Sven Knoth

Control Charts for Time Series: A Review

A state of the art survey in Statistical Process Control (SPC) for dependent data will be given. First papers about the influence on and the modification of standard SPC schemes are written some decades ago. After that, in the end of the 1980s and during the 1990s the consideration of dependence in the field of SPC became very popular. It turned out, that falsely assuming independence mostly leads to improper SPC schemes. Therefore, one has to be aware of dependence.

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.

Accurate ARL Calculation for EWMA Control Charts Monitoring Normal Mean and Variance Simultaneously

Abstract Exponentially weighted moving average (EWMA) control charts designed for monitoring the variance or the mean and the variance of a normally distributed variable are either based on the log transformation of the sample variance S 2 or provide only rough average run length (ARL) results. Gan (1995), as the most prominent example for the simultaneous case, calculated ARL values precisely for -ln S 2 EWMA schemes. The results in Knoth and Schmid (2002) for -S 2 ones are less accurate than the former one. The reason behind the lack of precision is that the methods usually applied for ARL calculation are not able to handle the restricted support of the chart statistic (S 2 and, of course, S and the range R are nonnegative random variables). While in Knoth (2005) this problem is treated for single variance monitoring by solving integral equations with collocation methods, this paper employs collocation and ideas similar to Gan (1995) in order to obtain accurate ARL values of -S 2 EWMA control charts. Additionally, the appropriate choice of the nonsymmetric control limits for the S 2 part of the scheme is addressed.

EWMA schemes with non-homogeneous transition kernels

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.

Misleading Signals in Simultaneous Residual Schemes for the Mean and Variance of a Stationary Process

The performance assessment of simultaneous surveillance schemes for the process mean (μ) and variance (σ2) requires a special performance measure, in addition to the average run length. It refers to two events which can be likely to happen when such schemes are at use: the individual chart for μ triggers a signal before the one for σ2, even though the process mean is on-target and the variance is off-target; the constituent chart for σ2 triggers a signal before the one for μ, although the variance is in-control and the process mean is out-of-control. These are called misleading signals since they correspond to a misinterpretation of a mean (variance) change as a shift in the process variance (mean) and can lead the quality control operator or engineer to a misdiagnosis of assignable causes and to deploy incorrect actions to bring the process back to target. This article discusses the impact of autocorrelation on the probability of misleading signals of simultaneous Shewhart and EWMA residual schemes for the mean and variance of a stationary process.

On the run length of the EWMA scheme: a monotonicity result for normal variables

Abstract We discuss the EWMA control chart for a stationary Gaussian process {Xt}. It is proved that in the in-control state the probability of no signal until a fixed time is a nondecreasing function in the autocorrelations of {Xt} provided that they satisfy a certain monotonicity assumption.

Simultaneous Shewhart-Type Charts for the Mean and the Variance of a Time Series

We introduce simultaneous control charts for the mean and the variance of a time series. Our schemes are extensions of the well-known Shewhart charts for independent variables. We consider a modified X-S2-chart, a modified X-R-chart, a residual X-S2-chart and a residual X-R-chart. A comparison of these schemes is made by means of the average run length. It turns out that residual schemes lead to better results. For nearly all parameter combinations the residual X-S2-chart is found to be the best control design.

The Case Against the Use of Synthetic Control Charts

The synthetic chart principle proposed by Wu and Spedding (2000) initiated a stream of publications in the control charting literature. Originally, it was claimed that the new chart has superior average run length (ARL) properties. Davis and Woodall (2002) indicated that the synthetic chart is nothing else than a particular runs-rule chart. Moreover, they criticized the design of the performance evaluation and advocated use of the steady-state ARL. The latter measure was used then, e.g., in Wu et al. (2010). In most of the papers on synthetic charts that actually used the steady-state framework, it was not rigorously described. See Khoo et al. (2011) as an exception, where it was revealed that the cyclical steady-state design was considered. The aim of this paper is to carefully analyze the steady-state (cyclical and the more popular conditional) for the synthetic chart, the original “2 of L + 1” (L < 1) runs-rule chart, and competing EWMA charts with two types of control limits. It turns out that the EWMA chart has a uniformly (over a large range of potential shifts) better steady-state ARL performance than the synthetic chart. Furthermore, the synthetic control chart exhibits the poorest performance among all considered competitors. Thus, we advise not applying synthetic control charts.

Run length quantiles of EWMA control charts monitoring normal mean or/and variance

Exponentially weighted moving average (EWMA) control charts are well-established devices for monitoring process stability. Typically, control charts are evaluated by considering their Average Run Length (ARL), that is the expected number of observations or samples until the chart signals. Because of the limitations of an average, various papers also dealt with the run length distribution and quantiles. Going beyond these papers, we develop algorithms for and evaluate the quantile performance of EWMA control charts with variance adjusted control limits and with fast initial response features, of EWMA charts based on the sample variance, and of EWMA charts simultaneously monitoring mean and variance. Additionally, for the mean charts we consider medium, late and very late process changes and their impact on appropriately conditioned run length quantiles. It is demonstrated that considering run length quantiles can protect from constructing distorted EWMA designs while optimising their zero-state ARL performance. The implementation of all the considered measures in the R package ‘spc’ allows any control chart user to consider EWMA schemes from the run length quantile prospective in an easy way.

EWMA p charts under sampling by variables

When it comes to manufacturing processes, the question of process quality is omnipresent. Statistical methods for monitoring quality measures are widespread and control charts are well established. Instead of checking parameters such as and , there are control charts to monitor the percent defective . While control charts under sampling by attributes are already illustrated in the literature, this paper develops and modifies different approaches of control charts for under sampling by variables. The results for a modified exponentially moving average chart, EWMA , are compared to those of several control charts. The new scheme is well suited for imperfect in-control situations, that is, the pre-run sample mean differs from a certain natural or target mean value of the monitored parameter. To calculate the most popular control chart performance measure, the average run length ( ), both the Markov chain approach and the collocation method are used. It turns out that the EWMA chart has properties similar to the usual EWMA and the CUSUM charts. Thus, it can be a helpful utility for practical applications.