Fisher's Legacy of Directional Statistics, and Beyond to Statistics on Manifolds

TL;DR

This paper reviews Fisher's pioneering work in directional statistics, discusses its evolution into statistics on manifolds, and introduces a new Fisher-type distribution motivated by machine learning applications.

Contribution

It provides a comprehensive overview of Fisher's contributions to directional statistics and introduces a novel Fisher-type distribution inspired by recent machine learning challenges.

Findings

Reanalysis of Fisher's geological data confirms distribution properties.

Introduction of a new Fisher-type distribution for machine learning applications.

Growth of directional statistics in diverse fields like life sciences and image analysis.

Abstract

It will not be an exaggeration to say that R A Fisher is the Albert Einstein of Statistics. He pioneered almost all the main branches of statistics, but it is not as well known that he opened the area of Directional Statistics with his 1953 paper introducing a distribution on the sphere which is now known as the Fisher distribution. He stressed that for spherical data one should take into account that the data is on a manifold. We will describe this Fisher distribution and reanalyse his geological data. We also comment on the two goals he set himself in that paper, and how he reinvented the von Mises distribution on the circle. Since then, many extensions of this distribution have appeared bearing Fisher's name such as the von Mises Fisher distribution and the matrix Fisher distribution. In fact, the subject of Directional Statistics has grown tremendously in the last two decades with new applications emerging in Life Sciences, Image Analysis, Machine Learning and so on. We give a recent new method of constructing the Fisher type distribution which has been motivated by some problems in Machine Learning. The subject related to his distribution has evolved since then more broadly as Statistics on Manifolds which also includes the new field of Shape Analysis. We end with a historical note pointing out some correspondence between D'Arcy Thompson and R A Fisher related to Shape Analysis.