Variable selection using penalised likelihoods for point patterns on a linear network.

Motivated by the analysis of a comprehensive database of road traffic accidents, we investigate methods of variable selection for spatial point process models on a linear network. The original data may include explanatory spatial covariates, such as road curvature, and ‘mark’ variables attributed to individual accidents, such as accident severity. The treatment of mark variables is new. Variable selection is applied to the canonical covariates, which may include spatial covariate effects, mark effects and mark-covariate interactions. We approximate the likelihood of the point process model by that of a generalised linear model, in such a way that spatial covariates and marks are both associated with canonical covariates. We impose a convex penalty on the log likelihood, principally the elastic-net penalty, and maximise the penalised loglikelihood by cyclic coordinate ascent. A simulation study compares the performances of the lasso, ridge regression and elastic-net methods of variable selection on their ability to select variables correctly, and on their bias and standard error. Standard techniques for selecting the regularisation parameter γ often yielded unsatisfactory results. We propose two new rules for selecting γ which are designed to have better performance. The methods are tested on a small dataset on crimes in a Chicago neighbourhood, and applied to a large dataset of road traffic accidents in Western Australia

Diffusion Smoothing for Spatial Point Patterns

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 thesmoothing kernel is the heat kernel on the spatial domain. This paper developsdiffusion smoothing into a practical statistical methodology for twodimensionalspatial point pattern data. We clarify the advantages and disadvantagesof diffusion smoothing over Gaussian kernel smoothing. Adaptivesmoothing, where the smoothing bandwidth is spatially-varying, can beperformed by adopting a spatially-varying diffusion rate: this avoids technicalproblems with adaptive Gaussian smoothing and has substantially betterperformance. We introduce a new form of adaptive smoothing using laggedarrival times, which has good performance and improved robustness. Applicationsin archaeology and epidemiology are demonstrated. The methods areimplemented in open-source R cod