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import sys
import numpy as np
from numpy import inf
from scipy import special
from scipy.stats._distribution_infrastructure import (
ContinuousDistribution, _RealDomain, _RealParameter, _Parameterization,
_combine_docs)
__all__ = ['Normal', 'Uniform']
class Normal(ContinuousDistribution):
r"""Normal distribution with prescribed mean and standard deviation.
The probability density function of the normal distribution is:
.. math::
f(x) = \frac{1}{\sigma \sqrt{2 \pi}} \exp {
\left( -\frac{1}{2}\left( \frac{x - \mu}{\sigma} \right)^2 \right)}
"""
# `ShiftedScaledDistribution` allows this to be generated automatically from
# an instance of `StandardNormal`, but the normal distribution is so frequently
# used that it's worth a bit of code duplication to get better performance.
_mu_domain = _RealDomain(endpoints=(-inf, inf))
_sigma_domain = _RealDomain(endpoints=(0, inf))
_x_support = _RealDomain(endpoints=(-inf, inf))
_mu_param = _RealParameter('mu', symbol=r'\mu', domain=_mu_domain,
typical=(-1, 1))
_sigma_param = _RealParameter('sigma', symbol=r'\sigma', domain=_sigma_domain,
typical=(0.5, 1.5))
_x_param = _RealParameter('x', domain=_x_support, typical=(-1, 1))
_parameterizations = [_Parameterization(_mu_param, _sigma_param)]
_variable = _x_param
_normalization = 1/np.sqrt(2*np.pi)
_log_normalization = np.log(2*np.pi)/2
def __new__(cls, mu=None, sigma=None, **kwargs):
if mu is None and sigma is None:
return super().__new__(StandardNormal)
return super().__new__(cls)
def __init__(self, *, mu=0., sigma=1., **kwargs):
super().__init__(mu=mu, sigma=sigma, **kwargs)
def _logpdf_formula(self, x, *, mu, sigma, **kwargs):
return StandardNormal._logpdf_formula(self, (x - mu)/sigma) - np.log(sigma)
def _pdf_formula(self, x, *, mu, sigma, **kwargs):
return StandardNormal._pdf_formula(self, (x - mu)/sigma) / sigma
def _logcdf_formula(self, x, *, mu, sigma, **kwargs):
return StandardNormal._logcdf_formula(self, (x - mu)/sigma)
def _cdf_formula(self, x, *, mu, sigma, **kwargs):
return StandardNormal._cdf_formula(self, (x - mu)/sigma)
def _logccdf_formula(self, x, *, mu, sigma, **kwargs):
return StandardNormal._logccdf_formula(self, (x - mu)/sigma)
def _ccdf_formula(self, x, *, mu, sigma, **kwargs):
return StandardNormal._ccdf_formula(self, (x - mu)/sigma)
def _icdf_formula(self, x, *, mu, sigma, **kwargs):
return StandardNormal._icdf_formula(self, x) * sigma + mu
def _ilogcdf_formula(self, x, *, mu, sigma, **kwargs):
return StandardNormal._ilogcdf_formula(self, x) * sigma + mu
def _iccdf_formula(self, x, *, mu, sigma, **kwargs):
return StandardNormal._iccdf_formula(self, x) * sigma + mu
def _ilogccdf_formula(self, x, *, mu, sigma, **kwargs):
return StandardNormal._ilogccdf_formula(self, x) * sigma + mu
def _entropy_formula(self, *, mu, sigma, **kwargs):
return StandardNormal._entropy_formula(self) + np.log(abs(sigma))
def _logentropy_formula(self, *, mu, sigma, **kwargs):
lH0 = StandardNormal._logentropy_formula(self)
with np.errstate(divide='ignore'):
# sigma = 1 -> log(sigma) = 0 -> log(log(sigma)) = -inf
# Silence the unnecessary runtime warning
lls = np.log(np.log(abs(sigma))+0j)
return special.logsumexp(np.broadcast_arrays(lH0, lls), axis=0)
def _median_formula(self, *, mu, sigma, **kwargs):
return mu
def _mode_formula(self, *, mu, sigma, **kwargs):
return mu
def _moment_raw_formula(self, order, *, mu, sigma, **kwargs):
if order == 0:
return np.ones_like(mu)
elif order == 1:
return mu
else:
return None
_moment_raw_formula.orders = [0, 1] # type: ignore[attr-defined]
def _moment_central_formula(self, order, *, mu, sigma, **kwargs):
if order == 0:
return np.ones_like(mu)
elif order % 2:
return np.zeros_like(mu)
else:
# exact is faster (and obviously more accurate) for reasonable orders
return sigma**order * special.factorial2(int(order) - 1, exact=True)
def _sample_formula(self, sample_shape, full_shape, rng, *, mu, sigma, **kwargs):
return rng.normal(loc=mu, scale=sigma, size=full_shape)[()]
def _log_diff(log_p, log_q):
return special.logsumexp([log_p, log_q+np.pi*1j], axis=0)
class StandardNormal(Normal):
r"""Standard normal distribution.
The probability density function of the standard normal distribution is:
.. math::
f(x) = \frac{1}{\sqrt{2 \pi}} \exp \left( -\frac{1}{2} x^2 \right)
"""
_x_support = _RealDomain(endpoints=(-inf, inf))
_x_param = _RealParameter('x', domain=_x_support, typical=(-5, 5))
_variable = _x_param
_parameterizations = []
_normalization = 1/np.sqrt(2*np.pi)
_log_normalization = np.log(2*np.pi)/2
mu = np.float64(0.)
sigma = np.float64(1.)
def __init__(self, **kwargs):
ContinuousDistribution.__init__(self, **kwargs)
def _logpdf_formula(self, x, **kwargs):
return -(self._log_normalization + x**2/2)
def _pdf_formula(self, x, **kwargs):
return self._normalization * np.exp(-x**2/2)
def _logcdf_formula(self, x, **kwargs):
return special.log_ndtr(x)
def _cdf_formula(self, x, **kwargs):
return special.ndtr(x)
def _logccdf_formula(self, x, **kwargs):
return special.log_ndtr(-x)
def _ccdf_formula(self, x, **kwargs):
return special.ndtr(-x)
def _icdf_formula(self, x, **kwargs):
return special.ndtri(x)
def _ilogcdf_formula(self, x, **kwargs):
return special.ndtri_exp(x)
def _iccdf_formula(self, x, **kwargs):
return -special.ndtri(x)
def _ilogccdf_formula(self, x, **kwargs):
return -special.ndtri_exp(x)
def _entropy_formula(self, **kwargs):
return (1 + np.log(2*np.pi))/2
def _logentropy_formula(self, **kwargs):
return np.log1p(np.log(2*np.pi)) - np.log(2)
def _median_formula(self, **kwargs):
return 0
def _mode_formula(self, **kwargs):
return 0
def _moment_raw_formula(self, order, **kwargs):
raw_moments = {0: 1, 1: 0, 2: 1, 3: 0, 4: 3, 5: 0}
return raw_moments.get(order, None)
def _moment_central_formula(self, order, **kwargs):
return self._moment_raw_formula(order, **kwargs)
def _moment_standardized_formula(self, order, **kwargs):
return self._moment_raw_formula(order, **kwargs)
def _sample_formula(self, sample_shape, full_shape, rng, **kwargs):
return rng.normal(size=full_shape)[()]
# currently for testing only
class _LogUniform(ContinuousDistribution):
r"""Log-uniform distribution.
The probability density function of the log-uniform distribution is:
.. math::
f(x; a, b) = \frac{1}
{x (\log(b) - \log(a))}
If :math:`\log(X)` is a random variable that follows a uniform distribution
between :math:`\log(a)` and :math:`\log(b)`, then :math:`X` is log-uniformly
distributed with shape parameters :math:`a` and :math:`b`.
"""
_a_domain = _RealDomain(endpoints=(0, inf))
_b_domain = _RealDomain(endpoints=('a', inf))
_log_a_domain = _RealDomain(endpoints=(-inf, inf))
_log_b_domain = _RealDomain(endpoints=('log_a', inf))
_x_support = _RealDomain(endpoints=('a', 'b'), inclusive=(True, True))
_a_param = _RealParameter('a', domain=_a_domain, typical=(1e-3, 0.9))
_b_param = _RealParameter('b', domain=_b_domain, typical=(1.1, 1e3))
_log_a_param = _RealParameter('log_a', symbol=r'\log(a)',
domain=_log_a_domain, typical=(-3, -0.1))
_log_b_param = _RealParameter('log_b', symbol=r'\log(b)',
domain=_log_b_domain, typical=(0.1, 3))
_x_param = _RealParameter('x', domain=_x_support, typical=('a', 'b'))
_b_domain.define_parameters(_a_param)
_log_b_domain.define_parameters(_log_a_param)
_x_support.define_parameters(_a_param, _b_param)
_parameterizations = [_Parameterization(_log_a_param, _log_b_param),
_Parameterization(_a_param, _b_param)]
_variable = _x_param
def __init__(self, *, a=None, b=None, log_a=None, log_b=None, **kwargs):
super().__init__(a=a, b=b, log_a=log_a, log_b=log_b, **kwargs)
def _process_parameters(self, a=None, b=None, log_a=None, log_b=None, **kwargs):
a = np.exp(log_a) if a is None else a
b = np.exp(log_b) if b is None else b
log_a = np.log(a) if log_a is None else log_a
log_b = np.log(b) if log_b is None else log_b
kwargs.update(dict(a=a, b=b, log_a=log_a, log_b=log_b))
return kwargs
# def _logpdf_formula(self, x, *, log_a, log_b, **kwargs):
# return -np.log(x) - np.log(log_b - log_a)
def _pdf_formula(self, x, *, log_a, log_b, **kwargs):
return ((log_b - log_a)*x)**-1
# def _cdf_formula(self, x, *, log_a, log_b, **kwargs):
# return (np.log(x) - log_a)/(log_b - log_a)
def _moment_raw_formula(self, order, log_a, log_b, **kwargs):
if order == 0:
return self._one
t1 = self._one / (log_b - log_a) / order
t2 = np.real(np.exp(_log_diff(order * log_b, order * log_a)))
return t1 * t2
class Uniform(ContinuousDistribution):
r"""Uniform distribution.
The probability density function of the uniform distribution is:
.. math::
f(x; a, b) = \frac{1}
{b - a}
"""
_a_domain = _RealDomain(endpoints=(-inf, inf))
_b_domain = _RealDomain(endpoints=('a', inf))
_x_support = _RealDomain(endpoints=('a', 'b'), inclusive=(True, True))
_a_param = _RealParameter('a', domain=_a_domain, typical=(1e-3, 0.9))
_b_param = _RealParameter('b', domain=_b_domain, typical=(1.1, 1e3))
_x_param = _RealParameter('x', domain=_x_support, typical=('a', 'b'))
_b_domain.define_parameters(_a_param)
_x_support.define_parameters(_a_param, _b_param)
_parameterizations = [_Parameterization(_a_param, _b_param)]
_variable = _x_param
def __init__(self, *, a=None, b=None, **kwargs):
super().__init__(a=a, b=b, **kwargs)
def _process_parameters(self, a=None, b=None, ab=None, **kwargs):
ab = b - a
kwargs.update(dict(a=a, b=b, ab=ab))
return kwargs
def _logpdf_formula(self, x, *, ab, **kwargs):
return np.where(np.isnan(x), np.nan, -np.log(ab))
def _pdf_formula(self, x, *, ab, **kwargs):
return np.where(np.isnan(x), np.nan, 1/ab)
def _logcdf_formula(self, x, *, a, ab, **kwargs):
with np.errstate(divide='ignore'):
return np.log(x - a) - np.log(ab)
def _cdf_formula(self, x, *, a, ab, **kwargs):
return (x - a) / ab
def _logccdf_formula(self, x, *, b, ab, **kwargs):
with np.errstate(divide='ignore'):
return np.log(b - x) - np.log(ab)
def _ccdf_formula(self, x, *, b, ab, **kwargs):
return (b - x) / ab
def _icdf_formula(self, p, *, a, ab, **kwargs):
return a + ab*p
def _iccdf_formula(self, p, *, b, ab, **kwargs):
return b - ab*p
def _entropy_formula(self, *, ab, **kwargs):
return np.log(ab)
def _mode_formula(self, *, a, b, ab, **kwargs):
return a + 0.5*ab
def _median_formula(self, *, a, b, ab, **kwargs):
return a + 0.5*ab
def _moment_raw_formula(self, order, a, b, ab, **kwargs):
np1 = order + 1
return (b**np1 - a**np1) / (np1 * ab)
def _moment_central_formula(self, order, ab, **kwargs):
return ab**2/12 if order == 2 else None
_moment_central_formula.orders = [2] # type: ignore[attr-defined]
def _sample_formula(self, sample_shape, full_shape, rng, a, b, ab, **kwargs):
try:
return rng.uniform(a, b, size=full_shape)[()]
except OverflowError: # happens when there are NaNs
return rng.uniform(0, 1, size=full_shape)*ab + a
class _Gamma(ContinuousDistribution):
# Gamma distribution for testing only
_a_domain = _RealDomain(endpoints=(0, inf))
_x_support = _RealDomain(endpoints=(0, inf), inclusive=(False, False))
_a_param = _RealParameter('a', domain=_a_domain, typical=(0.1, 10))
_x_param = _RealParameter('x', domain=_x_support, typical=(0.1, 10))
_parameterizations = [_Parameterization(_a_param)]
_variable = _x_param
def _pdf_formula(self, x, *, a, **kwargs):
return x ** (a - 1) * np.exp(-x) / special.gamma(a)
# Distribution classes need only define the summary and beginning of the extended
# summary portion of the class documentation. All other documentation, including
# examples, is generated automatically.
_module = sys.modules[__name__].__dict__
for dist_name in __all__:
_module[dist_name].__doc__ = _combine_docs(_module[dist_name])
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