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

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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

http://ima.udg.es/Activitats/CoDaWork03

http://ima.udg.es/Activitats/CoDaWork05

See Also

ilr,clr, alr,

Examples

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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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