# acomparith: Power transform in the simplex

### 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

`ilr`,`clr`, `alr`,

### Examples

 ```1 2 3 4 5 6 7 8``` ```acomp(1:5)* -1 + acomp(1:5) data(SimulatedAmounts) cdata <- acomp(sa.lognormals) plot( tmp <- (cdata-mean(cdata))/msd(cdata) ) class(tmp) mean(tmp) msd(tmp) var(tmp) ```

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