Pointpattern

Pointpattern refers to a configuration of points representing the locations of events within a region, typically in two dimensions. In spatial statistics, a point pattern is analyzed through the framework of a point process, where each event has a spatial coordinate and the collection may be unmarked or marked by attributes such as type or magnitude. The central goal is to describe and infer the underlying spatial distribution: whether events occur randomly, cluster, or exhibit regular spacing. Patterns can be homogeneous, with constant intensity across the region, or inhomogeneous, with intensity that varies spatially.

A canonical model for complete spatial randomness is the homogeneous Poisson process, while inhomogeneous Poisson processes allow the intensity function to vary over space. Beyond these, Cox processes and other cluster or inhibition models are used to represent more complex dependence structures. Key tasks include estimating the intensity or density of events, testing for randomness, and quantifying spatial dependence.

Common methods include descriptive and inferential tools. Descriptive methods encompass plotting, heat maps, and quadrat counts. Summary statistics feature the nearest-neighbor distance distribution, Ripley’s K-function and its L-transform, and the pair correlation function g(r), often with edge corrections. Kernel methods estimate the underlying intensity function. Inference frequently relies on Monte Carlo envelopes to assess deviations from CSR or to compare competing models.

Point patterns can be planar, temporal, or spatio-temporal, and may be extended to marked patterns where events carry auxiliary information. Applications appear in ecology (e.g., tree locations), epidemiology (case locations), criminology (crime events), seismology, and materials science. Software tools such as spatstat in R, along with Python libraries, implement these methods for estimation, hypothesis testing, and simulation.