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On properties driving diversity index selection

Martin Frohn, Kerry Manson

TL;DR

This work explores how the most suitable diversity indices can be found, and how formalizing the requirement for diversity indices to capture high levels of PD, or to maintain a scoring of taxa in the presence of uncertain edge lengths drives the selection of a suitable index.

Abstract

Phylogenies are commonly used to represent the evolutionary relationships between species, and often these phylogenies are equipped with edge lengths that indicate degrees of evolutionary difference. Given such a phylogeny, a popular measure for the biodiversity of a subset of the species is the phylogenetic diversity (PD). But if we want to focus conservation efforts on particular species, we may use a phylogenetic diversity index, a function that shares out the PD value of an entire phylogeny across all of its species. With these indices, various species-level conservation strategies can be evaluated. This work explores how the most suitable diversity indices can be found. In particular, how formalizing the requirement for diversity indices to capture high levels of PD, or to maintain a scoring of taxa in the presence of uncertain edge lengths drives the selection of a suitable index. Furthermore, we provide illustrations of these new mechanisms for diversity index selection in a case study. This analysis includes the comparison to popular phylogenetic indices from the conservation literature.

On properties driving diversity index selection

TL;DR

This work explores how the most suitable diversity indices can be found, and how formalizing the requirement for diversity indices to capture high levels of PD, or to maintain a scoring of taxa in the presence of uncertain edge lengths drives the selection of a suitable index.

Abstract

Phylogenies are commonly used to represent the evolutionary relationships between species, and often these phylogenies are equipped with edge lengths that indicate degrees of evolutionary difference. Given such a phylogeny, a popular measure for the biodiversity of a subset of the species is the phylogenetic diversity (PD). But if we want to focus conservation efforts on particular species, we may use a phylogenetic diversity index, a function that shares out the PD value of an entire phylogeny across all of its species. With these indices, various species-level conservation strategies can be evaluated. This work explores how the most suitable diversity indices can be found. In particular, how formalizing the requirement for diversity indices to capture high levels of PD, or to maintain a scoring of taxa in the presence of uncertain edge lengths drives the selection of a suitable index. Furthermore, we provide illustrations of these new mechanisms for diversity index selection in a case study. This analysis includes the comparison to popular phylogenetic indices from the conservation literature.

Paper Structure

This paper contains 12 sections, 4 theorems, 40 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The diversity difference
  3. The diversity index continuity property
  4. Robust biodiversity problems
  5. Exact solution algorithms for the RDDP and MCDP
  6. Combinatorial properties of diversity indices
  7. On the optimization aspects of the RDDP and MCDP
  8. A comparative study of the robust and compatible diversity indices
  9. Selecting diversity indices under uncertainty
  10. Selecting diversity indices for uncertain tree shapes
  11. Discussion and conclusions
  12. A polynomial time exact solution algorithm for the MISP

Key Result

Proposition 1

Let $\hat{T}=(T,\ell)$ be a rooted phylogenetic $X$-tree and let $\varphi_{\hat{T}}$ be a diversity index of $\hat{T}$. Then, $\varphi_{\hat{T}}$ is continuous if and only if

Figures (7)

  • Figure 1: A rooted phylogenetic $X$-tree for $X=\{x_1,x_2,\dots,x_{10}\}$ and edge lengths as shown.
  • Figure 2: The rooted phylogenetic $X$-tree from Figure \ref{['fig::simTree1']}. Dashed edges $e$ indicate that coefficients $\gamma(x,e)$, $x\in X$, in \ref{['DIfunction']} are identical across all consistent diversity indices. Red edges $\{e_2,e_3,e_4\}$ share a neutrality condition. The light blue taxa form a set $Y_5$ to which Proposition \ref{['prop::FM']} applies.
  • Figure 3: A Yule tree $\hat{T}$ with $n=16$ taxa with most edge lengths suppressed for readability and dimension $d_{\hat{T}}=4$. Coefficients $\gamma(x,e)$, $x\in X$, are constant for consistent diversity indices for all dashed edges $e$. Red edges $\{e_2,e_5,e_6\}$ share a neutrality condition and light blue taxa form a set $Y_9=W_{12}\cup\{x_{12}\}$ (see Appendix \ref{['appendix:MISP']}).
  • Figure 4: A rooted phylogenetic $X$-tree $\hat{T}$ with $n=22$ taxa representing different species of albatross from birdtree.org. The taxa labels and edge lengths are suppressed for readability and we have dimension $d_{\hat{T}}=7$. Coefficients $\gamma(x,e)$, $x\in X$, are constant for consistent diversity indices for all dashed edges $e$. Both straight red edges $\{e_1,e_4,e_8,e_{11}\}$ and dashed red edges $\{e_2,e_5,e_9\}$ share a neutrality condition each and light blue taxa form a set which fits into Proposition \ref{['prop::FM']}.
  • Figure 5: On the left the degree of discontinuity $\text{dd}(T,\ell,3,\theta,e)$ (rescaled by factor $10^{-1}$ for $e=e_1$) for the phylogenetic $X$-tree $(T,\ell)$ in Figure \ref{['fig::simTree2']}, edges $e\in\{e_1,e_2,e_3,e_4\}$ and parameter $\theta$ on the $x$-axis. The graph on the right shows $\text{dd}(T,\ell,7,\theta,e)$ for the same set of parameters.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Corollary 2
  • Definition 3
  • Proposition 3
  • Proposition 4
  • Example 1
  • Example 2
  • Example 3