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# Source code for torch.distributions.von_mises

# mypy: allow-untyped-defs
import math

import torch
import torch.jit
from torch.distributions import constraints
from torch.distributions.distribution import Distribution

__all__ = ["VonMises"]

def _eval_poly(y, coef):
coef = list(coef)
result = coef.pop()
while coef:
result = coef.pop() + y * result
return result

_I0_COEF_SMALL = [
1.0,
3.5156229,
3.0899424,
1.2067492,
0.2659732,
0.360768e-1,
0.45813e-2,
]
_I0_COEF_LARGE = [
0.39894228,
0.1328592e-1,
0.225319e-2,
-0.157565e-2,
0.916281e-2,
-0.2057706e-1,
0.2635537e-1,
-0.1647633e-1,
0.392377e-2,
]
_I1_COEF_SMALL = [
0.5,
0.87890594,
0.51498869,
0.15084934,
0.2658733e-1,
0.301532e-2,
0.32411e-3,
]
_I1_COEF_LARGE = [
0.39894228,
-0.3988024e-1,
-0.362018e-2,
0.163801e-2,
-0.1031555e-1,
0.2282967e-1,
-0.2895312e-1,
0.1787654e-1,
-0.420059e-2,
]

_COEF_SMALL = [_I0_COEF_SMALL, _I1_COEF_SMALL]
_COEF_LARGE = [_I0_COEF_LARGE, _I1_COEF_LARGE]

def _log_modified_bessel_fn(x, order=0):
"""
Returns log(I_order(x)) for x > 0,
where order is either 0 or 1.
"""
assert order == 0 or order == 1

# compute small solution
y = x / 3.75
y = y * y
small = _eval_poly(y, _COEF_SMALL[order])
if order == 1:
small = x.abs() * small
small = small.log()

# compute large solution
y = 3.75 / x
large = x - 0.5 * x.log() + _eval_poly(y, _COEF_LARGE[order]).log()

result = torch.where(x < 3.75, small, large)
return result

@torch.jit.script_if_tracing
def _rejection_sample(loc, concentration, proposal_r, x):
done = torch.zeros(x.shape, dtype=torch.bool, device=loc.device)
while not done.all():
u = torch.rand((3,) + x.shape, dtype=loc.dtype, device=loc.device)
u1, u2, u3 = u.unbind()
z = torch.cos(math.pi * u1)
f = (1 + proposal_r * z) / (proposal_r + z)
c = concentration * (proposal_r - f)
accept = ((c * (2 - c) - u2) > 0) | ((c / u2).log() + 1 - c >= 0)
if accept.any():
x = torch.where(accept, (u3 - 0.5).sign() * f.acos(), x)
done = done | accept
return (x + math.pi + loc) % (2 * math.pi) - math.pi

[docs]class VonMises(Distribution):
"""
A circular von Mises distribution.

This implementation uses polar coordinates. The loc and value args
can be any real number (to facilitate unconstrained optimization), but are
interpreted as angles modulo 2 pi.

Example::
>>> # xdoctest: +IGNORE_WANT("non-deterministic")
>>> m = VonMises(torch.tensor([1.0]), torch.tensor([1.0]))
>>> m.sample()  # von Mises distributed with loc=1 and concentration=1
tensor([1.9777])

:param torch.Tensor loc: an angle in radians.
:param torch.Tensor concentration: concentration parameter
"""

arg_constraints = {"loc": constraints.real, "concentration": constraints.positive}
support = constraints.real
has_rsample = False

def __init__(self, loc, concentration, validate_args=None):
batch_shape = self.loc.shape
event_shape = torch.Size()
super().__init__(batch_shape, event_shape, validate_args)

[docs]    def log_prob(self, value):
if self._validate_args:
self._validate_sample(value)
log_prob = self.concentration * torch.cos(value - self.loc)
log_prob = (
log_prob
- math.log(2 * math.pi)
- _log_modified_bessel_fn(self.concentration, order=0)
)
return log_prob

@lazy_property
def _loc(self):
return self.loc.to(torch.double)

@lazy_property
def _concentration(self):
return self.concentration.to(torch.double)

@lazy_property
def _proposal_r(self):
kappa = self._concentration
tau = 1 + (1 + 4 * kappa**2).sqrt()
rho = (tau - (2 * tau).sqrt()) / (2 * kappa)
_proposal_r = (1 + rho**2) / (2 * rho)
# second order Taylor expansion around 0 for small kappa
_proposal_r_taylor = 1 / kappa + kappa

def sample(self, sample_shape=torch.Size()):
"""
The sampling algorithm for the von Mises distribution is based on the
following paper: D.J. Best and N.I. Fisher, "Efficient simulation of the
von Mises distribution." Applied Statistics (1979): 152-157.

Sampling is always done in double precision internally to avoid a hang
in _rejection_sample() for small values of the concentration, which
starts to happen for single precision around 1e-4 (see issue #88443).
"""
shape = self._extended_shape(sample_shape)
x = torch.empty(shape, dtype=self._loc.dtype, device=self.loc.device)
return _rejection_sample(
self._loc, self._concentration, self._proposal_r, x
).to(self.loc.dtype)

[docs]    def expand(self, batch_shape):
try:
return super().expand(batch_shape)
except NotImplementedError:
validate_args = self.__dict__.get("_validate_args")
loc = self.loc.expand(batch_shape)
concentration = self.concentration.expand(batch_shape)
return type(self)(loc, concentration, validate_args=validate_args)

@property
def mean(self):
"""
The provided mean is the circular one.
"""
return self.loc

@property
def mode(self):
return self.loc

@lazy_property
def variance(self):
"""
The provided variance is the circular one.
"""
return (
1
- (
_log_modified_bessel_fn(self.concentration, order=1)
- _log_modified_bessel_fn(self.concentration, order=0)
).exp()
)


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