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
- If y is nonnegative: T = [(y + 1)^lambda − 1] / lambda for lambda ≠ 0; T = log(y + 1) for
- If y is negative: T = −[((−y + 1)^(2 − lambda) − 1) / (2 − lambda)] for lambda ≠ 2; T = −log(−y
The transformation is monotone in y for all lambda and has a closed-form inverse, enabling back-transformation
- For the nonnegative branch: y = [(lambda T) + 1]^(1/lambda) − 1 if lambda ≠ 0; y = exp(T) − 1 if
- For the negative branch: y = 1 − [1 − (2 − lambda) T]^(1/(2 − lambda)) if lambda ≠ 2; y = 1
Estimation and use: lambda is typically chosen to optimize some criterion, such as maximizing the normality
Limitations include potential interpretability challenges of the transformed scale and the fact that no transformation guarantees