On flexible random field models for spatial statistics: Spatial mixture models and deformed SPDE models

Sammanfattning: Spatial random fields are one of the key concepts in statistical analysis of spatial data. The random field explains the spatial dependency and serves the purpose of regularizing interpolation of measured values or to act as an explanatory model. In this thesis, models for applications in medical imaging, spatial point pattern analysis, and maritime engineering are developed. They are constructed to be flexible yet interpretable. Since spatial data in several dimensions tend to be large, the methods considered for estimation, prediction, and approximation are focused on reducing computational complexity. The novelty of this work is based on two main ideas. First, the idea of a spatial mixture model, i.e., a stochastic partitioning of the spatial domain using a latent categorically valued random field. This makes it possible to explain discontinuities in otherwise smoothly varying random fields. It also introduces a different perspective that of a spatial classification problem. This idea is used to model the spatial distribution of tissue types in the human head; an application important in reducing cell damage due to ionizing radiation in medical imaging. The idea is also used to introduce an extension of the popular log-Gaussian Cox process. This extension adds an extra layer of a latent random partitioning of the spatial domain. Using this model, it is possible to classify spatial domains based on observed point patterns. The second main idea of this thesis is that of spatially deforming a solution to a stochastic partial differential equation. In this way, a random field with a needed degree of non-stationarity and anisotropy can be acquired. A coupled system of two such stochastic partial differential equations is used to model the joint distribution of significant wave heights and wave periods in the north Atlantic. The model is used to assess risks in naval logistics.