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.