instance, if the matrix contains 10 descriptors, the number of plots to draw would be
equal to (10 Â 9)/2 ¼ 45. Such a series of scatter diagrams would allow neither to
bring out the most important structures of the data, nor to visualise the relationships
among descriptors (which, in general, are not linearly independent of one another).
The aim of ordination methods is to represent the data along a reduced number of
orthogonal axes, constructed in such a way that they represent, in decreasing order,
the main trends of variation of the data. These trends can then be interpreted visually
or in association with other methods like clustering or regression. Here we shall
describe five techniques. All these methods are descriptive: no statistical test is
provided to assess the significance of the structures detected. That is the role of
constrained ordination, a family of methods that will be presented in Chap. 6.
5.2.2 Ordination in Reduced Space
Most usual ordination methods (except NMDS) are based on the extraction of the
eigenvectors of an association matrix. They can be classified according to the
dissimilarity preserved among sites and to the type of variables that they can handle.
Legendre and Legendre (2012, Table 9.1, p. 426) provide a table showing their
domains of application.
The basic principle of ordination in reduced space is the following. Imagine an
n  p data set containing n objects and p variables. The n objects can be represented
as a cluster of points in the p-dimensional space. Now, this cluster is generally not
spheroid: it is elongated in some directions and flattened in others. These directions
are not necessarily aligned with a single dimension (¼ a single variable) of the
multidimensional space. The direction where the cluster is most elongated corresponds to the direction of largest variance of the cluster. This is the first axis that an
ordination will extract. Indeed, the direction of largest variance corresponds to the
strongest gradient present in the data: this is where the most important information
resides. The next axis to be extracted is the second most important in variance,
provided that it is orthogonal (linearly independent, scalar product of 0) to the first
one. The process continues until all axes have been computed.
When there are a few major structures in the data (gradients or groups) and the
method has been efficient at extracting them, then the few first axes contain most of
the useful information, i.e., they have extracted most of the variance of the data. In
that case, the distances among sites in the projection in reduced space (most often
two-dimensional) are relatively similar to the distances among objects in the
multidimensional space. Note, however, that an ordination can be useful even
when the first axes account for small proportions of the variance. This may happen
when there are some interesting structures in an otherwise noisy data set. The
question arising is then: how many axes should one retain and interpret? In other
words, how many axes represent interpretable structures? The answer depends on
the method and the data; several procedures will be explained in due course to help
users answer this question.
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equal to (10 Â 9)/2 ¼ 45. Such a series of scatter diagrams would allow neither to
bring out the most important structures of the data, nor to visualise the relationships
among descriptors (which, in general, are not linearly independent of one another).
The aim of ordination methods is to represent the data along a reduced number of
orthogonal axes, constructed in such a way that they represent, in decreasing order,
the main trends of variation of the data. These trends can then be interpreted visually
or in association with other methods like clustering or regression. Here we shall
describe five techniques. All these methods are descriptive: no statistical test is
provided to assess the significance of the structures detected. That is the role of
constrained ordination, a family of methods that will be presented in Chap. 6.
5.2.2 Ordination in Reduced Space
Most usual ordination methods (except NMDS) are based on the extraction of the
eigenvectors of an association matrix. They can be classified according to the
dissimilarity preserved among sites and to the type of variables that they can handle.
Legendre and Legendre (2012, Table 9.1, p. 426) provide a table showing their
domains of application.
The basic principle of ordination in reduced space is the following. Imagine an
n  p data set containing n objects and p variables. The n objects can be represented
as a cluster of points in the p-dimensional space. Now, this cluster is generally not
spheroid: it is elongated in some directions and flattened in others. These directions
are not necessarily aligned with a single dimension (¼ a single variable) of the
multidimensional space. The direction where the cluster is most elongated corresponds to the direction of largest variance of the cluster. This is the first axis that an
ordination will extract. Indeed, the direction of largest variance corresponds to the
strongest gradient present in the data: this is where the most important information
resides. The next axis to be extracted is the second most important in variance,
provided that it is orthogonal (linearly independent, scalar product of 0) to the first
one. The process continues until all axes have been computed.
When there are a few major structures in the data (gradients or groups) and the
method has been efficient at extracting them, then the few first axes contain most of
the useful information, i.e., they have extracted most of the variance of the data. In
that case, the distances among sites in the projection in reduced space (most often
two-dimensional) are relatively similar to the distances among objects in the
multidimensional space. Note, however, that an ordination can be useful even
when the first axes account for small proportions of the variance. This may happen
when there are some interesting structures in an otherwise noisy data set. The
question arising is then: how many axes should one retain and interpret? In other
words, how many axes represent interpretable structures? The answer depends on
the method and the data; several procedures will be explained in due course to help
users answer this question.
152
5 Unconstrained Ordination