February 2022 Diffusion Smoothing for Spatial Point Patterns
Adrian Baddeley, Tilman M. Davies, Suman Rakshit, Gopalan Nair, Greg McSwiggan
Author Affiliations +
Statist. Sci. 37(1): 123-142 (February 2022). DOI: 10.1214/21-STS825


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 R code.

Funding Statement

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 [29]. Software implementations of the algorithms described here are now included in spatstat. The simulations also used the packages spagmix [55] and doParallel [53]. Code scripts to perform all the calculations in this paper are available Supplementary Material.


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Adrian Baddeley. Tilman M. Davies. Suman Rakshit. Gopalan Nair. Greg McSwiggan. "Diffusion Smoothing for Spatial Point Patterns." Statist. Sci. 37 (1) 123 - 142, February 2022. https://doi.org/10.1214/21-STS825


Published: February 2022
First available in Project Euclid: 19 January 2022

MathSciNet: MR4372661
zbMATH: 07474201
Digital Object Identifier: 10.1214/21-STS825

Keywords: adaptive smoothing , bandwidth , heat kernel , Kernel estimation , lagged arrival method , Richardson extrapolation

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.37 • No. 1 • February 2022
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