Some topics in the analysis of spherical data.

This thesis is concerned with the statistical analysis of directions in 3 dimensions. An important reference is the book by Mardia (1972). At the time of publication of this book, the repertoire of spherical distributions used for modelling purposes was rather limited, and there was clearly a need to investigate other possibilities. In the last few years there has been some interest in the 8 parameter family of distributions mentioned by Mardia (1975), which is known as the Fisher-Bingham family. In Chapter 1 an outline of the thesis is given. The Fisher-Bingham family is discussed in Chapter 2, and an effective method for calculating the normalising constant is presented. Attention is then focussed on an interesting 6 parameter subfamily, and a simple rule is given for classifying the distributions in this subfamily according to type (unimodal, bimodal, ’closed curve'). Estimation and inference are then discussed, and the Chapter is concluded with a numerical example. In Chapter 3, the family of bimodal distributions presented in Wood (1982) is described. Other bimodal models are also mentioned briefly. The problem of simulating Fisher-Bingham distributions is considered in Chapter 4. Some inequalities are derived and then used to construct suitable envelopes so that an acceptance-rejection procedure can be used. In Chapter 5, the robust estimation of concentration for a Fisher distribution is considered, and L-estimators of the type suggested by Fisher (1982) are investigated. It is shown that the best of these estimators have desirable all-round properties. Indications are also given as to how these ideas can be adapted to other contexts. Possibilities for further research are mentioned in Chapter 6

Directional Distributions in Tracking of Space Debris

Directional distributions play an important role in describing uncertainty in spherical coordinates. A review is given of some standard distributions on the sphere which arise as special cases of the Fisher-Bingham distribution. A new distribution, called the “extreme FB5” istribution, is introduced to describe semi-concentrated behavior on the sphere, that is, patterns of data that are unimodal and concentrated near a great circle. This behavior is particularly relevant to tracking problems. Properties of the new distribution are discussed and methods are given for simulation and estimation. Two simple error propagation illustrations are given to demonstrate the usefulness of the new model

A New Unified Approach for the Simulation of a Wide Class of Directional Distributions

The need for effective simulation methods for directional distributions has grown as they have become components in more sophisticated statistical models. A new acceptance-rejection method is proposed and investigated for the Bingham distribution on the sphere using the angular central Gaussian distribution as an envelope. It is shown that the proposed method has high efficiency and is also straightforward to use. Next, the simulation method is extended to the Fisher and Fisher-Bingham distributions on spheres and related manifolds. Together, these results provide a widely applicable and efficient methodology to simulate many of the standard models in directional data analysis. An R package simdd, available in the online supplementary material, implements these simulation methods

An elliptically symmetric angular Gaussian distribution

We define a distribution on the unit sphere Sd−1 called the elliptically symmetric angular Gaussian distribution. This distribution, which to our knowledge has not been studied before, is a subfamily of the angular Gaussian distribution closely analogous to the Kent subfamily of the general Fisher–Bingham distribution. Like the Kent distribution, it has elliptical contours, enabling modelling of rotational asymmetry about the mean direction, but it has the additional advantages of being simple and fast to simulate from, and having a density and hence likelihood that is easy and very quick to compute exactly. These advantages are especially beneficial for computationally intensive statistical methods, one example of which is a parametric bootstrap procedure for inference for the directional mean that we describe