Spatial dependency between a linear network and a point pattern

Abstract

In this mini-dissertation we discuss the spatial relationship between point processes and a linear network. As a starting point, we discuss basic spatial point processes and tests for first-order homogeneity. Following that, we discuss second-order properties of point processes in the form of Ripley's K-function for unmarked point patterns and the cross-K function for marked point patterns. We then get to the main focus of this mini-dissertation, that is, the spatial relationship between points and linear structures, particularly linear networks. Recently developed is a method to characterise the spatial relationship between points and linear networks by Comas et al. [13], similar to Ripley's K-function for point-to-point relationships. The non-stationarity of a linear network is of particular interest in how it affects the measurement of this spatial relationship, which has not been explicitly investigated in the literature before. To investigate this we consider the Poisson line process and how one might simulate a non-stationary line process. Furthermore, we discuss a mechanism to extend tests of first-order homogeneity of point patterns to line patterns. The non-stationary line process is used to model linear networks in the simulations conducted to determine the effect of this non-stationarity on the developed method, which was not covered in the original article [13]. The methodology is developed and tested on a real data set.