Lecture Notes on Compositional Data Analysis - Sedimentology ...

8 Chapter 2. Sample space
The first two cases yield final masses of [53, 76, 14] gr, respectively [28, 41, 8] gr. In a
purely compositional data set, we do not know if we added or subtracted mass from
the sandstone to the sand. Thus, we cannot decide which of these cases really occurred.
Without further (non-compositional) information, there is no way to distinguish
between [53, 76, 14] gr and [28, 41, 8] gr, as we only have the value of the sand
composition after closure. Closure is a projection of any point in the positive orthant
of D-dimensional real space onto the simplex. All points on a ray starting at the
origin (e.g., [53, 76, 14] and [28, 41, 8]) are projected onto the same point of S D (e.g.,
[37, 53, 10]%). We say that the ray is an equivalence class and the point on S D a representant
of the class: Figure 2.2 shows this relationship. Moreover, if we change the
Figure 2.2: Representation of the compositional equivalence relationship. A represents the original
sandstone composition, B the final sand composition, F the amount of each part if feldspar was added
to the system (first hypothesis), and Q the amount of each part if quartz was depleted from the system
(second hypothesis). Note that the points B, Q and F are compositionally equivalent.
units of our data (for instance, from % to ppm), we simply multiply all our points by
the constant of change of units, moving them along their rays to the intersections with
another triangle, parallel to the plotted one.
Definition 2.5. Two vectors of D positive real components x,y ∈ R D + (x i, y i ≥ 0 for all
i = 1, 2, . . ., D), are compositionally equivalent if there exists a positive scalar λ ∈ R +
such that x = λ · y and, equivalently, C(x) = C(y).
It is highly reasonable to ask our analyses to yield the same result, independently of
the value of λ. This is what Aitchison (1986) called scale invariance: