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

## Arguments

- 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

## See also

Another implementation can be found in R package
`robCompositions`

.

## Author

René Lehmann and Johannes Ranke

## Examples

```
# Order matters
ilr(c(0.1, 1, 10))
#> [1] -1.628174 -2.820079
ilr(c(10, 1, 0.1))
#> [1] 1.628174 2.820079
# Equal entries give ilr transformations with zeros as elements
ilr(c(3, 3, 3))
#> [1] 0 0
# Almost equal entries give small numbers
ilr(c(0.3, 0.4, 0.3))
#> [1] -0.2034219 0.1174457
# Only the ratio between the numbers counts, not their sum
invilr(ilr(c(0.7, 0.29, 0.01)))
#> [1] 0.70 0.29 0.01
invilr(ilr(2.1 * c(0.7, 0.29, 0.01)))
#> [1] 0.70 0.29 0.01
# Inverse transformation of larger numbers gives unequal elements
invilr(-10)
#> [1] 7.213536e-07 9.999993e-01
invilr(c(-10, 0))
#> [1] 7.207415e-07 9.991507e-01 8.486044e-04
# The sum of the elements of the inverse ilr is 1
sum(invilr(c(-10, 0)))
#> [1] 1
# 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
#> [1] 4
ilr(c(b[1:3], 1 - sum(b[1:3]))) # Gives c(0.1, 0.3, 0.5)
#> [1] 0.1 0.3 0.5
```