A data-based power transformation for compositional data

Compositional data analysis is carried out either by neglecting the compositional constraint and applying standard multivariate data analysis, or by transforming the data using the logs of the ratios of the components. In this work we examine a more general transformation which includes both approaches as special cases. It is a power transformation and involves a single parameter�. The transformation has two equivalent versions. The �first is the stay-in-the-simplex version. This expression is the power transformation as de�fined by Aitchison (1986). The second version, which is a linear transformation of the stay-in-the-simplex, is a Box-Cox type transformation. We call the second version the isometric �alpha-transformation because of the multiplication with the Helmert sub-matrix. We discuss a parametric way of estimating the value of alpha�, which is maximization of its pro�le like-lihood (assuming multivariate normality of the transformed data) and the equivalence between the two versions is exhibited. Other ways include maximization of the correct classi�cation probability in discriminant analysis and maximization of the pseudo-R2 in linear regression. We examine the relationship between the transformation, the raw data approach and the isometric log-ratio transformation. Furthermore, we also de�fine a suitable family of metrics corresponding to the family of �alpha-transformation and consider the corresponding family of Fr�echet means.