This implementation is a special case of the class of isometric log-ratio transformations.

1 2 |

`x` |
A numeric vector. Naturally, the forward transformation is only sensible for vectors with all elements being greater than zero. |

The result of the forward or backward transformation. The returned components always sum to 1 for the case of the inverse log-ratio transformation.

RenĂ© Lehmann and Johannes Ranke

Peter Filzmoser, Karel Hron (2008) Outlier Detection for Compositional Data Using Robust Methods. Math Geosci 40 233-248

Another implementation can be found in R package `robCompositions`

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
# 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)
``` |

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