Regression for compositional data by using distributions defined on the hypersphere

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Spatial Regression Modeling for Compositional Data With Many Zeros (2013, background): “Scealy and Welsh (2011), building on work by Stephens (1982), employ a square-root transformation to convert the compositional data to directional data, where the occurrence of zeros is no longer an issue.”
Robust Principal Component Analysis for Power Transformed Compositional Data (2015, methodology): “…α = 1/2 maps compositional data onto the positive orthant of the hypersphere and, provided the data are not too close to the boundaries and have small enough variance, allows us to use ideas, distributions and estimators for directional data (Stephens 1982; Stanley 1990; Scealy and Welsh 2011).”
Aitchison’s Compositional Data Analysis 40 Years on: A Reappraisal (2022, methodology): “The square root transformation maps the proportions onto the multidimensional unit sphere in the positive orthant [84], and the Euclidean distances are then the same as the Hellinger distances, named after Ernst Hellinger [54].”
The mathematics of compositional analysis (2016, unknown): “Watson and Philip (1989), Wang, Liu, Mok, Fu, and Tse (2007) or Scealy and Welsh (2011), for example, prefer to apply the techniques characteristic of directional data, given that they take the positive orthant of the unit hypersphere centred at the origin as the sample space of the CoDa.”