Effects of Parameter Estimation on the Multivariate Distribution‐free Phase II Sign EWMA Chart

Multivariate nonparametric control charts can be very useful in practice and have recently drawn a lot of interest in the literature. Phase II distribution‐free (nonparametric) control charts are used when the parameters of the underlying unknown continuous distribution are unknown and can be estimated from a sufficiently large Phase I reference sample. While a number of recent studies have examined the in‐control (IC) robustness question related to the size of the reference sample for both univariate and multivariate normal theory (parametric) charts, in this paper, we study the effect of parameter estimation on the performance of the multivariate nonparametric sign exponentially weighted moving average (MSEWMA) chart. The in‐control average run‐length (ICARL) robustness and the out‐of‐control shift detection performance are both examined. It is observed that the required amount of the Phase I data can be very (perhaps impractically) high if one wants to use the control limits given for the known parameter case and maintain a nominal ICARL, which can limit the implementation of these useful charts in practice. To remedy this situation, using simulations, we obtain the “corrected for estimation” control limits that achieve a desired nominal ICARL value when parameters are estimated for a given set of Phase I data. The out‐of‐control performance of the MSEWMA chart with the correct control limits is also studied. The use of the corrected control limits with specific amounts of available reference sample is recommended. Otherwise, the performance the MSEWMA chart may be seriously affected under parameter estimation. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Control charts for monitoring the median parameter of Birnbaum-Saunders distribution

In this paper, we propose control charts for monitoring the Birnbaum-Saunders (BS) median parameter (scale parameter) on the basis of three estimators. Comparison of the control charts in terms of average run length using probability control limits and those based on asymptotic distribution of three estimators for the median parameter is developed. We also present guidelines for practitioners about the minimum sample size needed to match out-of-control average run length with the asymptotic control limits in function of the median parameter after an extensive simulation study. Numerical example illustrates the applied monitoring of BS median parameter.     

The Effect of Autocorrelation on the Hotelling T2 Control Chart

One of the basic assumptions for traditional univariate and multivariate control charts is that the data are independent in time. For the latter, in many cases, the data are serially dependent (autocorrelated) and cross‐correlated because of, for example, frequent sampling and process dynamics. It is well known that the autocorrelation affects the false alarm rate and the shift‐detection ability of the traditional univariate control charts. However, how the false alarm rate and the shift‐detection ability of the Hotelling T² control chart are affected by various autocorrelation and cross‐correlation structures for different magnitudes of shifts in the process mean is not fully explored in the literature. In this article, the performance of the Hotelling T² control chart for different shift sizes and various autocorrelation and cross‐correlation structures are compared based on the average run length using simulated data. Three different approaches in constructing the Hotelling T² chart are studied for two different estimates of the covariance matrix: (i) ignoring the autocorrelation and using the raw data with theoretical upper control limits; (ii) ignoring the autocorrelation and using the raw data with adjusted control limits calculated through Monte Carlo simulations; and (iii) constructing the control chart for the residuals from a multivariate time series model fitted to the raw data. To limit the complexity, we use a first‐order vector autoregressive process and focus mainly on bivariate data. © 2014 The Authors. Quality and Reliability Engineering International Published by 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.     

EWMA control chart limits for first‐ and second‐order autoregressive processes

Today's manufacturing environment has changed since the time when control chart methods were originally introduced. Sequentially observed data are much more common. Serial correlation can seriously affect the performance of the traditional control charts. In this article we derive explicit easy‐to‐use expressions of the variance of an EWMA statistic when the process observations are autoregressive of order 1 or 2. These variances can be used to modify the control limits of the corresponding EWMA control charts. The resulting control charts have the advantage that the data are plotted on the original scale making the charts easier to interpret for practitioners than charts based on residuals. Copyright © 2008 John Wiley & Sons, Ltd.     

The Use of Probability Limits of COM–Poisson Charts and their Applications

The conventional c and u charts are based on the Poisson distribution assumption for the monitoring of count data. In practice, this assumption is not often satisfied, which requires a generalized control chart to monitor both over‐dispersed as well as under‐dispersed count data. The Conway–Maxwell–Poisson (COM–Poisson) distribution is a general count distribution that relaxes the equi‐dispersion assumption of the Poisson distribution and in fact encompasses the special cases of the Poisson, geometric, and Bernoulli distributions. In this study, the exact k‐sigma limits and true probability limits for COM–Poisson distribution chart have been proposed. The comparison between the 3‐sigma limits, the exact k‐sigma limits, and the true probability limits has been investigated, and it was found that the probability limits are more efficient than the 3‐sigma and the k‐sigma limits in terms of (i) low probability of false alarm, (ii) existence of lower control limits, and (iii) high discriminatory power of detecting a shift in the parameter (particularly downward shift). Finally, a real data set has been presented to illustrate the application of the probability limits in practice. Copyright © 2012 John Wiley & Sons, Ltd.     

Use of median-based estimator to construct Phase II exponential chart

The Shewhart-type exponential control chart is a popular and extensively used among all time-between-events control charts for its simplicity. When the parameter is unknown, Phase II control limits are constructed, and the success of its implementation depends to an extent on the estimated value of the parameter, obtained from Phase I dataset. However, when the Phase I data are contaminated with spurious observations/outliers, the performance of the chart is suspected to deviate from what is normally expected. Traditionally, maximum likelihood estimator (MLE) and minimum variance unbiased estimator (MVUE) are used to estimate the unknown process parameter. Both of estimators are the functions of sample mean. In this paper, the median-based estimator (MBE) that is a function of sample median is used to construct Phase II control limits. Moreover, performance of the proposed chart is examined when Phase I sample consists of contaminated observations/outliers. It is found that the proposed chart outperforms the existing charts whether the sample is contaminated or not.     

Evaluation of Phase I analysis scenarios on Phase II performance of control charts for autocorrelated observations

Phase I analysis of a control chart implementation comprises parameter estimation, chart design, and outlier filtering, which are performed iteratively until reliable control limits are obtained. These control limits are then used in Phase II for online monitoring and prospective analyses of the process to detect out-of-control states. Although a Phase I study is required only when the true values of the parameters of a process are unknown, this is the case in many practical applications. In the literature, research on the effects of parameter estimation (a component of Phase I analysis) on the control chart performance has gained importance recently. However, these studies consider availability of complete and clean data sets, without outliers and missing observations, for estimation. In this article, we consider AutoRegressive models of order 1 and study the effects of two extreme cases for Phase I analysis; the case where all outliers are filtered from the data set (parameter estimation from incomplete but clean data) and the case where all outliers remain in the data set during estimation. Performance of the maximum likelihood and conditional sum of squares estimators are evaluated and effects on the Phase II use are investigated. Results indicate that the effect of not detecting outliers in Phase I can be severe on the Phase II application of a control chart. A real-world example is provided to illustrate the importance of an appropriate Phase I analysis.     

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.     

Phase II control charts for monitoring dispersion when parameters are estimated

Shewhart control charts are among the most popular control charts used to monitor process dispersion. To base these control charts on the assumption of known in-control process parameters is often unrealistic. In practice, estimates are used to construct the control charts and this has substantial consequences for the in-control and out-of-control chart performance. The effects are especially severe when the number of Phase I subgroups used to estimate the unknown process dispersion is small. Typically, recommendations are to use around 30 subgroups of size 5 each.

?We derive and tabulate new corrected charting constants that should be used to construct the estimated probability limits of the Phase II Shewhart dispersion (e.g., range and standard deviation) control charts for a given number of Phase I subgroups, subgroup size and nominal in-control average run-length (ICARL). These control limits account for the effects of parameter estimation. Two approaches are used to find the new charting constants, a numerical and an analytic approach, which give similar results. It is seen that the corrected probability limits based charts achieve the desired nominal ICARL performance, but the out-of-control average run-length performance deteriorate when both the size of the shift and the number of Phase I subgroups are small. This is the price one must pay while accounting for the effects of parameter estimation so that the in-control performance is as advertised. An illustration using real-life data is provided along with a summary and recommendations.     

Monitoring the Weibull Shape Parameter with Type II Censored Data

In this article, we introduce a method for monitoring the Weibull shape parameter β with type II (failure) censored data. The control limits depend on the sample size, the number of censored observations, the target average run length, and the stable value of β. The method assumes that the scale parameter α is constant during each sampling period, which is true under rational subgrouping. The proposed method utilizes the relationship between Weibull and smallest extreme value distribution. We propose an unbiased estimator of σ = 1/β as the monitoring statistic. We derive the control limits for one‐sided and two‐sided charts for several stable process average run lengths. We discuss two schemes, namely, the control‐limits‐only scheme and the control‐limits‐with‐warning‐lines scheme. The stable process average run length performance of the proposed charts is studied and compared with those of other charts for monitoring β under similar assumptions. Copyright © 2014 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.     

Modeling and Monitoring Ecological Systems—A Statistical Process Control Approach

Statistical process control monitoring of nonlinear relationships (profiles) has been the subject of much research recently. While attention is primarily given to the statistical aspects of the monitoring techniques, little effort has been devoted to developing a general modeling approach that would introduce ‘uniformity of practice’ in modeling nonlinear profiles (analogously with the three‐sigma limits of Shewhart control charts). In this article, we use response modeling methodology (RMM) to demonstrate implementation of this approach to statistical process control monitoring of ecological relationships. Using 10 ecological models that have appeared in the literature, it is first shown that RMM models can replace (approximate) current ecological models with negligible loss in accuracy. Computer simulation is then used to demonstrate that estimated RMM models and estimated data generating ecological models achieve goodness‐of‐fit that is practically indistinguishable from one another. A regression‐adjusted control scheme, based on control charts for the predicted median and for residuals variation, is developed and demonstrated for three types of ‘out of control’ scenarios. Copyright © 2013 John Wiley & Sons, Ltd.     

About Shewhart control charts to monitor the Weibull mean

In this paper, a new reparametrization expressed in terms of the process mean for Weibull distribution is studied; thus, the monitoring of the process mean can be made directly. Additionally, we call attention that the asymptotic control limits for control chart by central limit theorem (CLT) may lead to a serious erroneous decision. Definitively, they can only be used to signal small/medium shifts in the process mean but with a very very large sample size. We present guidelines for practitioners about the minimum sample size needed to match out‐of‐control average run length (ARL₁) with the exact and asymptotic control limits in function of the shape parameter after an extensive simulation study. The proposed schemes are applied to monitoring the Weibull mean parameter of the strength distribution of a carbon fibber used in composite materials.     

Monitoring k‐step‐ahead Controlled Processes

A dynamical system controlled by a k‐step‐ahead minimum variance controller is considered. Independent, identically distributed one‐step‐ahead process residuals are given for use in statistical process monitoring schemes. Problems encountered in the application of the monitoring schemes are discussed, particularly with respect to detecting process upsets. Upsets may occur in any of three ways, for which expressions are derived. It is shown that the mechanism by which upsets occur influences the ability of the residuals to detect the upsets. It is also shown that the effect of the disturbance on the residuals is independent of the process time delay k. The ability of the residuals to detect a change in the process dispersion is discussed. It is shown that the disturbance dynamics do not alter this ability. This information is useful in obtaining accurate estimates of control chart performance and directing the statistical process control practitioner in modifying the control chart design. Copyright © 2004 John Wiley & Sons, Ltd.     

POD–ISAT: An efficient POD‐based surrogate approach with adaptive tabulation and fidelity regions for parametrized steady‐state PDE discrete solutions

A combination of proper orthogonal decomposition (POD) analysis and in situ adaptive tabulation (ISAT) is proposed for the representation of parameter‐dependent solutions of coupled partial differential equation problems. POD is used for the low‐order representation of the spatial fields and ISAT for the local representation of the solution in the design parameter space. The accuracy of the method is easily controlled by free threshold parameters that can be adjusted according to user needs. The method is tested on a coupled fluid‐thermal problem: the design of a simplified aircraft air control system. It is successfully compared with the standard POD; although the POD is inaccurate in certain areas of the design parameters space, the POD–ISAT method achieves accuracy thanks to trust regions based on residuals of the fluid‐thermal problem. The presented POD–ISAT approach provides flexibility, robustness and tunable accuracy to represent solutions of parametrized partial differential equations.Copyright © 2013 John Wiley & Sons, Ltd.     

Monitoring the Weibull shape parameter by control charts for the sample range

In this paper, we propose control charts for monitoring changes in the Weibull shape parameter β. These charts are based on the range of a random sample from the smallest extreme value distribution. The control chart limits depend only on the sample size, the desired stable average run length (ARL), and the stable value of β. We derive control limits for both one‐ and two‐sided control charts. They are unbiased with respect to the ARL. We discuss sample size requirements if the stable value of βis estimated from past data. The proposed method is applied to data on the breaking strengths of carbon fibers. We recommend one‐sided charts for detecting specific changes in βbecause they are expected to signal out‐of‐control sooner than the two‐sided charts. Copyright © 2010 John Wiley & Sons, Ltd.     

Effectiveness of phase I applications for identifying randomly scattered out‐of‐control observations and estimating control chart parameters

Estimation of unknown process parameters with fixed‐size samples are studied in the following. The standard textbook approach for phase I control chart implementation with a Shewhart control chart is evaluated for the case of normally distributed independent observations with random sampling. The charts are simultaneously implemented by generating observations that have a given percentage of randomly scattered out‐of‐control observations. Simulating the phase I steps, where out‐of‐control samples are detected iteratively by determining trial control limits, identifying samples exceeding these limits, and revising the control limits, the standard practice is evaluated in terms of both detection performance and quality of parameter estimates. It is shown that standard phase I control chart implementations with 3‐σ‐limits may perform very poorly in identifying true out‐of‐control observations and providing a reference set of in‐control observations for estimation in some practical settings. A chart design with 2‐σ‐limits is recommended for a successful phase I analysis.     

On the empirical characteristic function process of the residuals in GARCH models and applications

The class of generalized autoregressive conditional heteroscedastic (GARCH) models has been proved to be particularly valuable in modeling financial data. This paper is devoted to study the empirical characteristic function process of the residuals. Specifically, it is shown that such process uniformly converges to the population characteristic function (CF) of the innovations in compact sets. The weak convergence of this empirical process, suitably normalized, is also studied. The limit depends on the population CF of the innovations, the equation defining the GARCH model and the parameter estimators employed to calculate the residuals. Applications of the obtained results for testing symmetry and goodness-of-fit to the law of the innovations are given.