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.