Confidence Regions for Trees

本研究建议使用几何和统计学来建立有意义的树状结构参数置信区域。二叉树作为参数出现在以下问题中：(1)层次聚类问题，例如当代遗传学的微阵列数据；(2)使用决策树的方法，如分类与回归树（CART）；(3)利用DNA数据构建系统发育树，用于物种分类，也用于构建“基因树”。将树估计嵌入到统计框架中，可以将树视为一种需要估计的特殊类型的参数。然而，开发的估计程序没有能力为这些参数构建置信区域或比较这些参数的各种估计量的方法。目前，生物学家通过对DNA序列进行简单的自举来干扰数据，然后通过将p值与共识树的分支相关联来总结树的集合，从而验证树的估计。这将问题简化为二项模拟的集合，丢失了同时出现的组的许多多元信息。这种新的构造几何方法使用了拓扑和代数组合学中可用的工具。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化的理解，从而可以更自然地定义“树多面体”中点之间的距离。树空间的拓扑研究对于邻域概念的定义至关重要，邻域概念允许对估计函数的连续性进行定义。树的空间是一个负曲率的空间。也可以在这个空间和凸包上定义测地线。可以定义置信区域的概念，并且可以组合从不同数据集构建的树或组合从不同基因获得的树，甚至将树与其他数据（例如生物地理）进行比较。理解“树空间”的几何结构有助于理解如何组合和比较树木。无论是为微阵列DNA数据构建的层次聚类树，还是从DNA序列构建的家谱树，二叉树数据在当今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负曲率空间的几何的最新结果，比如双曲面的表面。在这些空间中，平均和距离的概念必须重新定义，最难的是如何表示在我们通常的欧几里得空间中不能自然表征的东西。这些挑战既有计算上的，也有几何上的。

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.