Computational Statistics For Phylogenetic Trees

霍姆斯经典统计学发展了基于线性代数的各种平均值、投影和表示法。近年来，非数值数据和参数不断涌现。这个项目的目标是提供平均、建立置信域、运行蒙特卡罗算法、进行回归和测试有根二叉树模型的方法。目前，生物学家通过DNA序列列的简单引导来扰动数据，然后总结通过将p值与共识树的分支相关联而获得的树的集合，从而验证他们的系统发育树。这将问题简化为二项式参数的集合，丢失了大部分多变量信息。基于树空间中置信度区域的几何方法更好地克服了多重测试问题。这些项目使用相关树空间的完全几何构造，将二叉树的贝叶斯和频率推理过程扩展到非参数上下文。数学工具包括概率论、拓扑学和代数组合学。与Louis Billera和Karen Vogtmann(康奈尔数学系)增强了我们对树木空间的数学理解。树的空间是负曲率的，因此我们知道我们可以在这个空间上定义测地线以及凸壳。许多实际的距离计算可能具有指数复杂性，对于所考虑的应用来说，良好的近似算法是重要的。这项工作有助于将生物网络的统计视为树的泛化或混合。遗传学和分子生物学领域出现了一种新的数据类型，这些数据不是实数或载体，而是联系不同物种的树、家谱或系统发育树，以及根据不同表达模式联系不同基因的系统聚类树。这个项目提供了可视化和对这些新数据进行统计的程序，我们将为生物学家提供开源计算机包，他们可以使用这些包来分析自己的数据。例如，经典的线性回归是基于投影的；同样，如果我们想比较两组树，我们将使用距离和方法在树空间进行投影。该项目包括两个工作坊，一个是第一阶段的数学家研讨会，目的是宣传一些较难的公开问题，另一个是在去年的另一个研讨会，教生物学家如何在实际例子中使用用MatLab或R开发的几何工具。这笔赠款是在DMS/NIGMS联合倡议下提供的，以支持数学生物学领域的研究资助。这是一项由国家科学基金会数学科学部(DMS)和国立卫生研究院(NIGMS)国家普通医学研究所(NIGMS)主办的联合竞赛。

0241246Holmes 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.