• The Hellinger distance is a Euclidean distance between site vectors where the
abundance values are first divided by the site total abundance, and the result is
square-root transformed; this is called the Hellinger transformation. The
Hellinger transformation is also the chord transformation of square-root
transformed abundance data. The Hellinger transformation is obtained in one
step by decostand with the argument hellinger. The Hellinger distance
can also be computed in a single step by using function dist.ldc() of
package adespatial, whose argument hellinger is the default.
• The log-chord distance is a chord distance applied to log-transformed abundance
data. It can be obtained by first transforming the raw abundance data by ln(y + 1),
followed by the computation of the chord transformation or the chord distance, as
explained above. The log-chord distance can also be computed in a single step by
using function dist.ldc() of package adespatial with the argument
log.chord.
Legendre and Borcard (2018) have shown than the chord, Hellinger and logchord distances can be construed as chord distances computed on data transformed
by functions that are members of a series of Box-Cox normalizing transformations
(Eq. 3.1):
f y
ð Þ ¼ y
λ
À 1
À
Á =λ
ð3:1Þ
where λ ¼1 for the plain chord transformation, 0.5 for the Hellinger transformation
and 0 for the log-chord transformation. Note that λ ¼ 0 actually refers to the limit of f
( y) when λ approaches 0, which is ln( y), or ln(y + 1) for community composition
data (see Chap. 2). This sequence of transformations allows the normalization of
increasingly asymmetric frequency distributions. Any other exponent between 1 and
0, for example exponent 0.25 (double square root), could be used to pre-transform
the data before the chord transformation is applied.
# Load the required packages
library(ade4)
library(adespatial)
library(vegan)
library(gclus)
library(cluster)
library(FD)
# Source additional functions that will be used later in this
# Chapter. Our scripts assume that files to be read are in
# the working directory.
source("coldiss.R")
source("panelutils.R")
40
3 Association Measures and Matrices
abundance values are first divided by the site total abundance, and the result is
square-root transformed; this is called the Hellinger transformation. The
Hellinger transformation is also the chord transformation of square-root
transformed abundance data. The Hellinger transformation is obtained in one
step by decostand with the argument hellinger. The Hellinger distance
can also be computed in a single step by using function dist.ldc() of
package adespatial, whose argument hellinger is the default.
• The log-chord distance is a chord distance applied to log-transformed abundance
data. It can be obtained by first transforming the raw abundance data by ln(y + 1),
followed by the computation of the chord transformation or the chord distance, as
explained above. The log-chord distance can also be computed in a single step by
using function dist.ldc() of package adespatial with the argument
log.chord.
Legendre and Borcard (2018) have shown than the chord, Hellinger and logchord distances can be construed as chord distances computed on data transformed
by functions that are members of a series of Box-Cox normalizing transformations
(Eq. 3.1):
f y
ð Þ ¼ y
λ
À 1
À
Á =λ
ð3:1Þ
where λ ¼1 for the plain chord transformation, 0.5 for the Hellinger transformation
and 0 for the log-chord transformation. Note that λ ¼ 0 actually refers to the limit of f
( y) when λ approaches 0, which is ln( y), or ln(y + 1) for community composition
data (see Chap. 2). This sequence of transformations allows the normalization of
increasingly asymmetric frequency distributions. Any other exponent between 1 and
0, for example exponent 0.25 (double square root), could be used to pre-transform
the data before the chord transformation is applied.
# Load the required packages
library(ade4)
library(adespatial)
library(vegan)
library(gclus)
library(cluster)
library(FD)
# Source additional functions that will be used later in this
# Chapter. Our scripts assume that files to be read are in
# the working directory.
source("coldiss.R")
source("panelutils.R")
40
3 Association Measures and Matrices