Foundations of the wald space for phylogenetic trees

Jonas Lueg, Maryam K. Garba, Tom M. W. Nye*, Stephan F. Huckemann

*Corresponding author for this work

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    Abstract

    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 characterisation 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 Fréchet 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.
    Original languageEnglish
    Article numbere12893
    Pages (from-to)1-45
    Number of pages45
    JournalJournal of the London Mathematical Society
    Volume109
    Issue number5
    Early online date16 Apr 2024
    DOIs
    Publication statusPublished - 1 May 2024

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