A spatial point process is a random pattern of points on a space A ⊆ Rd. Typically A will be a d-dimensional box. Point processes on a plane have been well-studied. However, not much work has been done when it comes to modeling points on Sd -1 ⊂ Rd. There is some work in recent years focusing on extending exploratory tools on Rd to S d-1 , such as the widely used Ripley's K function.

In this dissertation, we propose a more general framework for modeling point processes on S2. The work is motivated by the need for generative models to understand the mechanisms behind the observed crater distribution on Venus. We start from a background introduction on Venusian craters. Then after an exploratory look at the data, we propose a suite of Exponential Family models, motivated by the Von Mises-Fisher distribution and its generalization. The model framework covers both Poisson-type models and more sophisticated interaction models. It also easily extends to modeling marked point process. For Poisson-type models, we develop likelihood-based inference and an MCMC algorithm to implement it, which is called MCMC-MLE. We compare this method to other procedures including generalized linear model fitting and contrastive divergence. The MCMC-MLE method extends easily to handle inference for interaction models. We also develop a pseudo-likelihood method (MPLE) and demonstrate that MPLE is not as accurate as MCMC-MLE.

In addition, we discuss model fit diagnostics and model goodness-of-fit. We also address a few practical issues with the model, including the computational complexity, model degeneracy and sensitivity. Finally, we step away from point process models and explore the widely used presence-only model in Ecology. While this model provides a different angle to approach the problem, it has a few notable defects.

The major contributions to spatial point process analysis are, 1) the development of a new model framework that can model a wide range of point process patterns on S2; 2) the development of a few new interaction terms that can describe both repulsive and clustering patterns; 3) the extension of Metropolis-Hastings algorithms to account for spherical geometry.