Object oriented data analysis: Sets of trees

TL;DR

This paper develops a new mathematical framework for analyzing populations of complex, tree-structured data objects, extending object oriented data analysis to non-Euclidean spaces.

Contribution

It introduces a novel approach for statistical analysis of tree-structured objects, bridging mathematics and statistics in non-Euclidean spaces.

Findings

Framework successfully models populations of trees

Enables statistical inference on complex data objects

Extends existing methods to non-Euclidean spaces

Abstract

Object oriented data analysis is the statistical analysis of populations of complex objects. In the special case of functional data analysis, these data objects are curves, where standard Euclidean approaches, such as principal component analysis, have been very successful. Recent developments in medical image analysis motivate the statistical analysis of populations of more complex data objects which are elements of mildly non-Euclidean spaces, such as Lie groups and symmetric spaces, or of strongly non-Euclidean spaces, such as spaces of tree-structured data objects. These new contexts for object oriented data analysis create several potentially large new interfaces between mathematics and statistics. This point is illustrated through the careful development of a novel mathematical framework for statistical analysis of populations of tree-structured objects.