A Note on the Aging Properties of the Run Length of Markov-Type Control Charts

Abstract

A change in a production process must be detected quickly so that a corrective action can be taken. Thus, it comes as no surprise that the run length (RL) is usually used to describe the performance of a quality control chart.

This popular performance measure has a phase-type distribution when dealing with Markov-type charts, namely, cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) charts, as opposed to a geometric distribution, when standard Shewhart charts are in use.

In this article, we briefly discuss sufficient conditions on the associated probability transition matrix to deal with run lengths with aging properties such as new better than used in expectation, new better than used, and increasing hazard rate.

We also explore the implications of these aging properties of the run lengths, namely when we decide to confront the in control and out-of-control variances of the run lengths of matched in control Shewhart and Markov-type control charts.     

Counting by Weighing: An Alternative Sampling Scheme

Abstract

In the last decades, various methods are applied for evaluating different performance measures of change-point detection schemes. In only some cases—recent examples are Chandrasekaran et al. (Chandrasekaran, S.; English, J.R.; Disney, R.L. Modeling and analysis of EWMA control schemes with variance-adjusted control limits. IIE Transactions 1995, 277, 282–290) and Steiner (Steiner, S.H. EWMA control charts with time-varying control limits and fast initial response. Journal of Quality Technology 1999, 31 (1), 75–86)—EWMA schemes with varying control limits are considered. However, EWMA charts with just these limits are sometimes more appropriate than those with fixed limits. Here, a computational approach is presented which allows to compute the usual performance measures with high precision. The main idea is connected to earlier results of Madsen and Conn (Madsen, R.W.; Conn, P.S. Ergodic behavior for nonnegative kernels. Ann. Probab. 1973, 1, 995–1013), Woodall (Woodall, W.H. The distribution of the run length of one-sided CUSUM procedures for continuous random variables. Technometrics 1983, 25, 295–301), and Waldmann (Waldmann, K.-H. Bounds for the distribution of the run length of geometric moving average charts. J. R. Stat. Soc., Ser. C, Appl. Stat. 1986a, 35, 151–158). Additionally, quantities as the steady-state ARL and the steady-state distribution of the chart statistic can be computed very precisely.     

Ewma charts for multivariate time series

In this paper a muIt,ivariat.e EWMA chart for time series is introduced. In principle, it is a generalization of the control scheme of Lowry et al. (I992) for multivarite indendent observations.

The autocovariances of the EWMA recursion are derived for stationary multivariate time series. IYsing tllese reslllts a co11t.rol chart hased or1 t11 illt.ivariate EWMA recursion is proposed. For a multivariate autoregressive process of order 1, a sufficient. condition is given such that the in-control average run length (ARL) is invariant, withrespect to the covariance of the White Noise process. This scheme is compared with the MEWMA control chart of Lowry et al. (1992) applied to the residuals.

By an extensive Monte carlo study the ARL of both charts are determined for several multivariate autoregressive processses.     

An ARL-unbiased thinning-based EWMA chart to monitor counts

Abstract

Shewhart control charts are known to be somewhat insensitive to shifts of small and moderate size. Expectedly, alternative control schemes such as the exponentially weighted moving average (EWMA) charts have been proposed to speed up the detection of such shifts. Unfortunately, applying the ordinary EWMA recursion to count data leads to a control statistic no longer with a fixed discrete range. Therefore, we propose a novel chart which relies on a EWMA control statistic where the usual scalar product is replaced by a thinning operation. Actually, we use the new fractional binomial thinning to avoid the typical over-smoothing ascribable to ceiling, rounding, and flooring operations. The properties of this discrete statistic are similar to the ones of its continuous EWMA counterpart and the run length (RL) performance of the associated chart can be computed exactly using the Markov chain approach for independent and identically distributed (i.i.d.) counts. Moreover, this chart is set in such way that: the average run length (ARL) curve attains a maximum in the in-control situation, i.e., the chart is ARL-unbiased; and the in-control ARL is equal to a pre-specified value. We use the R statistical software to provide compelling illustrations of this unconventional EWMA chart and to compare its RL performance with the ones of a few competing control charts for the mean of i.i.d. Poisson counts.     

A GLR Control Chart for Monitoring the Process Mean with Sequential Sampling

Abstract

Variable sampling rate (VSR) control charts have been used where the sampling rate changes as a function of the data from the process. In this article we investigate a generalized likelihood ratio (GLR) control chart based on sequential sampling (the SS GLR chart). The objective of process monitoring is assumed to be the effective detection of a wide range of two-sided shifts in the process mean. The performance of the SS GLR chart is evaluated and compared with other control charts. The SS GLR chart has much better performance than that of the fixed sampling rate GLR chart. It is also shown that the overall performance of the SS GLR chart is better than that of the variable sampling interval (VSI) GLR chart and the VSR CUSUM chart. The SS GLR chart has the additional advantage that it requires fewer parameters to be specified than other VSR charts. The optimal parameter choices are given in the article, and regression equations are provided to find the limits for the SS GLR chart.     

Evaluating properties of variable sampling interval control charts

Standard fixed sampling interval (FSI) control charts take samples from a process at fixed length sampling intervals for purposes of detecting changes in the peocess that may affect the quality of the output. Variable sampling interval (VSI) control charts vary the sampling interval as a function of what is observed from the process and can detect process changes faster than FSI control charts. Evaluation of properties of VSI control charts requires extensions of methods that have been developed for FSI control Control charts. A unified treatment of two widely used methods, the Markov chain method and the integral equation method. is given for VSI control charts.This unified treatment includes some results which are new in the FSI case. The new methods are used for the numerical evaluation of propertics of exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts.Some general optimality results for the choice fo the sampling intervals are also given.     

Shewhart Revisited

Abstract

The Shewhart control chart was first to monitor an ongoing process and raise an alarm when it appears that the level has changed. We show that the Shewhart chart is optimal for the criterion of maximizing the probability of detecting a change upon its occurrence subject to an average run length to false alarm. It is remarkable, particularly when the change is of moderate size, that Shewhart's procedure is better than cumulative sum (CUSUM). In the multivariate setting, applying the Shewhart procedure to each process separately is suboptimal. We create a generalized Shewhart procedure that is optimal for the aforementioned criterion. The results are illustrated in common settings.     

SEQUENTIAL METHODS FOR DETECTING CHANGES IN THE VARIANCE OF ECONOMIC TIME SERIES

In this paper EWMA charts and CUSUM charts are introduced for detecting changes in the variance of a GARCH process. The moments of the EWMA statistics are calculated. They permit a better understanding of the underlying control procedure. In an extensive simulation study all control schemes are compared with each other. “Optimal” smoothing parameters and “optimal” reference values are tabulated. It is shown how these charts can be applied to monitor stock market returns.     

Optimal Surveillance Based on Exponentially Weighted Moving Averages

Abstract

Statistical surveillance is used to detect an important change in a process as soon as possible after it has occurred. The exponentially weighted moving averages (EWMA) method is used in industry, economics, and medicine. Three optimality criteria of surveillance are studied. The average run length (ARL) criterion violates commonly accepted inference principles, and the drawbacks are demonstrated. The expected delay criterion is based on the minimal expected delay from change to detection. The full likelihood ratio method is optimal according to this criterion. Approximations of this method turn out to be modifications of the EWMA method. The approximations lead to a formula for the optimal value of the smoothing parameter of the EWMA statistic. The usefulness of this formula is shown. It is demonstrated that, for EWMA, the minimax criterion agrees well with that of the ED criterion but not with that of the ARL criterion.     

EWMA Charts for Monitoring the Mean and the Autocovariances of Stationary Gaussian Processes

Abstract

In this article simultaneous individual control charts for the mean and the autocovariances of a stationary process are introduced. All control schemes are EWMA (exponentially weighted moving average) charts. A multivariate quality characteristic is considered. It describes the behavior of the mean and the autocovariances. This quantity is transformed to a one-dimensional variable by using the Mahalanobis distance. The control statistic is obtained by exponentially smoothing these variables. Another control procedure is based on a multivariate EWMA recursion applied directly to our multivariate quality characteristic. After that the resulting statistic is transformed to a univariate random variable. Besides modified control charts we consider residual charts. For the residual charts the same procedure is used but the original observations are replaced by the residuals. In an extensive simulation study all control schemes are compared with each other. The target process is assumed to be an ARMA(1, 1) process.     

Use of risk-adjusted CUSUM and RSPRT charts for monitoring in medical contexts

In this paper we discuss the use of charts derived from the sequential probability ratio test (SPRT): the cumulative sum (CUSUM) chart, RSPRT (resetting SPRT), and FIR (fast initial response) CUSUM. The theoretical development of the methods is described and some considerations one might address when designing a chart, explored, including the approximation of average run lengths (ARLs), the importance of detecting improvements in a process as well as detecting deterioration and estimation of the process parameter following a signal. Two examples are used to demonstrate the practical issues of quality control in the medical setting, the first a running example and the second a fully worked example at the end of the paper. The first example relates to 30-day mortality for patients of a single cardiac surgeon over the period 1994-1998, the second to patient deaths in the practice of a single GP, Harold Shipman. The charts' performances relative to each other are shown to be sensitive to the definition of the 'out of control' state of the process being monitored. In light of this, it is stressed that a suitable means by which to compare charts is chosen in any specific application.     

Statistical Surveillance of the Parameters of a One-Factor Cox–Ingersoll–Ross Model

Abstract

In the paper, the tools of statistical process control, and in particular control charts, are applied to sequentially detect a change in the parameters of the Cox–Ingersoll–Ross term-structure model. Different types of multivariate EWMA control charts are introduced and constructed, taking into account the peculiarities of the Cox– Ingersoll–Ross model. The effectiveness of the control charts is analysed on the basis of extensive Monte Carlo simulations for a parameter constellation obtained from a data set on the term-structure in USA. In addition, the paper includes a practical example of a detection of a change in an initially estimated Cox–Ingersoll–Ross model.     

Multivariate Control Charts for Monitoring the Process Mean and Variability Using Sequential Sampling

Abstract

Traditional control charts for process monitoring are based on taking samples of fixed size at equally spaced sampling points. As an alternative to these traditional fixed sampling rate (FSR) control charts, variable sampling rate (VSR) control charts change the sampling rate as a function of the data from the process. With VSR control charts the sampling rate is increased whenever there is some indication of a problem with the process, and decreased when there is no indication of a problem. This paper investigates a type of VSR control chart based on sequential sampling in which the sample size used at a sampling point is a function of the data from the current and past sampling points. Sequential sampling is investigated in the context of simultaneously monitoring both the mean and variability of a multivariate normal process. Multivariate exponentially weighted moving average (MEWMA) control charts based on sample means and on the sum of the squared deviations from the target are considered. When sequential sampling is used in a combination of the MEWMA charts based on sample means and on squared deviations from the target, the average performance is much better than that of the standard FSR multivariate control charts that have traditionally been used.     

Assessing the Impact of Head Starts in the Performance of One-Sided Markov-Type Control Schemes

Abstract

Head start values—that is, nonzero initial values for the summary statistics of Markov-type control schemes such as the well-known cumulative sum (CUSUM) schemes—are throughly recommended in the literature and used by quality control practitioners. The rationale is as follows: if the process is operating in control, the summary statistic of the control scheme is soon brought to zero, so that the expected effect of the head start is minimal; otherwise, the operator is alerted to the out-of-control situation much sooner, which may prevent start-up problems.

This article assesses the impact of the adoption of a head start on the run length, indisputably the most popular performance measure of control schemes. It also brings stochastically monotone matrices into focus in the statistical process control setting and investigates their influence on the survival function, the alarm rate function and the aging character of the run length of one-sided Markov-type control schemes.

The derived properties, which may be of great interest to quality control practitioners, are illustrated through examples.     

Control charts applying a general sequential test at each sampling point

A control chart is proposed which applies a generalized sequential probability ratio test (SPRT) at fixed sampling intervals.the GSPRT chart, to monitor a parameter θ.which indexes a variable X with a genrnal distribution. Expression in terms of integral equations are developed for the moments of the zero-time and steady time to signal and number of observation to signal of the GSPRT chart. Comparsion are made between the performance of the GSPRT chart and the fixed sampling interval (FSI) and variable sampling interval (vsI) X and CUSUM charts. The GSPRT chart is found to be highly efficient and to have administrative advantages General Guidelines are provided for the design of GSPRT charts.     

A mixed control chart using process capability index

Control charts are considered as a compulsory tool for the monitoring and improvement of the quality of products. The commonly used attribute and the variable control charts are unsatisfactory for process monitoring. To overcome this di?culty, we proposed a monitoring scheme that is an improved technique based on the joint merits of the attribute as well as the variable control charts. The control chart coe?cients have been determined for in-control and out-of-control situations for specified average run length values using simulation. The e?ciency of the proposed control chart has been evaluated using average run length criteria for different values of the process settings. A practical example has been studied for implementation of the proposed technique. It has been observed that the proposed chart performs well for the mixed technique under the process capability index.     

Power of the MOSUM test for online detection of a transient change in mean

Abstract

In this article we discuss an online moving sum (MOSUM) test for detection of a transient change in the mean of a sequence of independent and identically distributed (i.i.d.) normal random variables. By using a well-developed theory for continuous time Gaussian processes and subsequently correcting the results for discrete time, we provide accurate approximations for the average run length (ARL) and power of the test. We check theoretical results against simulations, compare the power of the MOSUM test with that of the cumulative sum (CUSUM), and briefly consider the cases of nonnormal random variables and weighted sums.     

Improved process monitoring using the CUSUM and EWMA-based multiscale PCA fault detection framework

Process monitoring techniques are of paramount importance in the chemical industry to improve both the product quality and plant safety. Small or incipient irregularities may lead to severe degradation in complex chemical processes, and the conventional process monitoring techniques cannot detect these irregularities. In this study to improve the performance of monitoring, an online multiscale fault detection approach is proposed by integrating multiscale principal component analysis (MSPCA) with cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts. The new Hotelling's T² and square prediction error (SPE) based fault detection indices are proposed to detect the incipient irregularities in the process data. The performance of the proposed fault detection methods was tested for simulated data obtained from the CSTR system and compared to that of conventional PCA and MSPCA based methods. The results demonstrate that the proposed EWMA based MSPCA fault detection method was successful in detecting the faults. Moreover, a comparative study shows that the SPE-EWMA monitoring index exhibits a better performance with lower values of missed detections ranging from 0% to 0.80% and false alarms ranging from 0% to 21.20%.