Exponential Models on Manifolds

流形上的指数模型

• 批准号: RGPIN-2022-02945
• 负责人: Kim, Peter
• 金额: $ 1.31万
• 依托单位: University of Guelph
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759018
• 关键词: Exponential; Models; Manifolds

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.

方向数据的正式统计分析从von Mises-Fisher分布开始。该分布是描述平均方向以及浓度的一阶指数模型。为了包含二阶指数项，进行了几次尝试，但正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，该分布对于轴向数据很有用。在后者之后，人们试图同时将一阶和二阶指数结合起来，从而产生了Fisher-Bingham分布。在估计方面，主要使用数值最大似然法或矩量法。由于指数模型涉及许多高阶项，包括Fisher-Bingham分布，涉及复杂的归一化常数，这带来了挑战，因为它们需要近似和/或数值处理。因此，估计数必须以数字计算，而不能以封闭形式提供。作为一种方法，围绕这一回归为基础的方法制定了一致的非参数密度估计取代了归一化常数。这导致回归估计量渐近等价于最大似然估计量。这当然意味着，因为这是一个任意阶的正则指数模型，统计理论提供了渐近正态性，因此可以进行统计推断。超球面是流形的一个一般例子。我们的意思是，在每一点周围，都有一个邻域在拓扑上与某个欧几里得空间中的开单位球相同。方向数据分析的形式化，特别是通过球调和基，可以推广到一个流形。这可以通过将球谐函数理解为流形上拉普拉斯算子的本征函数来实现。虽然对超球面的研究是特定的，但大多数方法都可以推广到流形上，并沿着应用，将是本研究计划的主要内容。