Implementation of FB8, a generalization of the Kent (1982) and Bingham-Mardia (1978) distributions on a sphere

# 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.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 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. 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.

# Contributors

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

Tianlu Yuan

## Project details

This version 0.2.1 0.2.0 0.1.0