Traditional kernel methods for estimating the spatially-varying density of points in a spatial point pattern may exhibit unrealistic artefacts, in addition to the familiar problems of bias and over- or under-smoothing. Performance can be improved by using diffusion smoothing, in which the smoothing kernel is the heat kernel on the spatial domain. This paper develops diffusion smoothing into a practical statistical methodology for two-dimensional spatial point pattern data. We clarify the advantages and disadvantages of diffusion smoothing over Gaussian kernel smoothing. Adaptive smoothing, where the smoothing bandwidth is spatially-varying, can be performed by adopting a spatially-varying diffusion rate: this avoids technical problems with adaptive Gaussian smoothing and has substantially better performance. We introduce a new form of adaptive smoothing using lagged arrival times, which has good performance and improved robustness. Applications in archaeology and epidemiology are demonstrated. The methods are implemented in open-source code.
Funding was received from the Australian Research Council discovery grants DP130104470 (Baddeley) and DP130102322 (Baddeley, Rakshit, Nair); the Grains Research and Development Corporation and the University of Western Australia (Rakshit); and Royal Society of New Zealand Marsden Fund grants 15-UOO-192 and 19-UOO-191 (Davies).
Data analysis was performed in the R language using the contributed packages spatstat [4, 3] and sparr . Software implementations of the algorithms described here are now included in spatstat. The simulations also used the packages spagmix  and doParallel . Code scripts to perform all the calculations in this paper are available Supplementary Material.
"Diffusion Smoothing for Spatial Point Patterns." Statist. Sci. 37 (1) 123 - 142, February 2022. https://doi.org/10.1214/21-STS825