Additive log-ratio transform of a composition against a reference part

Formula: alr(x)_i = log( x_i / x_ref ), i != ref

Usage

Alr(x, ref = NULL)

Arguments

x

Composition with strictly positive parts.

ref

1-based index of the reference part; the default D uses the last part, as in Aitchison's own display.

Value

List with alr, ref, parts, 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). alr(x) = (ln(x_1/x_D), ..., ln(x_{D-1}/x_D)), the reference part being the last one. ref generalises that to any part; parts reports which 1-based indices the returned coordinates belong to, in their original order. The index is 1-based in BOTH language arms so the two agree.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Alr(V)
#> $alr
#> [1] -2.0794415 -1.3862944 -0.9808293 -0.6931472 -0.4700036 -0.2876821 -0.1335314
#> 
#> $ref
#> [1] 8
#> 
#> $parts
#> [1] 1 2 3 4 5 6 7
#> 
#> $D
#> [1] 8
#> 
#> $method
#> [1] "Additive log-ratio transform"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"