Inhomogeneous higher-order summary statistics for point processes on linear networks

Abstract

We introduce the notion of intensity reweighted moment pseudostationary point processes on linear networks. Based on arbitrary general regular linear network distances, we propose geometrically corrected versions of different higher-order summary statistics, including the inhomogeneous empty space function, the inhomogeneous nearest neighbour distance distribution function and the inhomogeneous Jfunction. We also discuss their non-parametric estimators. Through a simulation study, considering models with different types of spatial interaction, we study the performance of our proposed summary statistics. Finally, we make use of our methodology to analyse two datasets: motor vehicle traffic accidents and spider data. Fig 1: Left: Spider webs on a brick wall. Right: Motor vehicle traffic accidents in an area of Houston, US, during April, 1999. intensity estimators were proposed (Borruso; 2005, 2008; Xie and Yan; 2008). Later, other non-parametric kernel-based intensity estimators were defined (Okabe et al.; 2009; Okabe and Sugihara; 2012; McSwiggan et al.; 2017; Moradi et al.; 2018) which, although being statistically well-defined, tended to be computationally expensive on large networks. Moreover, Rakshit et al. (2019) proposed a fast kernel intensity estimator based on a two-dimensional convolution which can be computed rapidly even on large networks. With the aim of finding middleground between global and local smoothing, as well as an alternative to kernel estimation, Moradi et al. ( 2019) introduced their so-called resample-smoothing technique which they applied to Voronoi intensity estimators on arbitrary spaces. They showed that their estimation approach mostly performs better than kernel estimators, in terms of bias and standard error.Regarding second-order summary statistics and their estimation, Okabe and Yamada (2001) considered an analogue of Ripley's K-function for homogeneous linear network point processes, which was obtained by using the shortest-path distance instead of the Euclidean distance when measuring distance between points. However, this modification did not provide a well-defined K-function for linear network point processes since its behaviour depends on the topography of the network in question. As a remedy, Ang et al. (2012) introduced geometrically corrected second-order summary statistics which did not depend on the explicit geometry of the linear network under consideration and has a fixed known behaviour for Poisson processes. Hence, the geometrically corrected K-function and pair correlation function can be used e.g. for model selection, hypothesis testing and residual analyses. These summary statistics were later extended to the case of multitype and spatio-temporal point patterns by Baddeley et al. (2014) and . Surrounding theses papers, there appeared a discusimsart-sts ver.

Directional analysis for point patterns on linear networks

“…the distribution of X is invariant with respect to a family of rotations. These assumptions, however, are quite challenging when taking a linear network as the state space of an underlying point process X since there is no guarantee that points will still live on the underlying network after applying a transformation/rotation (Baddeley et al 2017;Cronie et al 2020).…”

“…Common statistical methodologies for point processes often assume that X is stationary, that is, the distribution of X is invariant with respect to a family of transformations/shifts, and isotropic, that is, the distribution of X is invariant with respect to a family of rotations. These assumptions, however, are quite challenging when taking a linear network as the state space of an underlying point process X since there is no guarantee that points will still live on the underlying network after applying a transformation/rotation (Baddeley et al, 2017; Cronie et al, 2020).…”

“…In this regard, Baddeley et al (2017) showed that popular procedures to construct point processes on linear networks give rise to models which are no longer stationary, if the considered metric is the shortest‐path distance. Notwithstanding, under different assumptions, geometrically corrected second‐order and higher‐order summary statistics are already built for inhomogeneous point processes on linear networks (Ang, Baddeley, & Nair, 2012; Cronie et al, 2020).…”

“…When looking at point patterns on linear networks, it is challenging to consider such assumptions due to the fact that it is not clear how such transformations or rotations should be done, since there is no nontrivial geometrical transformations that preserve the network (Baddeley, Nair, Rakshit, & McSwiggan, 2017; Cronie, Moradi, & Mateu, 2020), and besides, the structure of the underlying network itself is rarely homogeneous and/or isotropic. Thinking about street networks, we may see a concentration of streets in downtown, with other long highways in the countryside, according to which these streets/highways may further locate in particular directions with respect to an origin.…”

Gaussian Whittle-Matérn fields on metric graphs

“…Introduction. In many areas of applications, statistical models need to be defined on networks such as connected rivers or street networks (Okabe and Sugihara, 2012;Baddeley et al, 2017;Cronie et al, 2020). In this case, one wants to define a model using a metric on the network rather than the Euclidean distance between points.…”