Statistical Models for Complex Directional Data

Directional data are data that represent directions, e.g. the direction of wind or the direction in
which an animal moves. These data are represented as points on the unit circle or unit sphere, since
they have no "length". This renders their statistical treatment very delicate, as for instance the
simple average of two points on the sphere will result in a point lying inside the sphere and hence
not on the sphere. The classical mean thus cannot be used to describe such data, and more
generally all statistical procedures must be adapted to these special data types. This has led to the
research stream called Directional Statistics.
In the present project, I will propose statistical models, in particular probabilistic distributions, to
describe and deal with even more complex directional data. I will consider pairs of angular
directions, leading to data on the unit torus, pairs of direction and a linear component, leading to
data on cylinders, and toroidal/cylindrical data that moreover are spatially correlated, meaning that
there is an underlying dependence structure.
The new procedures allow me to tackle the data analysis in the following problems: 1) protein
structure prediction, where the pairs of dihedral angles form data on the torus, 2) earthquake
prediction, where the inter-occurrence time and magnitude form data on the cylinder, and 3)
wildfire prediction, where the fire orientation and size are cylindrical data. Spatial dependence is
present under 2) and 3).