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

ilr(x)
invilr(x)

## 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.

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.

## Examples

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