Previous approaches to modelling interval-censored data have often relied on assumptions of homogeneity in the sense that the censoring mechanism, the underlying distribution of occurrence times, or both, are assumed to be time-invariant. In this work, we introduce a model which allows for non-homogeneous behaviour in both cases. In particular, we outline a censoring mechanism based on a non-homogeneous alternating renewal process in which interval generation is assumed to be time-dependent, and we propose a Markov point process model for the underlying occurrence time distribution. We prove the existence of this process and derive the conditional distribution of the occurrence times given the intervals. We provide a framework within which the process can be accurately modelled, and subsequently compare our model to the homogeneous approach through a number of illustrative examples.

We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley‒Møller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges that do not contain triangles.

In this paper we describe methods for randomly thinning certain classes of spatial point processes. In the case of a Markov point process, the proposed method involves a dependent thinning of a spatial birth-and-death process, where clans of ancestors associated with the original points are identified, and where we simulate backwards and forwards in order to obtain the thinned process. In the case of a Cox process, a simple independent thinning technique is proposed. In both cases, the thinning results in a Poisson process if and only if the true Papangelou conditional intensity is used, and, thus, can be used as a graphical exploratory tool for inspecting the goodness-of-fit of a spatial point process model. Several examples, including clustered and inhibitive point processes, are considered.

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 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.

We show that a Poisson cluster point process is a nearest-neighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finite-range Markov point process in the sense of Ripley and Kelly [12]. Furthermore, when the parent Poisson process is replaced by a Markov or nearest-neighbour Markov point process, the resulting cluster process is also nearest-neighbour Markov, provided all clusters are non-empty. In particular, the nearest-neighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned.