T2 Chart Research Articles

Hotelling T2 control chart based on minimum vector variance for monitoring high‐dimensional correlated multivariate process

AbstractMultivariate control charts are practical tools that simultaneously monitor several correlated quality characteristics in a process. Monitoring high‐dimensional data structures is challenging because, in most cases, the process sample size for monitoring parameters is greater than the number of process characteristics. Many researchers have used the multivariate Hotelling's T2 chart to monitor high‐dimensional data using the maximum‐likelihood methods (MLM) to estimate the covariance matrices. However, the multivariate Hotelling's T2chart based on MLM suffers from low statistical performance. In this paper, we proposed a multivariate Hotelling's T2 chart based on the minimum vector variance (MVV) and some regularized methods for monitoring high‐dimensional data structures. The performance of the proposed chart is evaluated in terms of the average run length (ARL). The results reveal the superiority of the proposed MVV Hotelling's T2 chart over the existing Hotelling's T2 charts for high‐dimensional correlated processes.

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.

Phase I quality control framework for monitoring organ-at-risk dose

Objective. The aim of this work was to develop a Phase I control chart framework for the recently proposed multivariate risk-adjusted Hotelling’s T2 chart. Although this control chart alone can identify most patients receiving extreme organ-at-risk (OAR) dose, it is restricted by underlying distributional assumptions, making it sensitive to extreme observations in the sample, as is typically found in radiotherapy plan quality data such as dose-volume histogram (DVH) points. This can lead to slightly poor-quality plans that should have been identified as out-of-control (OC) to be signaled in-control (IC). Approach. We develop a robust iterative control chart framework to identify all OC patients with abnormally high OAR dose and improve them via re-optimization to achieve an IC sample prior to establishing the Phase I control chart, which can be used to monitor future treatment plans. Main Results. Eighty head-and-neck patients were used in this study. After the first iteration, P14, P67, and P68 were detected as OC for high brainstem dose, warranting re-optimization aimed to reduce brainstem dose without worsening other planning criteria. The DVH and control chart were updated after re-optimization. On the second iteration, P14, P67, and P68 were IC, but P40 was identified as OC. After re-optimizing P40’s plan and updating the DVH and control chart, P40 was IC, but P14* (P14’s re-optimized plan) and P62 were flagged as OC. P14* could not be re-optimized without worsening target coverage, so only P62 was re-optimized. Ultimately, a fully IC sample was achieved. Multiple iterations were needed to identify and improve all OC patients, and to establish a more robust control limit to monitor future treatment plans. Significance. The iterative procedure resulted in a fully IC sample of patients. With this sample, a more robust Phase I control chart that can monitor OAR doses of new plans was established.

Profile monitoring using synthetic T2 control chart

In this paper, we propose synthetic T2 chart for monitoring different parameters including Slope, Intercept and Error variance for simple and multiple linear regression profiles in Phase II. The construction and optimal design of the proposed synthetic T2 chart are discussed in detail. The steady-state performance of the proposed chart is evaluated by the Average Run Length (ARL) and Standard Deviation of Run Length (SDRL) and is compared with the existing charts in the literature for monitoring simple and multiple linear profiles. It turns out that the proposed synthetic T2 chart performs uniformly better than the traditional T2 chart in detecting different kinds of shifts in the parameters of the simple and multiple linear profiles under steady-state run length performance. An example with a real data set is also presented to explain the implementation of the proposed chart.

Control Chart T2Qv for Statistical Control of Multivariate Processes with Qualitative Variables

The scientific literature is abundant regarding control charts in multivariate environments for numerical and mixed data; however, there are few publications for qualitative data. Qualitative variables provide valuable information on processes in various industrial, productive, technological, and health contexts. Social processes are no exception. There are multiple nominal and ordinal categorical variables used in economics, psychology, law, sociology, and education, whose analysis adds value to decision-making; therefore, their representation in control charts would be useful. When there are many variables, there is a risk of redundant or excessive information, so the application of multivariate methods for dimension reduction to retain a few latent variables, i.e., a recombination of the original and synthesizing of most of the information, is viable. In this context, the T2Qv control chart is presented as a multivariate statistical process control technique that performs an analysis of qualitative data through Multiple Correspondence Analysis (MCA), and the Hotelling T2 chart. The interpretation of out-of-control points is carried out by comparing MCA charts and analyzing the χ2 distance between the categories of the concatenated table and those that represent out-of-control points. Sensitivity analysis determined that the T2Qv control chart performs well when working with high dimensions. To test the methodology, an analysis was performed with simulated data and with a real case applied to the graduate follow-up process in the context of higher education. To facilitate the dissemination and application of the proposal, a reproducible computational package was developed in R, called T2Qv, and is available on the Comprehensive R Archive Network (CRAN).

A Hotelling T2 chart for monitoring Gamma regression profile by considering estimation error

A Hotelling T2 chart for monitoring Gamma regression profile by considering estimation error

웨이블릿 에너지 스펙트럼 기반의 관리도를 통한 화력발전소 발전기의 상태기반보전 방법에 대한 연구

Purpose: Failing power-plant equipment can cause significant harm to both human life and expensive infrastructure. Hence, both effective process-monitoring and failure-prediction techniques are needed to prevent catastrophic equipment failure. Existing time-based maintenance-methods are not enough to prevent catastrophic failures. In this work, a condition-based maintenance (CBM) method, which can predict failures using wavelet-based process-monitoring methods, is studied.BRMethods: Both a wavelet spectrum analysis (with coefficients extracted from the wavelet transform) and a T² chart (with a slope and intercept based on a linear trend profile) are used in this study. The results are then embedded into a T² chart.BRResults: The proposed methodology in this study is validated using data signals coming from thermoelectric power plants. Overall, the proposed monitoring method has a higher prediction performance using breakdown signals than the traditional T² chart-based maintenance method.BRConclusion: The investigated failure-prediction method is not only more successful but also more specific when abnormal signal patterns are detected before failures. Therefore, using the proposed method for condition-based maintenance can help save resources and prevent human losses.

Are principal components with estimated parameters feasible in multivariate process control?

AbstractPrincipal components have been used in conjunction with Hotelling's T2 chart to monitor multivariate processes. It is known that prohibitively large sample sizes are needed to estimate the process parameters with enough precision to deploy the chart. We investigate whether principal components can be used to reduce the dimensionality of a process so that multivariate process control can be performed using estimated parameters. The chart based on the first k principal components, which we will refer to as the chart, is investigated. Specifically, we explore three research questions in this paper: (1) can the chart with estimated parameters be applied with moderate preliminary sample sizes?, (2) are there situations where the charts are able to detect a shift in the mean vector quicker than the T2 charts?, and (3) is it possible to exploit assumptions about the covariance matrix, such as equal covariances, to improve the performance of the chart. Using simulation, we find that for high dimensions, the chart with estimated parameters can be used to detect shifts in the direction of the first principal component. Otherwise, the chart requires very large preliminary sample sizes. When it is reasonable to assume equal covariances, the number of parameters to be estimated is substantially reduced and the performance of the chart is improved.

Non-Gaussian multivariate statistical monitoring of spatio-temporal wind speed frequencies to improve wind power quality in South Korea.

The spatial and temporal variability of renewable energy resources, particularly wind energy, should be statistically evaluated to achieve sustainable economic development to mitigate climate change. In this study, a non-Gaussian multivariate statistical monitoring approach is proposed to investigate the wind speed frequencies across different regions of South Korea. Anemometer data were first collected in 11 different provinces of South Korea with hourly resolution for one year. The best-of-fit for the corresponding distribution function was identified to characterize the behavior of the wind speed frequency at each region among more than 60 candidate functions using the chi-squared test. Furthermore, a non-Gaussian multivariate statistical monitoring method based on the Hotelling T2 chart was developed to spatially and temporally analyze the physical patterns of the wind speed frequencies using the estimated distribution parameters. Then determination rule of cut-in and cut-out speeds of wind turbine was suggested to improve the wind power quality across the regions. The results indicated that Weibull and Gamma distributions are best-of-fit functions of each province in South Korea; the physical patterns of wind including the average wind speed and gale can be identified by distribution parameters. Furthermore, the proposed non-Gaussian multivariate monitoring approach can elucidate the spatial and temporal variability of the regional wind speed frequencies, including the average wind speeds and extreme wind events across South Korea. Based on the statistically identified variability of wind behavior, the wind power quality of wind turbines can be improved by 12% on average by adjusting the cut-in and cut-off speed. Thus, the proposed non-Gaussian multivariate monitoring approach can provide practical guidelines for manufacturers to achieve reliable wind energy generation by considering the spatial and temporal wind behavior.

Monitoring profiles in multistage processes using the multivariate multiple regression model

AbstractMost recent production processes comprise multiple stages, where the final quality of products is determined by the upstream effects of output quality at each stage. Moreover, the quality of each stage is characterized by a functional relationship, referred to as a profile, between input variables and output quality variables. Therefore, the effective profile monitoring of the multiple stages is crucial in maintaining and improving the final output quality. A multistage process is separated into a series of single‐stage processes, and each single‐stage process is treated as a profile structure. In each stage, we implement the multivariate multiple linear regression (MMLR) model using the orthogonal design coding. The coefficient estimators of the MMLR are obtained as a matrix, whose column vectors indicate the coefficient vectors for given output quality variables and row vectors indicate the coefficients of output quality variables for given regression coefficients. Since the use of orthogonal design coding makes the regression coefficient estimator vectors for the intercept and the input variables mutually independent, it is possible to monitor the process using Hotelling's T2 charts separately. We explain the process of estimating and monitoring the regression coefficients from Phase I samples, and evaluate the performance of the proposed procedure in Phase II.

A nonlinear process monitoring strategy with a 2-phase fault diagnosis approach

The article delves into the development of Statistical Fault Detection and Diagnosis Strategies for an Integrated Steel Plant (ISP) taking into account the nonlinear relationship amongst the monitored Process and Feedstock Characteristics. The strategies being devised is based on Neural Network Fitting model cum Principal Component Analysis based technique (NNF-PCA) and Kernel Principal Component Analysis (KPCA) based technique. For detection of fault(s) Hotelling T2 chart based on (Principal Component Analysis) PCA and KPCA scores were employed and an ensemble strategy amalgamating KPCA and Self Organizing Map neural network has also been proposed for the detection of the out-of-control observations or faults. The article also proposes a 2-phase Fault Diagnosis approach christened as Preliminary Diagnosis phase and Specific Diagnosis phase. The Preliminary Diagnosis phase is based on Pattern Analysis of the control chart monitoring statistic observations and the Specific Diagnosis phase is based on the employment of appropriate Fault Diagnostic Statistic. The Preliminary Diagnosis reveals the broader source of assignable cause for the onset of the fault(s) and the Specific Diagnosis reveals the relative contribution of the individual Process and Feedstock characteristics. An in-depth comparative analysis between the NNF-PCA based strategy and KPCA based strategy w.r.t. three comparative aspects and four comparative parameters were carried out with their findings being duly highlighted which revealed the slight effectiveness of the KPCA based strategy with respect to the NNF-PCA based counterpart.

Multivariate Hotelling T2 Control Chart for Monitoring Some Quality Characteristics in Medium Density Fiberboard Manufacturing Process

Statistical process control tools are of great importance in terms of controlling manufacturing processes and improving product quality. In this study, the manufacturing process of medium density fiberboards manufactured in a company operating in the forest products industry was monitored by using multivariate Hotelling T2 statistical process control chart in terms of some quality characteristics. The T2 values of the signals detected by the Hotelling T2 control chart were also decomposed. By the decomposition of T2 values, it was determined which quality characteristics contributed more to each signal. It was seen that the process was not in control for Hotelling T2 control chart, which reveals the shift level in the mean of quality characteristics. As a result, the application of Hotelling T2 allowed fast detection of possible abnormalities in the process. The decomposition of T2 values successfully revealed which quality characteristics contributed significantly to the signals. Besides, it was concluded that, for monitoring, the Hotelling T2 chart was able to employ simultaneously different quality characteristics of medium density fiberboard. The current application study also contributed to the emergence of the root causes of the large shifts in the process. In conclusion, the findings of the study enabled the company to ensure the process stability and to facilitate decision-making on actions to be taken for quality improvement.

Design multivariate statistical process control procedure in the case of Ethio cement

PurposeThis study aims to design a multivariate control chart that improves the applicability of the traditional Hotelling T2 chart. This new type of multivariate control chart displays sufficient information about the states and relationships of the variables in the production process. It is used to make better quality control decisions during the production process.Design/methodology/approachMultivariate data are collected at an equal time interval and are represented by nodes of the graph. The edges connecting the nodes represent the sequence of operation. Each node is plotted on the control chart based on their Hotelling T2 statistical distance. The changing behavior of each pair of input and output nodes is studied by the neural network. A case study from the cement industry is conducted to validate the control chart.FindingsThe finding of this paper is that the points and lines in the classic Hotelling T2 chart are effectively substituted by nodes and edges of the graph respectively. Nodes and edges have dimension and color and represent several attributes. As a result, this control chart displays much more information than the traditional Hotelling T2 control chart. The pattern of the plot represents whether the process is normal or not. The effect of the sequence of operation is visible in the control chart. The frequency of the happening of nodes is recognized by the size of nodes. The decision to change the product feature is assisted by finding the shortest path between nodes. Moreover, consecutive nodes have different behaviors, and that behavior change is recognized by neural network.Originality/valueModifying the classical Hotelling T2 control chart by integrating with the concept of graph theory and neural network is new of its kind.

A novel efficient control chart to monitor the mean vector and variance-covariance matrix using support vector machine and tuned kernel function

Control chart is a popular statistical tool for continuous process monitoring and control which has been extensively extended to be applied. Hence current research addresses to a specific control chart based on one of the most important data-driven models, support vector machine. Such multivariate control chart is a useful learning system based on constrained optimisation theory that uses induction of structural error minimisation principle and results a general optimised answer. The novel recommended control chart gives the ability to detect out of control variable according to the appropriate decision functions. Computational results are conducted using Hotelling T2 chart employing average run length to determine the superiority of the proposed method. Implementing different shifts on process mean and variance in two and three variable processes reveals that the proposed chart has superior run length comparing by traditional Hotelling T2 chart. Furthermore, detecting the source of out of control variable is assessed.

A nonlinear process monitoring strategy for a Metal Forming process

The article deals with the development of a nonlinear statistical process monitoring strategy for a metal forming process. The developed strategy is based on Kernel Principal Component Analysis (KPCA) and has been adopted to address the issue of inherent nonlinearity amongst the measured characteristics. Fault Detection has been carried out by the employment of KPCA score based Hotelling T2 chart. The devised strategy was applied over a Rolling Mill Unit (RMU) which comes under Bulk Metal Forming process. Observations pertaining to the Process and Feedstock Characteristics (PFC) of the RMU have been considered for the analysis. The devised monitoring strategy has been found to be efficient in carrying out the fault detection.

Robust Multivariate Shewhart Control Chart Based on the Stahel-Donoho Robust Estimator and Mahalanobis Distance for Multivariate Outlier Detection

While researchers and practitioners may seamlessly develop methods of detecting outliers in control charts under a univariate setup, detecting and screening outliers in multivariate control charts pose serious challenges. In this study, we propose a robust multivariate control chart based on the Stahel-Donoho robust estimator (SDRE), whilst the process parameters are estimated from phase-I. Through intensive Monte-Carlo simulation, the study presents how the estimation of parameters and presence of outliers affect the efficacy of the Hotelling T2 chart, and then how the proposed outlier detector brings the chart back to normalcy by restoring its efficacy and sensitivity. Run-length properties are used as the performance measures. The run length properties establish the superiority of the proposed scheme over the default multivariate Shewhart control charting scheme. The applicability of the study includes but is not limited to manufacturing and health industries. The study concludes with a real-life application of the proposed chart on a dataset extracted from the manufacturing process of carbon fiber tubes.

An effective approach to detect the source(s) of out-of-control signals in productive processes by vector error correction (VEC) residual and Hotelling’s T2 decomposition techniques

One of the biggest challenges in the industrial sector is to simultaneously control quality characteristics. Multivariate control charts (MCC) are an efficient method to detect process variability, although they cannot signal the source responsible for variability to cause target deviation, especially in the presence of serial autocorrelation. Thus, this study proposes an approach to point out the source of variability of the process based on the vector error correction (VEC) residual and Hotelling’s T2 decomposition technique. The Design Research Methodology (DRM) was used to guide this study, and the accuracy and efficiency of the proposed approach were analyzed using simulated and real data from an extrusion machine operation, being: pressure (Pr), heating power (Hp), and temperature (Tp). Each of the monitored variables has 763 observations measured every 10 min of operation. An outlier was incorporated into each of the three variables: Pr - observation 91, Hp - observation 473, and Tp - observation 352. A VEC (3) model was fit and the residuals were plotted on a Hotelling’s T2 chart, which pointed the intentionally incorporated outliers. The Hotelling’s T2 decomposition of observation 91 showed thatT(Pr)2 = 14.4053 > 9.9187, and the variable Pr contributed significantly to the outlier. The variable Tp was identified as the source of variability for outlier 352 (p-value = 0.002). For observation 473, T(Hp)2=11.7202 > 9.9187; that is, Hp contributed significantly to the out-of-control point. Hotelling’s T2 chart was based on the VEC residual accurately identified the outliers, and the Hotelling’s T2 decomposition technique could identify the source of process variability.

Tensor-Based ECG Anomaly Detection toward Cardiac Monitoring in the Internet of Health Things.

Advanced heart monitors, especially those enabled by the Internet of Health Things (IoHT), provide a great opportunity for continuous collection of the electrocardiogram (ECG), which contains rich information about underlying cardiac conditions. Realizing the full potential of IoHT-enabled cardiac monitoring hinges, to a great extent, on the detection of disease-induced anomalies from collected ECGs. However, challenges exist in the current literature for IoHT-based cardiac monitoring: (1) Most existing methods are based on supervised learning, which requires both normal and abnormal samples for training. This is impractical as it is generally unknown when and what kind of anomalies will occur during cardiac monitoring. (2) Furthermore, it is difficult to leverage advanced machine learning approaches for information processing of 1D ECG signals, as most of them are designed for 2D images and higher-dimensional data. To address these challenges, a new sensor-based unsupervised framework is developed for IoHT-based cardiac monitoring. First, a high-dimensional tensor is generated from the multi-channel ECG signals through the Gramian Angular Difference Field (GADF). Then, multi-linear principal component analysis (MPCA) is employed to unfold the ECG tensor and delineate the disease-altered patterns. Obtained principal components are used as features for anomaly detection using machine learning models (e.g., deep support vector data description (deep SVDD)) as well as statistical control charts (e.g., Hotelling chart). The developed framework is evaluated and validated using real-world ECG datasets. Comparing to the state-of-the-art approaches, the developed framework with deep SVDD achieves superior performances in detecting abnormal ECG patterns induced by various types of cardiac disease, e.g., an F-score of 0.9771 is achieved for detecting atrial fibrillation, 0.9986 for detecting right bundle branch block, and 0.9550 for detecting ST-depression. Additionally, the developed framework with the T2 control chart facilitates personalized cycle-to-cycle monitoring with timely detected abnormal ECG patterns. The developed framework has a great potential to be implemented in IoHT-enabled cardiac monitoring and smart management of cardiac health.

PCA-based Hotelling's T2 chart with fast minimum covariance determinant (FMCD) estimator and kernel density estimation (KDE) for network intrusion detection

In this work, the combination between the Principal Component Analysis (PCA) and the Hotelling’s T2 chart is proposed to solve problems caused by the many highly correlated network traffic features and to reduce the computational time without reducing its accuracy detection. However, a new issue arises due to the difficulty of the network traffic observations to follow the multivariate normal distribution as required in Hotelling’s T2 chart. Consequently, many false alarms are found in inspecting network intrusion detection. To solve this issue, the Kernel Density Estimation (KDE) procedure is applied to obtain an optimum control limit. Also, to improve the accuracy detection, the Fast Minimum Covariance Determinant (FMCD) is employed to estimate the robust mean vector and covariance matrix. Experiments using the simulated dataset are conducted to assess the proposed chart’s performance in detecting the presence of outlier for the normal and non-normal of multivariate data. According to the simulation studies, the proposed chart yields higher accuracy and a high detection rate with a low false alarm rate. Further, the proposed Intrusion Detection System (IDS) is utilized in scanning attacks. The reputable KDD99 data is used as the benchmark to make a fair comparison between the proposed IDS and some algorithms. The monitoring outputs show that the proposed approach produces advancements in the speed of computational time with 87.42% of time efficiency. Compared to the other charts in detecting intrusion, the proposed chart produces the lower False Negative Rate (FNR). Also, compared to some classifiers the proposed chart yields a higher accuracy at about 0.9893.

Identifying Outliers in Response Quality Assessment by Using Multivariate Control Charts Based on Kernel Density Estimation

Abstract When monitoring industrial processes, a Statistical Process Control tool, such as a multivariate Hotelling T 2 chart is frequently used to evaluate multiple quality characteristics. However, research into the use of T 2 charts for survey fieldwork–essentially a production process in which data sets collected by means of interviews are produced–has been scant to date. In this study, using data from the eighth round of the European Social Survey in Belgium, we present a procedure for simultaneously monitoring six response quality indicators and identifying outliers: interviews with anomalous results. The procedure integrates Kernel Density Estimation (KDE) with a T 2 chart, so that historical “in-control” data or reference to the assumption of a parametric distribution of the indicators is not required. In total, 75 outliers (4.25%) are iteratively removed, resulting in an in-control data set containing 1,691 interviews. The outliers are mainly characterized by having longer sequences of identical answers, a greater number of extreme answers, and against expectation, a lower item nonresponse rate. The procedure is validated by means of ten-fold cross-validation and comparison with the minimum covariance determinant algorithm as the criterion. By providing a method of obtaining in-control data, the present findings go some way toward a way to monitor response quality, identify problems, and provide rapid feedbacks during survey fieldwork.