Fréchet平均在系统树空间中的极限行为
Limiting Behaviour of Fréchet Means in the Space of Phylogenetic Trees
摘要：
正如我们以前在${\boldsymbol T}_{4}$上的工作所证明的那样，具有四叶的系统发育树的空间、全局以及局部的空间拓扑结构在${\boldsymbol T}_{4}$上的概率分布的样本Fréchet均值的非经典限制行为中起着重要作用。然而，该论文中使用的技术是特定于${\boldsymbol T}_{4}$的，并且不能适用于分析具有$m(\geqslant5)$叶的系统发育树的空间${\boldsymbol T}_{m}$中的Fréchet平均值。为了研究后者，本文首先研究了${\boldsymbol T}_{m}$的对数映射，它是黎曼流形上指数映射的逆的推广。然后，在一个修改后的版本的日志地图，我们将Fréchet的手段在${\boldsymbol T}_{m}$，位于顶维或共维一层。我们推导出相应的样本Fréchet均值的极限分布，推广了我们以前的结果。特别是，结果表明，尽管它们与高斯分布有关，但极限分布所采取的形式取决于Fréchet平均值所在地层的余维。
英文摘要：
As demonstrated in our previous work on ${\boldsymbol T}_{4}$, the space of phylogenetic trees with four leaves, the global, as well as the local, topological structure of the space plays an important role in the non-classical limiting behaviour of the sample Fréchet means of a probability distribution on ${\boldsymbol T}_{4}$. Nevertheless, the techniques used in that paper were specific to ${\boldsymbol T}_{4}$ and cannot be adapted to analyse Fréchet means in the space ${\boldsymbol T}_{m}$ of phylogenetic trees with $m(\geqslant5)$ leaves. To investigate the latter, this paper first studies the log map of ${\boldsymbol T}_{m}$, a generalisation of the inverse of the exponential map on a Riemannian manifold. Then, in terms of a modified version of the log map, we characterise Fréchet means in ${\boldsymbol T}_{m}$ that lie in top-dimensional or co-dimension one strata. We derive the limiting distributions for the corresponding sample Fréchet means, generalising our previous results. In particular, the results show that, although they are related to the Gaussian distribution, the forms taken by the limiting distributions depend on the co-dimensions of the strata in which the Fréchet means lie.
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