A nonparametric triple exponentially weighted moving average sign control chart

Nonparametric control charts are used in process monitoring when there is insufficient information about the form of the underlying distribution. In this article, we propose a triple exponentially weighted moving average (TEWMA) control chart based on the sign statistic for monitoring the location parameter of an unknown continuous distribution. The run-length characteristics of the proposed chart are evaluated performing Monte Carlo simulations. We also compare its statistical performance with existing nonparametric sign charts, such as the cumulative sum (CUSUM), exponentially weighted moving average (EWMA), generally weighted moving average (GWMA), and double exponentially weighted moving average (DEWMA) sign charts as well as the parametric TEWMA-$\overline{X}$ chart. The results show that the TEWMA sign chart is superior to its competitors, especially for small shifts. Moreover, two examples are given to demonstrate the application of the new scheme.     

Simple and useful statistical control charts for monitoring count data

This paper studies control charts based on the BerG (which is a sum of Bernoulli and geometric random variables) process to deal with the cases of equidispersion, overdispersion, underdispersion, or zero inflation (or deflation). Its probability distribution function can be expressed in terms of the mean parameter and its cumulative distribution has a closed form, thus the construction of an $\overline{X}$ control chart to monitor the mean can be made easily. Additionally, we call attention that the asymptotic control limits for $\overline{X}$ control chart by central limit theorem (CLT) may lead to a serious erroneous decision. We present guidelines for practitioners about the minimum sample size needed to match out-of-control average run length (ARL₁) with the exact and asymptotic control limits in function of the shape parameter after an extensive simulation study. The proposed schemes are applied to monitoring the BerG mean parameter.     

Unconditional performance of the X¯ chart: Comparison among five standard deviation estimators

To design the $\overline{X}$ control chart, the in-control process mean and standard deviation must be estimated from historical samples, negatively affecting the chart's performance. The grand mean of a sample is the well-established estimator for the process mean. However, regarding the standard deviation, the chart's user has at least five different estimators available in the literature to choose from. In this paper, using intensive simulations, we study and compare the performance of the $\overline{X}$ chart (under normality assumption and three-sigma limits) among the most five used standard deviation estimators. The unconditional in-control run length (RL₀) is the most commonly used performance measure of a control chart, and many authors base their comparisons on the expectation of the RL₀ or on the mean square error of the estimators. In contrast, we based our comparison on the proportions of the RL₀ concentrated at some intervals that are usually considered undesired by the practitioner due to the high incidence of false alarms during the process monitoring (e.g., RL₀ between 1 and 200), occurring earlier than expected when compared with known parameter situations. From our results, all the studied standard deviation estimators generate similar performances. However, even with this similarity, we showed that one of the most recommended standard deviation estimators in the literature in control charts with estimated parameters may be the most inappropriate choice based on the undesired RL₀ proportions.     

A nonparametric double generally weighted moving average signed-rank control chart for monitoring process location

Most control charts have been developed based on the actual distribution of the quality characteristic of interest. However, in many applications, there is a lack of knowledge about the process distribution. Therefore, in recent years, nonparametric (or distribution-free) control charts have been introduced for monitoring the process location or scale parameter. In this article, a nonparametric double generally weighted moving average control chart based on the signed-rank statistic (referred as DGWMA-SR chart) is proposed for monitoring the location parameter. We provide the exact approach to compute the run-length distribution, and through an extensive simulation study, we compare the performance of the proposed chart with existing nonparametric charts, such as the exponentially weighted moving average signed-rank (EWMA-SR), the generally weighted moving average signed-rank (GWMA-SR), the double exponentially weighted moving average signed-rank (DEWMA-SR), and the double generally weighted moving average sign (DGWMA-SN) charts, as well as the parametric DGWMA- $\overline{X}$ chart for subgroup averages. The simulation results show that the DGWMA-SR chart (with suitable parameters) is more sensitive than the other competing charts for small shifts in the location parameter and performs as well as the other nonparametric charts for larger shifts. Finally, two examples are given to illustrate the application of the proposed chart.     

On monitoring the process mean of autocorrelated observations with measurement errors using the w-of-w runs-rules scheme

Runs-rules have been widely used since the 1950s in industrial and nonindustrial process monitoring applications to improve the performance of basic and other traditional monitoring schemes. However, none of the studies on runs-rules have accounted for a process with a combined effect of measurement errors and autocorrelation. Hence, in this paper, the use of the w-of-w runs-rules to improve the performance of the Shewhart $\overline{X}$ scheme using an additive model with a constant variance and a first-order autoregressive model is proposed. To reduce the combined negative effect of measurement errors and autocorrelation, we implement a sampling strategy based on rational subgroups in which (a) multiple measurements per item are taken (instead of a standard single measurement) and (b) non-neighboring observations are gathered. Moreover, the latter sampling strategy is incorporated into the values of probability elements of a Markov chain matrix which is used to derive some closed-form expressions for the zero- and steady-state run-length distribution. The main finding of this study is that, with respect to some overall performance measures, the proposed scheme outperforms the existing Shewhart $\overline{X}$ scheme by a significant margin. A real-life example is used to illustrate the practical implementation of the proposed scheme.     

Nonparametric triple exponentially weighted moving average signed-rank control chart for monitoring shifts in the process location

Nonparametric (or distribution-free) control charts are used for monitoring processes where there is a lack of knowledge about the underlying distribution. In this article, a triple exponentially weighted moving average control chart based on the signed-rank statistic (referred as TEWMA-SR chart) is proposed for monitoring shifts in the location parameter of an unknown, but continuous and symmetric, distribution. The run-length characteristics of the proposed chart are evaluated performing Monte Carlo simulations. A comparison study with other existing nonparametric control charts based on the signed-rank statistic, the TEWMA sign chart, and the parametric TEWMA-$\overline{X}$ chart indicates that the proposed chart is more effective in detecting small shifts, while it is comparable with the other charts for moderate and large shifts. Finally, two illustrative examples are provided to demonstrate the application of the proposed chart.     

Conditional median run length performance of the synthetic X¯ chart with unknown process parameters

Synthetic-type charts are efficient tools for process monitoring. They are easy to design and implement in practice. The properties of these charts are usually evaluated under the assumption of known process parameters. This assumption is sometimes violated in practice, and process parameters have to be estimated from different phase I data sets collected by different practitioners. This fact causes the between-practitioners variability among the properties of the synthetic-type charts designed for each practitioner. In fact, the shape of the run length distribution of the synthetic-type charts changes with the mean shift size. As a good alternative, the median run length $(MRL)$ metric is argued to evaluate the properties of different control charts. In this paper, the $MRL$ is used as a measure of the synthetic $\overline{X}$ chart's performance, and the conditional $MRL$ properties of the synthetic $\overline{X}$ chart with unknown process parameters are investigated. Both the average $MRL$ ( $AMRL$) and the standard deviation of $MRL$ ( $\text{◂⋅▸}SDMRL$) are used together to investigate the chart's properties when the process parameters are unknown. If the available number of phase I samples is not large enough to reduce the variability of the in-control $MRL$ values to an acceptable level, a bootstrap-type approach is suggested to adjust the control limits of the synthetic $\overline{X}$ chart and to further prevent many unwanted lower in-control $MRL$ values.     

A study on inspection schemes in optimal design of control charts for deteriorating processes

The non-uniform inspection scheme obtained by the constant integrated hazard procedure overcomes the uniform scheme economically in optimal design of control charts. The comperative study is generalized in this paper to an optimization problem which looks for the optimal sampling points among all possible sampling schemes. The objective function is simplified here by modelling sequential time intervals as a family of functions of the first sampling interval, which also has been induced by the constant integrated hazard approach. The study demonstrates the model implementation through the economic design of $\overline{X}$ and T²-Hotelling control charts, both under the two widely used process failure mechanisms, that is, Weibull and Chen distributions. A comprehensive numerical investigation illustrates the possibility of existence of sampling schemes which outperform the constant integrated hazard approach and emphasizes the necessity of further investigation into the solution procedure.     

The effect of measurement error on the performance of Mann-Whitney and Signed-Rank nonparametric control charts

Mann-Whitney and Signed-Rank control charts are two well-known nonparametric charts used for controlling the center of the process when the distribution of the process parameter is unknown or nonnormal. Considering the effect of measurement error on the performance of control charts, the mentioned effect with additive model is investigated on Mann-Whitney and Signed-Rank charts. Furthermore, a comparison is made between the two charts and a Shewhart-type $\overline{\mathrm{X}}$ chart (as a parametric one) in the presence of the error. To do so, a simulation program is used and average run length (ARL) of the charts are calculated under three distributions. The results for all three distributions show that the existence of measurement error weakens the performances of both nonparametric charts and larger values of the variance of the error will increase the effect. A numerical example is also discussed to show the effect on the performance of the charts. Multiple measurements is used as a way to decrease the effect of measurement error. Knowing the fact that it requires extra time and money, it can be used in real cases depending on the financial limitations of the user.     

A High-Performance Li–Mn–O Li-rich Cathode Material with Rhombohedral Symmetry via Intralayer Li/Mn Disordering

The search for new high-performance and low-cost cathode materials for Li-ion batteries is a challenging issue in materials research. Commonly used cobalt- or nickel-based cathodes suffer from limited resources and safety problems that greatly restrict their large-scale application, especially for electric vehicles and large-scale energy storage. Here, a novel Li–Mn–O Li-rich cathode material with $R\overline{3}m$ symmetry is developed via intralayer Li/Mn disordering in the Mn-layer. Due to the special atomic arrangement and higher $R\overline{3}m$ symmetry with respect to the C2/m symmetry, the oxygen redox activity is modulated and the Li in the Li-layer is preferentially thermodynamically extracted from the crystal structure instead of Li in the Mn-layer. The as-obtained material delivers a reversible capacity of over 300 mAh g⁻¹ at 25 mA g⁻¹ and rate capability of up to 260 mAh g⁻¹ at 250 mA g⁻¹ within 2.0–4.8 V. The excellent performance is attributed to its highly structural reversibility, mitigation of Jahn–Teller distortion, lower bandgap, and faster Li-ion 2D channels during the lithium-ion de/intercalation process. This material is not only a promising cathode material candidate but also raises new possibilities for the design of low-cost and high-performance cathode materials.     

Phase I process monitoring: The case of the balanced one-way random effects model

Phase I is crucial for the success of the overall statistical process control (SPC) and monitoring regime. Shewhart-type charts are recommended in this phase because of their broader shift detection ability. In this paper, a Phase I Shewhart-type $\overline{X}$ chart is considered for the balanced random effects (also called a variance components) model. The proposed methodology takes proper account of the effects of parameter estimation and uses the false alarm probability (FAP) metric to design the chart. In the sequel, the corrected (adjusted) charting constants are calculated and tabulated. The constant can be found, on demand, from an accompanying R package. Motivations and illustrations with some real data are provided. Performance of the chart is examined in terms of in-control robustness and detection of nonhomogeneity (out-of-control). The proposed chart is shown to be easily adaptable to more general models, with more variance components and nested factors, and can accommodate various estimators of variance. Thus, it enables a broader Phase I process monitoring strategy, under normality, which can be applied within the ANOVA framework applicable for many DOE models. A summary and some recommendations are provided.     

Maximum hybrid exponentially weighted moving average control chart in the presence of measurement error by using auxiliary information

Hybrid control charts have become part of statistical process control (SPC) but still, need more emphasis. Researchers are developing charts for joint monitoring of process mean and variance shifts just like Max-EWMA and their hybrid version using auxiliary information but are ignoring the effect of measurement error on the efficiency of charts. We propose maximum hybrid exponentially weighted moving average with measurement error using auxiliary information and name it Max-HEWMAMEAI control chart. The efficiency of this chart is proved through calculations of average run lengths ($ARLs$) and standard deviations of run lengths (SDRLs) using the Monte Carlo simulations method whereas, $ARLs$ and $\text{◂⋅▸}SDRLs$ are shown in tabular form. The effect of measurement error on the efficiency of the chart has been analyzed and the impact of multiple measurements to reduce the error effect has been studied using the covariate model. Real-life application is also part of this article to support the simulation results.     

Practitioners guide on parametric,nonparametric, and semiparametric profile monitoring

Profile monitoring is one of the methods used in statistical process control ($SPC$) to understand the functional relationship between response and explanatory variables by tracking this relationship and estimating parameters. SPC is done in two phases: In Phase $I$, a statistical model is created and its parameters estimated using historical data. Phase $II$ implements the statistical model and monitors the live ongoing process. Control charts are graphical tools used to monitor these functional relationships over time in both Phase $I$ and Phase $II$. This study provides a step-by-step application for parametric, nonparametric, and semiparametric methods in profile monitoring and creates an in-depth guideline with comparative analysis studies for novice practitioners. A comparative analysis under each distributional assumption is conducted for various control charts.