Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.

该论文对于在黎曼流形中处理方向性数据的方法进行了综述，从探索性分析、分布模型、推断方法、假设检验、回归、非参数曲线估计、降维、分类和聚类等多个方面进行了介绍，并概述了目前可用的分析方向数据的软件及未来发展的潜力。

方向统计学的最新进展

Recent advances in directional statistics

Directional or Circular statistics are pertaining to the analysis and
interpretation of directions or rotations. In this work, a novel probability
distribution is proposed to model multidimensional sparse directional data. The
Generalised Directional Laplacian Distribution (DLD) is a hybrid between the
Laplacian distribution and the von Mises-Fisher distribution. The
distribution's parameters are estimated using Maximum-Likelihood Estimation
over a set of training data points. Mixtures of Directional Laplacian
Distributions (MDLD) are also introduced in order to model multiple
concentrations of sparse directional data. The author explores the application
of the derived DLD mixture model to cluster sound sources that exist in an
underdetermined instantaneous sound mixture. The proposed model can solve the
general K x L (K<L) underdetermined instantaneous source separation problem,
offering a fast and stable solution.

本文提出了一种新的概率分布，即广义方向拉普拉斯分布（DLD），以建模多维稀疏方向数据，该分布是拉普拉斯分布和 von Mises-Fisher 分布之间的混合物。作者探讨了将导出的 DLD 混合模型应用于聚类存在于欠定瞬时声混合物中的声源，该模型可以解决通用的 K x L (K<L) 欠定瞬时源分离问题，提供快速和稳定的解决方案。

广义方向拉普拉斯分布：估计，混合模型和音频源分离

A Generalised Directional Laplacian Distribution: Estimation, Mixture  Models and Audio Source Separation

Distributions over permutations arise in applications ranging from
multi-object tracking to ranking of instances. The difficulty of dealing with
these distributions is caused by the size of their domain, which is factorial
in the number of considered entities ($n!$). It makes the direct definition of
a multinomial distribution over permutation space impractical for all but a
very small $n$. In this work we propose an embedding of all $n!$ permutations
for a given $n$ in a surface of a hypersphere defined in
$\mathbbm{R}^{(n-1)^2}$. As a result of the embedding, we acquire ability to
define continuous distributions over a hypersphere with all the benefits of
directional statistics. We provide polynomial time projections between the
continuous hypersphere representation and the $n!$-element permutation space.
The framework provides a way to use continuous directional probability
densities and the methods developed thereof for establishing densities over
permutations. As a demonstration of the benefits of the framework we derive an
inference procedure for a state-space model over permutations. We demonstrate
the approach with applications.

提出一种在超球面上嵌入全排列的方法，从而可以定义连续方向概率分布以及派生排列的密度，运用该方法实现了一个排列状态空间模型的推理过程并进行了相关应用。