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. This paper, through use of the $\alpha$-transformation, extends the classical $k$-$NN$ regression to what is termed $\alpha$-$k$-$NN$ regression, yielding a highly flexible non-parametric regression model for compositional data. The $\alpha$-$k$-$NN$ is further extended to the $\alpha$-kernel regression by adopting the Nadaray-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.