• Use these eigenvectors as spatial explanatory variables in multiple regression or
RDA.
In the classical PCNM method described above, the diagonal values of the
truncated distance matrix are 0, indicating that a site is connected to itself. However,
Dray et al. (2006) showed that setting the diagonal values to 4 Â thresh instead of
0 resulted in interesting properties. In particular, the eigenvalues of the resulting
spatial eigenvectors, now called dbMEM (acronym for distance-based Moran’s
eigenvector maps) are proportional to Moran’s I coefficient computed on these
eigenfunctions using the pairs of sites that remain connected after truncation (Legendre and Legendre 2012). This makes it easy to identify the eigenvectors that model
positive spatial correlation, which are the ones used in most ecological studies: they
are those with eigenvalues larger than Moran’s I expectation (Eq. 7.4) to within a
multiplicative constant. Note, however, that the eigenfunctions obtained by PCNM
and dbMEM are the same.
The dbMEM method presents great advantages over trend-surface analysis. It
produces orthogonal (linearly independent) spatial descriptors and covers a much
wider range of spatial scales. It allows the modelling of any type of spatial
structures, as Borcard and Legendre (2002) have demonstrated through extensive
simulations.
The dbMEM method can work for any sampling design, but the spatial variables
are easier to interpret in the case of regular designs, as will be seen below. When the
design is irregular, it may happen that a large truncation value must be chosen to
allow all site-points to remain connected on the map. A large truncation value means
a loss of the finest spatial structures. Therefore, ideally, even an irregular sampling
design should ensure that the minimum distance allowing all points to be connected
is as short as possible. The most commonly applied solution is to compute the
minimum spanning tree of a single-linkage clustering of the site coordinates (see
Sect. 4.3.1) and retain the largest edge value. In cases where this distance is too large,
which may happen if the sampling sites are clustered on the territory, Borcard and
Legendre (2002) suggested (1) to add a limited number of supplementary points to
the spatial data to cut down the threshold distance, (2) compute the dbMEM variables, and (3) remove the supplementary points from the dbMEM matrix. This
ensures that the finest scales are better modelled. The trade-off is that the resulting
dbMEM variables are no longer totally orthogonal to one another, but if the number
of supplementary points is small with respect to the number of true points, the
departure from orthogonality remains small. Another possibility arises when the
largest gap or gaps correspond to the separation between to or more distinct entities
in the study area (e.g. separate patches of forest; different islands in an archipelago).
This problem will be discussed in Sect. 7.4.3.5, which addresses nested sampling
designs.
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7 Spatial Analysis of Ecological Data
RDA.
In the classical PCNM method described above, the diagonal values of the
truncated distance matrix are 0, indicating that a site is connected to itself. However,
Dray et al. (2006) showed that setting the diagonal values to 4 Â thresh instead of
0 resulted in interesting properties. In particular, the eigenvalues of the resulting
spatial eigenvectors, now called dbMEM (acronym for distance-based Moran’s
eigenvector maps) are proportional to Moran’s I coefficient computed on these
eigenfunctions using the pairs of sites that remain connected after truncation (Legendre and Legendre 2012). This makes it easy to identify the eigenvectors that model
positive spatial correlation, which are the ones used in most ecological studies: they
are those with eigenvalues larger than Moran’s I expectation (Eq. 7.4) to within a
multiplicative constant. Note, however, that the eigenfunctions obtained by PCNM
and dbMEM are the same.
The dbMEM method presents great advantages over trend-surface analysis. It
produces orthogonal (linearly independent) spatial descriptors and covers a much
wider range of spatial scales. It allows the modelling of any type of spatial
structures, as Borcard and Legendre (2002) have demonstrated through extensive
simulations.
The dbMEM method can work for any sampling design, but the spatial variables
are easier to interpret in the case of regular designs, as will be seen below. When the
design is irregular, it may happen that a large truncation value must be chosen to
allow all site-points to remain connected on the map. A large truncation value means
a loss of the finest spatial structures. Therefore, ideally, even an irregular sampling
design should ensure that the minimum distance allowing all points to be connected
is as short as possible. The most commonly applied solution is to compute the
minimum spanning tree of a single-linkage clustering of the site coordinates (see
Sect. 4.3.1) and retain the largest edge value. In cases where this distance is too large,
which may happen if the sampling sites are clustered on the territory, Borcard and
Legendre (2002) suggested (1) to add a limited number of supplementary points to
the spatial data to cut down the threshold distance, (2) compute the dbMEM variables, and (3) remove the supplementary points from the dbMEM matrix. This
ensures that the finest scales are better modelled. The trade-off is that the resulting
dbMEM variables are no longer totally orthogonal to one another, but if the number
of supplementary points is small with respect to the number of true points, the
departure from orthogonality remains small. Another possibility arises when the
largest gap or gaps correspond to the separation between to or more distinct entities
in the study area (e.g. separate patches of forest; different islands in an archipelago).
This problem will be discussed in Sect. 7.4.3.5, which addresses nested sampling
designs.
316
7 Spatial Analysis of Ecological Data