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.

翻译：我们重新审视了针对成分数据的$α$--回归框架。将$α$--回归表述为非线性最小二乘问题，研究其渐近性质，并通过Levenberg--Marquardt算法实现高效估计。随后提出基于置换的假设检验方法，推导边际效应以辅助解释，并提供各预测变量效应的可视化诊断。进一步讨论鲁棒化版本、自然样条的引入以及成分预测变量的整合，这些改进为构建简洁的时间序列模型奠定基础。该框架通过四个模型扩展至空间场景：(a) $α$--空间滞后X回归模型，通过空间滞后协变量纳入空间溢出效应，并将其分解为直接效应和间接效应；(b) $α$--空间自回归模型，允许空间自相关存在；(c) 地理加权$α$--回归，允许系数随空间位置变化以捕捉局部关系；(d) $α$--特征向量空间滤波，通过核化距离矩阵的特征向量高效捕获空间依赖性。基于四个真实数据集的实验表明，所提模型性能与现有文献模型相当或更优。案例展示$α$--回归在不同场景下可优于多种竞争性回归模型，其空间扩展既能捕获依赖关系又能提升预测性能。总体上，实验结果证明对数比方法并非总能获得最优结果。