SIMPLICIAL INFERENCE
13
There are other classes of distributions on
8"-.
The fact that the Ld(J.L, I:.)
class is simply related to the Nd(J.L, I:.) class in
Jl::1
by way of the log ratio trans-
formation led
[A5]
to consider other transformations from
8"-
to
Jl::1
and defining
corresponding logistic-normal classes of distributions, the multiplicative and par-
titioned classes, which are directed at specific practical problems in compositional
data analysis. Also
[A4], [A5,
Section 13.4] extends the Ld(J.L,I:.} class by the in-
troduction of a single parameter to produce a generalization which includes both
the Dirichlet class and the Ld (J.L, I:.) logistic normal class. While this is a useful
extension, it is somewhat restricted by computational problems involving multiple
integrals. A more promising generalization, which is simpler computationally, is
an extension based on the recently introduced multivariate skew normal class of
distributions [Az-D]. In terms of a class of distributions on the simplex, a com-
position x can be said to have a logistic skew normal distribution if the log ratio
vector [log(x1 /xn), ... ,log(xd/xn)] has a multivariate skew normal distribution.
For recent applications of this class to compositional data problems, see [Mat-B-P],
[A-B].
For comparison with the fitting of parametric distributions to simplicial data
or for use when there is no satisfactory parametric class, resort may be made to a
non-parametric approach through kernel density estimation
[A-L].
10. Inferential principles: from theory to practice
So far in this paper we have concentrated on providing a sound logical basis for
study of data within the simplex, first by demonstrating that certain fundamental
aspects of compositional data analysis-scale invariance, subcompositional coher-
ence, and the central roles of the perturbation and power operations--shape the
nature of the appropriate algebra and geometry of the simplex sample space. Then
we have identified within the consequent study of probabilistic aspects, promising
parametric classes of distributions on the simplex. While, for this development
of the logical sequence of arguments, we have confined ourselves to developments
within the simplex it must be clear to statisticians, particularly in the relationship
of log ratio vectors to the multivariate normal distribution in the last section, that
a promising approach to compositional data analysis must be through the use of
transformation techniques along the same lines as logarithmic transformations or
Box-Cox transformations. This approach has formed the basis of what has come
to be known in some circles as log ratio analysis.
Basically the philosophy here is as follows. Any problem concerning composi-
tions can be expressed in terms of log ratios of the components of the composition.
Any convenient sufficient set of log ratios can be used. For example the set of final
divisor log ratios
(10.1}
Yi
=
log(xdxv),
i
=
1, ...
,d,
can be used with inverse transformation
(10.2)
Xi
=
exp(yi)/{exp(yt)
+ · · · +
exp(yd)
+
1},
i
=
1, ...
,d,
x
D
=
1/ { exp(yt)
+ .. · +
exp(yd)
+
1 }.
Note that transformations
(10.1)
and (10.3), which are asymmetric in the com-
ponents of the composition, form a mapping between the unit simplex
sd
and
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