The relative nature of compositional data lends itself to analysis with ratios. In his seminal publication, Aitchison (1986) introduced transformations based on logratios to utilise
these relative properties. Early work focussed on the additive logratio transformation (alr)
and the centred logratio transformation (clr). As the alr does not preserve distances, it was
primarily used for modelling purposes. The clr does preserve distances but leads to singular
covariance matrices and is thus used for techniques based on a metric. The other transformation of importance is the isometric logratio transformation (ilr); developed to overcome
the shortcomings of both the alr and the clr. The ilr is an association of coordinates with
compositions in an orthonormal system and results in an isometry between S D and RD−1 .
The additive logratio transformation alr : S D −→ RD−1 given by
ZD−1 (u)
Z1 (u)
, . . . , ln
(1)
alr(Z(u)) = X(u) = ln
ZD (u)
ZD (u)
embeds the D-simplex into (D − 1)-dimensional space and removes the constant sum constraint. In practice any one of the parts can be used as the divisor; and should be selected
so as to minimise any computational difficulties in later processes. Due to this, the alr
transformation results in D − 1 components; advantageous when considering the fitment of
the Linear model of coregionalisation to the resulting components. This property is also a
drawback as by changing the dividing component the alr transformation changes too; it is
defined as asymmetric. The major problem when applying the alr transfrom is that it is not
an isometric transformation from the simplex, with the Aitchison metric, onto the real alrspace with the Euclidean metric. The lack of isometry leads to difficulties in interpretation
of the transformed variables. The alr transform is invertible and its inverse is the additive
generalised logistic (agl) transform:
alg(X(u)) = Z(u) =
(exp(X1 (u)), . . . , exp(XD−1 (u)), 1)
PD−1
exp(Xi (u))
1 + i=1
The centred logratio transformation clr : S D −→ RD given by
Z1 (u)
ZD−1 (u)
clr(Z(u)) = X(u) = ln
, . . . , ln
g(Z(u))
g(Z(u))
where g(Z(u)) =
D
Y
(2)
(3)
! D1
Zi (u)
is the geometric mean.
(4)
i=1
The advantage of the clr is symmetry in the components but this is offset by a new constraint
on the transformed variable; the sum of the components has to be zero.
1