Dimension Reduction in Principal Component Analysis for Trees

TL;DR

This paper extends PCA methods to analyze complex tree-structured data, introducing new algorithms for dimension reduction and applying them to brain artery and organizational data to uncover structural differences.

Contribution

It generalizes PCA concepts to rooted and labeled trees, developing algorithms for forward and backward principal components in this space.

Findings

Identified structural differences in brain arteries related to aging and gender.

Revealed organizational structural variations across departments.

Proved path-independency between forward and backward PCA in tree space.

Abstract

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.