空间点分布的局部校准与无网格推断：闭式零分布、污染定律与可检测阈值
Locally calibrated and mesh-free inference for spatial point distributions: closed-form null, contamination law, and detectability threshold
- Institut Supérieur de Statistiques（高等统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种基于Tweedie--Miyasawa恒等式的无网格局部推断方法，推导闭式零分布与污染定律，实现免模拟校准，并应用于地震、星系等数据，显著降低计算成本。
AI中文摘要：
空间点分布的局部推断主要由蒙特卡洛校准主导。我们开发了一种基于经验贝叶斯的Tweedie--Miyasawa恒等式的替代方法，该恒等式将高斯核下点分布的局部加权矩与其在尺度空间中的对数强度导数联系起来。我们首先建立了一个刚性定理，表明这些恒等式的结构迫使采用高斯核。在完全空间随机性下，我们推导出有界尺度间对比度的闭式零分布，从而产生一个校准的、无模拟的点态检验。在实验中，在名义水平0.05下，测得的I型误差为0.070，对于相同的局部统计量，校准成本比蒙特卡洛小199倍。然后，我们推导了嵌入均匀背景中的维度为m、宽度为w的结构的污染定律，以及一个明确的检测阈值。对于三维中的丝状结构，阈值为16π。由此产生的尺度分辨局部维度估计器没有自由参数。应用于加利福尼亚地震活动、三叶结和10,071个SDSS星系，表明该方法无需空间网格即可跨尺度分离局部结构，并重现已发表的宇宙网分数。所有实验均可通过单个公共脚本重现。
英文摘要：
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.
