Getting started

pip install fb8

import numpy as np
from matplotlib import pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
from sphere.distribution import fb8


def grid(npts):
    return [_.flatten() for _ in np.meshgrid(np.linspace(0, np.pi, npts), np.linspace(0,2*np.pi, npts))]


def plot_fb8(fb8, npts):
    """
    Plot fb8 on 3D sphere
    """
    xs = fb8.spherical_coordinates_to_nu(*grid(npts))
    pdfs = fb8.pdf(xs)
    z,x,y = xs.T #!!! Note the ordering for xs here is used consistently throughout. Follows Kent's 1982 paper.

    fig = plt.figure(figsize=plt.figaspect(1.))
    ax = fig.add_subplot(111, projection='3d')
    ax.plot_surface(x.reshape(npts, npts),
                    y.reshape(npts, npts),
                    z.reshape(npts, npts),
                    alpha=0.5,
                    rstride=1, cstride=1,
                    facecolors=cm.plasma(pdfs.reshape(npts, npts)/pdfs.max()))
    ax.set_axis_off()
    plt.tight_layout(-5)
    plt.show()


plot_fb8(fb8(np.pi/16,-np.pi/3,0,10,10,-1,0.5,0.3), 200)

Basic information

Implements calculation of the density and fitting (using maximum likelihood estimate) of the FB8 distribution on a sphere, which is a generalization of the FB6, FB5 (Kent), and FB4 (Bingham-Mardia) distributions described below.

Implements the FB6 distribution that is first introduced in Rivest (1984).

Implements calculation of the density and fitting (using maximum likelihood estimate) of the Kent distribution based on Kent (1982). A unittest is performed if distribution.py is called from the command line.

Implements the Bingham-Mardia distribution whose mode is a small-circle on the sphere based on Bingham, Mardia (1978).

Also calculates directional, percentile levels which can be used to indicate the N% highest-posterior-density regions in the sky.

Additional references

Kent, Hussein, Jah, Directional distributions in tracking of space debris

Terdik, Jammalamadaka, Wainwright, Simulation and visualization of spherical distributions

Mardia, Jupp, Directional statistics

Notes

Currently the scipy.special.hyp2f1 is used and may exhibit inaccuracies for large parameters. See github issues.

Acknowledgements

This project was originally developed for the FB5 (Kent) distribution here.

Tianlu Yuan