Gimbal Regression: Orientation-Adaptive Local Linear Regression under Spatial Heterogeneity

TL;DR

Gimbal Regression is a geometry-aware local regression method designed to improve numerical stability and transparency in spatial heterogeneity analysis by explicitly constructing directional weights and providing auditable diagnostics.

Contribution

It introduces a deterministic, geometry-aware framework with explicit orientation objects and safeguards, enhancing stability and interpretability over existing local regression methods.

Findings

Demonstrates improved numerical stability compared to baseline methods.

Provides transparent diagnostics for local regression analysis.

Shows predictable computation and stability bounds in simulations and empirical examples.

Abstract

Local regression is widely used to explore spatial heterogeneity, but anisotropic or effectively low-dimensional neighborhoods can produce ill-conditioned local solves, causing coefficient variation driven by numerical artifacts rather than substantive structure. Such instability is often hidden when estimation relies on implicit tuning or optimization without exposing local diagnostics. This paper proposes Gimbal Regression (GR), a deterministic, geometry-aware local regression framework for stable and auditable estimation. GR constructs directional weights from neighborhood geometry using explicit orientation objects and deterministic safeguards, and computes local coefficients by a closed-form solve. Theoretical results are stated conditional on the realized neighborhood configuration, under which the estimator is a deterministic linear operator with finite-perturbation stability bounds. Simulations and empirical examples demonstrate predictable computation, transparent diagnostics, and improved numerical stability relative to common local regression baselines.