Kernel Density Estimation on a Linear Network

Abstract

© 2016 Board of the Foundation of the Scandinavian Journal of Statistics.This paper develops a statistically principled approach to kernel density estimation on a network of lines, such as a road network. Existing heuristic techniques are reviewed, and their weaknesses are identified. The correct analogue of the Gaussian kernel is the 'heat kernel', the occupation density of Brownian motion on the network. The corresponding kernel estimator satisfies the classical time-dependent heat equation on the network. This 'diffusion estimator' has good statistical properties that follow from the heat equation. It is mathematically similar to an existing heuristic technique, in that both can be expressed as sums over paths in the network. However, the diffusion estimate is an infinite sum, which cannot be evaluated using existing algorithms. Instead, the diffusion estimate can be computed rapidly by numerically solving the time-dependent heat equation on the network. This also enables bandwidth selection using cross-validation. The diffusion estimate with automatically selected bandwidth is demonstrated on road accident data.