Applies the inverse of the ll4 transformation.
Given y = ll4(x), this function calculates x = (base^(y * lambda) - 1)^(1/lambda).
Arguments
- y
A numeric vector or scalar of transformed values (output from
ll4).- lambda
A positive numeric scalar, the lambda parameter used in the original
ll4transformation. Must match the one used for the forward transform. Default is4.- base
A positive numeric scalar, the base of the logarithm used in the original
ll4transformation. Must match. Default is10.
Value
A numeric vector or scalar of the original, untransformed values.
Returns 0 when the intermediate quantity base^(y * lambda) - 1 is
negative (i.e., when y < 0 for base = 10 and even lambda), since
consumption cannot be negative.
Details
Domain and boundary behavior. The inverse LL4 transformation is defined
for y >= 0 (when base = 10 and lambda = 4). For y < 0, the
intermediate quantity base^(y * lambda) - 1 becomes negative, and raising
a negative number to the fractional power 1/lambda is undefined in real
arithmetic. In this case, the function returns 0 (consumption cannot be
negative).
This boundary condition arises in practice when a model predicts fitted values
below zero on the LL4 scale — typically for extrapolation to very high
prices. The mapping to zero is the natural floor because ll4(0) = 0 and
the LL4 transformation is monotonically increasing on [0, Inf).
Examples
original_values <- c(0, 1, 10, 100, 1000)
transformed_values <- ll4(original_values)
back_transformed_values <- ll4_inv(transformed_values)
print(data.frame(original_values, transformed_values, back_transformed_values))
#> original_values transformed_values back_transformed_values
#> 1 0 0.0000000 0
#> 2 1 0.0752575 1
#> 3 10 1.0000109 10
#> 4 100 2.0000000 100
#> 5 1000 3.0000000 1000
all.equal(original_values, back_transformed_values) # Should be TRUE or very close
#> [1] TRUE
# Negative y values are mapped to 0 (consumption floor)
ll4_inv(-0.5, lambda = 4, base = 10) # Returns 0
#> [1] 0