2

.JOHN AITCHISON

found in

[Wol], [Wo2].

For a detailed account of the history of such difficulties

and misinterpretations, see

[Al], [A2], [A5], [A12], [A13].

2. The nature of compositional and probability statement problems:

scale invariance

When we say that a problem is compositional we are recognizing that the

sizes

of our specimens are irrelevant. This trivial admission has far-reaching con-

sequences. A simple example can illustrate the argument. Consider two specimen

vectors

w

=

(1.6,

2.4, 4.0)

and

W

=

(3.0, 4.5, 7.5)

in R~ representing the weights

of the three parts (a, b, c) of two specimens of total weight 8 gm and 15 gm, re-

spectively. If we are interested in compositional problems, we recognize that these

are of the same composition, the difference in weight being taken account of by the

scale relationship

W

=

(15/B)w. More generally two specimen vectors

w

and

W

in

RIJ_

are compositionally equivalent, written

W "'w,

when there exists a positive

proportionality constant

p

such that

W

=

pw.

The fundamental requirement of

compositional data analysis can then be stated as follows: any meaningful func-

tion

f

of a specimen vector must be such that

f(W)

=

f(w)

when

W "' w,

or

equivalently

(2.1)

f(pw)

=

f(w)

for every

p

0.

In other words, the function

f

must be

invariant under the group of scale tmnsfor-

mations.

Since any group invariant function can be expressed as a function of any

maximal invariant h and since

h(w)

=

(wtfwv, ... , wv-dwv)

is such a maximal invariant we have the following important consequence.

Any meaningful (scale-invariant) function of a composition can

be expressed in terms of mtios of the components of the compo-

sition.

Note that there are many equivalent sets of ratios which may be used for the purpose

of creating meaningful functions of compositions. For example, a more symmetric

set of ratios such as

wj g(w),

where

g(w)

=

(w

1

•••

wv)

1fD is the geometric mean

of the components of

w,

would equally meet the scale-invariant requirement.

The choice of the unit simplex (1.1) as sample space is justified by the require-

ment of scale invariance since clearly the choice of units of measurement is irrel-

evant in a compositional problem. Statisticians will readily recognize the above

arguments in relation to probabilistic statements. It is clear that the standard

practice of measuring probabilities on the scale of

0

to 1 is merely a convention and

that any meaningful probabilistic statement can be expressed in terms of ratios,

equivalently odds.

3. Marginal concepts in the simplex: subcompositional and conditional

coherence

The marginal or projection concept for simplicial data is slightly more complex

than that for unconstrained vectors in RD, where a marginal vector is simply a

subvector of the full D-dimensional vector. For example, a geologist interested

only in the parts (Na20, KzO, Al203) of a ten-part major oxide composition of

a rock commonly forms the subcomposition based on these parts. Formally, the