynamic limiters for the duty cycle of each port are set. The proposed integrated TPC based PV-battery system is simulated using PSCAD/EMTDC software package to validate the analysis of different operation modes and to assess the dynamic performance of the system. Moreover, experimental results are presented to authenticate different operation modes of the proposed TPC topology and to evaluate the dynamic behavior of the integrated PV-battery system.

Phase II exponential charts for monitoring time between events data: performance analysis using exact conditional average time to signal distribution

In this paper, we study the performance properties of the phase II exponential chart with an unknown in-control (IC) rate parameter, used to monitor high-yield processes. The average time to signal (ATS) is used as the chart performance criterion instead of the usual average run length (ARL). Based on the IC conditional ATS (CATS) distribution, we examine the properties of both the equal-tailed and the ATS-unbiased exponential charts with estimated control limits and adjust the limits so that a nominal IC ATS performance is achieved. Two perspectives are investigated: the unconditional, under which the average of IC CATS distribution is set equal to a nominal ATS and, the conditional so that the IC CATS is set to or exceed a nominal ATS with a high probability. It is shown that the ATS-unbiased exponential chart under the conditional perspective has a better IC performance.

Outliers Detection Models in Shewhart Control Charts; an Application in Photolithography: A Semiconductor Manufacturing Industry

Shewhart control charts with estimated control limits are widely used in practice. However, the estimated control limits are often affected by phase-I estimation errors. These estimation errors arise due to variation in the practitioner’s choice of sample size as well as the presence of outlying errors in phase-I. The unnecessary variation, due to outlying errors, disturbs the control limits implying a less efficient control chart in phase-II. In this study, we propose models based on Tukey and median absolute deviation outlier detectors for detecting the errors in phase-I. These two outlier detection models are as efficient and robust as they are distribution free. Using the Monte-Carlo simulation method, we study the estimation effect via the proposed outlier detection models on the Shewhart chart in the normal as well as non-normal environments. The performance evaluation is done through studying the run length properties namely average run length and standard deviation run length. The findings of the study show that the proposed design structures are more stable in the presence of outlier detectors and require less phase-I observation to stabilize the run-length properties. Finally, we implement the findings of the current study in the semiconductor manufacturing industry, where a real dataset is extracted from a photolithography process.

Cumulative Sum Chart Modeled under the Presence of Outliers

Cumulative sum control charts that are based on the estimated control limits are extensively used in practice. Such control limits are often characterized by a Phase I estimation error. The presence of these errors can cause a change in the location and/or width of control limits resulting in a deprived performance of the control chart. In this study, we introduce a non-parametric Tukey’s outlier detection model in the design structure of a two-sided cumulative sum (CUSUM) chart with estimated parameters for process monitoring. Using Monte Carlo simulations, we studied the estimation effect on the performance of the CUSUM chart in terms of the average run length and the standard deviation of the run length. We found the new design structure is more stable in the presence of outliers and requires fewer amounts of Phase I observations to stabilize the run-length performance. Finally, a numerical example and practical application of the proposed scheme are demonstrated using a dataset from healthcare surveillance where received signal strength of individuals’ movement is the variable of interest. The implementation of classical CUSUM shows that a shift detection in Phase II that received signal strength data is indeed masked/delayed if there are outliers in Phase I data. On the contrary, the proposed chart omits the Phase I outliers and gives a timely signal in Phase II.

Conditional analysis of Phase II exponential chart for monitoring times to an event

ABSTRACTIt is now well known that the error in estimating parameter(s) adversely affects the performance of control charts. The charts with estimated control limits are usually assessed by some of characteristics of unconditional run length (URL) distribution. However, it is now under criticism because the unconditional analysis based on URL does not give the full impression of the chart’s performance conditioning on a given Phase I sample. In recent years, the focus has been shifted from unconditional performance analysis to conditional analysis which is carried out in terms of conditional run length (CRL) and its associated characteristics and provides deep understanding of conditional performance of the chart given a Phase I sample. In this article, Phase II exponential charts are considered to monitor times to an event and their performance are examined conditional on given Phase I sample by considering several indicators based on conditional average run length (CARL) distribution. In this respect, the exact expressions of the distribution functions of for the estimated exponential control charts are derived and several statistical constants are computed which could be helpful in evaluating and designing the chart.

Two perspectives for designing a phase II control chart with estimated parameters: The case of the Shewhart Chart

The impact of parameter estimation on control charts has been studied with great interest in the recent literature. The estimated control limits affect chart performance, often negatively, relative to the known parameter case. Guided by the need to design control charts with a specified in-control performance, so as to avoid excessive false alarms, two major perspectives have been advocated. Under the first, the so-called unconditional perspective, control limits are determined so that the in-control unconditional average run length equals a specified value. However, this perspective does not account for the so-called practitioner-to-practitioner variability inherent in control charts using parameter estimates. Hence, researchers have considered a second perspective, called the conditional perspective, under which the so-called exceedance probability criterion is used to calculate the control limits so that the in-control conditional average run-length is at least equal to a specified value with a given high probability. These perspectives, in turn, have led to adjustments to the traditional control limits, and various methods have been proposed to calculate the adjusted limits. In this article, we consider these two perspectives and examine the performance of the various proposed adjustments to the control limits for the Shewhart chart. We also provide a simple adjustment formula under the conditional perspective that can be used in practice without too many resources, which works well relative to the other methods. In addition, we derive an exact formula for calculating the adjusted limit coefficient (proposed in an earlier work using bootstrapping) which does not require any model fitting. For completeness, the adjusted limits calculated under one perspective are examined under the other in terms of performance. A summary and some practical recommendations are provided.

An Efficient Phase I Analysis of Linear Profiles with Application in Photo-Voltaic System

In many manufacturing industries, quality of an item depends on different characteristics and their relationship may be modeled by profiles. For instance, in photo-voltaic (PV) system, output voltage is dependent on the capacitance level of capacitors, used in the Z-source inverter. Control chart is a dynamic tool that works under two different phases namely Phase I and Phase II. The unknown parameters are estimated in Phase I, while Phase II focuses on monitoring of the process using estimated control limits from Phase I. In this study, we have investigated Phase I monitoring of linear profile parameters under several sampling approaches including ranked set sampling (RSS), double RSS, median RSS (MRSS), double MRSS, extreme RSS (ERSS) and double ERSS. A comparative analysis on the performance of existing and proposed schemes is performed in terms of probability to signal. The results advocate that the proposed methods under different ranked set schemes offer superior detection abilities as compared to the existing schemes for varying shifts in profile parameters (intercept, slope and error variance). Moreover, a real application related to PV system is included to show the practical demonstration of the profile monitoring (for the output voltage in relation to capacitance level) under proposed schemes.

기하분포에 기초한 관리도에서 베이즈추정량과 최대우도추정량 사용의 성능 비교

Charts based on geometric distribution are effective to monitor the proportion of nonconforming items in high-quality processes where the in-control proportion nonconforming is low. The implementation of this chart is often based on the assumption that in-control proportion nonconforming is known or accurately estimated. However, accurate parameter estimation is very difficult and may require a larger sample size than that available in practice for high-quality process where the proportion of nonconforming items is very small. An inaccurate estimate of the parameter can result in estimated control limits that cause unreliability in the monitoring process. The maximum likelihood estimator (MLE) is often used to estimate in-control proportion nonconforming. In this paper, we recommend a Bayes estimator for the in-control proportion nonconforming to incorporate practitioner knowledge and avoid estimation issues when no nonconforming items are observed in the Phase I sample. The effects of parameter estimation on the geometric chart and the geometric CUSUM chart are considered when the MLE and the Bayes estimator are used. The results show that chart performance with estimated control limits based on the Bayes estimator is generally better than that based on the MLE.

Performance of cumulative count of conforming chart of variable sampling intervals with estimated control limits

The control chart with cumulative count of conforming (CCC) items is very useful for high-quality processes. The current research of CCC chart performance with parameter estimation is focusing on the fixed sampling interval (FSI) scheme. In this paper, we investigate the effect of performance of CCC chart with variable sampling intervals with estimated control limits. The best design parameters of CCC chart with variable sampling intervals with known process nonconforming rate p0 are verified when such p0 is estimated with Bayesian estimator. The average time to signal (ATS) and its standard deviation (SDATS) are calculated when the high-quality process is in-control and out-of-control. Its efficiency is compared with that of FSI CCC chart, which shows a great cost reduction in practice.

The Negative Binomial Exponentially Weighted Moving Average Chart with Estimated Control Limits

The control chart based on the compound Poisson distribution (the negative binomial exponentially weighted moving average (EWMA) chart) has been shown to be more effective than the c‐chart to monitor the wafer nonconformities in semiconductor production process. The performance of the negative binomial EWMA chart is generally evaluated with the assumption that the process parameters are known. However, in many control chart applications, the process parameters are usually unknown and are required to be estimated. For an accurate parameter estimate, a very large sample size may be required, which is seldom available in the applications. This article investigates the effect of parameter estimation on the run length properties of the negative binomial EWMA charts. Using a Markov chain approach, we show that the performance of the negative binomial EWMA chart is affected when parameters are estimated compared with the known‐parameter case. We also provide recommendations regarding phase I sample sizes, smoothing constant and clustering parameter. The sample size must be quite large for the in‐control chart performance to be close to that for the known‐parameter case. Finally, a wafer process example has been used to highlight the practical implications of estimation error and to offer advice to practitioners when constructing/analysing a phase I sample. Copyright © 2013 John Wiley & Sons, Ltd.

Exponential CUSUM Charts with Estimated Control Limits

Exponential CUSUM charts are used in monitoring the occurrence rate of rare events because the interarrival times of events for homogeneous Poisson processes are independent and identically distributed exponential random variables. In these applications, it is assumed that the exponential parameter, i.e. the mean, is known or has been accurately estimated. However, in practice, the in‐control mean is typically unknown and must be estimated to construct the limits for the exponential CUSUM chart. In this article, we investigate the effect of parameter estimation on the run length properties of one‐sided lower exponential CUSUM charts. In addition, analyzing conditional performance measures shows that the effect of estimation error can be significant, affecting both the in‐control average run length and the quick detection of process deterioration. We also provide recommendations regarding phase I sample sizes. This sample size must be quite large for the in‐control chart performance to be close to that for the known parameter case. Finally, we provide an industrial example to highlight the practical implications of estimation error, and to offer advice to practitioners when constructing/analyzing a phase I sample. Copyright © 2013 John Wiley & Sons, Ltd.

Robust control charts for controlling location parameter

Control charts are very effective tools used for detecting whether there is any assignable cause of variation. They are usually developed under the assumption of independent and normally distributed data, an assumption rarely true in practice, and implemented with estimated control limits. But in general, we essentially want to control the process mean value and the process standard deviation, independently of the data distribution. In order to monitor these parameters, it thus seems sensible to advance with control charts based on robust statistics, because these statistics are expected to be more resistant to moderate changes in the underlying process distribution. In this paper, we will propose some alternatives control charts for controlling location parameters based on some robust estimators. We will also show the performance of the proposed control charts and compare them with some existing robust control chart.

The median chart with estimated parameters

A modified median chart with estimated control limits is proposed for monitoring the mean value of a normal process. The estimates for the nominal process parameters used to set up the control limits in Phase I are the average of the sample medians and the average of the sample ranges of m initial subgroups of size n. The run length properties of the chart are examined, and the obtained results lead us to conclude the proposed chart present an interesting RL performance (at least for an adequate choice of the chart parameter used to determine the control limits). Control chart parameters are provided for a specific in-control average run length of 370.4 for several combinations of n and m, which facilitates the use of the chart by practitioners, and an illustrative example of implementation of the chart is presented. Some recommendations are also given about the minimum number of initial subgroups m that must be considered to implement the chart with the constant parameter used in the case of known nominal values, and in order to obtain the expected performance.

Geometric Charts with Estimated Control Limits

The geometric control chart has been shown to be more effective than p and np‐charts for monitoring the proportion of nonconforming items, especially for high‐quality Bernoulli processes. When implementing a geometric control chart, the in‐control proportion nonconforming is typically unknown and must be estimated. In this article, we used the standard deviation of the average run length (SDARL) and the standard deviation of the average number of inspected items to signal, SDARL*, to show that much larger phase I sample sizes are needed in practice than implied by previous research. The SDARL (or SDARL*) was used because practitioners would estimate the control limits based on different phase I samples. Thus, there would be practitioner‐to‐practitioner variability in the in‐control ARL (or ARL*). In addition, we recommend a Bayes estimator for the in‐control proportion nonconforming to take advantage of practitioners' knowledge and to avoid estimation problems when no nonconforming items are observed in the phase I sample. If the in‐control proportion nonconforming is low, then the required phase I sample size may be prohibitively large. In this case, we recommend an approach to identify a more informative continuous variable to monitor. Copyright © 2012 John Wiley & Sons, Ltd.

Improved feature-based test statistic for assessing suitability of the preliminary samples for constructing control limits of $$ \bar{X} $$ chart

In spite of using a large number of subgroups (m) of small samples (n), the estimated control limits of \( \bar{X} \)chart in phase I can be erroneous unless the preliminary samples are drawn from a stable process. As a result, the performance of the chart in phase II can be significantly affected. The pattern in the \( \bar{X} \)chart, exhibited by the plots of the subgroup averages of the preliminary samples, will be different depending on stability and instability of the process while the preliminary samples were collected. Based on this concept, a new feature-based test statistic (FTS) is proposed for evaluating suitability of the preliminary samples for the designing of the \( \bar{X} \)chart. The FTS, for given m, approximately follows \( N[1,{\text{ SD(}}m{)]} \), where SD(m) is a function of m. The goodness of the approximation and effectiveness of the test are evaluated using simulated data. The results show that both are satisfactory for m > =48. The proposed statistic is also quite effective in detecting unstable process condition resulting in a cyclic pattern. The computation of FTS involves some complexities. However, now-a-days computers are widely available and so computation difficulty may not be a problem.

A test for suitability of the preliminary samples for constructing control limits of X¯ chart

Unless the preliminary m subgroups of small samples are drawn from a stable process, the estimated control limits of X¯ chart in phase I can be erroneous, due to which the performance of the chart in phase II can be significantly affected. In this work, a quantitative approach based on extraction of the shape features of control chart patterns in the X¯ chart is proposed for evaluating the stability of the process mean, while the preliminary samples were drawn and thus, the subjectivity associated with the visual analysis of the patterns is eliminated. The effectiveness of the test procedure is evaluated using simulated data. The results show that the proposed approach can be very effective for m≥48. The power of the test can be improved by identifying a new feature that can more efficiently discriminate the cyclic pattern of smaller periodicity from the natural pattern and by redefining the test statistic.

THE EXACT RUN LENGTH DISTRIBUTION AND DESIGN OF THE S2 CHART WHEN THE IN-CONTROL VARIANCE IS ESTIMATED

When monitoring the process variability, it is a common practice that a Phase I data set is used to estimate the unknown in-control process standard deviation σ0 or variance [Formula: see text] to set up the control limits, then monitoring proceeds. Once the process is considered to be in-control, the estimated control limits are assumed as fixed. This practice ignores the effect of estimating the unknown in-control process variance [Formula: see text]. In this paper, we derive the exact run length distribution of the S2 control chart when the in-control process variance [Formula: see text] is estimated and find that m = 200 or more Phase I samples are needed to neglect the effect of using estimated control limits. New control limits when m is small are also derived.

Design issues for adaptive control charts

AbstractAdaptive control charts allow the components of the quality‐monitoring scheme to vary in order to obtain improved performance over non‐adaptive control charts. Research has centered on components such as the sample size, time between samples, warning limits, and control limits and has recommended a variety of schemes, many of which are optimal in some sense. In practice, there are many other adaptive schemes that are near optimal, which will still yield considerable improvement over non‐adaptive control charts. In addition, the impact of parameter estimation on adaptive control chart performance must be taken into consideration. Based on the simulation results shown here, adaptive control charts should only be used for mature processes, where a sufficient amount of Phase I data have been obtained to ensure that the estimated control limits are accurate. When evaluating control chart performance, we consider initial state performance measures for simplicity and note that the conclusions obtained here apply to steady‐state performance measures. The evaluation of performance measures is easily handled by the Markov chain approach detailed in the Appendix. Copyright © 2008 John Wiley & Sons, Ltd.