Compositional data analysis by the square-root transformation: Application to NBA USG% data

Compositional data refers to data where the sum of the values of the components is a constant, hence the sample space is defined as a simplex making it impossible to apply statistical methods developed in the usual Euclidean vector space.A natural approach to overcome this restriction is to consider an appropriate transformation which moves the sample space onto the Euclidean space, and log-ratio typed transformations, such as the additive log-ratio (ALR), the centered log-ratio (CLR) and the isometric log-ratio (ILR) transformations, have been mostly conducted.However, in scenarios with sparsity, where certain components take on exact zero values, these log-ratio type transformations may not be effective.In this work, we mainly suggest an alternative transformation, that is the square-root transformation which moves the original sample space onto the directional space.We compare the square-root transformation with the log-ratio typed transformation by the simulation study and the real data example.In the real data example, we applied both types of transformations to the USG% data obtained from NBA, and used a density based clustering method, DBSCAN (density-based spatial clustering of applications with noise), to show the result.