Time-between-events Research Articles

Phase II control chart for monitoring Gumbel’s bivariate exponential distribution

Schemes for monitoring two or more time between events (TBE) data are becoming increasingly significant in statistical process monitoring. In practice, parameters must be estimated prior to the start of process monitoring. In this paper, we first propose a multivariate time between events (MTBE) control chart based on the one-sample log-likelihood ratio test for monitoring the Gumbel’s bivariate exponential (GBE) distribution, and then we analyze the effect of parameter estimation on this chart. To avoid the effect of parameter estimation, we then propose a method based on the two-sample log-likelihood ratio test for monitoring the shifts occurring in either the scale parameter or the dependence parameter or both for the GBE distribution process. The proposed method is designed with two-sample log-likelihood ratio test for univariate marginal distribution function and Gumbel copula function. A Max-type function-based approach is implemented for joint monitoring of margin distribution and copula-related structure. In contrast to the traditional schemes used to monitor bivariate processes, the strength of our proposed technique lies in the ability to identify exactly which component of the scale parameter or dependence parameter is responsible for the source of the signal. The performance of the proposed chart is analyzed via Monte Carlo simulations. Finally, two examples are mentioned to illustrate the implementation process of the proposed scheme.

Combined Shewhart–EWMA and Shewhart–CUSUM monitoring schemes for time between events

AbstractTo improve the detecting abilities of upward and downward parameter changes in high‐quality process, two combined schemes by integrating memory‐type, that is, exponentially weighted moving average (EWMA) or cumulative sum (CUSUM), and memoryless, that is, Shewhart, are proposed for monitoring the time between events (TBE), which is modeled as an exponential distributed variable. A Monte Carlo simulation method is employed to obtain the Run Length () properties, that is average run length (), of the proposed schemes for different parameter settings. The nearly optimal monitoring parameter combinations under different change size are obtained by minimizing the out‐of‐control with the satisfied in‐control . Using the designed parameters, the performance of the proposed monitoring schemes is compared with the existing EWMA and CUSUM TBE. The results show that the proposed combined Shewhart–EWMA or Shewhart–CUSUM TBE generally perform better than the corresponding EWMA or CUSUM TBE for large changes and they also show better performance than the Shewhart TBE for small changes. Finally, a real dataset of organic light‐emitting diode (OLED) failure time from Sumsung company is employed to indicate the usage and implementation of combined TBE schemes.

Reliability-oriented product manufacturing quality improvement approach based on R-KQC and DMAIC

ABSTRACT Reliability-oriented product quality improvement is of great significance to assurance the fitness of quality control and management activities in the manufacturing process. A novel reliability-oriented product manufacturing quality improvement approach based on DMAIC (i.e. define, measure, analysis, improvement, control) and driven by reliability-oriented key quality characteristic (R-KQC) is proposed to prevent reliability degradation and improve product fitness quality in the manufacturing process. First, with the manufacturing reliability based on RQR chain taken into consideration, the connotation of R-KQC is defined. Second, a reliability-oriented DMAIC process driven by R-KQC is proposed. The R-KQC identification method based on feature selection and Shaply additive explanations (SHAP) is proposed. In order to improve the quality of manufacturing process effectively, R-KQC improvement method of response surface and R-KQC control method based on time-between-events (TBE) control chart are proposed. Finally, an illustrated example of an insulating base is provided to verify the effectiveness of the proposed approach, the average failure rate of the latest batch is reduced by 37.8%.

A real‐time monitoring approach for bivariate event data

AbstractEarly detection of changes in the frequency of events is an important task in many fields, such as disease surveillance, monitoring of high‐quality processes, reliability monitoring, and public health. This article focuses on detecting changes in multivariate event data by monitoring the time‐between‐events (TBE). Existing multivariate TBE charts are limited because they only signal after an event occurred for each of the individual processes. This results in delays (i.e., long time‐to‐signal), especially when we are interested in detecting a change in one or a few processes with different rates. We propose a bivariate TBE chart, which can signal in real‐time. We derive analytical expressions for the control limits and average time‐to‐signal performance, conduct a performance evaluation and compare our chart to an existing method. Our findings showed that our method is an effective approach for monitoring bivariate TBE data and has better detection ability than the existing method under transient shifts and is more generally applicable. A significant benefit of our method is that it signals in real‐time and that the control limits are based on analytical expressions. The proposed method is implemented on two real‐life datasets from reliability and health surveillance.

Quality Decisions Based on Time between Events Data Analysis

Good decisions (Quality Decisions) depend on scientific analysis of data. Data are collected, generally, in two ways: 1) one sample of suitable size, 2) subsequent samples, at regular intervals of time. Often the data are considered normally distributed. This is wrong because the data must be analysed according to their distribution: Decisions are different. In several cases the data are exponentially distributed: we see how to scientifically deal with Control Charts (CC) to decide; this is opposite to what gives the T Charts that are claimed to be a good method for dealing with “rare events”: The Minitab Software (19 & 20 & 21) for “T Charts” is considered. The author will compare some methods, found in the literature with the author’s Theory RIT (Reliability Integral Theory): We will see various cases found in the literature. Classical Shewhart Control Charts and the TBE (Time Between Events) Control Charts have been considered: it appears that with RIT the future decisions will be both sounder and cheaper, for data is exponentially distributed. The novelty of the paper is in the scientific way of dealing with the Control Charts and their Control Limits, both with normally distributed data and with exponentially distributed data. In this way, a lot of wrong published papers on “Time Between Events” are to be discarded, even if their authors claim “We used Standard Statistical methods, typical in the vast literature of similar papers”. The author had to self-cite because it seems the only one that has been fighting for years for “Papers Quality”; he humbly asked the readers to inform him if some people did the same

Identifying Process Deterioration in Weighted Exponentially Distributed Time Between Events

In observational studies, the probability of selection of sampling units is not always equal. The recorded observations are biased in this scenario. The unweighted distributions in such situations are not useful until the inclusion probability of each item is same. The theory of weighted distributions offers a unifying approach for these types of conditions because it considers the adjustment bias. Failure to comply with such adjustment may lead to inappropriate results. In this article, an efficient mentoring scheme (Weighted-TBE chart) for time between events (TBE) using weighted exponential distribution has been proposed based on weighted variance (WV) method. A comparison has been established between CC based on weighted and unweighted probability distributions. The performance measure ARL has been calculated using Monte Carlo simulations. The Weighted-TBE chart has provided least values of ARL in the presence of unwanted process variations and proved to be more effective than the existing scheme. Further the proposed control chart has been applied to time between failures data to show its practical applicability.

Monitoring processes with multiple dependent production lines using time between events control charts

This article develops Time Between Events (TBE) control charts to monitor processes with multiple dependent production lines. To this end, a Shewhart-type and an EWMA-type TBE chart have been proposed. The copula approach is used to describe the dependence between production lines and the homogeneous Poisson process is considered to model the number of defectives. Performance of the proposed methods is evaluated using average time to signal metric. The numerical study showed that the EWMA-TBE chart uniformly performs better than the Shewhart-type chart. Eventually, the EWMA-TBE chart is applied to monitor two real-world processes with two and four production lines.

Enhanced fault detection models with real-life applications

<abstract> <p>Nonconforming events are rare in high-quality processes, and the time between events (TBE) may follow a skewed distribution, such as the gamma distribution. This study proposes one- and two-sided triple homogeneously weighted moving average charts for monitoring TBE data modeled by the gamma distribution. These charts are labeled as the THWMA TBE charts. Monte Carlo simulations are performed to approximate the run length distribution of the one- and two-sided THWMA TBE charts. The THWMA TBE charts are compared to competing charts like the DHWMA TBE, HWMA TBE, THWMA TBE, DEWMA TBE, and EWMA TBE charts at a single shift and over a range of shifts. For the single shift comparison, the average run length (ARL) and standard deviation run length (SDRL) measures are used, whereas the extra quadratic loss (EQL), relative average run length (RARL) and performance comparison index (PCI) measures are employed for a range of shifts comparison. The comparison reveals that the THWMA TBE charts outperform the competing charts at a single shift as well as at a certain range of shifts. Finally, two real-life data applications are presented to evaluate the applicability of the THWMA TBE charts in practical situations, one with boring machine failure data and the other with hospital stay time for traumatic brain injury patients.</p> </abstract>

Statistical design of ATS-unbiased charts with runs rules for monitoring exponential time between events

Time between events (TBE) control charts play an important role in monitoring high-yield processes. In this article, Phase II Shewhart-type tr-charts, with various runs rules, are considered for monitoring TBE exponential data with a known rate parameter and the charts’ performance is examined in terms of the average time to signal (ATS). The ATS metric is more suitable due to the possibly unequal times arising between the events. The tr-charts with runs rules are designed so that they are ATS-unbiased for a nominal ATS value. The performance of the proposed charts is compared with that of existing ATS-unbiased tr-charts without runs rules. The proposed charts are found to be more sensitive in detecting small downward and upward shifts. An illustration is provided based on real data, as well as conclusions and some topics for future research.

Enhancing the detection power of multivariate time between events control charts for Gumbel’s bivariate exponential distribution

One of the most important goals of statistical process control is quickly detecting any changes in the processes and consequently reducing process variability. Control charts are graphical tools that are widely used to achieve these goals. In this regard, monitoring methods based on the time between events (TBEs) have received much attention in recent years. These methods are widespread for monitoring high-quality processes and temporal quality characteristics. Although univariate TBE control charts have been extensively studied in the literature, multivariate TBE control charts need more investigation. In this paper, the performance of the multivariate exponentially weighted moving average (MEWMA), paired individual exponentially weighted moving average (PEWMA), and multivariate cumulative sum (MCUSUM) control charts for the bivariate TBEs which follow Gumbel's bivariate exponential (GBE) distribution are compared based on the average time to signal (ATS) metric. Also, the multivariate rate (MRA) control chart is proposed for enhancing the performance of the multivariate TBE control charts. Real-world examples are used to illustrate the applicability of the proposed monitoring scheme. The results of the performance evaluation show that the proposed approach can considerably enhance the performance of the multivariate TBE control charts.

A one-sided adaptive truncated exponentially weighted moving average scheme for time between events

In order to provide a balanced protection against a range of shift sizes in high-quality processes, a new one-sided adaptive truncated exponentially weighted moving average (ATEWMA) control chart with known and estimated parameters is developed for monitoring time-between-events (TBE) data. A dedicated Markov chain model is established for evaluating the run length properties in known and estimated parameters operating conditions. Furthermore, a two-stage optimal design procedure of the proposed scheme is developed based on the average run length (ARL) criteria. Simulation results show that the one-sided ATEWMA TBE scheme with known parameters is superior to its competitors in detecting both upward and downward shifts. Finally, two real data applications are employed to show the implementation of the recommended scheme in the monitoring of TBE data.

A number‐between‐events control chart for monitoring finite horizon production processes

AbstractIn this paper, we present a number‐between‐events (NBE) control chart for monitoring the fraction nonconforming in finite horizon production (FHP) processes and related specific performance measures. When monitoring fractions nonconforming in FHP processes, the common binomial ‐chart has two crucial limitations: the underlying distributional assumptions are violated when dealing with low‐volume production and a scarce efficiency in the case of processes characterized by a low fraction nonconforming. Thus, an efficient monitoring of FHP processes requires the selection of the correct underlying statistical model: in this case, a distribution from the hypergeometric family of discrete statistical distributions. An efficient statistical monitoring of processes with low fractions nonconforming can be achieved by means of discrete time‐between‐events (TBE) control charts, which count the number of units up to the appearance of a fixed number of nonconforming units in the sample. Here, we present a discrete TBE‐chart, denoted as NBE‐chart, based on the negative hypergeometric distribution that meets numerous requirements of efficient monitoring of FHP processes. The proposed control chart can be conveniently used for both low‐volume and mass production in processes with frequent changeovers.

A homogeneously weighted moving average control chart for monitoring time between events

AbstractIn high‐quality processes, non‐conforming events rarely occur, and the time between events (TBE) is more likely to follow a skewed distribution, such as gamma distribution. This paper proposes a gamma distribution‐based one‐sided homogeneously weighted moving average control chart for monitoring the TBE. The proposed control chart is known as the HWMA_TBE control chart. The HWMA_TBE control chart monitors the downward shifts in the process, that is, a decrease in inter‐arrival time that can lead to the process deterioration. The Monte Carlo simulation approach is used to compute specific performance indicators, such as average run length (), standard deviation run length , extra quadratic loss (), relative average run length (), and performance comparison index (). Based on these performance indicators, the HWMA_TBE control chart is compared to some competing control charts. The comparison suggests that the HWMA_TBE control chart is more efficient than the competing control charts. Moreover, this paper also investigates the robustness of the HWMA_TBE control chart, which indicates that the HWMA_TBE control chart is more robust than the competing control charts. In the end, two examples are provided, one with boring machine failure data and the other with breast cancer survival data, to evaluate the implementation of the HWMA_TBE control chart for practical situations.

A one-sided exponentially weighted moving average control chart for time between events

Exponentially weighted moving average (EWMA) control charts for time-between-events (TBE) are commonly suggested to monitor high-quality processes for the early detection of process deteriorations. In this study, an enhanced one-sided EWMA TBE scheme is developed for rapid detection of increases or decreases in the process mean. The use of the truncation method helps to improve the sensitivity of the proposed scheme in the process mean detection. Moreover, by taking the effects of parameter estimation into account, the proposed scheme with estimated parameters is also investigated. Both the average run length (ARL) and standard deviation of run length (SDRL) performances of the proposed scheme with known and estimated parameters are studied using the Markov chain method, respectively. Furthermore, an optimal design procedure is developed for the recommended one-sided EWMA TBE chart based on ARL. Numerical results show that the proposed optimal one-sided EWMA TBE chart is more sensitive than the existing optimal one-sided exponential EWMA chart in monitoring both upward and downward mean shifts. Meanwhile, it also performs better than the existing comparative scheme in resisting the effect of parameter estimation. Finally, two illustrative examples are considered to show the implementation of the proposed scheme for simulated and real datasets.

Aggregation levels when the time between events is Weibull distributed

Practical applications always require an assessment of an aggregation process that is best for early detection of any outbreak of events including sales, warrantee claims or disease outbreaks. This article provides a means of deciding on the level of temporal aggregation that best suits the needs of whoever aims to monitor a process or a service. This article aims to achieve a practical aggregation level for monitoring events when the in-control time between events (TBE) in the targeted process or service follows a non-homogeneous Weibull distribution. We analyze the impact of various aggregation levels on early detection of outbreaks with different magnitudes using several monitoring schemes including an adaptive exponentially weighted moving average (EWMA) plan and several simultaneous EWMA plans with differing amounts of temporal memory for TBE data. We also consider monitoring the related counting processes to address the problem of deciding on the temporal aggregation level. To the best of our knowledge, the effect of various levels of temporal aggregation on detecting outbreaks in TBEs, when they are Weibull distributed, is not studied thoroughly.

Time‐between‐events monitoring using nonhomogeneous Poisson process with power law intensity

AbstractThe traditional process monitoring techniques used to study high‐quality processes have several demerits, that is, high‐false alarm rate and poor detection, etc. A recent and promising idea to monitor such processes is the use of time‐between‐events (TBE) control charts. However, the available TBE control charts have been developed in a nonadaptive fashion assuming the Poisson process. There are many situations where we need adaptive monitoring, for example, health, flood, food, system, or terrorist surveillance. Therefore, the existing control charts are not useful, especially in sequential monitoring. This article introduces new adaptive TBE control charts for high‐quality processes based on the nonhomogeneous Poisson process by assuming the power law intensity. In particular, probability control limits are used to develop control charts. The proposed methodology allows us to get control limits that are dynamic and suitable for online process monitoring with an additional advantage to monitor a process where we believe the underlying failure rate may be changing over time. The average run length and coefficient of variation of the run length distribution are used to assess the performance of the proposed control charts. Besides simulation studies, we also discuss three examples to highlight the application of the proposed charts.

First passage time control charts assuming power law intensity for time to jointly monitor time and magnitude

AbstractControl charts are essential tools to monitor the quality of a process and commonly used in many industries besides industrial production. Due to the industrial revolution, many products have very low failure chances and labeled as the high‐quality products, where the time‐between‐events (TBE) charts are commonly used to monitor such processes. This study purposes control charts to monitor time and magnitude assuming the power law process for the TBE whereas the magnitude is assumed cumulative as well as noncumulative over time. Besides two illustrative examples, a comparison to the existing rate and first passage time (FPT) charts is also presented in this study. It is shown numerically that the proposed charts are more efficient than the existing FPT and rate charts.

Monitoring exponentially distributed time between events data: self-starting perspective

Time between events (TBE) control charts have been widely used to monitor high yield processes. Traditionally, an estimated in-control occurrence rate from a Phase I dataset is used to calculate the control limits when the rate is unknown. However, when Phase I analysis is time consuming or costly, the traditional Phase I/Phase II approach is not feasible. A self-starting method that sequentially updates the occurrence rate can be integrated to overcome this difficulty and leverages the use of TBE chart for cases of lack of in-control data. The motivation behind this study is to compare different self-starting TBE charts to investigate the contribution of such a sequential parameter update method. Our results indicate the potential of these schemes as they provide satisfactory performance when moderate and large sizes of base period before the shift observed. Additionally, time weighted schemes provided promising performance for relatively small sample sizes in the base period. However, our results also show a significant adverse effect in performance when the base periods are contaminated with out-of-control data.

Modified one-sided EWMA charts for monitoring time between events

Control charts based on Time Between Events (TBE) have been shown to be effective in high quality processes, where the defects or failures (events) occur at a very low probability. The occurrences of events are usually modeled as a homogeneous Poisson process. The TBE data is likely to follow a skewed distribution, such as the gamma distribution. In order to monitor high quality processes, this study focuses on the development of modified one-sided Exponentially Weighted Moving Average (EWMA) TBE charts. The Monte-Carlo simulation method is used to evaluate the Run Length (RL) properties of the proposed schemes. The results show that the proposed schemes are more sensitive to process shifts than some competitive procedures. Finally, a real data example is provided to illustrate the implementation of the proposed charts.

Temporal Changes in Central-Line–Associated Bloodstream Infection Time Between Events, 2017–2018 Versus 2015–2016

Background: The CDC NHSN surveillance coverage includes central-line–associated bloodstream infections (CLABSIs) in acute-care hospital intensive care units (ICUs) and select patient-care wards across all 50 states. This surveillance enables the use of CLABSI data to measure time between events (TBE) as a potential metric to complement traditional incidence measures such as the standardized infection ratio and prevention progress. Methods: The TBEs were calculated using 37,705 CLABSI events reported to the NHSN during 2015–2018 from medical, medical-surgical, and surgical ICUs as well as patient-care wards. The CLABSI TBE data were combined into 2 separate pairs of consecutive years of data for comparison, namely, 2015–2016 (period 1) and 2017–2018 (period 2). To reduce the length bias, CLABSI TBEs were truncated for period 2 at the maximum for period 1; thereby, 1,292 CLABSI events were excluded. The medians of the CLABSI TBE distributions were compared over the 2 periods for each patient care location. Quantile regression models stratified by location were used to account for factors independently associated with CLABSI TBE, such as hospital bed size and average length of stay, and were used to measure the adjusted shift in median CLABSI TBE. Results: The unadjusted median CLABSI TBE shifted significantly from period 1 to period 2 for the patient care locations studied. The shift ranged from 20 to 75.5 days, all with 95% CIs ranging from 10.2 to 32.8, respectively, and P < .0001 (Fig. 1). Accounting for independent associations of CLABSI TBE with hospital bed size and average length of stay, the adjusted shift in median CLABSI TBE remained significant for each patient care location that was reduced by ∼15% (Table 1). Conclusions: Differences in the unadjusted median CLABSI TBE between period 1 and period 2 for all patient care locations demonstrate the feasibility of using TBE for setting benchmarks and tracking prevention progress. Furthermore, after adjusting for hospital bed size and average length of stay, a significant shift in the median CLABSI TBE persisted among all patient care locations, indicating that differences in patient populations alone likely do not account for differences in TBE. These findings regarding CLABSI TBEs warrant further exploration of potential shifts at additional quantiles, which would provide additional evidence that TBE is a metric that can be used for setting benchmarks and can serve as a signal of CLABSI prevention progress.Funding: NoneDisclosures: None