Isometric/Orthonormal Log-Ratio Basis for Log-Transformed Compositions

Builds an isometric log-ratio (ilr) basis for a composition with k+1 parts, also called orthonormal log-ratio (olr) basis.

ilr_basis(dim, type = "default")

olr_basis(dim, type = "default")

Arguments

dim: An integer indicating the number of components. If a dataframe or matrix is provided, the number of components is inferred from the number of columns. If a character vector specifying the names of the parts is provided the number of component is its length.
type: Character string specifying the type of basis to generate. Options are "pivot", "cdp". Any other option will return the Helmert basis defined by Egozcue et al., 2013..

Value

A matrix representing the orthonormal basis.

Details

The basis vectors are constructed as: $$h_i = \sqrt{\frac{i}{i+1}} \log\frac{\sqrt[i]{\prod_{j=1}^i x_j}}{x_{i+1}}$$ for $i = 1, \ldots, k$.

Setting the type parameter to "pivot" (pivot balances) or "cdp" (codapack balances) allows generating alternative ilr/olr bases.

References

Egozcue, J.J., Pawlowsky-Glahn, V., Mateu-Figueras, G., & Barceló-Vidal, C. (2003). Isometric logratio transformations for compositional data analysis. Mathematical Geology, 35(3), 279–300.

Examples

ilr_basis(5)
#>          ilr1       ilr2       ilr3       ilr4
#> c1  0.7071068  0.4082483  0.2886751  0.2236068
#> c2 -0.7071068  0.4082483  0.2886751  0.2236068
#> c3  0.0000000 -0.8164966  0.2886751  0.2236068
#> c4  0.0000000  0.0000000 -0.8660254  0.2236068
#> c5  0.0000000  0.0000000  0.0000000 -0.8944272
ilr_basis(alimentation[,1:9])
#>          ilr1       ilr2       ilr3       ilr4       ilr5       ilr6       ilr7
#> RM  0.7071068  0.4082483  0.2886751  0.2236068  0.1825742  0.1543033  0.1336306
#> WM -0.7071068  0.4082483  0.2886751  0.2236068  0.1825742  0.1543033  0.1336306
#> E   0.0000000 -0.8164966  0.2886751  0.2236068  0.1825742  0.1543033  0.1336306
#> M   0.0000000  0.0000000 -0.8660254  0.2236068  0.1825742  0.1543033  0.1336306
#> F   0.0000000  0.0000000  0.0000000 -0.8944272  0.1825742  0.1543033  0.1336306
#> C   0.0000000  0.0000000  0.0000000  0.0000000 -0.9128709  0.1543033  0.1336306
#> S   0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 -0.9258201  0.1336306
#> N   0.0000000  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000 -0.9354143
#> FV  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000  0.0000000
#>          ilr8
#> RM  0.1178511
#> WM  0.1178511
#> E   0.1178511
#> M   0.1178511
#> F   0.1178511
#> C   0.1178511
#> S   0.1178511
#> N   0.1178511
#> FV -0.9428090