SIMPLICIAL INFERENCE 9

There are a number of useful and equivalent ways

[A5,

Chapter 4] in which to

summarize such a

sufficient

set of second-order moment characteristics: the

log

mtio covariance matrix

(6.7)

~(x)

=

[ai;]

=

cov{log(xdxn),log(x;fxn)],

using only the final component

xn

as the common ratio divisor; the

centered log

mtio covariance matrix

(6.8)

r(x)

=

[cov{log(xdg(x)}, log{x;/g(x)}];

or the variation matrix

(6.9)

with elements involving only pairs of components. These three dispersion char-

acteristics are equivalent: each can be derived from any other by simple matrix

operations

[A5,

Chapter 4]. Note that

Tis

symmetric, has zero diagonal elements,

and cannot be expressed as the standard covariance matrix of some vector, but has

the advantage of considering components only two at a time. We shall see later in

Section 8 that it has an important restricted negative definite property.

A first reaction to this variation matrix characterization is often surprise be-

cause it is defined in terms of variances only. The simplest statistical analogue is

in the use of a completely randomized block design in, say, an industrial experi-

ment From such a situation, information about

var(yi - Y;)

for all

i,

j is a sufficient

description of the variability for purposes of inference.

Note also that if

x

and y are independent compositions then, for any form

V

ofthe dispersion characteristic

V(x

o

y)

=

V(x)

+

V(y),

meeting, in particular, the

perturbation criterion (c), and

V(a·x)

=

a2 V(x),

meeting the power transformation

criterion (d).

Relative variances such as var{log(xdx;)} provide some compensation for de-

privation of the meaningless product-moment correlation interpretations. For ex-

ample,

Tij

=

0 means a perfect relationship between

Xi

and

x;

in the sense that

the ratio

xdx;

is constant, replacing the unusable idea of perfect positive corre-

lation between

Xi

and

x;

by one of perfect proportionality. Again, the larger the

value of

Tij

the more the departure from proportionality with

Tij

=

oo replacing

the unusable idea of zero correlation or independence between

Xi

and

x;.

Note

that if we are really interested in hypotheses of independence these are most ap-

propriately expressed in terms of independence of subcompositions. For example

independence of the (1,2,3)- and ( 4,5)-subcompositions would be reflected in the

following statements:

(6.10)

In the study of unconstrained variability in RD it is often convenient to have

available a measure of total variability, for example in principal component analysis

and in biplots. For such a sample space the trace of the covariance matrix is

the appropriate measure. Here we might consider trace{r(x)}, the trace of the

symmetric centered log ratio covariance matrix r(x). Equally, we might argue on

common sense grounds that the sum of all the possible relative variances in

T(x),

namely Lij var{log(xdx;)}, would be equally good. These two measures indeed

differ only by a constant factor and so we can define totvar(x), a measure of total