Evaluations of some Exponentially Weighted Moving Average methods

The need for statistical surveillance has been noted in many different areas, and examples of applications include the detection of an increased incidence of a disease, the detection of an increased radiation level and the detection of a turning point in a leading index for a business cycle. In all cases, preventive actions are possible if the alarm is made early. Several versions of the EWMA (Exponentially Weighted Moving Average) method for monitoring a process with the aim of detecting a shift in the mean are studied both for the one-sided and the two-sided case. The effects of using barriers for the one-sided alarm statistic are also studied. One important issue is the effect of different types of alarm limits. Different measures of evaluation, suitable in different types of applications, are considered such as the expected delay, the ARL¹, the probability of successful detection and the predictive value of an alarm, to give a broad picture of the features of the methods. Results from a large-scale simulation study are presented both for a fixed ARL0 and a fixed probability of a false alarm. It appears that important differences from an inferential point of view exist between the one- and two-sided versions of the methods. It is demonstrated that the method, usually considered as a convenient approximation, is to be preferred over the exact version in the overwhelming majority of applications.     

Aspects on the control of false alarms in statistical surveillance and the impact on the return of financial decision systems

In systems for online detection of regime shifts, a process is continually observed. Based on the data available an alarm is given when there is enough evidence of a change. There is a risk of a false alarm and here two different ways of controlling the false alarms are compared: a fixed average run length until the first false alarm and a fixed probability of any false alarm (fixed size). The two approaches are evaluated in terms of the timeliness of alarms. A system with a fixed size is found to have a drawback: the ability to detect a change deteriorates with the time of the change. Consequently, the probability of successful detection will tend to zero and the expected delay of a motivated alarm tends to infinity. This drawback is present even when the size is set to be very large (close to one). Utility measures expressing the costs for a false or a too late alarm are used in the comparison. How the choice of the best approach can be guided by the parameters of the process and the different costs of alarms is demonstrated. The technique is illustrated by financial transactions of the Hang Seng Index.     

Univariate fast initial response statistical process control with taut strings

We present a novel real-time univariate monitoring scheme for detecting a sustained departure of a process mean from some given standard assuming a constant variance. Our proposed stopping rule is based on the total variation of a nonparametric taut string estimator of the process mean and is designed to provide a desired average run length for an in-control situation. Compared to the more prominent CUSUM fast initial response (FIR) methodology and allowing for a restart following a false alarm, the proposed two-sided taut string (TS) scheme produces a significant reduction in average run length for a wide range of changes in the mean that occur at or immediately after process monitoring begins. A decision rule for when to choose our proposed TS chart compared to the CUSUM FIR chart that takes into account both false alarm rate and average run length to detect a shift in the mean is proposed and implemented. Supplementary materials are available online.     

A Process Monitoring Scheme Controlling False Discovery Rate

A new process monitoring scheme is proposed by using the Storey procedure for controlling the positive false discovery rate in multiple testing. For the 2-span control scheme, it is shown numerically that the proposed method performs better than X-bar chart in terms of the average run length. Some simulations are accomplished to evaluate the performance of the proposed scheme in terms of the average run length and the conditional expected delay. The results are compared with those of the existing monitoring schemes including the X-bar chart. The false discovery rate is also estimated and compared with the target control level.     

Run length,average run length and false alarm rate of shewhart x-bar chart: exact derivations by conditioning

The effects of estimation of the control limits on the performance of the popular Shewhart X-bar chart are examined via the average run length and the probability of a false alarm, when one or both of the process mean and variance are unknown. Exact expressions for the run length, the average run length (ARL) and the false alarm rate are obtained, in each case, using expectation by conditioning. Applying Jensen's inequality, together with expectation by conditioning, a simple lower bound to the ARL is obtained. This could be useful in designing the charts. The expressions for the exact ARL and the exact probabilities of false alarm are evaluated, using simulations, for various numbers of subgroups and shift sizes. The calculations throw new light on the performance of the Shewhart X-bar chart. Some recommendations are given.     

Reduction of control-chart signal variablity for high-quality processes

The design of a control chart is often based on the statistical measure of average run length (ARL). A longer in-control ARL is ensured by the design, but the variance run length distribution may also be large for such a design. In practical terms, the variability in false alarms and true signals may be large. If the sample size for plotting a point is not constant, then the focus is on the average number inspected as against the ARL. This article considers two well-known attribute control chart procedures for monitoring high quality based on the number inspected, and shows how the variability in false alarms and correct signals can be reduced.     

New control charts for simultaneous monitoring of the mean vector and covariance matrix of multivariate multiple linear profiles

Simultaneous monitoring of the mean vector and covariance matrix in multivariate processes allows practitioners to avoid the inflated false alarm rate that results from using two independent control charts. In this paper, we extend exponentially weighted moving average semicircle and generally weighted moving average semicircle control charts to monitor the mean vector and covariance matrix of multivariate multiple linear regression profiles in Phase II simultaneously. These new control charts are compared with the existing control charts in the literature in terms of the average run length criterion. Finally, a case is considered to show the application of the proposed charts.     

Business-Cycle Analysis With a Markov-Switching Model

This article explores the Hamilton Markov-switching model through an analysis of the business cycles of eight developed market economies. Forecasting and specification tests suggest only marginal improvements over linear autoregressive models. Yet filtered and smoothed conditional probabilities indicate turning points in business cycles that closely correlate with turning points from traditional methods. Tests regarding the asymmetry of business cycles reject the null of symmetry for most countries.     

Nonparametric control charts based on runs and Wilcoxon-type rank-sum statistics

In this article, we introduce three new distribution-free Shewhart-type control charts that exploit run and Wilcoxon-type rank-sum statistics to detect possible shifts of a monitored process. Exact formulae for the alarm rate, the run length distribution, and the average run length (ARL) are all derived. A key advantage of these charts is that, due to their nonparametric nature, the false alarm rate (FAR) and in-control run length distribution is the same for all continuous process distributions. Tables are provided for the implementation of the charts for some typical FAR values. Furthermore, a numerical study carried out reveals that the new charts are quite flexible and efficient in detecting shifts to Lehmann-type out-of-control situations.     

Distribution-free monitoring schemes based on order statistics: a general approach

In this article, we establish a new class of distribution-free Shewhart-type monitoring schemes based on order statistics. The setup of the proposed family of nonparametric control charts is presented in detail. Specific monitoring schemes, already introduced in the literature, are confirmed to be members of the new class. In addition, a new nonparametric monitoring scheme that belongs to the class is established, while explicit formulae for its basic characteristics are reached. The numerical study carried out reveals that the proposed scheme achieves adversarial in-control and out-of-control performance.KEYWORDS: Distribution-free monitoring schemes, average run length, false alarm rate, Lehmann alternatives, statistical process control, order statistics     

Nonparametric control charts based on order statistics: Some advances

ABSTRACT

In this article, we introduce new nonparametric Shewhart-type control charts that take into account the location of two order statistics of the test sample as well as the number of observations in that sample that lie between the control limits. Exact formulae for the alarm rate, the run length distribution and the average run length (ARL) are all derived. A key advantage of the new charts is that, due to its nonparametric nature, the false alarm rate (FAR) and in-control run length distribution is the same for all continuous process distributions. Tables are provided for the implementation of the proposed charts for some typical FAR and ARL values. Furthermore, a numerical study carried out reveals that the new charts are quite flexible and efficient in detecting shifts to Lehmann-type out-of-control situations, while they seem preferable from a robustness point of view in comparison with the distribution-free control chart of Balakrishnan et al. (2009).     

Properties and performance of the c-chart for attributes data

The effects of parameter estimation are examined for the well-known c-chart for attributes data. The exact run length distribution is obtained for Phase II applications, when the true average number of non-conformities, c, is unknown, by conditioning on the observed number of non-conformities in a set of reference data (from Phase I). Expressions for various chart performance characteristics, such as the average run length (ARL), the standard deviation of the run length (SDRL) and the median run length (MDRL) are also obtained. Examples show that the actual performance of the chart, both in terms of the false alarm rate (FAR) and the in-control ARL, can be substantially different from what might be expected when c is known, in that an exceedingly large number of false alarms are observed, unless the number of inspection units (the size of the reference dataset) used to estimate c is very large, much larger than is commonly used or recommended in practice. In addition, the actual FAR and the in-control ARL values can be very different from the nominally expected values such as 0.0027 (or ARL₀=370), particularly when c is small, even with large amounts of reference data. A summary and conclusions are offered.     

A class of distribution-free control charts

Summary.  A class of Shewhart-type distribution-free control charts is considered. A key advantage of these charts is that the in-control run length distribution is the same for all continuous process distributions. Exact expressions for the run length distribution and the average run length (ARL) are derived and properties of the charts are studied via evaluations of the run length distribution probabilities and the ARL. Tables are provided for implementation for some typical ARL values and false alarm rates. The charts proposed are preferable from a robustness point of view, have attractive ARL properties and would be particularly useful in situations where one uses a classical Shewhart   X -chart. A numerical illustration is given.     

A Portmanteau Test for ARMA Processes with Infinite Variance

When a control chart is used in practice, knowledge about several characteristics of the method is important for the judgement of which action is appropriate at an alarm. The probability of a false alarm, the delay of an alarm and the predictive value of an alarm are qualities (besides the usual ARL) which are described by a simulation study for the evaluations. Results for finite time are given for a shift in level of a Gaussian process.

Evaluations are made of the full likelihood ratio method that has two parameters and can be made optimal for both the size and the intensity of a shift     

Chi-Square quantile-based multivariate variance monitoring for individual observations

Under the normality assumption, some statistics for monitoring a multivariate process variance for individual observations can be used to detect a variance shift, but the distribution of their in-control run length has a high variance as well as the median that is extremely smaller than the mean, which leads to many false alarms in the in-control process. In this paper, we propose a chi-square quantile-based monitoring statistic which is free of the problems. The numerical experiments show that the proposed monitoring statistics outperform the existing monitoring statistics in terms of the detection of a shift for the variance.     

Determining Optimal Number of Samples for Constructing Multivariate Control Charts

Normally, an average run length (ARL) is used as a measure for evaluating the detecting performance of a multivariate control chart. This has a direct impact on the false alarm cost in Phase II. In this article, we first conduct a simulation study to calculate both in-control and out-of-control ARLs under various combinations of process shifts and number of samples. Then, a trade-off analysis between sampling inspection and false alarm costs is performed. Both the simulation results and trade-off analysis suggest that the optimal number of samples for constructing a multivariate control chart in Phase I can be determined.     

Guaranteed Conditional Performance of Control Charts via Bootstrap Methods

To use control charts in practice, the in‐control state usually has to be estimated. This estimation has a detrimental effect on the performance of control charts, which is often measured by the false alarm probability or the average run length. We suggest an adjustment of the monitoring schemes to overcome these problems. It guarantees, with a certain probability, a conditional performance given the estimated in‐control state. The suggested method is based on bootstrapping the data used to estimate the in‐control state. The method applies to different types of control charts, and also works with charts based on regression models. If a non‐parametric bootstrap is used, the method is robust to model errors. We show large sample properties of the adjustment. The usefulness of our approach is demonstrated through simulation studies.     

Signaling NBER turning points: a sequential approach

The dates of the U.S. business cycle are reported by the National Bureau of Economic Research with a considerable delay, so an early notion of turning points is of particular interest. This paper proposes a novel sequential classification approach designed for timely signaling these turning points, using the time series of coincident economic indicators. The approach exhibits a range of theoretical optimality properties for early signaling, moreover, it is transparent and easy to implement. The empirical study evaluates the signaling ability of the proposed methodology.     

Some statistical aspects of methods for detection of turning points in business cycles

Methods for online turning point detection in business cycles are discussed. The statistical properties of three likelihood-based methods are compared. One is based on a Hidden Markov Model, another includes a non-parametric estimation procedure and the third combines features of the other two. The methods are illustrated by monitoring a period of the Swedish industrial production. Evaluation measures that reflect timeliness are used. The effects of smoothing, seasonal variation, autoregression and multivariate issues on methods for timely detection are discussed.     

Some power aspects of methods for detecting different shifts in the mean

We study, by means of simulations, the performance of the Shewhart method, the Cusum method, the Shiryaev-Roberts method and the likelihood ratio method in the case when the true shift differs from the shift for which the methods are optimal. The methods are compared for a fixed expected time until false alarm. The comparisons are made with respect to some measures associated with power such as probability of alarm when the change occurs immediately, expected delay of true alarm and predictive value of an alarm.