This implementation is a special case of the class of isometric log-ratio transformations.
ilr(x)
invilr(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.
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.
# 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