# Phylogenetic Generalized Entropies

The classic measures reviewed in section “Generalized Entropies” were extended to incorporate phylogenetic distance between species. As mentioned in the Introduction and will be shown in section “Phylogenetic Hill numbers and related measures”, Faith's PD can be regarded as a phylogenetic generalization of species richness.

Rao's *quadratic entropy* takes account of both phylogeny and species abun-

dances (Rao 1982):

where *dij* denotes the phylogenetic distance (in years since divergence, number of DNA base changes, or other metric) between species *i* and *j*, and *pi* and *pj* denote the relative abundance of species *i* and *j*. This index measures the average phylogenetic distance between any two individuals randomly selected from the assemblage. Rao's *Q* represents a phylogenetic generalization of the Gini-Simpson index because in the special case of no phylogenetic structure (all species are equally related to one another), *dii* = 0 and *dij* = 1 (*i* ≠ *j*), it reduces to the Gini-Simpson index.

The *phylogenetic entropy HP* is a generalization of Shannon's entropy to incorporate phylogenetic distances among species (Allen et al. 2009):

where the summation is over all branches of a rooted phylogenetic tree, *Li* is the length of branch *i*, and *ai* denotes the summed relative abundance of all species descended from branch *i*.

For ultrametric trees, Faith's PD, Allen et al.'s *HP*, and Rao's *Q* can be united into a single parametric family of *phylogenetic generalized entropies* (Pavoine et al. 2009):

Here, *Li* and *ai* are defined in Eq. (2b) and *T* is the age of the root node of the tree. Then 0*I* = Faith's PD minus *T*; 1*I* is identical to Allen et al.'s entropy *HP* given in Eq. (2b); and 2*I* is identical to Rao's quadratic entropy *Q* given in Eq. (2a). In the special case that *T* = 1 (the tree height is normalized to unit length) and all branches have unit length, then the phylogenetic generalized entropy reduces to the classical generalized entropy defined in Eq. (1c), with species relative abundances {*p*1, *p*2, …, *pS*} as the tip-node abundances.

The abundance-sensitive (*q* > 0) phylogenetic generalized entropies provide useful information, but they do not obey the replication principle and thus have the same interpretational problems as their parent measures. This motivated Chao et al. (2010) to extend Hill numbers to phylogenetic Hill numbers, which obey the replication principle; see section “Phylogenetic Hill numbers and related measures”.