Multivariate Control Charts Research Articles

Functional EWMA control chart for phase II profile monitoring

Due to rapid developments in technology, profile monitoring is becoming challenging, as the profile data are a series of observations on a specific quality characteristic that are taken across some continuum. In this paper, a functional exponentially weighted moving average (FEWMA) control chart is developed for phase-II monitoring of nonlinear profiles based on the functional data approach. In particular, the profiles are estimated using cubic B-splines to solve the problem of high dimensionality that hinders the implementation of classical multivariate control charts. An extensive simulation study has been conducted to validate the effectiveness of the proposed control scheme for detecting changes in the functional mean of the process to its magnitude and assuming scenarios of independence and dependence between curves. The properties of the run-length distribution of the proposed scheme are discussed. The FEWMA control chart is also applied to the real-time data of the vertical density profile of particleboard manufacturing.

Monitoring and Interpretation of Process Variability Generated from the Integration of the Multivariate Cumulative Sum Control Chart and Artificial Intelligence

Monitoring and Interpretation of Process Variability Generated from the Integration of the Multivariate Cumulative Sum Control Chart and Artificial Intelligence

An image-based nonparametric multivariate EWMA control chart for metal additive manufacturing process monitoring

Additive manufacturing offers considerable promise in terms of manufacturing flexibility, but print quality remains a major obstacle to its widespread adoption. Consequently, timely and accurate detection of process variations or anomalies is critical for implementing corrective actions. Melt pool characteristics are crucial to final product quality, and current state-of-the-art studies on melt pool image flow have primarily focused on in-situ sensing, feature extraction, and their relationship with process settings and material properties. This study aims to develop a statistical process control approach to detect process variations immediately using a predefined distribution of monitoring statistics. Two main challenges are addressed: 1) the unknown distribution of melt pool image features, rendering traditional parametric control charts unreliable and 2) the autocorrelation of melt pool image flow features, violating the traditional control chart assumption of independent and identically distributed monitoring data. To overcome these challenges, we propose a multivariate nonparametric Exponential Weighted Moving Average (EWMA) control chart that can appropriately accommodate stationary serial data correlation to monitor process stability by tracking physical features extracted from the original images. Key features of the proposed control chart include its ability to eliminate serial correlation effects and its distribution-free nature, requiring no prior information about the data distribution. Comparative analysis of simulated and real data demonstrates that the proposed approach outperforms baseline methods in anomaly detection accuracy. An actual case study in laser metal deposition (LMD) further validates the effectiveness of the proposed method.

Classification of Quality Defects using Multivariate Control Chart with Ensemble Machine Learning Model

Multivariate control charts enable to monitor processes affected by more than one variable. But, when the process is out of control, it cannot detect which variable is causing it. It is an important requirement to know which variables in the process need corrective actions. In this study, a machine learning-based model is proposed to predict the variable/s that make the process out of control. For this purpose, ensemble algorithms, which are known to have higher prediction performance than single algorithms, were preferred. Because it is aimed to determine the variable(s) that cause the process to be out of control in the most accurate way. It is thought that a classification model in which ensemble algorithms are used together can increase the prediction accuracy. The model, which has not been encountered before in a quality control problem, was applied to a real problem and 98.06% classification accuracy was achieved. Another benefit is that it can predict the variable/variables that make the process uncontrolled without the need for multivariate control charts.

An efficient multivariate depth-based EWMA control chart for monitoring location

In multivariate statistical process control (MSPC), when there is a limited knowledge or there is no knowledge about the underlying process distribution, nonparametric control charts are useful. This paper introduces a new powerful multivariate depth-based control chart for online monitoring of location parameters in MSPC. Actually, by incorporating the exponentially weighted moving average control scheme, the weighted version of a well-known multivariate depth-based test statistic is used to formulate the new charting statistic. The methodology is based on adapting the intended charting statistic to the online sequential monitoring of the location parameters. The proposed control chart is affine-invariant and has a conditional distribution-free property over the class of elliptically symmetric distributions. This control chart is also very efficient in detecting small or moderate shifts in the process location parameter, especially for large dimensions.

Identification of the Out-of-Control Variable on a Multivariate Control Chart Signals in Iron Sheet Production Process

Monitoring and diagnosis of an out of control signals in manufacturing processes has become more challenging when it involves two or more correlated variables. In any production process and in the manufacturing industry, regardless of how well designed or carefully maintained it is, a certain amount of inherent or natural variability will always exist. The natural variability is the cumulative effect of many small, essentially unavoidable causes. A process that is operating with only chance causes of variation present is said to be in statistical control. In this paper we propose a method for approaching this problem based on principal components analysis. The principal component analysis method based on the ratios form was used to investigate which of the variable(s) were responsible for the out-of-control signal. A display of matrix of scatter plots was applied to investigate the correlation between the process variables and ellipses were used an approximate control limits. The principal component scores were used as an empirical reference distribution to establish a control region for the process to detect the variables causing the out-of-control signal. These insights derived from the principal component analysis offered valuable cues for identifying potential signals of an out-of-control process. The study findings indicated distinctive patterns of variance within the data, shedding light on potential signals for an out-of-control process.

Multivariate time between events control charts for Gumbel's bivariate exponential distribution with estimated parameters

The traditional approach of statistical process control is to estimate the in-control parameters from reference samples obtained in Phase I and then apply them to Phase II monitoring, which affects the performance of the monitoring scheme. We analyzed the effect of parameter estimation on three multivariate time between events control charts for monitoring the Gumbel's bivariate exponential distribution, namely the joint multivariate exponentially weighted moving average and multivariate rate, the joint multivariate cumulative sum and multivariate rate, and the joint paired individual exponentially weighted moving average and multivariate rate control charts. The simulation results show that the performance of the control charts is severely affected if control limits with known parameters are used, especially when only a few historical sample data are available. Mainly, the steady-state average time to signal was used as an evaluation index to analyze the affected degree of the three control charts, and the control limits were corrected. In addition, we evaluate the effect of model misspecification on the control charts. Finally, an example is given to illustrate the application of the proposed method.

Multivariate process capability analysis for evaluating metal additive manufacturing via electron beam melting

Abstract Electron beam melting (EBM) as one of the relatively new metal AM techniques showed promising and increasing applications. Therefore, there is a need to evaluate the quality of the EBM process using its critical quality characteristics. However, EBM and different AM process parts have many functionally or statistically correlated quality characteristics. Consequently, the quality characteristics of the EBM process should be evaluated together. Therefore, this research aims to evaluate the quality of the EBM process using a multivariate process capability index (MPCI). In this study, the dimensional accuracy in different directions is considered as a quality characteristics. The proposed methodology involves producing a large sample of small specimens of square shape using EBM technology. Three critical dimensions of the specimen in the X, Y, and Z axis are investigated as quality characteristics. The dimensions of quality characteristics are measured using a precise measurement device. The normality and stability assumptions of the collected data are investigated using skewness measure, and multivariate process control chart respectively. Then a large sample of the multivariate normal data is simulated using computer software to estimate the percent of nonconforming (PNC) from the established specification limits, which is used to estimate MPCI. Finally, the capable tolerance of the process is estimated and the sensitivity analysis of variation is investigated. The results show the capability of the EBM process under different specification limits designations. Estimating MPCI revealed that the EBM process is capable under very coarse limits only. Moreover, the sensitivity analysis showed that variation in quality characteristics data is very sensitive for MPCI estimation, especially variation in width quality characteristic.

Real-Time Monitoring of Human and Process Performance Parameters in Collaborative Assembly Systems using Multivariate Control Charts

With the rise in customized product demands, the production of small batches with a wide variety of products is becoming more common. A high degree of flexibility is required from operators to manage changes in volumes and products, which has led to the use of Human-Robot Collaboration (HRC) systems for custom manufacturing. However, this variety introduces complexity that affects production time, cost, and quality. To address this issue, multivariate control charts are used as diagnostic tools to evaluate the stability of several parameters related to both product/process and human well-being in HRC systems. These key parameters monitored include assembly time, quality control time, total defects, and operator stress, providing a more holistic view of system performance. Real-time monitoring of process performance along with human-related factors, which is rarely considered in statistical process control, provides comprehensive stability control over all customized product variants produced in the HRC system. The proposed approach includes defining the parameters to be monitored, constructing control charts, collecting data after product variant assembly, and verifying that the set of parameters is under control via control charts. This increases the system's responsiveness to both process inefficiencies and human well-being. The procedure can be automated by embedding control chart routines in the software of the HRC system or its digital twin, without adding additional tasks to the operator's workload. Its practicality and effectiveness are evidenced in custom electronic board assembly, highlighting its role in optimizing HRC system performance.

High-dimensional control charts with application to surveillance of grease damage in bearings of wind turbines

ABSTRACT High-dimensional data, characterized by having more attributes or variables than observations, presents unique challenges in industrial operations surveillance. Traditional multivariate control charts, like Hotelling’s T 2 chart, perform adequately with lower-dimensional data. However, they often fail to detect variations in process means as data dimensionality increases. This research proposes new control charts designed to enhance the detection of mean variations in both high and low-dimensional data. Specifically, Srivastava-Du (SD), Bai-Saranadasa (BS) and Dempster (DS) statistic-based charts are introduced, and their effectiveness is evaluated through simulations and real-life data applications. The performance of these charts is compared under various multivariate normal and non-normal distributions. Results indicate that DS and BS charts perform similarly, with the DS chart outperforming in low-dimensional normal distribution. Conversely, the SD chart outperformed in high-dimensional non-normal distributions. Additionally, the practical application of these proposed charts is illustrated through the monitoring of grease degradation in wind turbine bearings.

Statistical models assisted solvothermal synthesis of silver nanowires with controllable morphology

Silver nanowires (AgNWs) are conductive materials used in various applications such as solar panels and electronic skin. The morphology of AgNWs, including length and diameter, significantly affects conductivity. Despite various synthesis methods, a comprehensive understanding of how synthesis factors affect morphology is still lacking. This study focuses on solvothermal synthesis and investigates the influence of different nucleants on morphology. Through statistical analyses, including multivariate control charts and analysis of variance (ANOVA), this work demonstrates the joint effects of nucleants on AgNWs synthesis, thus setting the basis for morphology control.

Multiple exponentially weighted moving average control schemes for monitoring and diagnostics of correlated quality variables of different types with individual observations

In the era of big data analytics, more data, in terms of quantity and variety, are being collected for quality control purposes, thus leading to new challenges, especially for multivariate statistical process control (MSPC). One such challenge, which has received very little attention, is multivariate control chart (MCC) for monitoring correlated quality variables of different types. A recent study by Huang et al. (2023), which develops a Shewhart type MCC, is the first attempt to tackle such a challenge. The proposed chart of Huang et al. (2023), which utilizes the step-down multiple testing procedure of Holm (1979), not only can monitor correlated variables of different types, but also can provide instantaneous diagnostics of which parameters are out of control when the chart signals. In this study, we adapt and extend the methodology of Huang et al. (2023) to develop an exponentially weighted moving average (EWMA) chart specifically for the case when the sample size is one. The proposed chart is shown to be effective in detecting parameter changes as well as diagnosing which parameters are out of control when the chart signals.

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

AbstractPostoperative 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.

Group-wise monitoring of multivariate data with missing values

In this article, we aim to detect the changes in the process that generates multivariate data by monitoring their mean shift. To do this, we utilize a graphical tool known as a multivariate control chart. However, monitoring the mean of multivariate data poses two challenges: the grouped structure of individual observations and the presence of missing values. In this research, we introduce a novel method called HTC for monitoring group-wise multivariate data that includes missing values. HTC offers several advantages over existing methods. First, it is applicable to various types of dependence among individual observations within a group. Second, it provides a unique upper control limit (UCL) regardless of the missing data pattern. Lastly, HTC is computationally more efficient compared to resampling-based techniques. We conduct comprehensive numerical studies to evaluate the performance of the HTC method and compare it with the existing group-wise monitoring method, referred to as HTM. Compared to HTM, HTC achieves a higher true positive rate (TPR) while effectively controlling the in-control false alarm rate ( FAR 0 ) at a pre-determined level across various settings considered in our study. To illustrate its effectiveness, we applied HTC to monitoring multivariate environmental data collected from the manufacturing process of a semiconductor company in Korea.

Robust Multivariate Functional Control Chart

In modern Industry 4.0 applications, a huge amount of data is acquired during manufacturing processes and is often contaminated with outliers, which can seriously reduce the performance of control charting procedures, especially in complex and high-dimensional settings. In the context of profile monitoring, we propose a new framework that is referred to as robust multivariate functional control chart (RoMFCC) to monitor a multivariate functional quality characteristic while being robust to both functional casewise and componentwise outliers. In the former case, observations of the quality characteristic are contaminated in all functional variables or components, while, in the latter, the contamination affects one or more components independently. The RoMFCC relies on (I) a functional filter to identify componentwise outliers to be replaced by missing components; (II) a robust multivariate functional data imputation method; (III) a casewise robust dimensionality reduction; (IV) a monitoring strategy for the quality characteristic. Through a Monte Carlo simulation study, the RoMFCC is compared with competing schemes that have already appeared in the literature. A case study is finally presented where the proposed framework is used to monitor a resistance spot welding process in the automotive industry. RoMFCC is implemented in the R package funcharts, available online on CRAN.

Integration of the multivariate statistical control chart and machine learning to identify faults in the quality characteristics for polylactic acid with glass fiber composites in injection molding

Complex processing parameters need to be adjusted for expected qualities in injection molding processing. Once the process is abnormal, it is essential to spend time and human work on fault diagnosis. In this study, we focus on fault diagnosis of injection molding processing parameters for polylactic acid/glass fiber composites. The injection molding processing parameters include the melt temperature, injection speed, packing pressure, packing time, and cooling time. The qualities include tensile strength, hardness, impact strength, and flexure strength. When processing parameters deviate from the optimal process condition, the multivariate statistical control chart monitors downgraded qualities. The machine is operated at the optimal process conditions to generate normal samples and the corresponding four qualities of data are chosen as the historical data. Hotelling’s T2 is used to calculate the upper control limit (UCL) from the historical data to detect abnormal samples. If the T2 value exceeds the UCL, the corresponding sample is considered abnormal. Then, the residuals of qualities for abnormal samples are obtained by a residual control chart. They are chosen as the feature values for the backpropagation neural network (BPNN) to identify the abnormal processing parameters. The experimental results proved that the BPNN can achieve a 100% recognition rate for single-factor abnormal samples. For the single-/double-factor mixture, the accuracy rate of double-factor classification can reach 97.44%. This proposed study has the advantage of high stability, being non-destructive, high precision, and low cost, and can be widely promoted in injection molding industries.

Performance of T2-based PCA mix control chart with KDE control limit for monitoring variable and attribute characteristics

In this work, the mixed multivariate T2 control chart’s detailed performance evaluation based on PCA mix is explored. The control limit of the proposed control chart is calculated using the kernel density approach. Through simulation studies, the proposed chart’s performance is assessed in terms of its capacity to identify outliers and process shifts. When 30% more outliers are included in the data, the proposed chart provides a consistent accuracy rate for identifying mixed outliers. For the balanced percentage of attribute qualities, misdetection happens because of the high false alarm rate. For unbalanced attribute qualities and excessive proportions, the masking effect is the key issue. The proposed chart shows the improved performance for the shift in identifying the shift in the process.

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.

Generalized multivariate EWMA control chart supplemented with bivariate ranked set techniques

ABSTRACT Existing univariate exponential weighted moving average (EWMA) control charts with ranked set schemes are used for monitoring related characteristics independently. Also, existing multivariate EWMA control charts are generally applicable for simple random sampling (SRS) based processes. In this work, a generalized multivariate EWMA control chart is proposed with bivariate ranked set techniques (BVRST) for simultaneously monitoring. Furthermore, an extensive comparative analysis between proposed generalized multivariate EWMA and classical control chart is conducted. In this regard, we have used various performance measures, which cover average run length, median of run length, relative average run length, performance comparison index, and extra quadratic loss. The performance measures are obtained using the Monte Carlo simulation (MCS) approach. Results revealed that the proposed generalized multivariate EWMA control chart has an outstanding detection ability relative to the existing multivariate control charts. Additionally, existing multivariate control charts are found special cases of the proposed control chart. A real-life example from the agriculture field is included in which chemical properties (residual sodium and calcium-magnesium contents) of irrigation water are monitored through the control charts. The results revealed that the proposed control chart is proficient in diagnosing small variations in the chemical characteristics relative to the existing control charts.

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.