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We consider the problem of monitoring for the existence of serial dependence in real-valued and continuously distributed processes. These can exist when a process goes out-of-control. Besides, as many control charts are designed under the assumption of independent and identically distributed observations, the validity of this assumption needs to be checked. In the literature, the majority of studies handled this problem by using a specific case of autoregressive moving-average time series models. Also a moving-window approach is often considered, which leads to the drawback that at least a window length of observations need to be collected before an out-of-control situation can be detected. Here, we use ordinal patterns and propose charts that are fully nonparametric and distribution-free, have a unique chart design, and can be used almost instantaneously at the start of process monitoring. The proposed control charts do not require any model fitting, thus, eliminating the problems associated with estimation errors, but might be used as part of a Phase-I study. Through a comprehensive performance comparison study under various out-of-control scenarios, the effectiveness of the charts in uncovering serial dependence is shown and recommendations for selection are given. Implementations of the proposed charts are illustrated by using batch yield data from a chemical process. [ABSTRACT FROM AUTHOR]

Qualitative, more specifically, ordinal data generating processes are common in real-world process control implementations. In this study, a survey of control charts for the sample-based monitoring of independent and identically distributed ordinal data is provided together with critical comparisons of the control statistics, for memory-less Shewhart-type and for memory-utilizing exponentially weighted moving average (EWMA) and cumulative-sum types of control charts. New results and proposals are also provided for process monitoring. Using some real-world quality scenarios from the literature, a simulation study for performance comparisons is conducted, covering sixteen different types of control chart. It is shown that demerit-type charts used in combination with EWMA smoothing generally perform better than the other charts, which may rely on quite sophisticated derivations. A real-world data example for monitoring flashes in electric toothbrush manufacturing is discussed to illustrate the application and interpretation of the control charts in the study. [ABSTRACT FROM AUTHOR]

Linear models are typically utilized for time series analysis as these are often simple to implement and interpret, as well as being useful in modeling many practical phenomena. Hence, most of the literature on control charts for monitoring time series also consider linearity of the data generating processes (DGP). In practice, however, a nonlinear DGP can be modeled as if it is linear, either due to overlook or illusion, when it is approximately linear. This study quantifies the effects of nonlinear DGPs, misspecified and modeled as linear, on the performance of Shewhart‐type and cumulative sum (CUSUM) control charts for count time series data. Time series models for bounded and unbounded counts with several parametrizations are considered for studying the sensitivity of linear approximations to nonlinear DGPs. The Markov chain (MC) approach and simulations are used to compute the average run length (ARL) performance of the control charts. Robustness of the performance is evaluated with respect to the extent that the DGP violates linearity, with and without errors in parameter estimation. It is shown that the chart designs are, in general, robust to model misspecification when the parameters are specified. However, the CUSUM performance can be affected significantly if both the model is misspecified and the parameters are estimated. Real‐world data implementations are provided to illustrate the sensitivity of the control charts' performances to the type of model and to the estimation approach. [ABSTRACT FROM AUTHOR]

Monitoring stochastic processes with control charts is the main field of application in statistical process control. For a Poisson hidden Markov model (HMM) as the underlying process, we investigate a Shewhart individuals chart, an ordinary Cumulative Sum (CUSUM) chart, and two different types of log‐likelihood ratio (log‐LR) CUSUM charts. We evaluate and compare the charts' performance by their average run length, computed either by utilizing the Markov chain approach or by simulations. Our performance evaluation includes various out‐of‐control scenarios as well as different levels of dependence within the HMM. It turns out that the ordinary CUSUM chart shows the best overall performance, whereas the other charts' performance strongly depend on the particular out‐of‐control scenario and autocorrelation level, respectively. For illustration, we apply the HMM and the considered charts to a data set about weekly sales counts. [ABSTRACT FROM AUTHOR]

To improve the sensitivity of a Shewhart control chart, it is common among practitioners to use supplementary runs rules. The performance of such runs rules charts is studied in the presence of positive autocorrelation caused by a first-order discrete autoregressive process. This type of data-generating process allows to compute the chart's run length properties exactly and efficiently, by utilizing the finite Markov chain embedding technique. Explicit formulae are derived for common types of runs rules. Afterwards, a detailed performance study about runs rules charts under serial dependence is presented. [ABSTRACT FROM AUTHOR]

With the advances in measurement technologies, today products and processes may have hundreds of variables that can be monitored. As the number of variables to be monitored in a process increases, a cumbersome task is the design of control charts, especially when one needs to estimate unknown process parameters. In Phase-I control chart implementations, a set of samples that are ideally from an in-control process is formed by iteratively eliminating/retaining potentially out-of-control samples and this is then used in parameter estimation. Nevertheless, sampling variability and samples from an out-of-control process that are not eliminated from a Phase-I data set may have effects on the online process monitoring (Phase II) performance of a control chart due to the control chart design with estimation errors. To provide Phase-I control chart design guidelines, here we investigate in detail the iterative use of the individuals control chart (I-control chart) in Phase I for identifying potentially out-of-control process observations when the initial Phase-I data set consists of contaminated observations, which is then followed by another Phase-II I-control chart designed with the parameter estimates obtained in Phase I. Using the sample size and the width of I-chart's control limits as design factors in Phase I, process parameters are estimated, and these are used to design an I-chart with 3-sigma control limits for Phase II. In Phase II, expected value (AARL) and standard deviation (SDARL) of the average run lengths are evaluated. Six standard deviation estimators and three different contamination levels are considered. Guidelines for selecting the control limit width, sample size, and standard deviation estimator for the Phase I implementations are provided to yield desirable Phase II AARL performance with a small SDARL. Our contribution is neither in the theory of control charts nor in the individual techniques. Results of this study would be especially useful in automating control chart design processes where there are many control charts to be designed by well-trained operators in statistical process control. With transformations to Industry 4.0, which is a trend for automation and data exchange in manufacturing, utility of procedures for control chart design automation is expected to increase. [ABSTRACT FROM AUTHOR]

Control charts are commonly evaluated in terms of their average run length (ARL). However, since run length distributions are typically strongly skewed, the ARL gives a very limited impression about the actual run length performance. In this study, it is proposed to evaluate a control chart's performance using risk metrics, specifically the value at risk and the tail conditional expectation. When a control chart is evaluated for an in‐control performance, the risk is an early occurrence of a false alarm, whereas in an out‐of‐control state, there is a risk of a delayed detection. For these situations, risk metric computations are derived and exemplified for diverse types of control charts. It is demonstrated that the use of such risk metrics leads to important new insights into a control chart's performance. In addition to the cases of known process parameters for control chart design, these risk metrics are further used to analyze the estimation uncertainty in evaluating a control chart's performance if the design parameters rely on a phase 1 analysis. Benefits of the risk‐based metrics are discussed thoroughly, and these are recommended as supplements to the traditional ARL metric. [ABSTRACT FROM AUTHOR]

Shewhart control charts are known to be somewhat insensitive to shifts of small and moderate size. Expectedly, alternative control schemes such as the exponentially weighted moving average (EWMA) charts have been proposed to speed up the detection of such shifts. Unfortunately, applying the ordinary EWMA recursion to count data leads to a control statistic no longer with a fixed discrete range. Therefore, we propose a novel chart which relies on a EWMA control statistic where the usual scalar product is replaced by a thinning operation. Actually, we use the new fractional binomial thinning to avoid the typical over-smoothing ascribable to ceiling, rounding, and flooring operations. The properties of this discrete statistic are similar to the ones of its continuous EWMA counterpart and the run length (RL) performance of the associated chart can be computed exactly using the Markov chain approach for independent and identically distributed (i.i.d.) counts. Moreover, this chart is set in such way that: the average run length (ARL) curve attains a maximum in the in-control situation, i.e., the chart is ARL-unbiased; and the in-control ARL is equal to a pre-specified value. We use the R statistical software to provide compelling illustrations of this unconventional EWMA chart and to compare its RL performance with the ones of a few competing control charts for the mean of i.i.d. Poisson counts. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In designing a control chart for online process monitoring, a Phase I analysis is often conducted as a first step to estimate the unknown process parameters. It is based on historical data, where parameter estimation, chart design, and data filtering are iterated until a stable and reliable chart design is obtained. Researchers sometimes neglect the effects of Phase I analysis by assuming that process parameters are known but directly evaluate the performance of a control chart in Phase II (online process monitoring). In this research, the Phase I analysis of time-dependent count data stemming from Poisson INAR(1) and binomial AR(1) processes is considered. Due to data filtering, parameter estimation in Phase I analysis is challenging, especially when the data are autocorrelated. In this regard, solutions on how to modify the method of moments, least squares, and maximum likelihood estimation are presented. Performance of these estimators in Phase I as well as performance of designed control charts in Phase II are evaluated, and recommendations are provided. A real-world example on IP counts is used to illustrate the Phase I and Phase II control chart implementations. [ABSTRACT FROM AUTHOR]

The INARCH(1) model has been proposed in the literature as a simple, but practically relevant, twoparameter model for processes of overdispersed counts with an autoregressive serial dependence structure. In this research, we develop approaches for monitoring INARCH(1) processes for detecting shifts in the process parameters. Several cumulative sum control charts are derived directly from the log-likelihood ratios for various types of shifts in the INARCH(1) model parameters. We define zero-state (worst-state) and steady-state average run length metrics and discuss their computation for the proposed charts. An extensive study indicates that these charts perform well in detecting changes in the process. A real-data example of strike counts is used to illustrate process monitoring. [ABSTRACT FROM AUTHOR]

To monitor a binary process, exponentially weighted moving average (EWMA) control charts in different variations are proposed. The average run length performance of the EWMA approach both with standard and with skewness-corrected 3- σ control limits is investigated and design recommendations are derived. The proposed EWMA control charts are applied to medical diagnosis data taken from the diagnostic expert system SC in order to provide a semi-automatic component for monitoring the documentation behavior of different examiners. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Attributes control charts for counts generally assume that the process being monitored is independent and identically distributed in its in-control state. However, violation of this assumption in practice may significantly degrade a chart's performance and usefulness if the autocorrelation structure is not taken into account. To describe the autocorrelation structure of counts in an in-control process, integer-valued autoregressive moving average process models can be employed. This paper investigates the cumulative sum (CUSUM) control chart for monitoring autocorrelated processes of counts modeled by a Poisson integer-valued autoregressive model of order 1, namely Poisson INAR(1). The CUSUM chart is designed to detect assignable causes affecting the process mean, but also changes in the autocorrelation structure are considered. Exact numerical results obtained through a bivariate Markov chain approach are provided for sustained shifts in any or both of these process parameters. Some numerical results from a simulation study of the residuals' monitoring are also presented. It is shown that the considered CUSUM chart of observations has good overall performance in detecting assignable causes in autocorrelated count processes. [ABSTRACT FROM AUTHOR]