Compute the isometric log ratio transform of a (dataset of) composition(s), and its inverse.
ilr( x , V = ilrBase(x) ,...)
ilrInv( z , V = ilrBase(z=z),..., orig=gsi.orig(z))ilr gives the isometric log ratio transform,
ilrInv gives closed compositions with the given ilr-transforms
a composition, not necessarily closed
the ilr-transform of a composition
a matrix, with columns giving the chosen basis of the clr-plane
generic arguments. not used.
a compositional object which should be mimicked by the inverse transformation. It is especially used to reconstruct the names of the parts.
K.Gerald v.d. Boogaart http://www.stat.boogaart.de, Raimon Tolosana-Delgado
The ilr-transform maps a composition in the D-part Aitchison-simplex
isometrically to a D-1 dimensonal euclidian vector. The data can then
be analysed in this transformation by all classical multivariate
analysis tools. However the interpretation of the results may be
difficult, since there is no one-to-one relation between the original parts
and the transformed variables.
The isometric logratio transform is given by
$$ ilr(x) := V^t clr(x) $$
with clr(x) the centred log ratio transform and
\(V\in R^{d \times (d-1)}\) a matrix which columns form an orthonormal
basis of the clr-plane. A default matrix \(V\) is given by
ilrBase(D).
Egozcue J.J., V. Pawlowsky-Glahn, G. Mateu-Figueras and
C. Barcel'o-Vidal (2003) Isometric logratio transformations for
compositional data analysis. Mathematical Geology, 35(3)
279-300
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
https://ima.udg.edu/Activitats/CoDaWork03/
clr,alr,apt, ilrBase
(tmp <- ilr(c(1,2,3)))
ilrInv(tmp)
ilrInv(tmp) - clo(c(1,2,3)) # 0
data(Hydrochem)
cdata <- Hydrochem[,6:19]
pairs(ilr(cdata))
ilrBase(D=3)
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