Process monitoring of exponentially distributed characteristics through an optimal normalizing transformation

Many process characteristics follow an exponential distribution, and control charts based on such a distribution have attracted a lot of attention. However, traditional control limits may be not appropriate because of the lack of symmetry. In this paper, process monitoring through a normalizing power transformation is studied. The traditional individual measurement control charts can be used based on the transformed data. The properties of this control chart are investigated. A comparison with the chart when using probability limits is also carried out for cases of known and estimated parameters. Without losing much accuracy, even compared with the exact probability limits, the power transformation approach can easily be used to produce charts that can be interpreted when the normality assumption is valid.     

Normalization of the origin-shifted exponential distribution for control chart construction

This study demonstrates that a location parameter of an exponential distribution significantly influences normalization of the exponential. The Kullback–Leibler information number is shown to be an appropriate index for measuring data normality using a location parameter. Control charts based on probability limits and transformation are compared for known and estimated location parameters. The probabilities of type II error (β-risks) and average run length (ARL) without a location parameter indicate an ability to detect an out-of-control signal of an individual chart using a power transformation similar to using probability limits. The β-risks and ARL of control charts with an estimated location parameter deviate significantly from their theoretical values when a small sample size of n≤50 is used. Therefore, without taking into account of the existence of a location parameter, the control charts result in inaccurate detection of an out-of-control signal regardless of whether a power or natural logarithmic transformation is used. The effects of a location parameter should be eliminated before transformation. Two examples are presented to illustrate these findings.     

CUSUM Chart with Transformed Exponential Data

Time Between Events (TBE) charts were proposed to monitor the time between events occur based on exponential distribution, and have been shown to be more effective than monitoring the fraction non conforming directly. In this article, we consider monitoring the TBE data with CUSUM scheme by transformation. The idea behind it is to transform the TBE data to normal, and then apply the CUSUM scheme for the approximate normal data. Several simple transformation methods are examined. The calculation of Average Run Length (ARL) with Markov chain approach is described. Comparative studies on the ARL performance show that the transformed CUSUM is superior to the X-MR (Moving Range) chart with transformation, the Cumulative Quantity Control (CQC) chart, and have comparable performance with exponential CUSUM charts. The design procedures of optimal CUSUM chart are also presented. This study provides another possible alternative for monitoring TBE data with easy design procedures and relatively good performance.     

A new approach for monitoring process variance

ABSTRACT

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having non-normal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used in such circumstances. In this paper, we propose a new variance chart based on a simple statistic to monitor process variance shifts. We explore the sampling properties of the new monitoring statistic and calculate the average run lengths (ARLs) of the proposed variance chart. Furthermore, an arcsine transformed exponentially weighted moving average (EWMA) chart is proposed because the ARLs of this modified chart are more intuitive and reasonable than those of the variance chart. We compare the out-of-control variance detection performance of the proposed variance chart with that of the non-parametric Mood variance (NP-M) chart with runs rules, developed by Zombade and Ghute [Nonparametric control chart for variability using runs rules. Experiment. 2014;24(4):1683–1691], and the nonparametric likelihood ratio-based distribution-free exponential weighted moving average (NLE) chart and the combination of traditional exponential weighted moving average (EWMA) mean and EWMA variance (CEW) control chart proposed by Zou and Tsung [Likelihood ratio-based distribution-free EWMA control charts. J Qual Technol. 2010;42(2):174–196] by considering cases in which the critical quality characteristic has a normal, a double exponential or a uniform distribution. Comparison results showed that the proposed chart performs better than the NP-M with runs rules, and the NLE and CEW control charts. A numerical example of service times with a right-skewed distribution from a service system of a bank branch in Taiwan is used to illustrate the application of the proposed variance chart and of the arcsine transformed EWMA chart and to compare them with three existing variance (or standard deviation) charts. The proposed charts show better detection performance than those three existing variance charts in monitoring and detecting shifts in the process variance.     

Optimal Variable Sample Size and Sampling Interval MSE Chart Based on Improved Square Root Transformation

In the field of statistical process control (SPC), control charts for attributes are widely used to detect the out-of-control condition by checking the number of nondefective units or nondefective in a sample. In this article, we use the average time to signal (ATS) and the average number of observations to signal (ANOS) to evaluate the performance of the optimal variable sample size and sampling interval (VSSI) improved square root transformation (ISRT) mean square error (MSE) (VSSI_ ISRT_ MSE) control chart for attribute data. In addition, this control chart will be used to monitor: (1) the difference between the process mean and the target value, and (2) the process variance shifts. We found that the optimal VSSI_ ISRT_ MSE chart performs better than the specific VSSI, the optimal variable sampling interval (VSI), and the fixed parameters (FP) ISRT_MSE charts. An example is given to illustrate this new proposed approach.     

ARL-unbiased control charts for the monitoring of exponentially distributed characteristics based on type-II censored samples

In this paper, the problem of monitoring process data that can be modelled by exponential distribution is considered when observations are from type-II censoring. Such data are common in many practical inspection environment. An average run length unbiased (ARL-unbiased) control scheme is developed when the in-control scale parameter is known. The performance of the proposed control charts are investigated in terms of the ARL and standard deviation of the run length. The effects of parameter estimation on the proposed control charts are also evaluated. Then, we consider the design of the ARL-unbiased control charts when the in-control scale parameter is estimated. Finally, an example is used to illustrate the implementation of the proposed control charts.     

Synthetic exponential control charts with unknown parameter

The existing synthetic exponential control charts are based on the assumption of known in-control parameter. However, the in-control parameter has to be estimated from a Phase I dataset. In this article, we use the exact probability distribution, especially the percentiles, mean, and standard deviation of the conditional average run length (ARL) to evaluate the effect of parameter estimation on the performance of the Phase II synthetic exponential charts. This approach accounts for the variability in the conditional ARL values of the synthetic chart obtained by different practitioners. Since parameter estimation results in more false alarms than expected, we develop an exact method to design the adjusted synthetic charts with desired conditional in-control performance. Results of known and unknown in-control parameter cases show that the control limit of the conforming run length sub-chart of the synthetic chart should be as small as possible.     

Multivariate Control Chart Based on Multivariate Smirnov Test

Robust control charts are useful in statistical process control (SPC) when there is limited knowledge about the underlying process distribution, especially for multivariate observations. This article develops a new robust and self-starting multivariate procedure based on multivariate Smirnov test (MST), which integrates a multivariate two-sample goodness-of-fit (GOF) test based on multivariate empirical distribution function (MEDF) and the change-point model. As expected, simulation results show that our proposed control chart is robust to nonnormally distributed data, and moreover, it is efficient in detecting process shifts, especially large shifts, which is one of the main drawbacks of most robust control charts in the literature. As it avoids the need for a lengthy data-gathering step, the proposed chart is particularly useful in start-up or short-run situations. Comparison results and a real data example show that our proposed chart has great potential for application.     

Asymptotics of the run lengths of two control charts

The CUSUM control chart proposed by Page is a widely used in monitoring the quality of manufacturing processes. The Shiryayev-Roberts (S-R) control chart due to Shiryayev (1963) and Roberts (1988) is one of its competitors, This paper is concerned with the distribution properties of the run lengths of these two control charts. In context of continuous time, we first give the expansions of the higher moments of these run lengths. Then, we show that the asymptotic distributions of these run lengths are either some exponential distributions, or the distribution of the suprema of a standard Brownian motion, or some normal distributions, according to whether the μ<δ/2,μ =δ/2 and μ>δ/2. Here δ is the reference value of the above charts. Some similar results are also obtained in the context of discrete time.     

Designing of a control chart using belief statistic for exponential distribution

A new control chart is proposed by using the belief statistic for the exponential distribution. The structure of the proposed control chart is given to measure the average run length for the shifted process. The comparison of the proposed chart is given with the existing charts in terms of the average run lengths, which shows the outperformance of the proposed chart. The performance of the proposed control chart is also discussed with the help of simulated data.     

Statistical issues in the prospective monitoring of health outcomes across multiple units

Summary.  Following several recent inquiries in the UK into medical malpractice and failures to deliver appropriate standards of health care, there is pressure to introduce formal monitoring of performance outcomes routinely throughout the National Health Service. Statistical process control (SPC) charts have been widely used to monitor medical outcomes in a variety of contexts and have been specifically advocated for use in clinical governance. However, previous applications of SPC charts in medical monitoring have focused on surveillance of a single process over time. We consider some of the methodological and practical aspects that surround the routine surveillance of health outcomes and, in particular, we focus on two important methodological issues that arise when attempting to extend SPC charts to monitor outcomes at more than one unit simultaneously (where a unit could be, for example, a surgeon, general practitioner or hospital): the need to acknowledge the inevitable between-unit variation in 'acceptable' performance outcomes due to the net effect of many small unmeasured sources of variation (e.g. unmeasured case mix and data errors) and the problem of multiple testing over units as well as time. We address the former by using quasi-likelihood estimates of overdispersion, and the latter by using recently developed methods based on estimation of false discovery rates. We present an application of our approach to annual monitoring 'all-cause' mortality data between 1995 and 2000 from 169 National Health Service hospital trusts in England and Wales.     

On designing new optimal synthetic Tukey’s control charts

Tukey’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.     

One-Class Classification-Based Control Charts for Monitoring Autocorrelated Multivariate Processes

In recent years, statistical process control (SPC) of multivariate and autocorrelated processes has received a great deal of attention. Modern manufacturing/service systems with more advanced technology and higher production rates can generate complex processes in which consecutive observations are dependent and each variable is correlated. These processes obviously violate the assumption of the independence of each observation that underlies traditional SPC and thus deteriorate the performance of its traditional tools. The popular way to address this issue is to monitor the residuals—the difference between the actual value and the fitted value—with the traditional SPC approach. However, this residuals-based approach requires two steps: (1) finding the residuals; and (2) monitoring the process. Also, an accurate prediction model is necessary to obtain the uncorrelated residuals. Furthermore, these residuals are not the original values of the observations and consequently may have lost some useful information about the targeted process. The main purpose of this article is to examine the feasibility of using one-class classification-based control charts to handle multivariate and autocorrelated processes. The article uses simulated data to present an analysis and comparison of one-class classification-based control charts and the traditional Hotelling's T ² chart.     

Designing an effective exponential cusum chart without the use of nomographs

The development of control charts for monitoring processes associated with very low rates of nonconformities is increasingly becoming more important as manufacturing processes become more capable. Since the rate of nonconformities can typically be modeled by a simple homogeneous Poisson process, the perspective of monitoring the interarrival times using the exponential distribution becomes an alternative. Gan (1994) developed a CUSUM-based approach for monitoring the exponential mean. In this paper, we propose an alternative CUSUM-based approach based on its ease of implementation. We also provide a study of the relative performance of the two approaches.     

The gamma-exponentiated exponential distribution

In this paper, we introduce a new distribution generated by gamma random variables. We show that this distribution includes as a special case the distribution of the lower record value from a sequence of i.i.d. random variables from a population with the exponentiated (generalized) exponential distribution. The properties of this distribution are derived and the estimation of the model parameters is discussed. Some applications to real data sets are finally presented for illustration.     

关于单变量统计过程控制图某些研究结果简介

文章仅对一元连续变量的静态与动态控制图研究现状进行了简单的总结和介绍,并给出了较详细的参考文献,希望为国内开展此方向的研究抛砖引玉。     

Two-Sided Generalized Exponential Distribution

We derive a generalization of the exponential distribution by making log transformation of the standard two-sided power distribution. We show that this new generalization is in fact a mixture of a truncated exponential distribution and truncated generalized exponential distribution introduced by Gupta and Kundu [Generalized exponential distributions. Aust. N. Z. J. Stat. 41(1999):173–188]. The newly defined distribution is more flexible for modeling data than the ordinary exponential distribution. We study its properties, estimate the parameters, and demonstrate it on some well-known real data sets comparing other existing methods.     

A new t-chart using process capability index

Control charts play a vital role to enhance the efficiency of the manufacturing process. In many situations, the quality characteristic of interest to be monitored follows a non-normal distribution. In this article, we propose a new control chart using the process capability index when the quality characteristic follows the exponential distribution. The performance of the proposed chart is evaluated using the Monte Carlo simulation. Tables are presented for various values of specified average run length and sample size. The use of the proposed control chart is discussed with the help of an example.     

Transmuted generalized exponential distribution: A generalization of the exponential distribution with applications to survival data

In this article, we investigate the potential usefulness of the three-parameter transmuted generalized exponential distribution for analyzing lifetime data. We compare it with various generalizations of the two-parameter exponential distribution using maximum likelihood estimation. Some mathematical properties of the new extended model including expressions for the quantile and moments are investigated. We propose a location-scale regression model, based on the log-transmuted generalized exponential distribution. Two applications with real data are given to illustrate the proposed family of lifetime distributions.     

Acceptance control charts for non-normal data

Control charts are one of the most important methods in industrial process control. The acceptance control chart is generally applied in situations when an X¯ chart is used to control the fraction of conforming units produced by the process and where 6-sigma spread of the process is smaller than the spread in the specification limits. Traditionally, when designing control charts, one usually assumes that the data or measurements are normally distributed. However, this assumption may not be true in some processes. In this paper, we use the Burr distribution, which is employed to represent various non-normal distributions, to determine the appropriate control limits or sample size for the acceptance control chart under non-normality. Some numerical examples are given for illustration. From the presented examples, ignoring the effect of non-normality in the data leads to a higher type I or type II error probability.