Selected Topics for Further Research - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 471

Fig. 12.3 Ternary diagram
and log-contrast principal
axes for 23 aphyric Skye
lava compositions.
A: Alkali or Na 2 O+K 2 O;
F: Fe 2 O 3 ; M: MgO (Source:
Aitchison 1986 , Fig. 8.5)

6 example (Fig. 12.3 ) shows an example with AFM compositions of 23 aphyric
Skye lavas (original data from Thompson et al. 1972 ). Aitchison ( 1986 ) first applied
ordinary principal component analysis using the variance-covariance matrix.
This results in three principal components (PCs). The third PC has zero eigenvalue
and zero variance because it simply represents the closure of

the system

(A + F +M 1). PC1 and PC2 would plot as a cross in Fig. 12.3 represented by

two straight lines which are perpendicular to each other and pass through the center
of gravity of the cluster of input data. On the other hand, if the data are divided by
their geometric mean before logs are taken, the first two principal components
become as shown in Fig. 12.3 . Obviously PC1 (first axis) is meaningful in Fig. 12.3 .
It represents the magma composition spectrum. The second axis represents a noise
component that probably does not have any meaning genetically. Potentially this
result is of interest from a petrological point of view: from the curved pattern of
points in Fig. 12.3 , Thompson et al. ( 1972 , p. 235) had concluded that there were
two separate trends among the lavas (from relatively sodic, iron-rich hawaiite and
mugearite to benmoreite, and from hyperstene-normative basalt via relatively
potassic, iron-poor intermediate lavas to trachyte), whereas the clr method suggests
there is one trend only.

¼

12.2.1 Star Kimberlite Example

Grunsky and Kjarsgaard ( 2008 ) have classified discrete eruptive events of the Star
kimberlite, Saskatchewan, into five distinctive clusters using statistical methods
including compositional data analysis applied to whole rock geochemical data
(Fig. 12.4 ). The data set consisted of 270 kimberlite samples from 38 drill holes
that were analyzed for major oxides and trace -element geochemistry. Multivariate
techniques after a log-ratio transformation included principal component analysis
and discriminant analysis. Five distinct geochemical classes could be distinguished.
These are called the Cantiar, Pense, early Joli Fou (eJF), mid Joli Fou (mJF) and late