I'm recently diving into the field of compositional data analysis and I'm still wondering why the log transformations that Aitchison proposes work.
What I understand is that, having a $D$- part composition $\textbf{x} = (x_1, \dots, x_d)$ in the $D:=d-1$ dimensional simplex $\mathcal{S}^D$ means having the constraints that $x_1 > 0, \dots, x_d >0$ and $\sum_{i=1}^d x_i = 1$.
So what is the goal of using log-ratios-based transformations? I mean, why using logs and no other functions? My intuition suggests because of the beautiful properties that log function manifests, in particular that the log of ratio is the difference of logs, but I would like someone to formalize this intuition better.
To my understanding we also use it to modify the constraints specified above..
For example, considering the additive-log-ratio transform, we take the log-ratios of each coordinate with respect to a fixed one:
$$alr(\textbf{x}) : \mathcal{S}^D \ni (x_1, \, \dots \, , x_d) \, \mapsto \left[log\frac{x_1}{x_d}, \, \dots \, , log\frac{x_{d-1}}{x_d}\right] \in \mathbb{R}^D $$
So the constraint we have in the simplex at the beginning would become the following:
\begin{align} x_1+ \dots + x_d = 1 \, \, \, \overset{(?)}{\Rightarrow} \, log\frac{x_1}{x_d} + \dots + log\frac{x_{d-1}}{x_d} = 0 \end{align}
where the right part geometrically describes a hyperplane passing through the origin (even if I'm not sure that it's correct to apply the log function to above equation the way I did to illustrate the change of constraints).
EDIT:
Ok so In case of the $alr(\textbf{x})$ one totally removes the constraint going into a $D-1$ dimensional space (but then how can this transformation be described as an isomorphism?).
Instead another constraint gets introduced when one uses the centred log-ratio transform:
$$clr(\textbf{x}) : \mathcal{S}^D \ni (x_1, \, \dots \, , x_d) \, \mapsto \left[log\frac{x_1}{g(\textbf{x})}, \, \dots \, , log\frac{x_d}{g(\textbf{x})}\right] \in \mathbb{R}^D $$
with $g(\textbf{x}) = (\Pi_{i=1}^d x_i)^{1/d}$
It's actually using this second transformation that points get mapped to a $D$- dimensional hyperplane passing through the origin. But can anybody explain me better the motivation underlying these transformation using log-functions?
I remember that I've read somewhere that *the log-functions are the only respecting the scale-invariance principle at the base of CoDa..
