On the design of control charts with guaranteed conditional performance under estimated parameters

When designing control charts the in-control parameters are unknown, so the control limits have to be estimated using a Phase I reference sample. To evaluate the in-control performance of control charts in the monitoring phase (Phase II), two performance indicators are most commonly used: the average run length (ARL) or the false alarm rate (FAR). However, these quantities will vary across practitioners due to the use of different reference samples in Phase I. This variation is small only for very large amounts of Phase I data, even when the actual distribution of the data is known. In practice, we do not know the distribution of the data, and it has to be estimated, along with its parameters. This means that we have to deal with model error when parametric models are used and stochastic error because we have to estimate the parameters. With these issues in mind, choices have to be made in order to control the performance of control charts. In this paper, we discuss some results with respect to the in-control guaranteed conditional performance of control charts with estimated parameters for parametric and nonparametric methods. We focus on Shewhart, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) control charts for monitoring the mean when parameters are estimated.     

Design of a Phase II Control Chart for Monitoring the Ratio of two Normal Variables

In many industrial scenarios, on‐line monitoring of quality characteristics computed as the ratio of two normal random variables can be required. Potential industrial applications can include monitoring of processes where the correct proportion of a property between two ingredients or elements within a product should be maintained under statistical control; implementation of quality control procedures where the performance of a product is measured as a ratio before and after some specific operation, for example a chemical reaction following the introduction of an additive in a product and monitoring of a chemical or physical property of a product, which is itself defined and computed as a ratio. This paper considers Phase II Shewhart control charts with each subgroup consisting of n > 1 sample units. From one subgroup to another, the size of each sample unit, upon which a single measurement is made, can be changed. An approximation based on the normal distribution is used to efficiently handle the ratio distribution. Several tables are generated and commented to show the statistical performance of the investigated chart for known and random shift sizes affecting the in‐control ratio. An illustrative example from the food industry is given for illustration. Copyright © 2014 John Wiley & Sons, Ltd.     

A Generalized Procedure for Monitoring Right‐censored Failure Time Data

Right‐censored failure time data is a common data type in manufacturing industry and healthcare applications. Some control charting procedures were previously proposed to monitor the right‐censored failure time data under some specific distributional assumptions for the observed failure times and censoring times. But these assumptions may not be always satisfied in the real‐world data. Therefore, a more generalized control chart technique, which can handle different types of distributions of the data, is highly needed. Considering the limitations of existing methodologies for detecting changes of hazard rate, this paper develops a generalized statistical procedure to monitor the failure time data in the presence of random right censoring when abundant historical failure times are available. The developed method makes use of the one‐sample nonparametric rank tests without any specific assumptions of the data distribution. The operating characteristic functions of the control chart are derived on the basis of the asymptotic properties of the rank statistics. Case studies are presented to show the effectiveness of the proposed control chart technique, and its performance is investigated and compared with some Shewhart‐type control charts based on the conditional expected value weight. Copyright © 2014 John Wiley & Sons, Ltd.     

A Review and perspective on surveillance of Bernoulli processes

Bernoulli processes have been monitored using a wide variety of techniques in statistical process control. The data consist of information on successive items classified as conforming (nondefective) or nonconforming (defective). In some cases, the probability of obtaining a nonconforming item is very small; this is known as a high quality process. This area of statistical process control is also applied to health‐related monitoring, where the incidence rate of a medical problem such as a congenital malformation is of interest. In these applications, standard Shewhart control charts based on the binomial distribution are no longer useful. In our expository paper, we review the methods implemented for these scenarios and present ideas for future work in this area. We offer advice to practitioners and present a comprehensive literature review for researchers. Copyright © 2011 John Wiley & Sons, Ltd.     

Progressive mean control chart is not a special case of an exponentially weighted moving average control chart

The progressive mean (PM) statistic is based on a simple idea of accumulating information of each subgroup by calculating the average progressively. Its weighting structure is based on a subgroup number that changes arithmetically, which makes the PM chart unique and efficient compared with the existing classical memory control charts. In a recent article (see reference 1), it was claimed that the PM chart is a special case of the exponentially weighted moving average (EMWA) chart. In this article, it is shown that even though the PM statistic can be written in the form of an EWMA statistic, the variance of the PM statistic is different from that of the EWMA statistic. Consequently, the limits of the PM chart are different from that of the EWMA chart. Therefore, it is found that the PM chart is not a special case of the EWMA chart; hence, the claim in reference 1 is incorrect. Furthermore, it is pointed out in this paper that no adaptive property in the weighting parameter of the PM statistic exists, further contradicting the claim in reference 1.     

Two Charts: Not One

Difficulties can occur in the operation of traditional control charts. A principal reason for this is that the data coming from a typical operating process do not vary about a fixed mean. It is shown how by using a nonstationary model a continuously updated local mean level is provided. This can be used to produce (a) a bounded adjustment chart that tells you when to adjust the process to achieve maximum economy and (b) a Shewhart monitoring chart seeking assignable causes of trouble applied to the deviations from the local mean. Estimation of the mean and “standard deviation” are not required.     

Dynamic probability control limits for CUSUM charts for monitoring proportions with time-varying sample sizes

We consider the problem of monitoring a proportion with time-varying sample sizes. Control charts are generally designed by assuming a fixed sample size or a priori knowledge of a sample size probability distribution. Sometimes, it is not possible to know, or accurately estimate, a sample size distribution or the distribution may change over time. An improper assumption for the sample size distribution could lead to undesirable performance of the control chart. To handle this problem, we propose the use of dynamic probability control limits (DPCLs) which are determined successively as the sample sizes become known. The method is based on keeping the conditional probability of a false alarm at a predetermined level given that there has not been any earlier false alarm. The control limits dynamically change, and the in-control performance of the chart can be controlled at the desired level for any sequence of sample sizes. The simulation results support this result showing that there is no need for any assumption of a sample size distribution with the use of this proposed approach.     

A non-parametric double homogeneously weighted moving average control chart under sign statistic

In practical situations, the underlying process distribution sometimes deviates from normality and their distribution is partially or completely unknown. In that instance, rather than staying with/depending on the conventional parametric control charts, we consider non-parametric control charts due to their exceptional performance. In this paper, a new non-parametric double homogeneously weighted moving average sign control chart is proposed with the least assumptions. This chart is based on a sign test statistic for catching the smaller deviations in the process location. Run-length (RL) properties of the proposed chart are studied with the help of Monte Carlo simulations. Both in-control and out-of-control RL properties show that the proposed chart is a better contender as compared to some existing charts from the literature. A real-life application for practical consideration of the proposed chart is also provided.     

Improved p charts to monitor process quality

It is a common practice to monitor the fraction p of non-conforming units to detect whether the quality of a process improves or deteriorates. Users commonly assume that the number of non-conforming units in a subgroup is approximately normal, since large subgroup sizes are considered. If p is small this approximation might fail even for large subgroup sizes. If in addition, both upper and lower limits are used, the performance of the chart in terms of fast detection may be poor. This means that the chart might not quickly detect the presence of special causes. In this paper the performance of several charts for monitoring increases and decreases in p is analyzed based on their Run Length (RL) distribution. It is shown that replacing the lower control limit by a simple runs rule can result in an increase in the overall chart performance. The concept of RL unbiased performance is introduced. It is found that many commonly used p charts and other charts proposed in the literature have RL biased performance. For this reason new control limits that yield an exact (or nearly) RL unbiased chart are proposed.     

Robust kernel distance multivariate control chart using support vector principles

It is important to monitor manufacturing processes in order to improve product quality and reduce production cost. Statistical Process Control (SPC) is the most commonly used method for process monitoring, in particular making distinctions between variations attributed to normal process variability to those caused by ‘special causes’. Most SPC and multivariate SPC (MSPC) methods are parametric in that they make assumptions about the distributional properties and autocorrelation structure of in-control process parameters, and, if satisfied, are effective in managing false alarms/-positives and false-negatives. However, when processes do not satisfy these assumptions, the effectiveness of SPC methods is compromised. Several non-parametric control charts based on sequential ranks of data depth measures have been proposed in the literature, but their development and implementation have been rather slow in industrial process control. Several non-parametric control charts based on machine learning principles have also been proposed in the literature to overcome some of these limitations. However, unlike conventional SPC methods, these non-parametric methods require event data from each out-of-control process state for effective model building. The paper presents a new non-parametric multivariate control chart based on kernel distance that overcomes these limitations by employing the notion of one-class classification based on support vector principles. The chart is non-parametric in that it makes no assumptions regarding the data probability density and only requires ‘normal’ or in-control data for effective representation of an in-control process. It does, however, make an explicit provision to incorporate any available data from out-of-control process states. Experimental evaluation on a variety of benchmarking datasets suggests that the proposed chart is effective for process monitoring.     

Boxplot‐based phase I control charts for time between events

To monitor the quality/reliability of a (production) process, it is sometimes advisable to monitor the time between certain events (say occurrence of defects) instead of the number of events, particularly when the events occur rarely. In this case it is common to assume that the times between the events follow an exponential distribution. In this paper, we propose a one‐ and a two‐sided control chart for phase I data from an exponential distribution. The control charts are derived from a modified boxplot procedure. The charting constants are obtained by controlling the overall Type I error rate and are tabulated for some configurations. A numerical example is provided for illustration. The in‐control robustness and the out‐of‐control performance of the proposed charts are examined and compared with those of some existing charts in a simulation study. It is seen that the proposed charts are considerably more in‐control robust and have out‐control properties comparable to the competing charts. Copyright © 2011 John Wiley & Sons, Ltd.     

Monitoring of zero-inflated Poisson processes with EWMA and DEWMA control charts

The zero-inflated Poisson (ZIP) distribution is an extension of the ordinary Poisson distribution and is used to model count data with an excessive number of zeros. In ZIP models, it is assumed that random shocks occur with probability p, and upon the occurrence of random shock, the number of nonconformities in a product follows the Poisson distribution with parameter λ. In this article, we study in more detail the exponentially weighted moving average control chart based on the ZIP distribution (regarded as ZIP-EWMA) and we also propose a double EWMA chart with an upper time-varying control limit to monitor ZIP processes (regarded as ZIP-DEWMA chart). The two charts are studied to detect upward shifts not only in each parameter individually but also in both parameters simultaneously. The steady-state performance and the performance with estimated parameters are also investigated. The performance of the two charts has been evaluated in terms of the average and standard deviation of the run length, and compared with Shewhart-type and CUSUM schemes for ZIP distribution, it is shown that the proposed chart is very effective especially in detecting shifts in p when λ remains in control (IC) and in both parameters simultaneously. Finally, one real example is given to display the application of the ZIP charts on practitioners.     

A double exponentially weighted moving average control chart for monitoring COM‐Poisson attributes

The Conway‐Maxwell‐Poisson (COM‐Poisson) distribution is a two‐parameter generalization of the Poisson distribution, which can be used for overdispersed or underdispersed count data and also contains the geometric and Bernoulli distributions as special cases. This article presents a double exponentially weighted moving average control chart with steady‐state control limits to monitor COM‐Poisson attributes (regarded as CMP‐DEWMA chart). The performance of the proposed control chart has been evaluated in terms of the average, the median, and the standard deviation of the run‐length distribution. The CMP‐DEWMA control chart is studied not only to detect shifts in each parameter individually but also in both parameters simultaneously. The design parameters of the proposed chart are provided, and through a simulation study, it is shown that the CMP‐DEWMA chart is more effective than the EWMA chart at detecting downward shifts of the process mean. Finally, a real data set is presented to demonstrate the application of the proposed chart.     

Testing control chart subgroups for rationality

One of the key issues in statistical process control (SPC) is the forming ‘rational subgroups’. Rational subgroups are defined as

• 1 Subgroups displaying only random within-variation.
• 2 Subgroups having small within-variation to compare with variation between subgroups.

In a previous paper¹ we developed an approach of choosing the proper subgroup size for control charts for statistical process control. The approach is particularly appropriate for batch industries where some batch-to-batch variation is to be expected and should be accommodated. In this paper we will deal with the question of whether or not subgroups are rational. A randomness test can be used to verify rationality. The measure selected is the ratio of several different variance estimators. An example is provided to demonstrate the application of the measure.     

Geometric Standard Deviation - Reply to Bohidar

Bohidar (1) comments that a formula I proposed (2) for geometric standard deviation (GSD) is “absolutely incorrect”. This comment is based on erroneous interpretation of my proposal and to avoid further confusion it may be helpful to point this out. The issue addressed in ref(2) was how to summarize the extent of statistical variation in a sample of data drawn from the lognormal distribution, for which the underlying variation is multiplicative. This situation was contrasted with the normal, or Gaussian, distribution for which the underlying variation is additive. Normally distributed data are conveniently summarized by a mean plus or minus a standard deviation (SD) (or standard error, SE). For lognormal data the geometric mean was available as replacement for the ordinary mean, but no direct analogue existed for SD (or SE).     

Local approach concepts and the microstructures of steels

The paper describes cleavage fracture models that relate local fracture stresses and fracture toughness values to the sizes of brittle initiating particles. In a “quasi-homogeneous” steel, i.e. one possessing a smooth distribution of small particle sizes, a “typically coarse” particle is present in every sample tested and the failure stress or fracture toughness is single valued. When random experimental errors are included, the single valued function becomes a Gaussian distribution. Spatially heterogeneous microstructures produce quite different distributions and these are discussed with respect to extrapolations to low failure probabilities.     

Control chart to monitor circular data

This paper proposes a new Shewhart-type control chart for circular or directional data. This type of data is found in several fields and applications, such as wind direction, the arrival time of a patient in a hospital, and the route of animals. The proposal is based on the Jones–Pewsey distribution, which is a very flexible three-parameter distribution for modeling processes that can be represented as points on the circumference of the unit circle. We conduct an extensive Monte Carlo simulation study to evaluate and compare the performance of the proposed control chart with some competitors considering individual measurements and the mean direction (nonindividual observations). The results show that the Jones–Pewsey control chart outperforms the competitors in terms of run length distribution analysis. Finally, we present and discuss two applications based on actual datasets to show the applicability of the proposed control chart.     

The Effect of Autocorrelation on the Hotelling T2 Control Chart

One of the basic assumptions for traditional univariate and multivariate control charts is that the data are independent in time. For the latter, in many cases, the data are serially dependent (autocorrelated) and cross‐correlated because of, for example, frequent sampling and process dynamics. It is well known that the autocorrelation affects the false alarm rate and the shift‐detection ability of the traditional univariate control charts. However, how the false alarm rate and the shift‐detection ability of the Hotelling T² control chart are affected by various autocorrelation and cross‐correlation structures for different magnitudes of shifts in the process mean is not fully explored in the literature. In this article, the performance of the Hotelling T² control chart for different shift sizes and various autocorrelation and cross‐correlation structures are compared based on the average run length using simulated data. Three different approaches in constructing the Hotelling T² chart are studied for two different estimates of the covariance matrix: (i) ignoring the autocorrelation and using the raw data with theoretical upper control limits; (ii) ignoring the autocorrelation and using the raw data with adjusted control limits calculated through Monte Carlo simulations; and (iii) constructing the control chart for the residuals from a multivariate time series model fitted to the raw data. To limit the complexity, we use a first‐order vector autoregressive process and focus mainly on bivariate data. © 2014 The Authors. Quality and Reliability Engineering International Published by John Wiley & Sons Ltd.     

The dynamic histogram chart

This paper discusses a dynamic histogram control chart which combines some ideas of histograms, X charts, zone charts and chi-square charts. The chart is easy to use, easy to understand, and has quick response times. Another feature is that it does not require normal data as X charts and zone charts do. The chart is especially useful for controlling processes with low data accumulation rates such as chemical processes or for controlling processes involving short runs such as job shops.