Log Gaussian Cox Processes

Abstract

We view the locations and times of a collection of crime events as a space-time point pattern. So, with either a nonhomogeneous Pois-son process or with a more general Cox process, we need to specify a space-time intensity. For the latter, we need a random intensity which we model as a realization of a spatio-temporal log Gaussian process. Importantly, we view time as circular not linear, necessitating valid separable and nonseparable covariance functions over a bounded spatial region crossed with circular time. In addition, crimes are classified by crime type. Furthermore, each crime event is recorded by day of the year which we convert to day of the week marks. The contribution here is to develop models to accommodate such data. Our specifications take the form of hierarchical models which we fit within a Bayesian framework. In this regard, we consider model comparison between the nonhomogeneous Poisson process and the log Gaussian Cox process. We also compare separable vs. nonseparable covariance specifications. Our motivating dataset is a collection of crime events for the city of San Francisco during the year 2012. We have location, hour, day of the year, and crime type for each event. We investigate models to enhance our understanding of the set of incidences.

Likelihood based inference for partially observed renewal processes

“…As in Brix and Diggle (2001) and Diggle et al (2005), we assume that calls are made according to a log-Gaussian Cox process (Møller et al, 1998) and approximate the rate of calls during the ith day of the two-month period by…”

Inhomogeneity in Spatial Cox Point Processes – Location Dependent Thinning Is Not the Only Option

“…The two most often used classes of spatial Cox point processes are the log Gaussian Cox processes (Møller et al, 1998) and the shot noise Cox processes (Møller, 2003). Even though these can be generalized and unified in the framework of Lévy based Cox processes (Hellmund et al, 2008) for ease of exposition we stick to the two classical models.…”

Thinning spatial point processes into Poisson processes

“…Condition (4) implies X can be viewed as the superposition Q ∪ R of a Cox process Q driven by γ = λ − ρ and an independent Poisson process R with intensity function ρ. For example, γ may be log Gaussian and Q then a log Gaussian Cox process (Møller, Syversveen and Waagepetersen, 1998), or a shot noise process and Q then a shot noise Cox process (Møller, 2003).…”

“…For example, if γ is log Gaussian, a Langevin-Hastings algorithm can be used (Møller et al, 1998;Møller and Waagepetersen, 2004), and if γ is a shot-noise process, a birth-death Metropolis-Hastings algorithm applies (Møller, 2003;Møller and Waagepetersen, 2004). We run one of these Metropolis-Hastings algorithms until it is effectively in equilibrium, and then return an (approximate) simulation of λ conditional on X = x.…”