Non-parametric regression models for compositional data

Compositional data arise in many real-life applications and versatile methods for properly analyzing this type of data in the regression context are needed. When parametric assumptions do not hold or are difficult to verify, non parametric regression models can provide a convenient alternative method for prediction. To this end, we consider an extension to the classical $k$-$NN$ regression, termed $\alpha$-$k$-$NN$ regression, that yields a highly flexible non-parametric regression model for compositional data through use of the $\alpha$-transformation. Our model is further extended to the $\alpha$-kernel regression by adopting the Nadaraya-Watson estimator. Unlike many of the recommended regression models for compositional data, zeros values (which commonly occur in practice) are not problematic and they can be incorporated into the proposed models without modification. Extensive simulation studies and real-life data analyses highlight the advantage of using these non-parametric regressions for complex relationships between the compositional response data and Euclidean predictor variables. Both suggest that $\alpha$-$k$-$NN$ and $\alpha$-kernel regressions can lead to more accurate predictions compared to current regression models which assume a, sometimes restrictive, parametric relationship with the predictor variables. In addition, the $\alpha$-$k$-$NN$ regression, in contrast to $\alpha$-kernel regression, enjoys a high computational efficiency rendering it highly attractive for use with large scale, massive, or big data.