YeoJohnsontransformationer

Yeo-Johnson transformations are a family of monotone, parametric power transformations used to stabilize variance and approximate normality in data that include both positive and negative values. Named after Darren Yeo and David Johnson, they generalize the Box-Cox transformation by extending to non-positive data, enabling standard linear modeling and related analyses without forcing the data to be strictly positive.

For a real-valued y and a parameter lambda, the forward transformation T(y; lambda) is defined in a piecewise way:

- If y is nonnegative: T = [(y + 1)^lambda − 1] / lambda for lambda ≠ 0; T = log(y + 1) for lambda = 0.

- If y is negative: T = −[((−y + 1)^(2 − lambda) − 1) / (2 − lambda)] for lambda ≠ 2; T = −log(−y + 1) for lambda = 2.

The transformation is monotone in y for all lambda and has a closed-form inverse, enabling back-transformation of model predictions. The inverse is:

- For the nonnegative branch: y = [(lambda T) + 1]^(1/lambda) − 1 if lambda ≠ 0; y = exp(T) − 1 if lambda = 0.

- For the negative branch: y = 1 − [1 − (2 − lambda) T]^(1/(2 − lambda)) if lambda ≠ 2; y = 1 − exp(−T) if lambda = 2.

Estimation and use: lambda is typically chosen to optimize some criterion, such as maximizing the normality of residuals or the likelihood of a model assuming normal errors. The Yeo-Johnson transformation is widely used in regression, ANOVA, and other modeling contexts to stabilize variance and improve model fit, especially when data include zero or negative values.

Limitations include potential interpretability challenges of the transformed scale and the fact that no transformation guarantees normality or homoscedasticity for all datasets. The approach remains a standard tool in statistical preprocessing and data analysis.