Modelling the spatial structure of forest stands by multivariate point processes with hierarchical interactions

A stochastic model is applied to describe the spatial structure of a forest stand. We aim at quantifying the strength of the competition process between the trees in terms of interaction within and between different size classes of trees using multivariate Gibbs point processes with hierarchical interactions introduced in [Högmander, H., Särkkä, A., 1999. Multitype spatial point patterns with hierarchical interactions. Biometrics 55, 1051–1058]. The new model overcomes the main limitation of the traditional use of the Gibbs models allowing to describe systems with non-symmetric interactions between different objects. When analyzing interactions between neighbouring trees it is natural to assume that the size of a tree determines its hierarchical level: the largest trees are not influenced by any other trees than the trees in the same size class, while trees in the other size classes are influenced by the other trees in the same class as well as by all larger trees. In this paper, we describe a wide range of Gibbs models with both hierarchical and non-hierarchical interactions as well as a simulation algorithm and a parameter estimation procedure for the hierarchical models. We apply the hierarchical interaction model to the analysis of forest data consisting of locations and diameters of tree stems.