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Statistical methodology for analysing patterns of points on a network of lines, such as road traffic accident locations, often assumes that the underlying point process is "stationary" or "correlation-stationary." However, such processes appear to be rare. In this paper, popular procedures for constructing a point process are adapted to linear networks: many of the resulting models are no longer stationary when distance is measured by the shortest path in the network. This undermines the rationale for popular statistical methods such as the K-function and pair correlation function. Alternative strategies are proposed, such as replacing the shortest-path distance by another metric on the network.

Summary We propose a computationally efficient and statistically principled method for kernel smoothing of point pattern data on a linear network. The point locations, and the network itself, are convolved with a two‐dimensional kernel and then combined into an intensity function on the network. This can be computed rapidly using the fast Fourier transform, even on large networks and for large bandwidths, and is robust against errors in network geometry. The estimator is consistent, and its statistical efficiency is only slightly suboptimal. We discuss bias, variance, asymptotics, bandwidth selection, variance estimation, relative risk estimation and adaptive smoothing. The methods are used to analyse spatially varying frequency of traffic accidents in Western Australia and the relative risk of different types of traffic accidents in Medellín, Colombia.

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