树木主成分分析的降维方法
Dimension Reduction in Principal Component Analysis for Trees
摘要：
树型结构数据的统计分析是统计学中的一个新课题，有着广泛的应用前景。一些主成分分析（PCA）的想法是以前开发的二叉树空间。在这项研究中，我们将这些想法扩展到更一般的空间的根和标记的树。我们重新定义的概念，如树线和前向主成分树线这个更一般的空间，并推广的最佳算法，找到他们。然后，我们开发了一个模拟的经典降维技术在PCA的树空间。为了做到这一点，我们定义了携带树数据集的最小变化量的成分，称为向后主成分。我们提出了一个最优算法来找到它们。此外，我们调查这些前向主成分的关系，并证明了前向和后向技术之间的路径无关性。我们将我们的方法应用于98名受试者的脑动脉数据集。利用我们的技术，我们调查了衰老如何影响男性和女性的脑动脉结构。我们还分析了美国一家大型公司的组织结构数据集，并探讨了公司内部不同类型部门的结构差异。
英文摘要：
The statistical analysis of tree structured data is a new topic in statistics with wide application areas. Some Principal Component Analysis (PCA) ideas were previously developed for binary tree spaces. In this study, we extend these ideas to the more general space of rooted and labeled trees. We re-define concepts such as tree-line and forward principal component tree-line for this more general space, and generalize the optimal algorithm that finds them. We then develop an analog of classical dimension reduction technique in PCA for the tree space. To do this, we define the components that carry the least amount of variation of a tree data set, called backward principal components. We present an optimal algorithm to find them. Furthermore, we investigate the relationship of these the forward principal components, and prove a path-independency property between the forward and backward techniques. We apply our methods to a data set of brain artery data set of 98 subjects. Using our techniques, we investigate how aging affects the brain artery structure of males and females. We also analyze a data set of organization structure of a large US company and explore the structural differences across different types of departments within the company.
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