A Nonparametric Regression Model With Tree-Structured Response

Highly developed science and technology from the last two decades motivated the study of complex data objects. In this paper, 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 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, a fast algorithm with theoretical justification is developed. We implement the proposed method to analyze a data set 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