Spatial Mixture Models with Applications in Medical Imaging and Spatial Point Processes

Abstract: Finite mixture models have proven to be a great tool for both modeling non-standard probability distributions and for classification problems (using the latent variable interpretation). In this thesis we are building spatial models by incorporating spatially dependent categorical latent random fields in a hierarchical manner similar to that of finite mixture models. This allows for non-linear prediction, better interpretation of estimated model parameters, and the added possibility of addressing questions related to classification. This thesis consists of two papers. The first paper concerns a problem in medical imaging where substitutes of computed tomography (CT) images are demanded due to the risks associated with X-radiation. This problem is addressed by modeling the dependency between CT images and magnetic resonance (MR) images. The model proposed incorporates multidimensional normal inverse Gaussian distributions and a spatially dependent Potts model for the latent classification. Parameter estimation is suggested using a maximum pseudo-likelihood approach implemented using the EM gradient method. The model is evaluated using cross-validation on three dimensional data of human brains. The second paper concerns modeling of spatial point patterns. A novel hierarchical Bayesian model is constructed by using Gaussian random fields and level sets in a Cox process. The model is an extension to the popular log-Gaussian Cox process and incorporates a latent classification field in order to handle sudden jumps in the intensity surface and to address classification problems. For inference, a Markov chain Monte Carlo method based on the preconditioned Crank-Nicholson MALA method is suggested. Finally, the model is applied to a popular data set of tree locations in a rainforest and the results show the advantage of the proposed model compared to the log-Gaussian Cox process that has been applied to the very same data set in several earlier publications.