On adaptive kernel intensity estimation on linear networks

Abstract: In the analysis of spatial point patterns on linear networks, a critical statistical objective is estimating the first-order intensity function, representing the expected number of points within specific subsets of the network. Typically, non-parametric approaches employing heating kernels are used for this estimation. However, a significant challenge arises in selecting appropriate bandwidths before conducting the estimation. We study an intensity estimation mechanism that overcomes this limitation using adaptive estimators, where bandwidths adapt to the data points in the pattern. While adaptive estimators have been explored in other contexts, their application in linear networks remains underexplored. We investigate the adaptive intensity estimator within the linear network context and extend a partitioning technique based on bandwidth quantiles to expedite the estimation process significantly. Through simulations, we demonstrate the efficacy of this technique, showing that the partition estimator closely approximates the direct estimator while drastically reducing computation time. As a practical application, we employ our method to estimate the intensity of traffic accidents in a neighbourhood in Medellin, Colombia, showcasing its real-world relevance and efficiency.