The wrapping approach for circular data Bayesian modeling

Although circular data are special, they arise in many different contexts. Examples are found in earth sciences, meteorology, biology, physics, etc. Standard statistical techniques cannot be used to analyze circular data because of circular geometry of the sample space. There are different approaches to handle circular data. In the embedding approach the directions are treated as angles, while in the most popular intrinsic approach the directions are treated as unit complex number and modeled by von Mises distribution. An alternative, and more general class of distribution models can be obtained using the so called wrapping approach, in which the circular distributions are obtained wrapping the distributions on the real line onto the unit circle. In this thesis, after giving a general overview about circular data, we deeply analyze the wrapping approach showing the main drawback and advantages of this method. Focusing on wrapped Normal distribution, we provide an approximation for this circular distribution that turns out to be very useful to improve the inferential results. This approximation, in fact, is directly used into the Bayesian inference procedure allowing to overcome the main disadvantage, the identiability problem, and to show the flexibility and ease of applicability of this approach in model with complex structure as measurement error model and high dimensional spatial and spatiotemporal model. The main contribution of this work is substantially of overcoming the identiability problem with the consequently possibility to apply the standard in line inferential procedures and methods to circular data as well. In order to appreciate the flexibility and the ease of applicability and interpretability of the wrapping approach two original applications of measurement error model for circular data are presented: the first in a spatial context and the second in a dynamic spatiotemporal context. Some remarks and discussions about future developments conclude the thesis.