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

Michail Tsagris, Yannis Pantazis

TL;DR

The paper tackles regression for compositional data on the simplex, where zeros and the unit-sum constraint pose challenges for traditional approaches. It introduces the $\\alpha$-regression, a flexible framework that uses the $\\alpha$-transformation to interpolate between raw-data methods and log-rratio techniques, naturally handling zeros. The authors extend this framework to spatial contexts via $\\alpha$-SLX, $\\alpha$-SAR, and GW$\\alpha$R models, derive marginal effects, establish consistency and asymptotic normality, and demonstrate improved predictive performance on two real datasets. An R package, CompositionalSRcompositionalsr, implements these methods for practical use. The work provides a unified, interpretable approach to spatially aware regression for compositional data with zeros and offers tools for model selection and inference that are relevant to geography, economics, and environmental science.

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 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, provide efficient estimation via the Levenberg-Marquardt algorithm, and derive marginal effects for interpretation. The framework is extended to spatial settings through two 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.

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

TL;DR

The paper tackles regression for compositional data on the simplex, where zeros and the unit-sum constraint pose challenges for traditional approaches. It introduces the -regression, a flexible framework that uses the -transformation to interpolate between raw-data methods and log-rratio techniques, naturally handling zeros. The authors extend this framework to spatial contexts via -SLX, -SAR, and GWR models, derive marginal effects, establish consistency and asymptotic normality, and demonstrate improved predictive performance on two real datasets. An R package, CompositionalSRcompositionalsr, implements these methods for practical use. The work provides a unified, interpretable approach to spatially aware regression for compositional data with zeros and offers tools for model selection and inference that are relevant to geography, economics, and environmental science.

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 ace challenges regarding the unit-sum constraint and zero values. This paper revisits the --regression framework, which uses a flexible power transformation parameterized by to interpolate between raw data analysis and log-ratio methods, naturally handling zeros without imputation while allowing data-driven transformation selection. We formulate --regression as a non-linear least squares problem, provide efficient estimation via the Levenberg-Marquardt algorithm, and derive marginal effects for interpretation. The framework is extended to spatial settings through two models: the --spatially lagged X regression model, which incorporates spatial spillover effects via spatially lagged covariates with decomposition into direct and indirect effects, the --spatially autoregressive regression model and the geographically weighted --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.
Paper Structure (85 sections, 3 theorems, 172 equations, 1 figure, 4 tables)

This paper contains 85 sections, 3 theorems, 172 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. The $\alpha$--regression
  3. The $\alpha$--transformation
  4. The $\alpha$--regression
  5. Limiting case of $\alpha \rightarrow 0$
  6. Choosing $\alpha$
  7. Marginal effects
  8. Standard error of the MEs
  9. Advantages and limitations of the $\alpha$--regression
  10. Consistency and asymptotic normality
  11. Robust extensions of the $\alpha$--regression
  12. The $\alpha$--regression with continuous and compositional predictors
  13. The $\alpha$--SLX, $\alpha$--SAR and GW$\alpha$R models
  14. The $\alpha$--SLX model
  15. The contiguity matrix
  16. ...and 70 more sections

Key Result

Theorem 2.5

Under Assumptions ass:dgp--ass:continuity,

Figures (1)

  • Figure 1: The flood plain of the river Meuse in Netherlands.

Theorems & Definitions (10)

  • Theorem 2.5: Consistency
  • Remark 2.6: Necessity of Correct Specification in Consistency
  • Remark 2.7: Necessity of Bound on $y_{i,\alpha}$
  • Remark 2.8: Dominance and Uniform Convergence
  • Remark 2.9: Identification and Compactness
  • Theorem 2.13: Asymptotic Normality
  • Remark 2.14: Necessity of Correct Specification in Asymptotic Normality
  • proof
  • Theorem C.1: Asymptotic Normality
  • proof