Geoscience Reference
In-Depth Information
Fig. 5.14
Crossplots of Ni-Fe and Ni/SiO
2
-Fe/SiO
2
5.5.3.5 Transformations
The majority of analyses concerned with compositions apply
logratio transformations to the data prior to any exploratory
analysis. Aitchison et al. (
2002
) argue that compositions
provide data on a relative level, rather than on an absolute
level and logarithms of relative values expressed as ratios are
easier to statistically analyze than ratios themselves. Also,
logratio transformations do not affect the information con-
tent of the data. The reason one must work with ratios rather
than raw variables as in the ternry diagrams shown above is
those variables are not scale invariant: the subcompositions
shown are not coherent with the full composition.
The more common transforms are the additive logratio
(alr), the centered logratio (clr), the multiplicative logra-
tio (mlr, Aitchison
1986
), and the isometric logratio (ilr,
Egozcue et al.
2003
). Choice of transformation depends on
the problem being considered and the targeted properties of
results. The outcome of these transformations is a set of vec-
tors that exist in real space not constrained to the simplex.
Each component of the vectors refers to a coordinate. These
transforms are also one-to-one in that they map distinct val-
ues from one sample space to distinct values in the trans-
formed sample space.
The denominator
x
D
can be any one of the components of
x
,
but it must remain consistent when applying this transform to
a complete set of compositions, and
x
must be greater than 0.
The advantage is a new space free of constraints where clas-
sical multivariate analysis methods can be applied; however,
the space is not isometric. Coordinate axes are not orthogo-
nal, but are separated by 60 degrees (Pawlowsky-Glahn and
Egozcue
2006
).
This transformation is applied to the nickel laterite data
by dividing all components by the filler component
Z
of the
compositions, resulting in 7 variables. For visualization pur-
poses, the transformation was applied to the Ni-Fe-SiO
2
sub-
composition with silica as the divisor. A crossplot of the re-
sulting variables, log(Ni/SiO
2
) and log(Fe/SiO
2
) is shown on
the right in Fig.
5.14
. This can be compared with scatterplots
of the original variables Ni and Fe on the left. Prior to trans-
formation the data were constrained to positive real space
since they are expressed as percentages. Post-transformation
shows the data are unconstrained and that division by SiO
2
imposes a relationship.
5.5.3.7 Centered Logratio Transform
Unlike the alr transformation, clr results in orthogonal
axes which simplifies further multivariate computations.
The nature of this transformation results in vectors with
a zero sum meaning the subspace is actually a plane. This
zero sum property results in singular covariance matri-
ces (Pawlowsky-Glahn and Egozcue
2006
), but there are
methods to overcome this limitation (Quintana and West
1988
). The forward clr transform with
g
(
x
) the geometric
mean of
x
is:
5.5.3.6 Additive Logratio Transform
Forward and inverse alr transformations are expressed re-
spectively by the following equations:
†‡
x
y
=
log
i
,
i
=
1,...,
d
ˆ
Š‹
i
x
exp(
y
)
D
x
=
i
,
i
=
1,...,
d
i
d
†
x
‡
∑
exp(
y
)
+
1
i
y
=
log
,
i
=
1,...,
D
ˆ
‰
i
i
Š
g
()
x
‹
i
=
1
