Log-Logistic Transformation (LL4-like)

Applies a log-logistic like transformation, specifically log_base(x^lambda + 1) / lambda. This transformation is useful for compressing data that spans several orders of magnitude while handling zero values gracefully (as x=0 yields 0). It's a variation related to the Box-Cox transformation or a generalized logarithm.

Usage

ll4(x, lambda = 4, base = 10)

Arguments

x

A numeric vector or scalar of non-negative values to be transformed.

lambda

A positive numeric scalar, the lambda parameter of the transformation. Controls the curvature. Default is 4.

base

A positive numeric scalar, the base of the logarithm. Default is 10.

Value

A numeric vector or scalar of the transformed values. Returns NaN for x < 0 if lambda results in non-real numbers (e.g., even root of negative). However, the intended domain is x >= 0.

Examples

# Basic usage
ll4(0)
#> [1] 0
ll4(1)
#> [1] 0.0752575
ll4(10)
#> [1] 1.000011
ll4(c(0, 1, 10, 100, 1000))
#> [1] 0.0000000 0.0752575 1.0000109 2.0000000 3.0000000

# Using a different lambda or base
ll4(10, lambda = 2)
#> [1] 1.002161
ll4(10, base = exp(1)) # Natural log base
#> [1] 2.30261

# Typical workflow: transform -> fit -> back-transform
data(apt)
apt$y_ll4 <- ll4(apt$y)
# Verify round-trip: ll4_inv(ll4(x)) == x
all.equal(apt$y, ll4_inv(apt$y_ll4))
#> [1] TRUE