整理了工程与计算机常用的方向数据概率分布工具
A cheat sheet for probability distributions of orientational data
- 涵盖1~3自由度方向数据的分布建模方法
- 提供密度函数、数据拟合与采样实现方式
- 附带可直接使用的Python工具库，适合科研与工程应用
方向数据在工程与计算机科学中广泛存在，表现为角度、单位向量、旋转矩阵或四元数等形式。尽管方向统计学已发展出多种建模方法，但实际应用中使用较少。本文旨在作为方向数据概率分布的速查手册，系统讨论1-自由度、2-自由度和3-自由度方向数据的模型。针对每类模型，提供密度函数表达式、数据拟合方法及随机采样实现。论文在符号与术语上兼顾工程与统计视角，配套发布一个Python库，包含部分模型的实现函数。利用该库，文中展示了两个真实数据的应用案例。
原文摘要 · Abstract (English)
The need for statistical models of orientations arises in many applications in engineering and computer science. Orientational data appear as sets of angles, unit vectors, rotation matrices or quaternions. In the field of directional statistics, a lot of advances have been made in modelling such types of data. However, only a few of these tools are used in engineering and computer science applications. Hence, this paper aims to serve as a cheat sheet for those probability distributions of orientations. Models for 1-DOF, 2-DOF and 3-DOF orientations are discussed. For each of them, expressions for the density function, fitting to data, and sampling are presented. The paper is written with a compromise between engineering and statistics in terms of notation and terminology. A Python library with functions for some of these models is provided. Using this library, two examples of applications to real data are presented.
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