Distance-Based Functional Diversity Measures and Their Decomposition: A Framework Based on Hill Numbers

<div><p>Hill numbers (or the “effective number of species”) are increasingly used to characterize species diversity of an assemblage. This work extends Hill numbers to incorporate species pairwise functional distances calculated from species traits. We derive a parametric class of functional Hill numbers, which quantify “the effective number of equally abundant and (functionally) equally distinct species” in an assemblage. We also propose a class of mean functional diversity (per species), which quantifies the effective sum of functional distances between a fixed species to all other species. The product of the functional Hill number and the mean functional diversity thus quantifies the (total) functional diversity, i.e., the effective total distance between species of the assemblage. The three measures (functional Hill numbers, mean functional diversity and total functional diversity) quantify different aspects of species trait space, and all are based on species abundance and species pairwise functional distances. When all species are equally distinct, our functional Hill numbers reduce to ordinary Hill numbers. When species abundances are not considered or species are equally abundant, our total functional diversity reduces to the sum of all pairwise distances between species of an assemblage. The functional Hill numbers and the mean functional diversity both satisfy a replication principle, implying the total functional diversity satisfies a quadratic replication principle. When there are multiple assemblages defined by the investigator, each of the three measures of the pooled assemblage (gamma) can be multiplicatively decomposed into alpha and beta components, and the two components are independent. The resulting beta component measures pure functional differentiation among assemblages and can be further transformed to obtain several classes of normalized functional similarity (or differentiation) measures, including <i>N</i>-assemblage functional generalizations of the classic Jaccard, Sørensen, Horn and Morisita-Horn similarity indices. The proposed measures are applied to artificial and real data for illustration.</p></div

Comparison of various differentiation measures for three pairs of habitats in the real data analysis based on abundance and function (<i>A&F</i>), on function (<i>F</i>) only, and abundance (<i>A</i>) only.

<p> = 0.550 and  = 0.535 for the pair (EM, MO);  = 0.561,  = 0.537 for the pair (EM, TR);  = 0.574,  = 0.559 for the pair (MO, TR).</p>#<p>Differentiation measures are the abundance-based local differentiation measure (1−<i>C_qN</i>) and regional differentiation measure (1−<i>U_qN</i>) obtained from partitioning Hill numbers <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0100014#pone.0100014-Chiu1" target="_blank">[36]</a>;</p><p>--- No measures for <i>q</i> = 1 and <i>q</i> = 2 because species abundances are not considered for measures based on function (<i>F</i>) only.</p

Differentiation profiles for the functional differentiation measures (left panel) and (right panel) as a function of order <i>q</i> for three pairs of habitats (EM vs. MO, EM vs. TR and MO vs. TR.)

<p>Differentiation profiles for the functional differentiation measures (left panel) and (right panel) as a function of order <i>q</i> for three pairs of habitats (EM vs. MO, EM vs. TR and MO vs. TR.)</p

Diversity profiles as a function of order <i>q</i> for ordinary Hill numbers <i>^qD</i> (left panel), functional Hill numbers (the second panel from the left), mean functional diversity <i>^qMD</i>(<i>Q</i>) (the third panel from the left) and (total) functional diversity <i>^qFD</i>(<i>Q</i>) (right panel) for three habitats (TR, MO, and EM).

<p>All the profiles show a consistent diversity pattern about the ordering of the three habitats: TR>MO>EM.</p

Two major classes of distance-overlap (or similarity) measures and their special cases based on the functional beta diversity .

<p>The corresponding differentiation measures are the one-complements of the similarity measures. (The indices <i>i</i> and <i>j</i> are used to identify species, <i>i</i>, <i>j</i> = 1, 2, …, <i>S</i>, and the indices <i>k</i> and <i>m</i> are used to identify assemblages, <i>k</i>, <i>m</i> = 1, 2, …, <i>N</i>.)</p><p>Notation.</p><p><i>z_ik</i> = the abundance of the <i>i</i>th species in the <i>k</i>th assemblage, , , and ; see text for details.  = sum of the pairwise distances between species in the pooled assemblage;  = sum of <i>FAD</i> over all possible pairs of assemblages (there are <i>N</i>² pairs of assemblages). <i>S</i> = species richness in the pooled assemblage.  = average species richness per assemblage.</p

A framework for Hill numbers, functional Hill numbers, mean functional diversity and (total) functional diversity of a single assemblage.

<p>A framework for Hill numbers, functional Hill numbers, mean functional diversity and (total) functional diversity of a single assemblage.</p

Comparison of various differentiation measures for Matrix I (with = 0.48, = 0.47) and Matrix II (with = 0.167, = 0.102) based on abundance and function (<i>A&F</i>), on function (<i>F</i>) only, and abundance (<i>A</i>) only.

#<p>Differentiation measures are the abundance-based local differentiation measure (1−<i>C_qN</i>) and regional differentiation measure (1−<i>U_qN</i>) obtained from partitioning Hill numbers <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0100014#pone.0100014-Chiu1" target="_blank">[36]</a>;</p><p>--- No measures for <i>q</i> = 1 and <i>q</i> = 2 because species abundances are not considered for measures based on function (<i>F</i>) only.</p

rockfish-phylo_tree

The phylogenetic tree as a newick format, which is mirror from ELE_1344_sm_AppendixS4.txt (Pavoine et al. 2009)

Appendix D. Derivation of four classes of similarity measures and related properties.

Derivation of four classes of similarity measures and related properties

Appendix G. Additional analysis of Examples 2 and 3.

Additional analysis of Examples 2 and 3