The original PCNM method is now called distance-based MEM, or dbMEM.
dbMEM is actually a slightly modified form of PCNM; the difference is explained
below. dbMEM is a special case of a wider family of methods now called MEM
(Moran’s eigenvector maps; Dray et al. 2006). The acronym PCNM should
therefore be short-lived in the literature. We advocate the use of the generic
acronym MEM for the family, and dbMEM for the method that we originally
called PCNM.
7.4.2 Distance-Based Moran’s Eigenvector Maps (dbMEM)
and Principal Coordinates of Neighbour Matrices
(PCNM)
7.4.2.1 Introduction
Borcard and Legendre (2002) and Borcard et al. (2004) proposed to construct
eigenvectors of truncated matrices of geographical distances among sites. These
eigenvectors have interesting properties that make them highly desirable as
spatial explanatory variables. Let us briefly expose the principle of the original
PCNM method and see how it has been refined to become today’s dbMEM
technique.
The “classical” (first-generation) PCNM method worked as follows:
• Construct a matrix of Euclidean (geographic) distances among sites. A larger
distance means that the difficulty of communication between two sites increases.
• Truncate this matrix to retain only the distances among close neighbours. The
threshold thresh depends on the site coordinates. In most cases, it is chosen to be
as short as possible, but all points must remain connected by links smaller than or
equal to the truncation distance. Otherwise, different groups of eigenfunctions are
created, that model the spatial variation within separate subgroups of points but
not among these groups. How to choose the truncation threshold distance will be
described below. All pairs of points more distant than the threshold receive an
arbitrary “large” distance value corresponding to four times the threshold, i.e.,
4 Â thresh, or any larger multiplier.
• Compute a PCoA of the truncated distance matrix.
• In most studies, retain the eigenvectors that model positive spatial correlation
(Moran’s I larger than E(I), Eq. 7.4). This step requires the computation of
Moran’s I for all eigenvectors, since this quantity cannot be directly derived
from the eigenvalues.
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling
315
dbMEM is actually a slightly modified form of PCNM; the difference is explained
below. dbMEM is a special case of a wider family of methods now called MEM
(Moran’s eigenvector maps; Dray et al. 2006). The acronym PCNM should
therefore be short-lived in the literature. We advocate the use of the generic
acronym MEM for the family, and dbMEM for the method that we originally
called PCNM.
7.4.2 Distance-Based Moran’s Eigenvector Maps (dbMEM)
and Principal Coordinates of Neighbour Matrices
(PCNM)
7.4.2.1 Introduction
Borcard and Legendre (2002) and Borcard et al. (2004) proposed to construct
eigenvectors of truncated matrices of geographical distances among sites. These
eigenvectors have interesting properties that make them highly desirable as
spatial explanatory variables. Let us briefly expose the principle of the original
PCNM method and see how it has been refined to become today’s dbMEM
technique.
The “classical” (first-generation) PCNM method worked as follows:
• Construct a matrix of Euclidean (geographic) distances among sites. A larger
distance means that the difficulty of communication between two sites increases.
• Truncate this matrix to retain only the distances among close neighbours. The
threshold thresh depends on the site coordinates. In most cases, it is chosen to be
as short as possible, but all points must remain connected by links smaller than or
equal to the truncation distance. Otherwise, different groups of eigenfunctions are
created, that model the spatial variation within separate subgroups of points but
not among these groups. How to choose the truncation threshold distance will be
described below. All pairs of points more distant than the threshold receive an
arbitrary “large” distance value corresponding to four times the threshold, i.e.,
4 Â thresh, or any larger multiplier.
• Compute a PCoA of the truncated distance matrix.
• In most studies, retain the eigenvectors that model positive spatial correlation
(Moran’s I larger than E(I), Eq. 7.4). This step requires the computation of
Moran’s I for all eigenvectors, since this quantity cannot be directly derived
from the eigenvalues.
7.4 Eigenvector-Based Spatial Variables and Spatial Modelling
315