Lecture Notes on Compositional Data Analysis - Sedimentology ...

16 Chapter 3. Aitchison geometry
x1
n
x2
x3
Figure 3.3: Circles and ellipses (left) and perturbation of a segment (right) in S 3 .
Exercise 3.3. Apply the power transformation with α ranging from −3 to +3 in steps
of 0.5 to x 1 = C [0.7, 0.4, 0.8] and plot the resulting set of compositions. Join them by
a line. What do you observe?
Exercise 3.4. Perturb the compositions obtained in Ex. 3.3 by x 2 = C [0.2, 0.8, 0.1].
What is the result?
Exercise 3.5. Compute the Aitchison inner product of x 1 = C [0.7, 0.4, 0.8] and x 2 =
C [0.2, 0.8, 0.1]. Are they orthogonal?
Exercise 3.6. Compute the Aitchison norm of x 1 = C [0.7, 0.4, 0.8] and call it a. Apply
to x 1 the power transformation α ⊙ x 1 with α = 1/a. Compute the Aitchison norm of
the resulting composition. How do you interpret the result?
Exercise 3.7. Re-do Exercise 2.4, but using the Aitchison distance given in Definition
3.5. Is it subcompositionally dominant?
Exercise 3.8. In a 2-part composition x = [x 1 , x 2 ], simplify the formula for the Aitchison
distance, taking x 2 = 1 − x 1 (so, using κ = 1). Use it to plot 7 equally-spaced
points in the segment (0, 1) = S 2 , from x 1 = 0.014 to x 1 = 0.986.
Exercise 3.9. In a mineral assemblage, several radioactive isotopes have been measured,
obtaining [ 238 U, 232 Th, 40 K] = [150, 30, 110]ppm. Which will be the composition
after ∆t = 10 9 years? And after another ∆t years? Which was the composition ∆t
years ago? And ∆t years before that? Close these 5 compositions and represent them
in a ternary diagram. What do you see? Could you write the evolution as an equation?
(Half-life disintegration periods: [ 238 U, 232 Th, 40 K] = [4.468; 14.05; 1.277] · 10 9 years)