The starting point are data sets from Astrophysics. From pictures of the surface of Europa and Ganymede, two moons of Jupiter which are entirely covered with water and ice, the physicists extracted several thousand positions and orientations of cracks in the ice shield. Different theories about the properties and behaviour of the moons predict locally certain preferred orientations. Hence the task was to estimate preferred orientations at different locations on the surface from the data.

This task leads to the general topic of regression methods for directional and axial data. I will describe a possible approach for regression with observations (X,Y), where X is an arbitrary covariate vector, Y is a unit vector in d-dimensional space, and one assumes that the conditional distribution of Y given X is a von Mises-Fisher (vMF) distribution with parameters depending parametrically on X. This parametric generalised linear model can be adapted to nonparametric smoothing via local parametric modelling. Possible modifications to so-called Bingham distributions for axial response Y will be explained as well. (Axial response means that Y and -Y are considered as identical.)

In the third part we return to the specific data in which X corresponds to a point on the unit sphere in dimension 3, and Y describes an orientation at X, that is, a unit vector which is perpendicular to X. By means of stereographic projections and suitable parametrisations, one can solve the smoothing task via local linear or local quadratic generalised linear models with two-dimensional covariate vectors X' and two-dimensional axial response vectors Y'.

Personal website of Lutz Dümbgen