compositions (version 2.0-8)

alr: Additive log ratio transform

Description

Compute the additive log ratio transform of a (dataset of) composition(s), and its inverse.

Usage

alr( x ,ivar=ncol(x), ... )
          alrInv( z, ...,orig=gsi.orig(z))

Value

alr gives the additive log ratio transform; accepts a compositional dataset

alrInv gives a closed composition with the given alr-transform; accepts a dataset

Arguments

x

a composition, not necessarily closed

z

the alr-transform of a composition, thus a (D-1)-dimensional real vector

...

generic arguments. not used.

orig

a compositional object which should be mimicked by the inverse transformation. It is especially used to reconstruct the names of the parts.

ivar

The column to be used as denominator variable. Unfortunately not yet supported in alrInv. The default works even if x is a vector.

Author

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

Details

The alr-transform maps a composition in the D-part Aitchison-simplex non-isometrically to a D-1 dimensonal euclidian vector, treating the last part as common denominator of the others. The data can then be analysed in this transformation by all classical multivariate analysis tools not relying on a distance. The interpretation of the results is relatively simple, since the relation to the original D-1 first parts is preserved. However distance is an extremely relevant concept in most types of analysis, where a clr or ilr transformation should be preferred.

The additive logratio transform is given by $$ alr(x)_i := \ln\frac{x_i}{x_D} $$.

References

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

See Also

clr,ilr,apt, https://ima.udg.edu/Activitats/CoDaWork03/

Examples

(tmp <- alr(c(1,2,3)))
alrInv(tmp)
unclass(alrInv(tmp)) - clo(c(1,2,3)) # 0
data(Hydrochem)
cdata <- Hydrochem[,6:19]
pairs(alr(cdata),pch=".")