Computational Statistics For Phylogenetic Trees

Stanford University, Stanford CA

Investigators

Abstract

0241246 Holmes Classical statistics has developed various averages, projections and representations based on linear algebra. In recent years, non-numerical data and parameters have emerged. The object of this project is to provide ways of averaging, building confidence regions, running Monte Carlo algorithms, doing regression and testing models for rooted binary trees. Currently biologists validate their phylogenetic trees by perturbing the data through a simple bootstrap of the columns of DNA sequences and then summarizing the collection of trees obtained by associating p-values to the branches of a consensus tree. This reduces the problem to a collection of binomial parameters, losing much of the multivariate information. A more geometrical procedure based on confidence regions in tree space is preferable and overcomes the multiple testing problem. The projects extends both Bayesian and frequentist inferential procedures for binary trees to a nonparametric context using a complete geometric construction of the relevant tree space. The mathematical tools include probability theory, topology and algebraic combinatorics. Collaboration with Louis Billera and Karen Vogtmann (Cornell Mathematics Dept.) has enhanced our mathematical understanding of tree space. The space of trees has negative curvature, thus we know we can define geodesics on this space as well as convex hulls. Many of the actual distance computations can have exponential complexity, good approximation algorithms are important for the applications considered. This work helps think about the statistics of biological networks as generalizations or mixtures of trees. A new type of data has appeared in genetics and molecular biology, these data are not real numbers or vectors, but trees, family trees or phylogenetic trees relating different species and hierarchical clustering trees relating different genes according to their differing expression patterns. This project provides programs for visualizing and doing statistics on these new data, we will provide the biologists with open source computer packages that they can use to analyze their own data. For instance, classical linear regression is based on projections; in the same way if we want to compare two sets of trees, we will use distances and methods for projecting in tree space. The project includes two workshops, one for mathematicians in the first stage, to publicize some of the harder open problems, and another in the last year to teach biologists how to use the geometrical tools developed in matlab or R in practical examples. This grant is made under the Joint DMS/NIGMS Initiative to Support Research Grants in the Area of Mathematical Biology. This is a joint competition sponsored by the Division of Mathematical Sciences (DMS) at the National Science Foundation and the National Institute of General Medical Sciences (NIGMS) at the National Institutes of Health.