Holonomic Gradient Descent for the Fisher-Bingham Distribution on the $d$-dimensional Sphere

Asymptotic Matrix Variate von-Mises Fisher and Bingham Distributions with Applications

Probability distributions in Stiefel manifold such as the von-Mises Fisher and Bingham distributions find diverse applications in signal processing and other applied sciences. Use of these statistical models in practice is complicated by the difficulties in numerical evaluation of their normalization constants. In this letter, we derive asymptotical approximations to the normalization constants via recent results in random matrix theory. The derived approximations take simple forms and are reasonably accurate in regimes of practical interest. As an application, we show that the proposed analyt…

A new method to simulate the Bingham and related distributions in directional data analysis with applications

A new acceptance-rejection method is proposed and investigated for the Bingham distribution on the sphere using the angular central Gaussian distribution as an envelope. It is shown to have high efficiency and to be straightfoward to use. The method can also be extended to Fisher and Fisher-Bingham distributions on spheres and related manifolds.

Score matching estimators for directional distributions

One of the major problems for maximum likelihood estimation in the well-established directional models is that the normalising constants can be difficult to evaluate. A new general method of "score matching estimation" is presented here on a compact oriented Riemannian manifold. Important applications include von Mises-Fisher, Bingham and joint models on the sphere and related spaces. The estimator is consistent and asymptotically normally distributed under mild regularity conditions. Further, it is easy to compute as a solution of a linear set of equations and requires no knowledge of the nor…

Algebraic Analysis of Rotation Data

We develop algebraic tools for statistical inference from samples of rotation matrices. This rests on the theory of D-modules in algebraic analysis. Noncommutative Gröbner bases are used to design numerical algorithms for maximum likelihood estimation, building on the holonomic gradient method of Sei, Shibata, Takemura, Ohara, and Takayama. We study the Fisher model for sampling from rotation matrices, and we apply our algorithms for data from the applied sciences. On the theoretical side, we generalize the underlying equivariant D-modules from SO(3) to arbitrary Lie groups. For compact groups…

球面上加权经验测度的几何信息分解

Geometric Information Decomposition for Weighted Empirical Measures on the Sphere

研究单位球面上加权概率测度的方向不确定性，传统方法用von Mises-Fisher分布不完整，引入几何信息分解（GID），通过球面特征拟合最大熵投影序列，证明相关性质，实验展示不同情况下GID作用。

球面上的 Stein 矩方法

Stein's Method of Moments on the Sphere

本文利用 Stein 特征刻画为 Fisher-Bingham、von Mises-Fisher 和 Watson 三种球面分布提出新的矩型估计量，其渐近性质接近有效，并通过模拟和手写数字潜在表示的真实数据验证了其实用性。

重尾备择假设下高维均匀性检验的渐近分析

Asymptotic analysis of high-dimensional uniformity tests under heavy-tailed alternatives

针对高维均匀性检验问题，本文分析了Rayleigh和Bingham检验在重尾备择假设下的失效及packing检验的尺寸失真，利用三者渐近独立性提出基于Fisher组合技术的新检验，兼顾了各检验的最优性质与第一类错误控制。