The Exponentially Weighted Moving Average (EWMA) and Cumulative Sum (CUSUM) control charts have been used in profile monitoring to track drift shifts that occur in a monitored process. We construct Bayesian EWMA and Bayesian CUSUM charts informed by posterior and posterior predictive distributions using different loss functions, prior distributions, and likelihood distributions. A simulation study is performed, and the performance of the charts are evaluated via average run length (ARL), standard deviation of the run length (SDRL), average time to signal (ATS), and standard deviation of time to signal (SDTS). A sensitivity analysis is conducted using choices for the smoothing parameter, out-of-control shift size, and hyper-parameters of the distribution. Based on obtained results, we provide recommendations for use of the Bayesian EWMA and Bayesian CUSUM control charts.
Investigating the problem of setting control limits in the case of parameter uncertainty is more accessible when monitoring the variance because only one parameter has to be estimated. Simply ignoring the induced uncertainty frequently leads to control charts with poor false alarm performances. Adjusting the unconditional in-control (IC) average run length (ARL) makes the situation even worse. Guaranteeing a minimum conditional IC ARL with some given probability is another very popular approach to solving these difficulties. However, it is very conservative as well as more complex and more dif…
We show optimality, in a well-defined sense, using cumulative sum (CUSUM) charts for detecting changes in distributions. We consider a setting with multiple changes between two known distributions. This result advocates the use of non-restarting CUSUM charts with an upper boundary. Typically, after signalling, a CUSUM chart is restarted by setting it to some value below the threshold. A non-restarting CUSUM chart is not reset after signalling; thus is able to signal continuously. Imposing an upper boundary prevents the CUSUM chart rising too high, which facilitates detection in our setting. We…
This paper develops a new multivariate control charting method for vector autocorrelated and serially correlated processes. The main idea is to propose a Bayesian multivariate local level model, which is a generalization of the Shewhart-Deming model for autocorrelated processes, in order to provide the predictive error distribution of the process and then to apply a univariate modified EWMA control chart to the logarithm of the Bayes' factors of the predictive error density versus the target error density. The resulting chart is proposed as capable to deal with both the non-normality and the a…
In this work, we study the performance of two-sided EWMA charts for monitoring double bounded processes using individual observations. Specifically, the term double bounded refers to observations in the interval (0, 1) and thus, these charts are suitable for monitoring rates, proportions and percentages. There are several models that can be used to describe this kind of data (and the respective processes, as well) such as the Beta distribution, the Simplex distribution and the Unit Gamma distribution. For each of these three models, we provide the statistical design and the performance of the …
Cumulative sum (CUSUM) charts are typically used to detect changes in a stream of observations e.g. shifts in the mean. Usually, after signalling, the chart is restarted by setting it to some value below the signalling threshold. We propose a non-restarting CUSUM chart which is able to detect periods during which the stream is out of control. Further, we advocate an upper boundary to prevent the CUSUM chart rising too high, which helps detecting a change back into control. We present a novel algorithm to control the false discovery rate (FDR) pointwise in time when considering CUSUM charts bas…
This paper develops non-parametric rotation invariant CUSUMs suited to the detection of changes in the mean direction as well as changes in the concentration parameter of angular data. The properties of the CUSUMs are illustrated by theoretical calculations, Monte Carlo simulation and application to sequentially observed angular data from health science and astrophysics.
This article aims to consider a new univariate nonparametric cumulative sum (CUSUM) control chart for small shift of location based on both change-point model and Mann-Whitney statistic. Some comparisons on the performances of the proposed chart with other charts as well as the properties of the test statistic are presented. Simulations indicate that the proposed chart is sensitive in detection of the small mean shifts of the process by a high intensive accumulation of sample information when the underlying variable is completely distribution-free.
Nonparametric control charts that can detect arbitrary distributional changes are highly desirable due to their flexibility to adapt to different distributional assumptions and distributional changes. However, most of such control charts in the literature either involve some tuning parameter, which needs to be pre-specified, or involve intensive computation. In this paper, we propose a new nonparametric adaptive CUSUM chart for detecting arbitrary distributional changes. The proposed control chart does not depend on any tuning parameter and is efficient in computation. Its self-starting nature…
Many extensions and modifications have been made to standard process monitoring methods such as the exponentially weighted moving average (EWMA) chart and the cumulative sum (CUSUM) chart. In addition, new schemes have been proposed based on alternative weighting of past data, usually to put greater emphasis on past data and less weight on current and recent data. In other cases, the output of one process monitoring method, such as the EWMA statistic, is used as the input to another method, such as the CUSUM chart. Often the recursive formula for a control chart statistic is itself used recurs…
Control charts are important tools to monitor quality of products. One of useful applications is to monitor the proportion of non-conforming products. However, in practical applications, measurement error is ubiquitous and may occur due to false records or misclassification, which makes the observed proportion different from the underlying true proportion. It is also well-known that ignoring measurement error effects provides biases, and is expected that the resulting control charts may incur wrong detection. In this paper, we study this important problem and propose a valid method to correct …
We establish a simple connection between certain in-control characteristics of the CUSUM Run Length and their out-of-control counterparts. The connection is in the form of paired integral (renewal) equations. The derivation exploits Wald's likelihood ratio identity and the well-known fact that the CUSUM chart is equivalent to repetitive application of Wald's SPRT. The characteristics considered include the entire Run Length distribution and all of the corresponding moments, starting from the zero-state ARL. A particular practical benefit of our result is that it enables the in- and out-of-cont…
We study optimal design of the Exponentially Weighted Moving Average (EWMA) chart by a proper choice of the smoothing factor and the initial value (headstart) of the decision statistic. The particular problem addressed is that of quickest detection of an abrupt change in the parameter of a discrete-time exponential model. Both pre- and post-change parameter values are assumed known, but the change-point is not known. For this change-point detection scenario, we examine the performance of the conventional one-sided EWMA chart with respect to two optimality criteria: Pollak's minimax criterion a…
The behavior of CUSUM charts depends strongly on how they are initialized. Recent work has suggested that self-starting CUSUM methods retain some dependence on their very first readings, and introduced the concept of "conditional average run length" (CARL) -- the average run length conditioned on the first few process readings -- as a result of which is it claimed that different practitioners using the same methodology could experience different ARLs because of the random differences in their earliest readings. We cast doubt on whether CARL is relevant to practitioners who use self-starting me…
We argue against the use of generally weighted moving average (GWMA) control charts. Our primary reasons are the following: 1) There is no recursive formula for the GWMA control chart statistic, so all previous data must be stored and used in the calculation of each chart statistic. 2) The Markovian property does not apply to the GWMA statistics, so computer simulation must be used to determine control limits and the statistical performance. 3) An appropriately designed, and much simpler, exponentially weighted moving average (EWMA) chart provides as good or better statistical performance. 4) …
Most statistical process control programmes in healthcare focus on surveillance of outcomes at the final stage of a procedure, such as mortality or failure rates. Such an approach ignores the multi-stage nature of these procedures, in which a patient progresses through several stages prior to the final stage. In this paper, we develop a multi-stage control chart based on a multivariate exponentially weighted moving average (EWMA) test statistic derived from score equations. This allows simultaneous monitoring of all intermediate and final stage outcomes of a healthcare process, with adjustment…
A multivariate control chart is designed to monitor process parameters of multiple correlated quality characteristics. Often data on multivariate processes are collected as individual observations, i.e. as vectors one at the time. Various control charts have been proposed in the literature to monitor the covariance matrix of a process when individual observations are collected. In this study, we review this literature; we find 30 relevant articles from the period 1987-2019. We group the articles into five categories. We observe that less research has been done on CUSUM, high-dimensional and no…
Over the years, the most popularly used control chart for statistical process control has been Shewhart's $\bar{X}-S$ or $\bar{X}-R$ chart along with its multivariate generalizations. But, such control charts suffer from the lack of robustness. In this paper, we propose a modified and improved version of Shewhart chart, based on trimmed mean and winsorized variance that proves robust and more efficient. We have generalized this approach of ours with suitable modifications using depth functions for Multivariate control charts and EWMA charts as well. We have discussed the theoretical properties…
Standard Control Chart techniques to detect level shift in data streams assume independence between observations. As data today is collected with high frequency, this assumption is seldom valid. To overcome this, we propose to adapt the off-line test procedure for detection of outliers based on one-step prediction errors proposed by Tsay (1988) into an on-line framework by considering a moving window. Further, we present two algorithms, that in combination, estimate an appropriate test value for our control chart. We test our method on AR(1) processes exposed to level shifts of different sizes…
Modelling of functional profiles and explainable shape shifts detection: An approach combining the notion of the Fréchet mean with the shape invariant model
Hellenic Naval Academy（希腊海军学院）；Athens University of Economics and Business（雅典经济与商业大学）
