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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]

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]

Abstract: 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 x ¯ − s 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. [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]

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. [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]

We consider serially dependent binary processes, how they occur in several fields of practice. If such a process cannot be monitored continuously, because of process speed for instance, then one can analyze connected segments instead, where two successive segments have a sufficiently large time-lag. Nevertheless, the serial dependence has to be considered at least within the segments, i.e. the distribution of the segment sums is not binomial anymore. We propose the Markov binomial distribution to approximate the true distribution of the segment sums. Based on this distribution, we develop a Markov np chart and a Markov exponentially weighted moving average (EWMA) chart. We show how average run lengths (ARLs) can be computed exactly for both types of chart. Based on such ARL computations, we derive recommendations for chart design and investigate the out-of-control performance. A real-data example illustrates the application of these charts in practice. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The class of INARMA models is well suited to model the autocorrelation structure of processes with Poisson marginals arising in context of statistical quality control. After reviewing briefly the basic principles and important members of this broad family of models, we concentrate on the INAR(1) model, which is of particular relevance for quality control. We suggest four approaches to control such count processes, and compare their run length performance in a simulation study. Results show that only some of the out-of-control situations considered can be controlled effectively with the discussed control schemes. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]