A new Bayesian multivariate exponentially weighted moving average control chart for phase II monitoring of multivariate multiple linear profiles

Recently, statistical profile monitoring methods have become efficient tools for monitoring the quality of a product (or a production process) using control charts. The key idea is to describe the relationship between a response variable and a set of explanatory variables in the form of a statistical regression model, which called profile. Traditionally, those control charts are constructed with standard “frequentistic” regression models. Recently, it has been proposed to apply Bayesian regression models instead, and it has been empirically demonstrated that Bayesian regression models have the potential to perform significantly better. In this paper, we introduce a novel Bayesian multivariate exponentially weighted moving average control chart for monitoring multivariate multiple linear profiles in phase II. The key idea is to use the data from historical data sets to generate informative prior distributions for the regression models in phase II. The results of our empirical simulation studies show that the Bayesian multivariate multiple linear regression model is superior to its classical “frequentistic” counterpart in terms of the average run length. Our empirical findings are in agreement with findings reported in recently published articles. To shed more light onto the merit of the proposed Bayesian method, we carry out a sensitivity analysis, in which we investigate how the amount of phase I data influences the results. We also demonstrate the applicability and superiority of the proposed Bayesian method by a real‐world application.     

A predictive Bayesian approach to sequential time-between-events monitoring

A fundamental problem with all process monitoring techniques is the requirement of a large Phase-I data set to establish control limits and overcome estimation error. This assumption of having a large Phase-I data set is very restrictive and often problematic, especially when the sampling is expensive or not available, eg, time-between-events (TBE) settings. Moreover, with the advancement in technology, quality practitioners are now more interested in online process monitoring. Therefore, the Bayesian methodology not only provides a natural solution for sequential and adaptive learning but also addresses the problem of a large Phase-I data set for setting up a monitoring structure. In this study, we propose Bayesian control charts for TBE assuming homogenous Poisson process. In particular, a predictive approach is adopted to introduce predictive limit control charts. Beside the Bayesian predictive Shewhart charts with dynamic control limits, a comparison of the frequentist sequential charts, designed by using unbiased and biased estimator of the process parameter, is also a part of the present study. To assess the predictive TBE chart performance in the presence of practitioner-to-practitioner variability, we use the average of the average run length (AARL) and the standard deviation of the in-control run length (SDARL).     

Statistical process monitoring via image data using wavelets

Image data plays an important role in manufacturing and service industries because each image can provide a huge set of data points in just few seconds with relatively low cost. Enhancement of machine vision systems during the time has led to higher quality images, and the use of statistical methods can help to analyze the data extracted from such images efficiently. It is not efficient from time and cost point of views to use every single pixel in an image to monitor a process or product performance effectively. In recent years, some methods are proposed to deal with image data. These methods are mainly applied for separation of nonconforming items from conforming ones, and they are rarely applied to monitor process capability or performance. In this paper, a nonparametric regression method using wavelet basis function is developed to extract features from gray scale image data. The extracted features are monitored over time to detect process out‐of‐control conditions using a generalized likelihood ratio control chart. The proposed approach can also be applied to find change point and fault location simultaneously. Several numerical examples are used to evaluate performance of the proposed method. Results indicate suitable performance of the proposed method in detecting out‐of‐control conditions and providing precise diagnostic information. Results also illustrate suitable performance of our proposed method in comparison with a competitive approach.     

Bayesian monitoring to detect a shift in process mean

A Bayesian analogue of the Shewhart X‐bar chart is defined and compared with cumulative sum charts. The comparison identifies types of production process where the Bayesian chart has better expected performance than the cumulative sum chart. Implementing the Bayesian chart requires more detailed knowledge of the process structure than is required by the best‐known types of charts, but acquiring this information can yield tangible benefits. Copyright © 2007 John Wiley & Sons, Ltd.     

Phase II monitoring of general linear profiles in the presence of between‐profile autocorrelation

In this paper, an approach based on the U statistic is first proposed to eliminate the effect of between‐profile autocorrelation of error terms in Phase‐II monitoring of general linear profiles. Then, a control chart based on the adjusted parameter estimates is designed to monitor the parameters of the model. The performance of the proposed method is compared with the ones of some existing methods in terms of average run length for weak, moderate, and strong autocorrelation coefficients under different shift scenarios. The results show that the proposed method provides significantly better results than the competing methods to detect shifts in the regression parameters, while the competing methods perform better in detecting shifts in the standard deviation. At the end, the applicability of the proposed method is illustrated by an example. Copyright © 2015 John Wiley & Sons, Ltd.     

A conjugate Bayesian approach to control chart for multi-batch and low volume production

Control chart could effectively reflect whether a manufacturing process is currently under control or not. The calculation of control limits of the control chart has been focusing on traditional frequency approach, which requires a large sample size for an accurate estimation. A conjugate Bayesian approach is introduced to correct the calculation error of control limits with traditional frequency approach in multi-batch and low volume production. Bartlett’s test, analysis of variance test and standardisation treatment are used to construct a proper prior distribution in order to calculate the Bayes estimators of process distribution parameters for the control limits. The case study indicates that this conjugate Bayesian approach presents better performance than the traditional frequency approach when the sample size is small.     

Phase II monitoring of multivariate multiple linear regression profiles

In certain cases, the quality of a process or a product can be effectively characterized by two or more multiple linear regression profiles in which response variables are correlated. This structure can be modeled as multivariate multiple linear regression profiles. When linear profiles are monitored separately, then correlation between response variables is ignored and misleading results could be expected. To overcome this problem, the use of methods that consider the multivariate structure between response variables is inevitable. In this paper, we propose four methods to monitor this structure in Phase II. The performance of the methods is compared through simulation studies in terms of the average run length criterion. Furthermore, a method based on likelihood ratio approach is developed to determine the location of shifts and a numerical simulation is used to evaluate the performance of the proposed method. Finally, the use of the methods is illustrated by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

Statistical monitoring of nonlinear product and process quality profiles

In many quality control applications, use of a single (or several distinct) quality characteristic(s) is insufficient to characterize the quality of a produced item. In an increasing number of cases, a response curve (profile) is required. Such profiles can frequently be modeled using linear or nonlinear regression models. In recent research others have developed multivariate T² control charts and other methods for monitoring the coefficients in a simple linear regression model of a profile. However, little work has been done to address the monitoring of profiles that can be represented by a parametric nonlinear regression model. Here we extend the use of the T² control chart to monitor the coefficients resulting from a parametric nonlinear regression model fit to profile data. We give three general approaches to the formulation of the T² statistics and determination of the associated upper control limits for Phase I applications. We also consider the use of non‐parametric regression methods and the use of metrics to measure deviations from a baseline profile. These approaches are illustrated using the vertical board density profile data presented in Walker and Wright (Comparing curves using additive models. Journal of Quality Technology 2002; 34:118–129). Copyright © 2007 John Wiley & Sons, Ltd.     

Effect of Phase I Estimation on Phase II Control Chart Performance with Profile Data

This paper illustrates how phase I estimators in statistical process control (SPC) can affect the performance of phase II control charts. The deleterious impact of poor phase I estimators on the performance of phase II control charts is illustrated in the context of profile monitoring. Two types of phase I estimators are discussed. One approach uses functional cluster analysis to initially distinguish between estimated profiles from an in‐control process and those from an out‐of‐control process. The second approach does not use clustering to make the distinction. The phase II control charts are established based on the two resulting types of estimates and compared across varying sizes of sustained shifts in phase II. A simulated example and a Monte Carlo study show that the performance of the phase II control charts can be severely distorted when constructed with poor phase I estimators. The use of clustering leads to much better phase II performance. We also illustrate that the performance of phase II control charts based on the poor phase I estimators not only have more false alarms than expected but can also take much longer than expected to detect potential changes to the process. Copyright © 2014 John Wiley & Sons, Ltd.     

On improved monitoring of linear profiles under modified successive sampling

Control charts are often used for the monitoring of quality characteristics of interest. There may exist some other characteristics that are associated with the main quality characteristic. A model that quantifies the relationship between them is termed as a profile, for instance, the relation between charge and capacitance. The monitoring of the main variable that is linearly related with an associated explanatory variable is termed as simple linear profiles' monitoring. It is a common practice to use simple random sampling (SRS) for profile monitoring. This study intends to enhance profile monitoring by considering modified successive sampling (MSS) approach. The performance of the existing and the proposed schemes are evaluated using the well‐known metrics average run length. The comparative analysis revealed that the proposed structure outperforms the existing ones in terms of efficiency. A real application from electrical engineering is used to show the implementation of our proposal in practice.     

Modeling and monitoring the nonlinear profile of heat treatment process data by using an approach based on a hyperbolic tangent function

In this article, a new approach is proposed which uses the hyperbolic tangent function to model and monitor vacuum heat treatment process data. The proposed hyperbolic tangent function approach is compared to the smoothing spline approach. The latter serves as the benchmark when the vacuum heat treatment profile is investigated. The vector of the obtained parameter estimates is monitored by using Hotelling's method for the hyperbolic tangent function approach, and the metrics method used for the smoothing spline approach. For the purposes of verification, data from a real aluminum alloy heat treatment process is used to illustrate the proposed approach. In Phase I, the modified hyperbolic tangent function and the smoothing spline are first utilized to fit the process data. The proposed approach provides a better fitting result than the smoothing spline approach. In Phase II, the proposed approach produces a much better out-of-control average run length (ARL) performance than the smoothing spline approach when the heat treatment profile shows process abnormalities.     

Directed control charts for detecting the shape changes from linear profiles to quadratic profiles

In profile monitoring, control charts are constructed to detect any unanticipated departures from the statistical stability of product quality over time, where product quality is characterised by a function. In many situations, due to the characteristics of a system or an operation, certain process signals can be anticipated. Thus, when a kind of departure specifically feared is identified in advance, a directed process monitoring approach can be developed. Motivated by the monitoring of cylindrical surfaces, this paper focuses on quickly detecting the shape changes from a straight line to a second-order polynomial curve. Based on the hypothesis testing on the quadratic term, two directed control charts and a combined scheme are proposed to surveillance the sampled linear shape. The performance of our proposed methods is studied and compared with the alternative charts by numerical simulations. Simulation studies show that the two proposed directed charts are almost the same, and outperform the alternative methods in some cases. Moreover, the combined scheme is robust for all the parameter combinations.     

A Bayesian approach to diagnosing covariance matrix shifts

In addition to the quick detection of abnormal changes in a multivariate process, it is also critical to provide an accurate fault identification of responsible components following an out-of-control signal. In line with the work of Tan and Shi for diagnosing shifts in the mean vector, this paper develops a Bayesian approach for diagnosing shifts in the covariance matrix. The simulation comparisons favor the proposed approach. A real example is also presented to demonstrate the implementation of the proposed method.     

Monitoring nonlinear profiles using a wavelet-based distribution-free CUSUM chart

WDFTC is a wavelet-based distribution-free CUSUM chart for detecting shifts in the mean of a profile with noisy components. Exploiting a discrete wavelet transform (DWT) of the mean in-control profile, WDFTC selects a reduced-dimension vector of the associated DWT components from which the mean in-control profile can be approximated with minimal weighted relative reconstruction error. Based on randomly sampled Phase I (in-control) profiles, the covariance matrix of the corresponding reduced-dimension DWT vectors is estimated using a matrix-regularisation method; then the DWT vectors are aggregated (batched) so that the non-overlapping batch means of the reduced-dimension DWT vectors have manageable covariances. To monitor shifts in the mean profile during Phase II operation, WDFTC computes a Hotelling's T ²-type statistic from successive non-overlapping batch means and applies a CUSUM procedure to those statistics, where the associated control limits are evaluated analytically from the Phase I data. Experimentation with several normal and non-normal test processes revealed that WDFTC was competitive with existing profile-monitoring schemes.     

Phase I monitoring of social networks based on Poisson regression profiles

Nowadays, due to the increasing role of social networks in our daily life, monitoring and forecasting social trends have attracted the attention of many researchers. To the best of the authors' knowledge, the literature includes few studies of monitoring social networks. Existing researches have focused on analyzing only the existence of communications between people and have neglected to monitor the number of such communications. In this paper, first counts of communications between people are modeled using Poisson regression profiles. Then, 3 Phase I monitoring methods, extended T², F, and a standardized likelihood ratio test method is suggested to detect step changes, drift, and outliers in the parameters of Poisson regression profiles. The proposed methods are evaluated via simulation studies in terms of signal probability criterion. The results show that in most out‐of‐control situations the standardized likelihood ratio test method outperforms the T² and F methods. Then, a numerical example and a case study based on Enron email data are presented to illustrate the application of the extended methods.     

A Spatiotemporal Method for the Monitoring of Image Data

Machine vision systems are increasingly being used in industrial applications because of their ability to quickly provide information on product geometry, surface defects, surface finish, and other product and process characteristics. Previous research for monitoring these visual characteristics using image data has focused on either detecting changes within an image or between images. Extending these methods to include both the spatial and the temporal aspects of image data would provide more detailed diagnostic information, which would be of great value to industrial practitioners. Therefore, in this article, we show how image data can be monitored using a spatiotemporal framework that is based on an extension of a generalized likelihood ratio control chart. The performance of the proposed method is evaluated through computer simulations and experimental studies. The results show that our proposed spatiotemporal method is capable of quickly detecting the emergence of a fault. The computer simulations also show that our proposed generalized likelihood ratio control charting method provides a good estimate of the change point and the size/location of the fault, which are important fault diagnostic metrics that are not typically provided in the image monitoring literature. Finally, we highlight some research opportunities and provide some advice to practitioners. Copyright © 2012 John Wiley & Sons, Ltd.     

Novel Bayesian CUSUM and EWMA control charts via various loss functions for monitoring processes

In this work, both the cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts have been reconfigured to monitor processes using a Bayesian approach. Our construction of these charts are informed by posterior and posterior predictive distributions found using three loss functions: the squared error, precautionary, and linex. We use these control charts on count data, performing a simulation study to assess chart performance. Our simulations consist of sensitivity analysis of the out-of-control shift size and choice of hyper-parameters of the given distributions. Practical use of theses charts are evaluated on real data.     

A Bayesian control chart for monitoring the ratio of Weibull percentiles

This paper introduces a new Bayesian control chart to compare two processes by monitoring the ratio of percentiles of two quality characteristics that are assumed to be independent Weibull distributed random variables with the same and stable shape parameter larger than one. The chart analyses the sampling data directly, instead of transforming them in order to comply with the usual normality assumption, as many charts do. A real application in the wood industry and a wide simulation illustrate the features of the chart and its performance, depending on the number of training data, the quality of prior information, and the magnitude of the shift.     

A phase I multi-modelling approach for profile monitoring of signal data

Many industrial processes exhibit multiple in-control signatures, where signal data vary over time without affecting the final product quality. They are known as multimode processes. With regard to profile monitoring methodologies, the existence of multiple in-control patterns entails the study and development of novel monitoring schemes. We propose a method based on coupling curve classification and monitoring that inherits the so-called ‘multi-modelling framework’. The goal is to design a monitoring tool that is able to automatically adapt the control chart parameters to the current operating mode. The proposed approach allows assessing which mode new data belong to before applying a control chart to determine if they are actually in control or not. Contrary to mainstream multi-modelling techniques, we propose extending the classification step to include a novelty detection capability, in order to deal with the possible occurrence of in-control operating modes during the design phase that were not observed previously. The functional data depth paradigm is proposed to design both the curve classification and the novelty detection algorithm. A simulation study is presented to demonstrate the performances of the proposed methodology, which is compared against benchmark methods. A real case study is presented too, which consists of a multimode end-milling process, where different operating conditions yield different cutting force profile patterns.     

Multilevel spatial randomness approach for monitoring changes in 3D topographic surfaces

The 3D surface topography of finished products is a key characteristic for monitoring the quality of products and manufacturing processes. The topography has unique properties in which the topographic values are spatially autocorrelated with their neighbours and the locations of topographic values randomly change from one surface to another under the in-control process behaviour, making the online detection of local topographic changes challenging. Due to the complex structure of topographic data, the existing monitoring approaches lack the detection of local changes. Therefore, we develop a novel online monitoring approach for detecting local changes in 3D topographic surfaces. We introduce a multilevel surface thresholding algorithm for enhancing the representation of topographic values by slicing the 3D surface topography into cumulative levels in reference to the characteristics of the in-control surfaces. The spatial and random properties of topographic values are quantified at each surface level through the proposed spatial randomness profile. After obtaining the spatial randomness profile, an effective monitoring statistic based on the functional principal component analysis is developed for detecting anomaly surfaces. The proposed approach shows superior performance in identifying a wide range of fault patterns and outperforms the existing approaches in both simulated and real-life topographic data.