Salvador, Sara (2022). Bayesian inference and Monte Carlo methods for directional data. (Thesis). Universität Bern, Bern
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22salvador_s.pdf - Thesis Available under License Creative Commons: Attribution-Noncommercial-No Derivative Works (CC-BY-NC-ND 4.0). Download (9MB) | Preview |
Abstract
This thesis deals with directional distributions, hence distributions defined over Rp. We will consider in details the case of circular distributions, i.e. directional distributions in R2. We will focus on the study of an extension of the von Mises distribution (vM), namely the generalized von Mises distribution of order two (GvM2, or simply GvM). This distribution allows higher flexibility in terms of asymmetry and bimodality than the vM distribution. Two Bayesian tests are computed on the GvM. The first test concerns the symmetry of the GvM. Inference on symmetry is made via Bayes factors. A real circular data case is considered. The second test concerns the bimodality of the GvM. The problem is reduced to the study of the real roots of a quartic whose coefficients depend on the parameters of the model. A detailed analysis of the quartic is given and a region W of parameters that are associated to bimodality is obtained. Then, inference on bimodality is made via Bayes factors and via highest posterior density (HPD) credible sets. We use Markov Chain Monte Carlo methods (MCMC) to compute posterior probabilities of bimodality. Conclusions confirm that, when data are in accordance with the null hypotheses, Bayes factors are typically large. On the other hand, the HPD credible set is entirely contained inside W and bimodality is confirmed. Moreover, in this thesis we consider directional distributions over the unit sphere Sp−1 of Rp. These are called spherical distributions and here we manly focus on the generalized von Mises-Fisher (GvMF) distribution. This is an extension of the von Mises-Fisher distribution (vMF). We will give two methods to generate random variables from the GvMF using a conditional acceptance-rejection method and the Metropolis Hastings algorithm.
| Item Type: | Thesis |
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| Dissertation Type: | Single |
| Date of Defense: | 16 December 2022 |
| Subjects: | 500 Science > 510 Mathematics |
| Institute / Center: | 08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science |
| Depositing User: | Hammer Igor |
| Date Deposited: | 02 Feb 2023 18:00 |
| Last Modified: | 16 Dec 2023 23:25 |
| URI: | https://boristheses.unibe.ch/id/eprint/4069 |
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