composition_stats.alr_inv

composition_stats.alr_inv(mat, denominator_idx=0)

Performs 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

mat: numpy.ndarray

a matrix of alr-transformed data

denominator_idx: int

the index of the column (2D-composition) or position (1D-composition) of the output where the common denominator should be placed. By default denominator_idx=0 to specify the first column or position.

Returns

numpy.ndarray

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

Examples

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