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

Doctoral thesis
Authors Anders Hildeman
Date of public defense 2017-09-28
Opponent at public defense Associate Professor Johan Lindström, Centre for Mathematical Sciences, Lund University, Sweden
Publisher Chalmers University of Technolog
Place of publication Gothenburg
Publication year 2017
Published at Department of Mathematical Sciences
Language en
Links publications.lib.chalmers.se/record...
Keywords Non-Gaussian, Bayesian level set inversion, Point processes, Substitute CT, Finite mixture models, Spatial statistics, Gaussian fields
Subject categories Other Computer and Information Science, Probability Theory and Statistics

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.

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