Abstract
Modelling and inference are difficult but crucial tasks in spatial statistics. Exact likelihood based methods are desirable, but for clustered processes such as the log-Gaussian Cox process and Neyman-Scott processes, such methods are unavailable. This thesis makes important contributions to modelling and inference for spatial point patterns.
Intensity estimation, often achieved through kernel smoothing, is a key component of modelling spatial point patterns. Current methods of selecting the bandwidth, which controls the amount of smoothing, can lead to conflicting conclusions. We provide more reliable bandwidth selection methods for spatial intensity estimation by adapting methods previously developed for general multivariate density estimation.
In addition to estimating the intensity, we also wish to estimate the parameters that describe potential interaction between points. As the likelihood is intractable, one option is to approximate it using Monte Carlo methods. However, this is widely regarded as impractical, hence has rarely been implemented. We provide a comprehensive study of the computational properties of Monte Carlo maximum likelihood estimation for log-Gaussian Cox processes and compare it to other commonly used estimation methods.
The intractability of the likelihoods for clustered processes also leads to difficulties in inference. As a result, fundamental tasks such as a parametric test of clustering are out of reach. We leverage recent insights into Neyman-Scott processes to contribute formal hypothesis tests for such processes. Crucially, these tests can be used to conduct a test of clustering. A version of the likelihood ratio test can be performed using the Palm likelihood, however, the null distribution of the resulting test statistic does not behave as in the case of a true likelihood. In addition, when testing for clustering the test takes place on the boundary of the parameter space. We determine the appropriate adjustment for different hypotheses. These adjustments contain quantities that are intractable and must be replaced by estimates. We assess the performance of these adjustments empirically.
Overall, the contributions in this work advance and improve practical aspects of exploring and modelling spatial point patterns, and suggest avenues of future research.