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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.
+  }
+}
+\details{
+}
+\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{
+%% ~put references to the literature/web site here ~
+}
+\author{
+  René Lehmann and Johannes Ranke
+}
+\note{
+%%  ~~further notes~~
+}
+
+\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 ration 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 }