Modelling techniques for compositional data using distributions defined on the hypersphere

Abstract

Compositional data can be transformed to directional data by the square root transformation and then modelled using distributions defined on the hypersphere. The purpose of this thesis is to investigate new modelling techniques for a general p-dimensional compositional data vector using this square root transformation approach. One advantage is that zero components are catered for naturally in the models. The Kent distribution for directional data is a good candidate model because it has a sufficiently general covariance structure. We investigate the properties of the Kent distribution in p-dimensions and develop asymptotic normal theory results and an acceptance sampling algorithm. We propose a new regression model which models the mean direction of the Kent distribution as a function of a vector of covariates. Our estimators for the regression model parameters are motivated by asymptotic approximations and moment estimators and can be regarded as asymptotic maximum likelihood estimators. We show that these estimators perform well and are suitable for typical compositional datasets, including those with zero components. One advantage of our approach to estimation is that it is not computationally intensive and the parametric bootstrap is used for inference. Using two real example datasets we demonstrate a modelling strategy in practice, including how to check the model assumptions. A simulation study is also undertaken, confirming the value of our proposed estimators in typical cases. We also demonstrate in certain cases that our new approach for modelling compositional data should work much better than more traditional methods such as those based on logratio transformations.