Fast boundary-aware spatial intensity estimation on complex domains

2026-07-24

AI Summary

This work addresses the challenge of kernel density estimation in geospatial domains with complex boundaries—such as islands and coastlines—where conventional methods violate geometric constraints, and existing geometry-aware approaches suffer from high computational costs and poor adaptability to dynamic updates. The authors propose the Projected Diffusion Kernel Density Estimator (PDKDE), which expands a Neumann diffusion kernel in a truncated Laplacian eigenbasis to precompute domain geometry once; subsequent density estimates are generated efficiently using only spectral coefficients. PDKDE achieves, for the first time, reproducible, boundary-consistent density estimation on fixed complex domains with both the geometric fidelity of diffusion kernels and the scalability of spectral methods, supported by theoretical guarantees on mean integrated squared error and pointwise boundary consistency. Experiments demonstrate that PDKDE substantially suppresses boundary artifacts, significantly outperforms competing methods in speed, and enables second-scale generation of coastline-constrained spatiotemporal intensity maps, as illustrated in theft hotspot analysis on O‘ahu, Hawai‘i.

Abstract

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.

Problem

• spatial intensity estimation
• boundary-aware
• complex domains
• kernel density estimation
• diffusion smoothing

Innovation

• Projected Diffusion Kernel
• Boundary-aware Density Estimation
• Laplacian Eigenbasis
• Spatial Intensity Mapping
• Fixed-domain Spectral Methods