explanatory power, enters them one by one into the model and retains the model with
the lowest AIC c (corrected Akaike information criterion). When this is
done for all candidates, one retains the W matrix yielding the lowest AIC c .
The AIC c -based selection is but one possibility. One could also forward-select the
MEM within each candidate model using Blanchet et al. (2008a)’s double stopping
criterion and retain the model with the highest R
2
adj . This alternative, which had not
yet been devised when the Dray et al. (2006) paper was published, addresses the
concerns raised by these authors in their conclusion about the drawbacks of forward
selection procedures.
7.4.3.2 Generalized MEM Analysis of the Mite Data
Dray et al. (2006) used the oribatid mite data to illustrate MEM analysis. As an
example, we will duplicate their analysis, exploring some choices along the steps of
the method. Several packages will be used. The following example is based on
Stéphane Dray’s tutorial on MEM analysis, with our thanks to the author.
The functions available in adespatial and used below should make model
selection relatively easy. Of course, the final result depends upon a proper choice of a
class of model. The function test.W() is particularly useful as it combines
construction of MEM variables and model selection; examine the documentation
file of that function.
We will experiment with three classes of models:
• The first class is based on Delaunay triangulation with binary weights.
• The second class starts from the same connectivity matrix, to which weights are
added. The weighting function is based on Euclidean distances among the sites:
• f 2 ¼ 1 – (d/d max )
α where d is a geographic distance value and d max is the
maximum value in the distance matrix. This ensures that easiness of communication ranges from 1 (easy) to 0 (isolation).
• The third class evaluates a series of models based on a range of distances around
the points. All pairs of points within the distance considered are linked, the others
not. What should the range of distances be? This can be assessed by means of a
multivariate variogram of the response data. Variograms are plots of
semivariances against distance classes. Semivariance is a distance-dependent
measure of variation, which is used in the same way as in the correlograms
presented earlier (e.g. Bivand et al. 2013). A variant of this approach will weight
the links by the function of inverse distance that was used in the second model
class above. This last variant will duplicate the results presented in the Dray et al.
(2006) paper.
To be consistent with the dbMEM analysis, we shall restrict our model selection
to MEM with Moran’s I larger than its expectation (i.e., positive spatial correlation).
To compute an analysis with all MEM variables, change argument MEM.autocor
to "all" in all applications of function test.W().
First and second classes of MEM models: unweighted (binary) and distanceweighted Delaunay triangulation.
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling
335
the lowest AIC c (corrected Akaike information criterion). When this is
done for all candidates, one retains the W matrix yielding the lowest AIC c .
The AIC c -based selection is but one possibility. One could also forward-select the
MEM within each candidate model using Blanchet et al. (2008a)’s double stopping
criterion and retain the model with the highest R
2
adj . This alternative, which had not
yet been devised when the Dray et al. (2006) paper was published, addresses the
concerns raised by these authors in their conclusion about the drawbacks of forward
selection procedures.
7.4.3.2 Generalized MEM Analysis of the Mite Data
Dray et al. (2006) used the oribatid mite data to illustrate MEM analysis. As an
example, we will duplicate their analysis, exploring some choices along the steps of
the method. Several packages will be used. The following example is based on
Stéphane Dray’s tutorial on MEM analysis, with our thanks to the author.
The functions available in adespatial and used below should make model
selection relatively easy. Of course, the final result depends upon a proper choice of a
class of model. The function test.W() is particularly useful as it combines
construction of MEM variables and model selection; examine the documentation
file of that function.
We will experiment with three classes of models:
• The first class is based on Delaunay triangulation with binary weights.
• The second class starts from the same connectivity matrix, to which weights are
added. The weighting function is based on Euclidean distances among the sites:
• f 2 ¼ 1 – (d/d max )
α where d is a geographic distance value and d max is the
maximum value in the distance matrix. This ensures that easiness of communication ranges from 1 (easy) to 0 (isolation).
• The third class evaluates a series of models based on a range of distances around
the points. All pairs of points within the distance considered are linked, the others
not. What should the range of distances be? This can be assessed by means of a
multivariate variogram of the response data. Variograms are plots of
semivariances against distance classes. Semivariance is a distance-dependent
measure of variation, which is used in the same way as in the correlograms
presented earlier (e.g. Bivand et al. 2013). A variant of this approach will weight
the links by the function of inverse distance that was used in the second model
class above. This last variant will duplicate the results presented in the Dray et al.
(2006) paper.
To be consistent with the dbMEM analysis, we shall restrict our model selection
to MEM with Moran’s I larger than its expectation (i.e., positive spatial correlation).
To compute an analysis with all MEM variables, change argument MEM.autocor
to "all" in all applications of function test.W().
First and second classes of MEM models: unweighted (binary) and distanceweighted Delaunay triangulation.
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling
335