Underlying Process Distribution Research Articles

A distribution-free Max-EWMA scheme for multi-aspect process monitoring with industrial applications

In Statistical Process Monitoring, Distribution-free charting schemes play a vital role in Industry 4.0 for their superior effectiveness compared to parametric methods, especially in cases where the underlying process distribution is not well-defined or poses challenges in estimation. Traditional distribution-free charting schemes focus on surveilling only one process aspect: the location or scale parameter. The past two decades have also witnessed a rapid expansion of research on joint monitoring of the two aspects, location and scale, using combined bi-aspect charting schemes. However, there is a notable scarcity of methods that addressing complex industrial scenarios where multiple parameters (like location, scale, and shape) need simultaneous monitoring with high sensitivity and specificity. Our research is motivated by this challenge, and this paper describes a distribution-free tri-aspect monitoring procedure called the Maximum Exponentially Weighted Moving Average (Max-EWMA) scheme This scheme can effectively monitor shifts in three process characteristics, encompassing aspects related to process location, scale, and shape parameters. We utilize weighted versions of well-known statistical measures such as the Wilcoxon, Ansari-Bradley, and Savage-type statistics to construct the plotting statistics. An essential feature of our proposed Max-EWMA chart is its robustness under various continuous distributions, ensuring reliable performance in in-control situations. Competing schemes are compared via intensive computing techniques based on Monte Carlo using the Median Run Length metric, which evaluates the effectiveness of the proposed scheme. Finally, our proposed schemes are illustrated with two examples. Some concluding remarks and limitations of the study are noted.

Nonparametric precedence chart with repetitive sampling

AbstractIn most real‐world applications, such as production and manufacturing processes, the underlying process distribution does not always follow a normal distribution. In such cases, statistical process control literature recommends the use of nonparametric (or distribution‐free) control charts. This paper introduces a new distribution‐free precedence chart using repetitive sampling. The performance of the proposed chart is investigated in terms of the average run‐length ( ) profile. The expressions of the in‐control process probability and of the proposed chart are introduced using integral formulas. The out‐of‐control performance of the new chart is compared to that of the existing precedence charts with and without runs‐rules. A numerical example is provided using real‐life data to demonstrate the application and implementation of the new chart.

A new nonparametric adaptive EWMA procedures for monitoring location and scale shifts via weighted Cucconi statistic

In the context of distribution-free process monitoring and control, the Cucconi statistic has attracted more and more attention in recent years. The original Cucconi statistic was introduced for the two-sample location-scale problems. It is computationally more straightforward and is highly efficient. The Cucconi statistic is recently adopted for jointly monitoring the location and scale parameters of univariate continuous quality and service processes. It is distribution-free and does not require any stringent assumption on the underlying process distribution. Some recent development further revealed that the Cucconi statistic is the average of the squared standardised Wilcoxon statistic for monitoring location and the squared standardised Mood statistic for scale parameters — a fact unnoticed by the practitioners over five decades. The Cucconi statistic gives equal weight to shifts in location and scale. Therefore, this paper proposes a new adaptive nonparametric process monitoring scheme based on the weighted Cucconi statistic. The proposed monitoring scheme is designed to adapt the weight factor according to the estimated relative magnitude of mean and scale shift combined with an EWMA-type set-up. The numerical results based on the Monte-Carlo simulation study and a real data analysis reveal that the proposed monitoring scheme performs well across a wide range of shifts and compares favourably with prevailing distribution-free process monitoring schemes.

A comprehensive distribution‐free scheme for tri‐aspect surveillance of complex processes

AbstractDistribution‐free charting schemes for process monitoring are more robust and reliable than their parametric counterparts when the process distributions are unknown and complicated. Most of the existing control charts are uni‐aspect schemes because they can detect a shift in only one aspect of the process distribution, like location or scale. Research on schemes for simultaneous surveillance of location and scale parameters has been very active in recent years, leading to many bi‐aspect schemes. These schemes do not explicitly deal with the shape of the process distribution. However, a shift in the shape parameter with or without a location or scale shift can occur in practice, especially in production or time‐to‐event processes. Note that there are nonparametric schemes for detecting general shifts based on goodness‐of‐fit test statistics or empirical likelihood ratio statistics, which cannot isolate if there is a shift in location, scale, shape, or combination of these parameters. This article introduces a new distribution‐free process monitoring scheme that can detect a shift in the location, scale or shape parameters, or any combinations of them. The new scheme is based on a combined statistic designed via Euclidean distance of the standardized Wilcoxon statistic for location, the standardized Ansari–Bradley statistic for scale, and the standardized Savage‐type statistic for shape. We discuss the implementation design using the average run‐length as the performance metric and investigate the proposed scheme's in‐control performance. It is shown that the new charting scheme is in‐control robust irrespective of the underlying process distribution and therefore applicable to monitor any univariate continuous processes. An out‐of‐control performance comparison study of the new scheme with many existing schemes shows that the new scheme is preferable to the existing schemes. We illustrate the proposed chart with real data to monitor a passenger train's arrival delays in Italy's regional route. The proposed scheme is comprehensive because it integrates the follow‐up procedure via integrated sub‐charts for classifying the signaling component.

The SPRT sign chart for process location

This paper develops a nonparametric sequential probability ratio test (SPRT) control chart for monitoring the process location. The chart is based on a sequential sign test. Young’s algorithm to determine the properties of the SPRT for a parameter of discrete Koopman-Darmois density is used to obtain the exact statistical performance measures of the proposed chart. The proposed chart substantially outperforms the Shewhart sign, synthetic Shewhart sign, CUSUM sign, and EWMA sign charts in detecting a location shift of any size. It is a preferable alternative to the parametric SPRT chart for mean when the underlying process distribution is not precisely known. The implementation of the chart is illustrated through an example.

Distribution‐free double‐sampling precedence monitoring scheme to detect unknown shifts in the location parameter

AbstractIn most applications, parametric monitoring schemes are used to monitor the majority of industrial and nonindustrial processes in order to improve the quality of the outputs or services. However, parametric monitoring schemes are known to underperform when the normality assumption is not met or when there is not enough information about the symmetry or asymmetry nature of the process underlying distribution. Hence, in this paper, a new nonparametric Phase II Shewhart‐type double‐sampling (DS) monitoring scheme based on the precedence statistic is proposed in order to efficiently monitor quality processes when the underlying process distribution departs from normality. The performance is investigated using the average run length (ARL), standard deviation of the run length (SDRL), expected ARL (EARL) and expected average number of observations to signal (EANOS), and the average sample sizes (ASS) metrics. The latter metrics are computed using Monte Carlo simulation and exact formulae. In general, it is shown that the new DS precedence scheme outperforms the existing basic Shewhart precedence scheme with and without supplementary runs rules in many situations. A real‐life illustrative example based on a filling process of milk bottles is provided to demonstrate the application and implementation of the new DS precedence monitoring scheme.

A Phase II depth‐based variable dimension EWMA control chart for monitoring process mean

AbstractStatistical process control consists of tools and techniques that are useful for improving a process or ensuring that a process is in a stable and satisfactory state. In many modern industrial applications, it is critically important to simultaneously monitor two or more correlated process quality variables, thus necessitating the development of multivariate statistical process control (MSPC) as an important area of research for the new century. Nevertheless, the existing MSPC research is mostly based on the assumption that the process data follow a multinormal distribution or a known distribution. However, it is well recognized that in many applications the underlying process distribution is unknown. In practice, among a set of correlated variables to be monitored, there is oftentimes a subset of variables that are easy and/or inexpensive to measure, whereas the remaining variables are difficult and/or expensive to measure but contain information that may help more quickly detect a shift in the process mean. We are motivated to develop a Phase II control chart to monitor variable dimension (VD) mean vector for unknown multivariate processes. The proposed chart is based on the exponentially weighted moving average (EWMA) of a depth‐based statistic. The proposed chart is shown to lead to faster detection of mean shifts than the existing VDT2 and VD EWMAT2 charts studied in Aparisi et al. and Epprecht et al., respectively.

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

AbstractIn 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.

Nonparametric progressive sign chart for monitoring process location based on individual data

ABSTRACT Statistical process control (SPC) plays a vital role in the maintenances and improvements of quality outputs in manufacturing, industrial and service production processes. Control chart is an important SPC tool, used to detect noises and to improve process performance. When the underlying process distribution lacks the assumption of normality, nonparametric (NP) control charts become essential and particularly are useful because their in-control (IC) run length properties remain the same for every continuous distribution. This article develops the NP progressive mean sign (NPPM-SN) chart for monitoring the process target through 100% inspection by taking individual measurements from the process. The performance of the proposed NPPM-SN chart is examined under zero-state and steady-state scenarios. The IC and out-of-control run length properties of the proposed control chart are evaluated using Monte Carlo simulation. The proposed NPPM-SN chart is compared with traditional exponentially weighted moving average (EWMA), nonparametric EWMA sign (NPEWMA-SN) and traditional progressive mean (PM) control charts based on individual measurements using average run length and some other characteristics of the run-length distribution. The proposed NPPM-SN chart is found more robust (for all distributions) and efficient (for skewed distributions) as compared to its competitors. Along with a real-life example related to high voltage power supply, a simulated data example is also presented for the implementation of the proposed NPPM-SN chart.

A control chart for the generalized exponential distribution under time truncated life test

The usual assumption in the application of control chart is that the process follows a normal distribution. However, in practice, the underlying process distribution is either unknown or non-normal and the use of control chart assuming normality often leads to false alarm in the process. In this article, an attribute control chart under a truncated life test is proposed when the lifetime of a product follows a generalized exponential distribution assuming that the shape parameter is known and a shift in scale parameter has occurred. The control chart coefficient and the truncation constant are determined and the performance of the attribute control chart constructed on time truncated life test is evaluated in terms of the average run length (ARL). Tables of out-of-control ARL are also provided when the scale parameter are shifted. The application of the proposed chart is demonstrated using a simulated dataset for industrial use.

On Designing Distribution-Free Homogeneously Weighted Moving Average Control Charts

Abstract There are many practical situations in which the underlying process distribution is either completely unknown or partially known but deviates from normality. In such cases, the performance of the traditional parametric control charts deteriorates, and nonparametric control charts are considered as a robust alternative for monitoring the process. In this article, two new nonparametric homogeneously weighted moving average (NPHWMA) control charts are proposed for monitoring the deviations in process location from the target value. The proposed charts assign a specific weight to the current sample information, and the remaining weight is equally distributed among the previous samples. The proposed NPHWMA charts are based on sign test and Wilcoxon signed-rank test for process monitoring under the skewed and symmetric distributions, respectively. The Monte Carlo simulations are used to study the run length properties and compare the performance of the proposed charts with some existing control charts. It is found that the proposed charts outperform their existing counterparts. Empirical illustrations are provided for practical implementation of the proposed charts.

A nonparametric generally weighted moving average sign chart based on repetitive sampling

Most control charts that assume quality characteristics of interest follow a normal or specific distribution. However, in reality there is often limited or no information regarding the underlying process distribution. Therefore, in recent years, nonparametric techniques for quality control have been developed. In this paper, the nonparametric generally weighted moving average sign chart based on repetitive sampling (hereinafter RS-GWMA sign chart) is proposed to improve performance capability of existing charts in small process shifts. Simulation studies show that the nonparametric RS-GWMA sign chart with large design and adjustment parameters outperforms competing charts considered in this article. The fill volume of soft-drink beverage bottles is as industrial example used to illustrate the application of the proposed nonparametric RS-GWMA sign chart.

Comparisons of some distribution-free CUSUM and EWMA schemes and their applications in monitoring impurity in mining process flotation

Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) schemes are widely used in monitoring small and persistent shifts in specific process characteristics. Nonparametric (Distribution-free) CUSUM and EWMA schemes are useful in detecting such changes when the underlying process distribution is unknown or complicated. The CUSUM-Wilcoxon rank-sum (CUSUM-WRS), CUSUM-Precedence, EWMA-WRS, and EWMA-Precedence are well-known distribution-free CUSUM and EWMA schemes for Phase-II monitoring of a shift in the unknown location parameter of a process. In this article, we compare their performances with two robust CUSUM and EWMA schemes based on the Hogg-Fisher-Randle (HFR) type statistic. We investigate the accomplishments of these CUSUM and EWMA schemes in detecting shifts of different sizes for various process distributions and show that the proposed CUSUM and EWMA HFR schemes perform favourably for highly skewed distributions. In this article, we also discuss the efficacy of these nonparametric CUSUM and EWMA schemes when the test sample size is small. We also outline the implementation strategies various plans with an illustration using a dataset from the iron ore mining plant. We offer some concluding remarks and future research problems.

Distribution-free precedence schemes with a generalized runs-rule for monitoring unknown location

Nonparametric statistical process monitoring schemes are robust alternatives to traditional parametric process monitoring schemes, especially when the assumption of normality is invalid or when we do not have enough information about the underlying process distribution. In this paper, we propose to improve the well-known precedence scheme using the 2-of-(h + 1) supplementary runs-rules (where is a nonzero positive integer). The in-control and out-of-control performances of the proposed control schemes are thoroughly investigated using both Markov chain and simulation based approaches. We find that the proposed schemes outperform their competitors in many cases. A real-life example is given to illustrate the design and implementation of the proposed schemes.

On designing new optimal synthetic Tukey’s control charts

ABSTRACTTukey’s control chart is generally used for monitoring the processes where the measurement process physically damages the product. It is based on single observation and robust to outliers. In this paper, two optimal synthetic Tukey’s control charts are proposed by integrating the conforming run length chart with the Tukey’s control chart and its modification. The performance comparison of the proposed charts with the existing Tukey’s control charts is made by using out-of-control average run length and extra quadratic loss as performance metrics. The proposed charts offer better protection against the process shifts as compare to the existing Tukey’s control charts when the underlying process distribution is symmetric or asymmetric. Simulation studies also establish the supremacy of the proposed control charts over the existing Tukey’s control charts. In the end, an illustrative example based on a real data set of the combined cycle power plant is provided for practical implementation.

Robust skewness dispersion control charts

PurposeThe purpose of this paper is to explore the scope of robust dispersion control charts in a distribution-free environment, which is a specific case of non-normal control charts. These control charts are skewness-based structures designed to monitor skewed-type processes whilst equally performing under symmetric processes. Moreover, the choice of a suitable control chart for a particular non-normal situation is also suggested.Design/methodology/approachThe probability control limits approach is considered as an alternative way to determine the skewness-based structure of dispersion control charts. The proposals of five robust and two conventional Shewhart-type dispersion control charts are suggested as efficient competitors of skewness correction (SC) dispersion control charts. The evaluation of robust proposals and competing dispersion control charts is done through false alarm rate (FAR) and probability to signal (PTS) measures.FindingsThe proposed dispersion control charts are found robust and efficient alternatives of SC dispersion control charts in both normal and non-normal distributions. The FARs and PTS properties of proposed control charts are impressive in all studied cases, and a real-data example also verifies the dominance of proposed control charts.Originality/valueConventional dispersion control charts quickly lose their efficiency as underlying process distribution deviates from normality; however, robust control charts emerge as most suitable candidates in such situations. This paper proposes the idea of robust dispersion control charts under a distribution-free structure for the skewed-type process, which is not yet explored.

Optimizing joint location-scale monitoring – An adaptive distribution-free approach with minimal loss of information

Nonparametric statistical process monitoring (NSPM) schemes are very effective in monitoring various non-normal and complex processes. Nevertheless, there is one disadvantage of the traditional NSPM schemes. We often lose some information during scoring or ranking. For example, we sometimes ignore the information related to the shape or tail-weights of the underlying process distribution. In this paper, we introduce distribution-free adaptive Shewhart-Lepage (SL) type schemes for simultaneous monitoring of location and scale parameters using information about symmetry and tail-weights of the process distribution. We consider an adaptive SL type scheme, referred to as the LPA scheme, based on the three modified Lepage-type statistics. Using numerical results obtained via Monte-Carlo, and considering operational simplicity, we also propose a new adaptive SL type scheme, referred to as the MLPA scheme, with finite sample correction. These adaptive approaches use the Phase-I data to assess the tail-weights of the process distribution, and then select an appropriate SL type statistic for process monitoring. Consequently, these schemes correct the disadvantage of the traditional distribution-free schemes to a great extent. We compare the two adaptive schemes with the three individual SL type schemes, and also with the classical SL and Shewhart-Cucconi (SC) schemes in terms of the average, the standard deviation and some percentiles of the run length distribution. Numerical results establish that the MLPA scheme is superior for jointly monitoring the parameters of a broad class of process distributions belong to the location-scale family.

A double generally weighted moving average exceedance control chart

AbstractSince the inception of control charts by W. A. Shewhart in the 1920s, they have been increasingly applied in various fields. The recent literature witnessed the development of a number of nonparametric (distribution‐free) charts as they provide a robust and efficient alternative when there is a lack of knowledge about the underlying process distribution. In order to monitor the process location, information regarding the in‐control (IC) process median is typically required. However, in practice, this information might not be available due to various reasons. To this end, a generalized type of nonparametric time‐weighted control chart labeled as the double generally weighted moving average (DGWMA) based on the exceedance statistic (EX) is proposed. The DGWMA‐EX chart includes many of the well‐known existing time‐weighted control charts as special or limiting cases for detecting a shift in the unknown location parameter of a continuous distribution. The DGWMA‐EX chart combines the better shift detection properties of a DGWMA chart with the robust IC performance of a nonparametric chart, by using all the information from the start until the most recent sample to decide if a process is IC or out‐of‐control. An extensive simulation study reveals that the proposed DGWMA‐EX chart, in many cases, outperforms its counterparts.

A generally weighted moving average exceedance chart

ABSTRACTDistribution-free control charts gained momentum in recent years as they are more efficient in detecting a shift when there is a lack of information regarding the underlying process distribution. However, a distribution-free control chart for monitoring the process location often requires information on the in-control process median. This is somewhat challenging because, in practice, any information on the location parameter might not be known in advance and estimation of the parameter is therefore required. In view of this, a time-weighted control chart, labelled as the Generally Weighted Moving Average (GWMA) exceedance (EX) chart (in short GWMA-EX chart), is proposed for detection of a shift in the unknown process location; this chart is based on exceedance statistic when there is no information available on the process distribution. An extensive performance analysis shows that the proposed GWMA-EX control chart is, in many cases, better than its contenders.

An EWMA control chart based on the Wilcoxon rank-sum statistic using repetitive sampling

PurposeThe purpose of this paper is to develop an exponentially weighted moving average (EWMA) control chart based on the Wilcoxon rank-sum (WRS) statistic using repetitive sampling to improve the sensitivity of the EWMA control chart to process mean shifts regardless of the prior knowledge of the underlying process distribution.Design/methodology/approachThe proposed chart is developed without any distributional assumption of the underlying quality process for monitoring the location parameter. The authors developed formulae as well as algorithms to facilitate the design and implementation of the proposed chart. The performance of the proposed chart is investigated in terms of the average run-length, standard deviation of the run-length (RL), average sample size and percentiles of the RL distribution. Numerical examples are given as illustration of the design and implementation of the proposed chart.FindingsThe proposed control chart presents very attractive RL properties and outperforms the existing nonparametric EWMA control chart based on the WRS in the detection of the mean process shifts in many situations. However, the performance of the proposed chart relatively deteriorates for small phase I sample sizes.Originality/valueThis study develops a new control chart for monitoring the process mean using a two-sample test regardless of the nature of the underlying process distribution. The proposed control chart does not require any assumption on the type (or nature) of the process distribution. It requires a small number of subgroups in order to reach stability in the phase II performance.