Lecture Notes on Compositional Data Analysis - Sedimentology ...

14 Chapter 3. Aitchison geometry
3.3 Inner product, norm and distance
To obtain a linear vector space structure, we take the following inner product, with
associated norm and distance:
Definition 3.3. Inner product of x,y ∈ S D ,
〈x,y〉 a = 1 D∑ D∑
ln x i
ln y i
.
2D x j y j
Definition 3.4. Norm of x ∈ S D ,
‖x‖ a
= √ 1
2D
i=1
D∑
i=1
j=1
D∑
j=1
i=1
(
ln x i
x j
) 2
.
Definition 3.5. Distance between x and y ∈ S D ,
d a (x,y) = ‖x ⊖ x‖ a
= √ 1 D∑ D∑
(
ln x i
− ln y ) 2
i
.
2D x j y j
In practice, alternative but equivalent expressions of the inner product, norm and
distance may be useful. Two possible alternatives of the inner product follow:
(
〈x,y〉 a = 1 D−1
∑ D∑
ln x i
ln y D∑
i
= ln x i ln x j − 1 D
) (
∑ D
)
∑
ln x j ln x k ,
D x j y j D
i=1 j=i+1
i