compositions (version 0.9-11)

clr: Centered log ratio transform

Description

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

Usage

clr( x )
          clr.inv( z )

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)

Value

• clr gives the centered log ratio transform, clr.inv gives closed compositions with the given clr-transforms

Details

The clr-transform maps a composition in the D-part Aitchison-simplex isometrically to a D-1 dimensonal euclidian vector subspace: consequently, the transformation is not injective and only yields vectors which elements sum up to 0. 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 given by the vectors 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)))
clr.inv(tmp)
clr.inv(tmp) - clo(c(1,2,3)) # 0
data(Hydrochem)
cdata <- Hydrochem[,6:19]
pairs(clr(cdata))