Evaluation of CUSUM Charts for Finite-Horizon Processes

This article analyses and evaluates the properties of a CUSUM chart designed for monitoring the process mean in short production runs. Several statistical measures of performance that are appropriate when the process operates for a finite-time horizon are proposed. The methodology developed in this article can be used to evaluate the performance of the CUSUM scheme for any given set of chart parameters from both an economic and a statistical point of view, and thus, allows comparisons with various other charts.     

A Combination of CUSUM Charts for Monitoring a Zero-Inflated Poisson Process

The Zero-inflated Poisson distribution (ZIP) is used to model the defects in processes with a large number of zeros. We propose a control charting procedure using a combination of two cumulative sum (CUSUM) charts to detect increases in the parameters of ZIP process, one is a conforming run length (CRL) CUSUM chart and another is a zero truncated Poisson (ZTP) CUSUM chart. The control limits of the control charts are obtained using both Markov chain-based methods and simulations. Simulation experiments show that the proposed method outperforms an existing method. Finally, a real example is presented.     

CUSUM Control Charts for Multivariate Poisson Distribution

A cumulative sum control chart for multivariate Poisson distribution (MP-CUSUM) is proposed. The MP-CUSUM chart is constructed based on log-likelihood ratios with in-control parameters, Θ₀, and shifts to be detected quickly, Θ₁. The average run length (ARL) values are obtained using a Markov Chain-based method. Numerical experiments show that the MP-CUSUM chart is effective in detecting parameter shifts in terms of ARL. The MP-CUSUM chart with smaller Θ₁ is more sensitive than that with greater Θ₁ to smaller shifts, but more insensitive to greater shifts. A comparison shows that the proposed MP-CUSUM chart outperforms an existing MP chart.     

Dual Nonparametric CUSUM Control Chart Based on Ranks

In this article, we provide a sequential rank-based dual nonparametric CUSUM (DNC) control chart for detecting arbitrary magnitude of shifts in the location parameter. It is a self-starting scheme and thus can be used to monitor processes at the start-up stages. Moreover, we do not require any prior knowledge of the underlying distribution. A simulation study demonstrates that the proposed control chart not only performs robustly for different distributions, but also is efficient in detecting various magnitudes of shifts. An illustrative example is given to introduce the implementation of our proposed DNC control chart. It is easy to construct and fast to compute.     

关于单变量统计过程控制图某些研究结果简介

文章仅对一元连续变量的静态与动态控制图研究现状进行了简单的总结和介绍,并给出了较详细的参考文献,希望为国内开展此方向的研究抛砖引玉。     

CUSUM control schemes for Gaussian processes

A CUSUM control scheme for detecting a change point in a Gaussian process is derived. An upper and a lower bound for the distribution of the run length and for its moments is given. In a Monte Carlo study the average run length (ARL) of this chart is compared with the ARL of two other CUSUM procedures which are based on approximations to the sequential probability ratio, and, moreover, with EWMA schemes for autocorrelated data. Results on the optimal choice of the reference value are presented. Furthermore it is investigated how these charts behave if the model parameters are estimated.     

Formulas for the Design of CUSUM Quality Control Charts

ABSTRACT

In the design of CUSUM control charts, it is common to use charts, tables, or software to find an appropriate critical threshold (h). This article provides an approximate formula to calculate the threshold directly from prespecified values of the reference value (k) and the in-control average run length (ARL₀). Formulas are also provided for choosing k and h from prespecified values of the in-control and out-of-control average run lengths.     

The EWMA sign chart revisited: performance and alternatives without and with ties

The EWMA Sign control chart is an efficient tool for monitoring shifts in a process regardless the observations'' underlying distribution. Recent studies have shown that, for nonparametric control charts, due to the discrete nature of the statistics being used (such as the Sign statistic), it is impossible to accurately compute their Run Length properties using Markov chain or integral equation methods. In this work, a modified nonparametric Phase II EWMA chart based on the Sign statistic is proposed and its exact Run Length properties are discussed. A continuous transformation of the Sign statistic, combined with the classical Markov Chain method, is used for the determination of the chart''s in- and out-of-control Run Length properties. Additionally, we show that when ties occur due to measurement rounding-off errors, the EWMA Sign control chart is no longer distribution-free and a Bernoulli trial approach is discussed to handle the occurrence of ties and makes the proposed chart almost distribution-free. Finally, an illustrative example is provided to show the practical implementation of our proposed chart.     

Grey Predictive Process Control Charts

The present article intends to develop some imputation methods to reduce the impact of non response at both the occasions in two-occasion successive (rotation) sampling. Utilizing the auxiliary information, which is only available at the current occasion, estimators have been proposed for estimating the population mean at the current occasion. Estimators for the current occasion are also derived as a particular case when there is non response either on the first occasion or second occasion. Behaviors of the proposed estimators are studied and their respective optimum replacement policies are also discussed. To study the effectiveness of the suggested imputation methods, performances of the proposed estimators are compared in two different situations, with and without non response. The results obtained are demonstrated with the help of empirical studies.     

Multivariate CUSUM and EWMA Control Charts for Skewed Populations Using Weighted Standard Deviations

This article proposes a heuristic method of constructing multivariate cumulative sum and exponentially weighted moving average control charts for skewed populations based on the weighted standard deviation method which adjusts the variance–covariance matrix of quality characteristics and approximates the probability density function using several multivariate normal distributions. These control charts, however, reduce to the conventional control charts when the underlying distribution is symmetric. In-control and out-of-control average run lengths of the proposed control charts are compared with those of the conventional control charts for multivariate lognormal and Weibull distributions. Simulation results show that considerable improvements over the standard method can be achieved when the underlying distribution is skewed.     

A binomial CUSUM chart for detecting large shifts in fraction nonconforming

This article studies a unique feature of the binomial CUSUM chart in which the difference (d _t ?d ₀) is replaced by (d _t ?d ₀)² in the formulation of the cumulative sum C _t (where d _t and d ₀ are the actual and in-control numbers of nonconforming units, respectively, in a sample). Performance studies are reported and the results reveal that this new feature is able to increase the detection effectiveness when fraction nonconforming p becomes three to four times as large as the in-control value p ₀. The design of the new binomial CUSUM chart is presented along with the calculation of the in-control and out-of-control Average Run Lengths (ARL₀ and ARL₁).     

A a comparison of the markov chain and the integral equation approaches for evaluating the run length distribution of quality control charts

Two methods that are often used to evaluate the run length distribution of quality control charts are the Markov chain and integral equation approaches. Both methods have been used to evaluate the cumulative sum (CUSUM) charts and the exponentially weighted moving average (EWMA) control charts. The Markov chain approach involves "discretiz-ing" the possible values which can be plotted. Using properties of finite Markov chains, expressions for the distribution of the run length, and for the average run length (ARL), can be obtained. For the CUSUM and EWMA charts there exist integral equations whose solution gives the ARL. Approximate methods can then be used to solve the integral equation. In this article we show that if the product midpoint rule is used to approximate the integral in the integral equation, then both approaches yield the same approximations for the ARL. In addition we show that the recursive expressions for the probability functions are the same for the two approaches. These results establish the integral equation approach as preferable whenever an integral equation can be found     

Measurement error effect on the CUSUM control chart

The performance of the cumulative sum (CUSUM) control chart for the mean when measurement error exists is investigated. It is shown that the CUSUM chart is greatly affected by the measurement error. A similar result holds for the case of the CUSUM chart for the mean with linearly increasing variance. In this paper, we consider multiple measurements to reduce the effect of measurement error on the charts performance. Finally, a comparison of the CUSUM and EWMA charts is presented and certain recommendations are given.     

A Nonparametric Synthetic Control Chart Using Sign Statistic

This article presents a synthetic control chart for detection of shifts in the process median. The synthetic chart is a combination of sign chart and conforming run-length chart. The performance evaluation of the proposed chart indicates that the synthetic chart has a higher power of detecting shifts in process median than the Shewhart charts based on sign statistic as well as the classical Shewhart X-bar chart for various symmetric distributions. The improvement is significant for shifts of moderate to large shifts in the median. The robustness studies of the proposed synthetic control chart against outliers indicate that the proposed synthetic control chart is robust against contamination by outliers.     

Guaranteed Conditional Performance of Control Charts via Bootstrap Methods

To use control charts in practice, the in‐control state usually has to be estimated. This estimation has a detrimental effect on the performance of control charts, which is often measured by the false alarm probability or the average run length. We suggest an adjustment of the monitoring schemes to overcome these problems. It guarantees, with a certain probability, a conditional performance given the estimated in‐control state. The suggested method is based on bootstrapping the data used to estimate the in‐control state. The method applies to different types of control charts, and also works with charts based on regression models. If a non‐parametric bootstrap is used, the method is robust to model errors. We show large sample properties of the adjustment. The usefulness of our approach is demonstrated through simulation studies.     

CUSUM Hypothesis Tests and Alternative Response Probabilities for Finite Poisson Trials

This article develops a new cumulative sum statistic to identify aberrant behavior in a sequentially administered multiple-choice standardized examination. The examination responses can be described as finite Poisson trials, and the statistic can be used for other applications which fit this framework. The standardized examination setting uses a maximum likelihood estimate of examinee ability and an item response theory model. Aberrant and non aberrant probabilities are computed by an odds ratio analogous to risk adjusted CUSUM schemes. The significance level of a hypothesis test, where the null hypothesis is non-aberrant examinee behavior, is computed with Markov chains. A smoothing process is used to spread probabilities across the Markov states. The practicality of the approach to detect aberrant examinee behavior is demonstrated with results from both simulated and empirical data.     

EWMA Control Charts for Monitoring High Yield Processes

The main objective of this article is to scrutinize the efficiency and verify the performance superiority of the one-sided EWMA control chart on high-yield processes. The proposed control chart is designed to detect both upward and downward shifts of the fraction of non conforming products and is developed based on non transformed geometric counts. Its algorithmic function is theoretically established and numerous performance measures are extracted using analytical methods based on the Markov modeling of the chart. Comparisons with traditional high yield control charts are conducted. Optimality tables and nomograms are included to help graphical determination of the optimal chart parameters.     

A HEWMA-CUSUM control chart for the Weibull distribution

The Weibull distribution is one of the most popular distributions for lifetime modeling. However, there has not been much research on control charts for a Weibull distribution. Shewhart control is known to be inefficient to detect a small shift in the process, while exponentially weighted moving average (EWMA) and cumulative sum control chart (CUSUM) charts have the ability to detect small changes in the process. To enhance the performance of a control chart for a Weibull distribution, we introduce a new control chart based on hybrid EWMA and CUSUM statistic, called the HEWMA-CUSUM chart. The performance of the proposed chart is compared with the existing chart in terms of the average run length (ARL). The proposed chart is found to be more sensitive than the existing chart in ARL. A simulation study is provided for illustration purposes. A real data is also applied to the proposed chart for practical use.     

Determining Optimal Number of Samples for Constructing Multivariate Control Charts

Normally, an average run length (ARL) is used as a measure for evaluating the detecting performance of a multivariate control chart. This has a direct impact on the false alarm cost in Phase II. In this article, we first conduct a simulation study to calculate both in-control and out-of-control ARLs under various combinations of process shifts and number of samples. Then, a trade-off analysis between sampling inspection and false alarm costs is performed. Both the simulation results and trade-off analysis suggest that the optimal number of samples for constructing a multivariate control chart in Phase I can be determined.