On a Mixture Model for Directional Data on the Sphere

We consider mixtures of general angular central Gaussian distributions as models for multimodal directional data. We prove consistency of the maximum-likelihood estimates of model parameters and convergence of their numerical approximations based on an expectation-maximization algorithm. Then, we focus on mixtures of special angular central Gaussian distributions and discuss the details of a fast numerical algorithm, which allows to fit multimodal distributions to massive data, occurring, for example, in the study of the microstructure of materials. We illustrate the applicability with some data from fibre composites and from ceramic foams.