A Nonparametric Regression Model With Tree-Structured Response

Abstract

Developments in science and technology over the last two decades has motivated the study of complex data objects. In this article, we consider the topological properties of a population of tree-structured objects. Our interest centers on modeling the relationship between a tree-structured response and other covariates. For tree-structured objects, this poses serious challenges since most regression methods rely on linear operations in Euclidean space. We generalize the notion of nonparametric regression to the case of a tree-structured response variable. In addition, we develop a fast algorithm and give its theoretical justification. We implement the proposed method to analyze a dataset of human brain artery trees. An important lesson is that smoothing in the full tree space can reveal much deeper scientific insights than the simple smoothing of summary statistics. This article has supplementary materials online.

Keywords:

• Binary tree
• Data object
• Object-oriented data
• Smoothing

SUPPLEMENTARY MATERIALS

	Section A: introduces the cross-validation approach for the choice of bandwith in tree smoothing.
	Section B: states the algorithm for our proposed tree smoothing method at a fixed bandwidth.
	Section C: includes the simple regression and the classical smoothing results on the number of nodes of the cerebral artery tree versus age for all four cerebral artery systems.
	Sections D and E: more results from our proposed tree smoothing method.
	Section F: more results of the simulation study.

This article is the thesis work of Yuan Wang, written under the supervision of Haonan Wang. The research of Elizabeth Bullitt was partially supported by NIH grants R01EB000219-NIH-NIBIB and R01 CA124608- NIH-NCI. The research of J. S. Marron was partially supported by NSF grants DMS-0606577 and DMS-0854908, and NIH grant RFA-ES-04-008. The research of Burcu Aydın was partially supported by NSF grants DMS-0606577 and DMS-0854908, and NIH Grant RFA-ES-04-008. The research of Haonan Wang was partially supported by NSF grants DMS-0706761, DMS-0854903, and DMS-1106975, and by the Air Force Office of Scientific Research under contract number FA9550-10-1-0241.