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

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  • Yuichiro Otani

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.

Suggested Citation

  • Yuichiro Otani, 2026. "Gimbal Regression: Orientation-Adaptive Local Linear Regression under Spatial Heterogeneity," Papers 2603.10382, arXiv.org, revised Mar 2026.
    • Handle: RePEc:arx:papers:2603.10382
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    References listed on IDEAS

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    1. A. Stewart Fotheringham & Wenbai Yang & Wei Kang, 2017. "Multiscale Geographically Weighted Regression (MGWR)," Annals of the American Association of Geographers, Taylor & Francis Journals, vol. 107(6), pages 1247-1265, November.
    2. David Wheeler & Michael Tiefelsdorf, 2005. "Multicollinearity and correlation among local regression coefficients in geographically weighted regression," Journal of Geographical Systems, Springer, vol. 7(2), pages 161-187, June.
    3. A. Stewart Fotheringham & Taylor M. Oshan, 2016. "Geographically weighted regression and multicollinearity: dispelling the myth," Journal of Geographical Systems, Springer, vol. 18(4), pages 303-329, October.
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