AI 中文总结
针对地理受限区域点模式总结问题，提出投影扩散核密度估计器（PDKDE），通过在截断拉普拉斯特征基中扩展诺伊曼扩散核，兼具边界感知和几何保留特性，计算高效，减少伪影，在瓦胡岛应用中表现良好。
AI 中文摘要
空间强度图常用于总结地理受限区域上的点模式，如岛屿、海岸线等。标准核密度估计器存在问题，基于扩散的估计器虽考虑几何但数据变化时重新计算成本高。受夏威夷瓦胡岛盗窃事件热点映射启发，提出投影扩散核密度估计器（PDKDE）。它在截断拉普拉斯特征基中扩展诺伊曼扩散核，一次计算固定域几何，后续通过显式谱系数获得密度估计。证明了其一致性，模拟表明它减少了边界和障碍伪影，速度比其他方法快。在瓦胡岛应用中，该方法能快速生成受海岸线约束的描述性地图。
英文摘要
Spatial intensity maps are routinely used to summarize point patterns on geographically constrained regions, such as islands, coastlines, watersheds, ecological reserves, and administrative areas with physical barriers. In these settings, the domain is not a nuisance feature-- it determines where probability mass may be assigned and which locations should be smoothed together. Standard kernel density estimators can place mass outside the study region and smooth according to Euclidean distance, while diffusion-based estimators respect the geometry but are costly to recompute when the data subset, bandwidth, or evaluation grid changes. Motivated by repeated hotspot mapping of theft and larceny incidents on Oahu, Hawaii, we propose the projected diffusion kernel density estimator (PDKDE). PDKDE expands the Neumann diffusion kernel in a truncated Laplacian eigenbasis, so that the geometry of a fixed domain is computed once and subsequent density estimates are obtained through explicit spectral coefficients. The resulting estimator preserves the boundary-aware and geometry-respecting behavior of diffusion smoothing while making repeated estimation and least-squares cross-validation computationally practical. For the exact projected estimator, we prove MISE consistency and pointwise consistency up to the boundary. Controlled simulations show that PDKDE reduces boundary and barrier artifacts relative to Euclidean kernels and is substantially faster than the geometry-aware comparators considered after the one-time domain computation. In the Oahu application, the method produces coastline-constrained descriptive maps across time windows in seconds, illustrating its intended fixed-domain, changing-data use case.
