A Principal Component Analysis for Trees

TL;DR

This paper extends Principal Component Analysis to tree-structured data using tree-lines, providing fast algorithms and demonstrating their application on brain blood vessel trees.

Contribution

It introduces a novel PCA analog for trees based on tree-lines and offers efficient algorithms for their computation.

Findings

Successfully applied to brain vessel trees from 73 individuals.

Algorithms operate in linear time, enabling practical analysis.

Provides new tools for analyzing complex tree-structured data.

Abstract

The active field of Functional Data Analysis (about understanding the variation in a set of curves) has been recently extended to Object Oriented Data Analysis, which considers populations of more general objects. A particularly challenging extension of this set of ideas is to populations of tree-structured objects. We develop an analog of Principal Component Analysis for trees, based on the notion of tree-lines, and propose numerically fast (linear time) algorithms to solve the resulting optimization problems. The solutions we obtain are used in the analysis of a data set of 73 individuals, where each data object is a tree of blood vessels in one person's brain.