The Functional Distinctiveness of a species is the average functional
distance from a species to all the other in the given community. It is
computed as such:
$$
D_i(T) = 1 ~~ if ~~ T < min(d_{ij}), \\
D_i(T) =
\left( \frac{
\sum\limits_{j = 1 ~,
j \neq i ~,
d_{ij} \leq T}^S d_{ij} \times Ab_j
}{
\sum\limits_{
j = 1 ~,
j \neq i ~,
d_{ij} \leq T}^S Ab_j
} \right) \times \left(1 - \frac{
\sum\limits_{
j = 1 ~,
j \neq i ~,
d_{ij} \leq T}^S Ab_j
}{
N
} \right) ~~ if ~~ T \geq min(d_{ij}),
$$
with \(D_i\) the functional distinctiveness of species \(i\), \(N\)
the total number of species in the community and \(d_{ij}\) the
functional distance between species \(i\) and species \(j\). \(T\)
is the chosen maximal range considered. When presence-absence are used
\(Ab_j = 1/N\) and the term \( \left(1 - \frac{
\sum\limits_{
j = 1 ~,
j \neq i ~,
d_{ij} \leq T}^S Ab_j
}{
N
} \right)\) is replaced by 1.
IMPORTANT NOTE: in order to get functional rarity indices between 0
and 1, the distance metric has to be scaled between 0 and 1.