SIMPLICIAL INFERENCE

5

can always be expressed as a perturbation operation X

=

(X o x-

1) ox,

where

X

ox-1

is a perturbation in the group of perturbations in the unit simplex

Sd.

The

change from X to

x

is simply the inverse perturbation defined by (X o x-

1 )-1

=

x

o

x-

1.

Thus any measure of difference between compositions

x

and X must

be expressible in terms of one or other of these perturbations. A consequence

of this is that if we wish to define any

scalar measure of distance

between two

compositions

x

and

X,

say

~(x,X),

then we must ensure that it is a function of

the ratios x1/ X1, ... ,

xn/ Xn.

As we shall see later, this, together with attention

to

the need for scale invariance, subcompositional coherence and some other simple

requirements, has led

[All]

to advocate the use of

(4.4)

~(x,X)

=

(L)log(xi/x;)

-log(Xi/X;)}2Jll2

ij

as a simplicial metric, reinforcing an intuitive equivalent choice in

[A5,

Section 8.3].

In relation to probability statements the perturbation operation is a standard

process. Bayesians perturb the prior probability assessment

x

on a finite number

D

of hypotheses by the likelihood

p

to obtain the posterior assessment

X

through the

use of Bayes' formula ( 4.3). Again, in genetic selection, the population composition

x

of genotypes of one generation is perturbed by differential survival probabilities

represented by a perturbation

p

to obtain the composition

X

at the next genera-

tion, again by the perturbation probabilistic mechanism (4.3). In certain geological

processes, such as metamorphic change, sedimentation, crushing in relation to par-

ticle size distributions, change may be best modeled by such perturbation mecha-

nisms, where an initial specimen of composition xo is subjected to a sequence of

perturbations P1, ...

,pn

in reaching its current state

Xn:

X1

=

P1 o

Xo,

X2

=

P2 o X1, ... ,

Xn

=

Pn

o

Xn-1,

so that

(4.5)

Xn

=

(P1

o

P2

O • • • O

Pn)

0

Xo.

It is clear that in ( 4.5) we have the makings of some form of central limit theorem

but we delay consideration of this until we have completed the more mathematical

aspects of the simplex sample space.

A further role which perturbation plays in simplicial inference is in character-

izing imprecision or error. A simple example will suffice for the moment. In the

process of replicate analyses of aliquots of some specimen in an attempt to deter-

mine its composition,

e,

we may obtain different compositions X1, ...

,XN

because

of the imprecision of the analytic process. In such a situation we can model by

setting

(4.6)

Xr

=

e

0

Pr

r

=

1, ... 'N,

where the

Pr

are independent error compositions characterizing the imprecision.

4.3. Power transformation: another fundamental group operation in

the simplex. First we define the power operation and then consider its relevance

in simplicial inference. For any positive scalar

a

and any composition

X

E

sd'

we

define

(4.7)

a·x=C(x~,

...

,x0

)