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