关于多元控制图的若干问题研究

发布时间：2018-05-08 15:08

本文选题：多元统计过程控制 + Bootstrap　；参考：《华东师范大学》2016年博士论文

【摘要】：在多元统计过程控制中,当质量特性是连续性随机变量时,已有的控制图大多是基于正态分布建立的。而在实际中,正态性假定经常不成立,在这种情况下,基于正态分布建立的这些控制图的表现将会受到严重影响,因此需要发展一些非参或者稳健的多元控制图。当质量特性是属性变量时,其概率分布经常由列联表来刻画。传统的监控列联表数据的方法都是基于“格子数小样本量大”的情形。随着属性变量个数的增加,列联表中的格子数迅速增加,导致格子中的数目非常小或者为零,也就是所谓的稀疏列联表。在这种情况下,传统的方法已不再能使用,有必要发展一些新方法来监控稀疏列联表数据。当多元过程分布未知而仅有一些可利用的受控数据时,第二章把传统的多元累积和(MCUSUM)控制图的常数控制限扩展成一系列动态控制限,这些控制限可以由sprint长度给定下统计量的条件bootstrap分布确定。同传统的MCUSUM控制图相比,这种具有动态控制限的新控制图表现得更好。然而,它的计算比较繁琐,于是进一步我们使用sprint长度的连续函数来作为它的控制限,发展了一个更为便利的控制图。当过程均值漂移仅仅发生在很少的几个分量上时,第三章把传统的多元LASSO控制图扩展到一个稳健的版本中。新控制图在椭球方向分布族内是不依赖于分布的,这种不依赖于分布性指的是对椭圆方向分布族内的任何连续分布,其控制限是相同的,因此控制限可以由多元标准正态分布确定。模拟结果显示新提出的控制图对监控厚尾分布和偏斜分布中的稀疏漂移是非常有效的。当过程协方差漂移仅仅发生在一些元素上时,第四章把空间符号协方差阵和最大范数应用到指数加权移动平均(EWMA)方案中,构造了一个监控协方差阵的稳健控制图。性质研究表明新图在椭球方向分布族内是不依赖于分布的。对比研究表明新方法对监控厚尾分布下的稀疏漂移更有效,对监控偏斜分布下的稀疏漂移更稳健。在稀疏列联表下,第五章首先提出一个两阶段的group lasso方法对高维log-linear模型进行模型选择和参数估计,进而获得受控状态下的概率分布。然后基于一个修正的Pearson χ2统计量,提出一个新的EWMA控制图。与传统的基于Pearson χ2检验和似然比检验统计量的控制图相比,新控制图对模型系数的各种漂移是有效的,尤其在中小漂移下。
[Abstract]:In multivariate statistical process control, when the quality characteristic is a continuous random variable, most of the existing control charts are based on normal distribution. In practice, the assumption of normality is often not true, in which case, the performance of these control charts based on normal distribution will be seriously affected, so it is necessary to develop some non-parametric or robust multivariate control charts. When the quality property is an attribute variable, its probability distribution is often described by the column table. The traditional method of monitoring column data is based on the case of large sample size. With the increase of the number of attribute variables, the number of lattice in the column table increases rapidly, resulting in the number of the lattice is very small or zero, that is, the so-called sparse column table. In this case, the traditional method can no longer be used, it is necessary to develop some new methods to monitor the sparse column table data. When the distribution of multivariate processes is unknown and only some available controlled data are available, the constant control limits of the traditional multivariate cumulant and MCUSUM control charts are extended to a series of dynamic control limits in Chapter 2. These control limits can be determined by the conditional bootstrap distribution of the statistics given by the sprint length. Compared with the traditional MCUSUM control chart, the new control chart with dynamic control limit performs better. However, its calculation is rather cumbersome, so we further use the continuous function of sprint length as its control limit, and develop a more convenient control graph. When the mean shift of the process occurs on only a few components, the traditional multivariate LASSO control graph is extended to a robust version in Chapter 3. The new control graph does not depend on the distribution in the ellipsoidal distribution family, which means any continuous distribution in the ellipsoidal distribution family, and the control limit is the same. Therefore, the control limit can be determined by multivariate standard normal distribution. The simulation results show that the proposed control chart is very effective for monitoring the sparse drift in the thick tail distribution and skew distribution. When the process covariance drift occurs only on some elements, the fourth chapter applies the spatial symbol covariance matrix and the maximum norm to the exponential weighted moving average (EWMA) scheme, and constructs a robust control chart of the monitoring covariance matrix. It is shown that the new graph is not dependent on distribution in the ellipsoidal distribution family. The comparative study shows that the new method is more effective to monitor the sparse drift under the thick tail distribution and more robust to the sparse drift under the monitoring skew distribution. In the fifth chapter, a two-stage group lasso method is proposed to select the model and estimate the parameters of the high-dimensional log-linear model in the sparse list, and then the probability distribution in the controlled state is obtained. Then, based on a modified Pearson 蠂 2 statistic, a new EWMA control graph is proposed. Compared with the traditional control chart based on Pearson 蠂 2 test and likelihood ratio test, the new control chart is effective for all kinds of drift of model coefficients, especially for small drift.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：C8;TB114.2

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