Bayesian Modeling of Continuously Marked Spatial Point Patterns

Many analyses of continuously marked spatial point patterns assume that the density of points, with differing marks, is identical. However, as noted in the originative paper of Goulard et al. (1996), such an assumption is not realistic in many situations. For example, a stand of forest may have many more small trees than large, hence the model should allow for a higher density of points with small marks. In addition, as suggested by Ogata & Tanemura (1985), the interaction between points should be a function of their mark, allowing, for example, the range of interaction for large trees to exceed that of smaller trees. The aforementioned articles use frequentist inferential techniques, but interval estimation presents difficulties due to the complex distributional properties of the estimates. We suggest the use of Bayesian inferential techniques. Although a Bayesian approach requires a complex, computational implementation of (reversible jump) MCMC methodology, it enables a wide variety of inferences (including interval estimation). We demonstrate our approach by analyzing the well known Norway spruce dataset.