基于似然比检验的若干控制图
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摘要
随着经济全球化的深入,产品质量越来越受到重视。一个产品如果没有过硬的质量,就很难在激烈竞争的市场中立足。于是,关于提高和控制产品质量的技术和方法就得到了广泛的应用和发展。统计过程控制(Statistical Process Control,简称SPC)则在质量管理中扮演着重要角色,它能有效地检测生产过程是否可控,并能解决生产过程中出现的某些问题,从而提高产品质量。
质量控制发端于19世纪20年代,首张控制图由Shewhart博士于1924年提出,之后在二战中的英国和美国得到了广泛的研究应用和发展,但是在战后的和平时期却失去了它的重要性。然而,西方把SPC技术传到了日本,特别是W.E.Deming对50年代日本经济的发展起了重要作用。SPC在日本工业得到了广泛的应用和发展。事实证明,SPC不仅省钱还可以吸引更多的消费者,这一点可以从日本的工业产品能迅速的占领国际市场得到验证。可以说SPC为日本工业产品迅速占领国际市场起到了巨大的推动作用。目前,随着我国市场经济体制的完善以及全球经济的一体化,我国经济已融入了世界经济,如果我国工业产品不在质量上下功夫,就无法使我国工业产品走出国门,因此,我们必须要加强产品质量的监督与管理,而控制图在这方面扮演着重要的角色。
传统的质量控制图有三大类,其一是Shewhart控制图,它是提出最早、应用最多的对生产过程均值或标准差进行监测的方法。它的优点在于能够迅速的检测大的漂移,但是当过程漂移较小时,它的检验效果并不是很好。其原因在于它仅利用了当前样本的信息,而忽略了过去样本中的有用信息。为了克服这个缺陷,很多人提出了改进的Shewhart型控制图,如带有运行准则(Runs Rules)的Shewhart型控制图:联合(combined)Shewhart-CUSUM控制图和联合Shewhart-EWMA控制图,这些控制图都提高了Shewhart控制图检测小漂移的灵敏度。除此之外,另外两类常用的控制图是累计和控制图(Cumulative Sum,简记为CUSUM),由Page(1954)基于似然比导出,其二是指数加权移动平均控制图(Exponentially Weighted MovingAverage,简记为EWMA),由Roberts(1959)提出,这两个控制图的共同特点是它们不但利用当前样本的信息,而且还利用了前面所有样本的信息,从而对生产过程是否失控进行统计推断。理论和经验已经证明它们在检测小漂移的时候,都有很好的表现。
经过70多年的发展,SPC的研究范围已经非常广泛,新的控制图和新的理论不断涌现,并取得了相当丰富的成果。这些研究成果大致可分为如下几个方面:
1.独立数据的第一、第二阶段控制图的研究;
2.相关数据的第一、第二阶段控制图的研究;
3.多元数据控制图的研究;
4.基于非参数统计方法的控制图的研究;
5.动态控制图的研究;
6.控制图的经济设计研究;
7.从大样本角度研究某些常用控制图的优良性质。
最近,Woodall W.H.于2006年发表了一篇关于公共卫生保健(Public-HealthSurveillance)的文章,他强调可以把应用在工业生产中的SPC方法应用在监控卫生保健上,相反,也可以把监控卫生保健的一些方法用到工业生产巾来。这就为SPC的应用和研究领域提供了更为广阔的空间。
另外,关于Linear Profile控制图的研究与应用受到越来越多的关注,这是一个比较新的研究领域,可关于它的研究已日益加深。近几年来的主要工作集中在控制图方法的有效性、更广义曲线的研究以及偏离假定条件的影响研究等等,很多有意义的话题值得进一步研究。
本文的结构安排如下:
第一章为引言部分,简要介绍质量控制图的发展历史、研究方法和当前研究的一些热点问题,以及论文的结构安排和论文的贡献。
第二章,基于似然比检验的思想,我们提出了可用于同时监控过程均值和方差的综合控制图。很多学者都讨论过用一个综合图来同时检测这两个参数的漂移问题,但这些方法都有一定的缺陷。有的含有多个设计参数,而控制图的表现与这些参数的选取有很大关系,设计起来不易操作。有的不能够检测方差减小,而及时检测方差减小有时也是很必要的,因为这意味着产品质量的提高,技术的改进。另外,大多数的控制图都不能解决样本容量(Sample Size)为1(称为individual)的情况。本文中,我们提出的新的控制图与已有的控制图相比,新图不但可以有效的减少控制图的设计参数,显著减少工作量,还可以解决上面提到的方差减小与individual的情形,其检测效果就平均链长(Average Run Length,简写为ARL)而言,也具有很好的表现。
第三章,基于第二章的工作,我们考虑动态控制图的应用。这方面的研究成果可分为五类:VSI(Variable Sampling Interval);VSS(Variable Sample Size);VSSI(Variable Sample Size and Sampling Interval);VP(Variable Parameter):VSSIFT(Variable Sample Size and Sampling Interval at Fixed Time)。动态控制图的特点是其样本容量或抽样区间的大小与前一组样本的位置有关。本文中我们考虑VSI和VSS控制图。比较结果显示,经过这样的改进之后,其调整的平均报警时间(Adjust Average Time to Signal,简写为AATS)大幅度减少,从而加快了控制图的报警速度。
第四章,我们讨论自启动控制图(Self-Starting Chart,简写为SS-Chart)。在第二章中我们提出的控制图是假设过程参数在受控时是已知的,然而在很多实际问题当中,这些参数是未知的。一些学者建议用20-30个样本容量为4或5的样本去估计这些参数。事实表明,当样本数很小的时候,估计参数的这种变动性又会带来很多新的问题。一方面会产生受控ARL的严重的偏,另一方面又会降低图的灵敏度。另外一个方法就是增加样本容量。研究表明,至少需要200-300个样本才会产生与参数已知时相近的效果,而这在很多情况下,特别是在short-run下,是不可行的,因为这既费时又增加成本。因此,我们提出了自启动控制图。这就避免了由于估计参数而带来的一系列问题。比较结果显示,其检测效果同样具有很好的表现。
第五章,我们把研究范围推广到多变量情形。多变量控制图的研究要比单变量控制图的研究更为复杂。有的人认为,对有多个质量特征要求的质量管理问题,可以分别独立的进行观察和控制,想法可以理解,但这种做法却不可行,因为产品的质量特性在某种程度上存在着联系,一味的割断这种关系会产生处理中的硬伤,更何况多元统计理论研究表明,即使各个量相互之间保持着独立,对于同样的显著性水平,分别控制的信度并不等于同时联合控制的信度。在本文中,我们把似然比检验的思想推广到多元的情形,提出了用于同时监控过程均值向量和协方差阵的综合控制图。Hawkins and Masoudou-Tchao(2008)也提出过类似的用于控制协方差阵的多元控制图,有些情况下其ARL是有偏的,而我们提出的这种综合图既能检测协方差阵的漂移,又能检测均值向量的漂移,而且设计简单,操作方便,就ARL来讲,不再有偏,与其它的多个多元控制图相比,都具有很大的优势。
第六章,关于Linear Profile问题的研究与应用,近年来一直是一个热点问题。产品的质量有时是通过一个反应变量(response variable)与一个或多个解释变量(explanatory variable)之间的某种函数关系来描述的,很多情况下这种函数关系可以用简单的线性回归模型来表达。但是在有些情况下,则需要更为复杂的模型来刻划。关于Linear Profile控制图是Kang and Albin于2000年基于某些实际问题首先提出并加以研究的。到目前为止,所有关于Linear Profile控制图的假设模型仅有截距和斜率项,采用的方法多为对估计后的统计量做ShewhartX-bar或EWMA控制图;另外有人把似然比检验及Change-Point方法用于LinearProfile控制图的研究;为了避免区分一阶段与二阶段,也有人考虑过自启动的方法。在本章中我们把似然比的思想应用到Linear Profile的研究当中。比较结果显示,我们提出的控制图较其它控制图在检测方差的漂移上,无论是向上还是向下,都具有很大的优势,这在实际工业生产中是很重要的。此外,在其它参数的检测上也同样具有很好的表现。另外,我们把VSI的思想引入到此控制图中,大大减少了报警所需的平均时间。
最后对本文的主要结果和关键技术方法进行了总结,并提出了几个值得进一步研究的问题。
本文主要是基于似然比检验的思想,提出了某些控制图,并给出了这些图的设计,其中某些控制图的ARL可以用马氏链方法计算,但是对于有些控制图而言操作起来很是不便,因此我们用Monte Carlo随机模拟。
本文的创新主要体现在有以下几点:
1.基于似然比检验思想,本文提出了可用于同时检测过程均值与方差漂移的综合控制图,它不但可以检测方差增大还可以检测方差减小,此外还可以处理单个样本的情形。此图只含有两个参数,设计起来简单方便,另外我们进一步给出了在不同受控ARL下的控制图的设计方案。最后,我们推导了计算ARL的Markov-chain算法。
2.基于动态控制图在报警时间的高效性,我们提出了可变抽样时间与可变样本容量的动态控制图。给出了基于控制线的调整方案,对比了在相同抽样率的条件下,动态图和静态图在报警时间上的不同,并且与已知动态图进行比较,最后给出了ARL的Markov-chain算法。
3.本文提出了可用于同时检测过程均值与方差漂移的自启动控制图,这使得我们在使用控制图的初始阶段,没有必要收集大量的样本去估计参数。此外,我们给出了变点的似然比检验诊断方法,此方法能有效判定出是哪个参数或哪些参数发生了变化以及变点发生的位置。
4.本文提出了可用于同时检测多元均值向量与协方差阵漂移的综合控制图,考察了此图在不同维数以及不同漂移方式下的ARL表现,最后给出了在不同受控ARL及不同维数下控制图的设计参数。
5.基于似然比检验的思想,本文提出了用于控制简单线性模型(Linear Profile)的综合控制图。此图可用来同时检测三个参数的漂移情况,我们可以观测出哪个参数或哪些参数发生了漂移,而且在方差发生漂移的情况下,可以有效的判定方差变化的方向,同时给出了相应的动态控制图的设计方案。
With the development of economic globalization, more and more emphasis is placed upon product quality. A product without excellent quality is difficult to have its foothold in the highly competitive market. Thus, techniques and methods to improve the quality of products have been extensively used and developed. Statistical Process Control (SPC for short) is now recognized as playing a very important role in modern industry. It can effectively detect the variability of the production process, resolve certain problems and control product quality.
SPC was developed in 1920s and Walter A. Shewhart of Bell Telephone Laboratories developed the first sketch of a modern control chart in May 1924. SPC was used extensively in World War II both in the UK and in the USA, but lost its importance as industries converted to peacetime production. However, people in the West taught it to the Japanese, and W. E. Deming in particular made a big impact in Japan in the 1950s. Japanese industry applied SPC widely and proved that SPC saves money and attracts customers which can be verified by Japan's rapid industrial occupation of international market. At present, with the improvement of market economy system in China and the integration of the global economy, China's economy has been integrated into the world economy. If the quality of industrial products in China isn't paid much more attention, industrial products will not be exported abroad. Therefore, we must enhance product quality control and management.
In general, there are three kinds of control charts. One is the well-known Shewhart chart which is the earliest used to monitor the process mean and variance. The advantage of this chart is that it is more efficient in detecting large shifts of process than others, but as to detect small shifts, it is not the case. The reason is that it uses only the information of the current sample but ignores the former samples. In order to overcome this disadvantage, and increases the sensi- tivity of Shewhart chart in detecting the small shift, Woodall and Champ(1987) gave a scheme of Shewhart chart with supplementary runs rules and the combined Shewhart-CUSUM charts and combined Shewhart-EWMA charts were proposed by Lucas(1982), Klein(1996), respectively. In addition, Page (1954) proposed the famous cumulative sum (CUSUM chart for short) chart which based on the Wald test. Another important chart, i.e., exponential weighted moving average (EWMA for short) chart, was first proposed by Roberts (1959). Both CUSUM and EWMA charts have the same character that they will use not only the information of the current sample but also will use the former samples. It is possible to infer whether the process is in control from these two charts based on all the samples that have been collected over time. These two kinds of charts are very effective in detecting small shifts of the process.
After nearly seventy years development, the study range of SPC is therefore very broad and the new methods for the construction of the control chart are teeming out. The research results can be summarized briefly as follows:
1. the study of Phase I and phase II control chart for independent data;
2. the study of Phase I and phase II control chart for autocorrelated data;
3. the study of multivariate control chart;
4. the study of nonparametric control chart;
5. the study of dynamic control chart;
6. the economic design of control chart;
7. the fine property of some control charts from the view of large sample theory.
Recently, Woodall W. H. published a crucial paper in 2006 about the monitoring of public health-care. There are many applications of control charts in health-care monitoring and in public-health surveillance. He introduced some applications to industrial practitioners and discussed some of the ideas that arise that may be applicable in industrial monitoring. He considered the advantages and disadvantages of the charting methods proposed in the health-care and public-health areas. Some additional contributions in the industrial statistical process control literature relevant to this area are given in this paper. So there are many application and research opportunities available in the use of control charts for health-related monitoring.
Monitoring linear profiles and its applications have been drawn more and more attention. This is a relatively new area of research, but it is growing rapidly. Profile monitoring is very useful in an increasing number of practical applications. Much of the work in the past few years has focused on the use of more effective charting methods, the study of more general shapes of profiles, and the study of the effects of variations of assumptions. There are many promising research topics yet to be pursued given the broad range of profile shapes and possible models.
The structure of this dissertation is demonstrated as follows:
In Chapter 1, we introduce the outline of the development of SPC, including the study methods and the hot issues in current SPC research. Also, the structure of this dissertation and its attributions are listed.
In Chapter 2, we propose a new single chart which integrates the EWMA procedure with the likelihood ratio test statistics for jointly monitoring both the process mean and variance. Many authors have studied this problem. However, most of the work has their drawbacks. Some charts involve more parameters need to be determined and the performance of these control charts significantly depends on these parameters. So, its design is not easy. Others may not cope well with the monitoring of the decrease in the variance. This is a very important case in practice, because the variance decrease means that the quality of the product improved. In addition, most of the charts may not be appropriately used in the case that only an individual observation is available at one sampling point, which is quite common in many industrial processes. Our new chart has only tow parameters to be determined and it can be easily designed and constructed, and it has good average run length performance. It provides a quite satisfactory performance in various cases. In addition, it can solve the problems mentioned above effectively.
In Chapter 3, based on the work of Chapter 2, we develop an adaptive chart. The property of the adaptive control chart is that the sampling interval and sample size depends on what is being observed from the prior data. The results can be summarized as: VSI (Variable Sampling Interval); VSS (Variable Sample Size); VSSI (Variable Sample Size and Sampling Interval); VP (Variable Parameter); VSSIFT (Variable Sample Size and Sampling Interval at Fixed Time). The results show that the AATS does decrease significantly and the efficiency of the chart improved.
In Chapter 4, we discuss a self-starting control chart. In Chapter 2, we assume that the parameters of the process are known a prior. However, in most cases, these parameters are not known exactly a prior. Some authors have recommended using 20-30 samples with four to five observations each to estimate the process parameters for traditional control charts. They conclude that, when the number of reference samples is small, control charts with estimated parameters produce a large bias in the IC ARL and reduce the sensitivity of the chart in detecting process changes as measured by the out-of-control (OC) ARL. The obvious solution to this problem is to increase the Phase I sample size to reduce the variability in the sampling distribution of the estimates, especially the process variance. In fact that we need at least 200-300 samples to estimate the process parameters in order to get the similar performance as known parameters. In most cases, however, it may not be possible to wait for the accumulation of sufficiently large subgroups because the users usually want to monitor the process at the start-up stages and such huge samples are costly. Our new chart can solve the problems mentioned above, that is to say it is not necessary to assemble a large number of reference samples before the control scheme begins. The results show that the new chart has good performance.
In Chapter 5, we extend our method to the multivariate case. Multivariate control chart is more complicate than the univariate one. Someone think that we can operate several control charts for the process parameters simultaneously and separately. Maybe this idea is accessible, but it is not feasible, because these variables are correlated to some extent. Multivariate theory have shown that even the quality characteristics are mutually independent, for the same significance level, the separated reliability for each variable is not equal to the combined reliability. In this Chapter, we propose a new single control chart which integrates the exponentially weighted moving average (EWMA) procedure with the likelihood ratio test for jointly monitoring both the multivariate process mean vector and the covariance matrix. Hawkins and Masoudou-Tchao (2008) also considered such a method to monitor the covariance matrix, but there are some differences between ours. For their chart, it turned out to be ARL-biased under some circumstance. However, for our chart, it is not ARL-biased any more. From the comparison with other charts, we can see that our new chart can be easily designed and performs better than others for the case in which the quality characteristics are bivariate normal random variables.
In Chapter 6, we discuss the study and the applications of the monitoring of linear profiles. In many applications the quality of a process or product is best characterized and summarized by a functional relationship between a response variable and one or more explanatory variables. In some calibration applications, the profile can be represented adequately by a simple linear regression model, while in other applications more complicated model are needed. The monitoring of linear profile was firstly proposed and studied by Kand and Albin in 2000. By far, most of the models of linear profile contain only the intercept and the slope, and the monitoring methods are Shewhart X-bar control chart and the EWMA chart based on the estimation of the process parameters. In addition, the likelihood ratio test and the change-point methods were proposed for detecting changes in the parameters of a simple linear regression model. A self-starting method was proposed which avoids the distinction between Phase I and Phase II. In this Chapter, we propose a likelihood method which integrates the EWMA procedure. This chart performs better than others in detecting the process variance, including the increase and decrease which is very important in practical manufacturing production. Apart from this, it can also detect other parameters changes very well. At last, we apply the VSI procedure to our chart in order to reduce the time of detecting. It can be seen that the effect is remarkable.
At last in Chapter 7, we summarize the main results and the technical approaches, and suggest some promising research topics yet to be pursued.
In this dissertation, we proposed several creative control charts based on the likelihood ratio test statistics which integrates the EWMA procedure. The resulting ARL values can be obtained by Markov chain approximation for some charts, but others are obtained by Monte Carlo simulations.
The original and creative ideas are represented as follows:
1. We propose a creative single chart which integrates the EWMA procedure with the likelihood ratio test statistics for jointly monitoring both the process mean and variance, which can be easily designed. The new chart can cope well with the monitoring of the decrease and increase in the variance. Also, it can cope with the individual case. Further, we provide the design of the chart under different IC ARL. At last, we provide details on the Markov-chain approximation of ARL of the new chart.
2. Based on the efficiency of dynamic control chart, we address a VSI and VSS SLR chart based on the adjustment of control limits and the plotting statistics. We compare the difference between VSI and FSI control charts under the same sampling rate. Further, we provide details on the Markov-chain approximation of AATS of this chart.
3. We propose a self-starting control chart when the process parameters are not exactly known a prior, , and so there is no need to assemble a large number of reference samples before the control scheme begins. We propose a diagnostic aids based on the change-point method which can locate the change-point in the process and isolate the type of parameters change.
4. We propose a new single control chart for jointly monitoring both the multivariate process mean vector and the covariance matrix. The control limits are given and its performance is studied under different dimension, IC ARL and types of shifts. The design parameters are also given.
5. We propose a creative control chart for monitoring the linear profile based on the likelihood ratio test statistics. It can isolate the type of parameters change in a profile using some statistics in the charting statistics, especially the direction of the shift in the variance. We also considered the corresponding design of the VSI control chart for linear profile.
质量控制发端于19世纪20年代,首张控制图由Shewhart博士于1924年提出,之后在二战中的英国和美国得到了广泛的研究应用和发展,但是在战后的和平时期却失去了它的重要性。然而,西方把SPC技术传到了日本,特别是W.E.Deming对50年代日本经济的发展起了重要作用。SPC在日本工业得到了广泛的应用和发展。事实证明,SPC不仅省钱还可以吸引更多的消费者,这一点可以从日本的工业产品能迅速的占领国际市场得到验证。可以说SPC为日本工业产品迅速占领国际市场起到了巨大的推动作用。目前,随着我国市场经济体制的完善以及全球经济的一体化,我国经济已融入了世界经济,如果我国工业产品不在质量上下功夫,就无法使我国工业产品走出国门,因此,我们必须要加强产品质量的监督与管理,而控制图在这方面扮演着重要的角色。
传统的质量控制图有三大类,其一是Shewhart控制图,它是提出最早、应用最多的对生产过程均值或标准差进行监测的方法。它的优点在于能够迅速的检测大的漂移,但是当过程漂移较小时,它的检验效果并不是很好。其原因在于它仅利用了当前样本的信息,而忽略了过去样本中的有用信息。为了克服这个缺陷,很多人提出了改进的Shewhart型控制图,如带有运行准则(Runs Rules)的Shewhart型控制图:联合(combined)Shewhart-CUSUM控制图和联合Shewhart-EWMA控制图,这些控制图都提高了Shewhart控制图检测小漂移的灵敏度。除此之外,另外两类常用的控制图是累计和控制图(Cumulative Sum,简记为CUSUM),由Page(1954)基于似然比导出,其二是指数加权移动平均控制图(Exponentially Weighted MovingAverage,简记为EWMA),由Roberts(1959)提出,这两个控制图的共同特点是它们不但利用当前样本的信息,而且还利用了前面所有样本的信息,从而对生产过程是否失控进行统计推断。理论和经验已经证明它们在检测小漂移的时候,都有很好的表现。
经过70多年的发展,SPC的研究范围已经非常广泛,新的控制图和新的理论不断涌现,并取得了相当丰富的成果。这些研究成果大致可分为如下几个方面:
1.独立数据的第一、第二阶段控制图的研究;
2.相关数据的第一、第二阶段控制图的研究;
3.多元数据控制图的研究;
4.基于非参数统计方法的控制图的研究;
5.动态控制图的研究;
6.控制图的经济设计研究;
7.从大样本角度研究某些常用控制图的优良性质。
最近,Woodall W.H.于2006年发表了一篇关于公共卫生保健(Public-HealthSurveillance)的文章,他强调可以把应用在工业生产中的SPC方法应用在监控卫生保健上,相反,也可以把监控卫生保健的一些方法用到工业生产巾来。这就为SPC的应用和研究领域提供了更为广阔的空间。
另外,关于Linear Profile控制图的研究与应用受到越来越多的关注,这是一个比较新的研究领域,可关于它的研究已日益加深。近几年来的主要工作集中在控制图方法的有效性、更广义曲线的研究以及偏离假定条件的影响研究等等,很多有意义的话题值得进一步研究。
本文的结构安排如下:
第一章为引言部分,简要介绍质量控制图的发展历史、研究方法和当前研究的一些热点问题,以及论文的结构安排和论文的贡献。
第二章,基于似然比检验的思想,我们提出了可用于同时监控过程均值和方差的综合控制图。很多学者都讨论过用一个综合图来同时检测这两个参数的漂移问题,但这些方法都有一定的缺陷。有的含有多个设计参数,而控制图的表现与这些参数的选取有很大关系,设计起来不易操作。有的不能够检测方差减小,而及时检测方差减小有时也是很必要的,因为这意味着产品质量的提高,技术的改进。另外,大多数的控制图都不能解决样本容量(Sample Size)为1(称为individual)的情况。本文中,我们提出的新的控制图与已有的控制图相比,新图不但可以有效的减少控制图的设计参数,显著减少工作量,还可以解决上面提到的方差减小与individual的情形,其检测效果就平均链长(Average Run Length,简写为ARL)而言,也具有很好的表现。
第三章,基于第二章的工作,我们考虑动态控制图的应用。这方面的研究成果可分为五类:VSI(Variable Sampling Interval);VSS(Variable Sample Size);VSSI(Variable Sample Size and Sampling Interval);VP(Variable Parameter):VSSIFT(Variable Sample Size and Sampling Interval at Fixed Time)。动态控制图的特点是其样本容量或抽样区间的大小与前一组样本的位置有关。本文中我们考虑VSI和VSS控制图。比较结果显示,经过这样的改进之后,其调整的平均报警时间(Adjust Average Time to Signal,简写为AATS)大幅度减少,从而加快了控制图的报警速度。
第四章,我们讨论自启动控制图(Self-Starting Chart,简写为SS-Chart)。在第二章中我们提出的控制图是假设过程参数在受控时是已知的,然而在很多实际问题当中,这些参数是未知的。一些学者建议用20-30个样本容量为4或5的样本去估计这些参数。事实表明,当样本数很小的时候,估计参数的这种变动性又会带来很多新的问题。一方面会产生受控ARL的严重的偏,另一方面又会降低图的灵敏度。另外一个方法就是增加样本容量。研究表明,至少需要200-300个样本才会产生与参数已知时相近的效果,而这在很多情况下,特别是在short-run下,是不可行的,因为这既费时又增加成本。因此,我们提出了自启动控制图。这就避免了由于估计参数而带来的一系列问题。比较结果显示,其检测效果同样具有很好的表现。
第五章,我们把研究范围推广到多变量情形。多变量控制图的研究要比单变量控制图的研究更为复杂。有的人认为,对有多个质量特征要求的质量管理问题,可以分别独立的进行观察和控制,想法可以理解,但这种做法却不可行,因为产品的质量特性在某种程度上存在着联系,一味的割断这种关系会产生处理中的硬伤,更何况多元统计理论研究表明,即使各个量相互之间保持着独立,对于同样的显著性水平,分别控制的信度并不等于同时联合控制的信度。在本文中,我们把似然比检验的思想推广到多元的情形,提出了用于同时监控过程均值向量和协方差阵的综合控制图。Hawkins and Masoudou-Tchao(2008)也提出过类似的用于控制协方差阵的多元控制图,有些情况下其ARL是有偏的,而我们提出的这种综合图既能检测协方差阵的漂移,又能检测均值向量的漂移,而且设计简单,操作方便,就ARL来讲,不再有偏,与其它的多个多元控制图相比,都具有很大的优势。
第六章,关于Linear Profile问题的研究与应用,近年来一直是一个热点问题。产品的质量有时是通过一个反应变量(response variable)与一个或多个解释变量(explanatory variable)之间的某种函数关系来描述的,很多情况下这种函数关系可以用简单的线性回归模型来表达。但是在有些情况下,则需要更为复杂的模型来刻划。关于Linear Profile控制图是Kang and Albin于2000年基于某些实际问题首先提出并加以研究的。到目前为止,所有关于Linear Profile控制图的假设模型仅有截距和斜率项,采用的方法多为对估计后的统计量做ShewhartX-bar或EWMA控制图;另外有人把似然比检验及Change-Point方法用于LinearProfile控制图的研究;为了避免区分一阶段与二阶段,也有人考虑过自启动的方法。在本章中我们把似然比的思想应用到Linear Profile的研究当中。比较结果显示,我们提出的控制图较其它控制图在检测方差的漂移上,无论是向上还是向下,都具有很大的优势,这在实际工业生产中是很重要的。此外,在其它参数的检测上也同样具有很好的表现。另外,我们把VSI的思想引入到此控制图中,大大减少了报警所需的平均时间。
最后对本文的主要结果和关键技术方法进行了总结,并提出了几个值得进一步研究的问题。
本文主要是基于似然比检验的思想,提出了某些控制图,并给出了这些图的设计,其中某些控制图的ARL可以用马氏链方法计算,但是对于有些控制图而言操作起来很是不便,因此我们用Monte Carlo随机模拟。
本文的创新主要体现在有以下几点:
1.基于似然比检验思想,本文提出了可用于同时检测过程均值与方差漂移的综合控制图,它不但可以检测方差增大还可以检测方差减小,此外还可以处理单个样本的情形。此图只含有两个参数,设计起来简单方便,另外我们进一步给出了在不同受控ARL下的控制图的设计方案。最后,我们推导了计算ARL的Markov-chain算法。
2.基于动态控制图在报警时间的高效性,我们提出了可变抽样时间与可变样本容量的动态控制图。给出了基于控制线的调整方案,对比了在相同抽样率的条件下,动态图和静态图在报警时间上的不同,并且与已知动态图进行比较,最后给出了ARL的Markov-chain算法。
3.本文提出了可用于同时检测过程均值与方差漂移的自启动控制图,这使得我们在使用控制图的初始阶段,没有必要收集大量的样本去估计参数。此外,我们给出了变点的似然比检验诊断方法,此方法能有效判定出是哪个参数或哪些参数发生了变化以及变点发生的位置。
4.本文提出了可用于同时检测多元均值向量与协方差阵漂移的综合控制图,考察了此图在不同维数以及不同漂移方式下的ARL表现,最后给出了在不同受控ARL及不同维数下控制图的设计参数。
5.基于似然比检验的思想,本文提出了用于控制简单线性模型(Linear Profile)的综合控制图。此图可用来同时检测三个参数的漂移情况,我们可以观测出哪个参数或哪些参数发生了漂移,而且在方差发生漂移的情况下,可以有效的判定方差变化的方向,同时给出了相应的动态控制图的设计方案。
With the development of economic globalization, more and more emphasis is placed upon product quality. A product without excellent quality is difficult to have its foothold in the highly competitive market. Thus, techniques and methods to improve the quality of products have been extensively used and developed. Statistical Process Control (SPC for short) is now recognized as playing a very important role in modern industry. It can effectively detect the variability of the production process, resolve certain problems and control product quality.
SPC was developed in 1920s and Walter A. Shewhart of Bell Telephone Laboratories developed the first sketch of a modern control chart in May 1924. SPC was used extensively in World War II both in the UK and in the USA, but lost its importance as industries converted to peacetime production. However, people in the West taught it to the Japanese, and W. E. Deming in particular made a big impact in Japan in the 1950s. Japanese industry applied SPC widely and proved that SPC saves money and attracts customers which can be verified by Japan's rapid industrial occupation of international market. At present, with the improvement of market economy system in China and the integration of the global economy, China's economy has been integrated into the world economy. If the quality of industrial products in China isn't paid much more attention, industrial products will not be exported abroad. Therefore, we must enhance product quality control and management.
In general, there are three kinds of control charts. One is the well-known Shewhart chart which is the earliest used to monitor the process mean and variance. The advantage of this chart is that it is more efficient in detecting large shifts of process than others, but as to detect small shifts, it is not the case. The reason is that it uses only the information of the current sample but ignores the former samples. In order to overcome this disadvantage, and increases the sensi- tivity of Shewhart chart in detecting the small shift, Woodall and Champ(1987) gave a scheme of Shewhart chart with supplementary runs rules and the combined Shewhart-CUSUM charts and combined Shewhart-EWMA charts were proposed by Lucas(1982), Klein(1996), respectively. In addition, Page (1954) proposed the famous cumulative sum (CUSUM chart for short) chart which based on the Wald test. Another important chart, i.e., exponential weighted moving average (EWMA for short) chart, was first proposed by Roberts (1959). Both CUSUM and EWMA charts have the same character that they will use not only the information of the current sample but also will use the former samples. It is possible to infer whether the process is in control from these two charts based on all the samples that have been collected over time. These two kinds of charts are very effective in detecting small shifts of the process.
After nearly seventy years development, the study range of SPC is therefore very broad and the new methods for the construction of the control chart are teeming out. The research results can be summarized briefly as follows:
1. the study of Phase I and phase II control chart for independent data;
2. the study of Phase I and phase II control chart for autocorrelated data;
3. the study of multivariate control chart;
4. the study of nonparametric control chart;
5. the study of dynamic control chart;
6. the economic design of control chart;
7. the fine property of some control charts from the view of large sample theory.
Recently, Woodall W. H. published a crucial paper in 2006 about the monitoring of public health-care. There are many applications of control charts in health-care monitoring and in public-health surveillance. He introduced some applications to industrial practitioners and discussed some of the ideas that arise that may be applicable in industrial monitoring. He considered the advantages and disadvantages of the charting methods proposed in the health-care and public-health areas. Some additional contributions in the industrial statistical process control literature relevant to this area are given in this paper. So there are many application and research opportunities available in the use of control charts for health-related monitoring.
Monitoring linear profiles and its applications have been drawn more and more attention. This is a relatively new area of research, but it is growing rapidly. Profile monitoring is very useful in an increasing number of practical applications. Much of the work in the past few years has focused on the use of more effective charting methods, the study of more general shapes of profiles, and the study of the effects of variations of assumptions. There are many promising research topics yet to be pursued given the broad range of profile shapes and possible models.
The structure of this dissertation is demonstrated as follows:
In Chapter 1, we introduce the outline of the development of SPC, including the study methods and the hot issues in current SPC research. Also, the structure of this dissertation and its attributions are listed.
In Chapter 2, we propose a new single chart which integrates the EWMA procedure with the likelihood ratio test statistics for jointly monitoring both the process mean and variance. Many authors have studied this problem. However, most of the work has their drawbacks. Some charts involve more parameters need to be determined and the performance of these control charts significantly depends on these parameters. So, its design is not easy. Others may not cope well with the monitoring of the decrease in the variance. This is a very important case in practice, because the variance decrease means that the quality of the product improved. In addition, most of the charts may not be appropriately used in the case that only an individual observation is available at one sampling point, which is quite common in many industrial processes. Our new chart has only tow parameters to be determined and it can be easily designed and constructed, and it has good average run length performance. It provides a quite satisfactory performance in various cases. In addition, it can solve the problems mentioned above effectively.
In Chapter 3, based on the work of Chapter 2, we develop an adaptive chart. The property of the adaptive control chart is that the sampling interval and sample size depends on what is being observed from the prior data. The results can be summarized as: VSI (Variable Sampling Interval); VSS (Variable Sample Size); VSSI (Variable Sample Size and Sampling Interval); VP (Variable Parameter); VSSIFT (Variable Sample Size and Sampling Interval at Fixed Time). The results show that the AATS does decrease significantly and the efficiency of the chart improved.
In Chapter 4, we discuss a self-starting control chart. In Chapter 2, we assume that the parameters of the process are known a prior. However, in most cases, these parameters are not known exactly a prior. Some authors have recommended using 20-30 samples with four to five observations each to estimate the process parameters for traditional control charts. They conclude that, when the number of reference samples is small, control charts with estimated parameters produce a large bias in the IC ARL and reduce the sensitivity of the chart in detecting process changes as measured by the out-of-control (OC) ARL. The obvious solution to this problem is to increase the Phase I sample size to reduce the variability in the sampling distribution of the estimates, especially the process variance. In fact that we need at least 200-300 samples to estimate the process parameters in order to get the similar performance as known parameters. In most cases, however, it may not be possible to wait for the accumulation of sufficiently large subgroups because the users usually want to monitor the process at the start-up stages and such huge samples are costly. Our new chart can solve the problems mentioned above, that is to say it is not necessary to assemble a large number of reference samples before the control scheme begins. The results show that the new chart has good performance.
In Chapter 5, we extend our method to the multivariate case. Multivariate control chart is more complicate than the univariate one. Someone think that we can operate several control charts for the process parameters simultaneously and separately. Maybe this idea is accessible, but it is not feasible, because these variables are correlated to some extent. Multivariate theory have shown that even the quality characteristics are mutually independent, for the same significance level, the separated reliability for each variable is not equal to the combined reliability. In this Chapter, we propose a new single control chart which integrates the exponentially weighted moving average (EWMA) procedure with the likelihood ratio test for jointly monitoring both the multivariate process mean vector and the covariance matrix. Hawkins and Masoudou-Tchao (2008) also considered such a method to monitor the covariance matrix, but there are some differences between ours. For their chart, it turned out to be ARL-biased under some circumstance. However, for our chart, it is not ARL-biased any more. From the comparison with other charts, we can see that our new chart can be easily designed and performs better than others for the case in which the quality characteristics are bivariate normal random variables.
In Chapter 6, we discuss the study and the applications of the monitoring of linear profiles. In many applications the quality of a process or product is best characterized and summarized by a functional relationship between a response variable and one or more explanatory variables. In some calibration applications, the profile can be represented adequately by a simple linear regression model, while in other applications more complicated model are needed. The monitoring of linear profile was firstly proposed and studied by Kand and Albin in 2000. By far, most of the models of linear profile contain only the intercept and the slope, and the monitoring methods are Shewhart X-bar control chart and the EWMA chart based on the estimation of the process parameters. In addition, the likelihood ratio test and the change-point methods were proposed for detecting changes in the parameters of a simple linear regression model. A self-starting method was proposed which avoids the distinction between Phase I and Phase II. In this Chapter, we propose a likelihood method which integrates the EWMA procedure. This chart performs better than others in detecting the process variance, including the increase and decrease which is very important in practical manufacturing production. Apart from this, it can also detect other parameters changes very well. At last, we apply the VSI procedure to our chart in order to reduce the time of detecting. It can be seen that the effect is remarkable.
At last in Chapter 7, we summarize the main results and the technical approaches, and suggest some promising research topics yet to be pursued.
In this dissertation, we proposed several creative control charts based on the likelihood ratio test statistics which integrates the EWMA procedure. The resulting ARL values can be obtained by Markov chain approximation for some charts, but others are obtained by Monte Carlo simulations.
The original and creative ideas are represented as follows:
1. We propose a creative single chart which integrates the EWMA procedure with the likelihood ratio test statistics for jointly monitoring both the process mean and variance, which can be easily designed. The new chart can cope well with the monitoring of the decrease and increase in the variance. Also, it can cope with the individual case. Further, we provide the design of the chart under different IC ARL. At last, we provide details on the Markov-chain approximation of ARL of the new chart.
2. Based on the efficiency of dynamic control chart, we address a VSI and VSS SLR chart based on the adjustment of control limits and the plotting statistics. We compare the difference between VSI and FSI control charts under the same sampling rate. Further, we provide details on the Markov-chain approximation of AATS of this chart.
3. We propose a self-starting control chart when the process parameters are not exactly known a prior, , and so there is no need to assemble a large number of reference samples before the control scheme begins. We propose a diagnostic aids based on the change-point method which can locate the change-point in the process and isolate the type of parameters change.
4. We propose a new single control chart for jointly monitoring both the multivariate process mean vector and the covariance matrix. The control limits are given and its performance is studied under different dimension, IC ARL and types of shifts. The design parameters are also given.
5. We propose a creative control chart for monitoring the linear profile based on the likelihood ratio test statistics. It can isolate the type of parameters change in a profile using some statistics in the charting statistics, especially the direction of the shift in the variance. We also considered the corresponding design of the VSI control chart for linear profile.
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