ilr(x) invilr(x)
This implementation is a special case of the class of isometric log-ratio transformations.
Peter Filzmoser, Karel Hron (2008) Outlier Detection for Compositional Data Using Robust Methods. Math Geosci 40 233-248
# Order matters ilr(c(0.1, 1, 10))[1] -1.628174 -2.820079ilr(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 0.01invilr(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-01invilr(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] 4ilr(c(b[1:3], 1 - sum(b[1:3]))) # Gives c(0.1, 0.3, 0.5)[1] 0.1 0.3 0.5
robCompositions
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