SIMPLICIAL INFERENCE 15
(Figure 1) then converts the second-order approximation to
Z
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
k~ +k~
k~
+ ...
+kh
of the total variability of the compositional data set retained by
Z
2,
with
k2 ka
an obvious requirement.
Figure 1 consists of an
origin 0
which represents the center of the compositional
data set, a
vertex
at position
(k 1vn,k2Vi2)/(N
-1)!
for each of the parts, labeled
1, ...
,D,
and a
case marker
at position
(N
-1)!(u,.1,ur2)
for each of theN cases,
labeled c1
, ••• ,
CN.
We term the join of
0
to a vertex
i
the my
Oi
and the join of
two vertices i and j the
link
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
(11.1)
(11.2)
(11.3)
cos(iOj)
~
corr[log{xi/g(x)},log{x;/g(x)}].
It is tempting to imagine that (11.2) can be used to replace discredited
corr(xi,x;)
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
kl
intersect in
M,
then
(11.4)
cos(iMk)
~
corr[log(xdx;), log(xk/xl)],
A particular case of this is when the two links are at right angles so that cos(
iMk)
~
0, implying that there is zero correlation of the two log ratios. This is useful in
investigation of subcompositions for possible independence.
Subcompositional analysis.
The center
0
is the centroid (center of gravity) of
the
D
vertices 1, ... ,
D.
Since ratios are preserved under formation of subcom-
positions it follows that the biplot for any subcomposition
s
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
Xi/x;
is constant or nearly so.
Collinear vertices. If
a subset of vertices, say 1, ... ,
C
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 (
C-
2) log contrasts of the
components x
1
, ••• ,
xc.
Inspection of these constant log contrasts may then give
further insights into the nature of the compositional variability.
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