Monitoring the mean and the variance of a stationary process

We deal with the problem of how deviations in the mean or the variance of a time series can be detected. Several simultaneous control charts are introduced which are based on EWMA (exponentially weighted moving average) statistics for the mean and the empirical variance. The combined X − S² EWMA chart is extended to time series. Further simultaneous charts are considered. The comparision of these schemes shows that the residual attempt must be favored if a variance change is present.     

Run length distributions for estimated attributes charts

Attributes control charts, such as c and p charts, are popular methods for detecting out of control signals when it is practical only to obtain qualitative information about a process; in such cases, variables control charts, such as the , s and R charts, cannot be used. The run length distributions have previously been studied for variables charts when the control limits have been estimated. Little has been done in the case of attributes charts. In this paper, the run length distributions for the c chart and p chart are derived for the case when the control limits are estimated. It is shown that, as for variables charts, the effect of estimation on quantities such as the average run length (ARL) can be quite dramatic, but when the underlying process is in control, the ARL is potentially misleading as a basis for comparison. Received: September 1998     

Economic Design of EWMA Charts with Variable Sampling Intervals

Control charting is a graphical expression and operation of statistical hypothesis testing. In the present paper, we develop the economic design of the variable sampling intervals (VSI) exponentially weighted moving average (EWMA) charts to determine the values of the six test parameters of the charts (i.e., the sample size, the long sampling interval, the short sampling interval, the warning limit coefficient, the control limit coefficient, and exponential weight constant) such that the expected total cost is minimized. The genetic algorithm (GA) is applied to search for the optimal values of the six test parameters of the VSI EWMA chart, and an example is provided to illustrate the solution procedure. A sensitivity analysis is then carried out to investigate the effects of model parameters on the solution of the economic design.     

Run length analysis of Shewhart charts applied to drifting processes under an integrative SPC/EPC model

Engineering Process Controllers (EPC) are frequently based on parametrized models. If process conditions change, the parameter estimates used by the controllers may become biased, and the quality characteristics will be affected. To detect such changes it is adequate to use Statistical Process Control (SPC) methods. The run length statistic is commonly used to describe the performance of an SPC chart. This paper develops approximations for the first two moments of the run length distribution of a one-sided Shewhart chart used to detect two types of process changes in a system that is regulated by a given EPC scheme: i) changes in the level parameter; ii) changes in the drift parameter. If the drift parameter shifts, it is further assumed that the form of the drift process changes from a linear trend under white noise (the in-control drift model) into a random walk with drift model. Two different approximations for the run length moments are presented and their accuracy is numerically analyzed. Received: August 1998     

Surveillance of the mean behavior of multivariate time series

In this paper several cumulative sum (CUSUM) charts for the mean of a multivariate time series are introduced. We extend the control schemes for independent multivariate observations of crosier [ Technometrics (1988) Vol. 30, pp. 187–194], pignatiello and runger [ Journal of Quality Technology (1990) Vol. 22, pp. 173–186], and ngai and zhang [ Statistica Sinica (2001) Vol. 11, pp. 747–766] to multivariate time series by taking into account the probability structure of the underlying stochastic process. We consider modified charts and residual schemes as well. It is analyzed under which conditions these charts are directionally invariant. In an extensive Monte Carlo study these charts are compared with the CUSUM scheme of theodossiu [ Journal of the American Statistical Association (1993) Vol. 88, pp. 441–448], the multivariate exponentially weighted moving-average (EWMA) chart of kramer and schmid [ Sequential Analysis (1997) Vol. 16, pp. 131–154], and the control procedures of bodnar and schmid [ Frontiers of Statistical Process Control (2006) Physica, Heidelberg]. As a measure of the performance, the maximum expected delay is used.     

变样本容量比例控制图设计及应用

针对单边休哈特比例控制图失控状态下平均报警时间长的问题,提出一种单边变样本容量比例控制图来监控两个正态变量间比例的偏移。采用马尔可夫链方法计算了该控制图的平均运行链长。在保证过程处于受控状态的统计性能基础上,使失控状态下的平均运行链长最小,从而获得控制图的最优决策变量和最优性能指标。仿真结果表明,单边变样本容量比例控制图的性能明显优于单边休哈特比例控制图。最后将该方法应用于港口皮带机配煤比例监控的问题,通过仿真实验说明了该方法能够对煤炭配比过程中的比例偏移做出有效反应。     

Variables control chart limits and tests for special causes

Control charts are used to detect problems in control such as outliers, shifts in levels or excess variability in subgroup means that may have a special cause. This paper addresses itself to deriving control chart limits based on past data and based on initial samples in a current control situation. We present a general setting for control charts. Furthermore, an overview is given of tests for special causes. The tests are standardized so that the asymptotic type I error does not exceed a fixed level. The distributions of the run lengths of the tests and combinations of tests are also evaluated. We propose to use a low percen-tile of the run length distribution, instead of the average run length, to study the performance of the tests. These indicate that, in particular when tests are combined, the run length percentiles may be too small for practical purposes. It is shown that (nearly) exact control chart limits for observations from a normal distribution exist. The traditional limits differ considerably from the proposed ones and correspond to even smaller run length percentiles.     

Monitoring process mean level using auxiliary information

In this study, a Shewhart‐type control chart is proposed for the improved monitoring of process mean level (targeting both moderate and large shifts which is the major concern of Shewhart‐type control charts) of a quality characteristic of interest Y. The proposed control chart, namely the M_r chart, is based on the regression estimator of mean using a single auxiliary variable X. Assuming bivariate normality of (Y, X), the design structure of M_r chart is developed for phase I quality control. The comparison of the proposed chart is made with some existing control charts used for the same purpose. Using power curves as a performance measure, better performance of the proposedM_r chart is observed for detecting the shifts in mean level of the characteristic of interest.     

A less sensitive linear detector for the change point based on kernel smoothing method

In this paper, a new detecting procedure for the change point in the mean is proposed based on a linear kernel smoother. It uses a discontinuous kernel function and has a relative high constant efficiency in an interval of shift value and thus is less sensitive than the simple CUSUM, EWMA, FMA and Shiryayev-Roberts procedures. It also generalizes the FMA procedure.     

Performance monitoring of credit portfolios using survival analysis

We are interested in detecting changes in the performance of a credit portfolio quickly and robustly. The portfolio is dynamic: customers can either default or pay the full amount, and new customers can be taken on. Robust detection means that changing the number of new customers taken on should not lead to either a false or delayed signal. We investigate the performances of monitoring schemes via a simulation study that uses several scenarios. We consider monitoring based on default rates estimated through a gliding window, cumulative sum (CUSUM) charts based on default rates, CUSUM charts based on defaults within a given follow-up time after arrival, and a survival analysis CUSUM chart. We conclude that using a survival analysis approach is preferable to using the other approaches.     

On the monitoring of multi-attributes high-quality production processes

Over the last decade, there have been an increasing interest in the techniques of process monitoring of high-quality processes. Based upon the cumulative counts of conforming (CCC) items, Geometric distribution is particularly useful in these cases. Nonetheless, in some processes the number of one or more types of defects on a nonconforming observation is also of great importance and must be monitored simultaneously. However, there usually exist some correlations between these two measures, which obligate the use of multi-attribute process monitoring. In the literature, by assuming independence between the two measures and for the cases in which there is only one type of defect in nonconforming items, the generalized Poisson distribution is proposed to model such a problem and the simultaneous use of two separate control charts (CCC & C chats) is recommended. In this paper, we propose a new methodology to monitor multi-attribute high-quality processes in which not only there exist more than one type of defects on the observed nonconforming item but also there is a dependence structure between the two measures. To do this, first we transform multi-attribute data in a way that their marginal probability distributions have almost zero skewnesses. Then, we estimate the transformed mean vector and covariance matrix and apply the well-known χ² control chart. In order to illustrate the proposed method and evaluate its performance, we use two numerical examples by simulation and compare the results. The results of the simulation studies are encouraging.     

Economic design of T<Superscript>2</Superscript> control charts with the VSSI sampling scheme

T ² charts are used to monitor a process when more than one quality variable associated with process is being observed. Recent studies have shown that the T ² chart with variable sampling size and sampling interval (VSSI) detects a small shift in the process mean vector faster than the traditional T ² chart. The paper considers an economic design of the VSSI T ² chart, in which the expected hourly loss is constructed and regarded as an objective function for optimally determining the design parameters (i.e. the maximum/minimum sample size, the longest/shortest sampling interval, and the warning/action limits) in sampling-and-charting. Furthermore, the effects of process parameters and cost parameters upon the expected hourly loss and design parameters are examined.     

Monitoring the cross-covariances of a multivariate time series

In this paper sequential procedures are proposed for jointly monitoring all elements of the covariance matrix at lag 0 of a multivariate time series. All control charts are based on exponential smoothing. As a measure of the distance between the target values and the actual values the Mahalanobis distance is used. It is distinguished between residual control schemes and modified control schemes. Several properties of these charts are proved assuming the target process to be a stationary Gaussian process. Within an extensive Monte Carlo study all procedures are compared with each other. As a measure of the performance of a control chart the average run length is used. An empirical example about Eastern European stock markets illustrates how the autocovariance and the cross-covariance structure of financial assets can be monitored by these methods.     

基于Wilcoxon秩和检验的多元非参数控制图

在统计过程控制(SPC)中,对多元数据的监测仍然是一个重要且具有挑战性的问题。当缺乏或有限的关于潜在过程分布的认知时,特别是当过程测量是多变量的时候,非参数控制图在统计过程控制(SPC)中是有用的。文章基于Wilcoxon秩和检验结合广义加权移动平均(GWMA)控制方案来制定图表统计量,得到一个新的多元非参数控制图,用于监测多元数据的位置参数变化。文章的理论和数值研究表明,所提出的控制图能够为任意数据分布位置偏移检测提供令人满意的性能。     

Model-based control charts in phase 1 statistical process control

In this paper, a general method of constructing control charts for preliminary analysis of individual observations is presented, which is based on recursive score residuals. A simulation study shows that certain implementations of these charts are highly effective in detecting assignable causes.     

Hotelling’s <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> control chart with two adaptive sample sizes

Some quality control schemes have been developed when several related quality characteristics are to be monitored. The familiar multivariate process monitoring and control procedure is the Hotelling’s T ² control chart for monitoring the mean vector of the process. It is a direct analog of the univariate shewhart [`(x)]{\bar{x}} chart. As in the case of univariate, the ARL improvements are very important particularly for small process shifts. In this paper, we study the T ² control chart with two-state adaptive sample size, when the shift in the process mean does not occur at the beginning but at some random time in the future. Further, the occurrence time of the shift is assumed to be exponentially distributed random variable.     

Fuzzy Control Chart A Better Alternative for Shewhart Average Chart

This paper through a real illustrative example and a power test shows that designing a fuzzy control chart for process average of a continuous (variable) quality characteristic with a warning line is a better alternative to Shewhart chart in many respects, like providing better neural view to inspectors, offering different strategic options for company to choose, detecting the desire shifts more quickly, and more sensibility to small shifts without any complexity augmentation to the chart.     

CUMIN charts

Classical control charts are very sensitive to deviations from normality. In this respect, nonparametric charts form an attractive alternative. However, these often require considerably more Phase I observations than are available in practice. This latter problem can be solved by introducing grouping during Phase II. Then each group minimum is compared to a suitable upper limit (in the two-sided case also each group maximum to a lower limit). In the present paper it is demonstrated that such MIN charts allow further improvement by adopting a sequential approach. Once a new observation fails to exceed the upper limit, its group is aborted and a new one starts right away. The resulting CUMIN chart is easy to understand and implement. Moreover, this chart is truly nonparametric and has good detection properties. For example, like the CUSUM chart, it is markedly better than a Shewhart X-chart, unless the shift is really large.     

Self-adapting control charts

When the distributional form of the observations differs from normality, standard control charts are often prone to serious errors. Such model errors can be avoided by using (modified) nonparametric control charts. Unfortunately, these control charts suffer from large stochastic errors caused by estimation. In between these two types are the so-called parametric control charts. All three of them, as well as a combined chart, which chooses one of the three control charts according to the appropriate model assumption on the underlying distribution are discussed in this paper. The data indicate which of the three control charts to select. Readymade formulas for the several control charts are presented accompanied by an application on real data. Apart from bias removal, criteria based on exceedance probability and semi-variance are investigated.     

Sample size and the Binomial CUSUM Control Chart: the case of 100% inspection

The Binomial CUSUM is used to monitor the fraction defective (p) of a repetitive process, particularly for detecting small to moderate shifts. The number of defectives from each sample is used to update the monitoring CUSUM. When 100% inspection is in progress, the question arises as to how many sequential observations should be grouped together in forming successive samples. The tabular form of the CUSUM has three parameters: the sample size n, the reference value k, and the decision interval h, and these parameters are usually chosen using statistical or economic-statistical criteria, which are based on Average Run Length (ARL). Unlike earlier studies, this investigation uses steady-state ARL rather than zero-state ARL, and the occurrence of the shift can be anywhere within a sample. The principal finding is that there is a significant gain in the performance of the CUSUM when the sample size (n) is set at one, and this CUSUM might be termed the Bernoulli CUSUM. The advantage of using n=1 is greater for larger shifts and for smaller values of in-control ARL. First version: September 1998/Third revision: September 2000