skbio.stats.composition.alr_inv

skbio.stats.composition.alr_inv(mat, denominator_idx=0)

Perform inverse additive log ratio transform.

This function transforms compositions from the non-isometric real space of alrs to Aitchison geometry.

\[alr^{-1}: \mathbb{R}^{D-1} \rightarrow S^D\]

The inverse alr transformation is defined as follows:

\[alr^{-1}(x) = C[exp([y_1, y_2, ..., y_{D-1}, 0])]\]

where \(C[x]\) is the closure operation defined as

\[C[x] = \left[\frac{x_1}{\sum_{i=1}^{D} x_i},\ldots, \frac{x_D}{\sum_{i=1}^{D} x_i} \right]\]

for some \(D\) dimensional real vector \(x\) and \(D\) is the number of components for every composition.

Parameters:

matarray_like of shape (n_compositions, n_components - 1)

A matrix of alr-transformed data.

denominator_idxint

The index of the column (2-D matrix) or position (vector) of mat which should be used as the reference composition. Default is 0 which specifies the first column or position.

Returns:

ndarray of shape (n_compositions, n_components)

Inverse alr-transformed matrix or vector where rows sum to 1.

Examples

>>> import numpy as np
>>> from skbio.stats.composition import alr, alr_inv
>>> x = np.array([.1, .3, .4, .2])
>>> alr_inv(alr(x))
array([ 0.1,  0.3,  0.4,  0.2])