Editing Compositional data (section)

==== Isometric logratio transform ====
The isometric log ratio (ilr) transform is both an isomorphism and an isometry where <math>\operatorname{ilr}: S^D \rightarrow \mathbb{R}^{D-1} </math>

:: <math> \operatorname{ilr}(x) = \big[ \langle x, e_1 \rangle, \ldots, \langle x, e_{D-1} \rangle\big]
</math>

There are multiple ways to construct orthonormal bases, including using the [[Gram–Schmidt_process | Gram–Schmidt orthogonalization]] or [[singular-value decomposition]] of clr transformed data.  
Another alternative is to construct log contrasts from a bifurcating tree.  If we are given a bifurcating tree, we can construct a basis from the internal nodes in the tree.

[[File:Orthogonal-tree-basis.jpg|thumb|A representation of a tree in terms of its orthogonal components. l represents an internal node, an element of the orthonormal basis. This is a precursor to using the tree as a scaffold for the ilr transform]]

Each vector in the basis would be determined as follows

:: <math> e_\ell = C[\exp( \,\underbrace{0,\ldots,0}_k, \underbrace{a,\ldots,a}_r,\underbrace{b,\ldots,b}_s,\underbrace{0,\ldots,0}_t \, )]
</math>

The elements within each vector are given as follows

:: <math> a = \frac{\sqrt{s}}{\sqrt{r(r+s)}} \quad \text{and} \quad b = \frac{-\sqrt{r}}{\sqrt{s(r+s)}}
</math>

where <math>k, r, s, t</math> are the respective number of tips in the corresponding subtrees shown in the figure.  It can be shown that the resulting basis is orthonormal<ref>{{harvnb|Egozcue|Pawlowsky-Glahn|2005}}</ref>

Once the basis <math>\Psi</math> is built, the ilr transform can be calculated as follows

:: <math> 
\operatorname{ilr}(x) = \operatorname{clr}(x) \Psi^T
</math>

where each element in the ilr transformed data is of the following form

:: <math> b_i = \sqrt{\frac{rs}{r+s}} \log \frac{g(x_R)}{g(x_S)}  </math>

where <math> x_R</math> and <math> x_S</math> are the set of values corresponding to the tips in the subtrees <math> R</math> and <math> S</math>