YeoJohnsonTransformationen

The Yeo–Johnson transformation is a statistical method used to stabilize variance and improve normality of real-valued data. It extends the Box–Cox transformation to accommodate both positive and negative observations without the need to shift data. Introduced by Ishk I. K. Yeo and Raymond A. Johnson in 2000, it provides a continuous, monotone family of power transformations controlled by a single parameter lambda.

For a real-valued observation x, the transformed value T(x; lambda) is defined piecewise as follows:

- If x >= 0: T = ((x + 1)^lambda - 1) / lambda, for lambda not equal to 0; T = log(x + 1) when lambda = 0.

- If x < 0: T = -((-x + 1)^(2 - lambda) - 1) / (2 - lambda), for lambda not equal to 2; T = -log(-x + 1) when lambda = 2.

The parameter lambda is typically chosen to maximize normality or to stabilize variance, often via maximum likelihood, likelihood-based criteria, or cross-validation. In practice, the transformation is estimated from the data and then applied to obtain transformed values that are closer to normally distributed and suitable for downstream analyses such as linear regression, ANOVA, or other parametric methods.

Key advantages include handling of zero and negative values without pre-shifting, and the ability to approximate a normal distribution more flexibly than Box–Cox for real-valued data. Limitations involve interpretability of results on the original scale, dependence on the chosen lambda, and potential sensitivity to outliers during estimation. The Yeo–Johnson transformation remains a common preprocessing step in statistical modeling and data normalization workflows. See also Box–Cox transformation and normalizing transformations.