6
.JOHN AITCHISON
as the a-power transform of
x.
Such an operation · arises in simplicial inference in
two distinct ways. First it may be of relevance directly because of the nature of the
sampling process. For example, in grain size studies of sediments, sediment samples
may be successively sieved through meshes of different diameters and the weights
of these successive separations converted into compositions based on proportions
by weight. Thus, though separation is based on the linear measurement diameter,
the composition is based essentially on a weight, or equivalently, a volume measure-
ment, with a power transformation being the natural connecting concept. More
indirectly, the power transformation can be useful in describing regression relations
for compositions. For example, the finding of [A5, Section 7. 7] of the relationship
of a (sand, silt, clay) sediment
x
to depth d can be expressed in the form
(4.8)
x =
~
o {log d · 77) o p,
where 77 is a composition playing the counterpart of regression coefficients and
p
is
a perturbation playing the role of error in more familiar regression situations.
It must be clear that together the operations perturbation o and power · play
roles in the geometry of
sd
analogous to translation and scalar multiplication in
RD and indeed can be used to define a vector space in Sd. We can now complete
this geometric picture by considering how to arrive at a suitable metric.
5. A simplicial metric
For the simplex
sd
any metric A :
sd
X
sd
---+
R+
must satisfy not only
the usual axioms that
A(x, y)
~
0,
A(x, y)
= 0 if and only if
x
=
y, A(x,
y)
=
A(y, x)
(symmetry),
A(x, y)
+
A(y, z)
~
A(x,
z) (the triangular inequality) and
A(a·x, b·y)
=
ab A(x, y)
(the scalar relationship), but also certain important logical
requirements associated with compositional and probabilistic statement analysis.
For example, since, as we have seen above, the change from composition x to
y
is
the same as from similarly perturbed compositions
xop
and
yop,
namely
yox-
1
,
we require the metric to be perturbation invariant in the sense that
A(xop,yop)
=
A(x, y)
for every
x, y, p.
This perturbation invariance along with scale invariance
and the obvious requirement of permutation invariance led
[All]
to the adoption
of the following simple and equivalent forms for a simplicial metric:
(5.1) A(x,X)
[
D {
x;
X;
}2]
!
8
log g(x) -log g(X)
(5.2)
[~
L
{log
;i.
_log
~i.
}
2
]
!
ij
3 3
(5.3)
where y; = log(x;/xD),
Yi
= log(X;/XD), i = 1, ... ,d, H = [h;j], with h;j = 2
fori= j and h;j = 1 fori ::j; j, as in [A5, Section 4.7], and g(x) is the geometric
mean of the components of
x.
We may note here that this metric also satisfies
another obvious requirement that the distance between any two compositions must
be at least as great as the distance between any corresponding subcompositions of
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