Statistical Process Monitoring Research Articles

Nonparametric Threshold Estimation of Autocorrelated Statistics in Multivariate Statistical Process Monitoring

ABSTRACTMultivariate statistical process monitoring is commonly used to detect abnormal process behavior in real time. Multiple process variables are monitored simultaneously, and alarms are issued when monitoring statistics exceed a predetermined threshold. Traditional approaches use a parametric threshold based on the assumptions of independence and multivariate normality of the process data, which are often violated in complex processes with high sampling frequencies, leading to excessive false alarms. Some approaches for improved threshold selection have been proposed, but they assume independence of the monitoring statistics, which are often autocorrelated. In this paper, we compare the performance of nonparametric estimators for computing thresholds from autocorrelated monitoring statistics through simulation. The false alarm rate and in‐control average run length of each estimator under different distributions, sample sizes, and autocorrelation levels and types are found. Estimator performance is found to depend on sample size and the strength of autocorrelation. The class of kernel density estimation (KDE) methods tends to perform better than estimators that use bootstrapping, and the proposed adjusted KDE methods that account for autocorrelation are recommended for general use. A case study to monitor a wastewater treatment facility further illustrates the performance of nonparametric and parametric thresholds when applied to real‐world systems.

Statistical monitoring applied to data science in classification: continuous validation in predictive models

PurposeThis study aims to analyze the performance of quality indices to continuously validate a predictive model focused on the control chart classification.Design/methodology/approachThe research method used analytical statistical methods to propose a classification model. The project science research concepts were integrated with the statistical process monitoring (SPM) concepts using the modeling methods applied in the data science (DS) area. For the integration development, SPM Phases I and II were associated, generating models with a structured data analysis process, creating a continuous validation approach.FindingsValidation was performed by simulation and analytical techniques applied to the Cohen’s Kappa index, supported by voluntary comparisons in the Matthews correlation coefficient (MCC) and the Youden index, generating prescriptive criteria for the classification. Kappa-based control charts performed well for m = 5 sample amounts and n = 500 sizes when Pe is less than 0.8. The simulations also showed that Kappa control requires fewer samples than the other indices studied.Originality/valueThe main contributions of this study to both theory and practitioners is summarized as follows: (1) it proposes DS and SPM integration; (2) it develops a tool for continuous predictive classification models validation; (3) it compares different indices for model quality, indicating their advantages and disadvantages; (4) it defines sampling criteria and procedure for SPM application considering the technique’s Phases I and II and (5) the validated approach serves as a basis for various analyses, enabling an objective comparison among all alternative designs.

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.

Incipient fault detection for dynamic processes with canonical variate residual statistics analysis

In modern complex industrial operations, timely fault detection is imperative. While statistical process monitoring is widely used in practice, conventional approaches are usually insensitive to incipient faults (IFs) whose magnitudes are not obvious. To this end, an innovative approach is presented for IF detection in dynamic processes. To begin with, canonical variate residuals (CVRs) are generated by using the canonical variate dissimilarity analysis (CVDA) algorithm. The next step involves calculating statistics for the CVRs and arranging a corresponding statistic matrix. Afterward, the Mahalanobis distance index is constructed for fault detection purpose. The main reasons that this developed approach possesses high sensitivity to IFs in dynamic processes lie in the utilization of CVDA and the idea of monitoring extracted statistics rather than original residuals. Finally, its effectiveness and merits are demonstrated via a numerical example and a benchmark process.

An adaptive multivariate functional EWMA control chart

In many modern industrial scenarios, measurements of the quality characteristics of interest are often required to be represented as functional data or profiles. This motivates the growing interest in extending traditional univariate statistical process monitoring (SPM) schemes to the functional data setting. This article proposes a new SPM scheme, which is referred to as adaptive multivariate functional EWMA (AMFEWMA), to extend the well-known exponentially weighted moving average (EWMA) control chart from the univariate scalar to the multivariate functional setting. The proposed method distinguishes itself by adaptively selecting the weighting parameter in the calculation of the EWMA statistic to enhance the sensitivity of the AMFEWMA control chart across a spectrum of potential out-of-control scenarios. Such adaptability is essential in industrial processes, where multivariate functional quality characteristics are also subject to varying degrees of change. The favorable performance of the AMFEWMA control chart over existing methods is assessed via an extensive Monte Carlo simulation. Its practical applicability is demonstrated through a case study in monitoring the quality of a resistance spot welding (RSW) process in the automotive industry through online observations of dynamic resistance curves, which are associated with multiple spot welds on the same car body and are recognized as highly representative of the RSW process quality. The proposed method is implemented in the R package funcharts, available online on CRAN.

Real-time multivariate statistical monitoring of biopharmaceutical processes with no prior product-specific history

Biologics drug substance manufacturing normally comprises several processing steps including cell culture, purification, etc. Real-time Multivariate Statistical Process Monitoring (RT-MSPM) is an integral need in order to achieve early fault detection and prevention to guarantee robust and efficient process performance. As a part of RT-MSPM framework, Batch Evolution Models (BEMs), which take into account the transient phase of a batch prior to reaching full maturity (steady state), are the standard methods for real-time early fault detection in batch processes. Prior product-specific knowledge and training dataset are critical in order to model time (batch maturity) dependencies. Unlike traditional settings, modern biopharmaceutical manufacturing and clinical facilities use modular designs to enable simultaneous production of multiple medicines and quickly pivot from producing one medicine to another where no prior product-specific historical data might be available. In this article, first, a framework is proposed to detect the steady state operation of a manufacturing scale cell culture bioreactor. This is critical to achieve the ultimate goal to demonstrate feasibility of developing a continuous statistical process monitoring framework to monitor the steady state operation of a batch with no prior product specific history. Steady state time series data from previous step are processed following a systematic approach. The requirements for statistical properties of the processed time series in order to leverage data from different products to monitor a new product are outlined. The new framework establishes a guideline to develop models using datasets from other products with different recipes to monitor new products with no prior product-specific history. A novel software product is developed to test the framework. The framework is successfully applied for real-time monitoring of fast-rate sensors of a manufacturing cell culture bioreactor.

Integration of maintenance policy and statistical process monitoring for a cascade process with multiple assignable causes and random failures

This study develops an integrated model of statistical process monitoring and maintenance policy (MP) for a two-stage cascade process, involving two machines in series where the output of one stage affects the next stage. Instead of a single assignable cause (AC) affecting the quality characteristic, this production process experiences multiple ACs to more precisely simulate real conditions. Traditional and Hotelling’s T2 charts cannot detect the specific stage in which a shift occurs; therefore, this study employs a Cause-selecting control chart. The possibility of equipment failure and system stoppage are considered, with the process failure mechanism following a Weibull distribution. Depending on the conditions affecting the process or equipment, four MPs are considered to acheive the system’s initial condition, resulting in sixteen defined scenarios. The goal is to determine decision variables by minimizing a cost function subject to some statistical measures. This economic-statistical design can improve statistical and/or economic measures. However, it creates a more complex problem. Due to this complexity, a particle swarm optimization algorithm is used for optimization. The proposed model is implemented for a system that produces cotton yarn, with fiber length and skein strength are as the outputs of the first and second stages, respectively. Comparisons with two models validate the proposed model’s performance in cost reduction. A sensitivity analysis is performed for further investigations. The results indicate significant cost savings of the proposed model.

Designing an efficient adaptive EWMA model for normal process with engineering applications

Stability in process parameters is required to ensure the quality of the finished item. Control charts, as one of the critical parts of statistical process monitoring (SPM), have seen widespread use across many disciplines for detecting and responding to shifts in process parameters. The adaptive EWMA (AEWMA) control chartis a well-known scheme that detects both small and large process shifts by integrating the features of the Shewhart and classical EWMA charts. In this study, we propose an enhanced AEWMA chart, termed the EAEWMA chart, that efficiently monitors small and large process mean shifts simultaneously. The proposed EAEWMA chart enhances the existing AEWMA chart using the shift estimator, based on the hybrid EWMA (HEWMA) statistic. The Monte Carlo simulation approach is employed as the computational method to obtain the numerical findings for the various performance metrics. The EAEWMA chart is compared with various existing charts, including AEWMA, HEWMA, EWMA, ACSUM, and IACCUSUM, in zero- and steady-state scenarios. Conclusively, two practical applications of the EAEWMA chart are presented, demonstrating its value for practitioners and engineers and illustrating its efficacy in real-world scenarios.

Joint monitoring of location and scale for modern univariate processes

Autocorrelated sequences of individual observations arise in many modern-day statistical process monitoring (SPM) applications. Often times, interest involves jointly monitoring both process location and scale. To jointly monitor autocorrelated individuals data, it is common to first fit a time series model to the in-control process and subsequently use this model to de-correlate the observations so that traditional individuals and moving-range (I-MR) charts can be applied. If the time series model is correctly specified such that the resulting residuals are normal and independently distributed, then applying I-MR control charts to the residual process should work well. However, if the residual process deviates from normality and/or, due to time series model misspecification, contains levels of autocorrelation, the false alarm rate of such a strategy can dramatically rise. In this paper we propose improvements to a recently published distribution-free joint monitoring strategy that permits tighter control of the in-control average run length when model misspecification is a concern. We evaluate its average run length performance and conclude that the improved joint monitoring strategy proposed in this work is a very useful tool for today’s modern SPM practitioner. The proposed joint monitoring scheme is then applied to a real additive manufacturing process to illustrate its implementation in modern practice.

Long-Term Statistical Process Monitoring of an Ultrafiltration Water Treatment Process.

As water treatment technology has improved, the amount of available process data has substantially increased, making real-time, data-driven fault detection a reality. One shortcoming of the fault detection literature is that methods are usually evaluated by comparing their performance on hand-picked, short-term case studies, which yields no insight into long-term performance. In this work, we first evaluate multiple statistical and machine learning approaches for detrending process data. Then, we evaluate the performance of a PCA-based fault detection approach, applied to the detrended data, to monitor influent water quality, filtrate quality, and membrane fouling of an ultrafiltration membrane system for indirect potable reuse. Based on two short case studies, the adaptive lasso detrending method is selected, and the performance of the multivariate approach is evaluated over more than a year. The method is tested for different sets of three critical tuning parameters, and we find that for long-term, autonomous monitoring to be successful, these parameters should be carefully evaluated. However, in comparison with industry standards of simpler, univariate monitoring or daily pressure decay tests, multivariate monitoring produces substantial benefits in long-term testing.

Nonparametric control limits incorporating exceedance probability criterion for statistical process monitoring with commonly employed small to moderate sample sizes

This article aims to enhance the effectiveness of Phase I in statistical process monitoring by integrating the assessment of estimation uncertainty into control charts using the exceedance probability criterion. This criterion guarantees the desired in-control performance that a practitioner will achieve with a predefined high nominal coverage probability, which can help prevent high false alarm rates from occurring. In pursuit of this objective, we introduce two nonparametric approaches: one based on an analytical method and the other on a bootstrapping technique. Both approaches exhibit superior performance compared to the existing nonparametric method, particularly for Phase I, where small to moderate sample sizes are common. These proposed methodologies are especially advantageous for practitioners in real-world production environments.

Statistical process monitoring creates a hemodynamic trajectory map after pediatric cardiac surgery: A case study of the arterial switch operation.

Postoperative critical care management of congenital heart disease patients requires prompt intervention when the patient deviates significantly from clinician-determined vital sign and hemodynamic goals. Current monitoring systems only allow for static thresholds to be set on individual variables, despite the expectations that these signals change as the patient recovers and that variables interact. To address this incongruency, we have employed statistical process monitoring (SPM) techniques originally developed to monitor batch industrial processes to monitor high-frequency vital sign and hemodynamic data to establish multivariate trajectory maps for patients with d-transposition of the great arteries following the arterial switch operation. In addition to providing multivariate trajectory maps, the multivariate control charts produced by the SPM framework allow for assessment of adherence to the desired trajectory at each time point as the data is collected. Control charts based on slow feature analysis were compared with those based on principal component analysis. Alarms generated by the multivariate control charts are discussed in the context of the available clinical documentation.

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.

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.

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.

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.