A Comparison Between Two Multivariate EWMA Schemes

The signal issued by a control chart triggers the process professionals to investigate the special cause. Change point methods simplify the efforts to search for and identify the special cause. In this study, using maximum likelihood estimation, a multivariate joint change point estimation procedure for monitoring both location and dispersion simultaneously is proposed. After a signal is generated by the simultaneously used Hotelling's T ² and/or generalized variance control charts, the procedure starts detecting the time of the change. The performance of the proposed method for several structural changes for the mean vector and covariance matrix is discussed.     

Statistical control charts for monitoring the mean of a stationary process

Recently statistical process control (SPC) methodologies have been developed to accommodate autocorrelated data. A primary method to deal with autocorrelated data is the use of residual charts. Although this methodology has the advantage that it can be applied to any autocorrelated data it needs time series modeling efforts. In addition for a X residual chart the detection capability is sometimes small compared to the X chart and EWMA chart. Zhang (1998) proposed the EWMAST chart which is constructed by charting the EWMA statistic for stationary processes to monitor the process mean. The performance of the EWMAST chart the X chart the X residual chart and other charts were compared in Zhang (1998). In this paper comparisons are made among the EWMAST chart the CUSUM residual chart and EWMA residual chart as well as the X residual chart and X chart via the average run length.     

Applying State Space to SPC: Monitoring Multivariate Time Series

Monitoring cross-sectional and serially interdependent processes has become a new issue in statistical process control (SPC). In up-to-date SPC literature, Kalman filtering was reported to monitor univariate autocorrelated processes. This paper applies a Kalman filter or state-space method for SPC to monitoring multivariate time series. We use Aoki's approach to estimate the parameter matrices of a state-space model. Multivariate Hotelling T ² control charts are employed to monitor the residuals of the state-space. Examples of this approach are illustrated.     

Bootstrap-Based T 2 Multivariate Control Charts

Control charts have been used effectively for years to monitor processes and detect abnormal behaviors. However, most control charts require a specific distribution to establish their control limits. The bootstrap method is a nonparametric technique that does not rely on the assumption of a parametric distribution of the observed data. Although the bootstrap technique has been used to develop univariate control charts to monitor a single process, no effort has been made to integrate the effectiveness of the bootstrap technique with multivariate control charts. In the present study, we propose a bootstrap-based multivariate T ² control chart that can efficiently monitor a process when the distribution of observed data is nonnormal or unknown. A simulation study was conducted to evaluate the performance of the proposed control chart and compare it with a traditional Hotelling's T ² control chart and the kernel density estimation (KDE)-based T ² control chart. The results showed that the proposed chart performed better than the traditional T ² control chart and performed comparably with the KDE-based T ² control chart. Furthermore, we present a case study to demonstrate the applicability of the proposed control chart to real situations.     

One-Class Classification-Based Control Charts for Monitoring Autocorrelated Multivariate Processes

In recent years, statistical process control (SPC) of multivariate and autocorrelated processes has received a great deal of attention. Modern manufacturing/service systems with more advanced technology and higher production rates can generate complex processes in which consecutive observations are dependent and each variable is correlated. These processes obviously violate the assumption of the independence of each observation that underlies traditional SPC and thus deteriorate the performance of its traditional tools. The popular way to address this issue is to monitor the residuals—the difference between the actual value and the fitted value—with the traditional SPC approach. However, this residuals-based approach requires two steps: (1) finding the residuals; and (2) monitoring the process. Also, an accurate prediction model is necessary to obtain the uncorrelated residuals. Furthermore, these residuals are not the original values of the observations and consequently may have lost some useful information about the targeted process. The main purpose of this article is to examine the feasibility of using one-class classification-based control charts to handle multivariate and autocorrelated processes. The article uses simulated data to present an analysis and comparison of one-class classification-based control charts and the traditional Hotelling's T ² chart.     

A Class of Markov Chain Models for Average Run Length Computations for Autocorrelated Processes

The average run length (ARL) of conventional control charts is typically computed assuming temporal independence. However, this assumption is frequently violated in practical applications. Alternative ARL computations have often been conducted via time consuming and yet not necessarily very accurate simulations. In this article, we develop a class of Markov chain models for evaluating the run length performance of traditional control charts for autocorrelated processes. We show extensions from the univariate AR(1) model to the general multivariate VARMA(p, q) time series. The results of the proposed method are highly comparable to those of simulations and with significantly less computational overhead.     

The effect of parameter estimation on phase II monitoring of poisson regression profiles

ABSTRACT

The effect of parameters estimation on profile monitoring methods has only been studied by a few researchers and only the assumption of a normal response variable has been tackled. However, in some practical situation, the normality assumption is violated and the response variable follows a discrete distribution such as Poisson. In this paper, we evaluate the effect of parameters estimation on the Phase II monitoring of Poisson regression profiles by considering two control charts, namely the Hotelling’s T² and the multivariate exponentially weighted moving average (MEWMA) charts. Simulation studies in terms of the average run length (ARL) and the standard deviation of the run length (SDRL) are carried out to assess the effect of estimated parameters on the performance of Phase II monitoring approaches. The results reveal that both in-control and out-of-control performances of these charts are adversely affected when the regression parameters are estimated.     

Bivariate control charts for testing x data patterns

A new control chart, called the θ chart, for monitoring the mean of a process with bivariate quality characteristics is proposed. It can identify a rotation, shift or alternation between the subgroups of the process mean. The conventional application of X² chart to identify a sudden shift of the process mean is also expanded to identify a change of the process mean or a change of the process dispersion. Furthermore, when used together, the θ and X² charts could provide further insight into the process.     

SPC for short-run multivariate autocorrelated processes

This paper discusses the development of a multivariate control charting technique for short-run autocorrelated data manufacturing environment. The proposed approach is a combination of the multivariate residual charts for autocorrelated data and the multivariate transformation technique for i.i.d. process observations of short lengths. The proposed approach consists in fitting adequate multivariate time-series model of various process outputs and computes the residuals, transforming them into standard normal N(0, 1) data and then using standardized data as inputs to plot conventional univariate i.i.d. control charts. The objective for applying multivariate finite horizon techniques for autocorrelated processes is to allow continuous process monitoring, since all process outputs are controlled trough the use of a single control chart with constant control limits. Throughout simulated examples, it is shown that the proposed short-run process monitoring technique provides approximately similar shifts detection properties as VAR residual charts.     

The effect of Phase I sample size on the run length performance of control charts for autocorrelated data

Traditional control charts assume independence of observations obtained from the monitored process. However, if the observations are autocorrelated, these charts often do not perform as intended by the design requirements. Recently, several control charts have been proposed to deal with autocorrelated observations. The residual chart, modified Shewhart chart, EWMAST chart, and ARMA chart are such charts widely used for monitoring the occurrence of assignable causes in a process when the process exhibits inherent autocorrelation. Besides autocorrelation, one other issue is the unknown values of true process parameters to be used in the control chart design, which are often estimated from a reference sample of in-control observations. Performances of the above-mentioned control charts for autocorrelated processes are significantly affected by the sample size used in a Phase I study to estimate the control chart parameters. In this study, we investigate the effect of Phase I sample size on the run length performance of these four charts for monitoring the changes in the mean of an autocorrelated process, namely an AR(1) process. A discussion of the practical implications of the results and suggestions on the sample size requirements for effective process monitoring are provided.     

On the estimation of serial correlation in Markov-dependent production processes

In this paper, we present a study about the estimation of the serial correlation for Markov chain models which is used often in the quality control of autocorrelated processes. Two estimators, non-parametric and multinomial, for the correlation coefficient are discussed. They are compared with the maximum likelihood estimator [U.N. Bhat and R. Lal, Attribute control charts for Markov dependent production process, IIE Trans. 22 (2) (1990), pp. 181–188.] by using some theoretical facts and the Monte Carlo simulation under several scenarios that consider large and small correlations as well a range of fractions (p) of non-conforming items. The theoretical results show that for any value of p≠0.5 and processes with autocorrelation higher than 0.5, the multinomial is more precise than maximum likelihood. However, the maximum likelihood is better when the autocorrelation is smaller than 0.5. The estimators are similar for p=0.5. Considering the average of all simulated scenarios, the multinomial estimator presented lower mean error values and higher precision, being, therefore, an alternative to estimate the serial correlation. The performance of the non-parametric estimator was reasonable only for correlation higher than 0.5, with some improvement for p=0.5.     

A control chart based on sample ranges for monitoring the covariance matrix of the multivariate processes

For the univariate case, the R chart and the S ² chart are the most common charts used for monitoring the process dispersion. With the usual sample size of 4 and 5, the R chart is slightly inferior to the S ² chart in terms of efficiency in detecting process shifts. In this article, we show that for the multivariate case, the chart based on the standardized sample ranges, we call the RMAX chart, is substantially inferior in terms of efficiency in detecting shifts in the covariance matrix than the VMAX chart, which is based on the standardized sample variances. The user's familiarity with sample ranges is a point in favor of the RMAX chart. An example is presented to illustrate the application of the proposed chart.     

Control chart for monitoring the Weibull shape parameter under two competing risks

ABSTRACT

In this paper, we propose a control chart to monitor the Weibull shape parameter where the observations are censored due to competing risks. We assume that the failure occurs due to two competing risks that are independent and follow Weibull distribution with different shape and scale parameters. The control charts are proposed to monitor one or both of the shape parameters of competing risk distributions and established based on the conditional expected values. The proposed control chart for both shape parameters is used in certain situations and allows to monitor both shape parameters in only one chart. The control limits depend on the sample size, number of failures due to each risk and the desired stable average run length (ARL). We also consider the estimation problem of the target parameters when the Phase I sample is incomplete. We assumed that some of the products that fail during the life testing have a cause of failure that is only known to belong to a certain subset of all possible failures. This case is known as masking. In the presence of masking, the expectation-maximization (EM) algorithm is proposed to estimate the parameters. For both cases, with and without masking, the behaviour of ARLs of charts is studied through the numerical methods. The influence of masking on the performance of proposed charts is also studied through a simulation study. An example illustrates the applicability of the proposed charts.     

Mixed multivariate EWMA-CUSUM control charts for an improved process monitoring

Multivariate exponential weighted moving average and cumulative sum charts are the most common memory type multivariate control charts. They make use of the present and past information to detect small shifts in the process parameter(s). In this article, we propose two new multivariate control charts using a mixed version of their design setups. The plotting statistics of the proposed charts are based on the cumulative sum of the multivariate exponentially weighted moving averages. The performances of these schemes are evaluated in terms of average run length. The proposals are compared with their existing counterparts, including HotellingT², MCUSUM, MEWMA, and MC1 charts. An application example is also presented for practical considerations using a real dataset.     

Monitoring and change-point estimation for spline-modeled non-linear profiles in phase II

In some applications of statistical quality control, quality of a process or a product is best characterized by a functional relationship between a response variable and one or more explanatory variables. This relationship is referred to as a profile. In certain cases, the quality of a process or a product is better described by a non-linear profile which does not follow a specific parametric model. In these circumstances, nonparametric approaches with greater flexibility in modeling the complicated profiles are adopted. In this research, the spline smoothing method is used to model a complicated non-linear profile and the Hotelling T² control chart based on the spline coefficients is used to monitor the process. After receiving an out-of-control signal, a maximum likelihood estimator is employed for change point estimation. The simulation studies, which include both global and local shifts, provide appropriate evaluation of the performance of the proposed estimation and monitoring procedure. The results indicate that the proposed method detects large global shifts while it is very sensitive in detecting local shifts.     

Integration of classification algorithms and control chart techniques for monitoring multivariate processes

We propose new multivariate control charts that can effectively deal with massive amounts of complex data through their integration with classification algorithms. We call the proposed control chart the ‘Probability of Class (PoC) chart’ because the values of PoC, obtained from classification algorithms, are used as monitoring statistics. The control limits of PoC charts are established and adjusted by the bootstrap method. Experimental results with simulated and real data showed that PoC charts outperform Hotelling's T ² control charts. Further, a simulation study revealed that a small proportion of out-of-control observations are sufficient for PoC charts to achieve the desired performance.     

Bent‐cable regression with autoregressive noise

Motivated by time series of atmospheric concentrations of certain pollutants the authors develop bent‐cable regression for autocorrelated errors. Bent‐cable regression extends the popular piecewise linear (broken‐stick) model, allowing for a smooth change region of any non‐negative width. Here the authors consider autoregressive noise added to a bent‐cable mean structure, with unknown regression and time series parameters. They develop asymptotic theory for conditional least‐squares estimation in a triangular array framework, wherein each segment of the bent cable contains an increasing number of observations while the autoregressive order remains constant as the sample size grows. They explore the theory in a simulation study, develop implementation details, apply the methodology to the motivating pollutant dataset, and provide a scientific interpretation of the bent‐cable change point not discussed previously. The Canadian Journal of Statistics 38: 386–407; 2010 © 2010 Statistical Society of Canada     

Integration of support vector machines and control charts for multivariate process monitoring

Statistical process control tools have been used routinely to improve process capabilities through reliable on-line monitoring and diagnostic processes. In the present paper, we propose a novel multivariate control chart that integrates a support vector machine (SVM) algorithm, a bootstrap method, and a control chart technique to improve multivariate process monitoring. The proposed chart uses as the monitoring statistic the predicted probability of class (PoC) values from an SVM algorithm. The control limits of SVM-PoC charts are obtained by a bootstrap approach. A simulation study was conducted to evaluate the performance of the proposed SVM–PoC chart and to compare it with other data mining-based control charts and Hotelling's T ² control charts under various scenarios. The results showed that the proposed SVM–PoC charts outperformed other multivariate control charts in nonnormal situations. Further, we developed an exponential weighed moving average version of the SVM–PoC charts for increasing sensitivity to small shifts.     

Identifying the time of a step change with multivariate single control charts

Change point estimation procedures simplify the efforts to search for and identify special causes in multivariate statistical process monitoring. After a signal is generated by the simultaneously used control charts or a single control chart, add-on change point procedure estimates the time of the change. In this study, multivariate joint change point estimation performance for simultaneous monitoring of both location and dispersion is compared under the assumption that various single charts are used to monitor the process. The change detection performance for several structural changes for the mean vector and covariance matrix is also discussed. It is concluded that choice of the control chart to obtain a signal may affect the change point detection performance.     

Efficiency analysis of ten estimation procedures for quantitative linear models with autocorrelated errors

Many estimation procedures for quantitative linear models with autocorrelated errors have been proposed in the literature. A number of these procedures have been compared in various ways for different sample sizes and autocorrelation parameters values and for structured or random explanatory vaiables. In this paper, we revisit three situations that were considered to some extent in previous studies, by comparing ten estimation procedures: Ordinary Least Squares (OLS), Generalized Least Squares (GLS), estimated Generalized Least Squares (six procedures), Maximum Likelihood (ML), and First Differences (FD). The six estimated GLS procedures and the ML procedure differ in the way the error autocovariance matrix is estimated. The three situations can be defined as follows: Case 1, the explanatory variable x in the simple linear regression is fixed; Case 2,x is purely random; and Case 3x is first-order autoregressive. Following a theoretical presentation, the ten estimation procedures are compared in a Monte Carlo study conducted in the time domain, where the errors are first-order autoregressive in Cases 1-3. The measure of comparison for the estimation procedures is their efficiency relative to OLS. It is evaluated as a function of the time series length and the magnitude and sign of the error autocorrelation parameter. Overall, knowledge of the model of the time series process generating the errors enhances efficiency in estimated GLS. Differences in the efficiency of estimation procedures between Case 1 and Cases 2 and 3 as well as differences in efficiency among procedures in a given situation are observed and discussed.