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.

翻译：局部回归被广泛用于探索空间异质性，但各向异性或有效低维的邻域会导致局部求解存在病态问题，进而引发由数值伪影而非实质结构驱动的系数变异。当估计依赖于隐式调参或优化而不暴露局部诊断信息时，这种不稳定性常常被掩盖。本文提出Gimbal回归（GR）——一种确定性的、几何感知的局部回归框架，用于实现稳定且可审计的估计。GR通过使用显式方向对象和确定性保护措施，从邻域几何结构构建方向性权重，并通过闭式求解计算局部系数。理论结果基于实现邻域配置给出条件性陈述，在此条件下估计器是具备有限扰动稳定性界限的确定性线性算子。仿真与实证案例表明，与常见局部回归基准方法相比，该方法具有可预测的计算过程、透明的诊断特性以及更好的数值稳定性。