Exponential Models on Manifolds

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Formal statistical analysis of directional data begins with the von Mises-Fisher distribution. This distribution is a first order exponential model that describes a mean direction as well as concentration. To incorporate a second order exponential term, several attempts were made but the formal structure was assembled in the Bingham distribution. As this is a second order exponential model, this distribution is useful for axial data. Subsequent to the latter attempts to exponentially incorporate the first and second order simultaneously were made resulting in the Fisher-Bingham distributiion. In terms of estimation, numerical maximum likelihood or method of moment methods have been primarily used. As exponential models involving many higher-order terms, including the Fisher-Bingham distribution, involve complicated normalizing constants, this presents challenges as they would need to be approximately and/or numerically dealt with. Thus estimation would have to be calculated numerically and would not be available in closed form. As a way around this a regression based approach was formulated where a consistent nonparametric density estimator replaced the normalizing constant. This leads to a regression estimator that was asymptotically equivalent to the maximum likelihood estimator. This of course means that because this is formulated as a canonical exponential model of arbitrary order, statistical theory provides asymptotic normality and therefore statistical inference could be performed. The hypersphere is a generic example of a manifold. By this we mean that around every point, there is a neighbourhood that is topologically the same as the open unit ball in some Euclidean space. The formalization for directional data analysis, especially through the spherical harmonic basis can be generalized to a manifold. This can be achieved through understanding the spherical harmonics as the eigenfunctions of the Laplacian on a manifold. Although concentration to the hypersphere is specific, most of the methods developed can be extended to manifolds and along with applications, will be the main content of this research program.