Compute the pairwise log ratio transform of a (dataset of) composition(s), and its inverse.
pwlr( x, as.rmult=FALSE, as.data.frame=!as.rmult, ...)
pwlrInv( z, orig=gsi.orig(z))pwlr gives the pairwise log ratio transform; accepts a compositional dataset
pwlrInv gives a closed composition with the given wplr-transform; accepts a dataset
a composition, not necessarily closed
the pwlr-transform of a composition, thus a [D(D-1)/2]-dimensional real vector, or a matrix with such many columns
logical; should the output be produced as an rmult object?
logical; should be as a data.frame? if both are false, rmult will be taken
currently unused
the original composition, to check consistency and recover component names
R. Tolosana-Delgado http://www.stat.boogaart.de
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 $$ pwlr(x)_{ij} := \ln\frac{x_i}{x_j} $$.
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).
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. tools:::Rd_expr_doi("10.1007/s11004-009-9216-6").
clr,alr,apt,
http://ima.udg.es/Activitats/CoDaWork03/
(tmp <- pwlr(c(1,2,3)))
pwlrInv(tmp)
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