Functional Data Analysis of Tree Data Objects.

Area of Science:

• Computational Biology
• Statistical Analysis
• Data Science

Background:

• Analyzing data in non-Euclidean spaces, like tree structures, presents significant statistical challenges.
• Existing methods often struggle with the inherent complexity and sparsity of tree data.

Purpose of the Study:

• To establish a novel connection between tree data spaces and Functional Data Analysis (FDA).
• To develop and apply new statistical methods for exploring the properties of tree-structured data.
• To uncover new biological insights from tree-shaped data, specifically in the context of brain vasculature.

Main Methods:

• Utilized Dyck path and branch length representations to map trees into Euclidean spaces.
• Applied Functional Data Analysis (FDA) techniques to these representations for statistical exploration.
• Implemented a tree pruning technique to address data sparsity and focus on population structures, paralleling scale-space analysis.

Main Results:

• Demonstrated the utility of FDA for analyzing tree data through novel representations.
• Scale-space analysis of brain artery trees revealed deeper structure-age relationships.
• Identified the first statistically significant gender differences in brain artery structures using these methods.

Conclusions:

• The integration of FDA with tree representations offers a powerful framework for analyzing complex biological structures.
• Tree pruning and scale-space analysis provide effective tools for uncovering hidden patterns in sparse tree data.
• This research opens new avenues for understanding anatomical variations and their correlations with demographic factors.