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 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)
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