Fast Kernel Smoothing of Point Patterns on a Large Network using Two-dimensional Convolution

Abstract

We propose a computationally efficient and statistically principled method for kernel smoothingof point pattern data on a linear network. The point locations, and the network itself, are convolvedwith a two-dimensional kernel and then combined into an intensity function on the network. Thiscan be computed rapidly using the fast Fourier transform, even on large networks and for largebandwidths, and is robust against errors in network geometry. The estimator is consistent, and itsstatistical efficiency is only slightly suboptimal. We discuss bias, variance, asymptotics, bandwidthselection, variance estimation, relative risk estimation and adaptive smoothing. The methods areused to analyse spatially varying frequency of traffic accidents in Western Australia and the relativerisk of different types of traffic accidents in Medellín, Colombia.