#!/usr/bin/env python3
import math
from numbers import Number, Real
import torch
from torch.distributions import constraints, Chi2
from torch.distributions.exp_family import ExponentialFamily
from torch.distributions.kl import register_kl
from torch.distributions.utils import _standard_normal, broadcast_all
from gpytorch.distributions.distribution import Distribution
__all__ = ["QExponential"]
[docs]class QExponential(ExponentialFamily, Distribution):
r"""
Creates a q-exponential distribution parameterized by
:attr:`loc`, :attr:`scale` and :attr:`power`, with the following density
.. math::
p(x; \mu, \sigma^2) = \frac{q}{2}(2\pi\sigma^2)^{-\frac{1}{2}}
\left|\frac{x-\mu}{\sigma}\right|^{\frac{q}{2}-1}
\exp\left\{-\frac{1}{2}\left|\frac{x-\mu}{\sigma}\right|^q\right\}
Example::
>>> # xdoctest: +IGNORE_WANT("non-deterministic")
>>> m = QExponential(torch.tensor([0.0]), torch.tensor([1.0]))
>>> m.sample() # q-exponentially distributed with loc=0, scale=1 and power=2
tensor([ 0.1046])
Args:
loc (float or Tensor): mean of the distribution (often referred to as mu)
scale (float or Tensor): standard deviation of the distribution
(often referred to as sigma)
power (float or Tensor): power of the distribution
"""
arg_constraints = {"loc": constraints.real, "scale": constraints.positive, "power": constraints.positive}
support = constraints.real
has_rsample = True
_mean_carrier_measure = 0
@property
def mean(self):
return self.loc
@property
def mode(self):
return self.loc
@property
def stddev(self):
return self.scale
@property
def variance(self):
return self.stddev.pow(2)
@property
def rescalor(self):
return torch.exp((2./self.power*math.log(2) + torch.lgamma(0.5+2./self.power) - math.log(math.pi)/2.)/2.)
def __init__(self, loc, scale, power=torch.tensor(2.0), validate_args=None):
self.loc, self.scale = broadcast_all(loc, scale)
if isinstance(loc, Number) and isinstance(scale, Number):
batch_shape = torch.Size()
else:
batch_shape = self.loc.size()
self.power = power
super().__init__(batch_shape, validate_args=validate_args)
def confidence(self, alpha=0.05):
lower = self.icdf(torch.tensor(alpha/2))
upper = self.icdf(torch.tensor(1-alpha/2))
return lower, upper
def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(QExponential, _instance)
batch_shape = torch.Size(batch_shape)
new.loc = self.loc.expand(batch_shape)
new.scale = self.scale.expand(batch_shape)
new.power = self.power
super(QExponential, new).__init__(batch_shape, validate_args=False)
new._validate_args = self._validate_args
return new
def sample(self, sample_shape=torch.Size(), rescale=False):
shape = self._extended_shape(sample_shape)
with torch.no_grad():
eps = Chi2(1).sample(shape).to(self.loc.device)**(1./self.power) * _standard_normal(shape, dtype=self.loc.dtype, device=self.loc.device).sign()
if rescale: eps /= self.rescalor
return self.loc.expand(shape) + eps * self.scale.expand(shape)
def rsample(self, sample_shape=torch.Size(), rescale=False):
shape = self._extended_shape(sample_shape)
eps = _standard_normal(shape, dtype=self.loc.dtype, device=self.loc.device)
if self.power!=2: eps = eps.abs()**(2./self.power-1) * eps
if rescale: eps /= self.rescalor
return self.loc + eps * self.scale
def log_prob(self, value):
if self._validate_args:
self._validate_sample(value)
log_scale = (
math.log(self.scale) if isinstance(self.scale, Real) else self.scale.log()
)
scaled_diff = ((value - self.loc) / self.scale).abs()
res = -.5* ( scaled_diff**self.power + math.log(2 * math.pi) ) - log_scale
if self.power!=2: res += (self.power/2.-1)*scaled_diff.log() + torch.log(self.power/2.)
return res
def cdf(self, value):
if self._validate_args:
self._validate_sample(value)
scaled_diff = (value - self.loc) * self.scale.reciprocal()
if self.power!=2: scaled_diff *= scaled_diff.abs()**(self.power/2.-1)
return 0.5 * (
1 + torch.erf(scaled_diff / math.sqrt(2))
)
def icdf(self, value):
erfinv = torch.erfinv(2 * value - 1) * math.sqrt(2)
if self.power!=2: erfinv *= erfinv.abs()**(2./self.power-1)
return self.loc + self.scale * erfinv
def entropy(self, exact=False):
res = 0.5 + 0.5 * math.log(2 * math.pi) + torch.log(self.scale)
if self.power!=2: res += 0.5*(self.power/2.-1) *(2./self.power* Chi2(1).entropy() if exact else 0) - torch.log(self.power/2.)
return res
@property
def _natural_params(self):
if self.power!=2:
raise ValueError(f"Q-Exponential distribution with power {self.power} does not belong to exponential family!")
else:
return (self.loc / self.scale.pow(2), -0.5 * self.scale.pow(2).reciprocal())
def _log_normalizer(self, x, y):
if self.power!=2:
raise ValueError(f"Q-Exponential distribution with power {self.power} does not belong to exponential family!")
else:
return -0.25 * x.pow(2) / y + 0.5 * torch.log(-math.pi / y)
@register_kl(QExponential, QExponential)
def _kl_qexponential_qexponential(p, q, exact=False):
var_ratio = (p.scale / q.scale).pow(2)
t1 = ((p.loc - q.loc) / q.scale).pow(2)
res = 0.5 * ((var_ratio + t1).pow(q.power/2.) - 1 - var_ratio.log())
if q.power!=2: res += 0.5 * ( -(q.power/2.-1)*torch.log(var_ratio + t1) + (p.power/2.-1) * (-2./p.power*Chi2(1).entropy() if exact else 0) )
return res