Confidence Regions for Trees

该研究方案使用几何和统计来为树形结构参数建立有意义的置信域。二叉树在以下方面作为参数出现：(1)等级聚类问题，例如当代遗传学的微阵列数据；(2)使用决策树的方法，如分类和回归树(CART)；(3)根据DNA数据建立的系统发育树的估计，用于物种分类，也用于建立“基因树”。在统计框架中嵌入树估计允许将树视为一种特殊类型的待估计参数。然而，估计程序是在没有能力为这些参数构造置信域或比较这些参数的各种估计器的情况下开发的。目前，生物学家通过通过DNA序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集合，来验证所估计的树。这将问题简化为二项式模拟的集合，丢失了有关哪些组同时出现的大部分多变量信息。这种新的构造性几何方法使用了拓扑学和代数组合学的工具。与组合学家Louis Billera和拓扑学家Karen Vogtmann合作，对树空间进行更数学的理解，可以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，然后邻域概念允许定义估计函数的连续性。树的空间是一个负曲率空间。也可以在这个空间和凸壳上定义测地线。可以定义置信度区域的概念，并且可以组合来自不同数据集的树，或者组合从不同基因获得的树，甚至可以将树与其他数据(例如，生物地理)进行比较。理解树空间的几何有助于理解如何组合和比较树。无论是为微阵列DNA数据构建的层级聚类树，还是根据DNA序列构建的家谱树，二叉树数据在当今的基因组时代都是丰富的。目前的提案旨在将树木作为一个整体来考虑，而不是像目前所做的那样，将它们分解为兄弟姐妹关系。这个问题使用了拓扑学和具有负曲率的空间的几何的最新结果，如双曲面的曲面。在这些空间中，平均和距离的概念必须重新定义，最困难的是如何表示在我们通常的欧几里德空间中无法自然表示的东西。挑战既是计算上的，也是几何上的。

This research proposal uses geometry and statistics to build meaningful confidence regions for tree structured parameters. Binary trees appear as parameters in: (1) Hierarchical clustering problems, for instance for micro-array data from contemporary genetics; (2) Methods that use decision trees such as the Classification and Regression trees (CART); (3) Estimation of phylogenetic trees built from DNA data for the taxonomy of species, but also for building `gene trees'. The embedding of tree estimation in a statistical framework allows one to see trees as a special type of parameter to be estimated. However the estimation procedures have been developed without the ability to construct confidence regions for these paramters or ways of comparing various estimators of these parameters. Currently biologists validate the tree estimated by perturbing the data through a simple bootstrap of the 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 simulations, losing much of the multivariate information of which groups appear simultaneously. This new constructive geometrical approach uses the tools available from topology and algebraic combinatorics. Collaboration with a combinatorialist, Louis Billera, and a topologist, Karen Vogtmann, on a more mathematical understanding of tree space allows 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 then allows a definition of continuity of the estimation function. The space of trees is a space with negative curvature. It is also possible to define geodesics on this space and convex hulls. The notion of confidence region can be defined, and it is possible to combine trees from built from different datasets or to combine trees obtained from different genes or even compare trees with other data (biogeographic, for instance).Understanding the geometry of "tree space" helps understand how to combine and compare trees. Whether they are hierarchical clustering trees built for micro-array DNA data or family trees built from DNA sequences, binary tree data are abundant in today's genome era. The current proposal aims to consider trees as a whole, instead of breaking them down into just sibling relationships as is done currently. This problem uses recent results from topology and the geometry of spaces with negative curvature, like the surface of a hyperboloid. In these spaces notions of average and distance have to be redefined, the hardest being ways of representing what is not naturally representable in our usual euclidean space. The challenges are both computational and geometric.