YeoJohnsonTransformation

YeoJohnsonTransformation, commonly called the Yeo–Johnson transformation, is a family of power transformations that extends the Box–Cox approach to data that can take negative values. It introduces a single shape parameter, λ, to transform a real-valued variable y in a way that can stabilize variance and improve normality, which in turn can enhance the performance of statistical models that assume normality or homoscedasticity.

The transformation is defined piecewise for y ≥ 0 and y < 0:

- If y ≥ 0: y(λ) = [(y+1)^λ − 1]/λ for λ ≠ 0; and y(0) = ln(y+1).

- If y < 0: y(λ) = − [(-y+1)^(2−λ) − 1]/(2−λ) for λ ≠ 2; and y(2) = −ln(-y+1).

The Yeo–Johnson transformation is invertible for all real y and a suitable λ, with inverse formulas:

- For y ≥ 0: if λ ≠ 0, y = (1 + λz)^(1/λ) − 1; if λ = 0, y = exp(z) − 1, where z = y(λ).

- For y < 0: if λ ≠ 2, y = 1 − [1 − z(2−λ)]^(1/(2−λ)); if λ = 2, y = 1 − exp(−z).

Advantages include applicability to the full real line and the ability to approximate normality without data shifting. It is widely used in preprocessing for regression, ANOVA, and other statistical analyses where transformation of the response variable is beneficial. Limitations include sensitivity to the choice of λ and potential interpretability challenges of transformed scales.