Lecture Notes on Compositional Data Analysis - Sedimentology ...

40 Chapter 5. Exploratory data analysis
3. a box-plot, summarising the order statistics of each ilr coordinate.
Each coordinate is represented in a horizontal axis, which limits correspond to a certain
range (the same for every coordinate). The vertical bar going up from each one of these
coordinate axes represents the variance of that specific coordinate, and the contact point
is the coordinate mean. Figure 5.2 shows these elements in an illustrative example.
Given that the range of each coordinate is symmetric (in Figure 5.2 it goes from
−3 to +3), the box plots closer to one part (or group) indicate that part (or group) is
more abundant. Thus, in Figure 5.2, SiO2 is slightly more abundant that Al 2 O 3 , there
is more FeO than Fe 2 O 3 , and much more structural oxides (SiO2 and Al 2 O 3 ) than the
rest. Another feature easily read from a balance-dendrogram is symmetry: it can be
assessed both by comparison between the several quantile boxes, and looking at the
difference between the median (marked as “Q2” in Figure 5.2 right) and the mean.
In fact, a balance-dendrogram contains information on the marginal distribution of
each coordinate. It can potentially contain any other representation of these marginals,
not only box-plots: one could use the horizontal axes to represent, e.g., histograms or
kernel density estimations, or even the sample itself. On the other side, a balancedendrogram
does not contain any information on the relationship between coordinates:
this can be approximately inferred from the biplot or just computing the correlation
matrix of the coordinates.
5.6 Illustration
We are going to use, both for illustration and for the exercises, the data set X given
in table 5.1. They correspond to 17 samples of chemical analysis of rocks from Kilauea
Iki lava lake, Hawaii, published by Richter and Moore (1966) and cited by Rollinson
(1995).
Originally, 14 parts had been registered, but H 2 O + and H 2 O − have been omitted
because of the large amount of zeros. CO 2 has been kept in the table, to call attention
upon parts with some zeros, but has been omitted from the study precisely because of
the zeros. This is the strategy to follow if the part is not essential in the characterisation
of the phenomenon under study. If the part is essential and the proportion of zeros is
high, then we are dealing with two populations, one characterised by zeros in that
component and the other by non-zero values. If the part is essential and the proportion
of zeros is small, then we can look for input techniques, as explained in the beginning
of this chapter.
The centre of this data set is
g = (48.57, 2.35, 11.23, 1.84, 9.91, 0.18, 13.74,9.65,1.82,0.48, 0.22) ,
the total variance is totvar[X] = 0.3275 and the normalised variation matrix T ∗ is given
in Table 5.2.