12
JOHN AITCHISON
8. Log contrasts
In unconstrained multivariate analysis with sample space RD, substantial use
is made of properties of linear combinations (transformations) of the components
of vector observations, for example in all techniques involving eigen-analysis. In-
spection of the forms involved in the definitions of geometric center, dispersion
matrices, Mellin generating function, and the distribution emerging from the cen-
tral limit theorem suggest that the simplex analogue of a linear combination is a
log contrast [A3] of a composition
x
defined by
(8.1)
a1logx1
+ · · · +
av logxv,
where a
1
·
·+av
= 0. Just as linear combinations can be used to define subspaces
of the vector space RD by way of null spaces or range spaces, so log contrasts can
be used to identify subs paces of the already identified vector space Sd. We shall
see later the role that such log contrasts play in statistical analysis. The main
distributional result for log contrasts can be expressed as follows.
Property Ll:
If composition
X
has geometric center
e
and variation matrix
T,
then the vector£= (£1, ... ,£c) E R0
,
where
D
lr
=
l:ari
logx;,
r
=
1, ... ,C
i=1
has moment generating function G(t), where
t
= (t1, ... , tc), given by Gt(t) =
Mx(tAT),
where
A=
[ari]· A corollary of this result is that, if
x
follows a
.CP(e,
T)
distribution, then£ follows a
C0 (Aloge,-!ATAT)
distribution.
We may comment here on the negative signs that appear in this last result and,
for example, in the result
(7.1 ).
This is because of the nature of the variation matrix
T.
This can easily be shown to have a restricted form of negative definiteness in the
sense that, for any
u
E
Ud
as defined by
(6.12),
uTuT ~
0, so that the covariance
matrix
-!AT AT
in the above result is positive definite.
9. Parametric classes of distributions
The emergence of the logistic normal distribution LD-
1
(J1,
I:)
or LD(e, T) in
a central limit theorem ensures for this parametric class of distributions a central
role in the study of distributions on the simplex in a way similar to the multivariate
normal and lognormal distributions in
RD
and
Rf_.
In particular, in addition to
simple logistic normal subcompositional and conditional properties, this class of
distributions has the essential and useful properties of being closed under the basic
simplex operations of perturbation and power: see [A5, Chapter 6] for details. The
popular Dirichlet class Di(a) on the simplex with density function
!( ) /11
-1 Bv-1
x
ex: x
1

xrv
has so many drawbacks that it has virtually no role to play in simplicial inference.
For example, it has no simple perturbation or power transformation properties and
so is ill-suited to the basic group operations of the simplex. Moreover, it has so
many built-in independence properties that, apart from being a model of extreme
independence, it has almost no role to play in the investigation of the nature of the
dependence structure of compositional variability.
Previous Page Next Page