Aitchison norm of a composition

Formula: ||x||_a = sqrt( (1/D) sum_{i<j} log(x_i/x_j)^2 )

Usage

Compnorm(x)

Arguments

x

Composition with strictly positive parts.

Value

List with norm, norm2, 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). The norm associated with equation (10): ||x||_a^2 = <x, x>_a.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Compnorm(V)
#> $norm
#> [1] 1.861008
#> 
#> $norm2
#> [1] 3.46335
#> 
#> $D
#> [1] 8
#> 
#> $method
#> [1] "Aitchison norm"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"