Fisher’s legacy of directional statistics, and beyond to statistics on manifolds

Abstract

It is not 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 reanalyze his geological data. We also comment on the two goals he set himself in that paper, and on 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 distributions on manifolds which has been motivated by some problems in machine learning. The number of directional distributions has increased since then, including the bivariate von Mises distribution and we describe its connection to work resulting in the 2024 Nobel-winning AlphaFold (in Chemistry). Further, the subject has evolved as Statistics on Manifolds which also includes the new field of Shape Analysis, and finally, we end with a historical note pointing out some correspondence between D’Arcy Thompson and R.A. Fisher related to Shape Analysis.