jaccard {abdiv}  R Documentation 
Beta diversity for presence/absence data
Description
These functions transform the input vectors to binary or presence/absence format, then compute a distance or dissimilarity.
Usage
jaccard(x, y)
sorenson(x, y)
kulczynski_first(x, y)
kulczynski_second(x, y)
rogers_tanimoto(x, y)
russel_rao(x, y)
sokal_michener(x, y)
sokal_sneath(x, y)
yule_dissimilarity(x, y)
Arguments
x , y 
Numeric vectors 
Details
Many of these indices are covered in Koleff et al. (2003), so we adopt their
notation. For two vectors x
and y
, we define three quantities:

a
is the number of species that are present in bothx
andy
, 
b
is the number of species that are present iny
but notx
, 
c
is the number of species that are present inx
but noty
, and 
d
is the number of species absent in both vectors.
The quantity d
is seldom used in ecology, for good reason. For
details, please see the discussion on the "double zero problem," in section
2 of chapter 7.2 in Legendre & Legendre.
The Jaccard index of dissimilarity is 1  a / (a + b + c)
, or
one minus the proportion of shared species, counting over both samples
together. Relation of jaccard()
to other definitions:
Equivalent to R's builtin
dist()
function withmethod = "binary"
.Equivalent to
vegdist()
withmethod = "jaccard"
andbinary = TRUE
.Equivalent to the
jaccard()
function inscipy.spatial.distance
, except that we always convert vectors to presence/absence.Equivalent to
1  S_7
in Legendre & Legendre.Equivalent to
1  \beta_j
, as well as\beta_{cc}
, and\beta_g
in Koleff (2003).
The SÃ¸renson or Dice index of dissimilarity is
1  2a / (2a + b + c)
, or one minus the average proportion of shared
species, counting over each sample individually. Relation of
sorenson()
to other definitions:
Equivalent to the
dice()
function inscipy.spatial.distance
, except that we always convert vectors to presence/absence.Equivalent to the
sorclass
calculator in Mothur, and to1  whittaker
.Equivalent to
D_{13} = 1  S_8
in Legendre & Legendre.Equivalent to
1  \beta_{sor}
in Koleff (2003). Also equivalent to Whittaker's beta diversity (the second definition,\beta_w = (S / \bar{a})  1
), as well as\beta_{1}
,\beta_t
,\beta_{me}
, and\beta_{hk}
.
I have not been able to track down the original reference for the first and
second Kulczynski indices, but we have good formulas from Legendre &
Legendre. The first Kulczynski index is 1  a / (b + c)
, or
one minus the ratio of shared to unshared species.
Relation of kulczynski_first
to other definitions:
Equivalent to
1  S_{12}
in Legendre & Legendre.Equivalent to the
kulczynski
calculator in Mothur.
Some people refer to the second Kulczynski index as the KulczynskiCody index. It is defined as one minus the average proportion of shared species in each vector,
d = 1  \frac{1}{2} \left ( \frac{a}{a + b} + \frac{a}{a + c} \right ).
Relation of kulczynski_second
to other definitions:
Equivalent to
1  S_{13}
in Legendre & Legendre.Equivalent to the
kulczynskicody
calculator in Mothur.Equivalent to one minus the Kulczynski similarity in Hayek (1994).
Equivalent to
vegdist()
withmethod = "kulczynski"
andbinary = TRUE
.
The RogersTanimoto distance is defined as
(2b + 2c) / (a + 2b + 2c + d)
. Relation of rogers_tanimoto()
to other definitions:
Equivalent to the
rogerstanimoto()
function inscipy.spatial.distance
, except that we always convert vectors to presence/absence.Equivalent to
1  S_2
in Legendre & Legendre.
The RusselRao distance is defined
(b + c + d) / (a + b + c + d)
, or the fraction of elements not present
in both vectors, counting double absences. Relation of russel_rao()
to
other definitions:
Equivalent to the
russelrao()
function inscipy.spatial.distance
, except that we always convert vectors to presence/absence.Equivalent to
1  S_{11}
in Legendre & Legendre.
The SokalMichener distance is defined as
(2b + 2c) / (a + 2b + 2c + d)
. Relation of sokal_michener()
to
other definitions:
Equivalent to the
sokalmichener()
function inscipy.spatial.distance
, except that we always convert vectors to presence/absence.
The SokalSneath distance is defined as
(2b + 2c) / (a + 2b + 2c)
. Relation of sokal_sneath()
to other
definitions:
Equivalent to the
sokalsneath()
function inscipy.spatial.distance
, except that we always convert vectors to presence/absence.Equivalent to the
anderberg
calculator in Mothur.Equivalent to
1  S_{10}
in Legendre & Legendre.
The Yule dissimilarity is defined as 2bc / (ad + bc)
. Relation
of yule_dissimilarity()
to other definitions:
Equivalent to the
yule()
function inscipy.spatial.distance
, except that we always convert vectors to presence/absence.Equivalent to
1  S
, whereS
is the Yule coefficient in Legendre & Legendre.
Value
The dissimilarity between x
and y
, based on
presence/absence. The Jaccard, Sorenson, SokalSneath, Yule, and both
Kulczynski dissimilarities are not defined if both x
and y
have no nonzero elements. In addition, the second Kulczynski index and the
Yule index of dissimilarity are not defined if one of the vectors has no
nonzero elements. We return NaN
for undefined values.