Function to perform isometric log-ratio transformation

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

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ilr(x)
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-invilr(x)

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Arguments

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

Value

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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

Author

René Lehmann and Johannes Ranke

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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
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