## Fun with spherical distributions

I spend a good part of last year learning to do math on the sphere. Considering that the sphere is the shape of the retina, it is a little surprising that spherical geometry is typically not in the standard tool kit of visual neuroscientists.

The figure above is an illustration of the 5-parameter Fisher-Bingham distribution (FB5), which is also known as the Kent distribution. Wikipedia’s entry has an illustration but, man, it’s ugly. My version looks much nicer.

Just like the more familiar one-dimensional or two dimensional probability density functions, a spherical distribution assigns a probability density to each point in its domain, except that the domain is the unit sphere. I love spherical distributions because they really give an interesting twist to the concept of probability distributions. Before I had to do things with spherical distributions, it never occurred to me that probability distributions are defined on spaces with geometrical structures. Then I learned that distributions can be defined on klien bottles, toruses and other low-dimensional manifolds.

The FB5 distribution is joy to work with because its properties are very close to the familiar bivariate Gaussian. However, it can do some bizarre things if you are not careful. For example, it has a bimodal mode: