structure. The result is a rectangular sites-by-edges Table E where the sequence of
edges connecting each site to the “root” of the network receive code “1” and the
others get code “0”.
Legendre and Legendre (2012, Sect. 14.3) give an example for fish dispersal from
the river mouth in a group of lakes interconnected by a river arborescence. In other
cases, for instance a two-dimensional grid consisting of rows of sampling devices
placed across a large river or a marine current at regular or irregular intervals, each
sampling point may influence (and hence may be connected to) the one directly
downstream of it, plus the two adjacent to the latter. If the process is assumed to
originate upstream, an imaginary point “0” is created upstream of the sampling area,
representing the root of the process, with connections to each of the points in the first
row of sites. All links present in the network are numbered. In Table E, the rows (i)
are the sites and the columns ( j) are the edges. The construction rule for AEM is that
E(i,j) ¼ 1 for links j connecting site i to the root (or site 0) of the graph; otherwise, E
(i,j) ¼ 0.
The edges (columns) of Table E may be weighted if deemed necessary, e.g. if the
transmission of the directional effects are supposed to be more difficult through some
paths than others.
The next step consists in transforming Table E into eigenfunctions. This can be
done in different ways, but the simplest is to compute a PCA of Table E and use the
matrix of principal components in scaling type 1 as explanatory variables. The AEM
method produces n À 1 eigenvectors with positive eigenvalues and none with
negative eigenvalues. The corresponding eigenfunctions, however, are divided in
two groups depicting positive or negative spatial correlation, so that the selection of
significant variables must be run separately for these two groups, in the same way as
for MEM variables.
A more detailed explanation about AEM construction is provided by Blanchet
et al. (2008b). The authors address the various issues related to edge definition and
weighting, which can greatly influence the results of AEM analysis.
As a first example, let us construct a fictitious set of AEM variables based on the
river arborescence shown by Legendre and Legendre (2012 Sect. 14.3 p. 889). This
example shows how to construct AEM variables in the simplest case, when one can
easily produce a matrix of edges by hand. The 8 nodes are 6 lakes connected by a
river arborescence, plus two river junction points (Fig. 7.11). Consider that samples
have been taken at these 8 nodes. The construction is done by function aem() of the
package adespatial.
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7 Spatial Analysis of Ecological Data
edges connecting each site to the “root” of the network receive code “1” and the
others get code “0”.
Legendre and Legendre (2012, Sect. 14.3) give an example for fish dispersal from
the river mouth in a group of lakes interconnected by a river arborescence. In other
cases, for instance a two-dimensional grid consisting of rows of sampling devices
placed across a large river or a marine current at regular or irregular intervals, each
sampling point may influence (and hence may be connected to) the one directly
downstream of it, plus the two adjacent to the latter. If the process is assumed to
originate upstream, an imaginary point “0” is created upstream of the sampling area,
representing the root of the process, with connections to each of the points in the first
row of sites. All links present in the network are numbered. In Table E, the rows (i)
are the sites and the columns ( j) are the edges. The construction rule for AEM is that
E(i,j) ¼ 1 for links j connecting site i to the root (or site 0) of the graph; otherwise, E
(i,j) ¼ 0.
The edges (columns) of Table E may be weighted if deemed necessary, e.g. if the
transmission of the directional effects are supposed to be more difficult through some
paths than others.
The next step consists in transforming Table E into eigenfunctions. This can be
done in different ways, but the simplest is to compute a PCA of Table E and use the
matrix of principal components in scaling type 1 as explanatory variables. The AEM
method produces n À 1 eigenvectors with positive eigenvalues and none with
negative eigenvalues. The corresponding eigenfunctions, however, are divided in
two groups depicting positive or negative spatial correlation, so that the selection of
significant variables must be run separately for these two groups, in the same way as
for MEM variables.
A more detailed explanation about AEM construction is provided by Blanchet
et al. (2008b). The authors address the various issues related to edge definition and
weighting, which can greatly influence the results of AEM analysis.
As a first example, let us construct a fictitious set of AEM variables based on the
river arborescence shown by Legendre and Legendre (2012 Sect. 14.3 p. 889). This
example shows how to construct AEM variables in the simplest case, when one can
easily produce a matrix of edges by hand. The 8 nodes are 6 lakes connected by a
river arborescence, plus two river junction points (Fig. 7.11). Consider that samples
have been taken at these 8 nodes. The construction is done by function aem() of the
package adespatial.
350
7 Spatial Analysis of Ecological Data