First and Second Moments and Fractional Anisotropy of General von Mises-Fisher and Peanut Distributions
Alexandra Shyntar, Thomas Hillen
TL;DR
This work tackles the challenge of obtaining closed-form first and second moments for $n$-dimensional spherical distributions, specifically unimodal and bimodal von Mises–Fisher and the peanut distribution. By employing a divergence-theorem-based calculation on the unit ball, the authors derive explicit expressions for $\mathsf{E}[q]$ and $\mathsf{Var}[q]$ in arbitrary dimensions, and translate these into diffusion tensors $\mathbb{D}$ with clear eigenstructure. A key finding is that the peanut distribution yields bounded anisotropy ($R\le 3$ and $\text{FA}$ bounds), while the von Mises–Fisher distribution can realize the full range of anisotropy, making it more suitable for modeling directional diffusion. The explicit formulas enable faster simulations and PDE-based transport modeling in high dimensions and can extend to mixtures of von Mises–Fisher distributions, offering practical benefits for applications in biology and related fields.
Abstract
Spherical distributions, in particular, the von Mises-Fisher distribution, are often used for problems using or modelling directional data. Since expectation and variance-covariance matrices follow from the first and second moments of the spherical distribution, the moments often need to be approximated numerically by computing trigonometric integrals. Here, we derive the explicit forms of the first and second moments for an n-dimensional von Mises-Fisher and peanut distributions by making use of the divergence theorem in the calculations. The derived formulas can be easily used in simulations, significantly decreasing the computation time. Moreover, we compute the fractional anisotropy formulas for the diffusion tensors derived from the bimodal von Mises-Fisher and peanut distributions, and show that the peanut distribution is limited in the amount of anisotropy it permits, making the von Mises-Fisher distribution a better choice when modelling anisotropy.