系统发生树的信息几何
Information geometry for phylogenetic trees
摘要：
我们提出了一个新的空间的系统发育树，我们称之为瓦尔德空间。其动机是开发一个适合于遗传学统计分析的空间，但具有基于比现有空间更生物学原则假设的几何形状：在瓦尔德空间中，如果树在遗传序列数据上诱导相似的分布，则它们是接近的。作为一个点集，瓦尔德空间包含先前发展的Billera-Holmes-Vogtmann（BHV）树空间;它也包含不连通的森林，像边积（EP）空间，但没有EP空间的某些奇点。研究了wald空间上两个相关的几何。第一个是几何形状的Fisher信息度量的字符分布引起的两个国家的对称马尔可夫替代过程中的每棵树。无穷小地，度量与Kullback-Leibler散度成比例，或者等价地，如我们所示，与f -散度成比例。第二个几何类似地获得，但使用相关的连续值高斯过程中的每棵树，它可以被看作是协方差阵的仿射不变度量的迹度量。我们推导了一个梯度下降算法，从协方差阵的周围空间投影到Wald空间。对于这两种几何形状，我们推导出计算方法来计算测地线在多项式时间和数值显示，两个信息几何（离散和连续）是非常相似的。特别地，测地线被外推近似。与BHV几何的比较表明，我们的规范和生物动机的空间是实质上不同的。
英文摘要：
We propose a new space of phylogenetic trees which we call wald space. The motivation is to develop a space suitable for statistical analysis of phylogenies, but with a geometry based on more biologically principled assumptions than existing spaces: in wald space, trees are close if they induce similar distributions on genetic sequence data. As a point set, wald space contains the previously developed Billera-Holmes-Vogtmann (BHV) tree space; it also contains disconnected forests, like the edge-product (EP) space but without certain singularities of the EP space. We investigate two related geometries on wald space. The first is the geometry of the Fisher information metric of character distributions induced by the two-state symmetric Markov substitution process on each tree. Infinitesimally, the metric is proportional to the Kullback-Leibler divergence, or equivalently, as we show, any to f -divergence. The second geometry is obtained analogously but using a related continuous-valued Gaussian process on each tree, and it can be viewed as the trace metric of the affine-invariant metric for covariance matrices. We derive a gradient descent algorithm to project from the ambient space of covariance matrices to wald space. For both geometries we derive computational methods to compute geodesics in polynomial time and show numerically that the two information geometries (discrete and continuous) are very similar. In particular geodesics are approximated extrinsically. Comparison with the BHV geometry shows that our canonical and biologically motivated space is substantially different.
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