We consider the problem of estimating a latent point process, given the realization of another point process. We establish an expression for the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning, and augmentation with extra points. Our original analysis is based on an elementary and self-contained random measure theoretic approach. This simplifies and complements previous derivations given in Mahler (2003), and Singh, Vo, Baddeley and Zuyev (2009).

We introduce a class of Cox cluster processes called generalised shot noise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process that drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can be random. Thereby, a very large class of models for aggregated or clustered point patterns is obtained. Due to the structure of GSNCPs, a number of useful results can be established. We focus first on deriving summary statistics for GSNCPs and, second, on how to simulate such processes. In particular, results on first- and second-order moment measures, reduced Palm distributions, the J-function, simulation with or without edge effects, and conditional simulation of the intensity function driving a GSNCP are given. Our results are exemplified in important special cases of GSNCPs, and we discuss their relation to the corresponding results for SNCPs.

Shot noise Cox processes constitute a large class of Cox and Poisson cluster processes in ℝd, including Neyman-Scott, Poisson-gamma and shot noise G Cox processes. It is demonstrated that, due to the structure of such models, a number of useful and general results can easily be established. The focus is on the probabilistic aspects with a view to statistical applications, particularly results for summary statistics, reduced Palm distributions, simulation with or without edge effects, conditional simulation of the intensity function and local and spatial Markov properties.

We study the transformation of a non-stationary point process ξ on ℝn into a weakly stationary point process ͂ξ, with ͂ξ(B) = ξ(Φ-1(B)), where B is a Borel set, via a deformation Φ of the space ℝn. When the second-order measure is regular, Φ is uniquely determined by the homogenization equations of the second-order measure. In contrast, the first-order homogenization transformation is not unique. Several examples of point processes and transformations are investigated with a particular interest to Poisson processes.

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatial birth-and-death processes. We then investigate discrete-time Metropolis-Hastings samplers for point processes, and show how a variant which samples systematically from cells can be converted into a perfect version. An application is given to the Strauss point process.

In the Java sea, pelagic fish are exploited by purse seiners. The resource is dispersed and has to be aggregated before being fished. Light is used to reach this objective. The purse seiners are mobile and fish in ‘packs’. When concentrations of fish are found the fishing vessels gather in clusters. The spatial characterization of these clusters provides better understanding of the fishing strategy followed by the purse seiners. A point process approach is used to understand the clustering process and determine the life time of the clusters. The calculation of a density function defines the surface influenced by the cluster and estimates the intensity of the exploitation inside it. Clustering is the mark of the fishing phase that is defined by the significant divergence of the L-function from the Poisson process. In the fishery, such a result is observed when at least 15 purse seiners are gathered. Usually, the fishing phase lasts 4–6 days. The surface covered by the clusters represents a small part (15–25 %) of the surface that is accessible to the fleet. In these areas, the fishing intensity can reach high values (160 vessels × nautical mile–2).

A parametric family of completely random measures, which includes gamma random measures, positive stable random measures as well as inverse Gaussian measures, is defined. In order to develop models for clustered point patterns with dependencies between points, the family is used in a shot-noise construction as intensity measures for Cox processes. The resulting Cox processes are of Poisson cluster process type and include Poisson processes and ordinary Neyman-Scott processes.

We show characteristics of the completely random measures, illustrated by simulations, and derive moment and mixing properties for the shot-noise random measures. Finally statistical inference for shot-noise Cox processes is considered and some results on nearest-neighbour Markov properties are given.

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle.

Consider a spatial point pattern realized from an inhomogeneous Poisson process on a bounded Borel set , with intensity function λ (s; θ), where . In this article, we show that the maximum likelihood estimator and the Bayes estimator are consistent, asymptotically normal, and asymptotically efficient as the sample region . These results extend asymptotic results of Kutoyants (1984), proved for an inhomogeneous Poisson process on [0, T] , where T →∞. They also formalize (and extend to the multiparameter case) results announced by Krickeberg (1982), for the spatial domain . Furthermore, a Cramér–Rao lower bound is found for any estimator of θ. The asymptotic properties of and are considered for modulated (Cox (1972)), and linear Poisson processes.

A general way to look at kernel estimates of densities is to regard them as stochastic integrals with respect to a spatial point process. Under regularity conditions these behave asymptotically as if the point process were Poisson. However, this Poisson approximation may not work well if the data exhibits a lot of clustering. In this paper a more refined approximation to the characteristic functions of the integrals is developed. For clustered data, a ‘Gauss–Poisson’ approximation works better than the Poisson.

Starting from a suitable sequence of auto-Poisson lattice schemes, it is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit. Further pairwise-interaction processes are obtained as limits of sequences of auto-logistic lattice schemes.