10
JOHN AITCHISON
variability, as
{6.11) totvar(x)
=
trace{(r(x)}
=
{1/D) 'L:var{log(xdxi)}.
ij
We may also note here that the simplicial metric {5.1)-(5.3) is compatible with
the above definitions of covariance analogous to the compatibility of Euclidean dis-
tance with the covariance matrix of an unconstrained vector. As an illustration
of this consider how we might construct a measure of the total variability for a
compositional data set V as at (6.3). The definition at {6.9) suggests that we
may obtain such a total measure, totvar
1
say, by replacing each var{log(xdxi)}
in {6.11) by its standard estimate. An alternative intuitive measure of total vari-
ation is surely the sum of all the possible distances between the
N
compositions,
namely totvar2 =
Lrs
Ll2{xn
x
8
).
The easily established proportional relation-
ship totvar1 =
[D/{N(N-
1)}]totvar2 confirms the compatibility of the defined
covariance structures and scalar measures of distance for compositional variability.
6.3. Generating functions for simplicial distributions. The character-
istic and moment generating functions for distributions in
J1:1
are familiar useful
tools of distributional analysis. It is relatively easy to design the analogous tools
for the study of simplicial distributions in
Sd.
The transform which seems to be
most suited to this purpose is a multivariate adaptation of the Mellin transform.
Let
{6.12)
For a composition
X
E
sd
with density function
p(x),
define its Mellin generating
function
Mx : Ud
-+
14
by the relationship
(6.13)
Mx(u)
=
r
xr' ..
·x'iJDp(x)dx.
Jsd
Note that the restriction of the vector
u
to the hyperplane
ud
rather than RD is
dictated by the need to meet the requirement of scale invariance, here ensured by
the fact that integrand is expressible in terms of ratios of the components of
x.
The Mellin generating function has perturbation, power and limit properties
similar to additive and scale properties of characteristic and moment generating
functions for distributions in RD.
Property Ml: Mx(O)
= 1.
Property M2:
If
x
and y are independent compositions then
Mxoy(u)
=
Mx(u)My(u).
Property M3:
If
a
is a positive scalar, then
Ma-x(u)
=
Mx(au).
Property M4:
If
b
is a fixed perturbation, then
Mbox(u)
=
br' · · ·
b'fyD Mx(u).
Property M5:
Combining M2 and M3, if
x
andy are independent compositions,
then
Maxob-y(u)
=
Mx(au)My(bu).
Property M6: Moment genemting properties.
In a manner similar to the use
of moment-generating functions in RD, we can obtain expansions which produce
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