A cheat sheet for probability distributions of orientational data

TL;DR

This paper provides a comprehensive cheat sheet of probability distributions for orientational data, including models, formulas, and a Python library, bridging the gap between statistical theory and engineering applications.

Contribution

It introduces a unified overview of orientation probability distributions with practical tools and examples, tailored for engineering and computer science use cases.

Findings

Presented formulas for density functions of various orientation models

Provided a Python library for modeling and sampling orientations

Demonstrated applications on real data examples

Abstract

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.