Evolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. A geometry for the space of phylogenetic trees is necessary in order to properly quantify this uncertainty during the statistical analysis of collections of possible evolutionary trees inferred from biological data. Recently, the wald space has been introduced: a length space for trees which is a certain subset of the manifold of symmetric positive definite matrices. In this work, the wald space is introduced formally and its topology and structure is studied in detail. In particular, we show that wald space has the topology of a disjoint union of open cubes, it is contractible, and by careful characterization of cube boundaries, we demonstrate that wald space is a Whitney stratified space of type (A). Imposing the metric induced by the affine invariant metric on symmetric positive definite matrices, we prove that wald space is a geodesic Riemann stratified space. A new numerical method is proposed and investigated for construction of geodesics, computation of Fr\'echet means and calculation of curvature in wald space. This work is intended to serve as a mathematical foundation for further geometric and statistical research on this space.
翻译:物种之间的进化关系以植物基因树为代表,但这些关系由于演化的随机性质而具有不确定性。为了在从生物数据中推断出可能的进化树的收集的统计分析中适当量化这种不确定性,需要对植物基因树的空间进行几何测量。最近,引入了圆形空间:树木的长空间,这是对称正数确定矩阵的方位数的一部分。在这项工作中,正式引入了 wald空间,并详细研究了其表层和结构。特别是,我们表明 wald空间具有开放立方体脱节状态的表层学,这是可以比较的,通过对立方体界限的仔细定性,我们证明Wald空间是惠特尼的分层空间(A)。在对正数确定基数的方位矩阵上,我们证明 wald空间是一个地球德立方体空间,其表层和结构得到了详细研究。我们提出并调查了一种新的数字方法,用于构建大地学系,可以计算Fr\'echet的立方块界限,我们证明这是用于进一步进行空间的数学基础和计算。