# clr2ilr: Convert between clr and ilr, and between cpt and ipt. In compositions: Compositional Data Analysis

## Description

Compute the centered log ratio transform of a (dataset of) from isometric log-ratio transform(s) and its inverse. Equivalently, compute centered and isometric planar transforms from each other. Acts in vectors and in bilinear forms. For bilinear forms, transform between variation-form from clr-form.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```clr2ilr( x , V=ilrBase(x=x) ) ilr2clr( z , V=ilrBase(z=z), x=gsi.orig(z) ) clrvar2ilr( varx , V=ilrBase(D=ncol(varx)) ) ilrvar2clr( varz , V=ilrBase(D=ncol(varz)+1) ,x=NULL) clrvar2variation(Sigma) variation2clrvar(TT) is.clrvar(M, tol=1e-10) is.ilrvar(M, tol=1e-10) ```

## Arguments

 `x` the clr/cpt-transform of composition(s) (in the ilr2-routines provided only to give column names.) `z` the ilr/ipt-transform of composition(s) `varx, Sigma` variance or covariance matrix of clr/cpt-transformed compositions `varz` variance or covariance matrix of ilr/ipt-transformed compositions `V` a matrix with columns giving the chosen basis of the clr-plane `TT` variation matrix `M` a matrix, to check if it is a valid variance `tol` tolerance for the check

## Details

These functions perform a matrix multiplication with `V` in an appropriate way.

## Value

`clr2ilr` gives the ilr/ipt transform of the same composition(s),
`ilr2clr` gives the clr/cpt transform of the same composition(s),
`clrvar2ilr` gives the variance-/covariance-matrix of the ilr/ipt transform of the same compositional data set,
`ilrvar2clr` and `clrvar2variation` give the variance-/covariance-matrix of the clr/cpt transform of the same compositional data set.
`variation2clrvar` gives the variation matrix from the clr-covariance matrix
`is.*var` check if the given matrix satisfies the conditions to be an ilr-variance resp. a clr-variance

## Author(s)

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

## 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

`variation`, `ilr`, `ipt`, `clr`, `cpt`
 ```1 2 3 4 5 6 7``` ```data(SimulatedAmounts) ilrInv(clr2ilr(clr(sa.lognormals)))-clo(sa.lognormals) clrInv(ilr2clr(ilr(sa.lognormals)))-clo(sa.lognormals) ilrvar2clr(var(ilr(sa.lognormals)))-var(clr(sa.lognormals)) clrvar2ilr(var(cpt(sa.lognormals)))-var(ipt(sa.lognormals)) variation(acomp(sa.lognormals)) clrvar2variation(var(acomp(sa.lognormals))) ```