A Nonparametric Regression Model With Tree-Structured Response

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.