A Review of Flexible Transformations for Modeling Compositional Data

Vectors of non-negative components carrying only relative information, and often normalized to sum to one, are referred to as compositional data and their sample space is the simplex. Compositional data arise in many applications across a variety of disciplines such as ecology, geology, demography, and economics to name a few. For some time, log-ratio methods have been a popular approach for analyzing compositional data and have motivated much of the recent research in the area. In this paper, we consider two recently proposed transformations for data defined on the simplex. The first, referred to as the α-transformation, transforms the data from the simplex to a subset of Euclidean space while a more complex transformation, involving folding, results in data with Euclidean sample space. In both cases, the transformed data are assumed to follow a multivariate normal distribution and the parameter α provides flexibility compared to the traditional log-ratio transformations. Through an empirical study using several real-life data sets we illustrate that the α-transformation may be sufficient and preferred in practice compared to the α-folded model, and further that it is often needed over the log-ratio transformation.