composition_stats.sbp_basis

composition_stats.sbp_basis(sbp)

Builds an orthogonal basis from a sequential binary partition (SBP). As explained in [1], the SBP is a hierarchical collection of binary divisions of compositional parts. The child groups are divided again until all groups contain a single part. The SBP can be encoded in a \((D - 1) \times D\) matrix where, for each row, parts can be grouped by -1 and +1 tags, and 0 for excluded parts. The sbp_basis method was originally derived from function gsi.buildilrBase() found in the R package compositions [2]. The ith balance is computed as follows

\[b_i = \sqrt{ \frac{r_i s_i}{r_i+s_i} } \ln \left( \frac{g(x_{r_i})}{g(x_{s_i})} \right)\]

where \(b_i\) is the ith balance corresponding to the ith row in the SBP, \(r_i\) and \(s_i\) and the number of respectively +1 and -1 labels in the ith row of the SBP and where \(g(x) = (\prod\limits_{i=1}^{D} x_i)^{1/D}\) is the geometric mean of \(x\).

Parameters

sbp: np.array, int

A contrast matrix, also known as a sequential binary partition, where every row represents a partition between two groups of features. A part labelled +1 would correspond to that feature being in the numerator of the given row partition, a part labelled -1 would correspond to features being in the denominator of that given row partition, and 0 would correspond to features excluded in the row partition.

Returns

numpy.array

An orthonormal basis in the Aitchison simplex

References

1

Parent, S.É., Parent, L.E., Egozcue, J.J., Rozane, D.E., Hernandes, A., Lapointe, L., Hébert-Gentile, V., Naess, K., Marchand, S., Lafond, J., Mattos, D., Barlow, P., Natale, W., 2013. The plant ionome revisited by the nutrient balance concept. Front. Plant Sci. 4, 39, http://dx.doi.org/10.3389/fpls.2013.00039.

2

van den Boogaart, K. Gerald, Tolosana-Delgado, Raimon and Bren, Matevz, 2014. compositions: Compositional Data Analysis. R package version 1.40-1. https://CRAN.R-project.org/package=compositions.

Examples

>>> import numpy as np
>>> sbp = np.array([[1, 1,-1,-1,-1],
...                 [1,-1, 0, 0, 0],
...                 [0, 0, 1,-1,-1],
...                 [0, 0, 0, 1,-1]])
...
>>> sbp_basis(sbp)
array([[ 0.31209907,  0.31209907,  0.12526729,  0.12526729,  0.12526729],
       [ 0.36733337,  0.08930489,  0.18112058,  0.18112058,  0.18112058],
       [ 0.17882092,  0.17882092,  0.40459293,  0.11888261,  0.11888261],
       [ 0.18112058,  0.18112058,  0.18112058,  0.36733337,  0.08930489]])