Lecture Notes on Compositional Data Analysis - Sedimentology ...

20 Chapter 4. Coordinate representation
Although the clr coefficients are not coordinates with respect to a basis of the simplex,
they have very important properties. Among them the translation of operations
and metrics from the simplex into the real space deserves special attention. Denote ordinary
distance, norm and inner product in R D−1 by d(·, ·), ‖ · ‖, and 〈·, ·〉 respectively.
The following property holds.
Property 4.1. Consider x k ∈ S D and real constants α, β; then
clr(α ⊙ x 1 ⊕ β ⊙ x 2 ) = α · clr(x 1 ) + β · clr(x 2 ) ;
〈x 1 ,x 2 〉 a = 〈clr(x 1 ), clr(x 2 )〉 ; (4.5)
‖x 1 ‖ a = ‖clr(x 1 )‖ , d a (x 1 ,x 2 ) = d(clr(x 1 ), clr(x 2 )) .
4.4 Orthonormal coordinates
Omitting one vector of the generating system given in Equation (4.2) a basis is obtained.
For example, omitting w D results in {w 1 ,w 2 , . . .,w D−1 }. This basis is not orthonormal,
as can be shown computing the inner product of any two of its vectors. But a new basis,
orthonormal with respect to the inner product, can be readily obtained using the wellknown
Gram-Schmidt procedure (Egozcue et al., 2003). The basis thus obtained will
be just one out of the infinitely many orthonormal basis which can be defined in any
Euclidean space. Therefore, it is convenient to study their general characteristics.
Let {e 1 ,e 2 , . . .,e D−1 } be a generic orthonormal basis of the simplex S D and consider
the (D − 1, D)-matrix Ψ whose rows are clr(e i ). An orthonormal basis satisfies that
〈e i ,e j 〉 a = δ ij (δ ij is the Kronecker-delta, which is null for i ≠ j, and one whenever
i = j). This can be expressed using (4.5),
〈e i ,e j 〉 a = 〈clr(e i ), clr(e j )〉 = δ ij .
It implies that the (D − 1, D)-matrix Ψ satisfies ΨΨ ′ = I D−1 , being I D−1 the identity
matrix of dimension D − 1. When the product of these matrices is reversed, then
Ψ ′ Ψ = I D − (1/D)1 ′ D 1 D, with I D the identity matrix of dimension D, and 1 D a D-
row-vector of ones; note this is a matrix of rank D − 1. The compositions of the basis
are recovered from Ψ using clr −1 in each row of the matrix. Recall that these rows of
Ψ also add up to 0 because they are clr coefficients (see Definition 4.1).
Once an orthonormal basis has been chosen, a composition x ∈ S D is expressed as
D−1
⊕
x = x ∗ i ⊙ e i , x ∗ i = 〈x,e i 〉 a , (4.6)
i=1
where x ∗ = [ x ∗ 1, x ∗ 2, . . .,x ∗ D−1]
is the vector of coordinates of x with respect to the
selected basis. The function ilr : S D → R D−1 , assigning the coordinates x ∗ to x has