title: Functional Data Analysis of Tree Data Objects.
abstract: Data analysis on non-Euclidean spaces, such as tree spaces, can be challenging. The main contribution of this paper is establishment of a connection between tree data spaces and the well developed area of Functional Data Analysis (FDA), where the data objects are curves. This connection comes through two tree representation approaches, theDyck path representationand thebranch length representation. These representations of trees in Euclidean spaces enable us to exploit the power of FDA to explore statistical properties of tree data objects. A major challenge in the analysis is the sparsity of tree branches in a sample of trees. We overcome this issue by using atree pruningtechnique that focuses the analysis on important underlying population structures. This method parallels scale-space analysis in the sense that it reveals statistical properties of tree structured data over a range of scales. The effectiveness of these new approaches is demonstrated by some novel results obtained in the analysis of brain artery trees. The scale space analysis reveals a deeper relationship between structure and age. These methods are the first to find a statistically significant gender difference.
authors.affiliation: Department of Statistics and Operations Research, University of North Carolina at Chapel Hill Chapel Hill, NC 27599 ; Department of Biostatistics, University of North Carolina at Chapel Hill Chapel Hill, NC 27599.
authors.forename: Dan
authors.initials: D
authors.lastname: Shen
authors.affiliation: Department of Statistics and Operations Research, University of North Carolina at Chapel Hill Chapel Hill, NC 27599.
authors.forename: Haipeng
authors.initials: H
authors.lastname: Shen
authors.affiliation: Department of Statistics and Operations Research, University of North Carolina at Chapel Hill Chapel Hill, NC 27599.
authors.forename: Shankar
authors.initials: S
authors.lastname: Bhamidi
authors.affiliation: Research Institute, Beaumont Health System, Royal Oak, MI 48073.
authors.forename: Yolanda Muñoz
authors.initials: YM
authors.lastname: Maldonado
authors.affiliation: Department of Statistics, Seoul National University, South Korea.
authors.forename: Yongdai
authors.initials: Y
authors.lastname: Kim
authors.affiliation: Department of Statistics and Operations Research, University of North Carolina at Chapel Hill Chapel Hill, NC 27599 ; Lineberger Comprehensive Cancer Center, University of North Carolina at Chapel Hill Chapel Hill, NC 27599.
authors.forename: J S
authors.initials: JS
authors.lastname: Marron
keywords: DiProPerm; Dyck path; Support tree; Tree pruning; ranch length