medium scales, could be explained by unmeasured environmental variables,
although one cannot exclude the influence of past events that could still show their
marks in the mite community (Borcard and Legendre 1994). Fine-scale structures are
more likely explainable by spatial correlation produced by neutral biotic processes.
Neutral processes include ecological drift (variation in species demography due to
random reproduction and random survival of individuals due to competition,
predator-prey interactions, etc.) and random dispersal (migration in animals, propagule dispersion in plants). Controlling for spatial correlation by means of dbMEM
variables when testing species-environment relationships will be briefly addressed in
Sect. 7.4.3.4.
Finally, note that the broad and fine scale dbMEM variables have a non-null
intersection despite the fact that the dbMEM variables are orthogonal: fraction
[j + m + n + o] totals À0.7%. This occurs because other variables (environment
and trend), which are not orthogonal to the dbMEM, are involved in the partitioning,
and also because the variation partitioning procedure involves subtractions of R
2 that
have been adjusted on the basis of different numbers of explanatory variables.
7.4.3 MEM in a Wider Context: Weights Other than
Geographic Distances
7.4.3.1 Introduction
The dbMEM method provides an elegant way of constructing sets of linearly
independent spatial variables. Since its publication, it has gained a wide audience
and has been applied in many research papers. But it is not the end of the story.
Dray et al. (2006) have greatly improved the mathematical formalism of the
original PCNM analysis by showing that it is a particular case of a wider family of
methods that they called Moran’s eigenvector maps (MEM). They demonstrated the
link between the eigenvalues of the MEM eigenvectors and Moran’s spatial correlation index, I (Eq. 7.3).
They reasoned that the relationship among sites, which is the basis for any spatial
eigenvector decomposition, actually has two components: (1) a list of links among
objects, represented by a connectivity matrix, and (2) a matrix of weights to be
applied to these links. In the simplest case, the weights are binary (i.e., either two
objects are linked, or they are not). In more complex models, non-negative weights
can be placed on the links; these weights represent the easiness of exchange
(of organisms, energy, information, etc.) between the points connected by the
links. For instance, link weights can be made to be inversely proportional to the
Euclidean or squared Euclidean distances among sites.
Furthermore, Dray et al. (2006) showed that (1) by using similarities instead of
distances among sites, (2) setting the relationship of the sites with themselves to null
similarity, and (3) avoiding a square-root standardization of the eigenvectors within
the PCoA procedure, one obtains a family of flexible methods (MEM) that can be
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling
333
although one cannot exclude the influence of past events that could still show their
marks in the mite community (Borcard and Legendre 1994). Fine-scale structures are
more likely explainable by spatial correlation produced by neutral biotic processes.
Neutral processes include ecological drift (variation in species demography due to
random reproduction and random survival of individuals due to competition,
predator-prey interactions, etc.) and random dispersal (migration in animals, propagule dispersion in plants). Controlling for spatial correlation by means of dbMEM
variables when testing species-environment relationships will be briefly addressed in
Sect. 7.4.3.4.
Finally, note that the broad and fine scale dbMEM variables have a non-null
intersection despite the fact that the dbMEM variables are orthogonal: fraction
[j + m + n + o] totals À0.7%. This occurs because other variables (environment
and trend), which are not orthogonal to the dbMEM, are involved in the partitioning,
and also because the variation partitioning procedure involves subtractions of R
2 that
have been adjusted on the basis of different numbers of explanatory variables.
7.4.3 MEM in a Wider Context: Weights Other than
Geographic Distances
7.4.3.1 Introduction
The dbMEM method provides an elegant way of constructing sets of linearly
independent spatial variables. Since its publication, it has gained a wide audience
and has been applied in many research papers. But it is not the end of the story.
Dray et al. (2006) have greatly improved the mathematical formalism of the
original PCNM analysis by showing that it is a particular case of a wider family of
methods that they called Moran’s eigenvector maps (MEM). They demonstrated the
link between the eigenvalues of the MEM eigenvectors and Moran’s spatial correlation index, I (Eq. 7.3).
They reasoned that the relationship among sites, which is the basis for any spatial
eigenvector decomposition, actually has two components: (1) a list of links among
objects, represented by a connectivity matrix, and (2) a matrix of weights to be
applied to these links. In the simplest case, the weights are binary (i.e., either two
objects are linked, or they are not). In more complex models, non-negative weights
can be placed on the links; these weights represent the easiness of exchange
(of organisms, energy, information, etc.) between the points connected by the
links. For instance, link weights can be made to be inversely proportional to the
Euclidean or squared Euclidean distances among sites.
Furthermore, Dray et al. (2006) showed that (1) by using similarities instead of
distances among sites, (2) setting the relationship of the sites with themselves to null
similarity, and (3) avoiding a square-root standardization of the eigenvectors within
the PCoA procedure, one obtains a family of flexible methods (MEM) that can be
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling
333