from __future__ import annotations
import math
import typing
from abc import ABC, abstractmethod
from collections import defaultdict
from collections.abc import Iterable
from copy import deepcopy
from numbers import Real
from warnings import warn
import lxml.etree as ET
import numpy as np
from scipy.integrate import trapezoid
import openmc.checkvalue as cv
from .._xml import get_text
from ..mixin import EqualityMixin
_INTERPOLATION_SCHEMES = [
'histogram',
'linear-linear',
'linear-log',
'log-linear',
'log-log'
]
[docs]class Univariate(EqualityMixin, ABC):
"""Probability distribution of a single random variable.
The Univariate class is an abstract class that can be derived to implement a
specific probability distribution.
"""
@abstractmethod
def to_xml_element(self, element_name):
return ''
@abstractmethod
def __len__(self):
return 0
@classmethod
@abstractmethod
def from_xml_element(cls, elem):
distribution = get_text(elem, 'type')
if distribution == 'discrete':
return Discrete.from_xml_element(elem)
elif distribution == 'uniform':
return Uniform.from_xml_element(elem)
elif distribution == 'powerlaw':
return PowerLaw.from_xml_element(elem)
elif distribution == 'maxwell':
return Maxwell.from_xml_element(elem)
elif distribution == 'watt':
return Watt.from_xml_element(elem)
elif distribution == 'normal':
return Normal.from_xml_element(elem)
elif distribution == 'muir':
# Support older files where Muir had its own class
params = [float(x) for x in get_text(elem, 'parameters').split()]
return muir(*params)
elif distribution == 'tabular':
return Tabular.from_xml_element(elem)
elif distribution == 'legendre':
return Legendre.from_xml_element(elem)
elif distribution == 'mixture':
return Mixture.from_xml_element(elem)
[docs] @abstractmethod
def sample(n_samples: int = 1, seed: typing.Optional[int] = None):
"""Sample the univariate distribution
Parameters
----------
n_samples : int
Number of sampled values to generate
seed : int or None
Initial random number seed.
Returns
-------
numpy.ndarray
A 1-D array of sampled values
"""
pass
[docs] def integral(self):
"""Return integral of distribution
.. versionadded:: 0.13.1
Returns
-------
float
Integral of distribution
"""
return 1.0
[docs]class Discrete(Univariate):
"""Distribution characterized by a probability mass function.
The Discrete distribution assigns probability values to discrete values of a
random variable, rather than expressing the distribution as a continuous
random variable.
Parameters
----------
x : Iterable of float
Values of the random variable
p : Iterable of float
Discrete probability for each value
Attributes
----------
x : numpy.ndarray
Values of the random variable
p : numpy.ndarray
Discrete probability for each value
"""
def __init__(self, x, p):
self.x = x
self.p = p
def __len__(self):
return len(self.x)
@property
def x(self):
return self._x
@x.setter
def x(self, x):
if isinstance(x, Real):
x = [x]
cv.check_type('discrete values', x, Iterable, Real)
self._x = np.array(x, dtype=float)
@property
def p(self):
return self._p
@p.setter
def p(self, p):
if isinstance(p, Real):
p = [p]
cv.check_type('discrete probabilities', p, Iterable, Real)
for pk in p:
cv.check_greater_than('discrete probability', pk, 0.0, True)
self._p = np.array(p, dtype=float)
def cdf(self):
return np.insert(np.cumsum(self.p), 0, 0.0)
[docs] def sample(self, n_samples=1, seed=None):
rng = np.random.RandomState(seed)
p = self.p / self.p.sum()
return rng.choice(self.x, n_samples, p=p)
[docs] def normalize(self):
"""Normalize the probabilities stored on the distribution"""
norm = sum(self.p)
self.p = [val / norm for val in self.p]
[docs] def to_xml_element(self, element_name):
"""Return XML representation of the discrete distribution
Parameters
----------
element_name : str
XML element name
Returns
-------
element : lxml.etree._Element
XML element containing discrete distribution data
"""
element = ET.Element(element_name)
element.set("type", "discrete")
params = ET.SubElement(element, "parameters")
params.text = ' '.join(map(str, self.x)) + ' ' + ' '.join(map(str, self.p))
return element
[docs] @classmethod
def from_xml_element(cls, elem: ET.Element):
"""Generate discrete distribution from an XML element
Parameters
----------
elem : lxml.etree._Element
XML element
Returns
-------
openmc.stats.Discrete
Discrete distribution generated from XML element
"""
params = [float(x) for x in get_text(elem, 'parameters').split()]
x = params[:len(params)//2]
p = params[len(params)//2:]
return cls(x, p)
[docs] @classmethod
def merge(
cls,
dists: typing.Sequence[Discrete],
probs: typing.Sequence[int]
):
"""Merge multiple discrete distributions into a single distribution
.. versionadded:: 0.13.1
Parameters
----------
dists : iterable of openmc.stats.Discrete
Discrete distributions to combine
probs : iterable of float
Probability of each distribution
Returns
-------
openmc.stats.Discrete
Combined discrete distribution
"""
if len(dists) != len(probs):
raise ValueError("Number of distributions and probabilities must match.")
# Combine distributions accounting for duplicate x values
x_merged = set()
p_merged = defaultdict(float)
for dist, p_dist in zip(dists, probs):
for x, p in zip(dist.x, dist.p):
x_merged.add(x)
p_merged[x] += p*p_dist
# Create values and probabilities as arrays
x_arr = np.array(sorted(x_merged))
p_arr = np.array([p_merged[x] for x in x_arr])
return cls(x_arr, p_arr)
[docs] def integral(self):
"""Return integral of distribution
.. versionadded:: 0.13.1
Returns
-------
float
Integral of discrete distribution
"""
return np.sum(self.p)
[docs] def clip(self, tolerance: float = 1e-6, inplace: bool = False) -> Discrete:
r"""Remove low-importance points from discrete distribution.
Given a probability mass function :math:`p(x)` with :math:`\{x_1, x_2,
x_3, \dots\}` the possible values of the random variable with
corresponding probabilities :math:`\{p_1, p_2, p_3, \dots\}`, this
function will remove any low-importance points such that :math:`\sum_i
x_i p_i` is preserved to within some threshold.
.. versionadded:: 0.14.0
Parameters
----------
tolerance : float
Maximum fraction of :math:`\sum_i x_i p_i` that will be discarded.
inplace : bool
Whether to modify the current object in-place or return a new one.
Returns
-------
Discrete distribution with low-importance points removed
"""
cv.check_less_than("tolerance", tolerance, 1.0, equality=True)
cv.check_greater_than("tolerance", tolerance, 0.0, equality=True)
# Determine (reversed) sorted order of probabilities
intensity = self.p * self.x
index_sort = np.argsort(intensity)[::-1]
# Get probabilities in above order
sorted_intensity = intensity[index_sort]
# Determine cumulative sum of probabilities
cumsum = np.cumsum(sorted_intensity)
cumsum /= cumsum[-1]
# Find index which satisfies cutoff
index_cutoff = np.searchsorted(cumsum, 1.0 - tolerance)
# Now get indices up to cutoff
new_indices = index_sort[:index_cutoff + 1]
new_indices.sort()
# Create new discrete distribution
if inplace:
self.x = self.x[new_indices]
self.p = self.p[new_indices]
return self
else:
new_x = self.x[new_indices]
new_p = self.p[new_indices]
return type(self)(new_x, new_p)
[docs]class PowerLaw(Univariate):
"""Distribution with power law probability over a finite interval [a,b]
The power law distribution has density function :math:`p(x) dx = c x^n dx`.
.. versionadded:: 0.13.0
Parameters
----------
a : float, optional
Lower bound of the sampling interval. Defaults to zero.
b : float, optional
Upper bound of the sampling interval. Defaults to unity.
n : float, optional
Power law exponent. Defaults to zero, which is equivalent to a uniform
distribution.
Attributes
----------
a : float
Lower bound of the sampling interval
b : float
Upper bound of the sampling interval
n : float
Power law exponent
"""
def __init__(self, a: float = 0.0, b: float = 1.0, n: float = 0.):
self.a = a
self.b = b
self.n = n
def __len__(self):
return 3
@property
def a(self):
return self._a
@a.setter
def a(self, a):
cv.check_type('interval lower bound', a, Real)
self._a = a
@property
def b(self):
return self._b
@b.setter
def b(self, b):
cv.check_type('interval upper bound', b, Real)
self._b = b
@property
def n(self):
return self._n
@n.setter
def n(self, n):
cv.check_type('power law exponent', n, Real)
self._n = n
[docs] def sample(self, n_samples=1, seed=None):
rng = np.random.RandomState(seed)
xi = rng.random(n_samples)
pwr = self.n + 1
offset = self.a**pwr
span = self.b**pwr - offset
return np.power(offset + xi * span, 1/pwr)
[docs] def to_xml_element(self, element_name: str):
"""Return XML representation of the power law distribution
Parameters
----------
element_name : str
XML element name
Returns
-------
element : lxml.etree._Element
XML element containing distribution data
"""
element = ET.Element(element_name)
element.set("type", "powerlaw")
element.set("parameters", f'{self.a} {self.b} {self.n}')
return element
[docs] @classmethod
def from_xml_element(cls, elem: ET.Element):
"""Generate power law distribution from an XML element
Parameters
----------
elem : lxml.etree._Element
XML element
Returns
-------
openmc.stats.PowerLaw
Distribution generated from XML element
"""
params = get_text(elem, 'parameters').split()
return cls(*map(float, params))
[docs]class Maxwell(Univariate):
r"""Maxwellian distribution in energy.
The Maxwellian distribution in energy is characterized by a single parameter
:math:`\theta` and has a density function :math:`p(E) dE = c \sqrt{E}
e^{-E/\theta} dE`.
Parameters
----------
theta : float
Effective temperature for distribution in eV
Attributes
----------
theta : float
Effective temperature for distribution in eV
"""
def __init__(self, theta):
self.theta = theta
def __len__(self):
return 1
@property
def theta(self):
return self._theta
@theta.setter
def theta(self, theta):
cv.check_type('Maxwell temperature', theta, Real)
cv.check_greater_than('Maxwell temperature', theta, 0.0)
self._theta = theta
[docs] def sample(self, n_samples=1, seed=None):
rng = np.random.RandomState(seed)
return self.sample_maxwell(self.theta, n_samples, rng=rng)
@staticmethod
def sample_maxwell(t, n_samples: int, rng=None):
if rng is None:
rng = np.random.default_rng()
r1, r2, r3 = rng.random((3, n_samples))
c = np.cos(0.5 * np.pi * r3)
return -t * (np.log(r1) + np.log(r2) * c * c)
[docs] def to_xml_element(self, element_name: str):
"""Return XML representation of the Maxwellian distribution
Parameters
----------
element_name : str
XML element name
Returns
-------
element : lxml.etree._Element
XML element containing Maxwellian distribution data
"""
element = ET.Element(element_name)
element.set("type", "maxwell")
element.set("parameters", str(self.theta))
return element
[docs] @classmethod
def from_xml_element(cls, elem: ET.Element):
"""Generate Maxwellian distribution from an XML element
Parameters
----------
elem : lxml.etree._Element
XML element
Returns
-------
openmc.stats.Maxwell
Maxwellian distribution generated from XML element
"""
theta = float(get_text(elem, 'parameters'))
return cls(theta)
[docs]class Watt(Univariate):
r"""Watt fission energy spectrum.
The Watt fission energy spectrum is characterized by two parameters
:math:`a` and :math:`b` and has density function :math:`p(E) dE = c e^{-E/a}
\sinh \sqrt{b \, E} dE`.
Parameters
----------
a : float
First parameter of distribution in units of eV
b : float
Second parameter of distribution in units of 1/eV
Attributes
----------
a : float
First parameter of distribution in units of eV
b : float
Second parameter of distribution in units of 1/eV
"""
def __init__(self, a=0.988e6, b=2.249e-6):
self.a = a
self.b = b
def __len__(self):
return 2
@property
def a(self):
return self._a
@a.setter
def a(self, a):
cv.check_type('Watt a', a, Real)
cv.check_greater_than('Watt a', a, 0.0)
self._a = a
@property
def b(self):
return self._b
@b.setter
def b(self, b):
cv.check_type('Watt b', b, Real)
cv.check_greater_than('Watt b', b, 0.0)
self._b = b
[docs] def sample(self, n_samples=1, seed=None):
rng = np.random.RandomState(seed)
w = Maxwell.sample_maxwell(self.a, n_samples, rng=rng)
u = rng.uniform(-1., 1., n_samples)
aab = self.a * self.a * self.b
return w + 0.25*aab + u*np.sqrt(aab*w)
[docs] def to_xml_element(self, element_name: str):
"""Return XML representation of the Watt distribution
Parameters
----------
element_name : str
XML element name
Returns
-------
element : lxml.etree._Element
XML element containing Watt distribution data
"""
element = ET.Element(element_name)
element.set("type", "watt")
element.set("parameters", f'{self.a} {self.b}')
return element
[docs] @classmethod
def from_xml_element(cls, elem: ET.Element):
"""Generate Watt distribution from an XML element
Parameters
----------
elem : lxml.etree._Element
XML element
Returns
-------
openmc.stats.Watt
Watt distribution generated from XML element
"""
params = get_text(elem, 'parameters').split()
return cls(*map(float, params))
[docs]class Normal(Univariate):
r"""Normally distributed sampling.
The Normal Distribution is characterized by two parameters
:math:`\mu` and :math:`\sigma` and has density function
:math:`p(X) dX = 1/(\sqrt{2\pi}\sigma) e^{-(X-\mu)^2/(2\sigma^2)}`
Parameters
----------
mean_value : float
Mean value of the distribution
std_dev : float
Standard deviation of the Normal distribution
Attributes
----------
mean_value : float
Mean of the Normal distribution
std_dev : float
Standard deviation of the Normal distribution
"""
def __init__(self, mean_value, std_dev):
self.mean_value = mean_value
self.std_dev = std_dev
def __len__(self):
return 2
@property
def mean_value(self):
return self._mean_value
@mean_value.setter
def mean_value(self, mean_value):
cv.check_type('Normal mean_value', mean_value, Real)
self._mean_value = mean_value
@property
def std_dev(self):
return self._std_dev
@std_dev.setter
def std_dev(self, std_dev):
cv.check_type('Normal std_dev', std_dev, Real)
cv.check_greater_than('Normal std_dev', std_dev, 0.0)
self._std_dev = std_dev
[docs] def sample(self, n_samples=1, seed=None):
rng = np.random.RandomState(seed)
return rng.normal(self.mean_value, self.std_dev, n_samples)
[docs] def to_xml_element(self, element_name: str):
"""Return XML representation of the Normal distribution
Parameters
----------
element_name : str
XML element name
Returns
-------
element : lxml.etree._Element
XML element containing Watt distribution data
"""
element = ET.Element(element_name)
element.set("type", "normal")
element.set("parameters", f'{self.mean_value} {self.std_dev}')
return element
[docs] @classmethod
def from_xml_element(cls, elem: ET.Element):
"""Generate Normal distribution from an XML element
Parameters
----------
elem : lxml.etree._Element
XML element
Returns
-------
openmc.stats.Normal
Normal distribution generated from XML element
"""
params = get_text(elem, 'parameters').split()
return cls(*map(float, params))
[docs]def muir(e0: float, m_rat: float, kt: float):
"""Generate a Muir energy spectrum
The Muir energy spectrum is a normal distribution, but for convenience
reasons allows the user to specify three parameters to define the
distribution: the mean energy of particles ``e0``, the mass of reactants
``m_rat``, and the ion temperature ``kt``.
.. versionadded:: 0.13.2
Parameters
----------
e0 : float
Mean of the Muir distribution in [eV]
m_rat : float
Ratio of the sum of the masses of the reaction inputs to 1 amu
kt : float
Ion temperature for the Muir distribution in [eV]
Returns
-------
openmc.stats.Normal
Corresponding normal distribution
"""
# https://permalink.lanl.gov/object/tr?what=info:lanl-repo/lareport/LA-05411-MS
std_dev = math.sqrt(2 * e0 * kt / m_rat)
return Normal(e0, std_dev)
# Retain deprecated name for the time being
def Muir(*args, **kwargs):
# warn of name change
warn(
"The Muir(...) class has been replaced by the muir(...) function and "
"will be removed in a future version of OpenMC. Use muir(...) instead.",
FutureWarning
)
return muir(*args, **kwargs)
[docs]class Tabular(Univariate):
"""Piecewise continuous probability distribution.
This class is used to represent a probability distribution whose density
function is tabulated at specific values with a specified interpolation
scheme.
Parameters
----------
x : Iterable of float
Tabulated values of the random variable
p : Iterable of float
Tabulated probabilities
interpolation : {'histogram', 'linear-linear', 'linear-log', 'log-linear', 'log-log'}, optional
Indicate whether the density function is constant between tabulated
points or linearly-interpolated. Defaults to 'linear-linear'.
ignore_negative : bool
Ignore negative probabilities
Attributes
----------
x : numpy.ndarray
Tabulated values of the random variable
p : numpy.ndarray
Tabulated probabilities
interpolation : {'histogram', 'linear-linear', 'linear-log', 'log-linear', 'log-log'}, optional
Indicate whether the density function is constant between tabulated
points or linearly-interpolated.
"""
def __init__(
self,
x: typing.Sequence[float],
p: typing.Sequence[float],
interpolation: str = 'linear-linear',
ignore_negative: bool = False
):
self._ignore_negative = ignore_negative
self.x = x
self.p = p
self.interpolation = interpolation
def __len__(self):
return len(self.x)
@property
def x(self):
return self._x
@x.setter
def x(self, x):
cv.check_type('tabulated values', x, Iterable, Real)
self._x = np.array(x, dtype=float)
@property
def p(self):
return self._p
@p.setter
def p(self, p):
cv.check_type('tabulated probabilities', p, Iterable, Real)
if not self._ignore_negative:
for pk in p:
cv.check_greater_than('tabulated probability', pk, 0.0, True)
self._p = np.array(p, dtype=float)
@property
def interpolation(self):
return self._interpolation
@interpolation.setter
def interpolation(self, interpolation):
cv.check_value('interpolation', interpolation, _INTERPOLATION_SCHEMES)
self._interpolation = interpolation
def cdf(self):
c = np.zeros_like(self.x)
x = self.x
p = self.p
if self.interpolation == 'histogram':
c[1:] = p[:-1] * np.diff(x)
elif self.interpolation == 'linear-linear':
c[1:] = 0.5 * (p[:-1] + p[1:]) * np.diff(x)
else:
raise NotImplementedError('Can only generate CDFs for tabular '
'distributions using histogram or '
'linear-linear interpolation')
return np.cumsum(c)
[docs] def mean(self):
"""Compute the mean of the tabular distribution"""
if self.interpolation == 'linear-linear':
mean = 0.0
for i in range(1, len(self.x)):
y_min = self.p[i-1]
y_max = self.p[i]
x_min = self.x[i-1]
x_max = self.x[i]
m = (y_max - y_min) / (x_max - x_min)
exp_val = (1./3.) * m * (x_max**3 - x_min**3)
exp_val += 0.5 * m * x_min * (x_min**2 - x_max**2)
exp_val += 0.5 * y_min * (x_max**2 - x_min**2)
mean += exp_val
elif self.interpolation == 'histogram':
x_l = self.x[:-1]
x_r = self.x[1:]
p_l = self.p[:-1]
mean = (0.5 * (x_l + x_r) * (x_r - x_l) * p_l).sum()
else:
raise NotImplementedError('Can only compute mean for tabular '
'distributions using histogram '
'or linear-linear interpolation.')
# Normalize for when integral of distribution is not 1
mean /= self.integral()
return mean
[docs] def normalize(self):
"""Normalize the probabilities stored on the distribution"""
self.p /= self.cdf().max()
[docs] def sample(self, n_samples: int = 1, seed: typing.Optional[int] = None):
rng = np.random.RandomState(seed)
xi = rng.random(n_samples)
# always use normalized probabilities when sampling
cdf = self.cdf()
p = self.p / cdf.max()
cdf /= cdf.max()
# get CDF bins that are above the
# sampled values
c_i = np.full(n_samples, cdf[0])
cdf_idx = np.zeros(n_samples, dtype=int)
for i, val in enumerate(cdf[:-1]):
mask = xi > val
c_i[mask] = val
cdf_idx[mask] = i
# get table values at each index where
# the random number is less than the next cdf
# entry
x_i = self.x[cdf_idx]
p_i = p[cdf_idx]
if self.interpolation == 'histogram':
# mask where probability is greater than zero
pos_mask = p_i > 0.0
# probabilities greater than zero are set proportional to the
# position of the random numebers in relation to the cdf value
p_i[pos_mask] = x_i[pos_mask] + (xi[pos_mask] - c_i[pos_mask]) \
/ p_i[pos_mask]
# probabilities smaller than zero are set to the random number value
p_i[~pos_mask] = x_i[~pos_mask]
samples_out = p_i
elif self.interpolation == 'linear-linear':
# get variable and probability values for the
# next entry
x_i1 = self.x[cdf_idx + 1]
p_i1 = p[cdf_idx + 1]
# compute slope between entries
m = (p_i1 - p_i) / (x_i1 - x_i)
# set values for zero slope
zero = m == 0.0
m[zero] = x_i[zero] + (xi[zero] - c_i[zero]) / p_i[zero]
# set values for non-zero slope
non_zero = ~zero
quad = np.power(p_i[non_zero], 2) + 2.0 * m[non_zero] * (xi[non_zero] - c_i[non_zero])
quad[quad < 0.0] = 0.0
m[non_zero] = x_i[non_zero] + (np.sqrt(quad) - p_i[non_zero]) / m[non_zero]
samples_out = m
else:
raise NotImplementedError('Can only sample tabular distributions '
'using histogram or '
'linear-linear interpolation')
assert all(samples_out < self.x[-1])
return samples_out
[docs] def to_xml_element(self, element_name: str):
"""Return XML representation of the tabular distribution
Parameters
----------
element_name : str
XML element name
Returns
-------
element : lxml.etree._Element
XML element containing tabular distribution data
"""
element = ET.Element(element_name)
element.set("type", "tabular")
element.set("interpolation", self.interpolation)
params = ET.SubElement(element, "parameters")
params.text = ' '.join(map(str, self.x)) + ' ' + ' '.join(map(str, self.p))
return element
[docs] @classmethod
def from_xml_element(cls, elem: ET.Element):
"""Generate tabular distribution from an XML element
Parameters
----------
elem : lxml.etree._Element
XML element
Returns
-------
openmc.stats.Tabular
Tabular distribution generated from XML element
"""
interpolation = get_text(elem, 'interpolation')
params = [float(x) for x in get_text(elem, 'parameters').split()]
x = params[:len(params)//2]
p = params[len(params)//2:]
return cls(x, p, interpolation)
[docs] def integral(self):
"""Return integral of distribution
.. versionadded:: 0.13.1
Returns
-------
float
Integral of tabular distrbution
"""
if self.interpolation == 'histogram':
return np.sum(np.diff(self.x) * self.p[:-1])
elif self.interpolation == 'linear-linear':
return trapezoid(self.p, self.x)
else:
raise NotImplementedError(
f'integral() not supported for {self.inteprolation} interpolation')
[docs]class Legendre(Univariate):
r"""Probability density given by a Legendre polynomial expansion
:math:`\sum\limits_{\ell=0}^N \frac{2\ell + 1}{2} a_\ell P_\ell(\mu)`.
Parameters
----------
coefficients : Iterable of Real
Expansion coefficients :math:`a_\ell`. Note that the :math:`(2\ell +
1)/2` factor should not be included.
Attributes
----------
coefficients : Iterable of Real
Expansion coefficients :math:`a_\ell`. Note that the :math:`(2\ell +
1)/2` factor should not be included.
"""
def __init__(self, coefficients: typing.Sequence[float]):
self.coefficients = coefficients
self._legendre_poly = None
def __call__(self, x):
# Create Legendre polynomial if we haven't yet
if self._legendre_poly is None:
l = np.arange(len(self._coefficients))
coeffs = (2.*l + 1.)/2. * self._coefficients
self._legendre_poly = np.polynomial.Legendre(coeffs)
return self._legendre_poly(x)
def __len__(self):
return len(self._coefficients)
@property
def coefficients(self):
return self._coefficients
@coefficients.setter
def coefficients(self, coefficients):
self._coefficients = np.asarray(coefficients)
[docs] def sample(self, n_samples=1, seed=None):
raise NotImplementedError
def to_xml_element(self, element_name):
raise NotImplementedError
@classmethod
def from_xml_element(cls, elem):
raise NotImplementedError
[docs]class Mixture(Univariate):
"""Probability distribution characterized by a mixture of random variables.
Parameters
----------
probability : Iterable of Real
Probability of selecting a particular distribution
distribution : Iterable of Univariate
List of distributions with corresponding probabilities
Attributes
----------
probability : Iterable of Real
Probability of selecting a particular distribution
distribution : Iterable of Univariate
List of distributions with corresponding probabilities
"""
def __init__(
self,
probability: typing.Sequence[float],
distribution: typing.Sequence[Univariate]
):
self.probability = probability
self.distribution = distribution
def __len__(self):
return sum(len(d) for d in self.distribution)
@property
def probability(self):
return self._probability
@probability.setter
def probability(self, probability):
cv.check_type('mixture distribution probabilities', probability,
Iterable, Real)
for p in probability:
cv.check_greater_than('mixture distribution probabilities',
p, 0.0, True)
self._probability = probability
@property
def distribution(self):
return self._distribution
@distribution.setter
def distribution(self, distribution):
cv.check_type('mixture distribution components', distribution,
Iterable, Univariate)
self._distribution = distribution
def cdf(self):
return np.insert(np.cumsum(self.probability), 0, 0.0)
[docs] def sample(self, n_samples=1, seed=None):
rng = np.random.RandomState(seed)
# Get probability of each distribution accounting for its intensity
p = np.array([prob*dist.integral() for prob, dist in
zip(self.probability, self.distribution)])
p /= p.sum()
# Sample from the distributions
idx = rng.choice(range(len(self.distribution)), n_samples, p=p)
# Draw samples from the distributions sampled above
out = np.empty_like(idx, dtype=float)
for i in np.unique(idx):
n_dist_samples = np.count_nonzero(idx == i)
samples = self.distribution[i].sample(n_dist_samples)
out[idx == i] = samples
return out
[docs] def normalize(self):
"""Normalize the probabilities stored on the distribution"""
norm = sum(self.probability)
self.probability = [val / norm for val in self.probability]
[docs] def to_xml_element(self, element_name: str):
"""Return XML representation of the mixture distribution
.. versionadded:: 0.13.0
Parameters
----------
element_name : str
XML element name
Returns
-------
element : lxml.etree._Element
XML element containing mixture distribution data
"""
element = ET.Element(element_name)
element.set("type", "mixture")
for p, d in zip(self.probability, self.distribution):
data = ET.SubElement(element, "pair")
data.set("probability", str(p))
data.append(d.to_xml_element("dist"))
return element
[docs] @classmethod
def from_xml_element(cls, elem: ET.Element):
"""Generate mixture distribution from an XML element
.. versionadded:: 0.13.0
Parameters
----------
elem : lxml.etree._Element
XML element
Returns
-------
openmc.stats.Mixture
Mixture distribution generated from XML element
"""
probability = []
distribution = []
for pair in elem.findall('pair'):
probability.append(float(get_text(pair, 'probability')))
distribution.append(Univariate.from_xml_element(pair.find("dist")))
return cls(probability, distribution)
[docs] def integral(self):
"""Return integral of the distribution
.. versionadded:: 0.13.1
Returns
-------
float
Integral of the distribution
"""
return sum([
p*dist.integral()
for p, dist in zip(self.probability, self.distribution)
])
[docs] def clip(self, tolerance: float = 1e-6, inplace: bool = False) -> Mixture:
r"""Remove low-importance points from contained discrete distributions.
Given a probability mass function :math:`p(x)` with :math:`\{x_1, x_2,
x_3, \dots\}` the possible values of the random variable with
corresponding probabilities :math:`\{p_1, p_2, p_3, \dots\}`, this
function will remove any low-importance points such that :math:`\sum_i
x_i p_i` is preserved to within some threshold.
.. versionadded:: 0.14.0
Parameters
----------
tolerance : float
Maximum fraction of :math:`\sum_i x_i p_i` that will be discarded
for any discrete distributions within the mixture distribution.
inplace : bool
Whether to modify the current object in-place or return a new one.
Returns
-------
Discrete distribution with low-importance points removed
"""
if inplace:
for dist in self.distribution:
if isinstance(dist, Discrete):
dist.clip(tolerance, inplace=True)
return self
else:
distribution = [
dist.clip(tolerance) if isinstance(dist, Discrete) else dist
for dist in self.distribution
]
return type(self)(self.probability, distribution)
[docs]def combine_distributions(
dists: typing.Sequence[Univariate],
probs: typing.Sequence[float]
):
"""Combine distributions with specified probabilities
This function can be used to combine multiple instances of
:class:`~openmc.stats.Discrete` and `~openmc.stats.Tabular`. Multiple
discrete distributions are merged into a single distribution and the
remainder of the distributions are put into a :class:`~openmc.stats.Mixture`
distribution.
.. versionadded:: 0.13.1
Parameters
----------
dists : iterable of openmc.stats.Univariate
Distributions to combine
probs : iterable of float
Probability (or intensity) of each distribution
"""
# Get copy of distribution list so as not to modify the argument
dist_list = deepcopy(dists)
# Get list of discrete/continuous distribution indices
discrete_index = [i for i, d in enumerate(dist_list) if isinstance(d, Discrete)]
cont_index = [i for i, d in enumerate(dist_list) if isinstance(d, Tabular)]
# Apply probabilites to continuous distributions
for i in cont_index:
dist = dist_list[i]
dist.p *= probs[i]
if discrete_index:
# Create combined discrete distribution
dist_discrete = [dist_list[i] for i in discrete_index]
discrete_probs = [probs[i] for i in discrete_index]
combined_dist = Discrete.merge(dist_discrete, discrete_probs)
# Replace multiple discrete distributions with merged
for idx in reversed(discrete_index):
dist_list.pop(idx)
dist_list.append(combined_dist)
# Combine discrete and continuous if present
if len(dist_list) > 1:
probs = [1.0]*len(dist_list)
dist_list[:] = [Mixture(probs, dist_list.copy())]
return dist_list[0]