The $α$--regression for compositional data: a unified framework for standard, temporal and spatial regression models including compositional predictors

Abstract

We revisit the $α$--regression framework for compositional data. We formulate $α$--regression as a non--linear least squares problem, study its asymptotic properties, and provide efficient estimation via the Levenberg--Marquardt algorithm. We then propose a permutation--based hypothesis testing procedure, derive marginal effects for interpretation, and provide a visual inspection of the effect of each predictor. We further discuss robustified versions, the inclusion of natural splines, and the incorporation of compositional predictors, which further facilitate the formulation of a simple time series model. The framework is extended to spatial settings through four models. (a) The $α$--spatially--lagged X regression model, which incorporates spatial spillover effects via spatially--lagged covariates, with decomposition into direct and indirect effects. (b) The $α$--spatial autoregressive model that allows for spatial autocorrelation. (c) The geographically--weighted $α$--regression, which allows coefficients to vary spatially for capturing local relationships. (d) The $α$--eigenvector spatial filtering that is computationally efficient and captures spatial dependence via the eigenvectors of the kernelized distance matrix. Applications to four real datasets illustrate that the models perform on par with or outperform existing models in the literature. The examples showcase that $α$--regression can outperform various competing regression models under different scenarios and its spatial extensions capture the dependence and improve the predictive performance. Overall, the examples provide evidence that the log--ratio methodology does not always lead to the optimal results.