Locally calibrated and mesh-free inference for spatial point distributions: closed-form null, contamination law, and detectability threshold
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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BibTeX
@misc{lubukayi2026locally,
title = {{Locally calibrated and mesh-free inference for spatial point distributions: closed-form null, contamination law, and detectability threshold}},
author = {Henock Mwanza Lubukayi and Mechack Kabanga Ntolo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16497},
eprint = {2609.16497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16497},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Locally calibrated and mesh-free inference for spatial point distributions: closed-form null, contamination law, and detectability threshold
AU - Henock Mwanza Lubukayi
AU - Mechack Kabanga Ntolo
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16497
UR - https://arxiv.org/abs/2609.16497
AB - 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.
ER -