Space-time Multi Type Log Gaussian Cox Processes with a View to Modelling Weeds

Log Gaussian Cox processes as introduced in Moller et al. (1998) are extended to space-time models called log Gaussian Cox birth processes. These processes allow modelling of spatial and temporal heterogeneity in time series of increasing point processes consisting of different types of points. The models are shown to be easy to analyse yet flexible enough for a detailed statistical analysis of a particular agricultural experiment concerning the development of two weed species on an organic barley field. Particularly, the aspects of estimation, model validation and intensity surface prediction are discussed.     

Geometric Anisotropic Spatial Point Pattern Analysis and Cox Processes

We consider spatial point processes with a pair correlation function, which depends only on the lag vector between a pair of points. Our interest is in statistical models with a special kind of ‘structured’ anisotropy: the pair correlation function is geometric anisotropic if it is elliptical but not spherical. In particular, we study Cox process models with an elliptical pair correlation function, including shot noise Cox processes and log Gaussian Cox processes, and we develop estimation procedures using summary statistics and Bayesian methods. Our methodology is illustrated on real and synthetic datasets of spatial point patterns.     

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

Abstract. 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 ‘network K function’, analogous to Ripley's K function, for analysis of such data. However, values of the network K‐function depend on the network geometry, making interpretation difficult. In this study we propose a correction of the network K‐function that intrinsically compensates for the network geometry. This geometrical correction restores many natural and desirable properties of K, including its direct relationship to the pair correlation function. For a completely random point pattern, on any network, the corrected network K‐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’ network K‐function which compensates for a spatially varying intensity of points. We demonstrate applications to ecology (webs of the urban wall spider Oecobius navus) and criminology (street crime in Chicago).     

A J‐function for Inhomogeneous Spatio‐temporal Point Processes

We propose a new summary statistic for inhomogeneous intensity‐reweighted moment stationarity spatio‐temporal point processes. The statistic is defined in terms of the n‐point correlation functions of the point process, and it generalizes the J‐function when stationarity is assumed. We show that our statistic can be represented in terms of the generating functional and that it is related to the spatio‐temporal K‐function. We further discuss its explicit form under some specific model assumptions and derive ratio‐unbiased estimators. We finally illustrate the use of our statistic in practice. © 2014 Board of the Foundation of the Scandinavian Journal of Statistics