Thinning spatial point processes into Poisson processes

Thinning spatial point processes into Poisson processes

This paper describes 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 one simulates backwards and forwards in order to obtain the thinned process. In the case...

Transforming spatial point processes into Poisson processes

In 1986, Merzbach and Nualart demonstrated a method of transforming a two-parameter point process into a planar Poisson process of unit rate, using random stopping sets. Merzbach and Nualart's theorem applies only to a special class of point processes, since it requires two restrictive conditions: the (F4) condition of conditional independence and the convexity of the 1-compensator. The (F4) co...

Deterministic Thinning of Finite Poisson Processes

Let Π and Γ be homogeneous Poisson point processes on a fixed set of finite volume. We prove a necessary and sufficient condition on the two intensities for the existence of a coupling of Π and Γ such that Γ is a deterministic function of Π, and all points of Γ are points of Π. The condition exhibits a surprising lack of monotonicity. However, in the limit of large intensities, the coupling exi...

Spatial Point Processes

Property (iv) is called boundedly finite. So, when we use (iv) instead of (iii), we are interested in simple, boundedly finite spatial point processes with no fixed atoms. When we use (iii), the domain A can be an arbitrary set. When we use (iv), the domain A must have some notion of boundedness. This is no problem when A is a subset of Rd for some d. We just use the usual notion of boundedness...

Point processes , spatial – temporal