Bayesian modeling of continuously marked spatial point patterns

Abstract

Many analyses of continuously marked spatial point patterns assume that the density of points, with differing marks, is identical. However, as noted in the seminal paper of Goulard et al. (Scand J Stat 23:365---379, 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 and Tanemura (Biometrics 41:421---433, 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 extremely complex distributional properties of the estimates; it might be possible, however, to use parametric bootstrap methodology for such inferences (Baddeley et al. in J Roy Stat Soc Ser B 67:617---666, 2005). We suggest the use of Bayesian inferential techniques. Although a Bayesian approach requires a complex, computational implementation of (reversible jump) Markov Chain Monte Carlo methodology, it enables a wide variety of inferences (including interval estimation). We demonstrate our approach by analyzing the well known Norway spruce dataset.