SIMPLICIAL INFERENCE

19

14. Perturbation analysis

14.1. Perturbation analysis of simple change. A common application in

compositional data analysis is to investigate the way in which a specimen in an

initial state with composition x is altered by some process to a· specimen in a final

state with composition X. As we have seen such a change may be characterized

by an unknown perturbation

1r,

so that

X= x

o

1r.

There are two distinct forms of

data sets which may allow the investigation of such perturbations.

Paired compositions.

When each

Xr

is paired with an associated

Xr (

r

=

1, ... , N), for example when each

Xr

and Xn refer to the same physical unit, before

and after treatment. Then we have essentially a set of observed perturbations

Pr=Xrox;

1

(r=1, ... ,N),

and so we can analyze these as a compositional data set, for example, by fitting a

CD (

~,

T) distribution.

Sepamte

data

sets. Here we have a set of specimens with compositions {

Xr :

r

= 1, ... , N1} in a state regarded as initial, and also a completely separate set

of specimens with compositions {

Xr :

r

=

1, ... , N F} in a state regarded as fi-

nal. Modeling here may suppose that the

Xr

are identically and independently

distributed as

f:.D(~,

T). Then, a generic X may be supposed to be an unknown

perturbation

1r

of a generic

x,

say

X

=

x

o

1r,

so that the

Xr

are identically and

independently distributed as

f:.D(~

o

1r,

T). With this modeling we can construct

the likelihood

of(~,

T,

1r)

for a given data set, and so carry out any inference on

7r.

14.2. Differential perturbation processes. Consider a process which re-

sults in an observable D-part composition

x(t)

=

(x1 (t), ...

,xv(t))

which varies

with some ordered variable such as time t, although other variables such as temper-

ature or depth may be equally appropriate. Since processes are commonly assumed

to take place continuously over time we can attempt to describe such a process

in a differential way by relating the composition

x(t

+

dt)

at time

t

+

dt

to the

composition

x(t)

at previous time

t

in terms of a small perturbation, say

dp(t).

Since such an infinitesimal perturbation will be a slight departure from the identity

perturbation (1/ D, ... , 1/ D), it is most simply expressed in the form

(14.1)

dp(t)

= (1/ D)(1

+

1h (t)dt, ... ,

1

+

6v(t)dt),

where, for the sake of generality, we have allowed the perturbation to depend on t.

Sometimes it is convenient to assume that such a perturbation is in the D-simplex

but since the perturbation operation is invariant with respect to scale there is

strictly no need for such a requirement. As long as the perturbation densities

6i(t)

(i

=

1, ... , D) are small enough to ensure that the elements of

dp(t)

are positive,

and since we are dealing with differential perturbations this will always be the case.

Thus we define a differential perturbation process for the composition

x(t)

as

(14.2)

x(t

+

dt)

=

x(t)

o

(1/ D)(1

+

1h(t)dt, ... ,

1

+

6v(t)dt),

from which we obtain, in terms of log ratios,

(14.3) 1

(xi(t

+

dt))

=

lo

(xi(t))

lo ( 1

+

6 i(t)) .

og

x;(t

+

dt)

g

x;(t)

+

g 1

+

6 ;(t)