not present any directionality. In other words, the influence of any given point on its
surroundings does not depend on the direction.
There are other situations, however, where directionality matters. The most
obvious ones are the cases of streams or rivers. Consider community effects driven
by current: the physical process is geographically asymmetrical, the influence of a
site onto another following an upstream-downstream direction. Colonization of the
stream network by fish from the river mouth represents a different process, which
follows the opposite direction. dbMEM or MEM variables are computed on distance
or connectivity matrices where no directionality is specified. Therefore, information
about directionality is lost and the modelling, although adequate to reveal major
spatial structures, does not exploit all the potential of directional data. Trends must
not be extracted from the data prior to AEM analysis because directional processes
are expected to produce trends in the response data; so, a trend is a part of the
response data that one wants to model in AEM analysis.
This is the reason why Blanchet et al. (2008b) developed the Asymmetric
Eigenvector Maps modelling method. AEM is an eigenfunction-based technique
that uses information about the direction of the physical process, plus the same
information as MEM (spatial coordinates of sites, connection diagram, optional
weights) if needed. It works best on tree-like structures like river networks, on
two-dimensional sampling designs like series of cross-river traps, or on sampling
sites located in a large river or marine current. Depending on the process under
study, the origin(s), or root(s), in a river network may be located upstream (e.g. flow
of dissolved chemical substances, plankton dispersal) or downstream (fish invasion
routes).
For spatial transects or time series, AEM and MEM regression and canonical
models are very similar, and in most cases they explain the response data with
similar (although not strictly equal) R
2 . The AEM eigenfunctions are cosine-like, just
like MEM eigenfunctions, although the AEM have longer wavelengths than MEM
along transects. If the n observations are regularly spaced along the transect and the
sampling interval is s, the wavelength λ i of the AEM with rank i is λ i ¼ 2 ns/i. AEM
analysis should be preferred when modelling gradients and other spatial structures
generated by directional physical processes.
AEM analysis was devised for cases where physical forces drive the communities
in such a way that the causal relationships are directional. This is not the same as a
simple ecological gradient, where an ecological factor is spatially structured but the
communities can still interact in any direction. In the latter case, dbMEM and MEM
modelling are appropriate.
7.4.5.2 Principle and Application of the Method
The basic piece of information needed is a table where each site is described by the
connections (hereafter called “edges”, following graph-theory vocabulary) it has
with other sites located in the direction of the root(s) or origin(s) of the directional
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling
349
surroundings does not depend on the direction.
There are other situations, however, where directionality matters. The most
obvious ones are the cases of streams or rivers. Consider community effects driven
by current: the physical process is geographically asymmetrical, the influence of a
site onto another following an upstream-downstream direction. Colonization of the
stream network by fish from the river mouth represents a different process, which
follows the opposite direction. dbMEM or MEM variables are computed on distance
or connectivity matrices where no directionality is specified. Therefore, information
about directionality is lost and the modelling, although adequate to reveal major
spatial structures, does not exploit all the potential of directional data. Trends must
not be extracted from the data prior to AEM analysis because directional processes
are expected to produce trends in the response data; so, a trend is a part of the
response data that one wants to model in AEM analysis.
This is the reason why Blanchet et al. (2008b) developed the Asymmetric
Eigenvector Maps modelling method. AEM is an eigenfunction-based technique
that uses information about the direction of the physical process, plus the same
information as MEM (spatial coordinates of sites, connection diagram, optional
weights) if needed. It works best on tree-like structures like river networks, on
two-dimensional sampling designs like series of cross-river traps, or on sampling
sites located in a large river or marine current. Depending on the process under
study, the origin(s), or root(s), in a river network may be located upstream (e.g. flow
of dissolved chemical substances, plankton dispersal) or downstream (fish invasion
routes).
For spatial transects or time series, AEM and MEM regression and canonical
models are very similar, and in most cases they explain the response data with
similar (although not strictly equal) R
2 . The AEM eigenfunctions are cosine-like, just
like MEM eigenfunctions, although the AEM have longer wavelengths than MEM
along transects. If the n observations are regularly spaced along the transect and the
sampling interval is s, the wavelength λ i of the AEM with rank i is λ i ¼ 2 ns/i. AEM
analysis should be preferred when modelling gradients and other spatial structures
generated by directional physical processes.
AEM analysis was devised for cases where physical forces drive the communities
in such a way that the causal relationships are directional. This is not the same as a
simple ecological gradient, where an ecological factor is spatially structured but the
communities can still interact in any direction. In the latter case, dbMEM and MEM
modelling are appropriate.
7.4.5.2 Principle and Application of the Method
The basic piece of information needed is a table where each site is described by the
connections (hereafter called “edges”, following graph-theory vocabulary) it has
with other sites located in the direction of the root(s) or origin(s) of the directional
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling
349