Geometrically Corrected Second Order Analysis of Events on a Linear Network, with Applications to Ecology and Criminology

We study point patterns of events that occur on a network of lines, such as road accidents recorded on a road network. Okabe and Yamada developed a ‘networkKfunction’, analogous to Ripley'sKfunction, for analysis of such data. However, values of the networkK‐function depend on the network geometry, making interpretation difficult. In this study we propose a correction of the networkK‐function that intrinsically compensates for the network geometry. This geometrical correction restores many natural and desirable properties ofK, including its direct relationship to the pair correlation function. For a completely random point pattern, on any network, the corrected networkK‐function is the identity. The corrected estimator is intrinsically corrected for edge effects and has approximately constant variance. We obtain exact and asymptotic expressions for the bias and variance of under complete randomness. We extend these results to an ‘inhomogeneous’ networkK‐function which compensates for a spatially varying intensity of points. We demonstrate applications to ecology (webs of the urban wall spiderOecobius navus) and criminology (street crime in Chicago).