UCLA

A spatial point process is a random pattern of points on a space $A \subseteq \mathbb{R}^d$. 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 $\mathcal{S}^{d-1}\subset \mathbb{R}^d$. There is some work in recent years focusing on extending exploratory tools on $\mathbb{R}^d$ to $\mathcal{S}^{d-1}$, such as the widely used Ripley's $\mathcal{K}$ function.

In this dissertation, we propose a more general framework for modeling point processes on $\mathcal{S}^2$. 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 $\mathcal{S}^2$; 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.