Megan Owen

A Fast Algorithm for Computing Geodesic Distances in Tree Space

Comparing and computing distances between phylogenetic trees are important biological problems, especially for models where edge lengths play an important role. The geodesic distance measure between two phylogenetic trees with edge lengths is the length of the shortest path between them in the continuous tree space introduced by Billera, Holmes, and Vogtmann. This tree space provides a powerful tool for studying and comparing phylogenetic trees, both in exhibiting a natural distance measure and in providing a euclidean-like structure for solving optimization problems on trees. An important open problem is to find a polynomial time algorithm for finding geodesics in tree space. This paper gives such an algorithm, which starts with a simple initial path and moves through a series of successively shorter paths until the geodesic is attained.

Sticky central limit theorems on open books

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Frechet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine. We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky).

Polyhedral computational geometry for averaging metric phylogenetic trees

This paper investigates the computational geometry relevant to calculations of the Frechet mean and variance for probability distributions on the phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of probability measures on spaces of nonpositive curvature developed by Sturm. We show that the combinatorics of geodesics with a specified fixed endpoint in tree space are determined by the location of the varying endpoint in a certain polyhedral subdivision of tree space. The variance function associated to a finite subset of tree space has a fixed

Computing Geodesic Distances in Tree Space

C^\infty

A Hierarchical Scheme for Geodesic Anatomical Labeling of Airway Trees

algebraic formula within each cell of the corresponding subdivision, and is continuously differentiable in the interior of each orthant of tree space. We use this subdivision to establish two iterative methods for producing sequences that converge to the Frechet mean: one based on Sturms Law of Large Numbers, and another based on descent algorithms for finding optima of smooth functions on convex polyhedra. We present properties and biological applications of Frechet means and extend our main results to more general globally nonpositively curved spaces composed of Euclidean orthants.

Tree-Oriented Analysis of Brain Artery Structure

We present two algorithms for computing the geodesic distance between phylogenetic trees in tree space, as introduced by Billera, Holmes, and Vogtmann (2001). We show that the possible combinatorial types of shortest paths between two trees can be compactly represented by a partially ordered set. We calculate the shortest distance along each candidate path by converting the problem into one of finding the shortest path through a certain region of Euclidean space. In particular, we show there is a linear time algorithm for finding the shortest path between a point in the all positive orthant and a point in the all negative orthant of R^k contained in the subspace of R^k consisting of all orthants with the first i coordinates non-positive and the remaining coordinates non-negative for 0 <= i <= k.

Tree-space statistics and approximations for large-scale analysis of anatomical trees

We present a fast and robust supervised algorithm for labeling anatomical airway trees, based on geodesic distances in a geometric tree-space. Possible branch label configurations for a given tree are evaluated based on distances to a training set of labeled trees. In tree-space, the tree topology and geometry change continuously, giving a natural way to automatically handle anatomical differences and noise. The algorithm is made efficient using a hierarchical approach, in which labels are assigned from the top down. We only use features of the airway centerline tree, which are relatively unaffected by pathology.

Geodesic Atlas-Based Labeling of Anatomical Trees: Application and Evaluation on Airways Extracted From CT

Statistical analysis of magnetic resonance angiography (MRA) brain artery trees is performed using two methods for mapping brain artery trees to points in phylogenetic treespace: cortical landmark correspondence and descendant correspondence. The differences in end-results based on these mappings are highlighted to emphasize the importance of correspondence in tree-oriented data analysis. Representation of brain artery systems as points in phylogenetic treespace, a mathematical space developed in (Billera et al. Adv. Appl. Math 27:733–767, 2001), facilitates this analysis. The phylogenetic treespace is a rich setting for tree-oriented data analysis. The Fréchet sample mean or an approximation is reported. Multidimensional scaling is used to explore structure in the data set based on pairwise distances between data points. This analysis of MRA data shows a statistically significant effect of age and sex on brain artery structure. Variation in the proximity of brain arteries to the cortical surface results in strong statistical difference between sexes and statistically significant age effect. That particular observation is possible with cortical correspondence but did not show up in the descendant correspondence.

Central limit theorems for Fréchet means in the space of phylogenetic trees

Statistical analysis of anatomical trees is hard to perform due to differences in the topological structure of the trees. In this paper we define statistical properties of leaf-labeled anatomical trees with geometric edge attributes by considering the anatomical trees as points in the geometric space of leaf-labeled trees. This tree-space is a geodesic metric space where any two trees are connected by a unique shortest path, which corresponds to a tree deformation. However, tree-space is not a manifold, and the usual strategy of performing statistical analysis in a tangent space and projecting onto tree-space is not available. Using tree-space and its shortest paths, a variety of statistical properties, such as mean, principal component, hypothesis testing and linear discriminant analysis can be defined. For some of these properties it is still an open problem how to compute them; others (like the mean) can be computed, but efficient alternatives are helpful in speeding up algorithms that use means iteratively, like hypothesis testing. In this paper, we take advantage of a very large dataset (N=8016) to obtain computable approximations, under the assumption that the data trees parametrize the relevant parts of tree-space well. Using the developed approximate statistics, we illustrate how the structure and geometry of airway trees vary across a population and show that airway trees with Chronic Obstructive Pulmonary Disease come from a different distribution in tree-space than healthy ones. Software is available from http://image.diku.dk/aasa/software.php.

Geodesics in CAT(0) cubical complexes

We present a fast and robust atlas-based algorithm for labeling airway trees, using geodesic distances in a geometric tree-space. Possible branch label configurations for an unlabeled airway tree are evaluated using distances to a training set of labeled airway trees. In tree-space, airway tree topology and geometry change continuously, giving a natural automatic handling of anatomical differences and noise. A hierarchical approach makes the algorithm efficient, assigning labels from the trachea and downwards. Only the airway centerline tree is used, which is relatively unaffected by pathology. The algorithm is evaluated on 80 segmented airway trees from 40 subjects at two time points, labeled by three medical experts each, testing accuracy, reproducibility and robustness in patients with chronic obstructive pulmonary disease (COPD). The accuracy of the algorithm is statistically similar to that of the experts and not significantly correlated with COPD severity. The reproducibility of the algorithm is significantly better than that of the experts, and negatively correlated with COPD severity. Evaluation of the algorithm on a longitudinal set of 8724 trees from a lung cancer screening trial shows that the algorithm can be used in large scale studies with high reproducibility, and that the negative correlation of reproducibility with COPD severity can be explained by missing branches, for instance due to segmentation problems in COPD patients. We conclude that the algorithm is robust to COPD severity given equally complete airway trees, and comparable in performance to that of experts in pulmonary medicine, emphasizing the suitability of the labeling algorithm for clinical use.