An Affine Equivariant Multivariate Normal Score Transform for Compositional Data

The geostatistical treatment of continuous variables often includes a transformation to normal scores. In the case of analysing a composition, it has been suggested that standard methods can be applied to (isometric) logratio transformed compositions. Several logratio transformations are available and invariance of the final results under the choice of logratio transform is desirable. However, a geostatistical procedure which includes marginal normal scores transformations of the individual logratio scores via quantile matching will not have this invariance property, nor will the resulting vectors of scores show a joint multivariate normal distribution. In this paper an affine-equivariant normal score transform is proposed. The method is based on a continuous deformation of the underlying logratio space to a Gaussian space. The properties and performance of this method are illustrated and compared with existing alternatives using a simulated setting and a case study from a banded iron formation ore mining operation from Western Australia. The proposed method is also suitable for the study of other multivariate non-compositional cases.