A kernel density estimation method for networks, its computational method and a GIS‐based tool

Abstract

We develop a kernel density estimation method for estimating the density of points on a network and implement the method in the GIS environment. This method could be applied to, for instance, finding 'hot spots' of traffic accidents, street crimes or leakages in gas and oil pipe lines. We first show that the application of the ordinary two-dimensional kernel method to density estimation on a network produces biased estimates. Second, we formulate a 'natural' extension of the univariate kernel method to density estimation on a network, and prove that its estimator is biased; in particular, it overestimates the densities around nodes. Third, we formulate an unbiased discontinuous kernel function on a network, and fourth, an unbiased continuous kernel function on a network. Fifth, we develop computational methods for these kernels and derive their computational complexity. We also develop a plug-in tool for operating these methods in the GIS environment. Sixth, an application of the proposed methods to the density estimation of bag-snatches on streets is illustrated. Lastly, we summarize the major results and describe some suggestions for the practical use of the proposed methods.

Identifying Vehicular Crash High Risk Locations along Highways via Spatial Autocorrelation Indices and Kernel Density Estimation

“…However, in analyzing the hot spots of network-constraint events, the assumption of homogeneity of 2-D space does not hold and the relevant KDE methods may produce biased results [9]. Therefore, the planar KDE has been extended to the network KDE, which differs from the planar KDE in several aspects: (i) the network space is used as the point event context; (ii) both search bandwidth r and kernel function k are based on network distance (calculated as the shortest path distance in a network) instead of straight-line Euclidean distance; and (iii) density is measured per linear unit instead of area unit.…”

“…The cell size depends on the user choice and the dataset. Okabe et al [9] suggested using a cell size of (r/10) as a rule of thumb.…”

“…Therefore, the planar KDE has been extended to the network KDE, which differs from the planar KDE in several aspects: (i) the network space is used as the point event context; (ii) both search bandwidth r and kernel function k are based on network distance (calculated as the shortest path distance in a network) instead of straight-line Euclidean distance; and (iii) density is measured per linear unit instead of area unit. The network KDE is a 1-D measurement while the planar KDE is a 2-D measurement [9]. The network KDE is an extension of the planar 2-D KDE and it uses the following equation for the density estimation of network-constrained point events in a network space [8]:…”

“…However, this method has been criticized in relation to the fact that road crashes usually happen on the road links and need to be considered in a road network space represented by 1-D dimension. Therefore, some studies have extended the planar KDE to network spaces, which estimates the crash density over a distance unit in a 1-D measurement instead of an area unit [7] [8] [9]. However, a major weakness of the KDE methods is that it cannot be tested for statistical significance [8] [10].…”

Solving the median problem with continuous demand on a network

“…[18] suggests the use of general density distributions, from which random samples are generated, yielding a discrete approximation to the problem, which is the one which is later analyzed. Statistical kernel methods have also been recently proposed to model the demand, [20,23], though, as far as the authors know, no optimization has been carried out, excepting, as said above, discretization via simulation.…”