BoxCoxYeoJohnson

BoxCoxYeoJohnson is a family of power transformations used in statistics to stabilize variance and make data more normally distributed. It is designed to handle data that include positive, zero, and negative values by combining features of the Box-Cox transformation and the Yeo-Johnson transformation. The goal is to provide a single, monotonic transformation that can be applied across the entire real line, improving the suitability of data for parametric modeling.

Conceptually, BoxCoxYeoJohnson extends the Box-Cox approach, which requires strictly positive observations, by incorporating the Yeo-Johnson framework, which accommodates zero and negative values. In practice, the transformation is implemented as a piecewise function with a positive-domain branch and a negative-domain branch. The same shaping parameter governs the transformation (in many implementations a single lambda, with some variants allowing a separate parameter on one branch). On the nonnegative side, the positive branch resembles the Box-Cox form, while on the negative side it adopts a Yeo-Johnson-like form. In some formulations, two parameters or branch-specific adjustments are used to optimize normality.

Estimation and use: the transformation parameter is typically estimated by maximizing likelihood under the assumption of normality of the transformed data or through cross-validation. After transformation, standard preprocessing steps such as centering and scaling may follow. The transformed data can then be used in linear models, ANOVA, or other methods that assume approximate normality and stabilized variance.

See also: Box-Cox transformation; Yeo-Johnson transformation; data normalization in statistical modeling.