Phylogenetic diversity indices from an affine and projective viewpoint

TL;DR

This paper introduces more general phylogenetic diversity indices that can be derived from collections of subsets and collections of bipartitions of the given set of species, and provides a link between cluster-based and split-based phylogenetic diversity indices that uses a discrete analogue of the classical link between affine and projective geometry.

Abstract

Phylogenetic diversity indices are commonly used to rank the elements in a collection of species or populations for conservation purposes. The derivation of these indices is typically based on some quantitative description of the evolutionary history of the species in question, which is often given in terms of a phylogenetic tree. Both rooted and unrooted phylogenetic trees can be employed, and there are close connections between the indices that are derived in these two different ways. In this paper, we introduce more general phylogenetic diversity indices that can be derived from collections of subsets (clusters) and collections of bipartitions (splits) of the given set of species. Such indices could be useful, for example, in case there is some uncertainty in the topology of the tree being used to derive a phylogenetic diversity index. As well as characterizing some of the indices that we introduce in terms of their special properties, we provide a link between cluster-based and split-based phylogenetic diversity indices that uses a discrete analogue of the classical link between affine and projective geometry. This provides a unified framework for many of the various phylogenetic diversity indices used in the literature based on rooted and unrooted phylogenetic trees, generalizations and new proofs for previous results concerning tree-based indices, and a way to define some new phylogenetic diversity indices that naturally arise as affine or projective variants of each other.

Figures (11)

• Figure 1: (a) A rooted phylogenetic tree on the set $X = \{a,b,c,d,e\}$ of species. The root vertex is $r$ and all edges are weighted. The table gives the value $FP_r(x)$ of the fair proportion index on this rooted tree for each $x \in X$. (b) The unrooted phylogenetic tree with weighted edges on the same set $X$ of species obtained by suppressing the root of the tree in (a). The table gives the value $FP_u(x)$ of the fair proportion index on this unrooted tree for each $x \in X$.
• Figure 2: (a) The weighted clusters on $X$ corresponding to the edges of the rooted phylogenetic tree in Figure \ref{['fig:fp:ex:rooted:unrooted']}(a). (b) The matrix $\Gamma$ from Equation (\ref{['eq:matrix:linear:pdi']}) for the fair proportion index on $\mathcal{C}$, where $\mathcal{C}$ is the cluster system consisting of the clusters given in (a).
• Figure 3: Collapsing the edge with weight $6$ in the rooted phylogenetic tree $\mathcal{T}$ on $X=\{a,b,c,d\}$ yields the rooted phylogenetic tree $\mathcal{T}^*$ on $X$.
• Figure 4: This diagram depicts the relationship between a phylogenetic diversity index $\Phi$ on $\mathbb{L}(\mathcal{C})$ for a cluster system $\mathcal{C}$ on $X$ and a phylogenetic diversity index $\Phi'$ on $\mathbb{PD}(\mathcal{C})$ as described by Equation (\ref{['eq:commuting:phi:pd:phiprime']}).
• Figure 5: A diagram of the various maps we consider to study relationships between phylogenetic diversity indices. The left part of the diagram we have already seen in Figure \ref{['fig:diagram:first:part']}. In analogy to this, the right part of the diagram depicts phylogenetic diversity indices $\Psi$ and $\Psi'$ on $\mathbb{L}(\mathcal{S})$ and $\mathbb{PD}(\mathcal{S})$, respectively, where $\mathcal{S}$ is a split system on $X$. Finally $\tau$ associates with each weighting $\lambda$ of the splits in $\mathcal{S}$ a weighting $\omega = \tau(\lambda)$ of the clusters in a cluster system $\mathcal{C} = \mathcal{C}(\mathcal{S})$ that arises from $\mathcal{S}$ by (\ref{['eq:c:s']}).

Theorems & Definitions (11)

• Lemma 2.1
• Theorem 3.1
• Lemma 4.1
• Remark 4.2
• Theorem 4.3
• Corollary 4.4
• Theorem 5.1
• Theorem 6.1
• Corollary 6.2
• Theorem 6.3