本文为美国马萨诸塞大学阿默斯特分校（作者：Xuan Huang）的博士论文，共166页。

大多数业务流程在本质上是多变量和自相关的。高维性植根于同时考虑的多个变量，以提供更全面的图像过程。时间序列模型比独立同分布（I.I.D.）模型更可取，因为它们捕获了许多进程都有其过去的记忆这一事实。多元自相关的例子可以在业务领域的流程中找到，如运营管理、财务和营销。

统计控制的主题与质量管理最相关。虽然在统计过程控制（SPC）文献中，多变量I.I.D.过程和单变量自相关过程都受到了广泛关注，但在同时处理高维性和自相关方面的工作却很少。本文将单变量特殊原因图和共同原因图扩展到多变量情形，填补了这一空白。另外，将两种图表控制方案推广到非平稳过程，还提出了一类马尔可夫链模型，用于在过程自相关时提供精确的平均运行长度（ARL）计算。

本文的第二部分旨在设计一种自相关过程的降维方法。高维度常常掩盖了过程的真实底层组件。在传统的多元变量文献中，主成分分析（PCA）是降维的标准工具。然而，对于自相关过程，PCA没有考虑到自相关信息。因此，PCA是否是最佳选择值得怀疑。设计了一种多变量时间序列的两步降维方法。基于模拟实例和案例研究的比较表明，两步降维程序在提取真正的潜在因素方面更为有效。多变量时间序列的可视化有助于我们理解这个过程。本文的最后一部分提出了一个简单的三维图形来辅助PCA结果的可视化，它旨在补充现有的多变量时间序列数据的图形化方法。其思想是将多变量数据可视化为一个曲面，然后用PCA进行分解。开发的表面图用于统计过程分析，但也可以有助于经济数据的可视化，尤其是在协同集成方面。

Most business processes are, by nature,multivariate and autocorrelated. Highdimensionality is rooted in processeswhere more than one variable is considered simultaneously to provide a morecomprehensive picture. Time series models are preferable to an independentlyidentically distributed (I.I.D.) model because they capture the fact that manyprocesses have a memory of their past. Examples of multivariate autocorrelationcan be found in processes in the business fields such as Operations Management,Finance and Marketing. The topic of statistical control is most relevant toQuality Management. While both multivariate I.I.D. processes and univariateautocorrelated processes have received much attention in the StatisticalProcess Control (SPC) literature, little work has been done to simultaneously addresshigh-dimensionality and autocorrelation. In this dissertation, this gap isfilled by extending the univariate special cause chart and common cause chartto multivariate situations. In addition, two-chart control schemes are extendedto nonstationary processes. Further, a class of Markov Chain models is proposedto provide accurate Average Run Length (ARL) computation when the process isautocorrelated. The second part of this dissertation aims to devise a dimensionreduction method for autocorrelated processes. High-dimensionality oftenobscures the true underlying components of a process. In traditionalmultivariate literature, Principal Components Analysis (PCA) is the standardtool for dimension reduction. For autocorrelated processes, however, PCA failsto take into account the autocorrelation information. Thus, it is doubtful thatPCA is the best choice. A two-step dimension reduction procedure is devised formultivariate time series. Comparisons based on both simulated examples and casestudies show that the two-step procedure is more efficient in retrieving trueunderlying factors. Visualization of multivariate time series assists our understandingof the process. In the last part of this dissertation a simplethree-dimensional graph is proposed to assist visualizing the results of PCA.It is intended to complement existing graphical methods for multivariate timeseries data. The idea is to visualize multivariate data as a surface that inturn can be decomposed with PCA. The developed surface plots are intended forstatistical process analysis but may also help visualize economics data and, inparticular, co-integration.

1 引言

2 基于模型的自相关过程多变量监测图

3 一类用于自相关过程平均运行长度计算的马尔可夫链模型

4 多元时间序列的公因子与线性关系

5 多元过程数据主成分分析的可视化

6 结论

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