First and Second Moments of Spherical Distributions That Are Relevant for Biological Applications

Spherical distributions, in particular, the von Mises–Fisher distribution, are often used to analyze directional data. The first and second moments of these distributions are of central interest, as they describe mean orientations as well as anisotropic diffusion tensors. Finding these moments often requires a numerical approximation of complex trigonometric integrals. Instead, we apply the divergence theorem on suitable domains to derive explicit forms of the first and second moments for n-dimensional von Mises–Fisher and peanut distributions. Based on these new formulas, we characterize some meaningful characteristics of these distributions: fractional anisotropy and the anisotropy ratio. We find, surprisingly, that the peanut distribution has an upper bound on anisotropy, while the von-Mises Fisher distribution has no such bound. As a side benefit, we find different forms of some identities for Bessel functions.