The Exploration of Phylogenetic Tree Space through Combinatorics, Statistics and Geometry

我正在开发的研究方案旨在帮助生物学家了解所有系统发育树的空间特性。我建议结合三种方法来解决这个问题，组合方法、统计方法和几何方法。我已经通过组合方法解决了我从Persi Diaconis开始的问题，取得了一些良好的进展。我们在所有系统发育树的空间和完美匹配的空间之间建立了一个双射。这提供了新的树之间的距离，在树空间上进行随机行走的方法，以及建立系统发育树所涉及的$NP$完全优化过程的新方法。将树估计嵌入到统计框架中，可以将树视为一种特殊类型的被估计参数。下一步需要的是构建置信域的能力。几年前，我与Brad Efron和Betz Halloran一起完成了关于这个主题的第一项工作，并开始了这个项目。目前，生物学家通过扰动数据，然后简单地通过将p值与树的树枝相关联来总结，来验证估计的树。在这一领域，统计学家和生物学家之间的交流是一段艰难的历史。一个学派的分类学家一直鼓励使用最大似然法来估计树木。其他分类学家一直使用简约性作为他们的标准，并对他们所称的“统计学方法”提出了强烈的批评。其实他们自己的方法也是统计的，但简约方法是非参数方法，而最大似然方法是参数方法。最大似然估计模型是有限维的，然而，统计学家知道，如果允许参数的数量随着这个数字大于数据中包含的信息量而增加，那么该方法就变成了非参数方法。让生物学家知道这一解释将是一项有价值的工作，其结果将是统计学中的非参数-参数讨论可以启发简约-最大似然估计的辩论。理解树空间的几何将有助于理解如何将从扰动数据或同一物种上的不同基因获得的树组合在一起。我希望继续我与组合学家Louis Billera和拓扑学家Karen Vogtmann开始的合作，对树空间进行更数学的理解，允许更自然地定义树多面体中点之间的距离。树空间的这种拓扑研究在邻域概念的定义中是至关重要的，这将允许在估计函数中定义连续性。然后，可以定义置信域的概念，并且还可以组合来自不同数据集的树，或者将树与其他数据(例如，生物地理)组合。最后，我将通过一个名为“树空间的几何”的网站传播我的MatLab树分析程序和初步结果。该POWRE项目由公安部多学科活动办公室(OMA)提供支持。

The research proposal I am developing aims to help biologists understand the properties of the space ofall phylogenetic trees. I propose to combine three approaches to tackle this problem,combinatorial, statistical and geometrical.I have already made some good progress through a combinatorial approach to the problem that I have started with Persi Diaconis. We have built a bijection between the space of all phylogenetic trees and the space of perfect matchings. This has provided new distances between trees, ways of doing random walks on tree space, and a new approach to the $NP$ complete optimization procedures involved in building phylogenetic trees.The embedding of tree estimation in a statistical framework allows one to see trees as a special type of parameter to be estimated. The next step needed is the ability to construct confidence regions. My first work on this subject was done with Brad Efron and Betz Halloran a few years ago and started me on this project.Currently biologists validate the tree estimated by perturbing the data and then summarizing this simplyby associating p-values to the branches of the tree. Communication between statisticians and biologistshas a difficult history in this field. One school of taxonomists has been encouraging the use of maximum likelihood for the estimation of the tree.Other taxonomists have been using parsimony as their criterion, and have been highly critical ofwhat they called the `statistical approach'. In fact their own method is statistical also, but the parsimony method is a nonparametric approach, whereas the Maximum Likelihood is a parametric one. The MLE model is finite dimensional, however, statisticians know that if one allows the number of parameters to increase as this number becomes larger than the amount of information contained in the data, the method becomes nonparametric. Making this interpretation known to biologists would be a worthwhile enterprise, the consequence would be that nonparametric-parametric discussion in statistics could enlighten the parsimony-MLE debate.Understanding the geometry of the tree-space would help understand how to combine trees obtained from perturbed data or different genes on the same set of species. I hope to continue a collaboration I have started with a combinatorialist, Louis Billera and a topologist, Karen Vogtmann on a more mathematical understanding of tree space that would allow for a more natural definition of distances between points in the `tree polytope'. This topological study of tree space is crucial in the definition of a notion of neighborhood that would then allow a definition of continuity in the estimation function.The notion of confidence region could then be defined, and it would also be possible to combine trees from built from different datasets or to combine trees with other data (biogeographic, for instance). Finally I will diffuse my matlab tree analysis programs and preliminary results through a web site `Geometry of Tree Space'. This POWRE project is supported by the MPS Office of Multidisciplinary Activities (OMA).