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

Yuichiro Otani

TL;DR

Gimbal Regression is proposed, a deterministic, geometry-aware local regression framework for stable and auditable estimation that constructs directional weights from neighborhood geometry using explicit orientation objects and deterministic safeguards, and computes local coefficients by a closed-form solve.

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.

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

TL;DR

Gimbal Regression is proposed, a deterministic, geometry-aware local regression framework for stable and auditable estimation that constructs directional weights from neighborhood geometry using explicit orientation objects and deterministic safeguards, and computes local coefficients by a closed-form solve.

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.
Paper Structure (175 sections, 90 equations, 16 figures, 21 tables)

This paper contains 175 sections, 90 equations, 16 figures, 21 tables.

Table of Contents

  1. Introduction
  2. Method overview
  3. Stability viewpoint and interpretation
  4. Computational structure
  5. Paper organization
  6. Background and Positioning
  7. Local regression as a tool for spatial heterogeneity
  8. Directional weighting and anisotropic kernels
  9. Dependence modeling and prediction-focused alternatives
  10. Positioning of Gimbal Regression
  11. Summary
  12. Notation and Problem Setup
  13. Observations, locations, and neighborhoods
  14. Distances, bearings, and tangent-plane displacements
  15. Local design matrices and normal equations
  16. ...and 160 more sections

Figures (16)

  • Figure 8.1: Meuse: spatial distribution of solver and geometry diagnostics.
  • Figure 8.2: Rice paddies: spatial distribution of effective support and the local slope surface.
  • Figure :
  • Figure B.3: Spatial distributions
  • Figure :
  • ...and 11 more figures