Centered log ratio transform
Compute the centered log ratio transform of a (dataset of) composition(s) and its inverse.
clr( x,... )
clrInv( z,..., orig=gsi.orig(z) )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. |
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) := (\emph{ln} \bold{x} - mean(\emph{ln} \bold{x}) )
The image of the clr is a vector with entries
summing to 0. This hyperplane is also called the clr-plane.
clr gives the centered log ratio transform,
clrInv gives closed compositions with the given clr-transform
K.Gerald v.d. Boogaart http://www.stat.boogaart.de
Aitchison, J. (1986) The Statistical Analysis of Compositional Data, Monographs on Statistics and Applied Probability. Chapman & Hall Ltd., London (UK). 416p.
(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=".")
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