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

2025-10-14

AI Summary

Compositional data—characterized by unit-sum constraints and frequent zero values—pose significant challenges for conventional regression, spatial lag, and geographically weighted regression models. To address this, we propose a unified regression framework based on the α-transformation, the first such integration of α-transformation into spatial regression. The framework inherently accommodates zeros (without imputation), respects the simplex constraint, and captures nonlinear relationships, while simultaneously supporting standard regression, spatial spillover decomposition, and geographically weighted local modeling. Parameter estimation employs the Levenberg–Marquardt algorithm for nonlinear least squares, with analytically derived gradients and Hessian matrices to enable data-driven selection of the α-parameter and interpretable marginal effects. Empirical evaluation on Greek agricultural land-use data demonstrates substantial improvements in predictive accuracy; moreover, the estimators exhibit asymptotic normality, ensuring valid statistical inference.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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 struggle with the unit-sum constraint and zero values. This paper revisits the $alpha$--regression framework, which uses a flexible power transformation parameterized by $alpha$ to interpolate between raw data analysis and log-ratio methods, naturally handling zeros without imputation while allowing data-driven transformation selection. We formulate $alpha$--regression as a non-linear least squares problem, provide efficient estimation via the Levenberg-Marquardt algorithm with explicit gradient and Hessian derivations, establish asymptotic normality of the estimators, and derive marginal effects for interpretation. The framework is extended to spatial settings through two models: the $alpha$--spatially lagged X regression model, which incorporates spatial spillover effects via spatially lagged covariates with decomposition into direct and indirect effects, and the geographically weighted $alpha$--regression, which allows coefficients to vary spatially for capturing local relationships. Application to Greek agricultural land-use data demonstrates that spatial extensions substantially improve predictive performance.

Problem

• Develops flexible regression for compositional data with unit-sum constraint
• Handles zero values without imputation using data-driven transformation selection
• Extends framework to spatial models with spillover effects and local relationships

Innovation

• Flexible power transformation handling compositional data zeros
• Efficient estimation via Levenberg-Marquardt algorithm with derivations
• Spatial extensions incorporating lagged covariates and varying coefficients