Log-log transformation

Applies the stabilising transformation $$y = \log\!\bigl(\log\bigl(x/\alpha + 1\bigr) + 1\bigr).$$ Useful for very right-skewed non-negative data, particularly count processes with heavy tails. The +1 inside each log makes the transformation well-defined at zero.

Usage

loglog(x, alpha = 1)

Arguments

x

Numeric vector. Values must satisfy x / alpha > -1.

alpha

Positive scaling parameter. Default: 1.

Value

A numeric vector of transformed values.

See also

iloglog() for the inverse, shewhart_box_cox() for a data-driven transformation choice.

Examples

x <- c(0, 1, 5, 10, 100, 1000)
loglog(x)
#> [1] 0.000000 0.526589 1.026672 1.223156 1.725463 2.067970