Hierarchical spatial log Gaussian Cox process model for replicated point patterns
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We propose a conditional log Gaussian Cox process (LGCP) model to investigate the effect of a realization y of a point process Y on the intensity of a point process X. In the motivating forestry example, the point pattern y represents large trees and the point pattern x, a realization of X, seedlings. In the model, every point in Y has a parametric influence kernel or signal, which together form an influence field. Conditionally on the parameters, the influence field acts as a spatial covariate in the (log) intensity of the LGCP model, and the (log) intensity itself is a non-linear function of the parameters. Unlike in the typical unconditional LGCP situation, points of Y outside the observation window may affect the intensity of X inside the window. Therefore, we propose a simple edge correction method to account for this edge effect. The parameters of the model are estimated in a Bayesian framework using Markov chain Monte Carlo (MCMC) where a Laplace approximation is used for the Gaussian field of the LGCP model. Since forest data are often measured in small sample plots, we present the estimation procedure based on replicates. The proposed model is fitted to uneven-aged forest stands in Finland to study the effect of large trees on the success of regeneration.
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