Some topics in the analysis of spherical data.

Wood, Andrew T.A. (1985). Some topics in the analysis of spherical data. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0000f900

Abstract

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.