Selected Topics for Further Research - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 475

Fig. 12.8 Relationship
between logits and probits is
approximately linear for
logits within the [ 3, 3]
interval

4

Series1

3

2

1

0

-2 -1 0 1 2

-1

-2

-3

-4

are shown in Fig. 12.8 . More results including scores for other principal compo-
nents are provided by Grunsky and Kjarsgaard ( 2008 ). More than 90 % of the
variation of Si is accounted for by the first principal component, whereas 64.0 % of
the variability of Ti is accounted for in the second component. The actual contri-
butions provide a measure of how much each element contributes to each compo-
nent. In the first component, Si, Fe, Co and Ni contributions all exceed more than
10 %. Approximately 45 % of variability of the first component is accounted for by
these four elements.
Figure 12.8 is a plot of observations and element scores projected onto the first
two principal component axes. The variation of all the data in this diagram
describes 68.6 % of the overall variation in the data set. The ellipse boundaries
were constructed to encompass all observations for each eruptive and the size of an
ellipse is related to the degree of dispersion associated for that phase. Samples that
plot close to an element or group of elements are enriched in those elements relative
to other samples that plot farther away. It is evident that samples from the eJF show
relative enrichment in Si, Cr, Ni, Mg, Fe and Co as would be expected for olivine-
rich rocks such as kimberlite or kimberlite contaminated by lithospheric peridotite
(which is dominated by olivine). Average composition of two Canadian cratonic
mantle peridotite suites have extreme compositions relative to the kimberlite scores
and are not plotted in Fig. 12.8 , but the mantle contamination vector in this diagram
is defined by the scores of these two average compositions. Samples tending toward
the negative part of the C1 axis are certainly all mantle peridotite contaminated.
Further interpretations are given by Grunsky and Kjarsgaard ( 2008 ). The purpose of
this example was to show how compositional data analysis combined with classical
statistics can result in novel mineralogical and geochemical interpretations.
A final remark on compositional data analysis is as follows. In an interesting
paper, Filzmoser et al. ( 2009 ) argue that, for statistical analysis, logits ( cf . Sect. 5.2 )