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# skbio.stats.composition.ilr_inv

`skbio.stats.composition.``ilr_inv`(mat, basis=None, check=True)

  State: Experimental as of 0.4.0. Performs inverse isometric log ratio transform.

  This function transforms compositions from the real space to Aitchison geometry. The \(ilr^{-1}\) transform is both an isometry, and an isomorphism defined on the following spaces

  \(ilr^{-1}: \mathbb{R}^{D-1} \rightarrow S^D\)

  The inverse ilr transformation is defined as follows

  \[ilr^{-1}(x) = \bigoplus\limits_{i=1}^{D-1} x \odot e_i\]

  where \([e_1,\ldots, e_{D-1}]\) is an orthonormal basis in the simplex.

  If an orthonormal basis isn’t specified, the J. J. Egozcue orthonormal basis derived from Gram-Schmidt orthogonalization will be used by default.

  Parameters:

  - mat (numpy.ndarray, float) – a matrix of transformed proportions where rows = compositions and columns = components
  - basis (numpy.ndarray, float, optional) – orthonormal basis for Aitchison simplex defaults to J.J.Egozcue orthonormal basis
  - check (bool) – Specifies if the basis is orthonormal.

  Examples

      >>> import numpy as np
      >>> from skbio.stats.composition import ilr
      >>> x = np.array([.1, .3, .6,])
      >>> ilr_inv(x)
      array([ 0.34180297,  0.29672718,  0.22054469,  0.14092516])

  Notes

  If the basis parameter is specified, it is expected to be a basis in the Aitchison simplex. If there are D-1 elements specified in mat, then the dimensions of the basis needs be D-1 x D, where rows represent basis vectors, and the columns represent proportions.