# ipt: Isometric planar transform In compositions: Compositional Data Analysis

## Description

Compute the isometric planar transform of a (dataset of) composition(s) and its inverse.

## Usage

 ```1 2 3 4``` ``` ipt( x , V = ilrBase(x),... ) iptInv( z , V = ilrBase(z=z),...,orig=gsi.orig(z)) uciptInv( z , V = ilrBase(z=z),...,orig=gsi.orig(z) ) ```

## Arguments

 `x` a composition or a data matrix of compositions, not necessarily closed `z` the ipt-transform of a composition or a data matrix of ipt-transforms of compositions `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 ipt-transform maps a composition in the D-part real-simplex isometrically to a D-1 dimensonal euclidian vector. Although the transformation does not reach the whole R^{D-1}, resulting covariance matrices are typically of full rank.
The data can then be analysed in this transformation by all classical multivariate analysis tools. However, interpretation of results may be difficult, since the transform does not keep the variable names, given that there is no one-to-one relation between the original parts and each transformed variables. See `cpt` and `apt` for alternatives.

The isometric planar transform is given by

ipt(x) := V^t cpt(x)

with `cpt`(x) the centred planar transform and V a matrix which columns form an orthonormal basis of the clr-plane. A default matrix V is given by `ilrBase(D)`

## Value

`ipt` gives the centered planar transform, `iptInv` gives closed compositions with with the given ipt-transforms, `uciptInv` unconstrained iptInv does the same as iptInv but sets illegal values to NA rather than giving an error. This is a workaround to allow procedures not honoring the constraints of the space.

## Author(s)

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

## References

van den Boogaart, K.G. and R. Tolosana-Delgado (2008) "compositions": a unified R package to analyze Compositional Data, Computers & Geosciences, 34 (4), pages 320-338, doi: 10.1016/j.cageo.2006.11.017.

`ilr`,`ilrBase`, `cpt`
 ```1 2 3 4 5 6``` ```(tmp <- ipt(c(1,2,3))) iptInv(tmp) iptInv(tmp) - clo(c(1,2,3)) # 0 data(Hydrochem) cdata <- Hydrochem[,6:19] pairs(ipt(cdata)) ```