Exponential Models on Manifolds

方向性数据的正式统计分析始于冯·米塞斯-费希尔分布。这种分布是一个一阶指数模型，它描述了平均方向和浓度。为了结合二阶指数项，人们进行了几次尝试，但形式结构是在宾汉分布中组装的。由于这是一个二阶指数模型，这种分布对于轴向数据是有用的。在后者之后，试图将一阶和二阶同时指数地结合在一起，导致了Fisher-Bingham分布。在估计方面，主要采用数值极大似然法或矩方法。由于指数模型涉及许多高阶项，包括Fisher-Bingham分布，涉及复杂的正规化常数，这带来了挑战，因为它们需要进行近似和/或数值处理。因此，估计数必须以数字计算，不能以封闭形式提供。作为一种绕过这一问题的方法，提出了一种基于回归的方法，其中一致的非参数密度估计器取代了归一化常数。这导致了回归估计量与最大似然估计量渐近等价。这当然意味着，由于这是一个任意阶次的典型指数模型，统计理论提供了渐近正态，因此可以进行统计推断。超球体是流形的一个通用例子。这里我们的意思是，在每个点的周围，都有一个邻域在拓扑上与某些欧氏空间中的开单位球相同。定向数据分析的形式化，特别是球谐函数的形式化可以推广到流形上。这可以通过将球谐函数理解为流形上拉普拉斯函数的本征函数来实现。虽然对超球面的关注是特定的，但所发展的大多数方法都可以扩展到流形上，并将随着应用，将是本研究计划的主要内容。

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.