Lecture Notes on Compositional Data Analysis - Sedimentology ...

12 Chapter 3. Aitchison geometry
two operations which give the simplex a vector space structure. The first one is the
perturbation operation, which is analogous to addition in real space, the second one is
the power transformation, which is analogous to multiplication by a scalar in real space.
Both require in their definition the closure operation; recall that closure is nothing else
but the projection of a vector with positive components onto the simplex. Second, we
can obtain a linear vector space structure, and thus a geometry, on the simplex. We just
add an inner product, a norm and a distance to the previous definitions. With the inner
product we can project compositions onto particular directions, check for orthogonality
and determine angles between compositional vectors; with the norm we can compute
the length of a composition; the possibilities of a distance should be clear. With all
together we can operate in the simplex in the same way as we operate in real space.
3.2 Vector space structure
The basic operations required for a vector space structure of the simplex follow. They
use the closure operation given in Definition 2.3.
Definition 3.1. Perturbation of a composition x ∈ S D by a composition y ∈ S D ,
x ⊕ y = C [x 1 y 1 , x 2 y 2 , . . .,x D y D ] .
Definition 3.2. Power transformation of a composition x ∈ S D by a constant α ∈ R,
α ⊙ x = C [x α 1 , xα 2 , . . .,xα D ].
For an illustration of the effect of perturbation and power transformation on a set
of compositions, see Figure 3.1.
The simplex , (S D , ⊕, ⊙), with the perturbation operation and the power transformation,
is a vector space. This means the following properties hold, making them
analogous to translation and scalar multiplication:
Property 3.1. (S D , ⊕) has a commutative group structure; i.e., for x, y, z ∈ S D it
holds
1. commutative property: x ⊕ y = y ⊕ x;
2. associative property: (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z);
3. neutral element:
n = C [1, 1, . . ., 1] =
[ 1
D , 1 D , . . ., 1 D]
;
n is the baricenter of the simplex and is unique;