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.

方向数据的形式统计分析始于von mises-fisher分布。该分布是描述平均方向和浓度的一阶指数模型。为了纳入二阶指数术语，进行了几次尝试，但正式结构是在宾厄姆分布中组装的。由于这是二阶指数模型，因此此分布对于轴向数据很有用。在后者尝试同时纳入第一和第二阶的尝试之后，导致了Fisher-Bingham分布。就估计而言，主要使用了数值最大似然或力矩方法方法。由于涉及许多高阶术语（包括Fisher-Bingham分布）的指数模型涉及复杂的正常常数，因此这带来了挑战，因为它们需要大致和/或数字处​​理。因此，必须以数值计算估计，并且不会以封闭形式可用。作为围绕这种基于回归的方法的一种方式，在一致的非参数密度估计器取代了归一化常数的情况下。这导致回归估计量在渐近上等同于最大似然估计器。当然，这意味着，由于这是任意秩序的规范指数模型，因此统计理论提供了渐近态性，因此可以进行统计推断。超晶体是歧管的通用例子。我们的意思是，在每个点附近，都有一个邻居在某些欧几里得空间中与开放的单位球相同。方向数据分析的形式化，尤其是通过球形谐波基础的形式化可以推广到多种多样。这可以通过理解球形谐波作为laplacian的特征函数来实现。尽管集中到超晶体是特定的，但开发的大多数方法可以扩展到多种多样，并且与应用一起将是该研究计划的主要内容。