An Evaluation of Wheeler's Method for Monitoring the Rate of Rare Events

Its wide application in practice makes the monitoring of the rate of rare events a popular research topic. Recently a researcher proposed plotting the counts between events on an individuals X‐chart with an upper control limit to detect process improvement and plotting the reciprocals of the counts on an X‐chart to detect process deterioration. He also used the median as the center line and the median moving range to obtain control limits in both control charts to address the problem of the standard deviation estimate inflation caused by extreme values. In our paper, we investigated the statistical performance of the four proposed approaches using simulation. We find using the mean results in a high proportion of ineffective control limits, while using the median avoids the issue of ineffective control limits but produces an unacceptably high proportion of false alarms. Copyright © 2016 John Wiley & Sons, Ltd.     

The Design of the S2 Control Charts Based on Conditional Performance via Exact Methods

In this paper, we consider the conditional performance of the equal‐tailed and average run lengths (ARL)‐unbiased two‐sided S² charts when the in‐control variance of a normal process is estimated. We derive the exact probability distributions of the conditional ARL for the two S² charts. Then we evaluate the performance of each S² chart in terms of the percentiles, mean and standard deviation of the conditional in‐control ARL distribution. Because the parameter estimation seriously affects the conditional performance of these S² charts, we propose an exact method to design the equal‐tailed and ARL‐unbiased S² charts with desired conditional in‐control performance. The results indicate that the new ARL‐unbiased S² chart has far smaller standard deviation ARL values and the unconditional ARL values are more close to the desired value than the corresponding new equal‐tailed S² chart. Copyright © 2017 John Wiley & Sons, Ltd.     

Effect of estimation under nonnormality on the phase II performance of linear profile monitoring approaches

The number of studies about control charts proposed to monitor profiles, where the quality of a process/product is expressed as function of response and explanatory variable(s), has been increasing in recent years. However, most authors assume that the in‐control parameter values are known in phase II analysis and the error terms are normally distributed. These assumptions are rarely satisfied in practice. In this study, the performance of EWMA‐R, EWMA‐3, and EWMA‐3(d²) methods for monitoring simple linear profiles is examined via simulation where the in‐control parameters are estimated and innovations have a Student's t distribution or gamma distribution. Instead of the average run length (ARL) and the standard deviation of run length, we used average and standard deviation of the ARL as performance measures in order to capture the sampling variation among different practitioners. It is seen that the estimation effect becomes more severe when the number of phase I profiles used in estimation decreases, as expected, and as the distribution deviates from normality to a greater extent. Besides, although the average ARL values get closer to the desired values as the amount of phase I data increases, their standard deviations remain far away from the acceptable level indicating a high practitioner‐to‐practitioner variability.     

Fault Diagnosis in Multivariate Control Charts Using Artificial Neural Networks

Most multivariate quality control procedures evaluate the in‐control or out‐of‐control condition based upon an overall statistic, like Hotelling's T². Although T² is optimal for finding a general shift in mean vectors, it is not optimal for shifts that occur for some subset of variables. This introduces a persistent problem in multivariate control charts, namely the interpretation of a signal that often discourages practitioners in applying them. In this paper, we propose an artificial neural network based model to diagnose faults in out‐of‐control conditions and to help identify aberrant variables when Shewhart‐type multivariate control charts based on Hotelling's T² are used. The results of the model implementation on two numerical examples and one case of real world data are encouraging. Copyright © 2005 John Wiley & Sons, Ltd.     

Alternative Hotelling's T2 Charts using Winsorized Modified One‐Step M‐estimator

Hotelling's T² chart is a popular tool for monitoring statistical process control. However, this chart is sensitive in the presence of outliers. To alleviate the problem, this paper proposed alternative Hotelling's T² charts for individual observations using robust location and scale matrix instead of the usual mean vector and the covariance matrix, respectively. The usual mean vector in the Hotelling T² chart is replaced by the winsorized modified one‐step M‐estimator (MOM) whereas the usual covariance matrix is replaced by the winsorized covariance matrix. MOM empirically trims the data based on the shape of the data distribution. This study also investigated on the different trimming criteria used in MOM. Two robust scale estimators with highest breakdown point, namely S_n and T_n were selected to suit the criteria. The upper control limits for the proposed robust charts were calculated based on simulated data. The performance of each control chart is based on the false alarm and the probability of outlier's detection. In general, the performance of an alternative robust Hotelling's T² charts is better than the performance of the traditional Hotelling's T² chart. Copyright © 2012 John Wiley & Sons, Ltd.     

On continuous,discontinuous, mixed,and primal hybrid finite element methods for second‐order elliptic problems

Finite element formulations for second‐order elliptic problems, including the classic H¹‐conforming Galerkin method, dual mixed methods, a discontinuous Galerkin method, and two primal hybrid methods, are implemented and numerically compared on accuracy and computational performance. Excepting the discontinuous Galerkin formulation, all the other formulations allow static condensation at the element level, aiming at reducing the size of the global system of equations. For a three‐dimensional test problem with smooth solution, the simulations are performed with h‐refinement, for hexahedral and tetrahedral meshes, and uniform polynomial degree distribution up to four. For a singular two‐dimensional problem, the results are for approximation spaces based on given sets of hp‐refined quadrilateral and triangular meshes adapted to an internal layer. The different formulations are compared in terms of L²‐convergence rates of the approximation errors for the solution and its gradient, number of degrees of freedom, both with and without static condensation. Some insights into the required computational effort for each simulation are also given.     

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.     

Effective application of Q(R) charts in low‐volume manufacturing

Q charts provide means for statistical process control in low‐volume processes and start‐up phases of production. Concerns on their performance have led to research into different types of enhancements and much discussion on the appropriateness of these. Driven by the aim to implement control charts in the low‐volume production of advanced wafer steppers, we investigate the performance of additional run rules and tightening control limits on the traditional Q chart compared with an exponentially weighted moving average (EWMA). Furthermore, we develop an alternative QR chart based on the mean moving range as estimator of the process standard deviation and consider the economics of low‐volume processes by means of a specific cost model. The comparisons are based on the run length distributions after a permanent shift and trend, both with an onset early in the process. Real life examples are given for various important variables in wafer stepper production. It is concluded that the EWMA based on QR statistics provides the best performance throughout. Competing alternatives with almost equal performance are the EWMA of Q statistics and the combination of four tests of special causes (1‐of‐1, 2‐of‐3, 4‐of‐5 and 8‐of‐8) applied on either the Q or QR chart. Overall, the mean moving range performs better. Copyright © 1999 John Wiley & Sons, Ltd.     

New Synthetic Control Charts for Monitoring Process Mean and Process Dispersion

A statistical quality control chart is widely recognized as a potentially powerful tool that is frequently used in many manufacturing and service industries to monitor the quality of the product or manufacturing processes. In this paper, we propose new synthetic control charts for monitoring the process mean and the process dispersion. The proposed synthetic charts are based on ranked set sampling (RSS), median RSS (MRSS), and ordered RSS (ORSS) schemes, named synthetic‐RSS, synthetic‐MRSS, and synthetic‐ORSS charts, respectively. Average run lengths are used to evaluate the performances of the control charts. It is found that the synthetic‐RSS and synthetic‐MRSS mean charts perform uniformly better than the Shewhart mean chart based on simple random sampling (Shewhart‐SRS), synthetic‐SRS, double sampling‐SRS, Shewhart‐RSS, and Shewhart‐MRSS mean charts. The proposed synthetic charts generally outperform the exponentially weighted moving average (EWMA) chart based on SRS in the detection of large mean shifts. We also compare the performance of the synthetic‐ORSS dispersion chart with the existing powerful dispersion charts. It turns out that the synthetic‐ORSS chart also performs uniformly better than the Shewhart‐R, Shewhart‐S, synthetic‐R, synthetic‐S, synthetic‐D, cumulative sum (CUSUM) ln S², CUSUM‐R, CUSUM‐S, EWMA‐ln S², and change point CUSUM charts for detecting increases in the process dispersion. A similar trend is observed when the proposed synthetic charts are constructed under imperfect RSS schemes. Illustrative examples are used to demonstrate the implementation of the proposed synthetic charts. Copyright © 2014 John Wiley & Sons, Ltd.     

Memory‐Type Control Charts for Monitoring the Process Dispersion

Control charts have been broadly used for monitoring the process mean and dispersion. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are memory control charts as they utilize the past information in setting up the control structure. This makes CUSUM and EWMA‐type charts good at detecting small disturbances in the process. This article proposes two new memory control charts for monitoring process dispersion, named as floating T ? S² and floating U ? S² control charts, respectively. The average run length (ARL) performance of the proposed charts is evaluated through a simulation study and is also compared with the CUSUM and EWMA charts for process dispersion. It is found that the proposed charts are better in detecting both positive as well as negative shifts. An additional comparison shows that the floating U ? S² chart has slightly smaller ARLs for larger shifts, while for smaller shifts, the floating T ? S² chart has better performance. An example is also provided which shows the application of the proposed charts on simulated datasets. Copyright © 2013 John Wiley & Sons, Ltd.     

A bridge between projection methods and SIMPLE type methods for incompressible Navier–Stokes equations

A bridge is built between projection methods and SIMPLE type methods (Semi‐Implicit Method for Pressure‐Linked Equation). A general second‐order accurate projection method is developed for the simulation of incompressible unsteady flows by employing a non‐linear update of pressure term as Θⁿ?pⁿ⁺¹+(I?Θⁿ)?pⁿ, where Θⁿ is a coefficient matrix, which may depend on the grid size, time step and even velocity. It includes three‐ and four‐step projection methods. The standard SIMPLE method is written in a concise formula for steady and unsteady flows. It is proven that SIMPLE type methods have second‐order temporal accuracy for unsteady flows. The classical second‐order projection method and SIMPLE type methods are united within the framework of the general second‐order projection formula. Two iteration algorithms of SIMPLE type methods for unsteady flows are described and discussed. In addition, detailed formulae are provided for general projection methods by using the Runge–Kutta technique to update the convective term and Crank–Nicholson scheme for the diffusion term. Copyright © 2007 John Wiley & Sons, Ltd.     

On the performance of coefficient of variation charts in the presence of measurement errors

In the literature, coefficient of variation control charts have been introduced under the assumption of no measurement errors. However, measurement errors always exist in practice, and they do affect the performance of control charts in the detection of an out‐of‐control situation. In this paper, we therefore study the performance of a coefficient of variation Shewhart‐type control chart (Shewhart‐CV chart) and also one‐sided coefficient of variation exponentially weighted moving average–type control charts (EWMA‐γ² charts) using a model with linear covariates. Moreover, we propose and study the performance of a two‐sided EWMA‐γ² chart using a model with linear covariates. Several figures and tables are provided and analyzed to evaluate the statistical performance of these control charts for different sources of measurement errors. The obtained results show that the precision and accuracy errors significantly affect the performance of both the Shewhart‐CV and EWMA‐γ² control charts. An example illustrating the use of this study is finally presented.     

Statistical Performance of a Control Chart for Individual Observations Monitoring the Ratio of Two Normal Variables

Statistical Process Control monitoring of the ratio Z of two normal variables X and Y has received too little attention in quality control literature. Several applications dealing with monitoring the ratio Z can be found in the industrial sector, when quality control of products consisting of several raw materials calls for monitoring their proportions (ratios) within a product. Tables about the statistical performance of these charts are still not available. This paper investigates the statistical performance of a Phase II Shewhart control chart monitoring the ratio of two normal variables in the case of individual observations. The obtained results show that the performance of the proposed chart is a function of the distribution parameters of the two normal variables. In particular, the Shewhart chart monitoring the ratio Z outperforms the (p = 2) multivariate T² control chart when a process shift affects the in‐control mean of X or, alternatively, of Y and the correlation among X and Y is high and when the in‐control means of X and Y shift contemporarily to opposite directions. The sensitivity of the proposed chart to a shift of the in‐control dispersion has been investigated, too. We also show that the standardization of the two variables before computing their ratio is not a good practice due to a significant loss in the chart's statistical performance. An illustrative example from the food industry details the implementation of the ratio control chart. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Effect of Measurement Error with Linearly Increasing‐Type Variance on Simultaneous Monitoring of Process Mean and Variability

In most quality control applications, the errors generated from measurement system can adversely affect the ability of control charts in detecting out‐of‐control conditions. In this paper, the effect of measurement error with linearly increasing‐type variance on the performance of maximum exponentially weighted moving average and mean‐squared deviation (MAX‐EWMAMS) control chart is studied. For this purpose, different out‐of‐control scenarios including mean shifts, variance shifts, and simultaneous shifts in both are considered, and the detecting performance of the proposed approach is investigated through simulation study. The results of simulation study in terms of three criteria including average run length, standard deviation of run lengths, and the empirical distribution of run lengths prove that the measurement error with linearly increasing‐type variance can adversely affect the performance of MAX‐EWMAMS control. Copyright © 2015 John Wiley & Sons, Ltd.     

A nonparametric CUSUM chart for process dispersion

In the present article, we propose a nonparametric cumulative sum control chart for process dispersion based on the sign statistic using in‐control deciles. The chart can be viewed as modified control chart due to Amin et al, 6 which is based on in‐control quartiles. An average run length performance of the proposed chart is studied using Markov chain approach. An effect of non‐normality on cumulative sum S² chart is studied. The study reveals that the proposed cumulative sum control chart is a better alternative to parametric cumulative sum S² chart, when the process distribution is non‐normal. We provide an illustration of the proposed cumulative sum control chart.     

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.     

Sum of Squares Control Charts for Monitoring of Multivariate Multiple Linear Regression Profiles in Phase II

In this paper, we propose four control charts for simultaneous monitoring of mean vector and covariance matrix in multivariate multiple linear regression profiles in Phase II. The proposed control charts include sum of squares exponential weighted moving average (SS‐EWMA) and sum of squares cumulative sum (SS‐CUSUM) for monitoring regression parameters and corresponding covariance matrix and SS‐EWMARe and SS‐CUSUMRe control charts for monitoring mean vector and covariance matrix of residual. Proposed methods are able to identify the out‐of‐control parameter responsible for shift. The performance of the proposed control charts is compared with existing method through Monte‐Carlo simulations. Moreover, the diagnostic performance of the proposed control charts is evaluated through simulation studies. The results show better performance of the proposed control charts rather than competing control chart. Finally, the applicability of the proposed control charts is illustrated using a real case of calibration application in the automotive industry. Copyright © 2016 John Wiley & Sons, Ltd.     

Entanglement between two atoms interacting with a stochastic field

The entanglement property between two two-level atoms interacting with a stochastic field is studied. We analytically prove that when the correlation time κ ^?1 of the stochastic field is very short compared to the radiative lifetime γ ^?1 of the atom, the steady-state entanglement between the two atoms can appear. The concurrence characterizing the entanglement degree has also been calculated by using Monte Carlo simulation. It is shown that the correlation time has a strong effect on the entanglement. We also discuss the influence of strength of the stochastic process on the entanglement. The validity of the decorrelation approximation is also investigated.     

To shrink or not to shrink: Hotelling's T2 chart based on shrunken covariance estimates

Several authors have studied the effect of parameter estimation on the performance of Phase II control charts and shown that large in‐control reference samples are necessary for the Phase II control charts to perform as desired. For higher dimensional data, even larger reference samples are required to achieve stable estimation of the in‐control parameters. Shrinkage estimation has been widely studied as a method to achieve stable estimation of the covariance matrix for high‐dimensional data. We investigate the average run length (ARL) distribution of the Hotelling T² chart when using a shrunken covariance matrix. Specifically, we explore the following questions: (1) Does the use of a shrinkage estimator of the covariance matrix result in reduced variability in the ARL performance of the T² chart? (2) Does the use of a shrinkage estimator of the covariance matrix result in a reduced occurrence of “strictly multivariate” false alarms on the T²chart? (3) How does shrinkage of the covariance matrix affect the out‐of‐control performance of the T² chart? We use a simulation study to investigate the use of shrinkage estimation with the Hotelling T² chart in Phase II. Our results indicate that, while shrinkage estimation affects the ARL performance of the T² chart, the benefits are small and occur in fairly specific circumstances. The benefits of shrinking may not justify the use of more advanced techniques.     

Model‐based Multivariate Monitoring Charts for Autocorrelated Processes

Autocorrelation or nonstationarity may seriously impact the performance of conventional Hotelling's T² charts. We suggest modeling processes with multivariate autoregressive integrated moving average time series models and propose two model‐based monitoring charts. One monitors the predicted value and provides information about the need for mean adjustments. The other is a Hotelling's T² control chart applied to the residuals. The average run length performance of the residual‐based Hotelling's T² chart is compared with the observed data‐based Hotelling's T² chart for a group of first‐order vector autoregressive models. We show that the new chart in most cases performs well. Copyright © 2013 John Wiley & Sons, Ltd.