Statistics in Biosciences

, Volume 4, Issue 1, pp 132–156

New Approaches to Principal Component Analysis for Trees

Authors

    • HP Laboratories
  • Gábor Pataki
    • UNC at Chapel Hill
  • Haonan Wang
    • Colorado State University
  • Alim Ladha
    • UNC at Chapel Hill
  • Elizabeth Bullitt
    • UNC at Chapel Hill
  • J. S. Marron
    • UNC at Chapel Hill
Article

DOI: 10.1007/s12561-012-9055-8

Cite this article as:
Aydın, B., Pataki, G., Wang, H. et al. Stat Biosci (2012) 4: 132. doi:10.1007/s12561-012-9055-8

Abstract

Object Oriented Data Analysis is a new area in statistics that studies populations of general data objects. In this article we consider populations of tree-structured objects as our focus of interest. We develop improved analysis tools for data lying in a binary tree space analogous to classical Principal Component Analysis methods in Euclidean space. Our extensions of PCA are analogs of one dimensional subspaces that best fit the data. Previous work was based on the notion of tree-lines.

In this paper, a generalization of the previous tree-line notion is proposed: k-tree-lines. Previously proposed tree-lines are k-tree-lines where k=1. New sub-cases of k-tree-lines studied in this work are the 2-tree-lines and tree-curves, which explain much more variation per principal component than tree-lines. The optimal principal component tree-lines were computable in linear time. Because 2-tree-lines and tree-curves are more complex, they are computationally more expensive, but yield improved data analysis results.

We provide a comparative study of all these methods on a motivating data set consisting of brain vessel structures of 98 subjects.

Keywords

Binary treesObject oriented data analysisPrincipal component analysisTree-linesVessel structure

Copyright information

© International Chinese Statistical Association 2012