acomparith: Power transform in the simplex In compositions: Compositional Data Analysis

Description

The Aitchison Simplex with its two operations perturbation as + and power transform as * is a vector space. This vector space is represented by these operations.

Usage

 1 2 3 4 5 power.acomp(x,s) ## Methods for class "acomp" ## x*y ## x/y

Arguments

 x an acomp composition or dataset of compositions (or a number or a numeric vector) y a numeric vector of size 1 or nrow(x) s a numeric vector of size 1 or nrow(x)

Details

The power transform is the basic multiplication operation of the Aitchison simplex seen as a vector space. It is defined as:

(x*y)_i:= clo( (x_i^{y_i})_i )_i

The division operation is just the multiplication with 1/y.

Value

An "acomp" vector or matrix.

Note

For * the arguments x and y can be exchanged. Note that this definition generalizes the power by a scalar, since y or s may be given as a scalar, or as a vector with as many components as the composition in acomp x. The result is then a matrix where each row corresponds to the composition powered by one of the scalars in the vector.

Author(s)

K.Gerald v.d. Boogaart http://www.stat.boogaart.de

References

Aitchison, J. (1986) The Statistical Analysis of Compositional Data Monographs on Statistics and Applied Probability. Chapman & Hall Ltd., London (UK). 416p.

Aitchison, J, C. Barcel'o-Vidal, J.J. Egozcue, V. Pawlowsky-Glahn (2002) A consise guide to the algebraic geometric structure of the simplex, the sample space for compositional data analysis, Terra Nostra, Schriften der Alfred Wegener-Stiftung, 03/2003

Pawlowsky-Glahn, V. and J.J. Egozcue (2001) Geometric approach to statistical analysis on the simplex. SERRA 15(5), 384-398