Cumulative Sum Scheme Research Articles

The efficiency of CUSUM schemes for monitoring the multivariate coefficient of variation in short runs process

Current monitoring technologies emphasize and address the issue of monitoring high-volume production processes. The high flexibility and diversity of current industrial production processes make monitoring technology for small batch processes even more important. In multivariate process monitoring, a broader applicability exists in multivariate coefficients of variation (MCV) based monitoring schemes due to the lower restriction of the process. In view of the effectiveness of MCV monitoring and with the aim to achieve further performance improvement of current MCV monitoring schemes in a finite horizon production, we additionally introduce two one-sided cumulative sum (CUSUM) MCV schemes. In the case of deterministic and random shifts, the design parameters of the proposed schemes are obtained via an optimization procedure designed by the Markov chain method and the corresponding performance is analysed based on different run length (RL) characteristics, including the mean and the standard deviation. Simulation comparisons with existing exponentially weighted moving average (EWMA) MCV schemes show that the proposed CUSUM MCV schemes are more efficient in monitoring most of the shifts, including the deterministic and random shifts. Finally, to demonstrate the benefits of the new monitoring schemes, a comprehensive case study on monitoring a steel sleeve manufacturing process is conducted.

The Measurement Errors and Their Effects on the Cumulative Sum Schemes for Monitoring the Ratio of Two Correlated Normal Variables

Monitoring the ratio of two correlated normal random variables is often used in many industrial manufacturing processes. At the same time, measurement errors inevitably exist in most processes, which have different effects on the performance of various charting schemes. This paper comprehensively analyses the impacts of measurement errors on the detection ability of the cumulative sum (CUSUM) charting schemes for the ratio of two correlated normal variables. A thorough numerical assessment is performed using the Monte Carlo simulation, and the results indicate that the measurement errors negatively impact the performance of the CUSUM scheme for the ratio of two correlated normal variables. Increasing the number of measurements per set is not a lucrative approach for minimizing the negative impact of measurement errors on the performance of the CUSUM charting scheme when monitoring the ratio of two correlated normal variables. We consider a food formulation as an example that illustrates the quality control problems involving the ratio of two correlated normal variables in an industry with a measurement error. The results are presented, along with some suggestions for further study.

Designing Bayesian paradigm-based CUSUM scheme for monitoring shape parameter of the Inverse Gaussian distribution

Bayesian Control charts have gained prominence as highly effective statistical instruments in overseeing manufacturing processes and ensuring adept management of process variability. The Bayesian methodology proves especially adept in managing uncertainties related to parameters within the manufacturing sector. In this research endeavor, our focus lies in establishing the threshold for monitoring the shape parameter of the Inverse Gaussian distribution (IGD) in phase-II based on cumulative sum (CUSUM) chart in classical and Bayesian frameworks using a variety of loss functions for enhancing their utility and applicability and have compared them with the Shewhart and EWMA control charts. We have also compared the proposed IGD-based CUSUM charts with the Lomax-based CUSUM charts. The evaluation of the proposed approach hinges on performance measures such as average run length (ARL), standard deviation of run length (SDRL), and median of run length (MDRL). Through simulation, the study showcases the efficiency of the Bayesian approach-based CUSUM charts, highlighting their superiority over the conventional classical setup-oriented CUSUM charts. Additionally, the proposed approach is employed in practical scenarios using real-data case studies from the aerospace manufacturing sector. The outcomes derived from real-data analysis substantiate and reinforce the conclusions drawn from the simulated results.

Cumulative sum control charts for monitoring zero‐inflated COM‐Poisson processes

AbstractThe zero‐inflated Conway‐Maxwell Poisson (ZICMP) distribution models count data with many zero observations. ZICMP model has been developed assuming that zero observations exist with probability and the number of non‐conformities in a product unit follows the Conway‐Maxwell Poisson (COM‐Poisson) distribution with location parameter and dispersion parameter . This article presents four kinds of cumulative sum (CUSUM) charts for monitoring upward shifts in a ZICMP process. Three CUSUM schemes, namely ‐CUSUM, ‐CUSUM, and ‐CUSUM, have been designed to detect shift only in one parameter assuming that the other two are fixed and one CUSUM scheme, namely ‐CUSUM, has been designed to detect shifts in all the parameters. The performance of the proposed charts has been evaluated in terms of the average run‐length (ARL). Finally, a numerical example is given to demonstrate the application of the proposed charts.

Monitoring the Weibull Scale Parameter Based on Type I Censored Data Using a Modified EWMA Control Chart

In industrial production, the exponentially weighted moving average scheme is widely used to monitor shifts in product quality, especially small-to-moderate shifts. In this paper, we propose a modified one-sided EWMA scheme for Type I right-censored Weibull lifetime data for detecting shifts in the scale parameter with the shape parameter fixed. A comparative analysis with existing cumulative sum and exponentially weighted moving average results from the literature is provided. The zero-state and steady-state behaviour of the new scheme are considered with regard to the average run length, the standard deviation of the run length, and other performance measures. Our simulation shows stronger power in detecting changes in the censored lifetime data using the modified scheme than that using the traditional exponentially weighted moving average scheme, and the new scheme is superior to the cumulative sum scheme in most situations. A real-data example further demonstrates the effectiveness of the proposed method.

An optimally designed distribution‐free CUSUM procedure for tri‐aspect surveillance of continuous processes

AbstractIn the past few decades, research on nonparametric process monitoring schemes mainly dealt with the uni‐aspect or bi‐aspect schemes, focusing on monitoring process location or scale separately or jointly. Another critical process characteristic, namely, the process shape, is not explicitly dealt with in‐depth in most existing charting schemes. In classical hypothesis testing, some recent literature clearly showed that the multi‐aspect test statistics, although designed for very restrictive alternatives, often perform as well or better than many statistics for arbitrary distributional shifts. They are often better than Kolmogorov‐Smirnov or Cramér‐von Mises statistics and some empirical likelihood‐based test statistics. The current paper aims to use a tri‐aspect statistic to design a distribution‐free Phase‐II Cumulative Sum (CUSUM) charting scheme for monitoring any arbitrary process changes in the process. The proposal is nonparametric and is equivalent to an unknown standard case. It reflects, in addition, which parameters, among location, scale, or shape, are more responsible for a signal. The construction of the CUSUM scheme from an existing tri‐aspect Shewhart‐type chart is simple, so significant attention is devoted to determining a near‐optimal reference parameter of the chart in keeping an unknown shift type in mind. Comparisons of the optimal performance of various competitors are considered in terms of the median run length (MRL) metric. The proposed charting scheme designed with the tri‐aspect statistic compares highly favorably with many existing SPM schemes. The same is evident from our findings based on the Monte‐Carlo simulation. Finally, the proposed schemes are illustrated with a flow‐width measurement monitoring example.

Performance of the Shiryaev‐Roberts‐type scheme in comparison to the CUSUM and EWMA schemes in monitoring weibull scale parameter based on Type I censored data

AbstractThis paper introduces a process monitoring scheme based on the Shiryaev‐Roberts (SR) procedure for Type I right‐censored Weibull lifetime data for detecting shifts in the scale parameter for a fixed value of the shape parameter. The performance of the new charting scheme is evaluated under different censoring rates and sample sizes via Monte Carlo simulation. Comparison with two existing schemes, namely, the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) schemes, is performed in terms of average run length (ARL). Both zero‐state and steady‐state shifts are considered in this study for impartial comparisons. The numerical results based on simulation are highly encouraging in favour of the SR procedure. Our findings show that the new scheme outperforms the other two schemes in some cases. An illustrative example is given to illustrate the application of the proposed scheme.

New extended distribution-free homogenously weighted monitoring schemes for monitoring abrupt shifts in the location parameter.

A homogeneously weighted moving average (HWMA) monitoring scheme is a recently proposed memory-type scheme that gained its popularity because of its simplicity and superiority over the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) schemes in detecting small disturbances in the process. Most of the existing HWMA schemes are designed based on the assumption of normality. It is well-known that the performance of such monitoring schemes degrades significantly when this assumption is violated. Therefore, in this paper, three distribution-free monitoring schemes are developed based on the Wilcoxon rank-sum W statistic. First, the HWMA W scheme is introduced. Secondly, the double HWMA (DHWMA) W scheme is proposed to improve the ability of the HWMA W scheme in detecting very small disturbances in the location parameter and at last, the hybrid HWMA (HHWMA) W scheme is also proposed because of its flexibility and better performance in detecting shifts of different sizes. The zero-state performances of the proposed schemes are investigated using the characteristics of the run-length distribution. The proposed schemes outperform their existing competitors, i.e. EWMA, CUSUM and DEWMA W schemes, in many situations, and particularly the HHWMA W scheme is superior to these competitors regardless of the size of the shift in the location parameter. Real-life data are used to illustrate the implementation and application of the new monitoring schemes.

A new multivariate CUSUM chart for monitoring of covariance matrix with individual observations under estimated parameter

AbstractMultivariate charts for process dispersion detect changes in the variance‐covariance matrix of a process. Most of the existing multivariate charts for monitoring the dispersion of individual observations were designed based on exponentially weighted moving average (EWMA) charting schemes. However, an alternative to the EWMA scheme is the cumulative sum (CUSUM) control chart, which has proven to be better in some cases. In the last decades, few studies have been conducted on methods based on multivariate CUSUM (MCUSUM) schemes to monitor the covariance matrix of individual observations. Consequently, we propose a new MCUSUM dispersion chart. Besides, most of the existing methods have been developed by assuming that the process parameters are known and that the process distribution is normal; these assumptions are not always true in practice. Hence, we compare the performance of the proposed chart and its counterparts based on the estimation effects under normal and non‐normal distributions. The results show that the proposed chart outperforms the other charts in terms of minor shifts in the process. Similarly, the proposed chart is the most robust to the normality assumption among the compared charts. The average value of the conditional average run length was used as the performance measure. Finally, the proposed method was also implemented with a simulated dataset to support the stated proposal findings.

Two CUSUM schemes for simultaneous monitoring of unknown parameters of a shifted exponential process and its application in monitoring of call durations in telemarketing

ABSTRACT Telemarketing is crucial in sales promotion for products and services, and it is very popular in businesses and industries. The call duration during Telemarketing is an important feature that requires continuous surveillance. It is linked to project management related to coverage and outreach of the campaign, the overall time of the campaign, etc. In this paper, we model the call duration of a successful Telemarketing in a bank via the shifted exponential distribution and propose two Cumulative Sum (CUSUM) schemes for monitoring both the parameters of a shifted exponentially distributed process. The CUSUM SEMLE-Max and CUSUM SEMLE-Dis schemes, based, respectively, on max and distance statistics, are proposed by assuming that the standard values of the process parameters are unknown and estimated from a reference sample. The proposed schemes’ average run-length () and standard deviation of run-length () performances are evaluated using the Monte-Carlo simulation. The CUSUM SEMLE-Max and CUSUM SEMLE-Dis schemes offer impressive improvements over their Shewhart-type counterparts in terms of run-length properties. The practical application of the proposed procedures is illustrated by analyzing the call duration of telemarketing promotions in the Portuguese banking sector. A summary and some future research problems conclude the paper.

A parameter‐free exponential control chart based on Kullback‐Leibler information for time‐between‐events monitoring

AbstractThis paper proposes a Kullback‐Leibler information (KLI) control chart for monitoring the occurrence rate of a process that follows an exponential distribution. The chart is designed for detecting a decrease, an increase, or both in the occurrence rate. Due to the skewness of the plotted statistic, the performance of the chart is investigated separately when the occurrence rate is estimated by the maximum likelihood estimator, the uniformly minimum variance unbiased estimator, and the minimum mean squared error estimator. The KLI chart is effective for a wide range of shifts of the occurrence rate. Its performance is slightly behind cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts around sizes of shift where these two charts are optimized. It outperforms these two charts elsewhere. The relative mean index (RMI) values show that the KLI chart has better overall performance. A backward empirical sequential test is used for the plotted statistic of the chart. It is shown equivalent to the cumulative sum scheme and the change point method for the generalized likelihood ratio chart in terms of run length. The use of KLI chart does not require any design parameter other than the control limit.

Change points of real GDP per capita time series corresponding to the periods of industrial revolutions

Industrial revolutions can be linked to fundamental technological inventions that have provided mankind with subsidiary and complementary developments. Breakthrough changes have occurred in energy sources, telecommunication technologies, transportation methods, and production systems during the periods covered by industrial revolutions. Altogether these changes have accelerated the march of economic progress resulting in social transformations. While economies of frontier countries benefitted vastly from such changes, others had delays in following the advances. Therefore, observed effects of industrial revolutions varied among countries. In this study, it is assumed that there are identifiable economic changes during industrial revolutions. Real gross domestic product (GDP) per capita is used as an indicator to trace economic growth in the UK, the US, France, the Netherlands, Germany, and Turkey. The real GDP per capita time series of the countries mentioned are modelled by autoregressive integrated moving average models and a cumulative sum scheme is utilized to detect the change points in successive periods corresponding to industrial revolutions. Together with the levels of the real GDP per capita after a change, timings of the changes are compared for the countries studied. Results indicate that there are delays for some countries in benefiting from the prosperity offered by the developments.

Performance of Joint Quality Monitoring Schemes under Gaussian distribution

Jointly monitoring the process mean and variance has become a well-known topic in statistical quality control literature after it is considered as a bivariate problem. Many joint monitoring schemes have been proposed by using the Shewhart, cumulative sum and exponentially weighted moving average techniques. In this paper, best performing schemes from each technique has been selected and compared for their performance using average run length properties. It was found that selection of better joint monitoring scheme based on the shift in mean and variance to be detected quickly. In particular, the Shewhart distance joint monitoring scheme performs well when there is larger shifts in mean, variance or in both. In addition, the Shewhart distance joint monitoring scheme performs specific when there is no shift in mean and decrease in variance. For the smaller shifts in mean, variance or in both, cumulative sum and exponentially weighted moving average joint monitoring schemes can be recommended. At this scenario exponentially weighted moving average joint monitoring scheme performs marginally better than the cumulative sum scheme.

CUSUM control charts with variable sampling interval for monitoring the ratio of two normal variables

AbstractIn many industrial manufacturing processes, the ratio between two normal random variables plays a key role in ensuring quality of products. Thus, monitoring this ratio is an important task that is well worth considering. In this paper, we combine a variable sampling interval (VSI) strategy with a cumulative sum (CUSUM) scheme to create a new type of control chart for purpose of tracking the ratio between two normal variables. The average time to signal (ATS) and the expected average time to signal (EATS) criteria are used to evaluate the performance of the new VSI CUSUM RZ control chart. The numerical results show that the proposed control chart has much more attractive performance in comparison with the standard CUSUM‐RZ control chart and the VSI EWMA‐RZ control chart.

Comparisons of some distribution-free CUSUM and EWMA schemes and their applications in monitoring impurity in mining process flotation

Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) schemes are widely used in monitoring small and persistent shifts in specific process characteristics. Nonparametric (Distribution-free) CUSUM and EWMA schemes are useful in detecting such changes when the underlying process distribution is unknown or complicated. The CUSUM-Wilcoxon rank-sum (CUSUM-WRS), CUSUM-Precedence, EWMA-WRS, and EWMA-Precedence are well-known distribution-free CUSUM and EWMA schemes for Phase-II monitoring of a shift in the unknown location parameter of a process. In this article, we compare their performances with two robust CUSUM and EWMA schemes based on the Hogg-Fisher-Randle (HFR) type statistic. We investigate the accomplishments of these CUSUM and EWMA schemes in detecting shifts of different sizes for various process distributions and show that the proposed CUSUM and EWMA HFR schemes perform favourably for highly skewed distributions. In this article, we also discuss the efficacy of these nonparametric CUSUM and EWMA schemes when the test sample size is small. We also outline the implementation strategies various plans with an illustration using a dataset from the iron ore mining plant. We offer some concluding remarks and future research problems.

Power Quality Events Segmentation in a Voltage Waveform using Joint Triggering Point Detection Scheme

Accurate and timely detection of power quality (PQ) events is imperative for adequate corrective measures to be taken. This paper presents a method of PQ event detection development called joint triggering point detection (JTPD) scheme with a view to achieving a more accurate PQ event detection in a voltage waveform. The JTPD combines the advantage of cumulative sum (CUSUM) algorithm for the statistical distribution of a signal waveform and the discrete wavelet transform (DWT) for change-point detection. The performance of JTPD scheme using detection rate was compared with CUSUM and DWT schemes and based on the analysis, the detection rate of JTPD, CUSUM and DWT were 100%, 95% and 50% respectively. With this, the proposed approach outperformed CUSUM and DWT schemes by 5% and 50% respectively. Hence, the proposed approach is suitable for accurate detection of voltage dip, swell and interruption PQ events Keywords: Cumulative Sum, Discrete Wavelet Transform, Joint Triggering Point Detection, Power Quality, Voltage Waveform

A robust CUSUM scheme with a weighted likelihood ratio to monitor an overdispersed counting process

Abstract The Poisson cumulative sum (CUSUM) control chart is a method to monitor count data, which are commonly modeled by a Poisson distribution. However, the assumption of a Poisson distribution may not be valid in practice and the robustness of the Poisson CUSUM chart is shown to be poor when the volatility is greater than expected, a phenomenon known as overdispersion. Motivated by this, we propose an improved weighted Poisson CUSUM control chart based on the weighted log-likelihood ratio. For the purpose of discounting the unexplained volatility, smaller weights are assigned to the larger shifts. The continuity, monotonicity and convergence of the cumulative statistic proved in this paper make the design more practical. Compared with the conventional Poisson CUSUM and the Winsorized Poisson CUSUM control charts, the new weighted Poisson CUSUM scheme performs most robustly under overdispersion. In addition, the fast initial response (FIR) feature makes the chart more sensitive to start-up cases. A real-data example of monitoring daily orders on a start-up E-commerce company illustrates the superiorities of the proposed scheme.

Detection and Correction of Element Failures Using a Cumulative Sum Scheme for Active Phased Arrays

In this paper, we propose a novel technique that integrates the detection of element failures and the correction of damaged patterns into a systematic process for active phased arrays. The proposed technique uses the framework of mutual coupling method, and its kernel is based on a cumulative sum (CUSUM) scheme that monitors attenuators and phase shifters inside transmit/receive modules. The proposed method is organized into two phases. The phase one copes with the detection of failed components. By cumulating time-series data, the CUSUM can readily detect the defective elements even though very small shifts are examined. We test the effectiveness through a 448-element active phased array. The measured results show that no type I error is found, and the type II error of the most challenging scenario is significantly reduced. The phase two aims at the correction of patterns. By integrating the experimental result provided by the phase one and a least-squares method, the proposed technique can determine the new excitations leading to reduced sidelobe levels and a desired main beam characteristic. The capability is verified through three examples including linear and planar arrays. The results show that the proposed method can correct the damaged patterns with low computational complexity.

A CUSUM chart for detecting the intensity ratio of negative events

This paper developed a single cumulative sum (CUSUM) scheme, called the UCUSUM chart, for simultaneously detecting the size N and time interval T of an event. The new chart used the information of size and frequency of the event and the UCUSUM chart is carried out using the only one statistic U, which contains both T and N; on the other hand, the UCUSUM chart could allocate the detection power to the T shifts and the N shifts. The results present that the UCUSUM chart is significantly powerful compared to other charts which are in the current research with either the time interval T or with the size N. The UCUSUM chart could be applied in many areas including industries and non-industries and the performance of the new chart shows it is much effective in example.

State-Based General Gamma CUSUM for Modeling Heart Rate Variability Using Electrocardiography Signals

Traditional approaches based on short-term heart rate variability for cardiovascular disease diagnosis fail to capture the long-term dynamic information and individual effect from electrocardiography signals among subjects when examining the physiological condition. These shortages may lead to incorrect disease detection and weaken diagnosis performance. To address these problems, this paper proposes a new disease detection approach by considering the long-term dynamics and meanwhile the individual effect existing among subjects. Specifically, a multistate general Gamma cumulative sum (GGCUSUM) scheme is developed for signal state detection. Further, a backward elimination algorithm based on the exponential likelihood ratio test (ELRT) is proposed to reduce the risk of incorrect detection of change points. A general disease severity index is then designed based on our approach to satisfy the clinical requirement for disease diagnosis. A real clinical case from one of cardiovascular diseases, is given to validate the proposed approach, of which the result demonstrates the effectiveness with a satisfactory detection performance.