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
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