def VonMisesFisher_sample(phi0, theta0, sigma0, size=None):
""" Draw a sample from the Von-Mises Fisher distribution.
Parameters
----------
phi0, theta0 : float or array-like
Spherical-polar coordinates of the center of the distribution.
sigma0 : float
Width of the distribution.
size : int, tuple, array-like
number of samples to draw.
Returns
-------
phi, theta : float or array_like
Spherical-polar coordinates of sample from distribution.
"""
n0 = cartesian_from_polar(phi0, theta0)
M = rotation_matrix([0, 0, 1], n0)
x = numpy.random.uniform(size=size)
phi = numpy.random.uniform(size=size) * 2 * numpy.pi
theta = numpy.arccos(1 + sigma0**2 *
numpy.log(1 + (numpy.exp(-2 / sigma0**2) - 1) * x))
n = cartesian_from_polar(phi, theta)
x = M.dot(n)
phi, theta = polar_from_cartesian(x)
return phi, theta
def VonMisesFisher_distribution(phi, theta, phi0, theta0, sigma0):
""" Von-Mises Fisher distribution function.
Parameters
----------
phi, theta : float or array_like
Spherical-polar coordinates to evaluate function at.
phi0, theta0 : float or array-like
Spherical-polar coordinates of the center of the distribution.
sigma0 : float
Width of the distribution.
Returns
-------
float or array_like
log-probability of the vonmises fisher distribution.
Notes
-----
Wikipedia:
https://en.wikipedia.org/wiki/Von_Mises-Fisher_distribution
"""
x = cartesian_from_polar(phi, theta)
x0 = cartesian_from_polar(phi0, theta0)
norm = -numpy.log(4 * numpy.pi * sigma0**2) - logsinh(1. / sigma0**2)
return norm + numpy.tensordot(x, x0, axes=[[0], [0]]) / sigma0**2
def test_rotation_matrix():
numpy.random.seed(seed=0)
theta = numpy.random.rand(10, 2)*numpy.pi
phi = numpy.random.rand(10, 2)*2*numpy.pi
for (p1, p2), (t1, t2) in zip(phi, theta):
n1 = utils.cartesian_from_polar(p1, t1)
n2 = utils.cartesian_from_polar(p2, t2)
M = utils.rotation_matrix(n1, n2)
assert_allclose(M.dot(n1), n2)
assert_allclose(M.T.dot(n2), n1)
M = utils.rotation_matrix(n1, n1)
assert_allclose(M, numpy.identity(3))
def VonMises_std(phi, theta):
""" Von-Mises sample standard deviation.
Parameters
----------
phi, theta : array-like
Spherical-polar coordinate samples to compute mean from.
Returns
-------
solution for
..math:: 1/tanh(x) - 1/x = R,
where
..math:: R = || \sum_i^N x_i || / N
Notes
-----
Wikipedia:
https://en.wikipedia.org/wiki/Von_Mises-Fisher_distribution#Estimation_of_parameters
but re-parameterised for sigma rather than kappa.
"""
x = cartesian_from_polar(phi, theta)
S = numpy.sum(x, axis=-1)
R = S.dot(S)**0.5 / x.shape[-1]
def f(s):
return 1 / numpy.tanh(s) - 1. / s - R
kappa = scipy.optimize.brentq(f, 1e-8, 1e8)
sigma = kappa**-0.5
return sigma
def VonMises_mean(phi, theta):
""" Von-Mises sample mean.
Parameters
----------
phi, theta : array-like
Spherical-polar coordinate samples to compute mean from.
Returns
-------
float
..math::
\sum_i^N x_i / || \sum_i^N x_i ||
Notes
-----
Wikipedia:
https://en.wikipedia.org/wiki/Von_Mises-Fisher_distribution#Estimation_of_parameters
"""
x = cartesian_from_polar(phi, theta)
S = numpy.sum(x, axis=-1)
phi, theta = polar_from_cartesian(S)
return phi, theta
def test_cartesian_from_polar_array():
cart1 = utils.cartesian_from_polar(*test_polar.T)
assert_allclose(test_cartesian.T, cart1)
def test_cartesian_from_polar_scalar():
for ang, cart0 in zip(test_polar, test_cartesian):
cart1 = utils.cartesian_from_polar(*ang)
assert_allclose(cart0, cart1)
def g(phi, theta):
return f(phi, theta) * cartesian_from_polar(phi, theta)[i]