Locally calibrated and mesh-free inference for spatial point distributions: closed-form null, contamination law, and detectability threshold

Henock Mwanza Lubukayi; Mechack Kabanga Ntolo

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arXiv · 2609.16497

Locally calibrated and mesh-free inference for spatial point distributions: closed-form null, contamination law, and detectability threshold

Henock Mwanza Lubukayi·Mechack Kabanga Ntolo

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Abstract

Local inference for spatial point distributions is dominated by Monte Carlo calibration. We develop an alternative based on the Tweedie--Miyasawa identities of empirical Bayes, which relate locally weighted moments of a point distribution under a Gaussian kernel to derivatives of its log-intensity in scale space. We first establish a rigidity theorem showing that the structure of these identities forces the Gaussian kernel. Under complete spatial randomness, we derive a closed-form null distribution for a bounded inter-scale contrast, yielding a calibrated simulation-free pointwise test. In experiments, the measured type I error is 0.070 at a nominal level of 0.05, with a calibration cost 199 times smaller than Monte Carlo for the same local statistic. We then derive a contamination law for structures of dimension m and width w embedded in a uniform background, together with an explicit detectability threshold. For a filament in three dimensions, the threshold is 16 pi. The resulting scale-resolved local dimension estimator has no free parameters. Applications to California seismicity, a trefoil knot, and 10,071 SDSS galaxies show that the method separates local structures across scales without a spatial mesh and reproduces published cosmic web fractions. All experiments are reproducible from a single public script.

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Henock Mwanza Lubukayi, Mechack Kabanga Ntolo. 2026-09-15. Locally calibrated and mesh-free inference for spatial point distributions: closed-form null, contamination law, and detectability threshold. https://arxiv.org/abs/2609.16497

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