Location and Scale-Invariant Power Transformations for Transforming Data to Normality

Abstract Power transformations are used to stabilize variance and achieve normality of features, especially in methods assuming normal distributions such as ANOVA and linear discriminant analysis. However, the commonly used Box-Cox and Yeo-Johnson power transformation methods are sensitive to the location, scale, and presence of outliers in the data. Here we present location- and scale-invariant Box-Cox and Yeo-Johnson transformations to mitigate these issues. We derive maximum likelihood estimation criteria for optimizing transformation parameters and propose robust adaptations that reduce the influence of outliers. We also introduce an empirical test for assessing central normality of transformed features. In simulations and real-world datasets, robust location- and scale-invariant transformations outperform conventional variants, resulting in better transformations to central normality. In a machine learning experiment with 231 datasets with numerical features, integrating robust location- and scale-invariant power transformations into an automated data processing and machine learning pipeline did not result in a meaningful improvement or detriment to model performance compared to conventional variants. In conclusion, robust location- and scale-invariant power transformations can replace conventional variants.