A new class of alpha-transformations for the spatial analysis of Compositional Data

Georeferenced compositional data are prominent in many scientific fields and in spatial statistics. This work addresses the problem of proposing models and methods to analyze and predict, through kriging, this type of data. To this purpose, a novel class of alpha-transformations, named the Isometric alpha-transformation (alpha-IT), is proposed, which encompasses the traditional Isometric Log-Ratio (ILR) transformation. Similarly to other alpha-transformations existing in the literature, it is shown that the ILR is the limit case of the alpha-IT as alpha tends to 0 and that alpha corresponds to a linear transformation of the data. Unlike the ILR, the proposed transformation accepts 0s in the compositions when alpha>0. Maximum likelihood estimation of the parameter alpha is established. Prediction using kriging on alpha-IT transformed data is validated on synthetic spatial compositional data, using prediction scores computed either in the geometry induced by the alpha-IT, or in the simplex. Application to land cover data shows that the relative superiority of the various approaches w.r.t. a prediction objective depends on whether the compositions contained any zero component. When all components are positive, the limit cases (ILR or linear transformations) are optimal for none of the considered metrics. An intermediate geometry, corresponding to the alpha-IT with maximum likelihood estimate, better describes the dataset in a geostatistical setting. When the amount of compositions with 0s is not negligible, some side-effects of the transformation gets amplified as alpha decreases, entailing poor kriging performances both within the alpha-IT geometry and for metrics in the simplex.