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
)
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