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-\name{ilr}
-\alias{ilr}
-\alias{invilr}
-\title{
-  Function to perform isotropic log-ratio transformation
-}
-\description{
-  This implementation is a special case of the class of isotropic log-ratio transformations.
-}
-\usage{
-  ilr(x)
-  invilr(x)
-}
-\arguments{
-  \item{x}{
-    A numeric vector. Naturally, the forward transformation is only sensible for
-    vectors with all elements being greater than zero.
-  }
-}
-\value{
-  The result of the forward or backward transformation. The returned components always 
-  sum to 1 for the case of the inverse log-ratio transformation.
-}
-\references{
-  Peter Filzmoser, Karel Hron (2008) Outlier Detection for Compositional Data Using Robust Methods. Math Geosci 40 233-248
-}
-\author{
-  René Lehmann and Johannes Ranke
-}
-\seealso{
-  Other implementations are in R packages \code{compositions} and \code{robCompositions}.
-}
-\examples{
-# Order matters
-ilr(c(0.1, 1, 10))
-ilr(c(10, 1, 0.1))
-# Equal entries give ilr transformations with zeros as elements
-ilr(c(3, 3, 3))
-# Almost equal entries give small numbers
-ilr(c(0.3, 0.4, 0.3))
-# Only the ratio between the numbers counts, not their sum
-invilr(ilr(c(0.7, 0.29, 0.01)))
-invilr(ilr(2.1 * c(0.7, 0.29, 0.01)))
-# Inverse transformation of larger numbers gives unequal elements
-invilr(-10)
-invilr(c(-10, 0))
-# The sum of the elements of the inverse ilr is 1
-sum(invilr(c(-10, 0)))
-# This is why we do not need all elements of the inverse transformation to go back:
-a <- c(0.1, 0.3, 0.5)
-b <- invilr(a)
-length(b) # Four elements
-ilr(c(b[1:3], 1 - sum(b[1:3]))) # Gives c(0.1, 0.3, 0.5)
-}
-
-\keyword{ manip }