CUSUM Charts Research Articles

Adaptive EWMA and CUSUM charts with dynamic probability control limits

The utilization of dynamic probability control limits (DPCLs) can prove advantageous in the development of control charts, particularly in scenarios where sample sizes, areas of opportunity, or any other covariate values exhibit temporal fluctuations. In this study, we design the DPCLs-based two adaptive EWMA (AE) and adaptive CUSUM (AC) charts for monitoring the mean of a normally distributed process. We use Monte Carlo simulations to estimate the conditional false alarm rates, zero-state and steady-state (SS) average run-length profiles of the adaptive charts. It is found that the proposed AE and AC charts outperform the existing AE chart in terms of relative mean index. An example is given to demonstrate the implementation of the proposed charts.

Comparative evaluation of VAE-based monitoring statistics for real-time anomaly detection in AIS data

ABSTRACT Real-time tracking and anomaly detection in vessel navigation using AIS (Automatic Identification System) data is a critical issue in maritime logistics. However, the substantial volume, inherent noise, and irregular temporal intervals of AIS data pose challenges in applying traditional Statistical Process Monitoring (SPM) methods. To overcome these challenges, recent research has proposed the use of deep probabilistic latent variable models, specifically variational autoencoders (VAE). However, the monitoring statistics based on VAE employed in previous studies may vary. In this study, we conduct a comprehensive comparison and evaluation of various monitoring statistics based on VAE within the context of statistical process monitoring. Furthermore, we propose a new real-time monitoring method by integrating VAE-based monitoring statistics with the CUSUM chart for monitoring AIS data. By utilizing both simulated and real-world AIS data, our proposed method demonstrates superior detection performance and robustness compared to traditional methods.

New CUSUM and EWMA charts with simple post signal diagnostics for two‐parameter exponential distribution

AbstractThe two‐parameter exponential distribution (TPED) is often used to model time‐between‐events data. In this paper, we propose CUmulative SUM and exponentially weighted moving average charts for simultaneously monitoring the parameters (location and scale) of the TPED. A key feature of the proposed charts is their straightforward post‐signal diagnostics. Monte Carlo simulations are used to estimate the zero‐state and steady‐state average run‐length (ARL) profiles of the proposed charts. The ARL performances of existing and proposed charts are assessed in terms of expected weighted run‐length and relative mean index. It is found that the proposed charts outperform the existing charts. A real dataset is used to illustrate the implementation of the proposed charts.

Comment on A Method for Continuous Surgeon Improvement in Rectal Cancer: Observed Minus Expected (O-E) Chart Versus CUSUM Chart.

Comment on A Method for Continuous Surgeon Improvement in Rectal Cancer: Observed Minus Expected (O-E) Chart Versus CUSUM Chart.

948 Learning Curve for Single Surgeon Approach of Laparoscopic Peritoneal Dialysis Catheter Insertion Using Freehand Robotic Assistant

Abstract Introduction A single surgeon approach to laparoscopic peritoneal dialysis catheter insertion can offer improved efficiency to surgical lists as well as freeing up surgeons to complete other operations or ward jobs. Simple, low risk laparoscopic procedures which can be completed quickly with the aid of a robotic assistant can help hospitals improve efficiency of surgical departments. This study aims to assess the learning curve of a single surgeon approach using the Freehand robotic assistant in peritoneal dialysis catheter insertion. Method Seventy-two cases were analysed over a period of three years from 2020-2023, operations involving the Freehand robotic assistant were performed by a single consultant surgeon. The duration of the procedure was one of the main factors evaluated for assessing learning curve. Other factors considered were re-do procedures and post-operative complications. Results Total mean operating time was 30.23 minutes. The peak point of the CUSUM chart for operating time was found at case 30, from this point onward trend began to decrease. From the 72 cases 6 had post-operative complications due to peritonitis and 2 had to be re-done due to inadequate positioning. Conclusions CUSUM chart indicated proficiency was gained at operation 30. There were minimal replacements necessary and only 6 cases that had post-operative complications all within the first 25 operations. This implies a short learning curve with great potential for improving efficiency with the use of the Freehand robot in simple laparoscopic procedures such as peritoneal dialysis catheter insertion.

Ameliorating the diagnostic power of combined shewhart-memory-type control chart strategies for mean

A great deal of research in statistical process control (SPC) involves the integration of different ideas to achieve the optimal detection of anomalies in the location parameter of a process. Likewise, the prime rationale of the combined (Shewhart-Memory-type) strategies is to promptly detect the shift of any size: small, moderate or large. An effort, along the same lines, is made to detect shifts in the mean of a normally distributed process. The usual difference estimator of the process mean instead of the sample mean is employed in the Mixed EWMA-Crosier CUSUM Combined Shewhart Mixed EWMA-CUSUM and Combined Shewhart Mixed EWMA-Crosier CUSUM charts to make their upgraded versions. This enhances the sensitivity of these charts against any shift size. The performance of the charts is appraised and compared with some notable charts of the same ilk, with reference to the average run length (ARL), relative average run length, extra quadratic loss and performance comparison index. An illustrative example is also provided.

A Method for Continuous Surgeon Improvement in Rectal Cancer: Risk-Adjusted Cumulative Sum.

To develop and analyze a risk-adjusted cumulative sum (RA-CUSUM) chart as a potential method to monitor individual surgeon performance in robotic total mesorectal excision (TME) for rectal cancer. Currently, surgeons lack real-time tools to monitor and enhance their performance beyond residency completion. While national quality programs exist, granular, individual-level data is crucial for continuous improvement. Previous studies suggest CUSUM charts hold promise in identifying performance trends and outliers. This retrospective study analyzed data from 640 robotic TME cases performed by 12 surgeons at two institutions. RA-CUSUM charts were generated for three outcomes: complications, operative time, and length of stay. The overall RA-CUSUM curves for operative time and complications showed an initial learning phase followed by a plateau or downward slope, indicating proficiency or improvement. However, individual surgeon curves revealed significant heterogeneity. Three surgeons consistently excelled in operative time, while five minimized complications most effectively. Potential quality improvement could be implemented to drive performance toward positive outliers. No differences were found in unadjusted outcomes, including conversion, number of lymph nodes harvested, and positive circumferential margins. The RA-CUSUM chart is a promising method for identifying individual surgeon performance in robotic TME. It could help surgeons, teams, and leaders identify improvement areas and benchmark themselves against positive outliers. Further studies are needed to explore the potential of RA-CUSUM for implementing interventions to improve surgical quality.

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.

Navigating the challenges of minimally invasive mitral valve surgery: a risk analysis and learning curve evaluation

BackgroundThis study aimed to report the risk and learning curve analysis of a minimally invasive mitral valve surgery program performed through a right mini-thoracotomy at a single institution.MethodsFrom January 2013 through December 2019, 266 consecutive patients underwent minimally invasive mitral valve surgery in our department and were included in the current study. Multiple logistic regression analysis was used for the adverse event outcome. Distribution over time of perioperative complications, defined as clinical endpoints in the Valve Academic Research Consortium-2 (VARC-2) consensus document, as well as CUSUM charts for assessment of cardiopulmonary bypass and aortic cross-clamping duration over time, has been performed for learning curve assessment.ResultsOverall incidences of postoperative stroke (1.1%), myocardial infarction (1.1%), and thirty-day mortality (1.5%) were low. The mitral valve reconstruction rate in our series was 95%. Multivariable analysis revealed that concomitant tricuspid valve surgery (OR 4.44; 95%CI 1.61–11.80; p = 0.003) was significantly associated with adverse event outcomes. Despite a trend towards adverse event outcomes in patients with preexisting active mitral valve endocarditis (OR 2.69; 95%CI 0.81–7.87; p = 0.082), mitral valve pathology did not significantly impact postoperative morbidity and mortality. Distribution over time of perioperative complications, defined as clinical endpoints in the VARC-2 consensus document, showed a trend towards an improved complication rate after the initial 65–100 procedures.ConclusionsMitral valve surgery via right-sided mini-thoracotomy can be implemented safely with low perioperative morbidity and mortality rates. Careful patient selection regarding isolated mitral valve surgery in the presence of degenerative mitral valve disease may represent a significant safety issue during the learning curve.Trial Registration: The cantonal ethics commission of Zurich approved the study (registration ID 2020-00752, date of approval 24 April 2020).

Improving Sensitivity of the DEWMA Chart with Exact ARL Solution under the Trend AR(p) Model and Its Applications

The double exponentially weighted moving average (DEWMA) chart is a control chart that is a vital analytical tool for keeping track of the quality of a process, and the sensitivity of the control chart to the process is evaluated using the average run length (ARL). Herein, the aim of this study is to derive the explicit formula of the ARL on the DEWMA chart with the autoregressive with trend model and its residual, which is exponential white noise. This study shows that this proposed method was compared to the ARL derived using the numerical integral equation (NIE) approach, and the explicit ARL formula decreased the computing time. By changing exponential parameters that were relevant to evaluating in various circumstances, the sensitivity of AR(p) with the trend model with the DEWMA chart was investigated. These were compared with the EWMA and CUSUM charts in terms of the ARL, standard deviation run length (SDRL), and median run length (MRL). The results indicate that the DEWMA chart has the highest performance, and when it was small, the DEWMA chart had high sensitivity for detecting processes. Digital currencies are utilized to demonstrate the efficacy of the proposed method; the results are consistent with the simulated data. Doi: 10.28991/ESJ-2023-07-06-03 Full Text: PDF

Weighted adaptive CUSUM mean chart with variable sample size and sampling interval

The classical CUSUM (C) chart is designed by assuming that the size of the shift in the process, in which a quick detection is important, is known a priori. However, this assumption is often violated in practice, as the actual shift size is rarely known even though it may vary within a certain interval. The adaptive C (AC) chart gives better protection against process shifts in a certain interval than the classical C chart. In this paper, we propose a new weighted AC (WAC) chart with the variable sample size and sampling interval (VSSI) feature, which encompasses the fixed sampling rate (FSR), variable sample size (VSS) and variable sampling interval (VSI) based WAC charts. The Monte Carlo simulations are adopted in the run-length computations of the considered charts, where the newly developed (VSS and VSSI based) charts outperform the existing (FSR and VSI) WAC charts when detecting small shifts in the process mean. An illustrative example shows the implementation of the existing and proposed WAC charts.

A critical note on memory‐type charts with the Brownian statistic

AbstractIn the paper, we conduct an assessment of the conditional expected delay (CED) profiles of the EWMA, CUSUM, and dual CUSUM charts based on a statistic linked to Brownian motion. The study reveals a noteworthy observation: the CED curves for these control charts may exhibit an increasing trend over time. Moreover, in a steady‐state scenario, it becomes evident that these control charts could be outperformed by their established counterparts.

Efficient Monitoring of a Parameter of Non-Normal Process Using a Robust Efficient Control Chart: A Comparative Study

The control chart is a fundamental tool in statistical process control (SPC), widely employed in manufacturing and construction industries for process monitoring with the primary objective of maintaining quality standards and improving operational efficiency. Control charts play a crucial role in identifying special cause variations and guiding the process back to statistical control. While Shewhart control charts excel at detecting significant shifts, EWMA and CUSUM charts are better suited for detecting smaller to moderate shifts. However, the effectiveness of all these control charts is compromised when the underlying distribution deviates from normality. In response to this challenge, this study introduces a robust mixed EWMA-CUSUM control chart tailored for monitoring processes characterized via symmetric but non-normal distributions. The key innovation of the proposed approach lies in the integration of a robust estimator, based on order statistics, that leverages the generalized least square (GLS) technique developed by Lloyd. This integration enhances the chart’s robustness and minimizes estimator variance, even in the presence of non-normality. To demonstrate the effectiveness of the proposed control chart, a comprehensive comparison is conducted with several well-known control charts. Results of the study clearly show that the proposed chart exhibits superior sensitivity to small and moderate shifts in process parameters when compared to its predecessors. Through a compelling illustrative example, a real-life application of the enhanced performance of the proposed control chart is provided in comparison to existing alternatives.

Econometric modeling for the influence of economic variables on secondary copper production in India

The stocks and flows of primary metal resources, and the estimates of so generated secondary scrap metal flows are tracked and analysed utilizing the traditional analytical method of ‘material flow analysis’. However, the scrap availability and hence the secondary metal production is also influenced by fluctuating economic conditions. Here we utilize the autoregressive distributed lag approach to model the influence of selected economic variables on secondary metal production and hence the scrap availability in the country. We find electricity index, mining index, and electric power generation and distribution correlate with secondary copper production in long-run; and electricity index correlate with secondary copper production in short-run. A 1% increase in electricity index initially leads to 1.76% drop in secondary copper production in short-run, and 3.69% increase in secondary copper production in long-run. A 1% increase in both mining index, and electric power generation and distribution leads to 4.57% and 5.18% decrease in secondary copper production in long-run respectively. The CUSUM and CUSUMSQ charts show the developed ARDL model to be stable over time. Based on the model dependence, we suggest that the policy makers shall devise policies towards reducing copper mining activities thereby encouraging stakeholders to focus on effective collection and efficient recycling of copper towards conservation of natural metal resources.

Single and dual reference-free CUSCORE charts for detecting unknown patterned mean shifts

Most of the control charts in the literature focus on detecting constant mean shifts. In practice, dynamic and time-varying process mean shifts are prevalent in the monitoring of feedback-control and autocorrelated processes. The cumulative score (CUSCORE) charts have been proposed in the literature to detect dynamic and time-varying process shifts. One of such control charts is a reference-free CUSCORE (called RFCS-I) chart that may detect unknown patterned mean shifts effectively. In this paper, single and dual CUSCORE charts are developed to enable an efficient detection of unknown patterned mean shifts. The developed single CUSCORE chart, called the RFCS-II chart, is based on the control charting structure of the Crosier CUSUM (called CU-II) chart. Additionally, two RFCS-I charts are integrated with reflecting boundaries, while an integration of two RFCS-II charts is also made, to develop the dual RFCS-I (DRFCS-I) and dual RFCS-II (DRFCS-II) charts, respectively. In addition, the RFCS-I and RFCS-II charts are integrated to propose the mixed dual reference-free CUSCORE chart, called the MDRFCS chart. In essence, four CUSCORE charts, i.e. the RFCS-II, DRFCS-I, DRFCS-II and MDRFCS charts are suggested in this paper. The Monte Carlo simulation and a real application illustrate the superiority of these developed CUSCORE charts over their existing counterparts when detecting patterned mean shifts in terms of zero-state and steady-state relative mean indexes.

An optimization method for change-point monitoring in finite samples sequence

This article proposes a method of optimizing control chart (sequential test) to detect an abnormal change in a sequence of finite or even small samples with the unknown change-point and the unknown post-change probability distribution. We not only introduced a performance measure for a given charting statistic to evaluate the detection effect of a control chart, but also constructed an optimal control chart under the measure. The effect of optimization method was illustrated by numerical simulations of three optimized Shewhart, CUSUM and EWMA control charts, and a real data example.

Methylene blue sentinel lymph node biopsy for breast cancer learning curve in the COVID-19 era: How many cases are enough?

Background: Sentinel lymph node biopsy (SLNB) is now the gold standard for early breast cancer with clinically negative lymph nodes (N0). According to the Indonesian Board-Certified oncologist surgeon, the learning curve for evaluating fellow breast surgeons to achieve this competency could have been shorter due to the COVID-19 pandemic. This study aims to see if the learning curve for sentinel lymph node (SLN) identification can be shortened. Methods: Trainee breast surgeons were taught to perform SLNB on breast cancer patients. Intraoperative assessment and completion of axillary lymph node dissection (ALND) were performed in the first setting for standardization with the attending surgeon. Sentinel lymph node identification was plotted on cumulative sum chart (CUSUM) limitations for evaluating the variability competency between the attending and trainee surgeons based on a target identification rate of 85%. Results: We concluded that CUSUM charts are the best tools currently available for assessing psychomotor learning SLNB. According to a CUSUM chart based on a reasonable set of parameters, the learning curve for SLNB using methylene blue dye is reached after 4-5 consecutive positively detected SLN. Conclusion: CUSUM Chart showed that experienced breast surgeons have completed the SLNB learning curve after 4-5 successful methylene blue attempts. In the future, this learning curve analysis can be applied to trainee breast surgeons by utilizing a proxy measure for failure, such as failure to identify the SLN.

CUSUM learning curves: what they can and can't tell us.

There has been increased interest in assessing the surgeon learning curve for new skill acquisition. While there is no consensus around the best methodology, one of the most frequently used learning curve assessments in the surgical literature is the cumulative sum curve (CUSUM) of operative time. To demonstrate the limitations of this methodology, we assessed the CUSUM of console time across cohorts of surgeons with differing case acquisition rates while varying the total number of cases used to calculate the CUSUM. We compared the CUSUM curves of the average console times of surgeons who completed their first 20 robotic-assisted (RAS) cases in 13, 26, 39, and 52weeks, respectively, for their first 50 and 100 cases, respectively. This analysis was performed for prostatectomy (1094 surgeons), malignant hysterectomy (737 surgeons), and inguinal hernia (1486 surgeons). In all procedures, the CUSUM curve of the cohort of surgeons who completed their first 20 procedures in 13weeks demonstrated a lower slope than cohorts of surgeons with slower case acquisition rates. The case number at which the peak of the CUSUM curve occurs uniformly increases when the total number of cases used in generation of the CUSUM chart changes from 50 to 100 cases. The CUSUM analyses of these three procedures suggests that surgeons with fast initial case acquisition rates have less variability in their operative times over the course of their learning curve. The peak of the CUSUM curve, which is often used in surgical learning curve literature to denote "proficiency" is predictably influenced by the total number of procedures evaluated, suggesting that defining the peak as the point at which a surgeon has overcome the learning curve is subject to routine bias. The CUSUM peak, by itself, is an insufficient measure of "conquering the learning curve."

Optimized control charts using indifference regions

In statistical process monitoring, the CUSUM and EWMA control charts have received considerable attention because of their remarkable ability to detect small sustained shifts. In practice, small process variation and shifts are anticipated beforehand in many processes, so the focus should be on detecting a moderate to a large shift. The aforementioned charts identify minor changes in population parameters as out-of-control scenarios; thus, “small” and potentially practically insignificant shifts are producing signals. To counteract this, both charts are amended to accommodate an indifference region by optimizing the detection of a shift at the outer boundaries of the indifference region. The results show that the adapted CUSUM and EWMA monitoring schemes yield comparable results. On nearly all occasions, the CUSUM chart outperforms the EWMA chart, yet the EWMA chart seems more robust and is easier to interpret. Furthermore, we provide two practical examples to illustrate the use-case of optimized charts to mitigate small (unimportant) variations, such as seasonality and modest temporary shifts. Overall, this work provides a general approach tailored to practice in quality control, e.g., as prescribed by ISO standards. It also answers a recent call in statistical process monitoring literature to reconsider the design of control charts.

Monitoring aggregate warranty claims with dynamically designed CUSUM and EWMA charts

Statistical monitoring of warranty claims data using dynamic probability control limits has been shown to be effective in early detection of unforeseen reliability problems that emerge at the design and manufacturing phases. As the discrepancy between abnormal patterns and the normal pattern in aggregate warranty claims is usually small (especially at the early stage), we develop two new dynamic monitoring schemes that adopt CUSUM-type and EWMA-type statistics, named DyCUSUM and DyEWMA, respectively, to better address the warranty claims monitoring problem. Three effective algorithms – that is, the Monte Carlo simulation, Markov chain, and near-enumeration algorithms – are proposed to progressively determine control limits for the two schemes. In particular, comparison studies show that the near-enumeration algorithm can attain a higher approximation accuracy with a lower computational burden and is thus recommended. In-depth simulation experiments are then conducted to assess the performance of the schemes. We find that the DyEWMA scheme has superior and robust detection performance in various situations, whereas the DyCUSUM scheme is less effective and could even be ineffective in certain cases, compared with a Shewhart-type counterpart. Some specific suggestions are also provided to facilitate implementation of the proposed monitoring schemes. Improved schemes by combining the moving window approach to mitigate the ‘inertia’ problem is further discussed. Finally, a real case study is presented.