# skbio.stats.composition.ilr¶

skbio.stats.composition.ilr(mat, basis=None, check=True)[source]

State: Experimental as of 0.4.0. Performs isometric log ratio transformation.

This function transforms compositions from Aitchison simplex to the real space. The :math: ilr transform is both an isometry, and an isomorphism defined on the following spaces

$$ilr: S^D \rightarrow \mathbb{R}^{D-1}$$

The ilr transformation is defined as follows

$ilr(x) = [\langle x, e_1 \rangle_a, \ldots, \langle x, e_{D-1} \rangle_a]$

where $$[e_1,\ldots,e_{D-1}]$$ is an orthonormal basis in the simplex.

If an orthornormal basis isn’t specified, the J. J. Egozcue orthonormal basis derived from Gram-Schmidt orthogonalization will be used by default.

Parameters: mat: numpy.ndarray a matrix of proportions where rows = compositions and columns = components basis: numpy.ndarray, float, optional orthonormal basis for Aitchison simplex defaults to J.J.Egozcue orthonormal basis. check: bool Specifies if the basis is orthonormal.

Notes

If the basis parameter is specified, it is expected to be a basis in the Aitchison simplex. If there are D-1 elements specified in mat, then the dimensions of the basis needs be D-1 x D, where rows represent basis vectors, and the columns represent proportions.

Examples

>>> import numpy as np
>>> from skbio.stats.composition import ilr
>>> x = np.array([.1, .3, .4, .2])
>>> ilr(x)
array([-0.7768362 , -0.68339802,  0.11704769])
`