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