The Spherical Bingham Distribution edit page

The Bingham distribution on the sphere is an antipodal symmetric distribution (Bingham, 1974) with a probability density function given by

$p_{b}(\hat{x}\vert AKA^T) = \frac{1}{F(\kappa_{1},\kappa_{2},\kappa_{3})}\exp (\hat{x}^T AZA^T \hat{x})$

where $$A$$ is an orthogonal covariance matrix, and $$Z$$ a concentration matrix with $$\mathrm{diag}(\kappa_{1},\kappa_{2},\kappa_{3})$$ with $$\kappa_{1} < \kappa_{2} < \kappa_{3}$$.

In MTEX $$Z$$ is given by Z = [k1,k2,k3] with k3 = 0 and $$A$$ is given by three orthogonal vectors.

Bingham, C., An Antipodally Symmetric Distribution on the Sphere, The Annals of Statistics Vol. 2, No. 6 (Nov., 1974), pp. 1201-1225

## Meaning of $$Z$$

$$k1 = k2$$ defines a rotational symmetric point maximum and $$k2 = 0$$ defines a girdle distribution.

## Estimating a spherical Bingham distribution from discrete data

Given arbitrarily scattered data v on the sphere we can estimate the best fitting Bingham distribution by

Lets plot the fitted distribution with the data

Under the assumption of sufficiently many and sufficently concetrated data we may also estimate a confidence ellipse for the mean direction (default p = 0.95). The center of the ellipse is given by the largest principle vector stored in bs.a(3)

The orientation of the ellipse is specified by all the principle vectors bs.a and the a and b axes are computed by the command cEllipse