The $α$--regression for compositional data: a unified framework for standard, spatially-lagged, spatial autoregressive and geographically-weighted regression models

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.

💡 Research Summary

The paper introduces a unified framework for regression with compositional data—vectors of non‑negative components that sum to one—by exploiting the α‑transformation, a power‑based mapping that continuously bridges raw‑data analysis (RDA) and log‑ratio analysis (LRA). For any α∈