ilr_basis: Isometric/Orthonormal Log-Ratio Basis for Log-Transformed Compositions
Description
Builds an isometric log-ratio (ilr) basis for a composition with k+1 parts, also called orthonormal log-ratio (olr) basis.
Usage
ilr_basis(dim, type = "default")
olr_basis(dim, type = "default")
Value
A matrix representing the orthonormal basis.
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..
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.