Multitype spatial point patterns with hierarchical interactions

Abstract

Multitype spatial point patterns with hierarchical interactions are considered. Here hierarchical interaction means directionality: Points on a higher level of hierarchy affect the locations of points on the lower levels, but not vice versa. Such relations are common, for example, in ecological communities. Interacting point patterns are often modelled by Gibbs processes with pairwise interactions. However, these models are inherently symmetric, and the hierarchy can be acknowledged only when interpreting the results. We suggest the following trick allowing the inclusion of the hierarchical structure in the model. Instead of regarding the pattern as a realisation of a stationary multivariate point process, we build the pattern one type at a time according to the order of the hierarchy by using non-stationary univariate processes. As interactions connected to points x on a certain level are considered, the effect of the higher levels is interpreted as heterogeneity of the pattern x and the points on the lower levels are neglected owing to the hierarchical structure.