Kernel Density Estimation on a Linear Network
本文提出了一种基于热核的扩散估计方法，用于在道路网络等线性网络上进行核密度估计，解决了现有启发式方法的统计缺陷，并通过数值求解热方程实现快速计算和带宽选择。
Abstract 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.