compositions (version 2.0-4)

clr: Centered log ratio transform

Description

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

Usage

clr( x,... )
          clrInv( z,..., orig=gsi.orig(z) )

Value

clr gives the centered log ratio transform,

clrInv gives closed compositions with the given clr-transform

Arguments

x

a composition or a data matrix of compositions, not necessarily closed

z

the clr-transform of a composition or a data matrix of clr-transforms of compositions, not necessarily centered (i.e. summing up to zero)

...

for generic use only

orig

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

Author

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

Details

The clr-transform maps a composition in the D-part Aitchison-simplex isometrically to a D-dimensonal euclidian vector subspace: consequently, the transformation is not injective. Thus resulting covariance matrices are always singular.
The data can then be analysed in this transformation by all classical multivariate analysis tools not relying on a full rank of the covariance. See ilr and alr for alternatives. The interpretation of the results is relatively easy since the relation between each original part and a transformed variable is preserved.
The centered logratio transform is given by $$ clr(x) := \left(\ln x_i - \frac1D \sum_{j=1}^D \ln x_j\right)_i $$ The image of the clr is a vector with entries summing to 0. This hyperplane is also called the clr-plane.

References

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

See Also

ilr,alr,apt

Examples

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