Inverse additive log-ratio transform

Formula: alr^-1(y) = C( exp(y_1), ..., exp(y_{D-1}), 1 ) with the 1 inserted at the reference position

Usage

Alrinv(y, ref = NULL, total = 1)

Arguments

y: Additive log-ratio coordinates, length D - 1.
ref: 1-based index the reference part is restored to; the default is the last position D.
total: Constant kappa the closure sums to.

Value

List with composition, ref, total, D.

References

Aitchison, J. (1986), The Statistical Analysis of Compositional Data, Chapman and Hall, is the primary source; the definitions below are as printed in Mateu-Figueras, G., Pawlowsky-Glahn, V. and Egozcue, J. J., The normal distribution in some constrained sample spaces, arXiv:0802.2643 (published as SORT 37(1):29-56, 2013), Sect. 4.1, which prints them with equation numbers and attributes them to Aitchison (1982, 1986). Inverting alr puts a 1 back in the reference slot and closes: the reference part is the one whose log-ratio against itself is zero. ref must match the ref used in the forward transform for the round trip to return the original composition.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Alrinv(V)
#> $composition
#> [1] 0.0005764905 0.0015670636 0.0042597205 0.0115791209 0.0314753138
#> [6] 0.0855587737 0.2325728597 0.6321985783 0.0002120790
#> 
#> $ref
#> [1] 9
#> 
#> $total
#> [1] 1
#> 
#> $D
#> [1] 9
#> 
#> $method
#> [1] "Inverse additive log-ratio transform"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"