7.4.2.2 dbMEM Variables on Regular Sampling Designs
When the spatial coordinates correspond to points that are equispaced along a
transect or across a surface, the resulting dbMEM variables represent a series of
sinusoids of decreasing periods. For a transect with n regularly spaced points and
sampling interval s, the wavelength λ i of the eigenfunction with rank i is: λ i ¼ 2
(n + s)/(i + 1) (Guénard et al. 2010, Eq. 3)
1 . Let us construct and illustrate a
one-dimensional (Fig. 7.3) and a two-dimensional (Fig. 7.4) example, both
equispaced. Computation of the dbMEM will be done using function dbmem() of
package adespatial.
0
20
40
60
80
100
-1.5 0.0 1.5
X coordinate
dbMEM
1
0
20
40
60
80
100
-1.5
0.5
X coordinate
dbMEM
2
0
20
40
60
80
100
-1.5 0.5
X coordinate
dbMEM
4
0
20
40
60
80
100
-1.5 0.0
X coordinate
dbMEM
8
0
20
40
60
80
100
-1.5 0.0 1.5
X coordinate
dbMEM
15
0
20
40
60
80
100
-1.5
0.0
X coordinate
dbMEM
20
0
20
40
60
80
100
-1.5 0.0
X coordinate
dbMEM
30
0
20
40
60
80
100
-1.5 0.0
1.5
X coordinate
dbMEM
40
Fig. 7.3 Some of the 49 dbMEM variables with positive eigenvalues built from a transect with
100 equispaced points
1 A simple function to find the wavelength of rank i along a transect with n points for an intersite
distance s ¼ 1 is:
wavelength <À function(i, n) {2 * (n + 1) / (i + 1)}
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling
317