Perfect simulation of conditional and weighted models

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Abstract

This thesis is about probabilistic simulation techniques. Specifically we consider the exact or perfect sampling of spatial point process models via the dominated CFTP protocol. Fundamental among point process models is the Poisson process, which formalises the notion of complete spatial randomness; synonymous with the Poisson process is the Boolean model. The models treated here are
the conditional Boolean model and the area-interaction process. The latter is obtained by weighting a Poisson process according to the area of its associated Boolean model.

A fundamental tool employed in the perfect simulation of point processes are spatial birth-death processes. Perfect sampling algorithms for the conditional Boolean and area-interaction models are described. Birth-death processes are also employed in order to develop an exact omnithermal algorithm for the area-interaction process. This enables the simultaneous sampling of the process for a whole range of parameter values using a single realization. A variant of Rejection sampling, namely 2-Stage Rejection, and exact Gibbs samplers for the conditional Boolean and area-interaction processes are also developed here.

A quantitative comparison of the methods employing 2-Stage Rejection, spatial birth-death processes and Gibbs samplers is carried, the performance measured by actual run times of the algorithms. Validation of the perfect simulation algorithms is carried out via x2 tests.

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