18
JOHN AITCHISON
follows:
(12.1)
pr(t
~
llx)
~
I -
pr(t
~
Olx)
~
F (
ao
+
t,
a;logx;) , whe.e
t,
a;
~
0.
Hypotheses that the categorization depends only on a subcomposition, for exam-
ple on the subcomposition formed from parts 1, ... ,
C
is then simply specified by
ac+I
= · · · =
an
=
0,
and so the whole lattice of subcompositional hypotheses can
be readily and systematically investigated.
A striking example of the use of this technique is to be found in discriminating
between two types of limestone. Thomas and Aitchison
[T-A]
show that of the 17-
part (major-oxide, trace element) composition a simple major-oxide subcomposition
(CaO, Fe20a, MgO) provides excellent discrimination, equal to that of the full
composition.
13. Convex linear mixing problems
By a convex linear mixing problem we mean one in which some target
D-
composition y is visualized as arising from some convex linear combination
1r
=
{1r11
••• ,
7rc)
of
C
source or end member compositions x
1
, ••• , Xc,
as
(13.1)
One such problem is where information is available only on the target in the form of
a compositional data set and the objective is to attempt to find fixed end member
source compositions from which the target compositions could have arisen through
varying mixtures of these source compositions. For successful approaches to, and
algorithms for, this difficult problem, see [R] and [We].
A completely different problem arises if we have some information about the
sources. If there are
C
sources with precise compositions x
1
, ... ,
xc the problem
commonly posed is to what extent a target composition y can be expressed as a
convex linear combination {13.1) of the source compositions. The problem is often
made more complicated by the following features which increase substantially its
statistical nature.
(a) The source compositions are not precise but are defined only as an ob-
served cluster of compositions.
(b) The target composition is not precise but is defined only as a cluster of
compositions.
If we denote by y and x
1 , ••• ,
xc generic target and source compositions, then the
question of interest is the extent to which the data may support a relationship
of the form (13.1). A reasonable technique for resolving this problem
[A-B]
is
to use the
ith
source samples to assess in terms of
.CP(~i•
'li)
the compositional
variability of the
ith
source. Although the distributional problem of determining the
consequent variability of y in (13.1) cannot be resolved exactly, there are excellent
approximations in terms of logistic normal and skew logistic normal distributions
which allow the evaluation of the likelihood function for
7r
and hence the testing of
various hypotheses concerning
1r.
Aitchison and Bacon-Shone
[A-B]
also consider
other models of convex linear mixing, for example where
7r
may be variable and
where there may extra variability characterized by a perturbation of (13.1). Such
models are of interest in investigating the sources of pollution; see
[A-B]
for one
such application.
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