Nonparametric Modelling for Directional Data

In directional statistics, observations are directions or orientations. The three main types of directional data considered are circular, spherical and toroidal data, with corresponding support space on a circle, a hyper-sphere, and a torus, respectively. As conventional statistical tools and algorithms are mostly designed for data in an Euclidean space, it motivates us to develop special techniques for directional data to accommodate their general support on compact Riemannian manifolds. Additionally, by incorporating directional statistics, an alternative solution for usual statistical problems such as feature selection is provided. In this thesis, we present density estimators for the above three categories of directional observations under the framework of mixture models. Density estimation is a vital aspect in data analysis. It examines important properties for a random variable including multimodality, skewness and tail behavior. In particular, nonparametric density estimation is flexible and has a generally satisfactory performance. We adapt nonparametric and semiparametric mixtures developed for conventional data, and modify it for univariate and multivariate directional observations. Corresponding directional mixture components are selected with appropriate properties and support. The key issue of bandwidth selection is also addressed with some deterministic information criteria and simulation-based methods. The resultant mixture density estimators are compared with the commonly-used kernel smoothing method. Overall, nonparametric mixtures have highly competitive and sometimes superior performance to its competitors. Apart from estimating a reasonable density curve, an insightful data visualization is also indispensable. We concentrate on circular observations and propose a general formula to construct any type of circular plot, in an area-proportional manner. Finally, with the help of directional statistics, a new perspective for unsupervised feature selection is provided. In a nutshell, feature vectors are considered as directions on a hyper-sphere, and angles between such directions are measured to account for correlations. We propose a feature partitioning tree algorithm to divide features into well separable subgroups, which effectively guides the identification of a subset of relevant and non-redundant features.

Name thesis:degree_name

PhD

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Statistics

Grantor dc:publisher

ResearchSpace@Auckland

Year dc:date.issued

2022

Author dc:creator

• Xu, Danli

Advisors dc:contributor.advisor

• Wang, Yong
• Yee, Thomas

Statement dc:rights

• Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated.

Licence dc:rights.uri

• https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm

Handle dc:identifier.uri

https://hdl.handle.net/2292/61845

OAI identifier oai:identifier

oai:researchspace.auckland.ac.nz:2292/61845