Statistical Process Monitoring Research Articles

Economic modelling and optimisation of SPRT-based quality control schemes for individual observations

In many industrial processes, it may not be practical to obtain sample sizes larger than one due to various reasons. As a result, practitioners must rely on control charts based on individual observations for statistical process monitoring and control. The sequential probability ratio test (SPRT) chart is an appropriate solution for such situations. Existing literature predominantly delves into the statistical aspects of designing or optimising an SPRT chart, neglecting the economic and associated design aspects. However, past experience suggests that statistical design may not always be optimal from an economic perspective. This paper aims to bridge this research gap by proposing a two-stage Markov chain approach to develop an economic model for the optimal design of the SPRT chart with individual observations. Following the principles of economic-statistical designs, optimal parameters are determined by solving a mixed-integer non-linear programming problem that minimises the long-run cost per item while ensuring compliance with practical constraints. Through an extensive study, we find that the economic performance of the SPRT chart surpasses that of competing individual control charts in many cases. This indicates that the economic-statistical design of the SPRT chart can yield significant economic benefits while maintaining desirable statistical properties. Moreover, enhancements have been introduced to refine the optimal design of the SPRT chart, retaining excellent overall performance across a wide range of shifts. Finally, we provide two industrial examples to illustrate the effectiveness of the proposed SPRT chart.

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.

Closed-form expressions of the run-length distribution of the nonparametric double sampling precedence monitoring scheme

AbstractA significant challenge in statistical process monitoring (SPM) is to find exact and closed-form expressions (CFEs) (i.e. formed with constants, variables and a finite set of essential functions connected by arithmetic operations and function composition) for the run-length properties such as the average run-length ($$ARL$$ ARL ), the standard deviation of the run-length ($$SDRL$$ SDRL ), and the percentiles of the run-length ($$PRL$$ PRL ) of nonparametric monitoring schemes. Most of the properties of these schemes are usually evaluated using simulation techniques. Although simulation techniques are helpful when the expression for the run-length is complicated, their shortfall is that they require a high number of replications to reach reasonably accurate answers. Consequently, they take too much computational time compared to other methods, such as the Markov chain method or integration techniques, and even with many replications, the results are always affected by simulation error and may result in an inaccurate estimation. In this paper, closed-form expressions of the run-length properties for the nonparametric double sampling precedence monitoring scheme are derived and used to evaluate its ability to detect shifts in the location parameter. The computational times of the run-length properties for the CFE and the simulation approach are compared under different scenarios. It is found that the proposed approach requires less computational time compared to the simulation approach. Moreover, once derived, CFEs have the added advantage of ease of implementation, cutting off on complex convergence techniques. CFE's can also easily be built into mathematical software for ease of computation and may be recalled for further work.

Control charting methods for monitoring high dimensional data streams: A conceptual classification scheme

There are always challenges in various industrial or non-industrial processes in which the product quality/service is described by a large number of quality characteristics. Thus, statistical process monitoring (SPM) techniques for capturing the quality of high-dimensional processes are becoming increasingly important, and various control charts have been developed to monitor different types of quality characteristics in this area. In general, the construction of conventional control charts to monitor multivariate processes in a high-dimensional setting has some statistical limitations and leads to misleading interactions. As a result, novel control charting techniques have recently been suggested to ameliorate the efficiency of monitoring schemes under high-dimensional data streams. This paper undertakes a conceptual classification structure using content analysis to classify the existing literature in this context for the period 2004–2024. Furthermore, on the basis of 72 selected papers, the research gaps are identified and some directions to stimulate potential for future research are suggested.

A distribution-free Max-EWMA scheme for multi-aspect process monitoring with industrial applications

In Statistical Process Monitoring, Distribution-free charting schemes play a vital role in Industry 4.0 for their superior effectiveness compared to parametric methods, especially in cases where the underlying process distribution is not well-defined or poses challenges in estimation. Traditional distribution-free charting schemes focus on surveilling only one process aspect: the location or scale parameter. The past two decades have also witnessed a rapid expansion of research on joint monitoring of the two aspects, location and scale, using combined bi-aspect charting schemes. However, there is a notable scarcity of methods that addressing complex industrial scenarios where multiple parameters (like location, scale, and shape) need simultaneous monitoring with high sensitivity and specificity. Our research is motivated by this challenge, and this paper describes a distribution-free tri-aspect monitoring procedure called the Maximum Exponentially Weighted Moving Average (Max-EWMA) scheme This scheme can effectively monitor shifts in three process characteristics, encompassing aspects related to process location, scale, and shape parameters. We utilize weighted versions of well-known statistical measures such as the Wilcoxon, Ansari-Bradley, and Savage-type statistics to construct the plotting statistics. An essential feature of our proposed Max-EWMA chart is its robustness under various continuous distributions, ensuring reliable performance in in-control situations. Competing schemes are compared via intensive computing techniques based on Monte Carlo using the Median Run Length metric, which evaluates the effectiveness of the proposed scheme. Finally, our proposed schemes are illustrated with two examples. Some concluding remarks and limitations of the study are noted.

Optimal design of adaptive EWMA monitoring schemes for the coefficient of variation and performance evaluation with measurement errors

The coefficient of variation (CV) usually describes the relative dispersion in production or service processes and has been widely applied in various fields. Monitoring the CV has received great attention in statistical process monitoring. This paper develops two one-sided adaptive EWMA (AEWMA) CV schemes to enhance the existing CV schemes’ monitoring efficiency. This scheme can effectively defect various shift sizes by dynamically adjusting the smoothing parameter of the EWMA based on the residuals. The optimization models of the AEWMA CV are designed based on the Markov chain method from two different perspectives, (i) a pair of changes; (ii) a range of changes. The numerical and graphical comparisons confirm the superiority of the proposed schemes. Moreover, in the data collection process, measurement errors from devices and human operations may arise. To assess the impact of such measurement errors on the monitoring performance, a linear covariate error model is adopted. For illustration and validation, the proposed schemes are implemented to a real industrial example from the sintering process.

Variable contribution analysis in multivariate process monitoring using permutation entropy

Multivariate statistical process monitoring is widely used in industrial processes for performance monitoring and fault detection. Once a fault has been detected, fault diagnoses must be performed to identify the variables that are responsible for the fault or the deviation from normal operation. In this paper, we present a new data-driven method for variable contribution analysis to a fault or abnormal behaviour in a process. The method is based on permutation entropy (PE), which measures the order of fluctuations within a time series. The PE method for variable contribution analysis is independent of fault detection statistics, and is used to evaluate which variables experienced the greatest change in their expected behaviour over time, immediately after the fault has been identified. We show how PE can be used to identify and interpret the variables responsible for, or affected by faults in an industrial process. The well-known Tennessee Eastman Process (TEP) is used to illustrate the application of PE for variable contribution analysis. The results show that PE is an efficient analysis tool for specifying variable contributions to faults.

Statistical Process Monitoring from Industry 2.0 to Industry 4.0: Insights into Research and Practice

Industry 4.0 has emerged as an important era for process monitoring and improvement. Our expository paper provides a historical perspective on research and practice of statistical process monitoring (SPM) from the 1920s to the present to bring a high-level view of current practice and research directions. We focus on the Industry 4.0 era, which began around 2011 with the introduction of cyber-physical systems and the growth of the Internet of Things. These technological changes have brought tremendous challenges and opportunities to SPM that can only be met with new paradigms for the problems we aim to solve and the approaches we use to evaluate SPM methodology. We provide our perspective on these challenges, primarily focusing on industrial applications. We give recommendations on the evaluation and comparison of monitoring methods to improve the usefulness of research in this area.

Monitoring multistage healthcare processes using state space models and a machine learning based framework

Monitoring healthcare processes, such as surgical outcomes, with a keen focus on detecting changes and unnatural conditions at an early stage is crucial for healthcare professionals and administrators. In line with this goal, control charts, which are the most popular tool in the field of Statistical Process Monitoring, are widely employed to monitor therapeutic processes. Healthcare processes are often characterized by a multistage structure in which several components, states or stages form the final products or outcomes. In such complex scenarios, Multistage Process Monitoring (MPM) techniques become invaluable for monitoring distinct states of the process over time. However, the healthcare sector has seen limited studies employing MPM. This study aims to fill this gap by developing an MPM control chart tailored for healthcare data to promote early detection, confirmation, and patient safety. As it is important to detect unnatural conditions in healthcare processes at an early stage, the statistical control charts are combined with machine learning techniques (i.e., we deal with Intelligent Control Charting, ICC) to enhance detection ability. Through Monte Carlo simulations, our method demonstrates better performance compared to its statistical counterparts. To underline the practical application of the proposed ICC framework, real data from a two-stage thyroid cancer surgery is utilized. This real-world case serves as a compelling illustration of the effectiveness of the developed MPM control chart in a healthcare setting.

Optimal constrained design of control charts using stochastic approximations

In statistical process monitoring, control charts typically depend on a set of tuning parameters besides its control limit(s). Proper selection of these tuning parameters is crucial to their performance. In a specific application, a control chart is often designed for detecting a target process distributional shift. In such cases, the tuning parameters should be chosen such that some characteristic of the out-of-control (OC) run length of the chart, such as its average, is minimized for detecting the target shift, while the control limit is set to maintain a desired in-control (IC) performance. However, explicit solutions for such a design are unavailable for most control charts, and thus numerical optimization methods are needed. In such cases, Monte Carlo-based methods are often a viable alternative for finding suitable design constants. The computational cost associated with such scenarios is often substantial, and thus computational efficiency is a key requirement. To address this problem, a two-step design based on stochastic approximations is presented in this paper, which is shown to be much more computationally efficient than some representative existing methods. A detailed discussion about the new algorithm’s implementation along with some examples are provided to demonstrate the broad applicability of the proposed methodology for the optimal design of univariate and multivariate control charts. Computer codes in the Julia programming language are also provided in the supplemental material.

Process monitoring and fault diagnosis method combining bagging DPCA‐ICA with moving window Kolmogorov–Smirnov test

AbstractProcess monitoring and fault diagnosis systems are essential for ensuring the quality and safety of industrial processes. Real industrial process data always has dynamic features, autocorrelation features, and non‐Gaussian distributions, making many multivariate statistical process monitoring (MSPM) methods, such as principal component analysis (PCA), not work well. To solve the problem, a new process monitoring method (named DPCA‐ICA‐moving window Kolmogorov–Smirnov test [MWKS]) is proposed in the paper, which is a combination of DPCA and independent component analysis (ICA) with adaptive weight parameters and introduces a moving window technique and two‐sample K‐S test to obtain the K‐S distance of the commonly used Hotelling's and square predictive error as new monitoring parameters. Experimental results on the Tennessee Eastman chemical process demonstrate the effectiveness and accuracy of the proposed method, which shows it has better performance than existing commonly used methods. The average fault detection rate (FDR) is 93.131%, while the average false alarm rate (FAR) of the proposed method is only 0.1%.

Detecting outliers in the multivariate control charts for dispersion monitoring

AbstractOutlier detection is an important aspect of statistical process monitoring (SPM) because outliers affect the performance of control charts. SPM researchers study the negative impact of outliers on control charts for monitoring location parameters. However, there is little research on outlier detection in multivariate charts for monitoring process dispersion. This study aims to investigate the impact of outliers in multivariate control charts for monitoring covariance matrix of a process, and then to recommend techniques for detecting potential outliers present from Phase I samples. We propose a new multivariate dispersion chart that employs the determinant of logarithm of estimated covariance matrix as the monitoring statistic. Through Monte Carlo simulations, the results show how outliers from the first phase affect the overall performance of multivariate charts. The results also demonstrate that the minimum volume ellipsoid (MVE) estimator is effective in reducing the effect of outliers on the proposed control scheme than the other compared estimators.

PSINDy: Probabilistic sparse identification of nonlinear dynamics for temporal process modeling and fault detection

BackgroundMultivariate statistical process monitoring (MSPM) methods have been widely studied and applied for dynamic process monitoring. However, limitations exist in the extant literature, as numerous extant MSPM approaches fail to concurrently account for the governing dynamics and inherent non-linearity of systems. MethodsTo handle the above-mentioned issue, a novel probabilistic dynamic process monitoring algorithm named probabilistic sparse identification of nonlinear dynamics (PSINDy) is proposed. In this algorithm, the Expectation–Maximization (EM) method is employed to estimate the parameters. In the E-step, the particle filtering technique is adopted to calculate corresponding latent expectations whose posterior distributions are not Gaussian. Furthermore, Hotelling's T2 and two novel monitoring statistics, termed Mahalanobis-based predictive error (MPE) and dynamic predictive error (DPE), are designed and utilized for fault detection. Significant FindingsThe effectiveness of the proposed method is validated through the Tennessee Eastman (TE) chemical process and the three-phase flow facility (TPFF).

A monitoring framework for health care processes using Generalized Additive Models and Auto-Encoders

In recent years, there has been a considerable focus on developing effective methods for monitoring health care processes. Utilizing Statistical Process Monitoring (SPM) approaches, particularly risk-adjusted control charts, has emerged as a highly promising approach for achieving robust frameworks for this aim. Considering risk-adjusted control charts, longitudinal health care process data is typically monitored by establishing a regression relationship between various risk factors (explanatory variables) and patient outcomes (response variables). While the majority of prior research has primarily employed logistic models in risk-adjusted control charts, there are more intricate health care processes that necessitate the incorporation of both parametric and nonparametric risk factors. In such scenarios, the Generalized Additive Model (GAM) proves to be a suitable choice, albeit it often introduces higher computational complexity and associated challenges. Surprisingly, there are limited instances where researchers have proposed advancements in this direction. The primary objective of this paper is to introduce an SPM framework for monitoring health care processes using a GAM over time, coupled with a novel risk-adjusted control chart driven by machine learning techniques. This control chart is implemented on a data set encompassing two stroke types: ischemic and hemorrhagic. The key focus of this study is to monitor the stability of the relationship between stroke types and predefined explanatory variables over time within this data set. Extensive simulation results, based on real data from patients with acute stroke, demonstrate the remarkable flexibility of the proposed method in terms of its detection capabilities compared to conventional approaches.

Modified performance-enhanced PCA for incipient fault detection of dynamic industrial processes

Timely fault detection plays a critical role in modern complex industrial processes. While statistical process monitoring has gained significant practical application in recent years, traditional data-driven multivariate statistics often lack sensitivity in detecting incipient faults. To this end, this paper proposes a novel approach for fault detection of dynamic industrial processes. First, the process data are processed by the time series model, so as to separate and obtain corresponding static components. Then, a performance-enhanced principal component analysis method is employed to handle the static components. Besides, the detection index is further optimized to improve its detection performance. Finally, the proposed method’s effectiveness is illustrated through a numerical example and the continuous stirred tank reactor process. Moreover, the universality of the optimization algorithm which acts on the detection statistic is illustrated on the Tennessee Eastman process as well.

Real-time risk assessment and surveillance for early prediction of unplanned shutdown events

Statistical Process Monitoring (SPM) and Fault Detection & Diagnosis (FDD) are reactive methodologies by design. However, in some situations, the prompt detection ex post facto is totally unsatisfactory. This is the case of Unplanned Shutdown Events (USE), i.e., major disruptions that cause the process to be shutdown. In this article, a new proactive approach is proposed that monitors the risk of USE occurring in the near future, called Risk Assessment and Surveillance (RAS). Contrary to FDD/SPM, RAS is a risk-based approach that continuously predicts the probability of a major disruption taking place in the near future horizon. The predictive model is based only on high-resolution process data and its outcomes are classified in three zones: normal, warning (where countermeasures can prevent the USE), and critical (possibly irreversibly leading to shutdown). It is shown that, in some cases, the method can alert the operators several hours before they actually took any corrective actions. A process improvement diagnosis methodology is also presented for identifying conditions promoting USE’s, based on coarse-grained modeling of the Remaining Useful Life (RUL).

A change-point–based control chart for detecting sparse mean changes in high-dimensional heteroscedastic data

Because of the “curse of dimensionality,” high-dimensional processes present challenges to traditional multivariate statistical process monitoring (SPM) techniques. In addition, the unknown underlying distribution of and complicated dependency among variables such as heteroscedasticity increase the uncertainty of estimated parameters and decrease the effectiveness of control charts. In addition, the requirement of sufficient reference samples limits the application of traditional charts in high-dimension, low-sample-size scenarios (small n, large p). More difficulties appear when detecting and diagnosing abnormal behaviors caused by a small set of variables (i.e., sparse changes). In this article, we propose two change-point–based control charts to detect sparse shifts in the mean vector of high-dimensional heteroscedastic processes. Our proposed methods can start monitoring when the number of observations is a lot smaller than the dimensionality. The simulation results show that the proposed methods are robust to nonnormality and heteroscedasticity. Two real data examples are used to illustrate the effectiveness of the proposed control charts in high-dimensional applications. The R codes are provided online.

A Relevant Variable Selection and SVDD-Based Fault Detection Method for Process Monitoring

This study investigates the sample value imbalance problem of process monitoring. A fault detection approach based on variable selection and support vector data description (SVDD) is developed for efficient process monitoring. First, Kullback&#x2013;Leibler divergence serves as the variable selection algorithm, which highlights the most beneficial information about the concerned faults. The attained variables are segmented by block division to avoid faults information being covered in single space monitoring, so that the relevant variables and the most beneficial information are concentrated in the same block. Then, Kernel principal component analysis is applied in each block to address the challenge that variables may still be high-dimensional and nonlinear. After that, the monitoring result is given based on the proposed SVDD with a restructured radius index, which is more sensitive to the fault. As demonstrated from experimental results on the Tennessee Eastman process, this method is effective and outperforms counterparts with higher mean fault detection rate. <i>Note to Practitioners</i>&#x2014;Recently, multivariate statistical process monitoring (MSPM) has attracted much attention. In general, MSPM incorporates all variables for the large-scale process. However, only a small number of variables are fault-dependent. Namely, the sample value imbalance problem is encountered in application. In this scenario, the monitoring performance degrades and the online computational complexity increases. To this end, a SVDD-based fault detection method, which considers the fault-related variables, is proposed for process monitoring. The proposed method is verified by the Tennessee Eastman process and it is more sensitive to the concerned fault.

A re-evaluation of repetitive sampling techniques in statistical process monitoring

ABSTRACT In this article, we review the literature on the use of the repetitive sampling technique with quality control charts. We raise some important concerns and questions regarding its application and underlying assumptions. We show that a fixed sample size method that requires on average the same amount of sampling can be designed to have as good or better statistical properties when monitoring with data that are normally distributed. Repetitive sampling methods can be useful, however, when monitoring with count data.

Statistical Process Monitoring of Artificial Neural Networks

The rapid advancement of models based on artificial intelligence demands innovative monitoring techniques which can operate in real time with low computational costs. In machine learning, especially if we consider artificial neural networks (ANNs), the models are often trained in a supervised manner. Consequently, the learned relationship between the input and the output must remain valid during the model’s deployment. If this stationarity assumption holds, we can conclude that the ANN provides accurate predictions. Otherwise, the retraining or rebuilding of the model is required. We propose considering the latent feature representation of the data (called “embedding”) generated by the ANN to determine the time when the data stream starts being nonstationary. In particular, we monitor embeddings by applying multivariate control charts based on the data depth calculation and normalized ranks. The performance of the introduced method is compared with benchmark approaches for various ANN architectures and different underlying data formats.