SIMPLICIAL INFERENCE
11
moments of any order:
D 1
log Mx(u)
=LUi
log~i-
4.
uTuT
+terms of higher order,
i=l
where
~
and
T
are the center and variation matrix of the distribution. Moments
can also be found through a differentiation process.
If
we denote by
a
1
aui \
u;
differentiation with respect to
Ui,
after
u;
is expressed as
-Ekt-lUk,
then we have
the following useful results:
8logMx(0)/8ui \ u;
= log(~d~;),
ff2logMx(0)/8(ui \
u;)2 = var{log(xdx;)}.
M7: A limit property.
Let {xn} be a sequence
of compositions with density
functions {in} and Mellin transforms {Mn}·
If Mn(u) --* M(u)
and
M(u)
is the
Mellin transform of
f(x),
then
fn
converges in distribution
to
f.
7. Central limit theorem for compositions
An obvious question to ask about compositional variability
is whether there
is an analogue of the well-known limiting results for sequences of additive and
multiplicative changes leading to normal and lognormal variability through the
central limit theorems. As we have already noted the relationship ( 4.5) depicts
the result of a sequence of independent perturbations. In exactly the same way as
moment generating functions can be used to establish central limit theorems in RD
so we could use the above properties of the Mellin generating function to establish
a similar result for
Xn
in (4.5). A simple version for the case
where Pr
(r
= 1, 2, ... )
are independently and identically
distributed with cen(pr)
=
~
= (
6, ... ,
~D)
and
T(pr) = T(r =
1, ... ,
n)
leads to the following limiting Mellin
generating function
l
for
Yn
=
n
2
Xn:
(7.1)
M(u)
=
exp
(t,
ui
log~i- ~
uTuT).
Alternatively we can very simply relate (4.5) to an additive
central limit theorem
by rewriting it in terms of log ratios:
log(xndxnv)
= {log(pli/Pw) + · · · +
log(Pni/PnD)}
+
log(xodxov),
fori= 1, ... , d.
If
the perturbations are random then the sum within the brackets
will, under certain regularity conditions which need not divert us
here, tend for large
n towards a multivariate normal pattern of variability. It is a simple application of
distribution theory to deduce the form of the probability density function
f(x)
on
the unit simplex as
(7.2)
f(x) =
det(27rE)-!(x1

xd)-
1
exp
[-~{log(x-vfxv)-
JL}E- 1 {log(x-v/sv)-
JLV],
for
X
E
sD'
where
X-D
= (xl, ...
'xv_t), JL
is a d row vector and E is a positive
definite square matrix of order d. This is the
parametric class of additive logistic
normal distributions
Ld(JL,
E) described by [A-S].
This result differs from the
Mellin transform result only in the parameterization
of the parameters. We use the
notation
LD(~,
T)
to denote the distribution (7.3)
in this parameterization.
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