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.