Editing Compositional data (section)

==== Additive log ratio transform ====
The additive log ratio (alr) transform is an isomorphism where <math>\operatorname{alr}: S^D \rightarrow \mathbb{R}^{D-1} </math>.  This is given by

:: <math> \operatorname{alr}(x) = \left[ \log \frac{x_1}{x_D}, \cdots, \log \frac{x_{D-1}}{x_D} \right] </math>

The choice of denominator component is arbitrary, and could be any specified component.
This transform is commonly used in chemistry with measurements such as pH.  In addition, this is the transform most commonly used for [[multinomial logistic regression]].  The alr transform is not an isometry, meaning that distances on transformed values will not be equivalent to distances on the original compositions in the simplex.