ilr {compositions}

Isometric log ratio transform

Description

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

Usage

          ilr( x , V = ilrBase(x) ,...)
          ilrInv( z , V = ilrBase(z=z),..., orig=gsi.orig(z))
          

Arguments

x

a composition, not necessarily closed

z

the ilr-transform of a composition

V

a matrix, with columns giving the chosen basis of the clr-plane

...

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.

Details

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).

Value

ilr gives the isometric log ratio transform, ilrInv gives closed compositions with the given ilr-transforms

Author(s)

K.Gerald v.d. Boogaart http://www.stat.boogaart.de, Raimon Tolosana-Delgado

References

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/

See Also

clr,alr,apt, ilrBase

Examples

(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)