mushi.composition.clr

Performs centre log ratio transformation.

This function transforms compositions from Aitchison geometry to the real space. The \(clr\) transform is both an isometry and an isomorphism defined on the following spaces

\(clr: S^D \rightarrow U\)

where \(U= \{x :\sum\limits_{i=1}^D x = 0 \; \forall x \in \mathbb{R}^D\}\)

It is defined for a composition \(x\) as follows:

\[clr(x) = \ln\left[\frac{x_1}{g_m(x)}, \ldots, \frac{x_D}{g_m(x)}\right]\]

where \(g_m(x) = (\prod\limits_{i=1}^{D} x_i)^{1/D}\) is the geometric mean of \(x\).

Parameters:: mat (array_like, float) – a matrix of proportions where rows = compositions and columns = components
Returns:: clr transformed matrix
Return type:: numpy.ndarray

Examples

>>> import numpy as np
>>> import mushi.composition as cmp
>>> x = np.array([.1, .3, .4, .2])
>>> cmp.clr(x)
Array([-0.79451346,  0.30409883,  0.5917809 , -0.10136628], dtype=float64)