Adaptive CUSUM Charts Research Articles

Parametric and nonparametric versions of adaptive CUSUM charts for monitoring the location of individual measurements

Control charts are customarily developed under the assumption of normal quality characteristics to be monitored. However, in many real dataset applications, the normality assumption is not easy to hold. The in-control (IC) robustness of the charting schemes has significant importance for the valid out-of-control (OOC) performance. This article intends to investigate the IC robustness of the unbiased-function-based adaptive cumulative sum (ACUSUM) chart and nonparametric ACUSUM (NPACUSUM) chart against the non-normal process distributions including symmetrical, skewed, and contaminated normal distributions. The OOC run length profiles of the ACUSUM and the proposed NPACUSUM charts for the individual measurements against the non-normal distributions are also a part of this study. The run length profiles of the proposed charting schemes have been computed using the Monte Carlo simulation method. The artificial datasets have been taken from some symmetric and skewed distributions to implement the proposals. An electrical engineering dataset has also been taken for the implementation of the proposal on a real dataset.

New weighted adaptive CUSUM charts for monitoring the generalized variance of a bivariate normal process

In modern and high volume manufacturing environments, it is vital to early detect a small sustained shift in a short period of time as it may have a serious impact on a manufacturing process. Adaptive memory-type control charts have the ability to swiftly detect a range of the shift sizes, and one such control chart is a weighted adaptive CUSUM (abbreviated as C) chart. In this paper, using a simple likelihood ratio test-statistic, we propose two types of the C charts (with and without the normalizing transformation) using three different shift estimators for monitoring the generalized variance (GV) of a bivariate normally distributed production process. In addition, the sensitivities of these control charts are also enhanced with the variable sampling interval feature. Monte Carlo simulations are used to compute the zero-state and steady-state run-length characteristics of the proposed charts. Based on detailed run-length comparisons, it is observed that the proposed charts may uniformly and substantially outperform the existing charts when detecting different kinds of shifts in the GV. A real dataset is also selected to illustrate the implementation of the newly proposed charts.

Adaptive CUSUM chart with cautious parameter learning

AbstractThe classic CUSUM chart assumes that the in‐control (IC) mean and variance are known. In practice, these parameters are usually estimated from an IC Phase I sample. Recently, Capizzi and Masarotto proposed a cautious parameter learning scheme to incorporate Phase II IC observations in the estimation of the IC mean and variance to reduce the variation of conditional average run lengths (ARLs). In this paper, we develop a new cautious parameter learning scheme that can distinguish IC observations from out‐of‐control (OC) observations in the Phase II sample more effectively than Capizzi and Masarotto's scheme. As a result, our cautious parameter learning scheme can provide better estimation of the IC mean and variance. Combining the new cautious parameter learning scheme with an adaptive CUSUM chart, our proposed monitoring procedure is easy to implement, and is shown to have less variability in conditional ARLs and better overall performance for detecting different mean shifts than existing methods.

Maximum weighted adaptive CUSUM charts for simultaneous monitoring of process mean and variance

ABSTRACT In this paper, we propose maximum weighted adaptive Crosier CUSUM (MaxWACCUSUM) charts with three different unbiased estimators of the process mean and variance shifts for joint monitoring of the mean and variance of a normal process. The MaxWACCUSUM charts provide an overall good performance for detecting a range of joint mean and dispersion shift sizes. The run length characteristics of the proposed charts are computed using the Monte Carlo simulation method. A detailed comparative study is conducted to compare the run length performances of the proposed and existing charts. It is found that the MaxWACCUSUM charts are more sensitive than the maximum EWMA, double EWMA and CUSUM charts when detecting a range of joint shift sizes. Real datasets are considered to support the theory.

Nonparametric adaptive CUSUM chart for detecting arbitrary distributional changes

Nonparametric control charts that can detect arbitrary distributional changes are highly desirable due to their flexibility to adapt to different distributional assumptions and changes. However, most of the nonparametric control charts in the literature either can only detect location changes, or involve intensive computation. In this article, we propose a new nonparametric adaptive CUSUM chart. The proposed control chart can detect arbitrary distributional changes and is computationally efficient. Its self-starting nature makes the proposed control chart applicable to situations where no sufficiently large reference data are available. The proposed control chart also has a built-in post-signal diagnostics function that can identify what kind of distributional changes have occurred after an alarm. Our simulation study and real data analysis show that the proposed control chart performs well across a broad range of settings, and compares favorably with existing nonparametric control charts.

Enhanced adaptive CUSUM charts for process mean

ABSTRACTA statistical quality control chart is an important tool of the statistical process control, which is widely used to control and monitor a production process. The CUSUM chart is designed to detect a specific shift, provided that the shift size is known in advance. In practice, however, shift sizes are rarely known. It is then customary to use an adaptive CUSUM chart, which can effectively detect a range of shift sizes. In this paper, we enhance the sensitivities of the improved adaptive CUSUM mean charts using an auxiliary-information-based (AIB) mean estimator. The run length performances of the proposed charts are compared with those of the AIB adaptive and non-adaptive CUSUM charts in terms of the average run length (ARL), extra quadratic loss, and integral relative ARL. These run length comparisons reveal that the proposed charts are more sensitive than the existing charts when detecting different kinds of shift in the process mean. An example is given to demonstrate the implementation of existing and proposed charts.

New adaptive CUSUM charts for process mean

In this article, we develop an unbiased estimator of the mean shift, and then use it to enhance the sensitivity of the existing adaptive CUSUM (ACUSUM) chart. Moreover, a new ACUSUM chart is proposed using the structure of the Crosier CUSUM chart and the new mean shift estimator. A comprehensive zero-state run length comparison is conducted between the proposed and existing ACUSUM charts in terms of the average run length measures, which shows that the former uniformly outperform the latter. A real dataset is also used to explain the implementation of existing and proposed ACUSUM charts.

Optimal CUSUM and adaptive CUSUM charts with auxiliary information for process mean

ABSTRACTThe CUSUM chart is good enough to detect small-to-moderate shifts in the process parameter(s) as it can be optimally designed to detect a particular shift size. The adaptive CUSUM (ACUSUM) chart provides good detection over a range of shift sizes because of its ability to update the reference parameter using the estimated process shift. In this paper, we propose auxiliary-information-based (AIB) optimal CUSUM (OCUSUM) and ACUSUM charts, named AIB-OCUSUM and AIB-ACUSUM charts, using a difference estimator of the process mean. The performance comparisons between existing and proposed charts are made in terms of the average run length (ARL), extra quadratic loss and integral relative ARL measures. It is found that the AIB-OCUSUM and AIB-ACUSUM charts are more sensitive than the AIB-CUSUM and ACUSUM charts, respectively. Moreover, the AIB-ACUSUM chart surpasses the AIB-OCUSUM chart when detecting a range of mean shift sizes. Illustrative examples are given to support the theory.

An adaptive CUSUM chart for monitoring poisson rates with increasing population sizes

The CUSUM control chart is an effective tool for monitoring both variable and count data. However, it often requires the assumption of known process parameters. To get away from this assumption, the adaptive CUSUM (ACUSUM) procedure has been suggested, with primary focus on the monitoring of variable data. This paper extends the idea of ACUSUM control charts from the monitoring of variable data to the monitoring of count data based on an inhomogeneous Poisson model. The study is motivated by a real example of male thyroid cancer data in New Mexico. The comparison results show that the ACUSUM method generally performs better than the traditional Poisson CUSUM charts and other variations for monitoring a wide range of shifts in the Poisson mean. [Received 11 January 2014; Revised 15 July 2014; Accepted 25 August 2014]

Adaptive CUSUM charts for monitoring linear drifts in Poisson rates

There has been relatively less attention paid to the monitoring of linear drifts in Poisson rates. Although the conventional CUSUM charts can be used for monitoring linear drifts in Poisson rates, they often rely on the assumption of a known parameter. To get rid of this assumption, this paper develops adaptive CUSUM (ACUSUM) charts for monitoring drifts in Poisson rates. The basic idea is to first estimate the current Poisson mean level and then to dynamically update the likelihood ratio in the CUSUM chart based on the estimated mean. Three different mean estimators based on the exponentially weighted moving average (EWMA) schemes are discussed. The comparison results are shown to favor the ACUSUM chart, especially at small drifts.

An Adaptive CUSUM Chart with Single Sample Size for Monitoring Process Mean and Variance

This article proposes an adaptive absolute cumulative sum chart (called the adaptive ACUSUM chart) for statistical process control. The new development includes the variable sampling interval (VSI), variable sample size (VSS) and VSS and interval (VSSI) versions, all of which are highly effective for monitoring the mean and variance of a variable x by inspecting the absolute sample shift (where μ0 is the in‐control mean or target value of x). While the adaptive ACUSUM chart is a straightforward extension of the ABS CUSUM chart developed by Wu, et al., it is much more effective than all other adaptive CUSUM charts. Noteworthily, the superiority of VSI ACUSUM chart over the best adaptive CUSUM chart in literature is about 35% from an overall viewpoint. Moreover, the design and implementation of the adaptive ACUSUM chart are much simpler than that of all other adaptive CUSUM schemes. All these desirable features of the adaptive ACUSUM chart may be attributable to the use of a single sample size (n = 1). Another quite interesting finding is that the simpler VSI ACUSUM chart works equally well as the more complicated VSSI ACUSUM chart. Copyright © 2012 John Wiley & Sons, Ltd.

An enhanced adaptive CUSUM control chart

Adaptive CUSUM charts, referred to as ACUSUM charts, have attracted considerable attention from the research community. By adjusting the reference parameter k dynamically, an ACUSUM chart may achieve a better performance over a range of mean shifts than conventional CUSUM charts that are designed for maximal detection effectiveness at a particular level of process shift. This article studies a new feature of the ACUSUM chart related to an additional charting parameter w, i.e., the exponential of the sample mean shift in (x t – μ 0) w . The ACUSUM chart can be enhanced by adapting this parameter w according to the on-line estimated value of the mean shift, in conjunction with the reference parameter k. The testing cases reveal that this new adaptive CUSUM chart not only outperforms the earlier ACUSUM chart to a substantial degree, but also works as well as the most effective combined schemes consisting of a few CUSUM and/or charts. Furthermore, this enhanced ACUSUM chart is easier to design and implement in a computerized environment compared with those combined schemes. In addition, a general-purpose optimization algorithm is proposed to assist the designs of various CUSUM charts. This paper demonstrates that this algorithm can significantly improve the performance of many CUSUM charts over the entire process shift range. Moreover, a systematic performance comparison of eight CUSUM charts is presented. The findings from this comparison are useful aids for SPC practitioners to select an appropriate CUSUM chart for real applications. [Supplementary materials are available for this article. Go to the publisher's online edition of IIE Transactions for the following free supplementary resource: Appendix]

A CUSUM scheme with variable sample sizes and sampling intervals for monitoring the process mean and variance

AbstractThe adaptive control feature and CUSUM chart are two monitoring schemes that are much more effective than the traditional static Shewhart chart in detecting process shifts in mean and variance. However, the designs and analyses of the adaptive CUSUM chart are mathematically intractable and the operation is very laborious. This article proposes a VSSI WLC scheme, which is a weighted‐loss‐function‐based CUSUM (WLC) scheme using variable sample sizes and sampling intervals (VSSI). This scheme detects the two‐sided mean shift $\delta_{\mu}$ and increasing standard deviation shift $\delta_{\sigma}$ based on a single statistic WL (the weighted loss function). Most importantly, the VSSI WLC scheme is much easier to operate and design than a VSSI CCC scheme which comprises three individual CUSUM charts (two of them monitoring the increasing and decreasing mean shifts and one monitoring the increasing variance shift). Overall, the VSSI WLC scheme is much more effective than the static $\bar{X}$&S charts (by 72.36%), the VSSI $\bar {X}$&S charts (by 30.97%) and the static WLC scheme (by 50.94%) for detection. It is even more effective than the complicated VSSI CCC scheme for most cases. Copyright © 2006 John Wiley & Sons, Ltd.