The statistical analysis of a population of brain artery trees is considered. New representations of these tree structured data objects are developed, using ideas from topological data analysis. Specifically, a number of representations of each brain tree, using persistence diagrams that quantify branching and looping at multiple scales, are considered. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to improved correlations with covariates such as age and sex, relative to earlier analyses of this data set. Strikingly, these correlations remain strong even after controlling for more obvious geometric differences in the set of trees.

This is joint work with Alex Pieloch, J.S. Marron, Ezra Miller, and Sean Skwerer, and the dataset was obtained from the lab of Elizabeth Bullitt at UNC-CH.