Furtherance of modeling frameworks for multivariate directional statistics

Abstract

In this thesis, we propose multivariate directional models that serve to fill the gaps in literature and aim to develop innovative theoretical modeling frameworks for contemporary applications where either certain manifolds have been neglected or the use of directional statistics has been neglected. This thesis focuses on three different manifolds; the hyper-sphere, the disc and the poly-cylinder. For the multivariate circular observations we propose a family of distributions on the unit hyper-sphere obtained by considering normal mean mixture distributions from a transformation viewpoint. The resulting family of distributions, termed Mean Direction Mixture models, can account for symmetry, asymmetry, unimodality and bimodality. In addition to the multivariate circular domain, we consider the circular-linear domain. For the joint modeling of circular and linear observations we explore the disc manifold for the bivariate modeling of these observations and then delve into the multivariate domain of circular-linear observations by means of the poly-cylinder. A new class of bivariate distributions is proposed which has support on the unit disc in two dimensions that includes, as a special case, the existing M\"obius distribution on the disc. Applications of the proposed model for the use in wind description and wind energy analysis is presented. Furthermore, we propose a multivariate dependent modeling framework applicable to the 6D joint distribution of circular-linear data based on vine copulas. This framework is motivated by the analysis of the mechanical behavior of external fixators in the biomechanical domain. The work proposed in this thesis aims to play a part in addressing the larger need for multivariate models in directional statistics due to the increased amount of complex data containing angular observations.