skbio.stats.composition.alr#

skbio.stats.composition.alr(mat, ref_idx=0, axis=-1, validate=True)[source]#

Perform additive log ratio (ALR) transformation.

This function transforms compositions from a D-part Aitchison simplex to a non-isometric real space of D-1 dimensions. The argument ref_idx defines the index of the column used as the reference (a common denominator). The \(alr\) transformed data are amenable to multivariate analysis as long as statistics don’t involve distances.

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

The alr transformation is defined as follows

\[alr(x) = \left[ \ln \frac{x_1}{x_D}, \ldots, \ln \frac{x_{D-1}}{x_D} \right]\]

where \(D\) is the index of the part used as the reference.

Changed in version 0.7.0: The function now works on any dimension in arrays of any number of dimensions.

Parameters:
matarray_like of shape (…, n_components, …)

A matrix of proportions.

ref_idxint, optional

Index on the target axis which should be used as the reference composition (denominator). Default is 0 (the first position).

Changed in version 0.7.0: Renamed from denominator_idx. The old name is kept as an alias.

axisint, optional

Axis along which ALR transformation will be performed. Each vector along this axis is considered as a composition. Default is the last axis (-1).

Added in version 0.7.0.

validate: bool, default True

Check whether the input is positive, whether the mat is 2D.

Added in version 0.7.0.

Returns:
ndarray of shape (…, n_components - 1, …)

ALR-transformed data projected in a non-isometric real space of \(D - 1\) dimensions for a D-parts composition.

See also

alr_inv

Examples

>>> import numpy as np
>>> from skbio.stats.composition import alr
>>> x = np.array([.1, .3, .4, .2])
>>> alr(x)
array([ 1.09861229,  1.38629436,  0.69314718])
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