SIMPLICIAL INFERENCE 15
(Figure 1) then converts the second-order approximation to
given by the singular
value decomposition into a graphical display. As for standard multivariate biplots
the quality of the approximations is measured in terms of the proportion
of the total variability of the compositional data set retained by
an obvious requirement.
Figure 1 consists of an
which represents the center of the compositional
data set, a
for each of the parts, labeled
for each of theN cases,
, ••• ,
We term the join of
to a vertex
and the join of
two vertices i and j the
ij. These features constitute a biplot with the following
main properties for the interpretation of the compositional variability.
Links, mys and covariance structure.
The links and rays provide information
on the covariance structure of the compositional data set, in the sense that
It is tempting to imagine that (11.2) can be used to replace discredited
as a measure of the dependence between two components. Unfortunately, this
measure does not have subcompositional coherence.
A more useful result is the following. If links ij and
A particular case of this is when the two links are at right angles so that cos(
0, implying that there is zero correlation of the two log ratios. This is useful in
investigation of subcompositions for possible independence.
is the centroid (center of gravity) of
vertices 1, ... ,
Since ratios are preserved under formation of subcom-
positions it follows that the biplot for any subcomposition
is simply formed by
selecting the vertices corresponding to the parts of the subcomposition and taking
the center of the subcompositional biplot as the centroid of these vertices.
Coincident vertices. If
vertices i and j coincide or nearly so this means that
var(log(xifxj) is zero or nearly so, so that the ratio
is constant or nearly so.
Collinear vertices. If
a subset of vertices, say 1, ... ,
is collinear, then we know
from our comment on subcompositional analysis that the associated subcomposition
has a biplot that is one-dimensional, and then a technical argument leads us to the
conclusion that the subcomposition has one-dimensional variability. Technically,
this one-dimensionality is described by the constancy of (
2) log contrasts of the
, ••• ,
Inspection of these constant log contrasts may then give
further insights into the nature of the compositional variability.