Identification of patterns in control charts for processes with statistically correlated noise

In real industrial scenarios, if the quality characteristics of a continuous or batch production process are monitored using Shewhart control charts, there could be a large number of false alarms about the process going out of control. This is because these control charts assume that the inherent noise of the monitored process is normally, independently and identically distributed, although the assumption of independence is not always correct for continuous and batch production processes. This paper presents three control chart pattern recognition systems where the inherent disturbance is assumed to be stationary. The systems use the first-order autoregressive (AR(1)), moving-average (MA(1)) and autoregressive moving-average (ARMA(1,1)) models. A special pattern generation scheme is adopted to ensure generality, randomness and comparability, as well as allowing the further categorisation of the studied patterns. Two different input representation techniques for the recognition systems were studied. These gave nearly the same performance for the MA(1) and ARMA(1,1) models, while the raw data yielded the highest accuracies when AR(1) was used. The effect of autocorrelation on the pattern recognition capabilities of the developed models was studied. It was observed that Normal and Upward Shift patterns were the most affected.     

基于神经网络的SPC/EPC整合过程监测方法研究

为解决统计过程控制(SPC)/工程过程调整(EPC)整合引起的传统SPC控制图监测异常扰动效率低的问题,提出了采用神经网络技术监测SPC/EPC整合过程的策略,并对神经网络模型结构和参数设置进行分析,构建过程输入、过程输出及两者的协方差为输入参数,异常扰动发生与否为输出参数的3层神经网络模型。为验证该方法的性能,进行了大量的比较实验:即对相同的样本,分别采用Shewhar图、CUSUM图和上述神经网络模型进行监测。实验结果表明:神经网络模型能准确监测幅度大于2的阶跃扰动和大于2的过程漂移,平均运行步长(ARL)为1;传统SPC监测技术只能较准确地(监测率大于90%)监测幅度大于5的阶跃扰动和大于2的过程漂移,ARL大于2。与传统监测方法相比,该方法能快速有效地监测异常扰动的发生。     

Phase II monitoring of covariance stationary autocorrelated processes

Statistical process control charts are intended to assist operators in detecting process changes. If a process change does occur, the control chart should detect the change quickly. Owing to the recent advancements in data retrieval and storage technologies, today's industrial processes are becoming increasingly autocorrelated. As a result, in this paper we investigate a process‐monitoring tool for autocorrelated processes that quickly responds to process mean shifts regardless of the magnitude of the change, while supplying useful diagnostic information upon signaling. A likelihood ratio approach was used to develop a phase II control chart for a permanent step change in the mean of an ARMA (p, q) (autoregressive‐moving average) process. Monte Carlo simulation was used to evaluate the average run length (ARL) performance of this chart relative to that of the more recently proposed ARMA chart. Results indicate that the proposed chart responds more quickly to process mean shifts, relative to the ARMA chart, while supplying useful diagnostic information, including the maximum likelihood estimates of the time and the magnitude of the process shift. These crucial change point diagnostics can greatly enhance the special cause investigation. Copyright © 2010 John Wiley & Sons, Ltd.     

Parameter Selection for a Robust Tracking Signal

Tracking signals use past forecast errors to monitor and control a forecasting process. In this study, the cumulative‐sum tracking signal and the smoothed‐error tracking signal are evaluated on their ability to aid in shift (process upset) detection. The moving‐centerline EWMA control chart technique is coupled with these tracking signals to enhance the monitoring of autocorrelated processes. The analysis characterizes two prevalent time series models: AR(1) and ARMA(1,1). The goal of this paper is to explore the capabilities of the tracking signals and the moving‐centerline EWMA when the smoothing constants are varied and a shift is introduced into the process. The tracking signals are evaluated based on average run length (ARL) and false alarm rate (FA). Copyright © 2005 John Wiley & Sons, Ltd.     

A Two‐Sided Cumulative Sum Chart for First‐Order Integer‐Valued Autoregressive Processes of Poisson Counts

Count data processes are often encountered in manufacturing and service industries. To describe the autocorrelation structure of such processes, a Poisson integer‐valued autoregressive model of order 1, namely, Poisson INAR(1) model, might be used. In this study, we propose a two‐sided cumulative sum control chart for monitoring Poisson INAR(1) processes with the aim of detecting changes in the process mean in both positive and negative directions. A trivariate Markov chain approach is developed for exact evaluation of the ARL performance of the chart in addition to a computationally efficient approximation based on bivariate Markov chains. The design of the chart for an ARL‐unbiased performance and the analyses of the out‐of‐control performances are discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Design of Multiple Cause-Selecting Charts for Multistage Processes with Model Uncertainty

The cause-selecting chart (CSC) is an effective statistical process control tool for monitoring multistage processes. The multiple cause-selecting chart (MCSC) is the further development of the CSC, which deals with the case when the output measure is a function of multiple input measures. In practice, the model relating the input and output measures often needs to be estimated before the MCSC is implemented. However, the traditional design of MCSCs does not take parameter uncertainties into account when estimating the control limits. The actual false-alarm rate can substantially differ from what is expected. This article presents the design and implementation of MCSCs using prediction limits to account for parameter uncertainties. These limits are developed using two types of procedures: the least-squares estimation and principal component regression. The simulation results show that the prediction limits are quite effective in terms of maintaining a desired false-alarm rate.     

Automatic identification of time-series models from long autoregressive models

Identification is the selection of the model type and of the model order by using measured data of a process with unknown characteristics. If the observations themselves are used, it is possible to identify automatically a good time-series model for stochastic data. The selected model is an adequate representation of the statistically significant spectral details in the observed process. Sometimes, identification has to be based on many less than N characteristics of the data. The reduced statistics information is assumed to consist of a long autoregressive (AR) model. That AR model has to be used for the estimation of moving average (MA) and of combined ARMA models and for the selection of the best model orders. The accuracy of ARMA models is improved by using four different types of initial estimates in a first stage. After a second stage, it is possible to select automatically which initial estimates were most favorable in the present case by using the fit of the estimated ARMA models to the given long AR model. The same principle is used to select the best type of the time-series models and the best model order. No spectral information is lost in using only the long AR representation instead of all data. The quality of the model identified from a long AR model is comparable to that of the best time-series model that can be computed if all observations are available.     

Finite sample properties of ARMA order selection

The cost of order selection is defined as the loss in model quality due to selection. It is the difference between the quality of the best of all available candidate models that have been estimated from a finite sample of N observations and the quality of the model that is actually selected. The order selection criterion itself has an influence on the cost because of the penalty factor for each additionally selected parameter. Also, the number of competitive candidate models for the selection is important. The number of candidates is, of itself, small for the nested and hierarchical autoregressive/moving average (ARMA) models. However, intentionally reducing the number of selection candidates can be beneficial in combined ARMA(p,q) models, where two separate model orders are involved: the AR order p and the MA order q. The selection cost can be diminished by creating a nested sequence of ARMA(r,r-1) models. Moreover, not evaluating every combination (p,q) of the orders considerably reduces the required computation time. The disadvantage may be that the true ARMA(p,q) model is no longer among the nested candidate models. However, in finite samples, this disadvantage is largely compensated for by the reduction in the cost of order selection by considering fewer candidates. Thus, the quality of the selected model remains acceptable with only hierarchically nested ARMA(r,r-1) models as candidates.     

Monitoring Process Variability for Stationary Process Data

Processes that arise naturally, for example, from manufacturing or the environment, often exhibit complicated autocorrelation structures. When monitoring such a process for changes in variance, accounting for that structure is critical. While charts for monitoring the variance of processes of independent observations and some specific autocorrelated processes have been proposed in the past, the chart presented in this article can handle a general stationary process. The performance of the proposed chart was examined through simulations for the first‐order autoregressive and first‐order autoregressive‐moving average processes and demonstrated with examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Relative contributions of aleatory and epistemic uncertainty sources in time series prediction

This paper develops a novel computational framework to compute the Sobol indices that quantify the relative contributions of various uncertainty sources towards the system response prediction uncertainty. In the presence of both aleatory and epistemic uncertainty, two challenges are addressed in this paper for the model-based computation of the Sobol indices: due to data uncertainty, input distributions are not precisely known; and due to model uncertainty, the model output is uncertain even for a fixed realization of the input. An auxiliary variable method based on the probability integral transform is introduced to distinguish and represent each uncertainty source explicitly, whether aleatory or epistemic. The auxiliary variables facilitate building a deterministic relationship between the uncertainty sources and the output, which is needed in the Sobol indices computation. The proposed framework is developed for two types of model inputs: random variable input and time series input. A Bayesian autoregressive moving average (ARMA) approach is chosen to model the time series input due to its capability to represent both natural variability and epistemic uncertainty due to limited data. A novel controlled-seed computational technique based on pseudo-random number generation is proposed to efficiently represent the natural variability in the time series input. This controlled-seed method significantly accelerates the Sobol indices computation under time series input, and makes it computationally affordable.     

Time-series analysis if data are randomly missing

Maximum-likelihood (ML) theory presents an elegant asymptotic solution for the estimation of the parameters of time-series models. Unfortunately, the performance of ML algorithms in finite samples is often disappointing, especially in missing-data problems. The likelihood function is symmetric with respect to the unit circle for the estimated zeros of time-series models. As a consequence, the unit circle is either a local maximum or a local minimum in the likelihood of moving-average (MA) models. This is a trap for nonlinear optimization algorithms that often converge to poor models, with estimated zeros precisely on the unit circle. With ML estimation, it is much easier to estimate a long autoregressive (AR) model with only poles. The parameters of that long AR model can then be used to estimate MA and autoregressive moving-average (ARMA) models for different model orders. The accuracy of the estimated AR, MA, and ARMA spectra is very good. The robustness is excellent as long as the AR order is less than 10 or 15. For still-higher AR orders until about 60, the possible convergence to a useful model will depend on the missing fraction and on the specific properties of the data at hand.     

Shanks' Method for Dynamic Phasor Estimation

A new algorithm for phasor estimation is proposed. It is based on a signal model that allows amplitude and phase dynamic variations. An autoregressive moving average (ARMA) model is assumed for the oscillating signal. Its autoregressive part is fixed, and it is defined only by the nominal fundamental frequency. Its best moving average parameters are estimated with Shanks' method. These parameters provide the key information from which the phasor state vector is estimated through the partial fraction expansion of the ARMA rational polynomial. These estimates could be useful, not only for the monitoring and controlling of the power system, but also for discriminating between a fault and an oscillation state.     

Fatigue failure identification by time series model

In this paper an effective way of modeling stochastic strain data, for a fatigue failure identification based on the output response instead of the input is presented. Three steel specimens were tested to fatigue failure under stochastic loading, and autoregressive moving average (ARMA) models of the correlated strain response were determined. The objectives are to be able to identify the dynamics of the failure system, and to describe the current condition of the testpiece by the model parameters, which reflect the physical characteristic of the dynamic process itself for a more realistic assessment of fatigue damage.     

Cuscore control charts for generalized feedback‐control systems

The cumulative score (Cuscore) statistic is devised to ‘resonate’ with deviations or signals of an expected type. When a process signal subject to feedback control occurs, it results in a fault signature in the output error. In this paper, Cuscore statistics are designed to monitor process parameters and characteristics measured by a generalized minimum variance (GMV) feedback‐control system sensitive to the fault signature of a spike, step, and bump signal. In this study, the GMV considered is a first‐order dynamic system with autoregressive moving average (ARMA) noise. We show theoretically that the performance of Cuscore charts is independent of the amount of variability transferred from the output quality characteristic to the adjustment actions in the GMV control system. Simulation is used to test the performance using the Cuscore charts. In general, the Cuscore can detect signals over a broad range of system parameter values. However, areas of low detection capability occur for certain fault signatures. In these cases, a tracking signal test is combined with the Cuscore statistics to improve detection performance. This study provides several illustrations of the underlying behavior and shows how the methodology developed can be easily applied in practice. Copyright © 2006 John Wiley & Sons, Ltd.     

基于GRU神经网络的自相关过程残差控制图

为提升自相关过程监控的效率,提出基于门控循环单元(gated recurrent unit,GRU)神经网络的自相关过程残差控制图。采用受控下的自相关过程数据对GRU网络进行离线训练与测试,对预测误差进行监控,形成控制用残差控制图。采用训练好的GRU网络预测当前过程波动,利用控制用残差控制图判定当前过程是否失控。运用蒙特卡洛仿真法,与基于一阶自回归模型、BP神经网络以及支持向量回归构建的残差控制图进行性能对比。研究表明,过程受控时,所提残差控制图与其他3种的稳态平均运行链长相差不大,即4者的性能表现相当;而在均值偏移异常过程中,所提残差控制图的平均运行链长远小于其他3种,对自相关过程均值偏移具有较好的监控性能。     

On autoregressive model selection for the exponentially weighted moving average control chart of residuals in monitoring the mean of autocorrelated processes

With the development of automation technologies, data can be collected in a high frequency, easily causing autocorrelation phenomena. Control charts of residuals have been used as a good way to monitor autocorrelated processes. The residuals have been often computed based on autoregressive (AR) models whose building needs much experience. Data have been assumed to be first-order autocorrelated, and first-order autoregressive (AR(1) ) models have been employed to obtain residuals. But for a p th-order autocorrelated process, how the AR(1) model affects the performance of the control chart of residuals remains unknown. In this paper, the control chart of exponentially weighted moving average of residuals (EWMA-R) is used to monitor the p th-order autocorrelated process. Taking the mean and standard deviation of run length as performance indicators, two types of EWMA-R control charts, with their residuals obtained from the p th-order autoregressive AR(p) and AR(1) models, respectively, are compared. The results of the numerical experiment show that for detecting small mean shifts, EWMA-R control charts based on AR(1) models outperform ones based on AR(p) models, whereas for detecting large shifts, they are sometimes slightly worse. A practical application is used to give a recommendation that a large number of samples are necessary for determining an EWMA-R control chart before using it.     

随机过程数字仿真的ARMA法

文章讨论了采用ARMA 模型进行随机过程数字仿真问题。研究表明,可由AR 模型导出具有高计算效率的ARMA 模型。模型阶数和采样间隔的选取应保证相关函数在主要相关区段上的拟合。ARMA 模型阶数只要高于一定值,都可达到较好的仿真效果。     

A run-to-run control algorithm based on timely and delayed mixed-resolution information

In certain run-to-run (R2R) processes, timely accurate measurements are difficult to obtain due to slow laboratory measurement operations. Instead, only low-resolution categorical observations are observed online for important quality variables; continuous measurements for the same variables are provided after a specific amount of delay. Currently, most conventional R2R controllers cannot be applied if no continuous observations are available. It is therefore important to develop online algorithms for R2R process control based on mixed-resolution information that is partially timely and partially delayed. In this study, we take the lapping process in semiconductor manufacturing as an example and propose parameter estimation models with these mixed-resolution data for processes with the first-order autoregressive, AR(1), disturbance series. We also derive control strategies to generate recipes between production runs for better process control. The computational results of a performance evaluation show that the control performance of the proposed method is competitive compared to existing methods that are based on accurate measurements.     

A Multivariate EWMA Controller for Linear Dynamic Processes

Most research of run-to-run process control has been based on single-input and single-output processes with static input–output relationships. In practice, many complicated semiconductor manufacturing processes have multiple-input and multiple-output (MIMO) variables. In addition, the effects of previous process input recipes and output responses on the current outputs might be carried over for several process periods. Under these circumstances, using conventional controllers usually results in unsatisfactory performance. To overcome this, a complicated process could be viewed as dynamic MIMO systems with added general process disturbance and this article proposes a dynamic-process multivariate exponentially weighted moving average (MEWMA) controller to adjust those processes. The long-term stability conditions of the proposed controller are derived analytically. Furthermore, by minimizing the total mean square error (TMSE) of the process outputs, the optimal discount matrix of the proposed controller under vector IMA(1,?1) disturbance is derived. Finally, to highlight the contribution of the proposed controller, we also conduct a comprehensive simulation study to compare the control performance of the proposed controller with that of the single MEWMA and self-tuning controllers. On average, the results demonstrate that the proposed controller outperforms the other two controllers with a TMSE reduction about 32% and 43%, respectively.     

Model Inadequacy and Residuals Control Charts for Autocorrelated Processes

As a result of time series parameter estimation based on previous data, the probability content of residuals control charts may vary when standard control limits are used. In this paper, we consider the AR(1) process with the autoregressive parameter being estimated from a sample of observations. The performance of the exponentially weighted moving average (EWMA) control chart for residuals is investigated. Modified control limits that account for the uncertainty in the parameter estimate are provided. Comparisons through simulation signify the importance of the modified control limits. Copyright © 2004 John Wiley & Sons, Ltd.