Principal component analysis and the locus of the Fr,chet mean in the space of phylogenetic trees

• 成果类型: Article
• 署名作者: Nye, Tom M. W.; Tang, Xiaoxian; Weyenberg, Grady; Yoshida, Ruriko
• 署名单位: Newcastle University - UK; Texas A&M University System; Texas A&M University College Station; University of Hawaii System; University Hawaii Hilo; United States Department of Defense; United States Navy; Naval Postgraduate School
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asx047
• 发表日期: 2017
• 页码: 901922
• 关键词: algorithm; geometry; genes

摘要

Evolutionary relationships are represented by phylogenetic trees, and a phylogenetic analysis of gene sequences typically produces a collection of these trees, one for each gene in the analysis. Analysis of samples of trees is difficult due to the multi- dimensionality of the space of possible trees. In Euclidean spaces, principal component analysis is a popular method of reducing high- dimensional data to a low- dimensional representation that preserves much of the sample's structure. However, the space of all phylogenetic trees on a fixed set of species does not form a Euclidean vector space, and methods adapted to tree space are needed. Previous work introduced the notion of a principal geodesic in this space, analogous to the first principal component. Here we propose a geometric object for tree space similar to the kth principal component in Euclidean space: the locus of the weighted Frechet mean of k + 1 vertex trees when the weights vary over the k- simplex. We establish some basic properties of these objects, in particular showing that they have dimension k, and propose algorithms for projection onto these surfaces and for finding the principal locus associated with a sample of trees. Simulation studies demonstrate that these algorithms perform well, and analyses of two datasets, containing Apicomplexa and African coelacanth genomes respectively, reveal important structure from the second principal components.