Model‐based control chart for autoregressive and correlated data

In recent years, statistical process control for autocorrelated processes has received a great deal of attention. This is due in part to the improvements in measurement and data collection that allow processes to be sampled at higher frequency rates and, hence, data autocorrelation. A method for monitoring autocorrelated processes based on regression adjustment is presented in this paper. The performance of the residual‐based control chart in terms of the average run length is compared to observation‐based control charts via Monte Carlo simulations. In general, the observation‐based control charts perform very poorly when data are correlated over time. Under the assumption that the model is correct, the residual‐based control charts are superior for all cases considered here. This suggests using a residual‐based control chart to detect the mean shift. This is recommended particularly for chemical processes where there are often cascade processes with several inputs but only a few outputs, and where many of the variables are highly autocorrelated. Copyright © 2002 John Wiley & Sons, Ltd.     

Detecting mean increases in zero truncated INAR(1) processes

Count data with zero truncation are common in the production process. It's essential to monitor these data during production flow, production quality control and market management. Most of the previous studies were based on the independent observations assumption. In fact, serial dependence of count data which significantly affects the performance of the control charts exists extensively in practice. Motivated by this, several important first-order integer-valued autoregressive time series processes are used to model the autocorrelated count data with zero truncation. We investigate the effectiveness of three following charts, the combined jumps chart, the exponentially weighted moving average chart and the cumulative sum chart, to detect the upward shifts of the process mean based on these models. A bivariate Markov chain approach could be used to obtain the average run length of these charts. Design recommendations for achieving robustness are provided based on the computation study. An application to product quality complaints data is presented to demonstrate good performances of the charts.     

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.     

An AR‐Sieve Bootstrap Control Chart for Autocorrelated Process Data

There are two major approaches in dealing with autocorrelated process data in process control, that is, residual‐based approaches and methods that modify control limits to adjust for autocorrelation. We proposed a methodology for constructing control charts for autocorrelated process data using the AR‐sieve bootstrap. The simulation study illustrates the relative advantage of the AR‐sieve bootstrap control chart with respect to the in‐control and out‐of‐control run length and false alarm rate. The proposed methodology works even for small sample sizes and conditions of the near nonstationarity of the generating process. The proposed AR‐sieve bootstrap control chart presents the advantage of being distribution‐free for certain class of linear models as well as the tracking of actual process observations instead of model residuals, thus facilitating the implementation during actual plant operations. Copyright © 2011 John Wiley & Sons, Ltd.     

Improved CUSUM monitoring of Markov counting process with frequent zeros

There has been a growing interest in monitoring processes featuring serial dependence and zero inflation. The phenomenon of excessive zeros often occurs in count time series because of the advancement of quality in manufacturing process. In this study, we propose three control charts, such as the cumulative sum chart with delay rule (CUSUM‐DR), conforming run length (CRL)‐CUSUM chart, and combined Shewhart CRL‐CUSUM chart, to enhance the performance of monitoring Markov counting processes with excessive zeros. Numerical experiments are conducted based on integer‐valued autoregressive time series models, for example, zero‐inflated Poisson INAR and INARCH, to evaluate the performance of the proposed charts designed for the detection of mean increase. A real example is also illustrated to demonstrate the usability of our proposed charts.     

Exponentially weighted moving average chart with a likelihood ratio test for monitoring autocorrelated processes

In this article, an exponential weighted moving average chart based on a likelihood ratio test is developed to monitor the mean and variance shifts simultaneously for autocorrelated processes. A simple method is used to transform the positively autocorrelated data to the negatively autocorrelated data. The average run length of the proposed chart is derived from a simulation approach. The performance of our proposed chart is compared with some existing charts. The results show that the proposed chart provides better performance for detecting a wide range of shifts in the process mean and variance simultaneously. Additionally, the economic performance of different charts under the first-order autoregressive model is provided. A real example of a stepper motor in the heating, ventilation, and air conditioning module is used to demonstrate the application of the proposed method.     

Improved design of robust exponentially weighted moving average control charts for autocorrelated processes

Residual‐based control charts for autocorrelated processes are known to be sensitive to time series modeling errors, which can seriously inflate the false alarm rate. This paper presents a design approach for a residual‐based exponentially weighted moving average (EWMA) chart that mitigates this problem by modifying the control limits based on the level of model uncertainty. Using a Bayesian analysis, we derive the approximate expected variance of the EWMA statistic, where the expectation is with respect to the posterior distribution of the unknown model parameters. The result is a relatively clean expression for the expected variance as a function of the estimated parameters and their covariance matrix. We use control limits proportional to the square root of the expected variance. We compare our approach to two other approaches for designing robust residual‐based EWMA charts and argue that our approach generally results in a more appropriate widening of the control limits. Copyright © 2010 John Wiley & Sons, Ltd.     

Monitoring autocorrelated processes using variable sampling schemes at fixed‐times

Traditional control charts for process monitoring are based on taking samples of fixed size from the process using a fixed sampling interval. A recent development in control charts is the use of adaptive control charts, i.e. the variable sampling interval (VSI) and variable sampling size charts. This paper extends this idea to the autocorrelated process. We consider a time series model which is a first‐order autoregressive process plus a random error. With variable intervals, the sampling time may be inconvenient, so using only two intervals, referred to as ‘variable sampling interval at fix times’ makes the method easier to use in practice. The sampling rate can also be adjusted by the number of samples collected, VSRFT, for ‘variable sampling rate at fixed times’. We study what we call ‘variable sampling at fixed times’, VSFT, which includes both VSIFT and VSRFT schemes, using a Markov chain model and integral equations. We show that our methods detect process shifts faster, on average, than fixed sampling X‐bar charts, and at least comparable detection ability with the less practical VSI charts. Copyright © 2007 John Wiley & Sons, Ltd.     

Detecting changes in autoregressive processes with X¯ and EWMA charts

The traditional use of control charts necessarily assumes the independence of data. It is now recognized that many processes are autocorrelated thus violating the fundamental assumption of independence. As a result, there is a need for a broader approach to SPC when data are time-dependent or autocorrelated. This paper utilizes control charts with fixed control limits for residuals to monitor the performance of a process yielding time-dependent data subject to shifts in the mean and the autocorrelation structure. The effectiveness of the framework is evaluated by an average run length study of both X¯ and EWMA charts using analytical and simulation techniques. Average run lengths are tabulated for various process disturbance scenarios, and recommendations for the most effective monitoring tool are made. The findings of this research present motivation to extend the traditional paradigms of a shifted process (e.g., mean and/or variance). The results show that decreases in the underlying time series parameters are practically impossible to detect with standard control charts. Furthermore, the practitioner is motivated to employ runs rules since the runs are more likely with time-dependent observations.     

Identifying the time of polynomial drift in the mean of autocorrelated processes

Control charts are used to detect changes in a process. Once a change is detected, knowledge of the change point would simplify the search for and identification of the special ause. Consequently, having an estimate of the process change point following a control chart signal would be useful to process engineers. This paper addresses change point estimation for covariance‐stationary autocorrelated processes where the mean drifts deterministically with time. For example, the mean of a chemical process might drift linearly over time as a result of a constant pressure leak. The goal of this paper is to derive and evaluate an MLE for the time of polynomial drift in the mean of autocorrelated processes. It is assumed that the behavior in the process mean over time is adequately modeled by the kth‐order polynomial trend model. Further, it is assumed that the autocorrelation structure is adequately modeled by the general (stationary and invertible) mixed autoregressive‐moving‐average model. The estimator is intended to be applied to data obtained following a genuine control chart signal in efforts to help pinpoint the root cause of process change. Application of the estimator is demonstrated using a simulated data set. The performance of the estimator is evaluated through Monte Carlo simulation studies for the k=1 case and across several processes yielding various levels of positive autocorrelation. Results suggest that the proposed estimator provides process engineers with an accurate and useful estimate for the last sample obtained from the unchanged process. Copyright © 2009 John Wiley & Sons, Ltd.     

Performance of cusum charts from the viewpoint of change-point estimation in the presence of autocorrelation

Cusum charts have good performance on not only the detection of the process change, but also the estimation of the change point. In this paper we will discuss the performance of cusum charts from the viewpoint of change-point estimation in the presence of autocorrelation. Supposing a first-order autoregressive (AR(1)) model as a primitive autocorrelated model, we examine the performance of the change-point estimator in cusum charts. Then we apply a two dimensional Markov process to the representation of the cusum statistic. In addition, we examine the relationship between the performance of the change-point estimator in the AR(1) model and in other primitive autocorrelated models by using the autocorrelation function.     

Effects of Model Accuracy on Residual Control Charts

Residual control charts are acknowledged to be effective tools for statistical process control of multistage processes. In these monitoring procedures, the models on the stage‐wise correlation should be first derived before the control charts are implemented. Therefore, the monitoring performance is inevitably affected by the model fitting scheme. Most of the previous works are under the assumption that the derived models represent the process behavior perfectly. Far less is known about the effects of the model inaccuracy on the monitoring performance. To investigate the effects of the underlying models on the monitoring performance, residual control charts based on two different modeling schemes are compared in this paper. The results indicate that the charting performance is correlated with the model fitting schemes. That is, a more accurate model will significantly increase the detection power and decrease the false alarm rate as well. Copyright © 2015 John Wiley & Sons, Ltd.     

Poisson progressive mean control chart

In real life applications, many process‐monitoring problems in statistical process control are based on attribute data resulting from quality characteristics that cannot be measured on numerical or quantitative scales. For the monitoring of such data, a new attribute control chart has been proposed in this study, namely, the Poisson progressive Mean (PPM) control chart. The performance of the PPM chart is compared with the existing charts used for the monitoring of Poisson processes such as the Shewhart c‐chart, Poisson Exponentially Weighted Moving Average chart, Poisson double Exponentially Weighted Moving Average chart and the Poisson Cumulative Sum charts. The average run length comparison indicated the superior performance of the PPM chart in terms of shift detection ability. This study will help quality practitioners to choose an efficient attribute control chart.     

Monitoring multistage processes with autocorrelated observations

In multistage manufacturing processes, autocorrelations within stages over time are prevalent and the classical control charts are often ineffective in monitoring such processes. In this paper, we derive a linear state space model of an autocorrelated multistage process as a vector autoregressive process, and construct novel multivariate control charts, CBAM and Conditional-based MEWMA, for detecting the mean changes in a multistage process based on a projection scheme by incorporating in-control stage information. When in-control stages are unknown, finding in-control stages is a challenging issue due to the autocorrelations over time and the sequential correlations between stages. To overcome this difficulty, we propose a conditional-based selection that chooses stages with strong evidences of in-control stage using the cascading property of multistage processes. The information of selected stages is effectively utilised in obtaining powerful test statistics for detecting a mean change. The performance of the proposed charts is compared with other existing procedures under different scenarios. Both simulation studies and a real example show the effectiveness of the conditional-based charts in detecting a wide range of small mean shifts compared with the other existing control charts.     

Support Vector Regression control charts for multivariate nonlinear autocorrelated processes

Statistical process control charts are one of the most widely used techniques in industry and laboratories that allow monitoring of systems against faults. To control multivariate processes, most classical charts need to model process structure and assume that variables are linearly and independently distributed. This study proposes to use a nonparametric method named Support Vector Regression to construct several control charts that allow monitoring of multivariate nonlinear autocorrelated processes. Also although most statistical quality control techniques focused on detecting mean shifts, this research investigates detection of different parameter shifts. Based on simulation results, the study shows that, with a controlled robustness, the charts are able to detect the different applied disturbances. Moreover in comparison to Artificial Neural Networks control chart, the proposed charts are especially more effective in detecting faults affecting the process variance.     

Robust control charts for percentiles based on location‐scale family of distributions

In this paper, robust control charts for percentiles based on location‐scale family of distributions are proposed. In the construction of control charts for percentiles, when the underlying distribution of the quality measurement is unknown, we study the problem of discriminating different possible candidate distributions in the location‐scale family of distributions and obtain control charts for percentiles which are insensitive to model mis‐specification. Two approaches, namely, the random data‐driven model selection approach and weighted modeling approach, are used to construct the robust control charts for percentiles in order to effectively monitor the manufacturing process. Monte Carlo simulation studies are conducted to evaluate the performance of the proposed robust control charts for various settings with different percentiles, false‐alarm rates, and sample sizes. These proposed procedures are compared in terms of the average run length. The proposed robust control charts are applied to real data sets for the illustration of robustness and usefulness.     

Batch-means control charts for autocorrelated data

Modern statistical process control must often cope with large quantities of highly autocorrelated data. Alwan and Radson (1992) proposed the monitoring of autocorrelated processes by plotting the averages of small batches of data separated by skipping observations. Using results for the AR(1) process, we show that generally better performance can be achieved with no skipping and much larger batch sizes. The resulting batch-means charts derive from methods used in simulation output analysis and can be implemented easily with common digital control systems.     

Control Charts for Monitoring Correlated Poisson Counts with an Excessive Number of Zeros

The zero‐inflated Poisson distribution serves as an appropriate model when there is an excessive number of zeros in the data. This phenomenon frequently occurs in count data from high‐quality processes. Usually, it is assumed that these counts exhibit serial independence, while a more realistic assumption is the existence of an autocorrelation structure between them. In this work, we study control charts for monitoring correlated Poisson counts with an excessive number of zeros. Zero‐inflation in the process is captured via appropriate integer‐valued time series models. Extensive numerical results are provided regarding the performance of the considered charts in the detection of changes in the mean of the process as well as the effects of zero‐inflation on them. Finally, a real‐data practical application is given. Copyright © 2016 John Wiley & Sons, Ltd.     

Autocorrelated process monitoring using triggered cuscore charts

Some of the most widely‐investigated control charting techniques for autocorrelated data are based on time series residuals. If the mean shift in the autocorrelated process is a sudden step shift, the resulting mean shift in the residuals is time varying and has been referred to as the fault signature. Traditional residual based charts, such as a Shewhart, CUSUM, or EWMA on the residuals, do not make use of the information contained in the dynamics of the fault signature. In contrast, methods such as the Cuscore chart or Generalized Likelihood Ratio Test (GLRT) do incorporate this information. In order for the Cuscore chart to fully benefit from this, its detector coefficients should coincide with the fault signature. This is impossible to ensure, however, since the exact form of the fault signature depends on the time of occurrence of the mean shift, which is generally not known a priori. Any mismatch between the detector and the fault signature will adversely affect the Cuscore performance. This paper proposes a CUSUM‐triggered Cuscore chart to reduce the mismatch between the detector and fault signature. A variation to the CUSUM‐triggered Cuscore chart that uses a GLRT to estimate the mean shift time of occurrence is also discussed. It is shown that the triggered Cuscore chart performs better than the standard Cuscore chart and the residual‐based CUSUM chart. Examples are provided to illustrate its use. Copyright © 2002 John Wiley & Sons, Ltd.     

Nonparametric Estimator for the Time of a Step Change in the Trend of Random Walk Models with Drift

Living systems tend to have non‐normal behaviors, are autocorrelated, exhibit patterns of growth or decrement, and achieve states of dynamic equilibrium, making them hard to manage. One way to manage and improve these complex systems is by identifying assignable causes of variation whenever they occur, and control charts are one of the most known tool for those situations. However, in the presence of sustained changes, control charts are not capable of telling the initial moment of a change. This paper proposes a nonparametric estimator capable of dealing with non‐normal observations heterocedastic over time. The median test is used to estimate the time of a step change in Shewhart Control Charts using a relatively successful approach based on a binary segmentation approach. Furthermore, an application is developed to deal with changes on the trend of processes fitting a random walk with drift. Performance is evaluated with extensive Monte Carlo simulations, and results are compared with the maximum likelihood estimator for normal series of independent observations. Results showed acceptable performance when normality is not met and robustness under heterocedasticity. Copyright © 2012 John Wiley & Sons, Ltd.