4
JOHN AITCHISON
4. Group operations in the simplex
4.1. The role of group operations in statistics. For every sample space
there are basic group operations which, when recognized, dominate clear thinking
about data analysis. In RD the two operations, translation (or displacement) and
scalar multiplication, are so familiar that their fundamental role is often overlooked.
Yet the change from
y
to
Y
=
y
+
t
by the translation
t
or to
Y
=
ay
by the
scalar multiple
a
are at the heart of statistical methodology for RD sample spaces.
For example, since the translation relationship between y1 and
Yi
is the same as
that between y2 and
Y2
if and only if
Yi
and }2 are equal translations
t
of
y1
and
y2
,
any definition of a difference or a distance meaure must be such that
the meaure is the same for
(y1
,
Yt)
a for
(y1
+
t,
Y1
+
t)
for every translation
t.
Technically this is a requirement of invariance under the group of translations.
This is the reaon, though seldom expressed because of its obviousness in this
simple space, for the use of the mean vector
J.L
=
E(y)
and the covariance matrix
E
=
V(y)
=
E{(y- J.L)(y- J.L)T}
a meaningful meaures of central tendency and
dispersion. Recall also, for further reference, two baic properties: for a fixed
translation
t,
(4.1) E(y
+
t)
=
E(y)
+
t,
V(y
+
t)
=
V(y).
The second operation, that of scalar multiplication, also plays a substantial role
in, for example, linear forms of statistical analysis such a principal component
analysis, where linear combinations a 1y1
+ · · · +
anYn
with certain properties are
sought. Recall, again for further reference, that for a fixed scalar multiple,
(4.2) E(ay)
=
aE(y), V(ay)
=
a2 E(y).
Similar considerations of groups of fundamental operations are essential for
other sample spaces. For example, in the analysis of directional data, a in the study
of the movement of tectonic plates, it wa recognition that the group of rotations on
the sphere plays a central role and the use of a satisfactory representation of that
group that led
[C]
to the production of the essential statistical tool for spherical
regression.
4.2. Perturbation: a fundamental group operation in the simplex.
By analogy with the group operation arguments for RD, the obvious questions
for a simplex sample space are whether there is an operation on a composition
x,
analogous to translation in RD, which transforms it into X, and whether this
can be used to characterize 'difference' between compositions. The answer is to be
found in the perturbation operator a defined in [A5, Section 2.8]. If we define a
perturbation
p
a a differential scaling operator
p
= (p1
, .••
,pn)
E
Sd
and denote
by o the perturbation operation, then we can define the perturbation operation in
the following way. The perturbation p
=
(p1
, ••• ,
p n) applied to the composition
x
=
(x1
, •.•
,xn)
produces the composition
(4.3)
pox= (plxl,··· ,pnxn)f(plxl
+ ···
+pnxn).
Note that because of the nature of the scaling in this relationship it is not strictly
necessary for the perturbation p to be a vector in
sd.
In mathematical terms the set of perturbations in
sn
form a group with the
identity perturbation e
=
(1/ D, ... , 1/ D) and the inverse of a perturbation p being
the closure
C(p!
1
, ••.
,p£/).
The relation between any two compositions
x
and
X
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