modulated to optimize the construction of spatial variables. The MEM method
produces n - 1 spatial variables with positive and negative eigenvalues, allowing
the construction of a wide range of variables modelling positive and negative spatial
correlation. As in the special dbMEM case, the eigenvectors maximize Moran’s
I index, the eigenvalues being equal to Moran’s I multiplied by a constant. Therefore, the spatial structures of the data are extracted in such a way that the axes first
optimally display the positively autocorrelated structures in decreasing order of
importance, and then the negatively autocorrelated structures in increasing order.
The general MEM method consists in defining two matrices describing the
relationships among the sites:
• a binary connectivity matrix B defining which pairs of sites are connected (1) and
which are not (0);
• a weighting matrix A providing the intensity of the connections.
The final spatial weighting matrix W results from the Hadamard (i.e., term-byterm) product of these two matrices, B and A.
The connectivity matrix B can be constructed on the basis of distances
(by selecting a distance threshold and connecting all points that are within that
distance) or by other connection schemes such as Delaunay triangulation, Gabriel
graph or others (described by Legendre and Legendre 2012 Sect. 13.3). The connection matrix can of course be customized to fit special needs — for instance by
only allowing connections among sites along the littoral zone of a lake (not across
water) or along the shoreline of an island.
Matrix A is not mandatory, but it is often used to weight the connections
according to distance, e.g. by inverse distance or inverse squared distance, since it
is ecologically realistic to assume that a process influences a community with an
intensity decreasing with distance. The choice of both matrices is very important
because it greatly affects the structure of the spatial variables obtained. These variables, in turn, condition the results of the spatial analysis, especially in the case of
irregular sampling: “In the case of regular sampling (e.g., a regular grid), structures
defined by eigenvectors are roughly similar for different definitions of W. For
irregular distributions of sites, however, the number of positive/negative eigenvalues and the spatial structures described by their associated eigenvectors are
greatly influenced by the spatial relationships defined in W.” (Dray et al. 2006).
These authors provide the following general recommendations:
“The choice of the spatial weighting matrix W is the most critical step in spatial
analysis. This matrix is a model of the spatial interactions recognized among the
sites, all other interactions being excluded. In some cases, a theory-driven specification can be adopted, and the spatial weighting matrix can be constructed based
upon biological considerations [...]. In most situations, however, the choice of a
particular matrix may become rather difficult and a data-driven specification could
then be applied. Under this latter approach, the objective is to select a configuration
of W that results in the optimal performance of the spatial model.”
For data-driven model specification, the authors proposed a procedure starting
with a user-defined set of possible spatial weighting matrices. For each candidate,
one computes the MEM eigenfunctions, reorders them according to their
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7 Spatial Analysis of Ecological Data
produces n - 1 spatial variables with positive and negative eigenvalues, allowing
the construction of a wide range of variables modelling positive and negative spatial
correlation. As in the special dbMEM case, the eigenvectors maximize Moran’s
I index, the eigenvalues being equal to Moran’s I multiplied by a constant. Therefore, the spatial structures of the data are extracted in such a way that the axes first
optimally display the positively autocorrelated structures in decreasing order of
importance, and then the negatively autocorrelated structures in increasing order.
The general MEM method consists in defining two matrices describing the
relationships among the sites:
• a binary connectivity matrix B defining which pairs of sites are connected (1) and
which are not (0);
• a weighting matrix A providing the intensity of the connections.
The final spatial weighting matrix W results from the Hadamard (i.e., term-byterm) product of these two matrices, B and A.
The connectivity matrix B can be constructed on the basis of distances
(by selecting a distance threshold and connecting all points that are within that
distance) or by other connection schemes such as Delaunay triangulation, Gabriel
graph or others (described by Legendre and Legendre 2012 Sect. 13.3). The connection matrix can of course be customized to fit special needs — for instance by
only allowing connections among sites along the littoral zone of a lake (not across
water) or along the shoreline of an island.
Matrix A is not mandatory, but it is often used to weight the connections
according to distance, e.g. by inverse distance or inverse squared distance, since it
is ecologically realistic to assume that a process influences a community with an
intensity decreasing with distance. The choice of both matrices is very important
because it greatly affects the structure of the spatial variables obtained. These variables, in turn, condition the results of the spatial analysis, especially in the case of
irregular sampling: “In the case of regular sampling (e.g., a regular grid), structures
defined by eigenvectors are roughly similar for different definitions of W. For
irregular distributions of sites, however, the number of positive/negative eigenvalues and the spatial structures described by their associated eigenvectors are
greatly influenced by the spatial relationships defined in W.” (Dray et al. 2006).
These authors provide the following general recommendations:
“The choice of the spatial weighting matrix W is the most critical step in spatial
analysis. This matrix is a model of the spatial interactions recognized among the
sites, all other interactions being excluded. In some cases, a theory-driven specification can be adopted, and the spatial weighting matrix can be constructed based
upon biological considerations [...]. In most situations, however, the choice of a
particular matrix may become rather difficult and a data-driven specification could
then be applied. Under this latter approach, the objective is to select a configuration
of W that results in the optimal performance of the spatial model.”
For data-driven model specification, the authors proposed a procedure starting
with a user-defined set of possible spatial weighting matrices. For each candidate,
one computes the MEM eigenfunctions, reorders them according to their
334
7 Spatial Analysis of Ecological Data