Compositional data--vectors of non-negative components summing to unity--frequently arise in scientific applications where covariates influence the relative proportions of components, yet traditional regression approaches ace challenges regarding the unit-sum constraint and zero values. This paper revisits the $α$--regression framework, which uses a flexible power transformation parameterized by $α$ to interpolate between raw data analysis and log-ratio methods, naturally handling zeros without imputation while allowing data-driven transformation selection. We formulate $α$--regression as a non-linear least squares problem, study its asymptotic properties, provide efficient estimation via the Levenberg-Marquardt algorithm, and derive marginal effects for interpretation. The framework is extended to spatial settings through three models: the $α$--spatially lagged X regression model, which incorporates spatial spillover effects via spatially lagged covariates with decomposition into direct and indirect effects, the $α$--spatially autoregressive regression model and the geographically weighted $α$--regression, which allows coefficients to vary spatially for capturing local relationships. Applications to two real data sets illustrate the performance of the models and showcase that spatial extensions capture the spatial dependence and improve the predictive performance.

翻译：成分数据——即各分量非负且总和为一的向量——常见于协变量影响分量相对比例的科学应用中，然而传统回归方法在处理单位总和约束与零值时面临挑战。本文重新审视α回归框架，该框架通过参数α控制的灵活幂变换，在原始数据分析和对数比方法之间进行插值，无需插补即可自然处理零值，同时允许数据驱动的变换选择。我们将α回归表述为一个非线性最小二乘问题，研究其渐近性质，通过Levenberg-Marquardt算法提供高效估计，并推导用于解释的边际效应。该框架通过三种模型扩展至空间场景：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，并分解为直接与间接效应）、α-空间自回归回归模型，以及地理加权α回归（允许系数随空间变化以捕捉局部关系）。两个实际数据集的实证应用展示了模型的性能，并证明空间扩展能有效捕捉空间依赖性并提升预测表现。