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Tree Oriented Data Analysis

Complex data objects arise in many areas of modern science including evolutionary biology, nueroscience, dynamics of gene expression and medical imaging. Object oriented data analysis (OODA) is the statistical analysis of datasets of complex objects. Data analysis of tree data objects is an exciting research area with interesting questions and challenging problems. This thesis focuses on tree oriented statistical methodologies, and algorithms for solving related mathematical optimization problems. This research is motivated by the goal of analyzing a data set of images of human brain arteries. The approach we take here is to use a novel representation of brain artery systems as points in phylogenetic treespace. The treespace property of unique global geodesics leads to a notion of geometric center called a Fréchet mean. For a sample of data points, the Fréchet function is the sum of squared distances from a point to the data points, and the Fréchet mean is the minimizer of the Fréchet function. In this thesis we use properties of the Fréchet function to develop an algorithmic system for computing Fréchet means. Properties of the Fréchet function are also used to show a sticky law of large numbers which describes a surprising stability of the topological tree structure of sample Fréchet means at that of the population Fréchet mean. We also introduce non-parametric regression of brain artery tree structure as a response variable to age based on weighted Fréchet means.