Spatial Statistics along Networks

Abstract

Spatial statistics along networks is a branch of spatial statistics. Traditional spatial statistics deals with events occurring on a plane, referred to as planar spatial statistics . By contrast, spatial statistics along networks, referred to in parallel as network spatial statistics , deals with events occurring along networks. The events occurring “along” a network are of two types: those directly occurring on a network, for example, traffic accidents, and those occurring alongside a network, for example, stores facing a street. Networks may potentially include corridors, roads, railways, gas and oil pipelines, sewer systems, rivers, canals, electric wires, neurons, the Internet, social networks (e.g., SNS), and so on. Network spatial statistics is applied to various studies, for instance, pedestrian, car, and bicycle accidents, roadkill, street crimes, contamination of rivers, floods, facility locations, tree falls on a railway, etc. This article first describes the most basic hypothesis used in network spatial statistics, called the completely spatial randomness ( CSR ) hypothesis. Next, the article introduces nine methods used for network spatial statistics, that is, Voronoi diagrams, nearest‐neighbor distance method, K function method, spatial autocorrelation, point cluster analysis, kernel density estimation, spatial interpolation, and software for network spatial statistics.

Automated Detection of Missing Links in Bicycle Networks

“…There is a non‐trivial dependence of flow‐based centrality metrics on changes in network boundaries. This phenomenon, known as “network edge effect” (Okabe 2012) or “border effect” (Porta, Crucitti, and Latora 2006), also has relevant implications for equity considerations, since centrality metrics like betweenness have an inherent bias towards the center of the network. To account for this network edge effect, we introduced a cut‐off radius for the set of shortest paths, based on which the centrality metrics are computed (Gil 2017; Yamaoka, Kumakoshi, and Yoshimura 2021).…”

Self-exciting point process modelling of crimes on linear networks

“…Formally, a linear network L=∪i=1nli⊂ℝ2 is commonly taken as a finite union of line segments li⊂ℝ2 of positive length (Ang et al, 2012). The endpoints of the segments are called nodes and the degree of a node is the number of line segments that share the same node (Okabe and Sugihara, 2012). A line segment is defined as li=normalboldui,normalboldvi=knormalboldui+(1−k)normalboldvi:0≤k≤1, where normalboldui,normalboldvi∈ℝ2 are the endpoints of l i .…”

“…Concerning intensity estimation, several kernel-based methods (Borruso, 2005, 2008; Xie and Yan, 2008; Okabe et al, 2009; Okabe and Sugihara, 2012; McSwiggan et al, 2017; Moradi et al, 2018; Rakshit et al, 2019a) and a resample-smoothing technique applied to Voronoi intensity estimators (Moradi et al, 2019; Mateu et al, 2019) have been proposed.…”

“…Approaches based on sums of kernels (Xie and Yan, 2008) basically assume that the statistical basis for kernel estimation can be transferred from the real line to the linear network. However, several authors (Okabe et al, 2009; Okabe and Sugihara, 2012; McSwiggan et al, 2017) have highlighted that translating the kernel density estimate on the one-dimensional real line directly to the linear network, is a fallacious estimate because it does not conserve mass. This basically means that the kernel K⋅,xi=kdL⋅,xi is not a probability density on the linear network, and therefore the corresponding density estimate is not a a probability density.…”

“…As per path enumeration methods, Okabe et al (2009), Sugihara et al (2010) and Okabe and Sugihara (2012) considered computational algorithms which start with a kernel k on the real line, and progressively redistribute the mass of this kernel over the network. These authors proposed two kernel estimators satisfying many desired properties: (a) the ‘equal-split continuous’ rule, which has excellent properties (symmetric, conserves mass, and is unbiased when the true intensity is uniform) but it is extremely slow to compute; (b) the ‘equal-split discontinuous’ rule, which is faster, but has less desirable properties (see, Okabe and Sugihara (2012) and McSwiggan et al (2017) for the algorithms).…”