R: Pairwise log ratio transform

pwlr {compositions} | R Documentation

## Pairwise log ratio transform

### Description

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

### Usage

          pwlr( x, as.rmult=FALSE, as.data.frame=!as.rmult, ...)
          pwlrInv( z, orig=gsi.orig(z))

### Arguments

`x` | a composition, not necessarily closed
`z` | the pwlr-transform of a composition, thus a [D(D-1)/2]-dimensional real vector, or a matrix with such many columns
`as.rmult` | logical; should the output be produced as an rmult object?
`as.data.frame` | logical; should be as a data.frame? if both are false, rmult will be taken
`...` | currently unused
`orig` | the original composition, to check consistency and recover component names

### Details

The pwlr-transform maps a composition in the $D$-part Aitchison-simplex isometrically to a $D(D-1)/2$ dimensonal euclidian vector, computing each possible logratio (accounting for the fact that $log(A/B)=-log(B/A)$, and therefore only one of them is necessary). The data can then be analysed in this transformation by multivariate analysis tools not relying on the invertibility of the covariance function. The interpretation of the results is relatively simple, since each component captures the behaviour of the simple ratio between two party. However redundance between them is extremely high, and any of `alr`, `clr` or `ilr` transformations may be preferred in most applications.

The pairwise logratio transform is given by

pwlr(x)_{ij} := \ln(x_i/x_j)

.

The inverse requires some explanation, because of the redundance between pwlr scores. Note that for any three components $A,B,C$ it holds that $log(A/C)=log(A/B)+log(B/C)$. So, any vector of $D(D-1)/2$ coefficients will not be necessarily a valid pwlr-transformed composition: if these coefficients do not satisfy that kind of relations, the vector is, strictly speaking, not a pwlr and should not be inverted. Nevertheless, the function gives a least-squares inversion, as proposed by Tolosana-Delgado and von Eynatten (2009).

### Value

`pwlr` gives the pairwise log ratio transform; accepts a compositional dataset `pwlrInv` gives a closed composition with the given wplr-transform; accepts a dataset

### Author(s)

R. Tolosana-Delgado http://www.stat.boogaart.de

### References

Aitchison, J. (1986) The Statistical Analysis of Compositional Data Monographs on Statistics and Applied Probability. Chapman & Hall Ltd., London (UK). 416p.

Tolosana-Delgado, R. and H. von Eynatten (2009); Grain-size control on petrographic composition of sediments: compositional regression and rounded zeroes. Mathematical Geosciences: 41(8): 869-886. doi: 10.1007/s11004-009-9216-6.

### See Also

`clr`,`alr`,`apt`, http://ima.udg.es/Activitats/CoDaWork03/

### Examples

    (tmp <- pwlr(c(1,2,3)))
    pwlrInv(tmp)

[Package compositions version 2.0-4 Index]