统计过程监控中灵活而稳健的控制图
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摘要
在统计质量控制中,有两种类型的质量特性。可以数值测量的质量特性被称为变量。很多不能数值测量的质量特性却可以被分成很多类。这种类型的质量特性称为属性。本文对于有效的统计过程控制提出了灵活而稳健的控制图。计数数据可被用于记录在工业应用中不合格品的数量或者在卫生场所的某种疾病的发病率。泊松分布是用来刻画计数信息的常见分布,而关于计数数据的控制图也由此被建立起来。泊松分布是建立在隐含的等分散(均值和方差相等)假设上的。而在实际中,这个假设几乎不被满足,基于泊松分布的控制图对于过度分散(均值比方差小)或者不足分散(均值比方差大)的数据非常敏感。因此,我们需要能同时应用于不足分散和过度分散计数数据的灵活的控制图。本文提出了可以添加在统计过程控制(SPC)的工具包的单变量和多变量灵活的控制图结构。变量控制图一般是基于底层过程服从正态分布的假设。但是,这个假设条件在实际中也是很少被满足的。所以,我们需要可以用来同时有效的监控正态过程和非正态过程参数的控制图。本文提出了关于各种非正态分布的基尼图(一种过程变异图)的稳健性评估和设计。被提出的图结构设计旨在有效地监控分散计数数据集属性和提供监控变量的稳健过程。
Sellers (2012)提出了一种有效的和推广了的COM-泊松图来监控计数数据。她只是给出了3-σ区间。在本文中,我们推广了她的工作,得到了COM-泊松图的精确k-σ和真正的概率极限,用不同的度量方法对结果进行了分析。本文提出了基于COM-泊松分布的推广的灵活的几何指数加权移动平均(EWMA)控制图。三种累积总和(CUSUM)控制图也已被提出以更有效地监控分散计数数据小的变化。本文也对用来有效监控分散计数数据的灵活的多元常规型控制图进行了设计和研究学习。对于多种非正态过程的基尼图在本文中也做了研究。最后,本文做了沿X图的非正态分布的基尼图设计及提供了必要的系数和四分位点。所提出方法的表现情况会用一些有用的度量来评估,包括势函数(1-β),误报率,平均运行长度(ARL),中值运行长度(MDRL),运行长度分布的标准偏差(SDRL),信号平均数(ANOS).我们通过广泛的蒙特卡罗模拟和马尔可夫链方法研究和比较了所提出的不同控制图的表现情况。同时,我们在本文中也添加了一些实际的和模拟的数据集来突出所提出的方法在实际中的应用。另外,本文对于监控分散计数数据的控制图有一个全面的回顾和展望。
灵活的设计结构结果表明所有提出的图结构对于过度或不足分散计数数据是灵活的而且现有的属性图可作为推广后图结构的特殊情况。提出的灵活图结构比Sellers (2012)中的图在检测位移中的缺陷的平均数量方面更有效。而且,这些结构很容易实现,并提供了有关的过程控制状态的真实图像。稳健设计图结构显示了基尼图比其他已有的变量图对于非正态过程更稳健。基尼图在检测过程变异性转变方面比R和S图更强大有力。本文的结果对于理论工作者和实际工作者在设计属性图来监控分散计数数据和稳健变量控制图来监控变量数据方面都非常有帮助。
In statistical quality control, there are two types of quality characteristics. The quality characteristics that can be measured numerically are known as vari-ables. Many quality characteristics cannot be measured numerically but can be classified into various categories. Quality characteristics of this type are called attributes. This thesis proposed flexible and robust control charts for efficient statistical process monitoring. Count data may be used to measure the number of defective items in industrial applications or the incidence of a certain disease at a health facility. The Poisson distribution is a popular distribution used to describe count information, from which control charts involving count data have been es-tablished. The Poisson distribution is based on the underlying equi-dispersion (mean and variance are equal) assumption. In the reality, this assumption is seldom fulfill and the control charts based on the Poisson distribution are very sensitive to over-dispersed (mean is less than variance) or under-dispersed (mean is greater than variance) data. Therefore, flexible control charts are needed that can be used for under-dispersed or over-dispersed count data. This thesis con-tributes some univariate and a multivariate flexible control charting structures to be used as add-in for Statistical Process Control (SPC) toolkit. Control charts for variables are generally based on the assumption that the underlying process follow normal parent distribution. However, this assumption is also seldom ful-fill in the reality. Therefore, a robust control chart is needed that can be used to monitor the process parameter more efficiently for normal as well as non-normal processes. The robustness study and designing of Gini-Chart (a process variability chart) for various non-normal distributions is proposed in this thesis. The proposed charting structures are designed to monitor dispersed count data sets more efficiently in attributes and to provide a robust process monitoring in variables.
Sellers (2012) proposed a flexible and generalized COM-Poisson chart for monitoring count data. She developed the3-sigma limits only. In this thesis, we extended her work and developed the exact κ-sigma and the true probabil-ity limits of the COM-Poisson charts, and analyzed using different performance measures. A generalized and flexible Exponentially Weighted Moving Average (EWMA) control chart based on the COM-Poisson distribution has been pro-posed. Three kinds of cumulative sum (CUSUM) control charts have been also proposed to monitor small shifts in dispersed count data more efficiently. A flexible multivariate shewhart type control chart, for efficient monitoring of dis-persed count data is also designed and studied in this work. The robustness of the Gini-chart, a newly proposed variability chart, has been studied for various non-normal processes. Finally, the designing of the Gini-chart for non-normal distribution along the X chart is made and the necessary coefficients and quar-tile points are provided. The performance ability of the proposals is evaluated in terms of some useful measures including power function (1-β), False alarm rate, average run length (ARL), median run length (MDRL), standard deviation of run length (SDRL) distribution, average number of signals (ANOS). We have investigated and compared the performance of different proposed control charts using extensive Monte Carlo simulations and Markov chain approach. We have also included some real and simulated data sets in order to highlight, the practi-cal application of the proposals cover in this thesis. A comprehensive review and perspective of the control charts to monitor dispersed count data is also provided in this thesis.
The results of flexible design structures indicate that all the proposed chart-ing structures are flexible to under-or over-dispersed count data and generalize to the existing attributes charts as special cases. The proposed flexible charting structures are more efficient than the Sellers (2012) chart in detecting shifts in the average number of defects. Also, these structures are easy to implement and provide the real pictures about the state of process control. The results of the robust designed chart show that the Gini chart is very robust to non-normal processes than other existing variability charts. The Gini chart is more powerful than R and S charts in detecting the shift in the process variability. The results of this thesis are very helpful to the researchers and practioners in designing the attributes control chart to monitor dispersed count data and a robust variability control chart to monitor variable data.
Sellers (2012)提出了一种有效的和推广了的COM-泊松图来监控计数数据。她只是给出了3-σ区间。在本文中,我们推广了她的工作,得到了COM-泊松图的精确k-σ和真正的概率极限,用不同的度量方法对结果进行了分析。本文提出了基于COM-泊松分布的推广的灵活的几何指数加权移动平均(EWMA)控制图。三种累积总和(CUSUM)控制图也已被提出以更有效地监控分散计数数据小的变化。本文也对用来有效监控分散计数数据的灵活的多元常规型控制图进行了设计和研究学习。对于多种非正态过程的基尼图在本文中也做了研究。最后,本文做了沿X图的非正态分布的基尼图设计及提供了必要的系数和四分位点。所提出方法的表现情况会用一些有用的度量来评估,包括势函数(1-β),误报率,平均运行长度(ARL),中值运行长度(MDRL),运行长度分布的标准偏差(SDRL),信号平均数(ANOS).我们通过广泛的蒙特卡罗模拟和马尔可夫链方法研究和比较了所提出的不同控制图的表现情况。同时,我们在本文中也添加了一些实际的和模拟的数据集来突出所提出的方法在实际中的应用。另外,本文对于监控分散计数数据的控制图有一个全面的回顾和展望。
灵活的设计结构结果表明所有提出的图结构对于过度或不足分散计数数据是灵活的而且现有的属性图可作为推广后图结构的特殊情况。提出的灵活图结构比Sellers (2012)中的图在检测位移中的缺陷的平均数量方面更有效。而且,这些结构很容易实现,并提供了有关的过程控制状态的真实图像。稳健设计图结构显示了基尼图比其他已有的变量图对于非正态过程更稳健。基尼图在检测过程变异性转变方面比R和S图更强大有力。本文的结果对于理论工作者和实际工作者在设计属性图来监控分散计数数据和稳健变量控制图来监控变量数据方面都非常有帮助。
In statistical quality control, there are two types of quality characteristics. The quality characteristics that can be measured numerically are known as vari-ables. Many quality characteristics cannot be measured numerically but can be classified into various categories. Quality characteristics of this type are called attributes. This thesis proposed flexible and robust control charts for efficient statistical process monitoring. Count data may be used to measure the number of defective items in industrial applications or the incidence of a certain disease at a health facility. The Poisson distribution is a popular distribution used to describe count information, from which control charts involving count data have been es-tablished. The Poisson distribution is based on the underlying equi-dispersion (mean and variance are equal) assumption. In the reality, this assumption is seldom fulfill and the control charts based on the Poisson distribution are very sensitive to over-dispersed (mean is less than variance) or under-dispersed (mean is greater than variance) data. Therefore, flexible control charts are needed that can be used for under-dispersed or over-dispersed count data. This thesis con-tributes some univariate and a multivariate flexible control charting structures to be used as add-in for Statistical Process Control (SPC) toolkit. Control charts for variables are generally based on the assumption that the underlying process follow normal parent distribution. However, this assumption is also seldom ful-fill in the reality. Therefore, a robust control chart is needed that can be used to monitor the process parameter more efficiently for normal as well as non-normal processes. The robustness study and designing of Gini-Chart (a process variability chart) for various non-normal distributions is proposed in this thesis. The proposed charting structures are designed to monitor dispersed count data sets more efficiently in attributes and to provide a robust process monitoring in variables.
Sellers (2012) proposed a flexible and generalized COM-Poisson chart for monitoring count data. She developed the3-sigma limits only. In this thesis, we extended her work and developed the exact κ-sigma and the true probabil-ity limits of the COM-Poisson charts, and analyzed using different performance measures. A generalized and flexible Exponentially Weighted Moving Average (EWMA) control chart based on the COM-Poisson distribution has been pro-posed. Three kinds of cumulative sum (CUSUM) control charts have been also proposed to monitor small shifts in dispersed count data more efficiently. A flexible multivariate shewhart type control chart, for efficient monitoring of dis-persed count data is also designed and studied in this work. The robustness of the Gini-chart, a newly proposed variability chart, has been studied for various non-normal processes. Finally, the designing of the Gini-chart for non-normal distribution along the X chart is made and the necessary coefficients and quar-tile points are provided. The performance ability of the proposals is evaluated in terms of some useful measures including power function (1-β), False alarm rate, average run length (ARL), median run length (MDRL), standard deviation of run length (SDRL) distribution, average number of signals (ANOS). We have investigated and compared the performance of different proposed control charts using extensive Monte Carlo simulations and Markov chain approach. We have also included some real and simulated data sets in order to highlight, the practi-cal application of the proposals cover in this thesis. A comprehensive review and perspective of the control charts to monitor dispersed count data is also provided in this thesis.
The results of flexible design structures indicate that all the proposed chart-ing structures are flexible to under-or over-dispersed count data and generalize to the existing attributes charts as special cases. The proposed flexible charting structures are more efficient than the Sellers (2012) chart in detecting shifts in the average number of defects. Also, these structures are easy to implement and provide the real pictures about the state of process control. The results of the robust designed chart show that the Gini chart is very robust to non-normal processes than other existing variability charts. The Gini chart is more powerful than R and S charts in detecting the shift in the process variability. The results of this thesis are very helpful to the researchers and practioners in designing the attributes control chart to monitor dispersed count data and a robust variability control chart to monitor variable data.
引文
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