Inference and Visualization for Spatial Point Processes

Abstract

Spatial point processes are statistical models that describe the arrangement of objects that are randomly distributed in the plane or in space. In recent years, they have received sustained attention because researchers use them to model objects in ecology, biology, medicine and material science, to name a few. Inevitably, a goodness-of-fit test is needed to assess the fit of these models and to justify their choice. In this thesis, I propose a method, consisting of nested Monte Carlo simulations, which removes the bias of the resulting empirical level of the test. As a further contribution to statistical inference for the spatial point processes in this thesis, I introduce skew-elliptical cluster processes, where the clusters can have an anisotropic structure allowing the choice of a flexible covariance matrix and incorporating skewness or ellipticity parameters into the structure. Theses processes help to tackle the challenge arising with non-circular clusters, e.g., induced by a wind direction in the pattern. In particular, I formulate the construction of skew-elliptical-normal and skew-elliptical-t cluster processes. For the parameter estimation, I propose the minimum contrast method using an approximating pair correlation function to circumvent the complicated derivation of the maximum- or pseudo-likelihood and the computational complexity of the Bayesian approach or MCMC algorithm. The last contribution in this thesis is in diagnostics and influential measures for spatial point processes. I describe a method to define influential events of a spatial point pattern based on a parametric likelihood model or a second-order summary characteristic function if the likelihood model is difficult to derive. In particular, instead of deleting one observation/event at a time like in commonly-used approaches in detecting influential events, I add some noise to one event at a time. The perturbation provides a whole course of change of estimators based on which I quantify the influence. To visualize influential events, I use hair-plots and disc-plots to display the influence of each event. Those events with significantly high magnitude of influence can be considered as influential.