Biodiversity Conservation and Phylogenetic Systematics

158

1

1

DT HHT

z

z

z

PP k z

N

kk

bg()=-expl(),,a + og lo

æ

è

ç

ö

ø

÷

æ

è

ç

ö

ø

÷+

=

+

++

+

++

å gg,N

é

ë

ê

ù

û

ú

(10a)

where HP,γ and HP,α denote respectively the gamma and alpha phylogenetic entropy.
When the species importance measure zik represents the ith species relative abun-
dance in the kth current-time assemblage, then zz++kk== 11 ,,++Nz//zN++=. In
this special case, we have^1 DTbg()=-exp/ëé()HHPP,,a Tùû. Thus an additive
decomposition for phylogenetic entropy HP holds (Pavoine et al. 2009 ; Mouchet
and Mouillot 2011 ), as for ordinary Shannon entropy (Jost 2007 ).
(c) When q = 2, the phylogenetic beta diversity can be expressed as

2 1

2

2

DT

Lz

Lz

T

T

N

i

k

N

ik

ii

i

b()= N.

Î=

+

Î

åå

å

iB

B

In the special case of zz++k==1, + N, this phylogenetic beta diversity of order 2
can be linked to quadratic entropy as

(^2) DT 11 QT^1 QT^1
bg()=-()//()- a/,

• 
  
  
  
  • (10b)
  
  
  
  

where Qγ and Qα denote respectively the gamma and alpha quadratic entropy. The
above formula is also applicable to non-ultrametric trees by replacing all T with T,
the mean branch length in the pooled assemblage; see Chiu et al. ( 2014 , Appendix
C) for a proof.

Normalized Phylogenetic Similarity Measures

For traditional abundance-based diversity, the most commonly used similarity mea-
sures include N-assemblage generalizations of the Jaccard et al. ( 1966 ) and Morisita-
Horn (Morisita 1959 ) measures. The latter three measures were integrated into a
class of CqN measures by Chao et al. ( 2008 ). Jost ( 2006 , 2007 ), Chao et al. ( 2008 ,
2012 ), and Chiu et al. ( 2014 ) have demonstrated that all the above measures are
monotonic transformations of beta diversity based on the ordinary Hill numbers.
This is an advantage of using the framework of Hill numbers: a direct link exists
between diversity and similarity (or differentiation) among assemblages.
Chiu et al. ( 2014 ) extended this framework by proposing four classes of similar-
ity (or differentiation) measures that are monotonic functions of phylogenetic beta
diversity. The basic idea is that the phylogenetic beta diversity, a ratio of gamma and
alpha phylogenetic Hill numbers, is independent of alpha and measures the pure
differentiation among assemblages. The phylogenetic beta component always lies

A. Chao et al.