5.6 Nonmetric Multidimensional Scaling (NMDS)
5.6.1 Introduction
If the researcher’s priority is not to preserve the exact dissimilarities among objects
in an ordination plot, but rather to represent as well as possible the ordering relationships among objects in a small and specified number of axes, NMDS may be the
solution. Like PCoA, NMDS can produce ordinations of objects from any dissimilarity matrix. The method can cope with missing values, as long as there are enough
measures left to position each object with respect to a few others. NMDS is not an
eigenvalue technique, and it does not maximise the variability associated with
individual axes of the ordination. As a result, plots may arbitrarily be rotated or
inverted. The procedure goes as follows (very schematically; for details see Legendre and Legendre 2012, p. 512 et seq.):
• Specify the desired number m of axes (dimensions) of the ordination.
• Construct an initial configuration of the objects in the m dimensions, to be used as
a starting point of an iterative adjustment process. This is a tricky step, since the
end-result may depend upon the starting configuration. A PCoA ordination may
be a good starting point. Otherwise, try many independent runs with random
initial configurations.
• An iterative procedure tries to position the objects in the requested number of
dimensions in such a way as to minimize a stress function (scaled from 0 to 1),
which measures how far the dissimilarities in the reduced-space configuration are
from being monotonic to the original dissimilarities in the association matrix.
• The adjustment goes on until the stress value cannot be lowered, or until it reaches
a predetermined low value (tolerated lack-of-fit).
• Most NMDS programs rotate the final solution using PCA, for easier
interpretation.
For a given and small number of axes (e.g. m ¼ 2 or 3), NMDS often achieves a
less deformed representation of the dissimilarity relationships among objects than a
PCoA in the same number of dimensions. But NMDS is a computer-intensive
iterative technique exposed to the risk of suboptimal solutions. Indeed, the objective
stress function to be minimized often reaches a local minimum larger than the true
minimum.
5.6.2 Application to the Doubs Fish Data
NMDS can be performed in R with the elegant function metaMDS() of the vegan
package. metaMDS()accepts raw data or dissimilarity matrices. Let us apply it to
the fish abundances using the percentage difference index. metaMDS()uses random starts and iteratively tries to find the best possible solution. Species are added to
5.6 Nonmetric Multidimensional Scaling (NMDS)
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5.6.1 Introduction
If the researcher’s priority is not to preserve the exact dissimilarities among objects
in an ordination plot, but rather to represent as well as possible the ordering relationships among objects in a small and specified number of axes, NMDS may be the
solution. Like PCoA, NMDS can produce ordinations of objects from any dissimilarity matrix. The method can cope with missing values, as long as there are enough
measures left to position each object with respect to a few others. NMDS is not an
eigenvalue technique, and it does not maximise the variability associated with
individual axes of the ordination. As a result, plots may arbitrarily be rotated or
inverted. The procedure goes as follows (very schematically; for details see Legendre and Legendre 2012, p. 512 et seq.):
• Specify the desired number m of axes (dimensions) of the ordination.
• Construct an initial configuration of the objects in the m dimensions, to be used as
a starting point of an iterative adjustment process. This is a tricky step, since the
end-result may depend upon the starting configuration. A PCoA ordination may
be a good starting point. Otherwise, try many independent runs with random
initial configurations.
• An iterative procedure tries to position the objects in the requested number of
dimensions in such a way as to minimize a stress function (scaled from 0 to 1),
which measures how far the dissimilarities in the reduced-space configuration are
from being monotonic to the original dissimilarities in the association matrix.
• The adjustment goes on until the stress value cannot be lowered, or until it reaches
a predetermined low value (tolerated lack-of-fit).
• Most NMDS programs rotate the final solution using PCA, for easier
interpretation.
For a given and small number of axes (e.g. m ¼ 2 or 3), NMDS often achieves a
less deformed representation of the dissimilarity relationships among objects than a
PCoA in the same number of dimensions. But NMDS is a computer-intensive
iterative technique exposed to the risk of suboptimal solutions. Indeed, the objective
stress function to be minimized often reaches a local minimum larger than the true
minimum.
5.6.2 Application to the Doubs Fish Data
NMDS can be performed in R with the elegant function metaMDS() of the vegan
package. metaMDS()accepts raw data or dissimilarity matrices. Let us apply it to
the fish abundances using the percentage difference index. metaMDS()uses random starts and iteratively tries to find the best possible solution. Species are added to
5.6 Nonmetric Multidimensional Scaling (NMDS)
193