Markov Random Fields and Car models for mathematical and statistical data analysis

In the analysis of spatial phenomena closely related to the local context, the probabilistic model is commonly used by Markov random field, a random function that analyzes the influence of the immediately surrounding area, by appropriate probability distributions. In particular, chapter 1 suggests some elements of novelty represented by a possible classification of particular neighbourhood structures and an interesting "extension" in the space of an algorithm, the Gibbs sampler, widely used in the theory of stochastic processes and appropriately adapted to simulating maps. In chapter 2 and chapter 3, we propose two new models for areal data, the Spatial Temporal Conditional Auto-Regressive (Spatial Temporal CAR) model and the Markov Conditional Auto-Regressive (Markov CAR) model, which allow to handle the spatial dependence between sites as well as the temporal dependence among the realizations, in the presence of measurements recorded at each spatial location in a time interval.